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Quantum Chemistry & Spectroscopy A Guided Inquiry

Tricia D. Shepherd | Alexander Grushow

cover illustration credit Š Marina Koven/Shutterstock


Quantum Chemistry  &    Spectroscopy  

   

A  Guided  Inquiry                   Tricia D. Shepherd Professor of Chemistry Westminster College

Alexander Grushow Professor of Chemistry Rider University

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Acknowledgements While many physical chemistry experiments are performed in a vacuum, writing about physical chemistry is the result of many interactions the impact of which is a challenge to describe and quantify. First and formost on the list of contributors is Rick Moog (Franklin & Marshall College) and the POGIL Project for serving as chief catalyst and generous contributor of material from the first edition of physical chemistry activities. Many of those familiar with the first edition will see the imprint of that work on this publication. More subtle are the thoughts and insights derived from participating in numerous workshop and meetings organized through the POGIL Project through which the model of guided inquiry activities continues to develop and mature as more people and disciplines contruibute to this movement toward student centered learning. More directly, these activities have been used in a variety of contexts, and we appreciate the thoughtful comments and insights from their review: RenĂŠe Cole (University of Iowa), Marty Perry (Ouachita Baptist University), Robert Whitnell (Guilford College), Craig Teague (Cornell College), James Diamond (Linfield College), Valeria Molinero (University of Utah), Matthew Horn (Utah Valley University), Carl Salter (Moravian College) and Steve Singleton (Coe College). Both authors are also grateful for the patience and insightful comments received from the physical chemistry students in our classes at Westminster and Rider as they often worked through the earlier versions of these activities while trying to learn a challenging subject. And last, but not least, the other people who lived with this project just as much as we did, even though it was often not by choice. TS is indebted to an undergraduate teaching & research assistant Matthew Koc who (among many other things) worked diligently to create an electronic version of the solutions for exercises and problems. AG would like to thank Kira, Griffin and Donna for being patient while Daddy was glued to his computer. Alexander Grushow Rider University Lawrenceville, NJ

Tricia Shepherd Westminster College Salt Lake City, UT


To the Student Process Oriented Guided Inquiry Learning (POGIL) is a method of instruction where you as a student take an active role in the classroom. The contents of this “book� are specially designed guided inquiry activities intended for you to complete during your designated class period working with a small group of your peers. Each Activity introduces relevent Physical Chemistry content in a Model of experimental data, equations, or information. These are followed by a series of questions designed to lead you through the thought processes that will result in the development of course topics. Using this method you will learn to apply scientific concepts, analyze and evaluate scientific information, and communicate these ideas to others. The role of your instructor is to guide you through this process, facilitate discussion, and provide a context and conceptual framework for the course material. The authors are convinced that this process of learning is most similar to what you will experience beyond the traditional education system. You will learn to ask questions, read and evaluate information, learn to assess whether or not you understand new ideas, and you will learn to collaborate to develop new knowledge. For many students, this may be your first time experiencing this type of learning environment. While there is always of mix of reactions to this teaching approach, students will become very familiar with the process of exploring information, discovering new concepts, and solving problems in the context of a variety of perspectives and approaches through peer collaboration. It is also important to realize that the work done in the classroom must be followed by active work outside the classroom. This includes reading a textbook to provide further detail and context, reviewing the answers you develop to these activities and then confirming your understanding of the topics by application to homework problems assigned by your instructor. The concepts and ideas that are initially developed by using these activities in class will only become well-developed knowledge through repeated experience and exposure to the content materials.


To the Instructor The activities presented here address a wide variety of topics found in a typical undergraduate quantum physical chemistry course. As seen in the table of contents, the activities are divided into subsections titled: Introduction, Fundamental, Spectroscopy, Symmetry and Extension. The Introduction activities are just that; a set of activities that introduce some of the basic physical concepts underlying quantum mechanics or the historical motivation leading to the development of quantum mechanics. An instructor can choose any of these or none of these. They are recommended as a means to (a) get students familiar with a guided inquiry activity (if they are not already) and (b) “break the ice” in a course in quantum mechanics. The Fundamental activities are the core of the course. They should be used in the order presented without omission. They include the central topics that would be found in any quantum chemistry course. The Spectroscopy activities are included as a separate section to give instructors flexibility in terms of when they wish to introduce these topic in their course. They can be completed as a set after all the “Fundamental” topics have been covered. It is also possible to use these activities directly following the pre-requisite concepts have been introduced in a Fundamental activities. It may be helpful for students to consider the practical application of quantum chemistry in conjunction with the theoretical development. For ease of referencing, the Spectroscopy activities are numbered in accordance with the preceding Fundamental activity it could follow. That is, Spectroscopy.6 can be performed any time after Fundamental.6 has been completed. If there are more than one activity relating to a single Fundamental concept, a second number is introduced to differentiate the topics, e.g. Spectroscopy.8.1. The Symmetry activities serve as a basic introduction to the topics of symmetry that are typically found in an introductory physical chemistry text. The first Symmetry activity could be done at almost any point in the course, however the subsequent actitivities require knowledge of molecular orbtial theory. The Extension activities are designed exactly as they are named; extensions of the main topics of the course. Instructors will vary in terms of the details they wish to include on any topic depending on the context of the course they teach and their own interests. We have attempted to provide the necessary flexibility in these materials to allow instructors to extend the Fundamental material as they see fit in the course. As with the Spectroscopy activities, the Extensions are numbered with the Fundamental activity that they are connected to. Some Extensions are designed for use in the middle of a Fundamental activity and some at the end. Some Fundamental activities have multiple Extensions. Each activity begins with one or more Pre-Activity questions. These are to be completed by students before they come to class. An instructor could conceivably


collect these as homework or use them as a basis for a pre-activity quiz. As with any upper-level course, the content typically assumes prerequisite knowledge that students should be expected to review in order to be prepared for each class. The Pre-Activity questions are designed to review previous content material and/or make sure that they can perform a necessary calculation or mechanical manipulation that will be further explored in the activity. The Critical Thinking Questions are designed to guide students to discover physical insights related to the mathematical framework of quantum theory. For example, while students don’t “invent” the basic postulates of quantum mechanics, we hope they will “invent” conceptual understanding regarding the physical and chemical insights of microscopic systems that can be derived from simplistic models and approximations obtained from the solutions presented. There would, of course, be no physical chemistry without mathematics. Through our own experience in the classroom, however, we have discovered that many students may struggle with the mathematical manipulations required of a physical chemistry course. In a guided-inquiry setting this can often slow down the pace of small group discovery and invention. To counter this problem we have attempted to move as much of the mathematical manipulation into the Exercises and Extensions as possible. This gives an instructor the flexibility to explicitly cover only the portions of the mathematics they feel are most important for their course and their students. Some of the Exercises work well as direct application of the activity in the context of the class with students working as a group with the instuctor present as a guide. Instuctors could also assign problems from their favorite physical chemistry texts for students to complete outside of class after completing the concept invention part of the activity. The Problems that follow the Exercises in some activities often require synthesis of multiple concepts. The Instructor’s version of each activity provides a guide page at the beginning of each activity to assist in preparation for use in the classroom. Each guide starts with a Rationale for the activity and then the “prior knowledge” section. The Pre-requisite Concept (or Activity) is described to provide guidance of our expectations of students’ abilities before they can perform each activity. Most importantly are the lists of the content and process objectives describing what each student should acheive after completing the activity. Instructors may want to share this information with students and/or create assessment/evaluation materials that reinforce these stated objectives. The Instructor’s guide also indicates the appropriate Extension activities that might be used in conjunction with each Fundamental activity and implementation notes that with additional ideas or cautions regarding the use of the activities in the classroom. The authors always welcome additional insights/feedback/suggestions for material to be included in subsequent editions. Please feel free to contact us at tshepherd@westminstercollege.edu or grushow@rider.edu.


Table of Contents Introduction Activities Introduction.1 Essential Classical Mechanics Introduction.2 Blackbody Radiation Introduction.3 Photoelectric Effect Introduction.4 Introduction to Spectroscopy Introduction.5 The Energies of Molecules Introduction.6 The Bohr Model Fundamental Activities Fundamental.1 The Postulates of Quantum Mechanics Fundamental.2 The Particle-on-a-line Model Fundamental.3 Language of Quantum Mechanics Fundamental.4 Molecular Translation Fundamental.5 The Particle-on-a-Spring Model Fundamental.6 Molecular Vibration Fundamental.7 The Particle-on-a-Ring Model Fundamental.8 Molecular Rotation Fundamental.9 The Hydrogen Atom Fundamental.10 Hydrogen Radial Functions Fundamental.11 Hydrogen Orbitals Fundamental.12 Multielectron Atoms Fundamental.13 Electron Configurations Fundamental.14 Electron Spin Fundamental.15 Term Symbols Fundamental.16 The Born Oppenheimer Approximation Fundamental.17 Linear Combinations of Atomic Orbitals Fundamental.18 Diatomic Molecules


Spectroscopy Activities Spectroscopy.4 Population of Quantum States Spectroscopy.6 Vibrational Spectroscopy Spectroscopy.8.1 Rotational Spectroscopy Spectroscopy.8.2 Vibrational-Rotational Spectroscopy Spectroscopy.15 Electronic Spectroscopy I: Atoms Spectroscopy.18 Electronic Spectroscopy II: Molecules Symmetry Activities Symmetry.1 Group Theory Symmetry.2 Symmetry Adapted Molecular Orbitals Symmetry.3 Vibrational Modes Extension Activities Extension.1 Introduction to Operators and Eigenvalue Equations Extension.3.1 Hermitian Operators Extension.3.2 The Heisenberg Uncertainty Principle Extension.4 Separation of Variables Extension.5.1 Two Particle Systems Extension.5.2 Even & Odd Functions Extension.14 Slater Determinants Extension.16 Molecular Energy Integrals Extension.17 H端ckel Molecular Orbital Theory Extension.Sy2 Ethylene Molecular Orbitals Appendix


Introduction.1

Essential Classical Mechanics Model 1: Newton in a Nutshell The motion of a particle is represented by changes in position, x, as a function of time, t, defined as the instantaneous velocity, v=

dx = x dt

(1)

Note: The “dot notation” is used to signify a time derivative. Changes in the velocity as a function of time are defined as acceleration,

a=

dv d ⎛ dx ⎞ d 2 x = ⎜ ⎟= =  x dt dt ⎝ dt ⎠ dt 2

(2)

Newton’s first law of motion states that a body remains at rest or in uniform motion (constant speed and direction) unless acted on by a net external force. In response to a net external force, F , the acceleration of a body of mass m is a=

F m

(3)

Thus to describe the motion of any object, one must be willing and able to solve the following second-order differential equation,

d2x F = dt 2 m

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(4)


Critical Thinking Questions 1. Consider two different interpretations of the symbol x : a. What is the quantity represented?

b. Without using the word “derivative,” describe the operation represented.

2. Write two equivalent forms of Equation 4 using the dot notation, a. in terms of position: b. in terms of velocity: 3. In order to use Equation 4 to solve for x(t) what needs to be defined and why? (Note: mass is a fixed quantity.)

4. The elastic force of a spring can be modeled as a conservative force known as Hooke’s Law, F = −kx . when k is a constant. For a mass connected to a spring following Hooke’s law, describe in words how the following second order differential equation is derived. d2x k + x =0 dt 2 m


Introduction.1

Essential Classical Mechanics

Model 2: Energy Definitions The classical expression for the kinetic energy, T, of a moving particle (translational energy) is: T = 12 mv 2

(5)

For conservative forces (path independent), the one-dimensional net force is the negative slope of the potential energy, V, with repect to position F=−

dV dx

(6)

Critical Thinking Questions 5. Write an expression for kinetic energy in terms of x .

6. Derive an expression for the potential energy, V(x), of a spring modeled according to Hooke’s law. Assume integration limits for x = 0…x and V = 0…V.

7. Which energy equation (kinetic or potential) will uniquely define a variety of different types of systems at the same temperature? Present your group’s reasoning.


Model 3: Lagrangian Mechanics It is possible to model the motion of an object in more than one way. The Lagrangian Function, L, (or simply “the Lagrangian”) of a particle is defined as the difference between the kinetic energy and the potential energy, L( x˙ , x) = T( x˙ ) − V (x)

(7)

Lagrange’s equation of motion in one dimension is d ⎛ ∂L ⎞ ∂L ⎜ ⎟= dt ⎝ ∂x˙ ⎠ ∂x

(8)

Critical Thinking Questions 8. Why are partial derivatives denoted in equation 8?

9. Write the Lagrangian, Equation 7, for a mass connected to a spring following Hooke’s law as a function of x and x .

10. Show that Lagrange’s equation of motion, Equation 8, yields the same second order differential equation identified in CTQ 4 for a mass connected to a spring.


Introduction.1

Essential Classical Mechanics

11. How was the spring system defined differently in order to use Lagrange’s equation of motion in contrast to the more familiar Newton expression?

Model 4: Hamiltonian Mechanics The Hamiltonian Function, H, (or simply “the Hamiltonian”) of a particle is defined as the sum of the kinetic energy (in terms of momentum, p) and the potential energy (in terms of a general position coordinate, q).

H (q, p) = T ( p) + V (q)

(9)

Hamilton’s equations of motion in one dimension (q = x) are given as the following set of first order differential equations

∂H = x ∂p

(10a)

∂H = − p ∂x

(10b)

Critical Thinking Questions 12. What appears to be the most significant difference in the classical equations of motion defined by Hamilton in contrast to Lagrange’s representation?

13. The momentum, p , of an object depends on two physical quantities: the mass, m, and velocity, v , p = mv

Write an expression for kinetic energy in terms of momentum instead of velocity.


14. Write the Hamiltonian, Equation 9, for a mass connected to a spring following Hooke’s law as a function of p and x.

15. Show that Hamilton’s equations of motion yield the same second order differential equation as obtained in CTQ 4 for a mass connected to a spring.

16. Using grammatically correct English sentences, describe the similarities and differences between the three equivalent ways of representing a system using classical mechanics.


Introduction.2

Blackbody Radiation Pre-Activity Questions 1.

What is the general relationship between wavelength, λ, and frequency, ν, for light waves?

2.

Complete the following table referencing the general type of radiation and the specific frequency or wavelength of the radiation identified. Type of radiation

wavelength (λ) 2.5 cm

Gamma Ultraviolet

35 nm

frequency (ν) 12 GHz 4.5 × 1020 Hz

Information Whenever a charged particle changes its state of motion, electromagnetic radiation is emitted. Thus, every object in the Universe, including people, ice cubes and fire, emit radiation. The frequency and intensity of emission depends on the temperature of the body. For example, an electric hot plate first glows dark red and eventually gets brighter and becomes orange.

Model 1: An Idealized Blackbody Although the exact distribution of frequencies emitted by a solid depends on the type of substance, a general representation, called a “blackbody,” can be used as an idealized model of radiating material. Based on the assumption that a blackbody emits and absorbs all frequencies of radiation uniformly, Classical Mechanics predicts that the energy density, ρ, of radiated light frequencies, ν , at a given temperature, T, is given by the Rayleigh-Jeans formula:

ρ(ν ,T )dν =

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8π k BT ν 2 dν c3

(1)


where kB is the Boltzmann constant, and c is the speed of light. The quantity ρ(ν ,T)dν is interpreted as the total energy emitted per unit volume of radiation in the frequency range between ν and ν + dν . The rate of energy emitted per area, or intensity, is simply the product of the energy density, ρ(ν ,T)dν and the speed of light.

Critical Thinking Questions 1. Show that the units of intensity are are J/s⋅m2 as expected.

2. Draw a rough (qualitative) sketch of a intensity versus ν for two different temperatures (i.e., low & high) on the same plot based on Model 1.

3. Based on Equation 1, for an ideal blackbody at room temperature, what range of frequencies will be emitted with a greater intensity: microwave or ultraviolet?

4. If you were to sit near a fireplace for a few hours (i.e., in the presence of a hot metal object), identify the type of electromagnetic radiation you would be exposed to the most according to classical mechanics. Explain.


Introduction.2

Blackbody Radiation

5. In view of the a plot in CTQ 2, what is the classical prediction for the total amount of energy (the area under the curve) emitted by a blackbody? Provide justification for your group’s answer.

6. Does your answer to CTQ 5 depend on temperature?

7. Are the classical predictions of blackbody radiation result consistent with the experience of each group member? Explain why or why not.

Information While the Rayleigh-Jeans formula agrees well with experimental results for low frequencies (very long wavelengths), it states that the energy density of the radiation approaches infinity. The failure of this classical result to agree with experiment is called the ultraviolet catastrophe.

Model 3: Planck’s blackbody radiation law In 1901, Max Planck found that if he did not treat energy as a continuous quantity (the classical treatment), but instead, assumed it came in discrete units propotional to frequency, E = nhν .

(2)

where n is a positve integer (n = 0, 1, 2, …), he derived the following relationship for the energy density of a blackbody,

ρ(ν ,T)dν =

8πhν 3 1 dν 3 hν kB T c e −1

(3)


The constant of proportionality, h, (called Planck’s constant) was derived from a “fit” to the experimental data, and has been found to be h = 6.626 ×10−34 J ⋅ s .

Below is a plot of the normalized intensity for three different temperatures according to Equation 3. 7000 K

Relative Intensity

2000 K

1011

300 K

1012

1013 1014 Frequency !Hz"

1015

1016

Critical Thinking Questions 8. Why is Equation 3 a better representation of the emission of a hot object than Equation 1?

9. What assumption was required in order to obtain a more accurate energy density equation corresponding to the emission of a hot object?

10. What was Planck’s justification that energy was not a continuous quantity?


Introduction.2

Blackbody Radiation

11. What does this study of blackbody radiation suggest regarding the application of classical mechanics to describe the interaction of light and matter?

Problems 1. Show that the following expression for energy density as a function of wavelength is equivalent to Equation 3. (Note you may want to use mathematical software.) ρ( λ ,T )dλ =

8 πhc 1 dλ 5 hc λkB T λ e −1

2. Create a computer generated graph of λ max , which is the peak (or maximum) of the Equation 4 as a function of T. Determine the temperature of a tungsten light bulb (that has a maximum at 1035 nm.)


Introduction.3

Photoelectric Effect Model 1: A Phototube

Light Source

Conducting Metal Ring

Vacuum Chamber Metal Surface

e-

The emission of an electron by radiation from a metal suface is called the photoelectric effect. Any electrons leaving the metal surface and hitting the metal ring will register a current, a flow of electrons, from the surface to the ring.

Voltage Current Potential Meter

The natural oscillation of an electron in the metal surface is stimulated by the oscillating electromagnetic field of incident light. According to Classical Mechanics, the intensity of these oscillations is proportional to the square of the amplitude of the of the electric field.

Critical Thinking Questions 1. Describe how to induce a current in the circuit.

2. Based on Classical Mechanics, will the number of the ejected electrons from the surface depend more on the frequency of light used or the amplitude (intensity) of light? Explain.

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3. Based on Classical Mechanics, will the energy of ejected electrons from the surface depend more on the frequency of light used or the amplitude (intensity) of light? Explain.

Model 2: Stopping Voltage Using the apparatus shown in Model 1, electrons can be prevented from leaving the metal surface by applying an opposing voltage. The specific voltage at which no current flows is called the stopping voltage V0. In order to just barely prevent electrons from reaching the metal ring, the fastest electrons’ kinetic energy must have been converted entirely to electric potential energy. Thus the maximum kinetic energy of the electron Te = eV0 where e is the charge of an electron.

Photocurrent (relative units)

C B Stopping voltage A

0

1.0

2.0

3.0

4.0

Photocurrent (relative units)

C B A

Îť=500 nm

5.0

Retarding Voltage (V)

Figure 1. Photocurrent observed as a function of opposing voltage used. In each experiment the same intensity of light is used. The frequency of light is increased from experiment A to B to C.

0

1.0

2.0

3.0

4.0

5.0

Retarding Voltage (V)

Figure 2. Photocurrent observed as a function of opposing voltage used. In each experiment 550 nm light is used. The intensity of the light is increased from experiment A to B to C.


Introduction.3

Photoelectric Effect

Critical Thinking Questions 4. Based on the information in Model 2, what is the general relationship between the stopping voltage, V0, and the kinetic energy of an electron?

5. According to the experimental data, what is the general relationship between the frequency of the incident light and energy of the ejected electrons?

6. According to the experimental data, what is the general relationship between the intensity of the incident light and energy of the ejected electrons?

7. Are these experimental results consistent with the classical wave perspective of light as a continuous source of energy proportional to its amplitude (intensity)? Explain.

8. What does this study of the photoelectric effect suggest regarding the application of classical mechanics to describe the interaction of light and matter?


Model 3:

Kinetic Energy

The following is a plot of the kinetic energy of ejected electrons as a function of frequencies of incident light hitting two different metals.

Metal A

Metal B 0

for A

0

0

for B

Light Frequency

ΦA

ΦB

Critical Thinking Questions 9. Are electrons ejected from the metal surface when the frequency of incident light is greater than ν 0 or less than ν 0 ? Explain

10. Based on the data in Model 3, propose an equation relating frequency, ν , and kinetic energy, Te, of the ejected electrons.

a. What variable is the same for the two different metals?

b. What variable is different for the two different metals?


Introduction.3

Photoelectric Effect

Model 4: In 1905, Einstein used the relationship E = hν to explain the photoelectric effect,

Te = hν − Φ

(1)

where h is Planck’s constant 6.626x10-34 J⋅sec. Based on energy conservation considerations, this equation indicates that the kinetic energy of the ejected electron, Te, is the difference in the incident light energy (proportional to its frequency) and the binding energy of the electron in the solid, Φ (the work function) is defined as a positive number.

Critical Thinking Questions 11. Identify any limits with respect to the use of Equation 1, i.e. does this relationship produce physically meaningful results for all values of frequency? Explain.

Exercises 1. When a clean surface of silver metal is irradiated with 200-nm photons, the velocity of the ejected electrons is 7.42×105 m/s. a. Calculate the work function of silver in electron volts (eV). b. What is the maximum wavelength of light that will remove an electron from silver? 2. The following data were observed in an experiment on the photoelectric effect for potassium: Kinetic Energy (eV) Wavelength (nm)

2.80

1.93

1.18

0.836

0.437

0.194

250

300

350

400

450

500

Graphically evaluate these data to obtain values for the work function and Planck’s constant.


Introduction.4

Introduction to Spectroscopy Pre-Activity Questions Review the basic mathematic conversions between the energy, frequency and wavelength of electromagnetic radiation. 1.

Comparing two photons of wavelength 400 nm and 500 nm, which would have the highest frequency? Which would have the highest energy?

2.

If a single photon has wavelength of 550 nm, what is the energy (Joules) of that photon?

3.

Calculate the energy (kJ) of a mole of photons that have a frequency of 3.75 × 1014 sec-1.

4.

What happens to the energy and wavelength of a photon when its frequency is divided by two? You can either explain this with sample calculations or use the simple mathematical relationships between frequency, energy and wavelength.

Model 1 Consider a source of light. That light will have characteristics of a wave (a frequency and wavelength). It will also have particulate characteristics. Photons can be thought of as “particles of light”. Each photon also has a characteristic energy.

Critical Thinking Questions 1. A source of light is said to have an energy of 300 kJ/mole. What is the wavelength (in nm) of that light?

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2. Another common unit used in spectroscopy is the wavenumber, which is given by the unit cm–1. What mathematical operation(s) will convert a number with units of frequency to units of wavenumbers?

3. The C–O stretching vibration transition in formaldehyde occurs at 1750 cm–1. a. What is the energy of stretching vibration transition per molecule?

b. What is the energy per mole?

4. What happens to the energy of a photon as the wavenumber increases from 2200 cm-1 to 2300 cm-1?

Exercise 1. The photons discussed in the Pre-Activity questions are found in the visible part of the spectrum. The photons that will excite the C–O stretching vibration in formaldehyde are found in the infrared part of the spectrum. Photons from which part of the spectrum have the higher energies?


Introduction.4

Introduction to Spectroscopy

Model 2

Light Intensity (relative units)

Consider a halogen source that emits photons of all different wavelengths in the visible spectrum. That is, there are photons coming from this source that have discrete wavelengths (each photon has its own wavelength), but each photon might have a different wavelength (or color - in the visible spectrum). This is the nature of a white light source. Since the source does not emit the same number of photons at each wavelength, the “spectrum” of light that would be transmitted to a photodetector might look the one shown below.

400

500

600 Wavelength (nm)

700

800

Critical Thinking Questions 5. At approximately what wavelength does this light source emit the most photons?

6. Consider what would happen if a colored “filter” that absorbed photons that have a wavelength between 500 nm and 550 nm was placed between the halogen source and the photodetector. Create a sketch of the transmission spectrum of light transmitted to the photodetector in this situation.


7. Since most of the light is still transmitted in this case it might be more useful to plot a spectrum of the light that is absorbed by the filter, rather than the light that is allowed to pass through the filter. Create a sketch of what this absorption spectrum might look like.

8. Describe the relationship between a transmission spectrum and an absorption spectrum.

Model 3

400

500

656

486

434

410

Absorbance Absorbance (relative units)

If we were to take out the filter from between the source and detector and place a container of monatomic hydrogen gas, the absorption spectrum might look like the hypothetical spectrum shown below.

600 Wavelength (nm)

700

800


Introduction.4

Introduction to Spectroscopy

Critical Thinking Questions 9. Identify two qualitative differences between the model absorption spectrum of hydrogen and the spectrum you drew in CTQ 7.

10. Does a sample of hydrogen atoms absorb photons with an energy of 3.5 × 1019 J?

11. Given the model spectrum, what photon energies does atomic hydrogen absorb?

Model 4

Energy (J)

The energy level diagram for atomic hydrogen is given below. It is a fundamental observation in quantum mechanics that at a microscopic level all particles exhibit discrete energy states (called energy levels) and that each particle can exist at only one energy level at a time. Under certain conditions, it is possible for the energy of the system to change from one energy level to another through absorption or emission of the appropriate amount of energy. –4.5 x 10–20 –6.1 x 10–20 –8.7 x 10–20 –13.6 x 10–20 B

–24.2 x 10–20

A

–54.5 x 10–20


Critical Thinking Questions 12. What is the energy difference between energy levels marked A and B?

13. For a hydrogen atom to change from energy level A to B would the atom need to absorb energy or emit energy?

14. What wavelength of light is equal to the energy difference between A & B?

15. Is this photon of light absorbed or emitted for a hydrogen atom to change from energy level A to B?

16. Is this photon wavelength shown in the spectrum of atomic hydrogen?

17. For each peak displayed in the Model 3 absorption spectrum of atomic hydrogen, draw an arrow on the Model 4 energy level diagram corresponding to the appropriate energy transition. Label each arrow in a manner that clearly specifies the resultant peak.


Introduction.4

Introduction to Spectroscopy

Exercises 2. Consider the emission of photons corresponding to transitions from higher energy levels to energy level A. To which part of the electromagnetic spectrum do these wavelengths correspond? 3. Consider the emission of photons corresponding to transitions from higher energy levels to energy level B. To which part of the electromagnetic spectrum do these wavelengths correspond?


Introduction.5

The Energies of Molecules (How do Molecules Move?)

Model 1 A useful approximation is that the various motions that a particle has can be considered individually and independently, and the associated energies can be summed to give the total energy of the particle. That is, the presence (or absence) of one type of motion has no effect on the other types of motion. Table 1.

Energy Values for Various Particles

Particle

Molar Mass (g/mole)

H(g) C(g) O(g) F(g) Si(g) Ar(g) CO(g) HF(g) HCl(g)

1.01 12.01 16.00 19.00 28.09 39.95 28.01 20.01 36.46

Translational Rotational Vibrational Electronic Energy Energy Energy Energy 25°C 25°C 25°C 25°C (kJ/mole) (kJ/mole) (kJ/mole) (kJ/mole) 3.72 0 0 –1,312 3.72 0 0 –99,386 3.72 0 0 –197,182 3.72 0 0 –262,040 3.72 0 0 –761,102 3.72 0 0 –1,389,178 3.72 2.48 12.8 –297,654 3.72 2.48 23.7 –263,922 3.72 2.48 17.3 –1,211,909

Critical Thinking Questions 1. What correlation is there, if any, between the molar mass of a particle and

\

a)

translational energy?

b)

rotational energy?

c)

vibrational energy?

d)

electronic energy?

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2. What do all particles with no rotational energy have in common?

3. What do all particles with no vibrational energy have in common?

Information The center of mass of a diatomic molecule is defined as shown in Figure 1.

Figure 1.

Definition of the Center of Mass of a Diatomic Molecule. r r1

m1

r2

m2

center of mass

m1 r1 = m2 r2

Critical Thinking Questions 4. Is the center of mass of a diatomic molecule closer to the heavier atom or the lighter atom?


Introduction.5

The Energies of Molecules

Model 2: Translational Energy direction of motion of center of mass Translational Motion: Center of mass of molecule moves in some direction. center of mass for CO molecule 1

The translational energy of a molecule can be thought of as 2 mv2, where all of the mass of the molecule, m, is located at the center of mass of the molecule.

Critical Thinking Questions 5. Which atom is the carbon atom in Model 2? Present your group’s reasoning.

6. The internuclear distance (bond length) in CO is 113 pm. The triple-bond radius of a carbon atom is 60 pm, and the triple-bond radius of an oxygen atom is 53 pm. Does this mean that the center of mass of CO is located on the straight line connecting the two nuclei at 60 pm from the carbon nucleus?

Exercises 1. The molar mass of 12C is exactly 12 g/mole. The molar mass of 16O is 15.995 g/mole. Determine the mass of one carbon-12 atom (to four significant figures). Determine the mass of one oxygen-16 atom (to four significant figures). 2. Determine the location of the center of mass of 12C16O to four significant figures given that the internuclear distance is 113.1 pm.


Model 3: Rotational Energy time 3 time 2 time 1

time 4 etc.

Rotational Motion: Center of mass is fixed. Distance between nuclei does not change. Nuclei rotate around the center of mass.

The energy associated with the motion of two or more nuclei around the center of mass of a molecule is the rotational energy.

Critical Thinking Question 7. Why is the rotational energy of C(g), O(g), and Ar(g) equal to zero?

Model 4: Vibrational Energy. time 1 time 2 time 3 time 4 time 5

Vibrational Motion: Center of mass is fixed. Distance between nuclei does change.

The energy associated with the change of the distance between nuclei is the vibrational energy.

Critical Thinking Question 8. Why is the vibrational energy of C(g), O(g), and Ar(g) equal to zero?


Introduction.5

The Energies of Molecules

Model 5: Electronic Energy. The electronic energy is the sum of the following: • The coulombic interaction of all of the electrons in the molecule with all of the nuclei in the molecule (all of these are attractive forces and negative energies). • The coulombic interaction of all of the electrons in the molecule with each other (all of these are repulsive forces and positive energies). • The coulombic interaction of all of the nuclei in the molecule with each other (all of these are repulsive forces and are positive energies). • The magnetic interaction of all of the particles (electrons and nuclei) in the molecule. • The kinetic energy of the electrons. The motion of the center of mass of the molecule, the rotation of the nuclei around the center of mass, and the motion of the nuclei with respect to each other are not part of the electronic energy.

Critical Thinking Questions 9. What or whose law mathematically describes the potential energy of interaction between a positively charged particle and a negatively charged particle?

10. Use this law (answer to CTQ 9) to show that the potential energy of interaction between a positive and a negative charge is a negative quantity.


Information Figure 1.

Carbon monoxide has 14 electrons

The electronic energy, Uelec, of CO can be thought of as the ΔU for the following process: 14 e– (stationary) + C6+ (stationary) + O8+ (stationary) = CO (stationary) (1) Equation (1) can be written as the following thermodynamic sequence: Process C6+(g) + 1 e–(g) = C5+(g) C5+(g) + 1 e–(g) = C4+(g) C4+(g) + 1 e–(g) = C3+(g)

ΔU (kJ/mole)

Quantity Description*

–47,276

(–IE6 of carbon)

–37,829

(–IE5 of carbon)

–6,223

(–IE4 of carbon)

C3+(g) + 1 e–(g) = C2+(g) C2+(g) + 1 e–(g) = C1+(g) C1+(g) + 1 e–(g) = C(g) O8+(g) + 1 e–(g) = O7+(g)

–4,620

(–IE3 of carbon)

–2,352

(–IE2 of carbon)

–1,086

(–IE1 of carbon)

–84,074

(–IE8 of oxygen)

O7+(g) + 1 e–(g) = O6+(g) O6+(g) + 1 e–(g) = O5+(g) O5+(g) + 1 e–(g) = O4+(g) O4+(g) + 1 e–(g) = O3+(g)

–71,322

(–IE7 of oxygen)

–13,326

(–IE6 of oxygen)

–10,989

(–IE5 of oxygen)

–7,469

(–IE4 of oxygen)

O3+(g) + 1 e–(g) = O2+(g) –5,300 (–IE3 of oxygen) O2+(g) + 1 e–(g) = O1+(g) –3,388 (–IE2 of oxygen) O1+(g) + 1 e–(g) = O(g) –1,314 (–IE1 of oxygen) C(g) + O(g) = CO(g) –1,086 (–CO BE)† 14 e–(g) + C6+(g) + O8+(g) = CO(g) –297,654(electronic energy of CO)

*

IEi is the ith ionization energy The normal CO bond energy (BE) of 1074 kJ/mole has been corrected by 12 kJ/mole to account for the rotational, vibrational, and translational energy change between CO(g) and the separated atoms. †


Introduction.5

The Energies of Molecules

Critical Thinking Questions 11. Why is the value of i in IEi equal to 5 in the following equation? C4+(g) = C5+(g) + 1 e–(g) 12. Is energy required or released for the reaction in CTQ 11? Why?

13. Why is IE6 for carbon more positive than IE1 for carbon?

14. Why is IE1 for oxygen more positive than IE1 for carbon?

15. IE1 for oxygen is 1,314 kJ/mole. The IEi for oxygen increase gradually to 13,326 kJ/mole for IE6. Why is IE7 for oxygen so large, 71,322 kJ/mole?

16. Is energy required or released when C and O form CO? Explain.

17. According to the thermodynamic sequence given for equation (1), what is the value of the electronic energy of CO?

18.

According to the thermodynamic sequence given for equation (1), what is the value of the bond energy of CO?


19. What percent of the electronic energy of CO is due to the CO energy released when C(g) and O(g) react to form CO(g)? (Calculate a numerical value.)

20. The electronic energy for CO is a very large negative number. Therefore, which is stronger: the attractive forces between the electrons and the nuclei, or the repulsive forces (electron/electron and nuclei/nuclei)? Explain.

Exercises 3. Explain why the electronic energy of a molecule is a negative quantity whereas the translational, rotational, and vibrational energies are positive quantities. 4. Explain why, in general, the electronic energy of an atom or molecule becomes more negative as the molar mass increases. 5. Use the electronic energies of H(g) and O(g), see Model 1, to estimate the electronic energy of OH(g). Correct this value by taking into account the bond energy of OH, 459 kJ/mole. What fraction of the electronic energy of OH is due to the OH bond energy (calculate a numerical value)? 6. Predict which atom in each of the following pairs has the more negative electronic energy. Explain your reasoning. a. Be or Mg b. Si or K c. Ne or Na 7. Predict which molecule in each of the following pairs has the more negative electronic energy. Explain your reasoning. a. I2 or F2 b. CH4 or CF4 c. H2O or H2S


Introduction.5

The Energies of Molecules

Problems 1. a) Find the appropriate ionization values in the Handbook of Chemistry and Physics and calculate Uelec for Li(g). b) Ignore the bond energies and calculate Uelec for LiH(g) and LiF(g). [Hint: see Model 1.] 2. Using grammatically correct English sentences, define the center of mass of a diatomic molecule with atoms of mass m1 and m2.


Introduction.6

Bohr Model Can we use “classical mechanics” to describe atoms?

Pre-Activity Questions 1. What is the classical law that describes the interaction between two point charges? Identify both the potential energy and force equation form of this law clearly defining all variables and parameters (constants). 2. What is the classical force equation for motion in a circle?

Model 1: Bohr Model I. The electron in a hydrogen atom “stationary state” moves in a circle (orbit) around the nucleus obeying the laws of classical mechanics. II. An atom can take only certain distinct energies “stationary states” labled by integer n = 0…∞. III. A photon of light is emitted with transitions from higher to lower energy stationary states. En f − Eni = hν

(E

ni

> En f

)

IV. The allowed orbits are those for which the electron has an angular momentum. mevr = n

Critical Thinking Questions 1. For each “postulate” of the Bohr Model identified in Model 1, sketch an approriate figure illustrating each statement.

\

PGLSAG003007a


2. Write an expression equating the classical force equation for motion in a circle (i.e. centrfugal force) and the magnitude of the coulombic force expression between an electron with charge e and nucleus with charge Ze where Z is the number of protons. Note: This equation is the classical requirement for a stable orbit.

3. Using your answer to CTQ 2 and the assumption expressed in Bohr’s 4th postulate: a) derive an expression for the velocity of an electron according to the Bohr model of an atom.

b) derive an expression for the radius of an electron according to the Bohr model of an atom.

4. Write the classical Hamiltonian expression for energy, En (the sum of kinetic and potential energies) for Model 1. (Assume the nucleus does not move.)


Introduction.6

Bohr Model

5. Using the classical Hamiltonian expression for energy and appropriate expressions for velocity and radius in CTQ 3, derive an expression for the energy, En, for an arbitrary stationary state.

6. A crucial test of Bohr's model was whether it could correctly predict the wavelengths of light in the spectrum of hydrogen. Calculate the wavlengths of the light emitted by transitions from the n = 3 to n = 2 levels. How good is the agreement with the experimental value?

Exercises 1. Calculate the velocity of an electon in the first stationary state of a hydrogen atom (according to the Bohr model.) 2. By the end of the 19th century, it was known that atoms were on the order of a few tenths of a nm in diameter. Calculate the radius of the first Bohr orbit and tell whether Bohr's model predicts a reasonable size for the atom. 3. The ionization energy of hydrogen was known to be about 13.6 eV. This is the energy that must be supplied to pull the electron so far away from the atom that it becomes essentially free. In Bohr's model, the binding energy is the magnitude of the lowest energy state. Calculate that energy in eV and compare it with the known ionization energy. 4. Use the Bohr Model to calculate the value of the Rydberg constant. Identify the significance of this result.


Fundamental.1

The Postulates of Quantum Mechanics What are the basic assumptions of Quantum Mechanics?

Pre-Activity Questions Review the basic momentum and energy relationships defined by Classical Mechanics. 1.

The total energy of the system is made up of two parts. What are they?

2.

Write two expressions for kinetic energy, one in terms of velocity and one in terms of linear momentum.

3.

A conservative force is a function of position only. Write an equation defining the potential energy, V, of a system in terms of a one-dimensional force, F(x).

4.

Identify two important variables that can be used to specify the “state” of a classical system, e.g. what would you need to know in order to predict where to find a baseball in a stadium?

Information We start with the postulates of quantum mechanics because these are the basic tools we will use to develop our understanding of various systems. A postulate is nothing more than an assumption that is made without proof. The proof of these postulates lies in the fact that they work. That is, when appropriately applied to a chemical system, the results can be experimentally verified.

Postulate 1: Quantum States The state of a quantum mechanical system is completely specified by the wavefunction, ψ (q) , where q represents the set of spatial coordinates that the wavefunction depends upon. This mathematical function contains all the information (e.g. location and momentum) that can be known about a single particle or a system of particles. The wavefunction must be finite, continuous, and single valued for all possible values of q.

\

PGLSAG003008a


Critical Thinking Questions 1. Consider a single argon atom in a closed 3D container. Propose one possible set of spatial coordinates, q, that Ďˆ would be dependent upon.

2. Explain why the function f (x) = 1 / (2 − x) would not be an acceptable wavefunction for all values of x.

3. Clearly specify one set of conditions on the coordinate, x, under which the expression identified in CTQ 2 could be a valid wavefunction.

4. Draw an example of a function that is not an acceptable wavefunction.


Fundamental.1 The Postulates of Quantum Mechanics

Postulate 2: Quantum Properties The measureable properties or observables of a system (e.g, position, momentum, energy) are represented by mathematical operations defined by operators. Operators are denoted by a carat representing the mathematical operation, not the variable. All operators corresponding to physical observables will be derived from the expressions for position and momentum identified in Table 1. Note: the momentum operator contains the imaginary unit i which is defined as the square root of –1 and  (called h-bar) is Planck’s constant, h, divided by 2π. Table 1: One-Dimensional operators for physical properties Observable

Operator Symbol

Operation

Description

position

x⋅

multiply by x

momentum

pˆ x

−i

d dx

take the first derivative with respect to x and multiply by −i

Critical Thinking Questions 5. Using grammatically correct English sentences, describe the difference between x and xˆ .

6. What is the result of the momentum operator acting on the function f (x) = x ?

ˆ ? 7. What is the result of carrying out xy


8. Write the classical expression for kinetic energy in terms of momentum in the x-direction. Create the corresponding operator, Tˆx by replacing the momentum variable with the corresponding operator identified in Table 1.

9. The procedure described above, can be used to create any operator corresponding to a physical observable.* In the third row of Table 1, enter the appropriate information based on your group’s answer to CTQ 8.

Postulate 3: Quantum Measurements For a single measurement of a property with associated operator Aˆ , all measurable quantities are given by the eigenvalue equation, Aˆψ (q) = aψ (q)

(1)

where a is a constant. ψ(q) is said to be an eigenfunction of the operator Aˆ and a is its associated eigenvalue corresponding to the value of the measureable quantity, i.e. the observable.

Critical Thinking Questions 10. Is ψ (x) = x an eigenfunction of the momentum operator? justification for your group’s answer.

*

A summary of useful Quantum operators are provided in Appendix A.3

Provide the


Fundamental.1 The Postulates of Quantum Mechanics

11. Is it possible to identify the measurable value of momentum corresponding to a state represented as ψ (x) = x ? Present your group’s rationale.

12. Give one example of a function that is an eigenfunction of the momentum operator and identify the corresponding observable.       13. If necessary, revise your group’s eigenfunction in CTQ 12 such that the observable (eigenvalue) is real and not an imaginary number.         14. Based on Postulate 3, what information is necessary to determine the kinetic energy of an argon atom according to Quantum Mechanics?

Postulate 4: Quantum Equation The wavefunction specifying the state of a system is the solution to the timeindependent Schrödinger equation,†

Hˆ ψ ( q ) = Eψ ( q )

(2)

where Hˆ is the Hamiltonian operator corresponding to the total energy, E, of the state ψ(q) †

A more complete treatment of a Quantum Mechanical system also includes time dependence.


Critical Thinking Questions 15. Is the Schrรถdinger Equation an eigenvalue equation? Why or why not.

16. In order to use Equation 2 to solve for the wavefunction representing the state of an argon particle, what must be defined?

17. Consider a hypothetical particle of mass m, confined to a single dimension, i.e. a line. Assume that there are no forces acting on this particle. a. What is the classical expression for kinetic energy (in terms of momentum) of this system?

b. What is the classical expression for potential energy, V, of this system?

c. What is the classical expression for the total energy, E, of this system?

d. Using the information given in Postulate 2, write the general functional form of the quantum Hamiltonian operator for this system.

e. Use the previous answer to write the corresponding Schrรถdinger equation for this system.


Fundamental.1 The Postulates of Quantum Mechanics

18. Assuming you were to “solve” the Schrödinger equation identified above, what would you expect to get? A number, or function, or both? Present your group’s rationale for this answer.

19. Based on the postulates presented, summarize the basic steps necessary to calculate an experimentally observed property of a system.

Exercises 1. Identify the operation corresponding to pˆ x2 . 2. Show that the function eα x is an eigenfunction of the pˆ x2 operator. What is the associated eigenvalue? 3. Is the function xeα x an eigenfunction of the pˆ x operator? If so, what is the associated eigenvalue?


Fundamental.2

The Particle-on-a-Line Model What can we learn from a simple quantum model?

Pre-Activity Question 1. Consider an argon atom moving at a speed of 300 m/s in a vacuum. According to Classical Mechanics, what is the energy of this particle? 2. Suppose you wanted to restrict the motion of an object to a certain region of space. Create two different plots of potential energy as a function of position, one corresponding to a weak force and the second a strong force trapping an object to move only in a confined region. Explain your reasoning.

Model 1 Consider a particle (e.g. Ar atom) with mass, m, restricted to motion in a onedimensional (1D) region of space of length, a. The potential energy, V(x), is infinite for all values of x ≤ 0 and x ≥ a . Between x = 0 and x = a the potential energy is zero. ∞

V(x)

0 0

\

PGLSAG003009a

x

a


Critical Thinking Questions 1. Although it is not physically realistic for a system to have infinite potential energy: a. what is the magnitude of the corresponding force (in essence) for positions x ≤ 0 and x ≥ a ?

b. clearly describe the region of space where the particle may be found and where it will not be present.

2. Is there energy associated with the moving particle between x = 0  and x = a ?     If  so,  what type of energy? Present your group’s justification for this answer.

3. Assume Classical Mechanics is used to characterize Model 1: a. What is the smallest energy possible for this particle confined to   a   line? Record your group’s justification for this answer.

b. Within a physical range of particle velocities, are there any restrictions to the amount of energy associated with this particle confined to   a   line? Explain.


Fundamental.2

The Particle-on-a-Line Model

4. Assume Quantum Mechanics is used to characterize Model 1: a. Within the limits x = 0 and   x = a , write the functional form of the Hamiltonian operator, Hˆ .   b. Use the functional form of the Hamiltonian to write an expression for the corresponding Schrödinger equation for the particle-on-a-line model.  

Model 2 The following is a general solution to the particle-on-a-line model Schrödinger equation,

ψ (x) = Asin kx + B cos kx

(1)

with

E=

k 22 2m

(2)

Critical Thinking Questions 5. Show that the eigenfunction and energy eigenvalue identified in Model 2 is a solution to the Schrödinger equation your group proposed in CTQ 4.b.


6. Although Equation 1 is a solution to the Schrödinger equation, it must also satisfy any limits or “boundaries” defined in Model 1. a. Recall that the wavefunction represents all the information that can be known about the particle defined by Model 1. In view of CTQ 1b, explain why it is appropriate that ψ (x) = 0 when x ≤ 0 .

b. Does Equation 1 satisfy this boundary condition, i.e. ψ (0) = 0 , for any value of A & B? If not identify the appropriate value(s) required for the wavefunction to be a true solution.

7. Now ensure that the function ψ (x) also satisfies the boundary condition ψ (a) = 0 . a. Assuming n is any positive integer, propose an expression that identifies all values of α that make sin α = 0 a true statement.

b. Propose a general equation for k (as a function of n) such that sin ka = 0 .

8. In view of the boundary condition requirements for B and k: a. Rewrite the solution of the particle-on-a-line wavefunction, ψ (x) in terms of n and length a.

b. Write a general expression for the energy, E, of a particle on a line.


Fundamental.2

The Particle-on-a-Line Model

9. Consider the value of Ďˆ (x) when n = 0 . a. What would this solution imply about the particle? (Recall Postulate 1 in Fundamental.1)

b. Should this be considered a possible solution describing a particle-on-aline? Explain.

10. Write an expression identifying all allowed values of n that are referred to as the quantum numbers for the particle-on-a-line model.

11. Thinking about the questions in Model 2, how did this unique set of integers, i.e. quantum numbers, evolve as an important requirement of the particle-ona-line solutions defined in CTQ 8?

12. The quantum state with the overall smallest value of energy is called the ground state. Identify the value of the quantum number for the particle-on-aline ground state.

13. The value of the ground state energy is called the zero point energy. Is the zero point energy for the particle-on-a-line model equal to zero? Justify this answer.


14. Based on the Quantum Mechanical solution describing a particle confined to a 1D region of space: a. How many unique states can be defined for a single particle confined to a line.

b. How is each unique state identified?

c. Are there any restrictions to the amount of energy associated with this particle confined to a line? Provide your group’s justification for this answer.

15. What happens to the energy of the particles as n increases?

16. What happens to the energy of the particle for the state n=1 if the length of the line is increased to 2a?


Fundamental.2

The Particle-on-a-Line Model

17. Use the results of the particle-on-a-line model to explain the following statement: An electron in a π -system, such as 1,3 butadiene or benzene, is spread out (delocalized) over a large region of space (compared to an electron bound only to a single atom or to an electron confined to two atoms in a normal covalent bond). It is well-known that the electrons in a π -system are at a lower energy than in a normal π bond (the energy is said to be lowered by delocalization).

18. For 1D confined motion (kinetic energy only) between two boundaries, summarize the unique features of the quantum description of energy that differ from a classical description of this system.

Exercises 1. For n = 1…4, provide an expression for En as function of m and a. 2. Develop expressions for the difference in energy between E1 & E2, E2 & E3, and E3 & E4 as function of m and a. How does the difference in energy between two sequential states change as the mass of the particle increases? Describe the relationship between the difference in energy between two sequential states and the length of the line. 3. Derive an expression for the difference in energy between two sequential states, ΔE n →n +1 = E n +1 − E n . How does this difference in energy change as n increases?


4. Using expressions from Exercise 3, compare the difference in energy between two sequential states for an argon atom confined to a line of length: a. 300 pm (the diameter of a C70 molecule). b. 1 cm (the diameter of a test tube) 5. Compare the calculated zero-point energies according to Quantum Mechanics and Classical Mechanics for a xenon atom confined to a 10 nm line.

Problems 1. Electronic transitions of the molecule butadiene, CH2=CH-CH=CH2, can be approximated using the particle-on-a-line model, if one assumes an electron in butadiene can span the entire four-carbon chain. a. If a butadiene molecule absorbs a photon with a wavelength 2170 Ă… and an electron transitions from the n = 2 to the n = 3 state, what is the approximate length of the C4H6 molecule? b. Describe why it is quite remarkable that the predicted length of butadiene based on the particle-on-a-line model is close to the experimental value of 4.8 angstroms. 2. Instead of the boundary conditions defined at x = 0 to a, assume that the limits for x = -a/2 to +a/2. Derive acceptable wave functions for this particle-on-aline. What are the quantized energies for the particle?


Fundamental.3

The Language of Quantum Mechanics What does the wavefunction tell us about a system?

Pre-Activity Question ⎛ nπ x ⎞ 1. Sketch the  particle-­‐on-­‐a-­‐line  wavefunction,   ψ n (x) = Asin ⎜ , for   ⎝ a ⎟⎠ values  of  the  quantum  number   n = 1…4 .  (Note:  assume  A  =  1)

1(x)

2(x)

0

a 4

x

a 2

3a 4

a

3(x)

0

a 4

x

a 2

3a 4

a

4(x)

0

a 4

x

a 2

3a 4

a

0

a 4

x

a 2

3a 4

a

Model 1 The wavefunction is a mathematical function describing the state of a system. 2 The physical significance, however, is found in the square of this function, ψ , which is proportional to the probability of finding the particle at a particular location.

\

PGLSAG003010a


Critical Thinking Questions 1. Use the functions drawn in the pre-activity question to sketch the square of the wavefunction for the n =1 state, and identify the position(s) where the particle is most likely found.

2. Consider the n = 2 state. Sketch the square of this wavefunction and identify the position(s) where the particle will most likely be found.

3. Based on your answer to CTQ 2, predict the average value of x for a particle described by the n =2 state.

4. For the n =2 state, a. Identify any position(s) where the wavefunction changes sign. positions are referred to as nodes.

Such

b. What is the probability of finding a particle at this location?

c. What is the general relationship between the number of node and the quantum number, n?


Fundamental.3

The Language of Quantum Mechanics

5. Imagine a sketch of ψ for n=1 and n=5 and n=500. As n increases, where does it become increasingly probable to find the particle in comparison to lower energy states? 2

Term 1: Probability For a 1D system, the probability, P, of finding a particle in the region between x1 and x 2 can be evaluated quantitatively as

P=

x2 x1

ψ dx = 2

x2 x1

ψ * (x)ψ (x)dx

(1)

* where ψ 2 is the probability density and ψ (x) is the complex conjugate of ψ (x)

obtained by reversing the sign of any imaginary component. For example, the complex conjugate of eix is e-ix. If the function ψ (x) is real, then ψ * (x) = ψ (x) . Generally, wavefunctions are scaled or normalized such that the integral (i.e. sum) of probabilities over the full coordinate space (all space) is equal to one, 2 ∫ ψ dτ = 1 where dτ is the appropriate differential volume element defined by all space

the coordinate space. For a 1D system dτ = dx .

Critical Thinking Questions 6. Identify the limits of integration, x1 and x2, representing the full coordinate space (all space) for the particle-on-a-line model.

7. Using the particle-on-a-line wavefunction, set up a specific expression that when evaluated would indicate whether or not this wavefunction is normalized.


⎛ nπ x ⎞ a sin 2 ⎜ dx = , what specific ⎟ 0 2 ⎝ a ⎠ modification could be made to the wavefunction such that the expression in CTQ 7 is one?

8. Assuming the following integral solution:

a

9. As discovered in CTQ 8, a multiplicative constant is typically found such that Aψ (x) is the normalized wavefunction. In Set up a general expression to solve for the normalization constant A in terms of ψ (x) .

Exercises 1. Describe how the curvature (steepness of the slope) of the wavefunction changes as the energy of the particle increases. 2. Using an argument based on the general form of the Schrödinger Equation, explain why if ψ (x) is a solution to the Schrödinger equation, then Aψ (x) must also be a solution if A is a constant. 3. Using the normalized particle-on-a-line wavefunction, evaluate the probability a of finding the particle in the region 0 ≤ x ≤ . 2 4. Using the particle-on-a-line model, determine the probability of finding the a a a particle in the region 0 ≤ x ≤ and ≤ x ≤ . Does this answer depend on 4 4 2 the value of n? Does the form of the probability make sense?


Fundamental.3

The Language of Quantum Mechanics

Term 2: Orthonormality While the normalization integral represents the product of two wavefunctions of the same state, the overlap of two different states can also be considered. An important consequence of the fact that the eigenvalues of quantum mechanical operators must represent real physical observables, is that the eigenfunctions representing two different states, i and j, must be orthogonal

−∞

ψ *iψ j dτ = 0

when i ≠ j

(2)

Wavefunctions that are both normalized and orthogonal form an orthonormal set such that

⎧⎪ 1 i = j * ψ ψ d τ = ⎨ ∫−∞ i j ⎪⎩ 0 i ≠ j ∞

(3)

Critical Thinking Questions 10. Provide one example of two functions that are orthogonal.

11. If two functions are orthogonal, what is the net overlap of the functions over all space?

12. Make a sketch of the n = 1 and n = 2 wavefunctions for the particle-on-a-line model on the same plot.

a. How would the sum of the product of these two wavefunctions for x < 2a compare to the sum of the product of these two wavefunctions for x > 2a ?

b. Are these two states orthogonal?


Exercise 5. For the particle-on-a-line model, set up and evaluate the appropriate integral expression to show that the n = 1 and the n = 2 wavefunctions are indeed orthogonal.

Term 3: Average Values For a system described by the normalized wavefunction, ψ (q) , the average value of the observable a, also referred to as the expectation value, is given by

a =

ψ * Aˆψ dτ

(4)

all space

where Aˆ is the operator associated with the observable.

Critical Thinking Questions 13. Using a grammatically correct English sentence, describe what the symbol x represents.

14. Set up the integral to find the expectation value of x for the particle-on-a-line using a normalized particle-on-a-line wavefunction. Simplify your expression as much as possible, but do not evaluate it.


Fundamental.3

The Language of Quantum Mechanics

15. Is the answer your group predicted for CTQ 3 consistent with the following ⎛ nπ x ⎞ 2 a a integral solution: ∫ x sin 2 ⎜ dx = ? Explain. ⎟ a 0 2 ⎝ a ⎠

16. Does x depend on quantum number n? Explain why this result is consistent with the ψ . 2

17. Although it is possible to determine the average position for a particle-on-aline, is it possible to determine its exact position, i.e., an observable representing a measureable value for x? (Hint: Recall Postulate 3 in Fundamental.1)

Exercises 6. Using the normalized particle-on-a-line wavefunction, calculate the average momentum of the particle p . Is this result consistent with the value you would predict? Explain why or why not.


7. For the particle-on-a-line model, consider the probability of finding the particle between between x = 0 and x = a . a. According to classical mechanics is there any point on the line that the particle would more likely be found? Sketch a plot of P(x) for a classical system in this region. b. Is the probability of finding a particle at any particular point on the line more similar to the classical result for the n =2 state or the n =50 state? Explain. 8. For the particle-on-a-line model, consider the average value of the particle’s position between x = 0 and x = a . a. According to classical mechanics what is the average position of the particle in this region? Justify this result. b. Is the average position of the particle more similar to the classical result for the n =2 state or the n =50 state? Explain. 9. Consider the function ( 30 a 5 )

1/ 2

(ax − x ) 2

a. Is this function normalized? b. Is this function a possible solution to the Schrödinger equation for the particle-on-a-line model? Why or why not? 10. Propose a more general form of equation (3) such that ψ (x) does not necessarily need to be normalized.


Fundamental.4

Molecular Translation How do quantum solutions change if we increase the dimensions of the system?

Pre-Activity Questions 1. What is meant by the term “translational energy”? 2. Write the energy corresponding to a 1D particle-on-a-line in terms of Planck’s constant, h, rather than  .

Model 1 Consider a particle (e.g. argon atom) with mass, m, restricted to a twodimensional region of space 0 ≤ x ≤ a and 0 ≤ y ≤ b for which V(x, y)=0. Outside this region the potential energy is infinite.

Critical Thinking Questions 1. Draw the region of space specified for the system in Model 1. In what way(s) does the boundary differ for this 2D wavefunction, ψ ( x, y ) in comparison to the particle-on-a-line model.

2. Given the model conditions, write the Hamiltonian for a particle confined to a particular value of y and allowed to move freely in the x-direction. What type of system does this Hamiltonian define?

\

PGLSAG003011a


3. Given the model conditions, write the Hamiltonian for a particle confined to a particular value of x and allowed to move freely in the y direction.

4. Is the motion of the particle in the x direction dependent upon the motion in the y direction? Record your group’s rationale for this answer.

5. Use the answers from CTQ 2 & 3 to formulate a Hamiltonian for a particle confined to a plane.

Model 2 The following are the normalized solutions to the Schrödinger equations for a particle confined to a plane defined in Model 1.

ψ n x ,n y (x, y) =

n πy 4 n πx sin x sin y ab a b

En n (x, y) = x y

nx2 h 2 8ma 2

+

ny2 h 2 8mb 2

The contour plots below represent three different wavefunctions for a system such that a = b corresponding to the length of the horizontal x axis and vertical y axis, respectively. b

b

b –

y

+

y

+

y +

0

0

x

(i)

a

0

0

+

+

x

(ii)

a

0

0

x

(iii)

a


Fundamental.4

Molecular Translation

Critical Thinking Questions 6. How many quantum numbers are necessary to specify the state of a particleon-a-plane?

7. Compare the functional form of the wavefunction derived for a particle confined to a 2D plane, ψ nx ,ny , to one confined to a 1D line, ψ nx or ψ ny . Does there appear to be a general relationship between the two solutions? If so, specifically identify how they are related.

8. Compare the expression derived for translational energy for a particle confined to a 2D plane, Enx ,ny , to one confined to a 1D line, Enx or Eny . Does there appear to be a general relationship between the two solutions? If so, specifically identify how they are related.

9. Recall that a node is defined where the wavefunction changes sign. a. Draw where the node(s) exist for wavefunction (iii). How do the nodes of a 2D wavefunction differ from the 1D particle-on-a-line model?

b. Identify the similarities and differences for the node(s) corresponding to wavefunctions (ii) and (iii).


10. Which contour plot represents ψ 1,1 ? Provide your group’s justification for this answer.

11. Identify the position(s) where the particle corresponding to ψ 2,2 is most likely found.

12. Identify the values of the quantum numbers nx and ny for the state represented by contour plot (iii).

13. Rank the three wavefunctions identified in Model 2 in terms of increasing energy.

14. Based on the energy equation, is it possible for a state with different values of nx and ny to have the same energy as the state represented by contour plot (iii)? Provide justification for your group’s answer.

15. Draw the corresponding contour plot for the state identified in CTQ 14 with the same energy as the state represented by contour plot (iii).


Fundamental.4

Molecular Translation

Exercise 1. Assume one more dimension, 0 ≤ z ≤ c , is added to represent a 3D particle-inbox. In view of your answers to CTQ 7 and 8, write expressions for both the wavefunction and energy for this model.

Information It is very common for two or more states to have the same energy. Different states with the same energy are said to be degenerate. The number of states that have the same energy is called the degeneracy of that state.

Critical Thinking Questions 16. Draw and label an energy diagram depicting the eight lowest energy states for the 2D system described in Model 2. Be sure to clearly indicate any degenerate states.

17. How many unique values of energy (or energy levels) are represented by these eight unique energy states?

18. If the length a ≠ b , would the two states identified in CTQ 14 be degenerate? Present justification for your group’s answer.


Exercises 2. For a particle in a 3D cubic box, a = b = c, the energy corresponding to the state ψ n x ,n y ,n z is: E n x ,n y ,n z

2 2 n z2 h 2 n x2 h 2 n y h = + + 8ma 2 8mb 2 8mc 2

a. Which state is lower in energy, ψ 2,2,2 or ψ 3,2,1 ? b. Determine the degeneracy of the ψ 3,3,3 state. 3. For a particle in a 3D rectangular box where a = b and c = 2a, rank each of the following states according to their energy (lowest energy to highest energy): ψ 2,1,2 , ψ 2,2,1 , ψ1,2,2 . 4. The average translational energy of one mole of CO molecules at 25°C is 3.72 kJ. a. What is the average translational energy of one CO molecule at 25°C? b. For a box of dimensions x = 10 cm, y = 10 cm, and z = 10 cm, let nx = ny = nz = n, and calculate the value of n for a CO molecule which has the average translational energy at 25°C.


Fundamental.5

The Particle-on-a-Spring Model What might be the molecular equivalent to a spring?

Pre-Activity Questions 1. Write an equation describing the potential energy of an object connected to a spring following Hookeâ&#x20AC;&#x2122;s Law. Sketch a plot (clearly labeled) comparing the potential energy as a function of the position for this object attached to a stiff spring and a loose spring. 2. Based on classical mechanics, describe how the total energy, E, of an object attached to a spring changes with position. How do potential, V, and kinetic, T, energies change with position? On the potential energy plot, clearly indicate the minimum and maximum positions assuming a total energy equal to E for the object.

Model 1 Consider a particle with mass, m, attached to a spring with a restoring force proportional to the displacement, x, of a spring (i.e. Hookeâ&#x20AC;&#x2122;s Law). The proportionality constant, k is called the force constant of the spring.

Critical Thinking Questions 1. Identify one similarity and one difference in the motion of a particle connected to a spring described in Model 1 in comparison to the translational motion defined by the particle-on-a-line model.

PGLSAG003012a


2. In general, how will the Hamiltonian operator for Model 1 differ from the particle-on-a-line model?

3. For this particle-on-a-spring model, write the functional form of the Quantum Hamiltonian operator.

4. Based on a sketch of the potential energy (pre-activity question 1), identify the two boundary conditions the wavefunction, ψ ( x ) , must satisfy.

Model 2 The following are the solutions to the particle-on-a-spring Schrödinger equation: 1/2 1⎞ 1 ⎛ k⎞ ⎛ En = hν ⎜ n + ⎟ ν = n = 0,1, 2, 3,... ⎜ ⎟ ⎝ 2⎠ 2π ⎝ m ⎠

ψ n (x) = cn H n (ξ )e

− 12 α x 2

1/2

⎛ km ⎞ where α = ⎜ 2 ⎟ and ξ = α 1/2 x is the argument of ⎝ ⎠ the Hermite polynomial, Hn. cn is the normalization constant which is given by 1/ 4 1 ⎛α⎞ cn = ⎜ ⎟ 2 n n! ⎝ π ⎠

H n (ξ ) is known as the

Hermite polynomial series.

H 0 (ξ ) = 1 H 1 (ξ ) = 2ξ H 2 (ξ ) = 4ξ 2 − 2 H 3 (ξ ) = 8ξ 3 − 12ξ


Fundamental.5

The Particle-on-a-Spring Model

Critical Thinking Questions 5. Write expressions for the first two Hermite polynomials in terms of the model parameters (m, k) used in the particle-on-a-spring wavefunction.

6. Write an expression (in terms of α) for the normalized wavefunction corresponding to the two lowest energy states for the particle-on-a-spring model.

7. The energy solution can also be written in terms of the angular frequency, ω , as En = ω ( n + 12 ) . What is the relationship between the two representations of frequency?

8. Identify the quantum number and zero point energy corresponding to the ground state for the particle-on-a-spring model.


9. Identify the difference in energy, ΔE , between n = 0 and n = 1 for a particleon-a-spring.

10. Draw and label an energy level diagram depicting the four lowest energy states for a particle-on-a-spring.

11. In what way does the energy level diagram for an object connected to a spring qualitatively differ from an object confined to a line?

Exercises 1. Show that the n = 0 eigenfunction and corresponding energy eigenvalue identified in Model 2 is a solution to the particle-on-a-spring Schrödinger equation. 2. Set up and evaluate the integral necessary to determine the normalization constant for ψ 1 (x) . What are the appropriate integration limits for x, the displacement of the spring from equilibrium? 3. Show that ψ 0 (x) and ψ 2 (x) are orthogonal.


Fundamental.6

Molecular Vibration Is a molecule a harmonic oscillator?

Pre-Activity Questions 1. Consider a diatomic molecule like CO. Offer a physical explanation as to why the potential energy of this system increases as the bond is stretched longer than its equilibrium bond length. Why does the potential energy of this system increase as the bond is compressed smaller than its equilibrium bond length? 2. Based on quantum mechanics, write a general expression required to calculate the average bond length, r , of CO assuming ψ (r) is known. Describe how you would identify the most probable bond length.

Model 1 A Harmonic Oscillator m1

m2

r

Consider two atoms with mass, m1 and m2 connected by a chemical bond with an equilibrium bond length, re. If the restoring force on the atoms displaced from equilibrium is approximated by Hooke’s Law, this molecule is referred to as a Harmonic Oscillator. The solution to the Schrödinger equation for this diatomic system is the same as the particle-on-a-spring model with the exception that x = r mm - re and the single particle mass, m, is replaced by the reduced mass, µ = 1 2 . m1 + m2

PGLSAG003013a


V=

1 2 kx 2

n 3

2

3

2

E

2

2

1

2

1

0

2

0

x=0 x

Figure 1: A sketch of the energy levels and Ď&#x2C6; for the first few vibrational states. 2

Critical Thinking Questions 1. On the diagram above, draw a vertical line representing the equilibrium bond length, re, corresponding to the four lowest energy states. 2. Based on the sketch of Ď&#x2C6; , what is the average bond length of the molecule for the n = 0 vibrational state? 2


Fundamental.6

Molecular Vibration

3. Does the average bond length of the molecule depend on the quantum number, n? Provide your groupâ&#x20AC;&#x2122;s rationale for this answer.

4. What changes in the molecular bond length are observed with increasing n?

5. Write an expression that identifies the number of nodes as a function of the quantum number n. Using CO as an example, describe the physical significance of a node for this model.

6. On the diagram in Model 1, identify the point(s) where V(x) = E for the n = 0 state. According to quantum mechanics, is it possible for the molecule to have a bond length for which the potential energy is greater than the total energy, V(x) > E? Why or why not?

7. Based on classical mechanics, is it possible for the potential energy of a particle to be greater than the total energy, V(x) > E? Present your groupâ&#x20AC;&#x2122;s rationale for this answer. Â


Information The penetration of a particle into a classically forbidden region is called tunneling.

Critical Thinking Questions 8. For each state displayed in Figure 1, shade in the area corresponding to the total probability a particle may be observed in a classically forbidden region. How would your group decide if tunneling was more likely for the n = 0 state or the n = 3 state?

9. What effect does the possibility of tunneling have on the observed bond length of a molecule?

10. Consider the difference in how potential energy approaches infinity for a particle-on-a-line and a harmonic oscillator. How does the shape of the potential energy surface determine the probability that a system will be found in a classically forbidden region?

11. Based on the potential energy function for this model, discuss whether or not the two atoms can ever completely separate. If not, propose how your group might modify the potential energy function to account for â&#x20AC;&#x153;bond breaking.â&#x20AC;? Â


Fundamental.6

Molecular Vibration

Model 2: The Morse Potential In order to account for a chemical bond breaking event, the vibrational motion must be anharmonic, where the restoring force is no longer proportional to the displacement. An analytic expression that accounts for the anharmonicity of real bonds was proposed by Morse in 1929.

V (r) = De ⎡⎣1 − e−α (r − re ) ⎤⎦

2

(1)

The Morse potential is a empirical function that uses only three parameters: • • •

De, the dissociation energy relative to the bottom of the potential re, the equilibrium bond length α, which is related to the force constant and the dissociation energy k α= 2De The following is the energy eigenvalue solution to the Schrödinger equation using the Morse potential energy function

En = hν ( n +

1 2

) − hν x ( n + )

1 2 2

Energy

n=6 n=5 n=4 n=3 n=2

x=

hν 4De

(2)

De

n=1 n=0 Internuclear Separation (r)

Figure 2: Comparison of a Harmonic and Morse potential including corresponding energy states


Critical Thinking Questions 12. According to Figure 2, which state(s) would the harmonic oscillator model most accurately represent? Explain.

13. According to Figure 2, what happens to the molecule in the n = 14 state?

14. What is the effect of anharmonicity on the energy spacing between vibrational states as n increases?

15. Although De is referred to as the dissociation energy, is this the amount of energy required to break the bond? If not identify the actual amount of energy needed.

16. Assuming the wavefunctions derived based on the Morse potential look similar to the harmonic solutions, would the average bond length of the molecule depend on the quantum number, n? If so, how?

17. Summarize when it is appropriate to model the vibrational energy of a molecule as a harmonic oscillator and when anharmonicity correction should be included.


Fundamental.6

Molecular Vibration

Exercises 1. Rationalize whether O=O or Câ&#x2030;ĄO would have a larger ground state vibrational energy. 2. The force constant for 12C16O is 1902 N/m. Determine the value of the zero point energy (kJ/mol) of CO.

Problem 1. Determine the vibrational quantum numbers at which the Morse vibrational energy level spacing of H2 and CO deviate 1% and 10% from the harmonic energy level spacing. Comment on the significance of these results.


Fundamental.7

The Particle-on-a-Ring Model What type of molecules could be appropriately modeled as a ring?

Pre-Activity Questions Review the basic force and momentum relationships for rotating bodies based on classical physics. 1.

Describe the difference between linear momentum and angular momentum.

2.

Write an expression for the kinetic energy of a rotating object in terms of angular momentum, L, and the moment of inertia, I.

3.

According to classical mechanics, are there any restrictions on the possible values of angular momentum of a rotating body? Explain.

Model 1 Particle-on-a-Ring Consider a particle with mass, m, confined to move on a ring in the x-y plane at a fixed radius, ro. The potential energy is zero on the ring and infinite otherwise. z

Cylindrical Coordinates 0 ≤ r ≤ ∞ , 0 ≤ φ ≤ 2π , −∞ ≤ z ≤ ∞ Transformation Equations x = r cos φ , y = r sin φ , z = z

ro

x

Differential Volume Element dτ = rdr ⋅ dφ ⋅ dz Laplacian ∂2 1 ∂ 1 ∂2 ∂2 ∇2 = 2 + + 2 2+ 2 ∂r r ∂r r ∂φ ∂z

The Laplacian operator is a differential operator that is used to represent the kinetic energy of the defined system, 2 2 ˆ T =− ∇ 2m

PGLSAG003014a

(1)


Critical Thinking Questions 1. Identify the changing Cartesian coordinate(s) the function Ď&#x2C6; would depend on describing the state of the rotating particle illustrated in Model 1.

2. What type(s) of energy will be included in the Hamiltonian for this system?

3. In view of your answers to CTQ 1 & 2, describe how this model is similar but distinct from a particle-on-a-plane.

4. Identify the changing cylindrical coordinate(s) that the function Ď&#x2C6; would depend on to describe the state of the rotating particle illustrated in Model 1.

5. Suggest why using cylindrical coordinates for Model 1 is preferred over the Cartesian coordinate space.

6. In view of your groupâ&#x20AC;&#x2122;s answer to CTQ 4, which derivatives in the Laplacian operator are not necessary to include for the system identified in Model 1?


Fundamental.7

The Particle-on-a-Ring Model

7. Given the model conditions, write the Hamiltonian for a particle confined to a ring.

Model 2 The following is a general (normalized) solution to the particle-on-a-ring model Schrödinger equation,

ψ (φ ) =

eiml φ

( 2π )1/2

(2)

with

E=

ml2  2 2I

(3)

where ml is a constant, and I is the moment of inertia, I = mr02

Critical Thinking Questions 8. Identify the symbol for the quantum number identifying each unique state for the particle-on-a-ring model.

9. Since the wavefunction must be single valued, it must satisfy a cyclic boundary condition ψ (φ ) = ψ (φ + 2π ) . Identify all allowed values of ml such that the wavefunction satisfies this boundary condition. (Note: eix = cos(x) + i sin(x) ).


10. Sketch a plot of the probability density ψ as a function of φ for the lowest energy state of a particle-on-a-ring. Does this result depend on φ ? 2

11. Draw and label an energy level diagram depicting the five lowest energy states for a particle-on-a-ring. Be sure to clearly indicate any degenerate states and the value of the zero point rotational energy.

Model 3 Angular Momentum A particle-on-a-ring has a non-zero component of angular momentum perpendicular to the x-y plane, Lz. The quantum operator for the z-component of ∂ angular momentum is Lˆz = −i . ∂φ

Critical Thinking Questions 12. Is the particle-on-a-ring wavefunction given in Model 2 an eigenfunction of the Lˆz operator? If so, what are observable values of angular momentum for this model?


Fundamental.7

The Particle-on-a-Ring Model

13. Based on the result identified in CTQ 12 and the Pre-Activity questions: a. Describe two ways the observable values of angular momentum differ significantly for the classical and quantum models.

b. What is the best physical interpretation of the corresponding positive and negative values of the quantum number ml?

Problem 1. The organic molecule benzene, C6H6, has a cyclic structure where the carbon atoms make a hexagon. The π electrons in the cyclic molecule can be approximated as having two-dimensional rotational motion. Calculate the diameter of this “electron ring” if it is assumed that a transition occurring at 260.0 nm corresponds to an electron going from ml =3 to ml =4.


Fundamental.8

Molecular Rotation Is there spinning and tumbling on a molecular level?

Pre-Activity Questions Review the basic angular momentum relationships from physics 1. In the picture, draw in Lz.

z

L

x

y

2. Is it possible for Lz to have a magnitude greater than L? Explain why or why not.

Model 1: The Rigid Rotor Consider a diatomic molecule with a fixed bond length,  r0,  freely  rotating  in   three-­‐dimensional  space.    Using a center of mass coordinate system, this problem is equivalent to a single reduced mass, µ , moving on the surface of a sphere of radius  r0. Spherical Coordinates z 0 ≤ r ≤ ∞ , 0 ≤ θ ≤ π , 0 ≤ φ ≤ 2π € Differential Volume Element dτ = r 2 sin θ drdθ dφ θ

r0

y

φ x

PGLSAG003015a

Laplacian & Legendrian ∂2 2 ∂ 1 ∇2 = 2 + + 2 Λ2 ∂r r ∂r r 1 ∂2 1 ∂ ∂ Λ2 = 2 + sin θ 2 sin θ ∂φ sin θ ∂θ ∂θ


Critical Thinking Questions 1. Identify the additional changing coordinate that distinguishes this model from the particle-on-a-ring.

2. Which derivatives in the Laplacian operator are not necessary to include for the system identified in Model 1?

3. For the rigid rotor model, write the functional form of the Quantum Hamiltonian operator.

4. How many quantum numbers does your group expect ψ (θ , φ ) to depend on? Record your group’s rationale for this answer.

5. Are the motions or position of the particle in the φ direction dependent upon the motions in the θ direction?

6. How likely is a real molecule to be a rigid rotor? Present your group’s justification for this answer.


Fundamental.8

Molecular Rotation

Model 2 The wavefunctions for the rigid rotor model can be written as a product of two functions, Φ ml , a function of f, and Θl, ml , a function of q;

ψ l, ml (θ , φ ) = Θl, ml (θ ) ⋅ Φ ml (φ ) l = 0,1, 2,... ml = 0, ±1, ±2,..., ±l where Φ ml (φ ) =

eiml φ

( 2π )1/2

(1)

is the same solution derived for the particle-on-a-ring

model (Fundamental.7) and Θl, ml (θ ) are the associated Legendre polynomials. Table 1: The normalized associated Legendre polynomials and corresponding polar plots. These plots are two dimensional representations of three dimensional functions. Some features, such as nodes, may be missing from the given representation. l

ml

Θl, ml (θ )

0

0

1 2 2

1

0 €

1 6 cosθ 2

1

1 3 sin θ 2

€ 0

2

€ 1 €

2

1 10 (cos2 θ −1) 4 1 15 sin θ cos θ 2 1 15 sin 2 θ 4

l=0 m =0

l=1 m =0

l=1 |m|= 1

l=2 m =0

l=2 |m|=1

l =2 |m| =2

The€3D rotational energies are given by, €

where I = µr2. €

El =

2 l(l + 1) 2I

(2)


Critical Thinking Questions 7. Identify an appropriate expression for the ground state wavefunction, ψ 0,0 and the corresponding energy, E0 for the rigid rotor. €

8. If the angular momentum quantum number l = 1, what are the possible values of the magnetic quantum number ml?

9. List all the states with the same energy as ψ 2,1 .

10. Derive an expression that gives the number of degenerate states for a given value of l.

11. Describe the general relationship between the number of nodes in the rotational wavefunction and the corresponding rotational energy.

12. Will an H2 or I2 molecule have a greater rotational energy corresponding to the l = 3 energy level? Record your group’s rationale for this answer.


Fundamental.8

Molecular Rotation

Exercises 1. Show that ψ 0,0 is normalized. 2. Draw and label an energy level diagram depicting the five lowest energy states for a rigid rotor. Be sure to clearly indicate any degenerate states and the value of the zero point rotational energy.

Model 3 Angular Momentum The total angular momentum is defined by the quantum number  where

Lˆ2ψ l, ml (θ , φ ) =  (  + 1)  2ψ l, ml (θ , φ )

(3)

Recall from physics that angular momentum is a vector quantity. Thus the vector L, orbital angular momentum, has a magnitude and a direction. The direction cannot be completely specified. We can however determine the z component of the angular momentum using the operator Lˆ z .

Lˆ zψ l, ml (θ , φ ) = ml ψ l, ml (θ , φ )

(4)

Critical Thinking Questions 13. What is the value (eigenvalue) of Lz for the wavefunction ψ 1,−1 ? give your answer in terms of .

You may

14. What is the value of the magnitude of L = L2 , for the wavefunction ψ 2,1 ? You may give your answer in terms of h– . €

15. Using grammatically correct English sentences, describe how general shape (curvature) and number of nodes of the wavefunction changes with increasing angular momentum.


16. In view of the Pre-Activity questions and the expressions for L and Lz considered above, explain why the quantum number ml can only have values from ď Ź to â&#x20AC;&#x201C;ď Ź.

Problems 1. The bond distance for HCl is 1.29 angstroms. In its lowest rotational state, the molecule is not rotating, and so the rigid rotor equations indicate that its rotational energy is zero. a. What is the energy and angular momentum (Lz) corresponding to the first nonzero rotational state? Explain your analysis clearly and state any assumptions you make. b. Using the expression for 2-D rotational energy, construct a generic expression for the energy difference between two adjacent levels, E(ml+1) - E(ml). c. For HCl, E(1)-E(0)=20.7cm-1. Calculate E(2)-E(1), assuming HCl acts as a 2-D rigid rotor. d. The energy difference you calculated in part (c) is determined experimentally to be 41.4 cm-1. How good would you say a 2D model is for this system? 2. Repeat parts (b), (c) and (d) for 3-D rotational motion. appropriate to treat HCl as a 2-D or 3-D rigid rotor? Explain.

Is it more


Fundamental.9

The Hydrogen Atom One proton and one electron, how hard could it be?

Pre-Activity Questions 1. What is the classical physics expression that describes the force between two point charges? Define all variables and parameters (constants). 2. Using the force defined in Question 1, derive an expression for the potential energy corresponding to the interaction between two point charges.

Model 1 The hydrogen atom can be described in terms of a positive nucleus and an electron interacting according to Coulomb’s law.

+

r

This two-particle system can be defined in terms of a single reduced mass, mm µ = p e , moving at a variable distance, r, from a fixed center of mass. m p + me

Critical Thinking Questions 1. Explain why for a hydrogen atom, the reduced mass µ ≈ me .

2. Where is the center of mass for this system? Recall, XCOM =

PGLSAG003016a

m1 x1 + m2 x2 . m1 + m2


3. Identify the spherical coordinate(s) (Model 1 Fundamental.8) that the wavefunction, Ď&#x2C6; , would depend upon for the system described in Model 1.

4. In what way is this set of coordinates similar to the rigid rotor model? In what way does it differ?

5. In general, describe the parts of the Hamiltonian operator for the hydrogen atom that will be similar to the rigid rotor model.

6. In general, describe the additional terms needed for the Hamiltonian operator describing a hydrogen atom that are not present in the rigid rotor model.

7. Write an expression corresponding to potential energy for a hydrogen atom.

8. Since the position of the proton (nucleus) is fixed at the coordinate origin, this same model can be used to describe any hydrogen-like single electron system such as He+ or Li2+. What modification to the potential energy is necessary to generalize this model for any atom with one electron and Z protons in the nucleus?


Fundamental.9 The Hydrogen Atom

9. Identify the sign (positive or negative) for the potential energy of a hydrogen atom and identify whether it corresponds to an attractive or repulsive force.

10. Write out the Schrödinger equation for a hydrogen-like single electron system.

11. Are the motions of the electron in each of the three directions r, φ , and θ independent?

12. How many quantum numbers does your group expect ψ (r, θ , φ ) to depend on?

Model 2 The wavefunctions for the hydrogen atom can be written as a product of three functions:

ψ n,l, ml (r,θ , φ ) = Rn,l (r)Θl, ml (θ )Φ ml (φ )

n = 1, 2, 3,... l = 0,1, 2,...n − 1 ml = 0, ±1, ±2,... ± l

(1)


where Φ ml and Θl, ml are the same functions as found for the rigid rotor model (Fundamental.8). The allowed energy states are

En = −

Z 2 me e 4 n 2 2(4πε 0 )2

(2)

Quantum chemists typically report energies in hartrees defined as

mee 4 1 hartree ≡ = 27.211 eV = 4.35944 ×10 −18 J 2 (4πε 0 )

Critical Thinking Questions 13. Identify all quantum numbers corresponding to the ground state of a hydrogen atom electron.

14. Explain why ψ 2,2,2 is not a valid hydrogen wavefunction.

15. In what way are the values of l and ml similar to the rigid rotor model? In what way do they differ?

16. Identify all states (wavefunctions) that are degenerate to ψ 2,1,0 .

17. What is the “lowest” energy value (in hartrees) corresponding to the ground state of a hydrogen atom?


Fundamental.9 The Hydrogen Atom

18. Which state of the hydrogen atom is “higher” in energy: ψ 2,1,0 or ψ 4,3,2 ? Explain.

19. Is it possible for the energy of a hydrogen atom to be zero? If so, under what conditions?

20. The ionization energy is the energy needed to completely remove an electron from an atom. From which state of the hydrogen atom is it easier to remove the electron: ψ 2,1,0 or ψ 4, 3,2 ? Explain.

21. Is it harder to remove an electron from He+ or Li2+? Explain.

22. How does the difference in energy between two sequential states change as n increases?


23. Draw and label an energy level diagram (approximately to scale) depicting the three lowest energy states for a hydrogen atom. Be sure to clearly indicate any degenerate states.

24. If a hydrogen atom electronic transition occurs from ψ 4, 3,2 to ψ 2,1,0 is a photon absorbed by the atom or released by the atom? Explain.

25. The set of emissions from excited hydrogen atoms to the ground state (n = 1) is referred to as the "Lyman series" and the set of emissions from excited hydrogen atoms to the n = 2 state is referred to as the "Balmer series". Which series includes higher energy photons, the Lyman or the Balmer series? Explain without calculating the precise energy of any photons.  

Exercises 1. Calculate the energy (in joules) of the photon released for the lowest energy transition for the Lyman series and for the Balmer series. 2. Calculate the energy (in joules), the frequency, and wavelength (in nm), and the wavenumber (in cm–1) for the first five lines of the Balmer series of the hydrogen atom spectrum. In what range(s) of the electromagnetic spectrum are these lines found?


Fundamental.10

Hydrogen Radial Functions How far is an electron from the nucleus?

Pre-Activity Question 1. Consider a generic probability function, ρ(r), which only depends on a distance r. Sketch one example of a function for which the most probable radius would not equal the average radius.

Model 1 Recall that the wavefunction solutions for the hydrogen atom are written as a product of three functions:

ψ n,l, ml (r,θ , φ ) = Rn,l (r)Θl, ml (θ )Φ ml (φ )

n = 1, 2, 3,... l = 0,1, 2,...n − 1 ml = 0, ±1, ±2,... ± l

(1)

Θ n,l (θ ) and Φ ml (φ ) are the same functions as found for the rigid rotor model (see Fundamental.8). The product of the two angular functions is typically replaced with the appropriate spherical harmonic function: Yl ml = Θl, ml Φ ml . The radial functions, Rn,l, are products of a decaying exponential function and a Laguerre polynomial in the dimensionless variable r/a0. The first few normalized radial functions for a hydrogen-like atom with one electron and Z protons are given below:

⎛Z⎞ R10 = 2 ⎜ ⎟ ⎝a ⎠

3/2

e− Zr/a0

0

R20 = R21 =

1 ⎛Z⎞ 8 ⎜⎝ a0 ⎟⎠

3/2

1 ⎛Z⎞ 24 ⎜⎝ a0 ⎟⎠

⎛ Zr ⎞ − Zr/2a0 ⎜⎝ 2 − a ⎟⎠ e 0

3/2

⎛ Zr ⎞ − Zr/2a0 ⎜⎝ a ⎟⎠ e 0

The Bohr radius, a0, is defined as 1 bohr ≡ a0 =

PGLSAG0030017a

(4πε 0 ) 2 = 52.918 pm . me e 2


Critical Thinking Questions 1. For each of the hydrogen (where Z = 1) radial functions defined in Model 1, identify all values of r (including 0 and ∞ ) for which the function is zero. a) R10 b) R20 c) R21 2. Is it possible for the radial function to be less than zero for certain values of r? Provide justification for your group’s answer.

3. A radial node occurs when the radial function changes sign. Identify the radial function(s) defined in Model 1 that has(have) a radial node.

4. Using grammatically correct English sentences, describe the radial functions, making particular note of the values of r = 0 and as r → ∞ , and the location of any radial nodes. A response for R10 has been provided as an example. R10: This function is positive at r = 0, has no radial nodes, and approaches 0 as r → ∞ . It is positive at all values of r. a) R20:

b) R21:


Fundamental.10

Hydrogen Radial Functions

Information When normalizing a wavefunction, calculating probabilities, or determining average values for two- or three-dimensional systems, the integral must be evaluated over all space or all possible values of the wavefunction. This requires solving double or triple integrals using the appropriate volume element. For example, the integral over all space of a 3D separable function in Cartesian coordinates is

∫∫∫ f (x, y,z)dτ = ∫

all space

X(x)dx ∫−∞ Y (y)dy ∫−∞ Z(z)dz −∞

(2)

where dτ is the infinitesimal volume element of a cube.

€ Critical Thinking Questions

€ 5. The hydrogen atom wavefunction is a separable function in r, θ , and φ . What is the volume element dτ in spherical polar coordinates (recall Fundamental.8)?

€ 6. Carefully setup an integral expression in the form of Equation 2, that when evaluated will show that the hydrogen atom wavefunction ψ (r,θ , φ ) = R(r)Θ(θ )Φ(φ ) is normalized, i.e. the total probability is one.

7. In view of your answer to CTQ 6, assuming each individual function R(r), Θ(θ ) and Φ(φ ) is normalized, would the wavefunction ψ (r,θ , φ ) be normalized? Provide your group’s justification for this answer.

8. In view of CTQ 6 & 7, identify the integral expression that must be evaluated to determine the probability of finding an electron anywhere in a spherical shell between r1 and r2.


Model 2 The probability of finding an electron between r and r + dr in a thin spherical shell 2 2 centered at the nucleus is given by r 2 Rn,l is called the dr . The function r 2 Rn,l radial distribution function and is plotted for several states of hydrogen below. 10

r2 !R10 2 "1000

8 6 4 2 0 0

100

r !in pm" 200

300

400

Critical Thinking Questions 9. Sketch a plot of r2 vs. r and a plot of R102 vs. r. What feature of the radial distribution plot results from the multiplication of these two functions?

10. How do the radial distribution function r 2 R102 and the probability density R102 differ at small distances from the nucleus?

11. What is significant about the radius at which the radial distribution function is a maximum?


Fundamental.10

Hydrogen Radial Functions

12. Set up a general expression and describe how your group would calculate the most probable radius of an electron from the nucleus for R10.

13. Set up a general expression and describe how your group would calculate the average radius of an electron from the nucleus for R10.

14. In view of Model 2, how will the average radius of the electron from the nucleus compare to the most probable radius for an electron described by the function R10 ? Provide your groupâ&#x20AC;&#x2122;s justification for this answer.

Exercises 1. Show that the most probable radius for an electron described by the function R10 is the Bohr radius, a0. 2. Calculate the average value of r,

r , for an electron described by the

function R10 . Compare this value to the most probable radius and rationalize this result based on the plot of the radial distribution function. 3. Find the location (in units of a0) of the radial node for the 2s orbital in the He+ ion and Li2+ ion. How does the location of the radial node change as the nuclear charge increases?


Fundamental.11

Hydrogen Orbitals Is there a difference between an orbital and a wavefunction?

Pre-Activity Question 1. Using the appropriate 3D rigid rotor solutions for Yl ml = Θl, ml Φ ml and the radial solution Rn,l (r) defined in Fundamental.10, show that

ψ 100

1 ⎛Z⎞ = (π )1/2 ⎜⎝ a0 ⎟⎠

3/2

e

− Zr/a0

, and ψ 210

⎛Z⎞ 1 = 1/2 ⎜ 4(2π ) ⎝ a0 ⎟⎠

3/2

Zr − Zr/2a0 e cosθ . a0

Model 1: 1s orbital The one-electron wavefunctions, ψ n,, m ( r,θ , φ ) , are referred to both as the hydrogen atom eigenfunctions and the hydrogen atom orbitals. The letters s, p, d, and f are often used to denote l = 0, 1, 2, and 3, respectively. Thus, ψ100 is referred to as the 1s orbital or wavefunction and all three wavefunctions with n = 2 and l = 1 are referred to as 2p orbitals. The angular portion of the wavefunction determines the characteristic shape of an orbital. The shape is defined by a surface of constant probability density that encloses 90% of the probability of finding the electron within the contour.

z 1 ⎛Z⎞ ψ 1s = (π )1/2 ⎜⎝ a0 ⎟⎠ 2

2

3/2

e− Zr/a0 +

2 1/2

r = (x + y + z )

x

PGLSAG003018a

y


Critical Thinking Questions 1. Make a sketch of ψ 1s versus r (0 to ∞). Be sure to show where the function is positive, negative, and zero.

2. Based on your answer to CTQ 1, why is a positive sign found in the graphical representation depicted in Model 1?

3. Why is a circle or a sphere used in the graphical representation of ψ 1s depicted in Model 1?

4. When l = 0, the angular function is Y00 (θ , φ ) =

1 . Will the shape of a 1s 4π

orbital differ from a 3s orbital? Explain.

5. Recall that R20 has a radial node at 2a0. Sketch a graphical representation for a 2s orbital similar to the depiction in Model 1.


Fundamental.11

Hydrogen Orbitals

6. Which geometric term best describes a radial node at a specific non-zero value of r: point, line, plane, spherical shell, cubical shell, conical surface?

Model 2: The ψ 210 wavefunction ψ 210

⎛Z⎞ 1 = 1/2 ⎜ 4(2π ) ⎝ a0 ⎟⎠ z = r cosθ

3/2

Zr − Zr/2a0 e cosθ a0

r = (x2 + y2 + z2)1/2

Critical Thinking Questions 7. For the hydrogen atom, determine the type value (positive, negative or zero) of ψ 210 at a. x = 0, y = 0, z = 0 b. x = 0, y = 0, z = a0 c. x = 0, y = 0, z = –a0 d. x = 0, y = 0, z = 2a0 e. x = 0, y = 0, z = –2a0 f. x = 0, y = 0, as z approaches ∞?

g. x = 0, y = 0, as z approaches –∞?


8. For the hydrogen atom, make a sketch of ψ 210 versus z (–∞ to ∞) on the axes below. Be sure to show where the function is positive, negative, and zero. z

ψ210

z=0

9. Based on your answer to CTQs 7 & 8, does this wavefunction, ψ 210 have any nodes? If so, is it a radial node? Explain.


Fundamental.11

Model 3: 2pz orbital

Hydrogen Orbitals

z

+

y

x

Critical Thinking Questions 10. Why is the wavefunction in Model 3 labeled 2pz, as opposed to 2px or 2py?

11. Based on Models 2 and 3 and your answers to previous questions, a. Why is a positive sign found in the graphical representation of ψ2pz ?

b. Why is a negative sign found in the graphical representation of ψ2pz ?

c. Why does the graphical representation of ψ2pz have the shape of a dumbbell?

12. What geometric term best describes this angular node in space: point, line, plane, spherical shell, cubical shell, conical surface?


Information Both the angular and radial parts of the wavefunction have nodes. It can be shown that there are n-l-1 radial nodes and l angular nodes associated with each hydrogen atom wavefunction.

Critical Thinking Questions 13. Confirm that the 2pz has the correct number of angular and radial nodes.

Exercises 1. Every wavefunction for a one-electron atom is a product of two functions (or three, depending on how you count them)—a radial function and an angular function. Show that ψ2pz is a product of R21 and Y10 . 2. Calculate the value of ψ1s for the hydrogen atom, the He+ ion, and the Li2+ ion at r = 0, r = ao, and r = 4ao. Rationalize the trends in the values that you obtain. 3. Compare the number of angular and radial nodes for the 4s and 4d orbitals. 4. Carefully set up the integral that when evaluated will yield the probability of finding a 1s electron of the hydrogen atom between r = 0 and r = rmp. 5. Examine graphical representations of the 3d orbitals in a textbook. For each orbital, what geometric term best describes the angular nodes in space: point, line, plane, spherical shell, cubical shell, conical surface?

Problem 1. When ml ≠ 0 the spherical harmonics, Yl ml have imaginary components to them. Write out the spherical harmonics Y11 , and Y1−1 , and show that their 1 1 Y11 − Y1−1 are real functions. Y11 +Y1−1 ) and linear combinations, ( 2i 2

(

)


Fundamental.12

Multielectron Atoms Is there anything additional we learn by simply adding another electron?

Pre-Activity Question 1. Calculate the zero-point energy of a single electron He+ ion in atomic units (hartrees).

Model 1: The Helium Atom r12

r1

r2

+2

The Schrödinger equation for a helium atom is given by

⎡ ⎛  2 2  2 2 ⎞ ⎛ 2e2 2e2 e2 ⎞ ⎤ − ∇ + ∇ + − − + ⎢ ⎜ ⎥ψ (1,2) = Eψ (1,2) 1 2 2me ⎟⎠ ⎜⎝ 4πε 0 r1 4πε 0 r2 4πε 0 r12 ⎟⎠ ⎦ ⎣ ⎝ 2me

(1)

We will use the notation ψ (1, 2,..., n) = ψ (r1θ1φ1 , r2θ 2φ2 ,..., rnθ nφn ) to represent an n-electron wavefunction.

Critical Thinking Questions 1. In Equation 1, circle the terms corresponding to kinetic energy and explain why there is more than one.

PGLSAG003019a


2. Each potential energy term describes the interaction between two particles. Identify the two particles resulting in the following expression: a) −

2e2 4πε o r2

b)

e2 4πε o r12

3. Why is there a “2” in the first two potential energy terms in Equation 1 but not in the third?

4. Why are the first two potential energy terms in Equation 1 negative but not in the third?

5. Identify the term(s) in Equation 1 that would be used to describe a single electron He+ ion.

6. Is it possible to separate the Hamiltonian operator for helium into two hydrogen-like (He+) Hamiltonians? Identify any missing or extra terms not present in the two single electron Hamiltonian operators.

Information Mathematicians, chemists, physicists, and other industrious individuals have tried to find a function that is a solution to Equation 1. None has succeeded. The reason for this is that this equation cannot be solved exactly. There are no functions that are eigenfunctions of the Hamiltonian operator in Equation 1.

Critical Thinking Question 7. Is this course over?


Fundamental.12

Multielectron Atoms

Model 2: A simple approximation For the helium atom, we have two Hamiltonians that are identical to a hydrogen atom except for a different nuclear charge and the additional “r12 term” due to electrostatic repulsion of the two electrons at a distance r12. As a naïve zero-order approximation we can simply ignore the “r12 term”.

Critical Thinking Question 8. Assuming that the Hamiltonian operator for helium is simply two separate hydrogen-like (He+) Hamiltonians write a general expression for: a. the wavefunction of helium ψ He (1,2) terms of a single electron He+ wavefunction, ψ He (1) . +

b. the energy of helium, EHe , in terms of the single electron He+ energy, EHe+

9. Using your answer to the Pre-Activity question, identify the energy of He in atomic units, neglecting the r12 term.

10. Compare the energy identified for a helium atom in CTQ 9 to the experimental value of -2.9033 hartrees. What affect does ignoring interelectronic repulsion have on the energy of helium? Explain this result using Coulomb’s Law.


Information: Variational Theory Variational theory is a common approach used to calculate the approximate ground state energy of a multi-electron system. This method is summarized below: • Write an appropriate Hamiltonian, Hˆ , corresponding to a system of interest. •

Make an educated guess of the ground state wavefunction—called the trial solution, ψtrial. The trial wavefunction must obey the boundary conditions of the defined system. The proposed solution typically contains one or more adjustable variational parameters that can be systematically optimized.

Determine the average value of energy using the following expression:

Eψ trial

* ˆ trial H

∫ψ = ∫ψ

ψ trial dτ

(2)

* trial

ψ trial dτ

The variational theorem guarantees that Eψ trial will always be greater than the true ground state energy. The lower the energy (e.g. more negative) of the trial solution, the closer the trial solution is to the real solution of the Schrödinger equation. Thus, we can minimize Eψ trial with respect to the variational parameter € to obtain the best ψtrial.

Model 3: Trial Function for Helium Atom € 3

1⎛ ζ ⎞ − ψtrial(1,2) = ψtrial(1) ψtrial (2) = ψ1s(1) ψ1s(2) = ⎜ ⎟ e π ⎝ a0 ⎠

Table 1:

ζ r1 ζr − 2 a0 a0

e

(3)

Average Energy for Helium Atoms Using Trial Functions with Different Values of ζ

ζ 2.25 2.00 1.75 1.50 1.25

Energy (hartrees) -2.528 -2.747 -2.840 -2.809 -2.653


Fundamental.12

Multielectron Atoms

Critical Thinking Questions 11. Provide a rationale for use of Equation 3 as the trial wavefunction for the helium atom.

12. Which value of ζ in Table 1 provides a function that is the best approximation to the ground state of the He atom? What is the basis for your answer?

13. When the trial wavefunction (Equation 3) is used in Equation 2, the following expression for the average value of energy (atomic units) is obtained: Eψ trial = ζ 2 −

27 ζ 8

a) Assuming ζ is a variational parameter, set up an equation to determine the value of ζ for which Eψ is minimized. trial

b) Compare the energy of He according to variational theory to the experimental value identified in CTQ 10.

14. What physical explanation can you provide for the result that the "best" value of ζ does not correspond to the nuclear charge Z =2?


Exercises 1. The following is a possible trial function for a 1D particle-on-a-line, ψ trial = B [x(a – x)]. a) Show that the proposed trial wavefunction satisfies the boundary conditions. b) Using variational theory, calculate the energy consistent with the trial solution. c) Compare the approximate energy and the actual solution for n = 1. Express the energy difference as a percent. 2.

Assuming that the true ground state wavefunction is not known and a trial wavefunction for the hydrogen atom system is defined as 2

φ ( λ ) = e− λ r . When this trial wavefunction is used in Equation 3, the following expression for the average value of energy is obtained: 1

3e2 λ ao e2 λ 2 Eφ ( λ ) = − 3 8πε o ε oπ 2 2 Determine the best value for λ and the corresponding variational energy. How does this value compare with the true energy?


Fundamental.13

Electron Configurations How are multiple electrons arranged in an atom?

Pre-Activity Question 1. Define the term â&#x20AC;&#x153;effective nuclear chargeâ&#x20AC;? and give an example of its use in the context of multi-electron atoms.

Model 1: Relative Energy Levels of Isolated H, He, and He+ H atom 4s 3s

4p 3p

2s

2p

4s 3d

4p

3s

3p

2s

2p

1s

Energy

He+ ion

He atom 3d 4s

4p

3s

3p

2s

2p

1s

1s

PGLSAG003020a

3d


Critical Thinking Questions 1. What is the nuclear charge of H? He? He+?

2. How many electrons are there in H? He? He+?

3. Why is the 1s energy level for He+ lower than the 1s energy level for H?

4. Why is the 1s energy level for a He higher than the 1s energy level for He+?

5. According to Model 1, do the quantum numbers representing degenerate states differ for a single-electron and multi-electron atom? If so how?

6. An electron configuration specifies the values of n and l for each electron. The ground state electron configuration for helium is 1s2 where the superscript designates the number of electrons characterized by a 1s wavefunction. Identify one possible electron configuration corresponding to an excited state for He.

7. Is 4p1 a possible electron configuration for hydrogen? Explain.


Fundamental.13

Electron Configurations

8. The radial distribution functions corresponding to ψ1s , ψ2s, and ψ2p are shown below. Provide a rationale as to why a 1s electron more effectively shields an electron in a 2p orbital than an electron in a 2s orbital from the full nuclear charge. 10

1s

r2 ¥R 2 ¥1000

8 6

2p 4

2s

2 0

0

200

400 rHpmL

600

800

9. In view of your group’s answer to CTQ 8, explain why the 1s12s1 electron configuration for helium is lower in energy than 1s12p1 electron configuration.

10. The electron configuration 1s13s1 for helium is lower in energy than the configuration 1s13d1. What inference can be drawn regarding the relative shielding of electrons in the 3s and 3d orbitals by an electron in the 1s orbital?


11. Why are the 3s, 3p, and 3d energy levels of He+ at the same energy?

12. Recall the effective nuclear charge on an electron in the 1s orbital of He is +1.68e. Based on Model 1, which is probably closer to the effective charge experienced by an electron in the 2s orbital of He in the configuration 1s12s1, +2e; +1.9e; +1.5e; +1.1e? Explain your reasoning.

Information For atoms (and ions) with large numbers of electrons, the ordering of the energy levels gets quite complicated. Increased nuclear charge results in a lowering of energy for all orbitals, but the magnitude of the effect varies depending on both n and ď Ź. One reason for this is the variation in the effectiveness of the shielding for these different orbitals. For example, an electron in a 4s orbital is much less effectively screened from the nucleus than one in a 3d orbital because the 4s wavefunction has a much larger amplitude close to the nucleus. As atomic number increases, the energy of the 4s orbital is thus expected to become approximately equal to that of the 3d at some point.

Exercises 1. Carefully explain why the 2p energy level in Li is at a higher energy than the 2s level in Li. That is, explain why the configuration 1s22s1 is lower in energy than 1s22p1. 2. Briefly explain why the 2s and 2p energy levels are at the same energy for Li2+.


Fundamental.13

Electron Configurations

3. Explain why the ground state wavefunction of ψ 123 = φ1s (1)φ1s (2)φ2s (3) and not ψ 123 = φ1s (1)φ1s (2)φ2 p (3) .

lithium

is

4. Comment on the validity of this statement: To some extent, every electron shields the nuclear charge from every other electron. 5. Write the expression for the classical electronic energy of a Be atom. Write the Hamiltonian operator for a Be atom. Write a trial function for the electrons in a Be atom.


Fundamental.14

Electron Spin Is it possible for an electron to get dizzy?

Pre-Activity Questions 1. The total angular momentum operator is Lˆ2 = Λ2 . Explain why the angular momentum eigenvalue for a particle-on-a-sphere (Fundamental.8) is the same for a one-electron atom. € 2. Show that the value of the magnitude of the total angular momentum, L = L2 , for any s orbital is zero.

Information €

When a beam of atoms with one electron in an s orbital passes through an inhomogeneous magnetic field (this is known as a Stern-Gerlach experiment), the beam splits into two components of equal intensity, but opposite in direction. This implies that there are two equal and opposite magnetic moments possible for the electron in an s orbital. Since the nonrelativistic Schrödinger version of quantum mechanics predicts that the angular momentum of an electron in an s orbital is zero, the existence of electron spin is an additional postulate.

Model 1: Spin angular momentum We define the spin angular momentum operator, Sˆ 2 , its component along the z axis, Sˆz and their eigenfunctions α and β by the following:

Sˆ 2α = s(s + 1) 2α  Sˆzα = ms α = α 2

Sˆ 2 β = s(s + 1) 2 β  Sˆz β = ms β = − β 2

(1)

where ms = −s, −s + 1,..., s − 1, s . Experimentally it has been shown that the quantum number s is equal to 1/2 for electrons.

PGLSAG003021a


Critical Thinking Questions 1. How do the eigenvalue equations for spin angular momentum compare with the eigenvalue equations for orbital angular momentum? (Recall Model 3 Fundamental.8)

2. Identify the four quantum numbers necessary to distinguish each electron of an atom.

3. Identify all possible values of the quantum number ms.

4. What is the value of the magnitude S = S 2 for the spin function α ? Give your answer in terms of .

Model 2: Spin-orbital wavefunctions The complete wavefunction for an n-electron system, ψ (1, 2,..., n) must include a 1 (&α for ms = 2 (* spatial part, φn,l, ml (r1 ,θ1 , φ1 ) , and a spin part, χ ms (σ 1 ) = ' for each 1+ ()β for ms = − 2 (, electron. The one-electron wavefunctions are called spin orbitals and can be represented by the shorthand notation ψ 100 1 = φ1s (1)α (1) or 1s(1)α (1) . 2


Fundamental.14

Electron Spin

Critical Thinking Questions 5. Give the shorthand notation for ψ 2,1,0,− 1 . 2

6. Using the shorthand notation, identify all possible wavefunctions for the ground state of hydrogen.

7. How does the additional spin coordinate change the degeneracy of hydrogen atom ground state?

8. For a two-electron system, propose four possible unique spin functions, e.g. χ 1 , 1 = α (1)α (2) . 2 2

9. According to the Pauli Exclusion principle, no two electrons in an atom can have the same values for all four quantum numbers, n, l, ml, ms. Assuming the electron configuration for helium is 1s2, which of the proposed spin functions in CTQ 8 are not possible.


10. Since no experiment is known to differentiate between electrons, they are thus indistinguishable and cannot have unique labels. What does this imply about the remaining two spin functions?

Model 3: Antisymmetry In order to satisfy indistinguishability, either the spin or spatial functions can be written in terms of linear combinations with all possible electron labels. For example, linear combination of the spin functions for two electrons gives

χ + = 12 [α (1)β (2) + α (2)β (1)] and χ − = 12 [α (1)β (2) − α (2)β (1)] . When the electron labels are exchanged the wavefunction will either be even (symmetric) or odd (antisymmetric) ψ (1, 2) = ψ (2,1) symmetric (even) ψ (1, 2) = −ψ (2,1) antisymmetric (odd) Experimentally, the complete wavefunction (including both spatial and spin coordinates) must be antisymmetric (odd) with respect interchange of the coordinates of any two particles. This can be satisfied in two ways:

ψ antisymmetric = φodd χ even ψ antisymmetric = φeven χ odd

Critical Thinking Questions 11. For each spin function χ + and χ − identified in Model 3, determine whether it is symmetric or antisymmetric.


Fundamental.14

Electron Spin

12. Identify an appropriate spatial component for the ground state of helium and determine if this function is symmetric or antisymmetric.

13. Write the full (approximate) antisymmetric wavefunction for the ground state of helium atom.

Exercises 1. Identify four relationships between α and β showing that they are orthonormal functions. Note that dτ for the electronic spin coordinate is written as dσ. 2. Identify four possible antisymmetric wavefunctions for the 1s12s1 excited state configuration of helium.


Fundamental.15

Term Symbols How are quantum numbers assigned for multi-electron atoms?

Pre-Activity Question 1. How many different ways are there to place a single electron in three degenerate 2p orbitals? Identify the unique set of quantum numbers {n, l, ml, ms} for each state.

Model 1: Experimentally obtained energy level diagram for a nitrogen atom with the electron configuration: 1s22s22p3. Level C

28839 cm–1

Level B

19229 cm–1

Level A

0

Energy relative to the groundstate (in spectroscopic units of cm–1) All three energy levels have the electron configuration: 1s2 2s2 2p3

Critical Thinking Questions 1. What arrangement of the three electrons in the three 2p orbitals do you expect results in the lowest energy (the ground state; energy level A)?

PGLSAG003022a


2. Propose a reason as to why a different arrangement of the three electrons in the three 2p orbitals would have a higher energy.

Model 2: The p3 electron configuration There are 20 ways to place three electrons in three degenerate p orbitals. This corresponds to 20 microstates for the p3 electron configuration. Three example microstates are shown below. All 20 microstates are not degenerate, nor does each microstate have a unique energy.

ml =

+1

0

–1

ML = 0 MS =

3 2

ML = 2 MS =

1 2

ML = 1 MS =

1 2

When several electrons are present, the total orbital and spin angular momentum can be determined from the vector sum of the electrons in unfilled subshells. L is the quantum number that represents the total orbital angular momentum and S represents the total spin angular momentum of a particular electronic state. The values of quantum numbers L and S are consistent with the sum of the scalar components ml and ms for each electron,

M L = ∑ mli and M S = ∑ msi i

i

The component of L along the z-axis has the possible values of M L  where M L = −L, −L + 1,..., L − 1, L . Similarly, the component of S along the z-axis has the possible values of M S  where M S = −S, −S + 1,..., S − 1, S .


Fundamental.15

Term Symbols

Critical Thinking Questions 3. How does the symbol representing the orbital angular momentum quantum number differ for a single electron and a multielectron atom?

4. Show that the MS and ML values in Model 2 are correct.

5. Draw a microstate that will give the maximum value of ML for three electrons in the p3 configuration. Explain why this value also corresponds to the largest value of the quantum number L for this configuration.

6. Based on the value of L determined in CTQ 5, what other values of ML are consistent with this state?

7. Identify the max MS possible for a microstate with the max ML value identified in CTQ 5. Explain why this value also corresponds to the largest value of the quantum number S for this configuration.

8. Based on the value of S determined in CTQ 7, what other values of MS are consistent with this state?


9. In view of your groupâ&#x20AC;&#x2122;s answers to CTQ 6 and CTQ 8, draw three other microstates corresponding to the same value of L and S identified above.

10. How many total microstates (different combinations of MS, ML) correspond to the same value of L and S identified above?

Exercises 1. Identify the maximum value of MS for three electrons in the p3 configuration. What value of L must correspond to this microstate? How many total microstates correspond to these same values of L and S. 2. Prepare a table similar to the one in Model 2 that shows all 20 microstates for the p3 configuration. Give the MS and ML values for each microstate. 3. How many microstates are possible for the p5 configuration? (Fluorine, for example.) Give the MS and ML values for each microstate.

Model 3

Term Symbols

Although an electron configuration is a useful way to represent the arrangement of electrons in an atom, it does not specify a multi-electron quantum state. All states with the same L and S values are referred to as a term represented by the symbol 2S+1L . The upper left superscript, 2S +1 is referred to as the spin multiplicity. A multiplicity of 1 (S = 0) is called a singlet. A multiplicity of 2 is a doublet, 3 is a triplet, and so on. Letters are used to represent the total orbital angular momentum of an electronic state, i.e. L = 0, 1, 2, 3, 4, . . . are associated with letters S, P, D, F, G, . . .


Fundamental.15

Term Symbols

Critical Thinking Questions 11. Identify the values of L and S corresponding to all states associated with the term 3P.

12. Write the term symbol representing the 10 microstates identified in CTQ 10.

13. Consider a single electron in the p1 configuration. Draw a microstate that will give the maximum value of MS and the max ML possible for this value of MS. Write the term symbol(s) consistent with these values.

14. Based on the values of L and S, how many total states are represented by a 2P term? Provide your groupâ&#x20AC;&#x2122;s justification for this answer.

15. What is the term symbol for any configuration with a completely filled set of orbitals (a closed electronic shell)?

Exercise 4. How many microstates exist for the electron configuration 2p1 3s1 (an excited state of carbon)? Both the 3P and 1P terms arise for this electron configuration. Do these terms symbols account for the appropriate number of microstates?


Model 4: The Ground State Term Symbol (Hundâ&#x20AC;&#x2122;s Rules) 1.

For the lowest energy electron configuration, the term with the largest value of S (maximum multiplicity) represents the lowest energy state.

2.

If multiple terms have the same value of S, the state with the largest value of L is the lowest energy state.

Critical Thinking Questions 16. It can be shown that there are three terms associated with the p3 configuration for nitrogen, 4S, 2D, and 2P. According to Model 4, what is the term symbol for the ground state of nitrogen (Level A in Model 1)?

17. Is this term consistent with your groupâ&#x20AC;&#x2122;s prediction in CTQ 1? Explain.

18. Identify the lowest energy term for oxygen.

19. Summarize how to determine the ground state term for a multielectron atom.


Fundamental.15

Term Symbols

Exercises 5. Show that the terms identified for the p3 configuration for nitrogen (CTQ 16) account for the 20 possible microstates. 6. Determine the ground-state term symbol for each of the following: He; Ni; Clâ&#x20AC;&#x201C; 7. What are the lowest energy terms for the ions Ti2+, Mn2+, and Fe3+, if the first two electrons lost by the corresponding neutral atoms are the two 4s electrons? 8. Identify the lowest energy term symbol for the 3s1 configuration of sodium. Find the lowest energy term symbol for the 3s0 3p1 configuration of sodium. Make an energy level diagram for these two term symbols. In a flame, ground sodium atoms get promoted to an excited state. The electron then returns to the ground state and emits a yellow photon. Use an arrow to indicate this transition on your energy level diagram.


Fundamental.16

Born Oppenheimer Approximation What assumptions are needed to model a molecule?

Information For simplicity, quantum equations are often expressed in atomic units rather than SI units. The following table is a brief list of some atomic units. Quantity

Atomic Unit

SI Equivalent

mass

m=1

9.1091x10-31 kg (electron mass)

charge

|e|=1

1.6021x10-19 C (proton charge)

angular momentum

=1

1.0545x10-34 J⋅sec

length

ao=1

5.29167x10-11 m (Bohr radius)

permittivity

4πεo=1

energy

me e 4 =1 (4πε 0 )2

1.1126x10-10 C2⋅J–1⋅m–1 4.35944×10-18 J

Pre-Activity Questions 1.

The Hamiltonian for the helium atom is given by ⎛ 2 2 2 2 ⎞ 2e2 2e2 e2 Hˆ = − ⎜ ∇1 + ∇2 ⎟ − − + 2me ⎠ 4πε o r1 4πε o r2 4πε o r12 ⎝ 2me Rewrite this Hamiltonian using atomic units. Identify the specific particle interaction that each coulombic term represents. Explain the sign on each potential energy term.

2.

Based on knowledge from previous chemistry courses, why do bonds form to create molecules? Use Coulomb’s law in your rationale.

PGLSAG003023a


Model 1: H2+ molecule â&#x20AC;&#x201C; r1A

r1B

+ RAB

+

Critical Thinking Questions 1. Based on the diagram in Model 1, write a Hamiltonian in atomic units that accounts for the kinetic energy of all the species in the hydrogen molecular ion and all the coulombic interaction potentials.

2. How do you expect the magnitude of the nuclei velocity compare to the motion of electrons? Explain.

Information Because of the very different timescales for nuclear and electronic motions, the two motions can be separated using the Born-Oppenheimer approximation. Using this approximation we can solve the SchrĂśdinger equation at a fixed nuclear distance. If this procedure is repeated for many values of the internuclear separation, RAB, we can determine the electronic energy as a function of internuclear distance E(R).

Critical Thinking Questions 3. Using the Born-Oppenheimer approximation, identify the Hamiltonian operator (in atomic units) that is used to determine the electronic energy of a molecule for a particular value of RAB.


Fundamental.16

Born Oppenheimer Approximation

4. Starting at a very large value of RAB, as the nuclei move closer together, would you assume the energy of the system increases or decreases? Present your group’s rationale.

5. As the distance between nuclei decreases to very small values of RAB, does the energy of the system increase or decrease? Explain.

6. Propose an approximate sketch of the electronic energy of the hydrogen molecular ion as a function of different values of RAB.

7. Does the energy curve proposed in CTQ 6 suggest the formation of a “bond” between the two nuclei? Explain what feature of this curve would be indicative of bond formation.

8. Using the Hamiltonian operator identified in CTQ 3, how many times must we solve the Schrödinger equation to obtain the favored electronic energy of a molecule?

9. Although it is possible to exactly solve the Schrödinger equation for this single electron hydrogen molecule ion, it cannot be solved exactly for any multielectron molecules. If we were to use variational theory to determine an approximate value of the energy of the hydrogen molecule ion depicted in Model 1, what would your group propose as a trial wavefunction?


Model 2 While this is not the only possible trial function, it is generally assumed that the electron distribution in a molecule is best approximated as a linear combination of the wavefunctions of the individual atoms. For the hydrogen molecular ion system we assume:

ψ trial = cAφ1sA + cBφ1sB

(1)

The coefficients cA and cB are constants, but their values are unknown at this point. The average energy is defined as:

E

* trial

∫ψ = ∫ψ

Hˆ ψ trial dτ

* trial

ψ trial dτ

(2)

Using the trial wavefunction defined by Equation 1, the average energy expression can be expanded:

E =

cA2 H AA + cB2 H BB + 2cA cB H AB cA2 SAA + cB2SBB + 2cA cBSAB

(3)

* where H ij = ∫ φi*Hˆ φ j dτ and the overlap integral is abbreviated Sij = ∫ φi φ j dτ .

Critical Thinking Questions 10. Will the value of HAA differ from HBB for a homonuclear diatomic molecule? Explain.

11. Write an expression corresponding to the overlap integral, SAA, and identify the value of this integral.


Fundamental.16

Born Oppenheimer Approximation

12. The one-electron function φ1sA is a hydrogen-like spherically symmetric wavefunction centered about nucleus A. Sketch φ1sA as a function of r, the distance of the electron from nucleus.

increasing r

A r =0

increasing r

13. Consider nuclei A and B nuclei at a fixed distance RAB. On the graph below, sketch φ1sA and φ1sB as a function of the distance of the electron from each nucleus.

0

A

B

14. In view of your answer to CTQ 13, identify the values of the overlap integral, SAB, for each of the following limits of RAB: a) RAB = 0?

b) RAB = ∞ ?


Model 3 The coefficients cA and cB can be viewed as variational parameters. When the energy is minimized with respect to each parameter, the resultant equations are obtained:

∂ E = 0 = cA ( H AA − E ) + cB ( H AB − SAB E ∂cA

)

(4)

∂ E = 0 = cA ( H AB − E SAB ) + cB ( H BB − E ∂cB

)

(5)

For homonuclear diatomic molecules, it can be shown that these two equations require cB = ±cA . The magnitude of these constants can be found through normalization.

Critical Thinking Questions

15. Why is it desired to minimize the energy with respect to cA and cB?

16. Write the variational solutions for the two ground state wavefunctions of the hydrogen molecular ion based on Model 3 and Trial Solution Equation 1.

17. Which wavefunction in CTQ 16 best represents the sketch your group drew to answer CTQ 13?


Fundamental.16

Born Oppenheimer Approximation

18. On the graph below, sketch the other possible wavefunction for the hydrogen molecular ion and identify any significant differences.

0

A

B

19. Compare the probability of finding an electron between the two nuclei for the two molecular wavefunction solutions. How do they differ?

20. Based on your answer to Pre-Activity Question 2, which molecular wavefunction would you expect to have a higher energy? Explain.

21. Would you expect the electronic energy curve as a function of RAB (CTQ 6) to be the same shape for both states? If not, illustrate how they might differ.


Exercises

1. Using the information in Model 3, show that cA = ±cB . 2. Using the wavefunction for which cB = cA, setup an appropriate expression to solve for the magnitude of cA through normalization. Simplify your expression to show how the value cA depends on the magnitude of the overlap integral SAB. 3. Assuming cA = –cB, show that the average energy expression (Equation 3) H AA − H AB reduces to E = . 1− S AB


Fundamental.17

Linear Combinations of Atomic Orbitals How many different types of orbitals are there?

Pre-Activity Question 1.

Recall that the complete wavefunction for an n-electron system must include both a spatial and spin part for each electron. Write an expression for the full antisymmetric wave function for the ground state of H2 molecule.

Model 1 A molecular orbital is formed from a linear combination of atomic orbitals using the following principles: 1) Only atomic orbitals of similar energy and the same ml values will combine to make molecular orbitals. 2) Molecular orbitals are labeled according to their ml values. The Greek letters σ , π , and δ are used to denote ml = 0, 1, and 2 , respectively. 3) The inversion symmetry of the orbital with respect to the center of the molecule, i.e. coordinate transformation from (x, y, z) to (-x, -y, -z), is denoted as g or gerade (even) for symmetric functions and u or ungerade (odd) for antisymmetric functions. 4) Antibonding molecular orbitals have a nodal plane perpendicular to the internuclear axis between two nuclei. They are labeled with an asterisk. As an example, the two molecular orbitals assumed for the hydrogen molecular ion system are:

PGLSAG003024a

σ g 1s =

1 2 + 2SAB

σ u*1s =

1 2 − 2SAB

(φ (φ

1sA

+ φ1sB

1sA

− φ1sB

) )

(11)


Critical Thinking Questions 1. For each of the following, identify whether or not two atomic orbitals corresponding to nuclei A and B of a homonuclear diatomic molecule can be used to create a molecular orbital according to Model 1. a) A 2s of atom A and 2s of atom B b) A 1s of atom A and 2s of atom B c) A 2pz of atom A and 2pz of atom B d) A 2pz of atom A and 2px of atom B e) A 2s of atom A and 2pz of atom B 2. A pictorial representation of the σ g 1s orbital is given below: 1sA

A

+

1sB

g1s

B

Draw a similar representation of the σu*1s molecular orbital and explain why the symmetry designation is correct.

3. Describe how a σ g 2s orbital will differ from a σ g 1s orbital?

4. Propose a pictorial representation of the π u molecular orbital similar to CTQ 2.


Fundamental.17

Linear Combinations of Atomic Orbitals

Exercises + 1. The electronic configurations for H2 and H2 are マト1s and マト21s respectively. Which will have the stronger bond? Why?

Model 2

y

y

-z

z x

A

Atomic Orbitals

-z

z

B

x

Molecular Orbitals


Critical Thinking Questions

5. What do the two different shades of the orbitals in Model 2 represent?

6. Identify the appropriate ml values for the two different types of p orbital combinations represented in Model 2. Explain your reasoning.

7. Identify the three antibonding MOs in Model 2 and label each based on the information in Model 1. (e.g. σu*)

8. Provide an example of an antibonding molecular orbital that is the result of a positive linear combination of atomic orbitals.

9. Label each of the remaining bonding molecular orbitals based on the information in Model 1. (e.g. σg)

10. Is it possible for a bonding molecular orbital to have a node? Explain.


Fundamental.17

Linear Combinations of Atomic Orbitals

11. Are all bonding molecular orbitals symmetric with respect to the center of inversion? If not, provide an example.

Model 3

x N A

N B

z

y

ψ i = c1φ2sN + c2φ2 p N + c3φ2 p N + c4φ2 p N + c5φ2sN + c6φ2 p N + c7φ2 p N + c8φ2 p N A

x

A

y

A

z

A

B

x

B

y

B

z

B

(12)

Table 1. Hartree-Fock STO-3G* orbital coefficients and energies corresponding to the valence orbitals of N2. Coefficients for each MO (ψi) c1 c2 c3 c4

ψ1 0.500 0.000 0.000 0.230

ψ2 0.747 0.000 0.000 0.253

ψ3 0.000 0.630 0.000 0.000

ψ4 0.000 0.000 0.630 0.000

ψ5 0.400 0.000 0.000 0.604

ψ6 0.000 0.823 0.000 0.000

ψ7 0.000 0.000 0.823 0.000

ψ8 -1.095 0.000 0.000 1.163

c5 c6

0.500 0.000

-0.747 0.000

0.000 0.630

0.000 0.000

0.400 0.000

0.000 -0.823

0.000 0.000

1.095

0.000 0.000

0.630 0.000

0.000 -0.604

0.000 0.000

-0.823 0.000

0.000 0.000 1.163

MO Energies (in Hartrees) -1.408 -0.728 -0.549

-0.549

-0.530

0.265

0.265

1.041

c7 c8

*

0.000 -0.230

0.000 0.253

The Hartree Fock method uses variational theory (Fundamental.12) to solve for the coefficients of the ground-state wave function and ground-state energy of a molecule. STO-3G (a minimal basis set) defines the form of the initial trial wavefunction.


Critical Thinking Questions 12. How many atomic orbitals contribute to the first molecular orbital ψ 1 identified in Table 1?

13. How many different molecular orbitals were obtained from the linear combination of the eight atomic orbitals defined in Equation 2?

14. MO ψ1 is drawn below. –

+

+

+

+

Make note of the coordinate system used in Model 3 and rationalize both the size and signs shown in the figure.

15. Using the notation introduced in Model 1, what type of molecular orbital is represented by ψ3.

16. Explain why ψ3 and ψ4 are degenerate.

17. Using the information in Model 3, draw an energy level diagram for N2. Label each state using the appropriate molecular orbital notation as described in Model 1.


Fundamental.17

Linear Combinations of Atomic Orbitals

18. How many electrons can be assigned to each molecular wavefunction? Explain.

19. Based on the number valence electrons for N2 and the fact that two electrons occupy each molecular orbital, what is the highest occupied molecular orbitalâ&#x20AC;&#x201D;the HOMO?

20. Identify the amount of energy required to remove an electron from the HOMO of N2 molecule (i.e., the first ionization energy)?


Fundamental.18

Diatomic Molecules What can we lean about molecules based on MO theory?

Pre-Activity Question 1. Identify the HOMO for H −2 . Will the H −2 ion have a stronger or weaker bond than H2? Explain.

Model 1: Relative Energies of Valence Orbitals of Diatomic Molecules For any homonuclear diatomic molecule in the second period, say B2 or F2, the same eight atomic orbitals are used. The energy levels for B2 would be at somewhat higher energies, and MOs for F2 would be somewhat lower energies, but the eight MOs would be the same. The relative energy levels for homonuclear molecules, and almost homonuclear molecules (CO; CN–, and so on) of the second period are given below. O2, F2 & Heteronuclear Diatomics

Atomic Number Z < 8 (Li2, Be2, B2, …)

* u * g

* u * gu

E

* g

E u

u

* gu

g u

u

g

PGLSAG003025a

* u

* u

g

g


Critical Thinking Questions 1. For homonuclear diatomic molecules with atomic number Z < 8, which molecular orbital is moved up in energy relative to the molecular orbitals for O2 and F2?

2. Consider the valence electrons of the O2 molecule. a. How many electrons will occupy the MOs shown in Model 1? b. Appropriately place these electrons on the diagram in Model 1. c. Identify the highest occupied molecular orbital, HOMO, for O2. 3. Experiments indicate that the ground state of O2 is paramagnetic which results from having at least one unpaired electron. a) Assess whether or not the Lewis structure of O2 accurately predicts this experimental property.

b) Assess whether or not molecular orbital theory accurately predicts the magnetic properties of molecular oxygen.


Fundamental.18

Diatomic Molecules

Information Electrons in bonding molecular orbitals attract the atomic nuclei together, while electrons in antibonding molecular orbitals destabilize this bond. The bond order is given by half the number of electrons in bonding molecular orbitals minus half the number of electrons in antibonding molecular orbitals. The larger the bond order, the â&#x20AC;&#x153;strongerâ&#x20AC;? the bond, i.e. the more energy it takes to break the bond. Molecules with BO < 1 are very unstable.

Critical Thinking Questions 4. Determine the O2 bond order based on MO theory. consistent with the Lewis structure of O2?

Is the bond order

5. Would it take more energy to break the bond in O2 or O2+ ion? Explain.

6. For each of the following explain why the following relationship is observed: a. The larger the bond order, the shorter the bond length.

b. The smaller the bond order, the smaller the force constant.


Model 2: Energy level diagram for heteronuclear diatomic HF H

HF

F

1s 2p

2s

Critical Thinking Questions 7. Why is the energy of the 1s atomic orbital of hydrogen greater than the 2s atomic orbital of fluorine?

8. Specifically identify the 2p atomic orbital of fluorine that combines with the 1s orbital of hydrogen.

9. Explain why the lowest energy molecular orbital of HF is referred to as a nonbonding MO.


Fundamental.18

Diatomic Molecules

10. Label each molecular orbital for HF identified in Model 2 and sketch an approximate shape for each. (Recall Fundamental.17 Model 1)

11. Explain why the u and g notation is not used for heteronuclear MOs.

Exercises 1. For the following molecules, use the diatomic MO energy level diagrams in Model 1 to a) determine the bond order, b) determine whether the molecule is diamagnetic or paramagnetic, and c) write the electron configuration of the molecule: B2; C2; CN; BF. 2. For each of the following pairs of molecules (or ions) use the diatomic MO energy level diagram in Model 1 to determine which species has the stronger bond: N2 and CNâ&#x20AC;&#x201C;; CO and CO+; NO and NO+. 3. Draw an MO energy diagram and predict the bond order of Be2+ and Be2â&#x20AC;&#x201C;


Spectroscopy.4

Population of Quantum States What would a molecular census reveal?

Pre-Activity Questions 1.

Derive an expression for the difference in energy between two sequential states En and En+1 for a particle-on-a-line. How does the energy difference depend on the length of the line?

Information According to classical mechanics, any value of energy is allowed for a system resulting in a continuous energy spectrum. In a quantum mechanical system, only certain values of energy are allowed resulting in a discrete energy spectrum.

Model 1: Consider a single helium atom confined to a 1D region of space of length, a. At equilibrium, the total energy of this atom will fluctuate within the range of ΔE thermal = k bT where kb is Boltzmann’s constant and T is the temperature. The difference in energy between two sequential quantum states is denoted by ΔE n →n +1 . € €

Critical Thinking Questions 1. According to Model 1, is it correct to assume that the energy of a helium atom h2 in the ground state at room temperature is E = ? Explain why or why 8ma 2 not.

PGLSAG003026a


2. Draw and label an approximate energy level diagram representing the first four energy states corresponding to Model 1. On your diagram, indicate an approximate range of energies corresponding to Î&#x201D;Ethermal for the n = 2 and n = 3 state.

3. If the range of energies Î&#x201D;Ethermal associated with the n = 2 state overlaps with the n = 3 state, would you characterize the possible energies of helium as continuous or discrete?

4. For each of the following, which system is more likely to exhibit a continuous energy spectrum? Present your groupâ&#x20AC;&#x2122;s justification. a. A helium atom confined to a short or long line

b. A helium atom at high or low temperatures

c. A helium atom or an oxygen molecule confined to the same 1D space


Spectroscopy.4

Population of Quantum States

5. When ΔEthermal > ΔEn→n+1 , is it more appropriate to use quantum or classical mechanics to describe this system? Explain.

Model 2 Consider a large number of helium atoms at room temperature, T =300 K confined to a 1D region of space. Assuming the available energy states are described by the particle-on-a-line model, the following data table identifies relative number of helium atoms in state i in comparison to the ground state, , N1. Table 1: Helium modeled as a 1D particle-in-a-box

th

i state 2 3 4 5 € 6 7 8 9 10

3 Angstrom 1D-box E i − E1 Ni k bT N1 0.067 0.936 0.177 0.837 0.333 0.717 € 0.532 0.587 € 0.776 0.460 1.064 0.345 1.397 0.247 1.774 0.170 2.195 0.111

10 Angstrom 1D-box E i − E1 Ni k bT N1 0.006 0.994 0.016 0.984 0.030 0.971 € 0.048 0.953 0.070 0.933 0.096 0.909 0.126 0.882 0.160 0.852 0.198 0.821

Critical Thinking Questions 6. Based on the data in Model 2, do all helium atoms have the same energy at 300 K? Record your group’s justification for this answer.


7. For each 1D-box, identify the energy state with the largest number of helium atoms.

8. Are more helium atoms found in the n = 2 state in a 3 angstrom box or a 10 angstrom box? Explain.

9. How does the population of states for the 10-angstrom box differ from the 3angstrom box for the 6 lowest energy states?

10. Based on Model 2, if kbT is much larger than the energy difference Ei â&#x2C6;&#x2019; E1 what does this imply about the number of particles in state Ei?

11. If a similar number of helium atoms populate a large number of states, would a continuous or discrete energy model seem more appropriate? (Hint: compare your answers to CTQ 10 and CTQ 5)


Spectroscopy.4

Population of Quantum States

Information: Boltzmann Distribution The Boltzmann distribution describes the distribution, i.e. ratio of particles between two different energy states i and j at a given temperature,

N i gi −(E i −E j )/ k bT = e N j gj

(1)

where kb = 1.38 ×10 −23 J / K , Ni is the number of particles with energy Ei and degeneracy g€i and Nj is the number of particles with energy Ej and degeneracy gj.

Critical Thinking Questions 12. Based on the Boltzmann's equation, which state(s) are populated at T = 0 K? Provide your group’s justification.

13. Describe the population distribution as T → ∞ . (Hint: What does the value of Ni imply?) N1

14. If

ΔE > 1 is a quantum or classical approach more appropriate to describe the kBT

energies of this system?

15. Identify the general characteristics of a particle confined to a 1D line for which a Quantum Mechanical description may be necessary.


Exercises 1. Consider a hypothetical molecule with only two energy states available: State 1

4.55 × 10 −21 J State 0 a. For one mole of molecules, predict the number of molecules in state 0 and state 1 at T = 0, 300, and ∞ K. b. Using the Boltzmann Distribution determine the number of molecules in state 0 and state 1 at T = 0, 300, and ∞ K and compare these results with your predictions in part a. c. How does the number of molecules in state 0 and state 1 at T = 0, 300, and ∞ K change if state 1 is two-fold degenerate? 2. At what temperature is the zero point energy indistinguishable from zero (i.e., the fluctuations in the particle energy is the same as the lowest energy 1D particle-in-a-box energy) in a 10.0 nm wide box for a) a helium atom b) an argon atom? Explain the significance of this result.


Spectroscopy.6

Vibrational Spectroscopy Can we actually see molecules vibrating?

Pre-Activity Questions 1.

Recall, when a photon is emitted or absorbed by an atom or molecule, it changes from one energy state to another. The energy of the photon is equal to the energy difference between the states. If the energy difference between two successive vibrational states is ~10-20 J, calculate the photon wavelength commensurate with this energy level spacing. Identify the corresponding region of the electromagnetic spectrum.

2.

Determine the typical ratio of molecules in the first excited vibrational state at room temperature, 300K. (Hint: Review Spectroscopy.4 Boltzmann Distribution).

3.

A common quantity used in spectroscopy is the wavenumber with units of cm–1. A tilde over a symbol indicates that the quantity is expressed in wavenumbers. The vibrational frequency, ν , of 1H35Cl is 8.963×1013 s-1. What is the value of ν ?

Information So far we have investigated wave functions that describe the stationary states or time-independent states of atoms and molecules. The absorption and emission of electromagnetic radiation, however, is a time-dependent process. In order to absorb a photon, a molecule must have some aspect or feature that permits coupling with the oscillating electric (or magnetic) field of the photon.

PGLSAG003027a


Model 1: The Effect of an Oscillating Electric Field on a CO molecule and on an O2 Molecule.

– C

O +

C

O

+

C

– O

O +

O

O

+ O

O

O –

Critical Thinking Questions 1. If a CO molecule is placed between two oppositely charged plates, why does the carbon atom tend to orient toward the negative plate and the oxygen atom toward the positive plate?

2. Now, assume the charge on the metal plates varies sinusoidally with time. Does the dipole moment of the CO molecule change in response to an oscillating electric field? Why or why not?

3. In view of Model 1, why is a vibrating CO molecule able to absorb a photon, while a vibrating O2 molecule will not?


Spectroscopy.6

Vibrational Spectroscopy

4. Consider the following two possible bond vibrations of a linear CO2 molecule.

O A

C

O

O

C

B

O

a. Does a CO2 molecule have a permanent dipole moment? Explain.

b. Does a CO2 molecule have a time-dependent dipole moment? Explain

c. In view of Model 1, which vibrational state of a CO2 molecule represented above will change with the absorption of a photon? Explain.

Model 2: Selection Rules It can be shown that the probability of a transition between two quantum states described by ψ a and ψ b will occur is proportional to: •

the intensity of the radiation at the proper wavelength,

the number of molecules in the absorbing (or emitting) state,

the transition (dipole) moment

µ ba ≡ ∫ ψ b*µˆψ a dτ

where µˆ =∑Qi ri

(1)

i

In equation (1) µˆ is the electric dipole-moment operator and ri is the displacement vector of charge Qi from the origin. If µ ba is non-zero, the transition is said to be an allowed transition. If µ ba is zero, the transition is said to be forbidden, which means that the frequency corresponding to this transition is either absent from the spectrum or extremely weak.


Critical Thinking Question 5. If a transition between the vibrational ground state and excited state of CO is not observed spectroscopically, what is the value of the transition moment, µ ba ?

6. For a heteronuclear diatomic molecule modeled as a harmonic oscillator, the transition moment is zero unless Δn = ±1 . Sketch an approximate energy level diagram for a harmonic oscillator and draw an arrow representing all allowed transitions from the ground vibrational state.

7. In view of the spectroscopic selection rules and your answer to Pre-Activity question 2, explain why vibrational transitions from n = 1 to n = 2 are not typically observed at or near room temperature.

8. Assuming molecules occupied the two lowest vibrational energy states, how many different absorption frequencies would be observed? Present your group’s justification for this answer.


Spectroscopy.6

Vibrational Spectroscopy

9. The fundamental vibrational transition is from n = 0 to n = 1. Based on the molecular parameters µ and k, describe how the frequency of this transition depends on a.

bond type (single, double or triple bonds)

b. molecular mass

10. The fundamental transition energy is 4160 cm-1 for H2 and 2900 cm-1 for D2. What molecular property is most likely the reason for this significant difference? Explain.

Model 3: Overtone Transitions For a real diatomic molecule, the energy states are approximately defined as En = ν e (n + 12 ) − ν e x e (n + 12 )2 hc

(2) 1/2

1 !k$ where ν e is an experimental parameter which is almost equal to # & 2π c " µ % (recall Fundamental.5) and ν e x e is an experimental parameter called the anharmonicity constant. Since real diatomic molecules are more accurately modeled using an anharmonic potential, it is possible to observe vibrational overtone transitions originating from the n=0 state for which Δn = ±2, ±3,... These transitions are much weaker, but still observable since the Δn = ±1 selection rule is only rigorously obeyed for a harmonic oscillator. The following table lists the experimentally observed vibrational transition energies of CO.


Transition 0→1 0→2 0→3 0→4 0→5

Description fundamental first overtone second overtone third overtone fourth overtone

ν obs (cm–1) 2143 4260 6350 8414 10452

Table 1: Observed vibrational transition energies for CO

Critical Thinking Questions 11. If the vibrational energy levels for a CO molecule were equally spaced, what would be the energy (in cm–1) of the first overtone transition based on the data provided in Table 1? Provide your group’s justification for this answer.

12. According to Model 3, what is the actual energy of separation (in cm–1) between: a)

E2 and E1?

b)

E3 and E2?

c)

E4 and E3?

13. As the vibrational energy increases, does the energy level spacing increase or decrease?


Spectroscopy.6

Vibrational Spectroscopy

14. Based on the data in Model 3, identify two ways the harmonic oscillator model does not accurately predict the vibrational spectrum of a real molecule.

Exercises 1. Explain how the relative bond strengths can be inferred from the observed vibrational frequencies ν obs , of the C=O and C=C stretches. Clearly state any assumptions that you make. 2. The force constant of HF is 880 N m–1. At what wave number is the fundamental vibrational absorption expected? Where would the corresponding absorption of DF be expected? [Appropriate atomic masses (g/mole) are: 1H, 1.0078; 2H, 2.014; 19F, 18.998] 3. The fundamental and first overtones of the vibrational spectrum of 1H35Cl are at 2885.9 cm -1 and 5668.0 cm -1 respectively. Use these data to calculate the vibrational constants ( ν e and ν e x e ) of this molecule.


Spectroscopy.8.1

Rotational Spectroscopy How can light be used to determine the bond length of a molecule?

Pre-Activity Question 1.

The energy difference between two successive rotational states is typically 10-23 J. Calculate the photon wavelength commensurate with this energy level spacing and identify the corresponding region of the electromagnetic spectrum.

2.

Determine the typical ratio of molecules in the first excited rotational state at room temperature, 300K.

Model 1: Rotational Selection Rules for a 3D Rigid Rotor. For historical reasons, the angular momentum quantum number, l, is typically replaced by J when describing spectroscopic transitions between rotational states.

EJ =

h2 J(J + 1) = Be J(J + 1) 8π 2 I e where

J = 0,1, 2,...

(1)

re is the equilibrium internuclear distance Ie is the equilibrium moment of inertia, I e = µre2 , and Be is the rotational constant

The selection rules corresponding to spectroscopic transitions between two rotational states of a molecule are summarized as follows: 1. The molecule must have a permanent dipole moment. 2. For a heteronuclear diatomic molecule modeled as a rigid rotor, the transition moment is zero unless ΔJ = ±1 .

PGLSAG003028a


Critical Thinking Questions 1. Give an example of a diatomic molecule that will produce a rotational spectrum.

2. Give an example of a diatomic molecule that will not produce a rotational spectrum.

3. Is the following rotational transition for a molecule such as CO allowed or forbidden: J = 7 → J = 6?

4. What molecular information can be obtained from the rotational constant, Be?

5. Derive a general expression for the difference in energy between two sequential rotational states, ΔEJ → J +1 (in terms of Be and J).

6. In view of your answer to CTQ 5, propose a sketch of a rotational absorption spectrum assuming only the first four rotational energy levels are populated. Indicate the corresponding J transition for each line.


Spectroscopy.8.1

Rotational Spectroscopy

Model 2: A Simulated Rotational Spectrum for a Diatomic Molecule The degeneracy of each energy level has been omitted for clarity in the diagram.

EJ

30 Be

J=5

20 Be

J=4

12 Be

J=3

6 Be

J=2

2 Be 0

J=1 J=0 0 1 1 2

2 3

3

4

4 5 Spectrum

Energy

Critical Thinking Questions 7. What does each line of the rotational spectrum correspond to in terms of the energy level diagram?


8. The transition J = 0 â&#x2020;&#x2019; J = 1 occurs at 2Be. Label the photon energy for each of the other transitions on the Spectrum in Model 2. 9. Does the amount of energy absorbed by the molecule change as J increases? If so, by how much?

10. Using grammatically correct English sentences, explain what the difference in energy between two spectral lines corresponds to in terms of the energy level diagram.

11. Although the difference in energy between adjacent rotational states is always increasing, the difference in energy between adjacent rotational spectral lines is always the same. Explain how this is possible.

12. Using the Boltzmann distribution, explain why the intensities of the first four peaks increases as energy increases.


Spectroscopy.8.1

Rotational Spectroscopy

13. Based on the Boltzmann distribution, explain why the intensities eventually start to decrease as energy increases.

14. Propose a sketch of the spectrum corresponding to Model 2 at a lower temperature.

15. What molecular information can be obtained from the spacing of the spectral lines? Present your groupâ&#x20AC;&#x2122;s justification for this answer.


Exercises 1. The J = 0 → J = 1 transition of carbon monoxide has been measured at 3.842 cm–1. a. Convert this value to joules. b. Calculate the wavelength of the photon. c. In what region of the electromagnetic spectrum is this photon? d. If a molecule in the J = 1 state loses energy and rotates with the energy given by J = 0, what is the energy, in joules, of the emitted photon? e. What is the moment of inertia of the CO molecule? f. If the absorption of this photon is from a molecule comprised of 12C and 16O, what is the bond length of the CO molecule? 2. Estimate, as precisely as you can, the frequency, in cm–1, for the radiation absorbed in the J = 0 → J = 1 transition for 13C16O. State/justify any assumptions made for this calculation. 3. The high temperature microwave spectrum of KCl vapor shows an absorption at a frequency of 15,376 MHz. This frequency represents a photon with energy of 10.19 x 10–24 J. This peak has been identified with the J = 1 → J = 2 transition of 39K35Cl. Given that the atomic masses of 39K and 35Cl are 38.96 and 34.97 amu, respectively, calculate the internuclear distance of 39K35Cl in meters and pm. 4. The spacing between the lines in the microwave spectrum of 1H35Cl is 6.350 × 105 MHz. Calculate the internuclear distance of 1H35Cl in meters and pm. The atomic mass of 1H is 1.008 amu. Compare this bond length to the bond length of 39K35Cl. (See Exercise 3.) Does the comparison make sense?


Spectroscopy.8.2

Vibrational-Rotational Spectroscopy Don’t real molecules spin and vibrate simultaneously?

Pre-Activity Question 1.

Using the information in Model 1, create an energy level diagram for a 12C16O molecule including states n = 0 & 1 and J = 0, 1, 2 & 3. Clearly label the energy and quantum numbers associated with each state.

Model 1: Rigid Rotor-Harmonic Oscillator model When a molecule absorbs energy to change its vibrational state, it will also be able to change its rotational state. When diatomic molecules absorb infrared radiation, the new state is defined by both n and J according to the selection rules Δn = ±1 and ΔJ = ±1. The energy levels for a rigid rotor harmonic oscillator are given by: En,J 1⎞ ⎛ = ve ⎜ n + ⎟ + B e J(J + 1) ⎝ hc 2⎠

Molecule 12

C16O

where n = 0,1,2,... and J = 0,1,2,...

ν e / cm −1 2169.8

(1)

B e / cm −1 1.931

Critical Thinking Questions 1.

Consider a 12C16O molecule in the state n=0, J=0 modeled as a rigid rotor harmonic oscillator: a. To which state(s) (n, J) are infrared transitions allowed?

b. What will be the wavenumber corresponding to each transition? Write out the equation you used to determine this value.

PGLSAG003029a


2.

Consider a 12C16O molecule in the state n=0, J=1 modeled as a rigid rotor harmonic oscillator: a. To which state(s) (n, J) are infrared transitions allowed?

b. Without performing a calculation, predict which transition identified above will correspond to a larger wavenumber. Explain.

c. Consider your answer to the PreActivity question and draw an arrow representing the allowed transitions identified in CTQ 2a. For each transition, predict (without calculation) whether it will be observed at a larger or smaller wavenumber than determined in CTQ 1b.

d. Calculate the wavenumber corresponding to each transition and compare this result to your prediction.

3.

For any given vibrational-rotational transition assuming Î&#x201D;n = +1 will the Î&#x201D;J = +1 transition be observed at a smaller or larger wavenumber than the Î&#x201D;J = -1 transition?

4.

According to Model 1, will a 12C16O molecule absorb infrared radiation corresponding to 2169.8 cm-1? Explain.


Spectroscopy.8.2

Vibrational-Rotational Spectroscopy

Model 2: A Simulated Rotational-Vibrational Spectrum for a Diatomic Molecule. J=5 J=4 J=3 J=2 J=1 J=0

n=1

J=5 J=4

n=0

J=3 J=2 J=1 J=0

Spectrum

Frequency

Critical Thinking Questions 5.

Those transitions which have ∆J = –1 are called P-branch transitions and ∆J = +1 transitions are called the R-branch transitions. Identify the P and R branches on the absorption spectrum above. Indicate the starting J value for each transition in parenthesis, e.g. P(2) is the ∆J=–1 transition originating at J=2.


6.

The lines in the spectrum in Model 2 appear to be equally spaced except for a gap in the middle of the spectrum. What is the reason for this gap?

7.

If the spectrum above were in terms of wavelength rather than frequency which transitions would correspond to the P-branch? Present your group’s reasoning.

8.

The spectrum in Model 2 corresponds to 298 K. Draw a similar (approximate) spectrum of 12C16O at 350 K indicating how you expect this spectrum to change with temperature.

Model 3: More accurate representation of RotationalVibrational Energy for diatomic molecules The harmonic oscillator & rigid rotor models are only an approximate representation for diatomic molecules. A more complete expression for the vibrational-rotational energy levels of a real molecule is En, J 1 1 1 = ve (n + ) − ve xe (n + )2 + B e J(J + 1) − α e (n + )J(J + 1) hc 2 2 2

Molecule 12

C16O

ν e / cm −1 2169.8

(2)

ve xe / cm −1

B e / cm −1

α e / cm −1

13.3

1.931

0.018


Spectroscopy.8.2

Vibrational-Rotational Spectroscopy

Critical Thinking Questions 9.

Circle the two additional terms in Equation 2 included to more accurately represent the energy levels of 12C16O.

10. Explain why the first correction term is needed to describe the vibrationalrotational energy of a real molecule.

11. In view of the quantum number dependence of the second correction factor, explain why the second additional term is needed to describe the vibrationalrotational energy of a real molecule.

12. What information can be obtained about a molecule, i.e. molecular parameters, based on the spectral peaks observed in a vibrational-rotational spectrum?

Exercises 1. Derive the following equation for the R branch frequencies, ν˜ R , corresponding to ΔJ = +1 or J i → J i + 1 labeled by the initial J i state. Identify expressions for vo , B˜ o , and B˜1. €

ν R (J i ) = ν 0 + B1 (J i + 1)(J i + 2) − B 0 J i (J i + 1)

€ € 2. Derive the following equation for the P branch frequencies, ν˜ P , corresponding to ΔJ = −1 or J i → J i −1 labeled by the initial J i state. ν P (J i ) = ν 0 + B1 (J i − 1)J i − B 0 J i (J€ i + 1)


Spectroscopy.15

Electronic Spectroscopy I: Atoms Pre-Activity Question 1. Show that the lowest energy terms for the 2s22p2 ground state configuration and the 2s2p3 excited state configuration of carbon shown in the diagram below are correct.

Information The selection rules for electronic transitions in atoms are:

Δl = ±1, ΔL = 0, ±1 (except 0 ↔ 0) and ΔS = 0

Model 1: Energy Levels of Carbon 3

P

2s2 2p 3s

3 D 1 P 3

P

80,000 70,000 60,000 50,000

2s 2p3

40,000 5

S

1

S

2s2 2p 2

20,000

1

10,000

3

0

D P

PGLSAG003030a

30,000

Spectroscopic energy (in units of cm–1)


Critical Thinking Questions 1. Identify the number of triplet P states represented in Model 1. Why is each associated with a different energy?

2. The ground state electron configuration of carbon is 2s22p2. Based on the selection rule Î&#x201D;l = Âą1 , give one example of an excited state electron configuration corresponding to an allowed transition.

3. Is it possible for a carbon atom in the 2s22p2 1D state of Model 1 to absorb a photon and change to the 2s22p2 1S state of Model 1? Explain.

4. Is it possible for a carbon atom in the 2s22p2 1S state of Model 1 to change to the 2s2p3 5S state of Model 1? Explain.

5. Based on Model 1, identify the approximate energies corresponding to the two lowest transitions from the 2s22p2 3P state.


Spectroscopy.15

Electronic Spectroscopy I: Atoms

6. Based on Model 1, how many different absorption peaks would your group expect to observe originating from the ground state electronic configuration of carbon? Identify each transition.

Model 2: Spin-orbit coupling Since it is possible for the spin and orbital magnetic moments to combine, the total angular momentum is defined as the vector sum of its spin and orbital momenta. It is represented by the quantum number J with values J = L + S, L + S − 1,... L − S . The component of J along the z-axis has the possible values of M J  where M J = −J, −J + 1,..., J − 1, J . All states with the same L, S and J values are represented by the symbol 2S+1LJ .

Critical Thinking Questions 7. Identify all possible J values corresponding to the 3P term.

8. List all possible values of MJ (states) for each term identified in CTQ 7. How many total states are represented by a 3P term?

9. Based on the values of L and S, how many total states are represented by a 3P term? Provide your group’s justification for this answer.


Model 3: Fine Structure Additional spectral lines, called fine structure, are observed due to spin-orbit coupling. These transitions follow the selection rule ΔJ = 0, ±1 (except 0 ↔ 0 ). 3 3

P

P0

3

P2

3

P1

3 3

D

2s2 2p 3s

1

P

3

P

2s 2p 3

P1

64087 cm –1 61981 cm

1

P2

3

P1

3

P0

1

1

1

3

P

D3

3

1

D

–1

D1

3

5

S

–1

75254 cm

64091 cm –1 64090 cm

3

5

S

2s 2 2p 2

D2

–1

75256 cm –1 75255 cm

S2 S0 D2

3

P2

3

P1

3

P0

–1 –1

60393 cm

–1

60352 cm

–1

60333 cm

–1

33735 cm

–1

21648 cm

–1

10192 cm

–1

43 cm –1 16 cm –1

0 cm

Note: the vertical axis is not to scale. Use the given values to determine the relative position of the energy levels.


Spectroscopy.15

Electronic Spectroscopy I: Atoms

Critical Thinking Questions 10. Based on Model 3, how many different absorption peaks would your group expect to observe for a carbon atom in the 2s22p2 3P state to the 2s22p3s 3P state? Identify each transition.

11. Identify the value of the two lowest energy transitions identified in CTQ 10.

12. Describe two ways the electronic spectrum of an atom changes based on spinorbit coupling.

Exercises 1. Identify the term symbol for each of the following electron configurations of a hydrogen atom: the 1s1 configuration, the 2s1 configuration, and the 2p1 configuration. Determine whether or not the following transition is allowed or forbidden: 1s1 to 2s1? 2s1 to 1s1? 2p1 to 1s1? 2. For aluminum, the transition [Ne]3s23p1 → [Ne]3s24s1 has two lines, ν~1 = 25354.8 cm -1 and ν~2 = 25242.7 cm -1 . The transition from the ground -1 state → [Ne]3s23d1 has three lines, ν~ = 32444.8 cm , ν~ = 32334.0 cm -1 , 3

4

-1 and ν~5 = 32332.7 cm . Sketch the energy-level diagram of the states involved and explain the source of all the lines.


Spectroscopy.18

Electronic Spectroscopy II: Molecules Pre-Activity Question 1. The typical energy gap between electronic states is 10-18 J. Calculate the wavelength of the radiation commensurate with the energy level spacing for electronic energies and identify the corresponding region of the electromagnetic spectrum. Is it possible to obtain vibrational information from an electronic spectrum? Explain 2. Identify all filled molecular orbitals corresponding to the valence electrons of N2 molecule. Identify the value of ml associated with each molecular orbital.

Model 1: Molecular Term Symbols for Diatomic Molecules Similar to atoms, molecular term symbols are labeled according to Ml, the sum of the ml for all of the electrons. The Greek letters Σ, Π, and Δ are used to denote M l = 0, 1, and 2 , respectively. The superscript corresponds to the multiplicity, 2S +1. In addition, the overall symmetry is indicated based on the product of the symmetry of all filled molecular orbitals. Recall, an even function times an even function results in an even function, an odd function times an odd function results in an even function, and an odd function times an even function results in an odd function. For example, the term symbol corresponding to the ground electronic state of N2 is 1Σg.

Critical Thinking Questions 1. Based on your answer to Pre-Activity Question 2 and the information in Model 1, confirm that the term symbol for the ground state of N2 is 1Σg.

PGLSAG003031a


2. Identify the term symbol for the ground state of O2.    

Model 2: The Franck-Condon Principle Prior to an electronic transition, most molecules are in the n = 0 vibrational state. Since an electron is much lighter than any nucleus, it subsequently moves much faster. Thus, during the time when an electronic transition occurs, there is no appreciable change in the internuclear distances in the molecule—this is called the Franck-Condon Principle. More precisely, the transition probability is given by the overlap integral of the initial and final vibrational wave functions at the same internuclear distance.

Energy

Internuclear distance


Spectroscopy.18

Electronic Spectroscopy II: Molecules

Critical Thinking Questions 3. On the diagram in Model 2, draw a line corresponding to the lowest vibrational level of the excited state. Offer a physical explanation as to why the equilibrium internuclear distance for the n = 0 vibrational excited state is larger than the n = 0 vibrational ground state.

4. Based on the Franck-Condon principle, explain why it is unlikely to observe a transition from re (n = 0) of the electronic ground state to re (n ' = 0) of the electronic excited state.

5. Recall the Boltzmann distribution for vibrational energy levels and explain why at room temperature the most likely electronic transition is from the n = 0 state of the electronic ground state (rather than the n = 1 state of the electronic ground state).

6. Assuming the transition illustrated in Model 2 is allowed, would a vibrational selection rule of ホ馬 = ツア1 apply? Why or why not.


Information Selection Rules for Electronic Transitions in Molecules: • The spin quantum number must not change, ΔS = 0. • Other selection rules depend on the symmetry of the molecule. For molecules with a center of symmetry, electronic transitions between ungerade and gerade states are allowed (u ↔ g); transitions between gerade states (g ↔ g) and transitions between ungerade states (u ↔ u) are forbidden. • There is no restriction on the change in the vibrational quantum number. In general, there is no restriction on changes in orbital angular momentum.

Model 3:

Fluorescence.

When a ground state molecule absorbs a photon (say from n = 0 in the ground state to n' = 5 in the excited electronic state) several subsequent processes can result: • •

a fast emission of a UV/VIS energy photon and return to the ground state. emission of one or more infrared photons (as the system goes to υʹ′ ≤ 4 of the excited electronic state) followed by emission of a UV/VIS energy photon and return to the ground electronic state. a radiationless transition to a lower vibrational state (to υʹ′ ≤ 4 of the excited electronic state) may occur. The vibrational energy lost can be transferred (a) to another molecule in a collision, (b) to a different vibrational mode within the same molecule, (c) to rotational motion within the same molecule, or (d) any combination of the above. Once the molecule is in a lower vibrational energy state (υʹ′ ≤ 4 of the excited electronic state), the molecule can emit a UV/VIS energy photon and return to the ground state. This is the predominant fluorescence process for molecules in liquid and solid phases.


Spectroscopy.18

Electronic Spectroscopy II: Molecules

All radiative transitions to the ground electronic state for which the spin quantum number, S, does not change, is called fluorescence. Typically, fluorescence occurs about 10–6 seconds after absorption of a photon.

0

1

2

3

4

Energy

1

0

Σg

1

Internuclear Distance

Critical Thinking Questions 7. Based on the following term symbols, label the excited state in Model 3 by the only term representing an allowed ground state to excited state transition. 1Δ 1Σ g g

1Σ u

3Σ g


8. According to Model 3, will all the emitted photons have the same wavelength? Explain.

9. Based on Model 3, explain why the energy of the emitted photon (fluorescence) is generally lower than the energy of the absorbed photon.

Model 4:

Phosphorescence.

Another possibility exists for an electronically excited molecule. The excited molecule can undergo a collision (radiationless transition) that causes the spin quantum number to change (usually, this is from an S = 0 state to an S = 1 state); this is called intersystem crossing. The transition back to the ground state is now forbidden and the molecule is trapped in the excited electronic state. It will, of course, lose vibrational energy and most likely wind up in the Ď&#x2026; = 0 vibrational level of the excited electronic state. Eventually, the electronically excited state molecule releases a photon in a radiative transition to the ground electronic state and the spin quantum number, S, does change, the molecule is said to phosphoresce and the process is called phosphorescence. Phosphorescence typically occurs at 10â&#x20AC;&#x201C;3 to 10 seconds after absorption of a photon.


Spectroscopy.18

Electronic Spectroscopy II: Molecules

1

Σu

3Σ u

0

Energy

1

2

3

1

0

Σg

1

Internuclear distance

Critical Thinking Questions 10. For each of the following processes, draw and label an arrow depicting: absorption, intersystem crossing, vibrational relaxation and phosphorescence.


11. Why is the intersystem crossing a radiationless transition?

12. According to Model 4, is the energy of the emitted photon (phosphorescence) higher or lower than the energy of the absorbed photon?


Symmetry.1

Group Theory What is the benefit of characterizing group properites for molecules?

Model 1 Symmetry Elements & Operators 180째

Consider a box. When you rotate the box by 180째 the resulting figure looks indistinguishable (but not identical) to the original arrangement. The rotation of the rectangle to generate an equivalent rectangle is referred to as a symmetry operation. The axis about which the rotation occurs is the corresponding symmetry element of this operation. An axis of symmetry is denoted Cn, where n is the number of times the operation has to be repeated in order for the object to return to its original arrangement. Many objects contain more than one symmetry axis. The Cn axis with the largest n is the principle axis.

Critical Thinking Questions 1. Why is the rectangle indistinguishable but not identical after the 180째 rotation?

2. Is there more than one symmetry axis for a rectangle? If so, how many?

3. Identify the name of each unique symmetry axis for a rectangle.

PGLSAG003032a


4. If this rectangle had four equal sides (a square) is there more than one type of symmetry axis for this object? Specify the name of each unique axis and identify the principle axis.

5. Identify the symmetry operation that is equivalent to two successive Cˆ 4 operations, i.e., Cˆ 2 . 4

6. Does a rectangle contain a reflection plane of symmetry, σ , reflecting every point of the rectangle through the plane like a mirror? If so, identify the number of horizontal mirror planes, σ h (perpendicular to a principle axis) and vertical mirror planes, σ v (which contain the principle axis).

Information Symmetry Elements

Symmetry Operations

E

Identity

No change

Cn

n-Fold rotation axis

Cˆ n

rotate about axis by 360°/ n

σ

reflection (mirror) plane

σˆ

reflect through a plane

i

center of inversion

(x, y,  z)  →  (–x,  –y,  –z)  

Sn

n-Fold rotation-reflection (improper rotation) axis

Sˆn

Rotate by 360°/n followed by reflection through a plane perpendicular to the rotation axis


Symmetry.1

Group Theory

Critical Thinking Questions 7. Identify each possible symmetry element for the rectangle depicted in Model 1.

8. Some operations are not unique, i.e. they can be equivalently expressed by more than one operator. For each of the following, specify an equivalent operator: a) Cˆ1 b) Sˆ1 c) Sˆ2 9. What type of molecule would have a C∞ rotational axis?

Exercises 1. Identify all unique symmetry elements for each of the following molecules: a) H2O b) CH2Cl2 c) NH3 d) BF3


Model 2 Symmetry Groups C2

If two symmetry operations are symmetry elements of an object, then their product is also a symmetry element. Any set of operations for which any product of that set is also a member of the set is called a group. For example, a water molecule has four symmetry elements E, C2, and two σ v which we will designate σ v (xz) and σ v! (yz) . These four symmetry elements form the C2v point group

σv

σ v,

z

x

y

Critical Thinking Questions 10. Using H2O, pictured above, as an example, identify the symmetry element that results from the following sequential operations: a) Cˆ 2 Eˆ b) Cˆ 2Cˆ 2 c) Cˆ 2σˆ v d) Cˆ 2σˆ v! 11. What is the relationship between the result of each operation performed in CTQ 10 and the symmetry elements defined for this group?


Symmetry.1

Group Theory

12. Complete the following multiplication table multiplying each row operation by the designated column operation and identifying the resulting symmetry operation in the table. The results of CTQ 10 are already included.

σˆ v (xz)

σˆ v! (yz)

σˆ v!

σˆ v

Cˆ 2

Eˆ Cˆ 2

Cˆ 2

Cˆ 2

σˆ v (xz)

σˆ v!

σˆ v! (yz)

σˆ v

13. What is the relationship between the result of each operation performed in CTQ 12 and the symmetry elements defined for this group?


Model 3 Flowchart for Determining Point Groups Is it linear? No

Yes Is there an inversion center?

Is there more than one axis that is greater than C2 ? Yes

Does it have a Cn axis?

Is the highest axis a C3 axis?

No

No

Yes

Is the highest axis a C4 axis? No Are there n C2 axes to the Cn axis?

I, Ih

O, Oh

Is there a mirror plane?

No

Yes

Yes

Yes

No

Cs

Is there an inversion center? Yes

Is there a horizontal mirror plane? Yes

Is there a horizontal mirror plane? No

Are there n vertical mirror planes? Yes

Cnv Are there n vertical mirror planes?

Dnh

Yes Dnd

No Dn

C1

Ci

No

Cnh

Yes

No

No Is there an S 2n axis? Yes S 2n

No Cn

No

Yes

No

Dâ&#x2C6;&#x17E;h

Câ&#x2C6;&#x17E;v

Yes

T, Th , Td


Symmetry.1

Group Theory

Critical Thinking Questions 14. Using H2O as an example, describe the decision-making process that classifies this molecule as belonging to the C2v point group.    

Exercises 2. Using the flow chart, identify the point group for each of the following molecules: a) CH2Cl2 b) NH3 c) BF3 d) C6H6 (benzene)

Model 4: Character Tables For each group, the symmetry operations can be represented by numbers that obey the multiplication table of a group. For example, the operators of the C2v group can be represented by different combinations of the numbers -1 and 1 such that the multiplication table is satisfied. While each group has an infinite number of different possible representations, the most fundamental are called the irreducible representations, Γ , identified by its character table. For example, there are four different irreducible representations for the C2v point group defined by the following character table: Cˆ 2

σˆ v (xz)

σˆ v! (yz)

Functions

Γ1 = A1

Eˆ 1

1

1

1

z, x2, y2, z2

Γ 2 = A2

1

1

-1

-1

xy

Γ 3 = B1

1

-1

1

-1

x, xz

Γ 4 = B2

1

-1

-1

1

y, yz


Critical Thinking Questions 15. Using the B1 irreducible representation as an example, show the product of the numbers representing each operation leads to the correct representation (number) of the resulting symmetry element determined in CTQ 12: a) Cˆ 2 Eˆ b) Cˆ 2Cˆ 2 c) Cˆ 2σˆ v c) Cˆ 2σˆ v! 16. Each representation can also be labeled as either symmetric or asymmetric with respect to rotation. Based on this definition, identify the two symmetric irreducible representations.

17. The last column of the character table specifies the coordinate that is consistent with the representation. For example since the x coordinate changes sign upon a Cˆ 2 operation, it must be represented by either B1 or B2. Justify why the x-axis is associated with the B1 representation.


Symmetry.1

Group Theory

Exercises 3. Create a multiplication table to prove that the symmetry elements E, C2, i, and Ď&#x192; h form a group. 4. Determine the point groups for each of the following planar molecules: a)

b) H

H C

Cl

c) H

C

Cl C

Cl

Cl

H

C

Cl C

H

H

C Cl


Symmetry.2

Symmetry Adapted Molecular Orbitals Pre-Activity Questions 1. Propose a trial wavefunction corresponding to the valence electrons of a water molecule. (Similar to Fundamental.17 Model 3) Do you think each atomic wavefunction will contribute equally to a single molecular orbital of water? Why or why not? 2. Using the same coordinate system designated in Model 1, create a sketch of the valence atomic orbitals associated with a single water molecule and identify each orbital with a unique label, e.g., 1sH A .

Model 1 C2

σv

σ v,

z

x

PGLSAG003033a

y

Cˆ 2

σˆ v (xz)

σˆ v′ (yz)

A1

1

1

1

1

A2

1

1

-­‐1

-­‐1

B1

1

-­‐1

1

-­‐1

B2

1

-­‐1

-­‐1

1


Critical Thinking Questions

1. Which of the following relationships best describes the effect of a C2 operation on the 2py oxygen wavefunction of water? Present your group’s reasoning. a) Cˆ 2 2 py(O) = 2 py(O) b) Cˆ 2 2 py(O) = 2 px(O) c) Cˆ 2 2 py(O) = −2 py(O)

2. Which of the following relationships best describes the effect of a C2 operation on one of the 1s hydrogen orbitals associated with water? a) Cˆ 2 1s(H A ) = −1s(H A ) b) Cˆ 2 1s(H A ) = 1s(HB ) c) Cˆ 2 1s(H A ) = −1s(HB ) 3. Give an example of a symmetry operator (aside from the identity operator) that when performed would leave the 2pz orbital unchanged after one operation.


Symmetry.2

Symmetry Adapted Molecular Orbitals

4. Complete the following table indicating the result of the following operation on the designated atomic orbital. The operations identified in the previous CTQs are included as an example.

1s(H A )

1s(H B )

2s(O)

2 px(O)

2 py(O)

2 pz(O)

−2 py(O)

2 pz(O)

Eˆ Cˆ 2

1s(H B )

σˆ v (xz) σˆ v′ (yz)

5. According to group theory, each orbital belongs to a particular representation of the point group based on how it is transformed by each symmetry operation. a) Consider how each symmetry operator affects the three 2p orbitals of oxygen. In view of the character table in Model 1, does it appear that each orbital would belong to the same representation? Why or why not.

b) Assuming the 2pz orbital belongs to the A1 representation, identify all other orbitals that are transformed by each symmetry operation in a similar manner.

6. Which representation would you associate the 2py orbital? Explain.


Model 2 Symmetry Adapted Atomic Orbitals (SO) We will use the following procedure to specify all unique symmetry adapted atomic orbitals (SO) and their symmetry representation. For each irreducible representation of the molecule’s point group, we will multiply the character of the corresponding symmetry operation by the orbital specified for a single column of the table created in CTQ 4. The sum of n products is multiplied by the normalization factor 1/n, resulting in the SO with the symmetry properties of the corresponding irreducible representation. Using the first column as an example and the character table displayed in Model 1, we identify two nonzero SOs for water. 1 1 ψ A1 = (1s(HA ) +1s(HB ) +1s(HB ) +1s(HA ) ) = (1s(HA ) +1s(HB ) ) 4 2 1 ψ A2 = (1s(HA ) +1s(HB ) −1s(HB ) −1s(HA ) ) = 0 4 1 ψ B1 = (1s(HA ) −1s(HB ) +1s(HB ) −1s(HA ) ) = 0 4 1 1 ψ B2 = (1s(HA ) −1s(HB ) −1s(HB ) +1s(HA ) ) = (1s(HA ) −1s(HB ) ) 4 2

Critical Thinking Questions 7. Based on the procedure outlined in Model 2 and the A2 symmetry example, explain why two of the atomic orbitals are “added” and two are “subtracted” for this linear combination.

8. Determine whether or not the 2 py(O) column of the CTQ 4 table belongs to the A1 representation using the Model 2 procedure. Present your group’s work.


Symmetry.2

Symmetry Adapted Molecular Orbitals

9. In addition to the Ď&#x2C6; A1 identified in Model 2, determine the two other unique nonzero SOs that belong to the A1 representation for water and write them below.

10. Are there any nonzero SOs that belong to the A2 representation? If so, identify the resulting linear combination of orbitals.

11. Identify all nonzero SOs that belong to the B1 representation for water.

12. Identify all nonzero SOs that belong to the B2 representation for water.

13. Identify the total number of SOs that will be used to create molecular orbitals of water. How does a trial wavefunction using symmetry adapted molecular orbital of water differ from the one proposed in Pre-Activity 1?


14. Recall that the overlap integral between two atomic orbitals i and j is Sij = ∫ φi*φ j dτ . If the value of this integral corresponding to two SOs is zero, will they both contribute to the same molecular orbital of water? Explain.

Model 3 Symmetry product functions Consider the following integral,

∫ψ

* Γi

ψ Γ j dτ

(1)

where Γ i and Γ j correspond to the irreducible representation indicating the symmetry of each wavefunction. Each irreducible representation of a point group ˆ defined by the character is an n-dimensional vector with components χ i ( R) representing each symmetry operation, Rˆ . According to group theory, Equation 1 is nonzero if the corresponding direct product of the representation vectors of each function is totally symmetric,

Γ i ⊗ Γ j = Γ A1

(2)

This direct product is an element-by-element multiplication that forms a new vector: Γ i ⊗ Γ j = ( χ i ( Rˆ1 ), χ i ( Rˆ 2 ), χ i ( Rˆ 3 ),...) ⊗ ( χ j ( Rˆ1 ), χ j ( Rˆ 2 ), χ j ( Rˆ 3 ),...) = ( χ i ( Rˆ1 ) χ j ( Rˆ1 ), χ i ( Rˆ 2 ) χ j ( Rˆ 2 ), χ i ( Rˆ 3 ) χ j ( Rˆ 3 ),...)

(3)

For example,

Γ B1 ⊗ Γ B2 = (1, −1,1, −1) ⊗ (1, −1, −1,1) = (1,1, −1, −1) = Γ A2

(4)


Symmetry.2

Symmetry Adapted Molecular Orbitals

Critical Thinking Questions 15. Complete the following multiplication table identifying the resulting direct product of each pair of irreducible representations. The result from Model 3 is already included.

Γ A1

Γ A2

Γ B1

Γ B2

Γ A1 Γ A2 Γ B1 Γ B2

Γ A2 Γ A2

16. Based on Model 3 and your group’s answer to CTQ 15, what is required of each wavefunction such that the overlap integral is nonzero?

17. Will a 2 py(O) SO combine with (1sH1 +1sH 2 ) SO to form a water molecular orbital? Explain.

18. Identify all SO wavefunctions that will have non-zero contribution with 2 pz(O) to form a water molecular orbital. Write one possible trial wavefunction specifying only the SO(s) that will contribute to molecular orbital (i.e., nonzero coefficients) and indicate the number of possible linear combinations that belong to this same representation.


19. Write one possible trial wavefunction corresponding to molecular orbital that belong to the B2 representation. Propose a sketch of this MO and indicate whether your group expects this orbital to be higher or lower in energy than the MO in CTQ 18 and why.

Model 4 Symmetry Adapted Linear Combinations (SALCs)

All symmetry adapted atomic orbitals (SOs) of the same symmetry and similar energy will contribute to some extent to the overall molecular orbital wavefunction. The degree of contribution is defined by the magnitude of the molecular orbital coefficients that can be calculated using a variational theory approach, e.g. Hartree-Fock theory. Table 1. Hartree-Fock STO-3G wavefunctions for water Coefficients for each MO ψ i Atomic Orbital Energy (a.u.) 2s(O) 2 px(O) 2 py(O) 2 pz(O) 1s(H1 ) 1s(H2 )

ψ A1

ψ B2

ψ A1

ψ B1

ψ A1

ψ B2

-1.258

-0.5939

-0.4598

-0.3926

0.5820

0.6929

0.8444 0.0000 0.0000

0.0000 0.0000 0.1229

-0.5383 0.0000 0.0000

0.0000 1.0000 0.0000

-0.8204 0.0000 0.0000

0.0000 0.0000 0.9599

0.1229 0.1556 0.1556

0.0000 -0.4492 0.4492

0.7559 0.2950 0.2950

0.0000 0.0000 0.0000

-0.7636 0.7693 0.7693

0.0000 0.8148 -0.8148

Critical Thinking Questions 20. Are the coefficients presented in Table 1 consistent with your group’s predictions in CTQ 18 & 19? Rationalize any differences.


Symmetry.2

Symmetry Adapted Molecular Orbitals

21. Identify the highest occupied molecular orbital (HOMO) for water.

22. Is the sign for each AO coefficient belonging to the A1 representation always the same? Suggest how this observation can be used to explain the reason for the energy differences between these MOs.

23. Consider the two MOs belonging to the B2 representation. Propose a sketch for each of these two MOs and provide an explanation for their significant difference in energy.

24. Describe one advantage of using symmetry (group theory) for proposing potential trial wavefunctions for polyatomic molecules. Â


Symmetry.3

The Symmetry of Vibrational Modes Model 1: Vibrational Degrees of Freedom Let the number of nuclei in a molecule equal N. One can specify the position of all of the nuclei with 3 N coordinates (xi,yi,zi for each nucleus). The molecule is said to have 3N degrees of freedom. Alternatively, one could specify the center of mass of the entire molecule with three coordinates (Xi,Yi,Zi). Because the translational motion of a molecule is expressed through the motion of the center of mass of the molecule, a molecule is said to have three translational degrees of freedom. If the molecule is linear, the orientation of the nuclei about its center of mass requires two coordinates (usually θ and φ). A diatomic molecule is said to have two rotational degrees of freedom. If the molecule is not linear, the orientation of the nuclei about its center of mass requires three coordinates, and the molecule is said to have three rotational degrees of freedom. The remaining coordinates, 3N – 5 for a linear molecule and 3N – 6 for a nonlinear molecule, are assigned to the vibrational motion of the molecule and are called vibrational degrees of freedom.

Translational degrees of freedom Rotational degrees of freedom Vibrational degrees of freedom

Linear 3 2 3N – 5

Nonlinear 3 3 3N – 6

Critical Thinking Questions 1. Why does SO2 have three degrees of rotational freedom whereas CO2 has only two degrees of rotational freedom?

2. Identify the number of vibrational degrees of freedom for H2O.

PGLSAG003034a


Exercise 1. Complete the following Table.

Molecule N2 HCl CS2 H2 O CH4 benzene H2NCH2COOH

Total 6

Degrees of Freedom Translational Rotational 3 2

Vibrational 1

Model 2: The Vibrational Modes of Water It is beyond the scope of this course, but it can be shown (for the harmonic oscillator model) that the number of vibrational degrees of freedom of a molecule, Nvib, is expressed as Nvib harmonic oscillators or normal modes in the molecule. For example, the water molecule has Nvib = 3 and water has three vibrational normal modes; each of the modes has an oscillating dipole moment.

Symmetric Stretch = 3650 cm–1

Antiymmetric Stretch = 3760 cm –1

Bend = 1600 cm–1

Each vibrational mode can be classified according to its symmetry with respect to the effect of the operations of the point group. A value of one is assigned if the direction of the motion unchanged, and a value of negative one is assigned if the motion is opposite in direction after the operation.


Symmetry.3

The Symmetry of Vibrational Modes

Critical Thinking Questions 3. Which vibrational mode(s) of water remain unchanged after preforming a C2 operation?

4. For each of the following symmetry operations, indicate whether or not the operation will result in a sign change with respect to the collective motion of the atoms for the antisymmetric stretch of water. (Refer to Symmetry.2 Model 1) a) Eˆ b) Cˆ 2 c) σˆ v (xz) d) σˆ v′ ( yz) 5. In view of how each vibrational mode is transformed by each symmetry operation of the C2v point group identify the representation each mode belongs to. a) Symmetric stretch b) Asymmetric stretch c) Bend


Model 3: Infrared Activity of normal modes Recall that the probability of a transition between two quantum states described by Ψ a and Ψ b is given by the integral * ∫Ψ a uˆΨ b dτ

where µˆ is the transition moment operator. According to group theory, this integral is nonzero if the direct product of the representation vectors of each function is totally symmetric. (Recall Symmetry.2 Model 3) For vibrational transitions, µˆ is the electric dipole moment with symmetry corresponding to irreducible representations of either x, y, or z. Since the ground state vibrational wavefunction is symmetric, if the symmetry species of a normal mode is the same as any of the symmetry species of x, y, or z, then the mode is infrared active.

Critical Thinking Questions 6. Which axis (x, y, or z) does not change sign following a C2 operation? 7. Based on how each axis is transformed by the symmetry operations of the C2v point group identify the representation each axis belongs to. a) x b) y c) z

8. Would you see a peak in the IR spectra of water corresponding to a vibrational normal mode of water belonging to the A2 representation? Why or why not.

9. Which normal modes of water IR active?


Extension.1

Operators and Eigenvalue Equations Do you really know your algebra?

Information An operator is a symbol that defines a specific transformation or operation to be performed on a function resulting in new function. Operators are typically denoted by a carat. Operator Mˆ

Operation 7

Description multiply by 7

d dx

take derivative with respect to x

take the square root

The sum of two operators is defined by:

ˆ f (x) ≡ Af ˆ (x) + Bf ˆ (x) ( Aˆ + B) The product of two operators is defined by:

ˆ (x) ≡ Aˆ ⎡ Bf ˆ ⎤ Aˆ Bf ⎣ (x) ⎦

Critical Thinking Question 1. Determine the result of applying the following operators from above:

(

Dˆ x 2 + 5x

(

Dˆ α eα x

PGLSAG003035a

)

)

( )

( Mˆ + Dˆ ) 3x

Sˆ −x 4

Dˆ 2 cos(2x)

Sˆ x 3 − 3x

(

)


Model 1 Linear Operators In quantum mechanics, only linear operators are used such that

ˆ (x) + Af ˆ (x) Aˆ [ f1 (x) + f2 (x)] = Af 1 2

(1)

Critical Thinking Questions 2. Consider the operators introduced in CTQ 1. Identify one that is an example of a linear operator. Provide your group’s justification for this answer.

3. Give an example from CTQ 1 of a nonlinear operator.

Model 2 Eigenfunctions and Eigenvalues Frequently we will examine problems that take the form of an eigenvalue equation.

Aˆφ (x) = aφ (x)

(2)

where a is a constant. φ(x) is said to be an eigenfunction of the operator Aˆ and a is the eigenvalue associated with φ(x).


Extension 1

Operators and Eigenvalue Equations

Critical Thinking Questions 4. Which example in CTQ 1 represents an eigenvalue equation for the Dˆ operator? Identify the corresponding eigenvalue.

5. For any given operator, do all functions have a corresponding eigenvalue? Present your group’s justification for this answer.

6. Can an operator have more than one eigenfunction? If so, give an example.

7. Despite the form of Equation 2, describe why it is incorrect to assume that Aˆ = a ?

8. Write an eigenvalue equation for Sˆ and the function, φ(x)=1. Identify the associated eigenvalue.


9. Is it possible to write an eigenvalue equation for the operator Dˆ when φ(x)=1? If so, identify the corresponding eigenvalue.

10. The eigenvalue equations shown in CTQ 8 and 9 are called trivial solutions since the eigenvalue is either zero or one. Provide your group’s rationale for calling these trivial solutions.

11. In view of your answers to CTQ 4-6, determine whether the following statement is true or false and provide the group’s rationale for this answer. For a particular operator and function, there will exist many values that are eigenvalues. Indeed every function will have at least one eigenvalue for each operator.


Extension.3.1

Hermitian Operators Model 1 Quantum mechanical operators that correspond to physical properties must be linear and Hermitian. A Hermitian operator, Aˆ , is one which satisfies the equation

f * Aˆ gdτ =

all space

all space

( )

ˆ *dτ g Af

(1)

where f and g are well behaved functions as defined in Postulate 1 of Fundamental.1.

Critical Thinking Questions 1.

Assume f and g are the same normalized function, ψ , and this function is an eigenfunction of Aˆ with eigenvalue a. Based on these assumptions, rewrite the first integral in Equation 1 and identify its result.

2.

Using a grammatically correct English sentence, describe the significance of the integral proposed in CTQ 1.

PGLSAG003036a


*

3.

Based on your answer in CTQ 1, propose an integral expression for A in terms of the function, ψ , and show that it is equivalent to the second integral assuming f and g are the same function.

4.

Identify the result of CTQ 3 assuming ψ is a normalized eigenfunction of Aˆ with eigenvalue a.

5.

Based upon the equality of the two integrals in Model 1, what do the results of CTQ 1 and 4 imply about observables in quantum mechanics?

6.

Why must the operators used in Quantum Mechanics be Hermitian operators?

7.

Now assume f and g represent two different normalized eigenfunctions, ψ n and ψ of the Hermitian operator, Aˆ . Identify an appropriate eigenvalue m

equation for each function assuming a unique observable corresponds to each state.


Extension.3.1

Hermitian Operators

8.

Based on the assumptions identified in CTQ 7, rewrite the first integral of Model 1 and simplify the expression using the appropriate eigenvalue relationship.

9.

Based on the assumptions identified in CTQ 7, rewrite the second integral of Model 1 and simplify the expression using the appropriate eigenvalue relationship.

10. Subtract the simplified expression in CTQ 9 from the expression in CTQ 8 and identify the result based in Model 1.

11. If each state function has a unique eigenfunction, what does this imply about the value of the integral in CTQ 10?

12. What does this result indicate must be true of the eigenfunctions of a Hermitian operator?


Extension.3.2

The Heisenberg Uncertainty Principle Can we be certain of anything?

Pre-Activity Questions 1. According to classical mechanics, how are observable values of momentum calculated? Is it possible to simultaneously know the value of position and momentum, i.e., can you predict the exact measureable value of momentum for any observable value of position? Explain. 2. According to quantum mechanics, how are observable values of momentum calculated? 3. Characterize how uncertainty in a laboratory measurement might arise and explain how that uncertainty might be reduced.

Term 1: Simultaneous measurements According to Quantum Mechanics, measurable quantities corresponding to a particular state (represented by a wavefunction) are the eigenvalues of the operator representing the physical observable. (Recall Fundamental.1 Postulate 3). Thus the measured observables are obtained mathematically by operating on a wavefunction. Measurable values of two different observables can be obtained by performing two sequential operations: Bˆ ( Aˆ ψ (x)) or Aˆ ( Bˆ ψ (x)) where the operation in parenthesis is performed first. The values of two different observables (e.g. position and momentum) can only be known simultaneously if the measurement doesn’t change the state of the system. € €

Critical Thinking Questions 1. Recall that ψ (x) represents the state of a system. If the state of a system changes, what does this imply about the wavefunction describing the state?

PGLSAG003037a


2. Assume ψ (x) , is an eigenfunction of the operator, Aˆ . Is it possible to determine the value of the measured observable a without changing the state of the system? Identify your group’s justification for this answer. €

3. For the same system described in CTQ 2, assume ψ (x) is also an eigenfunction of the operator, Bˆ , with corresponding observable b. Is it possible to simultaneously measure a and b? Present your group’s rationale for this answer. €

Term 2: The Commutator The commutator of two operators is an operator defined as Aˆ , Bˆ ≡ Aˆ Bˆ − Bˆ Aˆ . For an arbitrary function, f(x),

[ ]

ˆ B] ˆ f (x) = A( ˆ Bf ˆ (x)) − B( ˆ Af ˆ (x)) = gf (x) [ A, €

(1)

If the value of the commutator g = 0, the operators Aˆ and Bˆ commute.

Critical Thinking Questions 4. For the system described in CTQ 3, do operators Aˆ and Bˆ commute? Justify your group’s answer.


Extension.3.2

The Heisenberg Uncertainty Principle

5. What is the value of the commutator corresponding to the position, x, and momentum, px, operators? Do these operators commute?

6. According to quantum mechanics, is it possible to know the values of position, x, and momentum, px, simultaneously? Identify your group’s reasoning.

Term 3: Variance and Uncertainty 2 Consider a measurable quantity with an average value a . The variance, σ , of the measurement is defined as

σ 2 = a2 − a

2

(2) € € The standard deviation, σ , quantifies the uncertainty of the measurement. The following inequality is a general statement of the Heisenberg uncertainty principle: € 1 ˆ ˆ € σ aσ b ≥ A, B (3) 2

[ ]

where σ a and σ b represent the inherent uncertainty in the simultaneous measurement of observables a and b corresponding to operators Aˆ and Bˆ , € respectively. €

Critical Thinking Questions 7. Identify the three different mathematical operations represented symbolically in Equation 3.


8. According to Equation 3, if two operators commute, what is the inherent (uncontrollable) uncertainty of the simultaneous measurement of the two corresponding observables?

9. What does Equation 3 suggest regarding the inherent uncertainty of two measurements for which their corresponding operators do not commute?

10. As a consequence of Equation 3, if two operators do not commute, how does the inherent uncertainty in the measurement of one observable change as the uncertainty in the measurement of the second observable decreases? Identify your groupâ&#x20AC;&#x2122;s rationale for this answer.

11. In view of your answer to CTQ 5, what is the minimum uncertainty in the simultaneous measurement of position and momentum? (Note:

a + bi = a 2 + b 2 )

12. According to quantum mechanics, if the position, x, of a particle is known exactly, how well can the momentum, px, of the particle be known?


Extension.3.2

The Heisenberg Uncertainty Principle

Problems 1. Consider the n = 1 state of a particle confined to a line of length a: Calculate x  and   x 2  as  a  function  of  a.    

a.

b.€ According to Equation 2, what is the value of σ x ? c. Calculate   p  and   p 2  as  a  function  of  a. € € d. Based on the expectation values calculated above, what is the value of € σ xσ p ? € € e. Is the uncertainty principle (Equation 3) satisfied for the n = 1 state? Justify your answer in view of CTQ 11.   €

2. Consider the n = 2 state of a particle confined to a line of€length a: a.

Calculate x ,   x 2 ,  and   σ x  as a function of a.    

b.€ Calculate p ,   p 2 ,  and   σ p  as a function of a. c. How does the position uncertainty change by going from the n = 1 state € € € to the   n = 2 state? Rationalize this result by comparing plots of the probability distributions for these two states. € € € d. Calculate the inherent uncertainty, σ xσ p for the n =2 state. How does € this result compare to the n = 1 state? Describe the significance of this result. €


Extension.4

Separation of Variables Do I really need to know multivariate calculus?

Model 1 If the Hamiltonian operator can be written as the sum of operators that are dependent upon independent coordinates, q1, q2, …, qn :

Hˆ = Hˆ 1 q1 + Hˆ 2 q2 + … + Hˆ n qn

( )

( )

( )

then the wavefunction can be written as the product of functions of each coordinate:

ψ = f1 (q1 ) f2 (q2 )... fn (qn )

Critical Thinking Questions 1. Consider a particle confined to the x-y plane. Define Hˆ 1 and Hˆ 2 for this model.

2. Write the Schrödinger equation for a particle confined to a plane but replace ψ ( x, y ) ≡ X(x) ⋅ Y (y) .

PGLSAG003038a


3. Simplify the expression in CTQ 2 as much as possible. (Note: X(x) is not a function of y and Y(y) is not a function of x, so some functions remain unaffected by certain derivatives.)

4. Divide the whole expression from CTQ 3 by X(x)â&#x2039;&#x2026;Y(y) and simplify.

a. Consider both sides of this equation and explain why the constant, E, can be viewed as having an independent contribution from each coordinate.

b. Considering that E is a constant and that X(x) and Y(y) are independent 1 d 2 X(x) functions, will the function be a constant or have a X(x) dx 2 variable value? Â


Extension.4

Separation of Variables

5. Based on your group’s analysis in CTQ 4, write two separate 1D differential equations such that E = E x + E y . Do they look familiar?

6. How many quantum numbers are needed to define this 2D system? Propose the appropriate variable(s) and define all possible values.

7. In view of Model 1, write a general expression for the wavefunction, ψ ( x, y ) for a particle confined to a two-dimensional region of space 0 ≤ x ≤ a and 0 ≤ y ≤ b.

8. Write a general expression for the energy, E for a particle confined to a plane.

9. What is the primary advantage of using the separation of variables technique when solving the Schrödinger equation?


Extension.5.1

Two-Particle Systems Model 1 m1

m2

x1

x2

For the two-body system illustrated above, it is convenient to specify the position of the center-of-mass, XCOM, and a relative or internal coordinate xr defined by X COM ≡

m1 x1 + m2 x 2 M

where the total mass M = m1 + m2. €

and x r ≡ x 2 − x1

(1)

Critical Thinking Questions 1. On the diagram in Model 1, indicate the approximate position of the center-ofmass assuming m1 > m2. 2. If the two particles are connected by a spring, is the position of particle one, x1, dependent on the position of particle two, x2? Record your group’s rationale for this answer.

3. In view of your answer to CTQ 2, is it possible to express the Hamiltonian operator for this system as a function of two independent coordinates: Hˆ (x , x ) = Hˆ (x ) + Hˆ (x ) ? Explain. 1

PGLSAG003039a

2

1

2


Model 2 Using the definitions presented in Model 1, the general form of the Hamiltonian operator for a two-body system can be written as a function of the center-of-mass coordinate, XCOM, and an internal coordinate xr, 2 2 2 d 2 d ˆ H( XCOM , xr ) = − − + V (xr ) 2 2M dXCOM 2 µ dxr2

m1m2 . The introduction of center-of-mass and m1 + m2 internal coordinates recasts the Hamiltonian operator in terms of a fictitious particle of mass M, located at the system’s center-of-mass coordinate, XCOM, and a second fictitious particle of mass µ , subject to a potential energy described by the system’s internal€coordinate, xr.

where the reduced mass µ =

Critical Thinking Questions 4. Is it possible to express the Hamiltonian operator as a function of two ˆ ˆ ˆ independent coordinates: H (XCOM , xr ) = H (XCOM ) + H (xr ) ? Explain.

ˆ 5. Consider the form of the Schrödinger equation corresponding to H (XCOM ) . Is it possible to identify the solution to this equation based on a previous model for which the solutions are known? Explain.

6. Specifically describe the type of energy corresponding to the expression identified in CTQ 5.


Extension.5.1

Two-Particle Systems

ˆ 7. Write the Schrödinger equation corresponding to H (xr ) , assuming V (xr ) = 12 kxr2 . Is it possible to identify the solution to this equation based on a previous model for which the solutions are known? Explain.

  8. Specifically describe the type of energy corresponding to the expression identified in CTQ 7.

9. In view of your answers above, what is the total energy of this two-particle system?

10. Describe the advantage of changing the coordinates of Model 1 from Hˆ (x1 , x2 ) to Hˆ (XCOM , xr ) .

Problems 1. Based on the information in Model 1, derive the following expression: m x1 = X COM − 2 x r M 2 2 ˆ x , x ) = px1 + px2 + V (x , x ) is equivalent to 2. Verify that H( 1 2 1 2 2m1 2m2 € p X2 px2 ˆ COM H( XCOM , xr ) = + r + V (xr ) . 2M 2µ


Extension.5.2

Even and Odd Functions Terms A function is said to be even if

f (x) = f (−x) A function is said to be odd if

f (x) = − f (−x)

Critical Thinking Questions 1. Given the two functions graphed below which is an even function and which is an odd function?

x=0

PGLSAG003040a

x=0


2. Consider the functions drawn in CTQ 1. When integrating the even function, a) will the integral

xo 0

f ( x ) dx (where xo is arbitrary) result in a positive or

negative number?

b) how will the result of

0 − xo

f ( x ) dx , compare to the

xo 0

f ( x ) dx ?

3. Now consider the odd functions drawn in CTQ 1. When integrating this function, how will the result of

0 − xo

f ( x ) dx , compare to the

xo 0

f ( x ) dx ?

4. Upon integration from –∞ to +∞ of the functions drawn in CTQ 1 which will be zero? What does this imply about the integration of odd functions.


Extension.5.2

Even and Odd Functions

5. Using the functions drawn in CTQ 1, sketch the result of each of the following operations and decide whether or not the resulting function is even or odd: a) Multiplication of two even functions

b) Multiplication of two odd functions

c) Multiplication of an even and an odd function

6. Consider the wavefunctions corresponding to the two lowest energy states for a harmonic oscillator. (See Fundamental.5 CTQ 6) Determine whether the function is even or odd and record your group’s justification.           7. Based on your answers to CTQ 5 & 6, assess whether or not the harmonic oscillator wavefunctions ψ 0 ( x ) and ψ1 ( x ) are orthogonal. Present your group’s reasoning.


Extension.14

Slater Determinants Information In addition to the existence of spin, another relativistic result confirmed by experiment is that the complete wavefunction (including both spatial and spin coordinates) must be antisymmetric (odd) with respect to the interchange of the coordinates of any two particles, ψ (1, 2) = −ψ (2,1) . This requirement is satisfied by representing mulitelectron wavefunctions as a Slater Determinant,

φ1 (1)α (1) 1 φ1 (2)α (2) ψ (1, 2,...n) =  n! φ1 (n)α (n)

φ1 (1)β (1) ... φ m (1)α (1) φ1 (2)β (2) ... φ m (2)α (2)   φ1 (n)β (n) ... φ m (n)α (n)

(1)

A unique one-electron spin-orbital is represented for each column, and a different electron is represent for each row.

Critical Thinking Questions 1. Why is the same spatial function, φ1 specified in the first two columns?

2. In what way does the construction of this matrix ensure electron indistinguishability?

PGLSAG003041a


3. Write the wavefunction for helium in the Slater Determinant form.

4. A second order determinant is defined by

a b = ad â&#x2C6;&#x2019; bc . Explicitly c d

solve for the determinant and compare this result to your answer to CTQ 13 of Fundamental.14.

5. It can be shown that if any two rows or columns are identical in a matrix, the determinant is zero. With this in mind, propose a Slater Determinant representing the 1s22s1 electron configuration of lithium atom.

6. Is there more than one possible Slater Determinant representing this electron configuration for lithium atom? Explain.


Extension.16

Molecular Energy Integrals What integrals need to be evaluated to determine the energy of a molecule?

Model 1 Based on the Born Oppenheimer approximation, the Hamiltonian describing the electronic energy of a molecule of a single electron molecule with nuclei labeled A & B is

1 1 1 Hˆ = − 12 ∇12 − − + rA1 rB1 RAB

(1)

The two variational molecular wavefunctions generated from the two 1s atomic wavefunctions centered at the nuclei are:

ψg =

1 2+2SAB

ψu =

1 2−2SAB

(φ (φ

1sA

+ φ1sB )

1sA

− φ1sB )

(2)

The corresponding energies can be expressed as

H AA + H AB 1+ SAB H − H AB Eu = AA 1− SAB Eg =

where Hij = ∫ φi* Hˆ φ j dτ and Sij = gerade (even) and ungerade (odd).

PGLSAG003042a

(3)

* i j

∫ φ φ dτ .

The g and u refer to the German


Critical Thinking Questions

 1. Using the information in Model 1, explicitly show that HAA is the sum of four integral expressions.

2. Two of the four integrals can be combined such that they describe a single electron atom for which we know the solution, EA. Circle these two integrals in CTQ 1.

3. One of the two remaining integrals in CTQ 1 is also known. Identify its corresponding value.

4. The final integral is referred to as the Coulomb integral, J. What coulombic interaction is represented by this integral?


Extension.16

Molecular Energy Integrals

5. Summarize your results to CTQs 1-4 writing out an expression for HAA equivalent to CTQ 1, but without any explicit integrals.

6. Using the information in Model 1, explicitly show that HAB is the sum of three integral expressions. (Combine the one kinetic & potential energy term such that it is similar to CTQ 2.)

7. Show how two of the three integrals in CTQ 6 differ only by a factor of SAB from the known values determined for HAA.

8. The remaining integral is referred to as the Exchange integral, K. Identify how this integral differs from the Coulomb integral defined in CTQ 4.

9. Summarize your results to CTQs 6-9 writing out an expression for HAB equivalent to CTQ 6, but without any explicit integrals.


10. Compare CTQ 5 & 9. Which quantity HAA or HAB better represents the nonbonded energy of a molecule? Explain.

11. Next we will consider the difference in energy, ΔEg between a molecule in a state described by ψ g and the energy of a hypothetical “nonbonded” state. Show that ΔEg =

− K + JSAB . 1+ SAB

12. Similarly it can be shown that the difference in energy, ΔEu between a molecule in a state described by ψ u and the energy of a hypothetical “nonbonded” state is ΔEu =

K − JSAB . When R = Re, the magnitude of the 1− SAB

exchange integral is larger than JSAB . For this internuclear distance, draw an energy diagram relating Eg and Eu and Haa.

13. For a hydrogen molecular ion, which trial function best describes the ground state of the electron? Explain.


Extension.17

Hückel Molecular Orbital Theory (Is HMO a health management organization?)

Model 1: The π-bond in Ethylene/HMO. H

H C1

C2

H

H

+

+

2p orbital on carbon 1 = p1

p2 = 2p orbital on carbon 2

The trial wave function is written as a linear combination of atomic orbitals, LCAO, that comprise the π system. For ethylene, we use the 2p orbitals normal to the plane of the molecule.

ψ trial = c1φ p1 + c2φ p2

(1)

The expression for the average value of the energy can be written: <ε> =

c12H11 + c22H22 + c1c2H12 + c1c2H21 c12 + c22 + c1c2S12 + c1c2S21

where Hij = ∫ φi* Hˆ φ *j dτ Sij =

PGLSAG003043a

* i

∫ φ φ dτ j

(2) (3) (4)


The coefficients c1 and c2 can be treated as parameters, and the energy can be minimized with respect to each parameter. The resultant equations, (5) and (6) are known as secular equations. ∂<ε> = 0 = (H11 – <ε>S11)c1 + ∂c1

(H12 – <ε>S12)c2

(5)

∂<ε> = 0 = (H21 – <ε>S21)c1 + ∂c2

(H22 – <ε>S22)c2

(6)

The assumptions of HMO theory are: •

Sij = 0 when i ≠ j

(that is, all overlap integrals equal zero)

Sii = 1

(that is, all atomic orbitals are normalized)

Hij = β = the exchange integral when i and j are adjacent atoms

Hij = 0

When the two atoms are not adjacent, we assume that the value of Hij is sufficiently small that it can be set to zero.

Hii = α = the Coulomb integral

when i and j are not adjacent atoms

Generally, the Coulomb integral is viewed as the negative of the first IE of the isolated atom under consideration; for example, the first ionization energy of carbon is 1.09 MJ/mole and α = –1.09 MJ/mole. Both α and β have energy values that are negative.


Extension.17

Hückel Molecular Orbital Theory

Critical Thinking Questions 1. Use the assumptions of HMO to rewrite equations (5) and (6). Show that the result is: 0 = (α – ε)c1 + β c2 0 = β c1

+

(α – ε)c2

where ε = <ε >

2.

Solve these equations by the "brute force" technique. That is, solve for c1 (in terms of a, b, ε , and c2) with equation (7). Use this value in equation (8) to solve for ε . [Hint: you will obtain two values for ε .]

3. Solve these equations by a "more elegant" technique. That is, solve for ε by setting the determinant of the coefficients of the ci values in equation (7) and (8) equal to zero. You should obtain the same values for ε .

(7) (8)


Exercise 1. In the HMO method, what is the trial function for the π-system of 1,3butadiene?

Model 2: The Energy Levels of the p-bond of Ethylene. There are two electrons in the π-bond of ethylene.

Critical Thinking Questions

4. Why is the energy level α + β at a lower energy than α − β ? [Hint: what is the sign of α ? Of β ?]

5. Which case (A or B) in Model 2 would be called the ground state? The excited state?


Extension.17

Hückel Molecular Orbital Theory

6. How much energy is required to move the electron from the ground state to the excited state (in terms of b)?

Exercise 2. The electronic transition from the ground to excited state in ethylene occurs at 165 nm. Calculate a value for β (in kJ/mole) based on the absorption in ethylene.

Information The coefficients c1 and c2 are determined by using the normalization and orthogonality requirements of the wave functions. (Here we assume ψ is real for simplicity.) First, solve for the general relationship between c1 and c2.

ψ = c1φ p1 + c2φ p2

∫ψ 1 =

2

=1 = 2 1

∫cφ

2 p1

(9)

∫ (c φ

dτ +

1 p1

2 2

∫cφ

2 p2

+ c2φ p2 ) (c1φ p1 + c2φ p2 )d τ dτ +

∫ccφ 1 2

p1

φ p2 d τ +

∫ccφ 1 2

p2

φ p1 d τ

1 = c12 + c22 + c1c2S12 + c1c2S21 = c12 + c22 1 = c12 + c22

(10)

Next, solve for the coefficients of the wave function associated with <ε> = α + β. Either one of the secular equations can be used—(7) or (8). Here, equation (7) is used. 0 = (α – ε)c1 + β c2 = (α – α – β)c1 + β c2

βc1 = β c2 c1 = c2

(11)


Combination of equations (10) and (11) yields: c1 =

1 = c2 2

Thus,

επ = α + β , ψ π = 12 φ p + 12 φ p 1

(12)

2

Similarly, if ε = α – β (higher energy state), then c1 = –c2:

επ* = α – β , ψπ = 12 φ p − 12 φ p *

1

2

Model 3: The Energy Levels and Orbitals of the π-bond of Ethylene.

Critical Thinking Questions

7. For the π orbital, examine the wavefunction, equation (12), and the diagram in Model 3. Why are the two positive signs found on the same side of the molecular plane in Model 3?

(13)


Extension.17

Hückel Molecular Orbital Theory

8. For the π * orbital, examine the wave function, equation (13), and the diagram in Model 3. Why are the two positive signs found on the opposite sides of the molecular plane in Model 3?

9. Point B is equidistant from the two carbon nuclei. Suppose that φ p1 has a value of 4.2 × 10–6 pm–3/2 at point B. a)

What is the value of ψπ * at point B?

b)

What is the value of ψπ * at any point equidistant from the two nuclei?

10. How many nodes are there in ψπ ? In ψπ * ? What is the relationship between the number of nodes and the energy of the MO?


Model 4: 1,3-butadiene.

H

H

H

H

H

C1

C2

C3

C4

+

+

+

+

p1

p2

p3

p4

H

The trial function uses these four atomic orbitals.

ψ trial = c1φ p1 + c2φ p2 + c3φ p3 + c4φ p4 As a result, there are four coefficients in the trial function. The trial function must be minimized with respect to each of the four coefficients. ∂<ε> = 0 = (H11 – <ε>S11)c1 + (H12 – <ε>S12)c2 + (H13 – <ε>S13)c3 + ∂c1 (H14 – <ε>S14)c4 ∂<ε> = 0 = (H21 – <ε>S21)c1 + (H22 – <ε>S22)c2 + (H23 – <ε>S23)c3 + ∂c2 (H24 – <ε>S24)c4 ∂<ε> = 0 = (H31 – <ε>S31)c1 + (H32 – <ε>S32)c2 + (H33 – <ε>S33)c3 + ∂c3 (H34 – <ε>S34)c4 ∂<ε> = 0 = (H41 – <ε>S41)c1 + (H42 – <ε>S42)c2 + (H43 – <ε>S43)c3 + ∂c4 (H44 – <ε>S44)c4

(14)


Extension.17

Hückel Molecular Orbital Theory

Four secular equations are generated. 0 = (α – ε) c1

+

0 =

β c1

+

0 =

0

+

0 =

0

+

β c2 (α – ε) c2 β c2 0

+

0

+

0

+

β c3

+

0

+

(α – ε) c3

+

+

β c3

+

β c4 (α – ε) c4

There are four solutions for the energy.

ε1 ε2 ε3 ε4

= α + 1.6180 β = α + 0.6180 β = α – 0.6180 β = α – 1.6180 β

Four wave functions are generated.

ψ1 =

0.3717 φ p1 + 0.6015 φ p2 + 0.6015 φ p3 + 0.3717 φ p4

ψ2 =

0.6015 φ p1 + 0.3717 φ p2 –

0.3717 φ p3 – 0.6015 φ p4

ψ3 =

0.6015 φ p1 –

0.3717 φ p2 –

0.3717 φ p3 + 0.6015 φ p4

ψ4 =

0.3717 φ p1 –

0.6015 φ p2 + 0.6015 φ p3 – 0.3717 φ p4

Critical Thinking Questions 11. Using the information in Model 4, sketch an energy level diagram for 1,3butadiene. Create an approximate sketch of each corresponding molecular orbital.


12. If an electron is placed into ψ 2 , is the electron more likely to be found on carbon atom 1 or carbon atom 2? Explain.

13. The magnitude of the coefficients on carbon atoms 1 and 4 have the same value in any one molecular orbital, ψ i . Why must this be the case?

14. ψ 1 does not have any nodes (other than the node in the molecular plane). How many nodes does ψ 2 have? ψ 3 ? ψ 4 ?

15. What is the apparent relationship between the number of nodes of the orbital and the energy level of the orbital?

16. For ψ 1 , there appears to be bonding over all four carbon atoms. For ψ 2 , there appears to be bonding carbon atoms 1 and 2, bonding between carbon atoms 3 and 4, and a node between carbon atoms 2 and 3. What type of bonding or nodes seem to be evident between the carbon atoms when an electron is described by ψ 4 ?


Extension.17

Hückel Molecular Orbital Theory

17. The electron configuration of the ground state of 1,3-butadiene can be written as ψ 12 ψ 22. Write the electron configuration of the lowest-energy excited state of 1,3-butadiene?

Model 5: π-electron Charge Densities and π-electron Charges. The square of a coefficient of an atomic orbital is taken as the fraction of an electron found near that atom (by each electron in an occupied molecular orbital). That is, for one electron in the molecular orbital

ψ2 =

0.6015 φ p1 + 0.3717 φ p2 – 0.3717 φ p3 – 0.6015 φ p4

0.3618 of the electron is found at atom 1, 0.1382 at atom 2, and so on. Thus, the π-electron charge density (a misnomer), q, at atom r is given by

qr =

∑ ni (cir)2 i

where ni is the number of electrons in the ith molecular orbital , cir is the coefficient of atom r in the ith molecular orbital, and the sum is over all molecular orbitals containing one or more electrons. The π-electron charge (also a misnomer) on atom r is taken as the number of electrons that an atom contributes to the π system minus the charge density, qr. For example, if a carbon atom contributes one electron to the π system and has a π-electron charge density of 1.023, then the charge on the carbon atom is –0.023. On the other hand, if a pyrrole-like nitrogen contributes two electrons to the π system and has a π-electron charge density of 1.988, then the charge on the nitrogen atom is +0.012.

(15)


Critical Thinking Questions 18. Give a rationale for taking the square of a coefficient of an atomic orbital as the fraction of an electron found near that atom (by each electron in an occupied molecular orbital)?

19. What is the sum of the squares of all of the coefficients for ψ 2 ? Explain why this makes sense.

20. Why does the number of electrons in the orbital, ni, appear in equation (15)?

Exercises 3. Calculate the π-electron charge density on atom 1, q1, for the ground state of 1,3-butadiene. Calculate the π-electron charge density on atom 2, q2. Without doing any calculations, what is the π-electron charge density on atom 3? On atom 4? 4. According to your energy level diagram for 1,3-butadiene, what is the Δε for the lowest energy electronic transition (in terms of β)? Use your value of β, calculated in Exercise 2 to determine the wavelength of this transition. The experimental value for 1,3-butadiene is 217 nm. How well does your calculated value compare to the experimental value?


Extension.17

Hückel Molecular Orbital Theory

Model 6: π-electron Bond Orders. Bonding takes place when there is a significant sharing of electron density between two atoms. For an occupied orbital, the electron sharing is large when the wave function is large between the adjacent atoms. Thus, the bond order between adjacent atoms is considered to be large when the coefficients on the adjacent atoms are large and have the same sign. The π-electron bond order is another defined quantity. For adjacent atoms r and s, the bond order is given by prs =

∑ ni (cir cis) i

where cir and cis are the coefficients of atom r and s, respectively, in the ith molecular orbital, and the sum is taken over all orbitals.

Critical Thinking Questions 21. Suppose the coefficients for the p orbitals on two adjacent atoms are 0.50 and 0.01, respectively. Suppose the coefficients for the p orbitals between a second set of adjacent atoms is 0.25 and 0.26. Give a rationale for taking the products of the coefficients rather than the sum of the coefficients as a measure of the bond strength.

21. Suppose the coefficients for the p orbitals on two adjacent atoms are –0.25 and –0.26. Give a rationale for taking the products of the coefficients rather than the sum of the coefficients as a measure of the bond strength.

(16)


23. Suppose the coefficients for the p orbitals on two adjacent atoms are 0.25 and –0.26. Does this represent bonding between the two atoms, or will there be a node between the two atoms?

24.

Why does the number of electrons in the orbital, ni, appear in equation 16?

Exercises 5. For the ground state of 1,3-butadiene, calculate the π-electron bond order p12. Calculate the π-electron bond order p23. Without doing any calculations, give the π-electron bond order p34. 6. What is the sum of the number of p bonds in the lowest energy resonance structure of 1,3-butadiene? What is the sum of the number of p bonds in the HMO description of 1,3-butadiene? That is, what is the sum of p12, p23, p34? Explain why the amount of p-bonding is greater using HMO than with the lowest energy resonance structure.


Extension.Sy2

Ethylene Molecular Orbitals Pre-Activity Question 1. Review Extension.17. Draw and label an energy level diagram including a sketch of each corresponding MO of ethylene as predicted by Hückel theory.

Model 1: The Energy Levels of Ethylene using MOPAC/AM1. 6

H3

H C1

C2

5

z The molecular plane is the xy-plane

H4

H

y

x

Coordinate system on each atom

ψ trial = c1φ2 sC1 + c2φ2 pxC1 + c3φ2 pyC1 + c4φ2 pzC1 + c5φ2 sC2 + c6φ2 pxC2 +c7φ2 pyC2 + c8φ2 pzC2 + c9φ2 sH3 + c10φ2 sH 4 + c11φ2 sH 5 + c12φ2 sH 6 Table 1.

(1)

Hartree-Fock STO-3G orbital coefficients and energies corresponding to the valence orbitals ethylene. Coefficients for each MO (ψi) ψ3 ψ4 ψ5

Atomic Orbital 2sC1 2pxC1 2pyC1 2pzC1

ψ1

ψ2

0.4680 0.1195 0.0000 0.0000

-0.4100 0.2033 0.0000 0.0000

0.0000 0.0000 0.3969 0.0000

-0.0169 -0.5000 0.0000 0.0000

0.0000 0.0000 -0.3934 0.0000

0.0000 0.0000 0.0000 0.6320

2sC2 2pxC2 2pyC2 2pzC2

0.4680 -0.1195 0.0000 0.0000

0.4100 0.2033 0.0000 0.0000

0.0000 0.0000 0.3969 0.0000

-0.0169 0.5000 0.0000 0.0000

0.0000 0.0000 0.3934 0.0000

0.0000 0.0000 0.0000 0.6320

PGLSAG003044a

ψ6


1sH3 1sH4 1sH5 1sH6 MO Energy (eV)

0.1120 0.1120 0.1120 0.1120

0.2236 0.2236 -0.2236 -0.2236

0.2566 -0.2566 -0.2566 0.2566

-26.87

-20.29

-16.48

ψ7

ψ8

0.2170 0.2170 0.2170 0.2170

0.3506 -0.3506 0.3506 -0.3506

0.0000 0.0000 0.0000 0.0000

-12.46

-9.129

Coefficients for each MO (ψi) ψ9 ψ10 ψ11

ψ12

-14.70

Atomic Orbital 2sC1 2pxC1 2pyC1 2pzC1

0.0000 0.0000 0.0000 -0.8176

0.0000 0.0000 0.6983 0.0000

0.8271 -0.5210 0.0000 0.0000

1.0560 -0.2300 0.0000 0.0000

0.0000 0.0000 0.9584 0.0000

0.9300 1.1792 0.0000 0.0000

2sC2 2pxC2 2pyC2 2pzC2

0.0000 0.0000 0.0000 0.8176

0.0000 0.0000 0.6983 0.0000

0.8271 0.5210 0.0000 0.0000

-1.0560 -0.2300 0.0000 0.0000

0.0000 0.0000 -0.9584 0.0000

-0.9300 1.1792 0.0000 0.0000

1sH3 1sH4 1sH5 1sH6 MO Energy (eV)

0.0000 0.0000 0.0000 0.0000

-0.6263 0.6263 0.6263 -0.6263

-0.6313 -0.6313 -0.6313 -0.6313

0.6463 0.6463 -0.6463 -0.6463

0.6123 -0.6123 0.6123 -0.6123

-0.1174 -0.1174 0.1174 0.1174

19.14

25.79

27.97

8.934

16.84

19.04

Note: 1 eV = 1.6022 × 10–19 J

Critical Thinking Questions 1. How does the number of atomic orbitals used in the trial function for ethylene compare to the number of MOs presented in Table 1?


Extension.Sy2

Ethylene Molecular Orbitals

2. Identify the symmetry point group for ethylene.

3. In the ground state, what is the HOMO (ψ n ) for ethylene? Record your group’s reasoning.

4. Based on the atomic orbital coefficients in Table 1, make a sketch of the HOMO and LUMO of ethylene. How do these functions compare to the MOs obtained from Hückel theory?

5. For each C2 rotation and the inversion symmetry operator of the point group identified in CTQ 2, indicate the result of the operation on the designated molecular orbital and determine the representation to which they belong. a) HOMO

b) LUMO


6. Based on the atomic orbital coefficients in Table 1, make a sketch of ψ 1 and ψ3 .

a) Do both of these MOs contribute to bonding between the two carbons? If so, describe any significant similarities or differences to the C-C bonding predicted using Hückel theory.

b) Are there other type(s) of bonding present in these MOs that is not accounted for in Hückel theory.

c) Do these two molecular orbitals belong to the same representation? Explain.

d) Based on symmetry, present your group’s justification as to why the 2s orbital does not overlap with the 2py orbital.


Extension.Sy2

Ethylene Molecular Orbitals

7. Assume we are interested in the wavelength of a photon that will promote an electron from the HOMO to the LUMO. a) Is it possible to calculate this result based on Hartree Fock theory? If so, explain how it is obtained and whether or not it depends on any experimental parameters.

b) Is it possible to calculate this result based on Hückel theory? If so, explain how it is obtained and whether or not it depends on any experimental parameters.

8. What information do you obtain from both theories (Hartree Fock and Hückel) and why is it useful?

9. Which theory (Hartree Fock or Hückel) provides more information – and why is it useful?

Exercise 1. Using SALC MO theory, prove that c3, c4, c7 and c8 should be zero for ψ 1 . Hint: you only need to consider the atomic orbitals associated with these four coefficients and prove that their linear combination is zero for the symmetry representation of ψ 1 .


Appendix Table A.1

Fundamental Constants

Atomic mass constant Avogadro’s constant Boltzmann’s constant Electron charge Electron rest mass Ideal Gas constant Permittivity of vacuum Planck’s constant Rydberg’s constant Speed of light in a vacuum

Table A.2

Energy Conversions

joule (SI unit) erg rydberg electron volt inverse centimeter calorie (thermochemical) liter atmosphere

PGLSAG003045a

amu = 1.6605402×10-27 kg Na = 6.0221367×1023 mol-1 kb = 1.380658×10–23 J K-1 qe = 1.6021892×10-19 C me = 9.1093897×10-31 kg R = 8.3145101 J K-1 mol-1 ε0 = 8.854187816×10-12 C2 J-1 m-1 h = 6.6260755×10-34 J s  = 1.05457266×10-34 J s RH = 109677.581 cm-1 R∞ = 10973731.534 m-1 c = 2.99792458×108 m s-1

J erg = 10-7 J Ry = 2.17987×10-18 J eV = 1.60218×10-19 J cm-1 = 1.98645×10-23 J Cal = 4.184 J L × atm = 101.325 J


Table A.3 Extended List of Operators Representing Physical Quantities. Observable

me (mass of electron)

Operator Symbol

Operation

me (or 9.109 × 10–31 kg) x z r ∂ −i ∂x 2  ∂2 − 2m ∂x 2  2 ⎛ ∂2 ∂2 ∂2 ⎞ − + + 2m ⎜⎝ ∂x 2 ∂ y 2 ∂z 2 ⎟⎠

x z r px

mˆ e xˆ zˆ rˆ pˆ x

Tx

Tˆx

T (Cartesian coordinates)

T (spherical coordinates)

Lx (Cartesian coordinates)

Lˆx

Lx (spherical coordinates)

Lˆx

⎛ ∂ ∂⎞ −i ⎜ − sin φ − cot θ cosφ ⎟ ∂θ ∂φ ⎠ ⎝

Ly (Cartesian coordinates)

Lˆ y

⎛ ∂ ∂⎞ −i ⎜ z − x ⎟ ∂z ⎠ ⎝ ∂x

Ly (spherical coordinates)

Lˆ y

⎛ ∂ ∂⎞ −i ⎜ cosφ − cot θ sin φ ⎟ ∂θ ∂φ ⎠ ⎝

Lz (Cartesian coordinates)

^ L z

⎛ ∂ ∂⎞ −i ⎜ x −y ⎟ ∂x ⎠ ⎝ ∂y

Lz (spherical coordinates)

Lˆz z

L2

(spherical coordinates)

Lˆ2

⎫ ⎧∂⎛ ∂⎞ 1 ⎡∂ ( sin θ ) ∂ ⎤⎥ + 1 ∂ ⎬ ⎨ ⎜r ⎟ + ⎢ ⎝ ⎠ 2m r ⎩ ∂r ∂r sin θ ⎣ ∂θ ∂θ ⎦ sin θ ∂φ ⎭ 

2

1

2

2

2

⎛ ∂ ∂⎞ −i ⎜ y − z ⎟ ∂y⎠ ⎝ ∂z

−i

∂ ∂φ

⎧ 1 ⎡ ∂ ∂ ⎤ 1 ∂2 ⎫ − 2 ⎨ sin θ + ( ) ⎬ ⎢ ∂θ ⎥⎦ sin 2 θ ∂φ 2 ⎪⎭ ⎩⎪ sin θ ⎣ ∂θ

2

2


Appendix

Table A.4

F = ma

Useful Relationships F=

dp dt

F =−

dV dx

F = −kx

N i gi −(E i −E j )/ k bT = e N j gj € € € mm p2 T = 12 mv 2 = µ= 1 2 p = mv m1 + m2 2m

F=

q1q2 4πε 0 r 2

c = λν

E = hν

Table A.5 1D Particle-in-a-Box € En =

n2h2 8ma 2

2 # nπ x & sin % $ a (' a

ψn =

n = 1, 2, 3....

Table A.6 Harmonic Oscillator & Morse Potential

1⎞ ⎛ En = hν ⎜ n + ⎟ ⎝ 2⎠

1 ⎛ k⎞ ν= ⎜ ⎟ 2π ⎝ m ⎠

ψ n (x) = cn H n (ξ )e

1/2

n = 0,1, 2, 3,...

− 12 α x 2

1/2

⎛ km ⎞ where α = ⎜ 2 ⎟ and ξ = α 1/2 x is the argument of ⎝ ⎠ the Hermite polynomial, Hn. cn is the normalization constant which is given by 1/ 4 1 ⎛α⎞ cn = ⎜ ⎟ 2 n n! ⎝ π ⎠

V (r) = De ⎡⎣1 − e−α (r − re ) ⎤⎦

En = hν ( n +

1 2

2

α=

) − hν x ( n + )

1 2 2

k 2De

x=

hν 4De

H 0 (ξ ) = 1 H 1 (ξ ) = 2ξ H 2 (ξ ) = 4ξ 2 − 2 H 3 (ξ ) = 8ξ 3 − 12ξ


Table A.7 2D Rotation z

Cylindrical Coordinates 0 ≤ r ≤ ∞ , 0 ≤ φ ≤ 2π , ro

x

−∞ ≤ z ≤ ∞

Transformation Equations x = r cos φ y = r sin φ

z=z

Differential Volume Element

dτ = rdrdφdz Laplacian

∇2 = Em =

m22 2I

∂2 1 ∂ 1 ∂2 ∂2 + + + ∂r 2 r ∂r r 2 ∂φ 2 ∂z 2

1 imφ e m = 0, ±1, ±2, ±3... 2π

ψ m (φ ) =

Table A.8 3D Rotation Spherical Coordinates 0 ≤ r ≤ ∞ , 0 ≤ θ ≤ π , 0 ≤ φ ≤ 2π Transformation Equations x = r sin θ cosφ y = r sin θ sin φ z = r cosθ Differential Volume Element

z

dτ = r 2 sin θdrdθdφ

θ

Laplacian & Legendrian r0 y x

φ

2 E l = l(l + 1) 2I

∂2 2 ∂ 1 + + 2 Λ2 2 r ∂r r ∂r 1 ∂2 1 ∂ ∂ Λ2 = 2 + sin θ 2 sin θ ∂φ sin θ ∂θ ∂θ

∇2 =

ψ l,m (θ, φ ) = Θ l,m (θ )Φm (φ ) €

l = 0,1,2,…

m = 0,±1,±2,…,±l


Appendix

l

m

Θ l,m (θ )

0

0 €

1 2 2

1

0 €

1 6 cosθ 2 1 3 sin θ 2

1

€ 2

0

€ 1 €

2

l=0 m =0

1 10 (cos2 θ −1) 4 1 15 sin θ cos θ 2 1 15 sin 2 θ 4

€ Table A.9

l=1 m =0

l=1 |m|= 1

l=2 m =0

l=2 |m|=1

l =2 |m| =2

Vibrational-Rotational Spectroscopy

En, J€ 1 1 1 = ve (n + ) − ve xe (n + )2 + B e J(J + 1) − α e (n + )J(J + 1) hc 2 2 2

Table A.10 Hydrogen Atom

m Z 2 e4 En = − e2 2 2 8ε 0 h n

ψ n,l, ml (r,θ , φ ) = Rn,l (r)Θl, ml (θ )Φ ml (φ )

⎛Z⎞ R10 = 2 ⎜ ⎟ ⎝ a0 ⎠

n = 1, 2, 3,... l = 0,1, 2,...n − 1 ml = 0, ±1, ±2,... ± l

3/2

e− Zr/a0 3/2

R20 =

1 ⎛Z⎞ 8 ⎜⎝ a0 ⎟⎠

R21 =

1 ⎛Z⎞ 24 ⎜⎝ a0 ⎟⎠

⎛ Zr ⎞ − Zr/2a0 ⎜⎝ 2 − a ⎟⎠ e 0

3/2

⎛ Zr ⎞ − Zr/2a0 ⎜⎝ a ⎟⎠ e 0

1 bohr ≡ a0 =

(4πε 0 ) 2 = 52.918 pm me e 2


Table A.11 Indefinite Integrals 1 1. ∫ sinbx cosbx dx = sin 2 bx 2b 1 2. ∫ sinbx dx = − cosbx b 1 1 3. ∫ ( sin 2 bx ) dx = x − sin 2bx 2 4b 1 x 4. ∫ x ( sinbx ) dx = 2 sinbx − cosbx b b 2x b2 x 2 − 2 cosbx 5. ∫ x 2 ( sinbx ) dx = 2 sinbx − b b3 x 2 x sin 2bx cos 2bx 2 − 6. ∫ x sin bx dx = − 4 4b 8b 2 x3 ⎛ x2 1 ⎞ x cos 2bx 7. ∫ x 2 ( sin 2 bx ) dx = − ⎜ − 3 ⎟ sin 2bx − 6 ⎝ 4b 8b ⎠ 4b 2 1 9. ∫ cosbx dx = sinbx b 1 1 10. ∫ ( cos 2 bx ) dx = x + sin 2bx 2 4b 1 x 11. ∫ x ( cosbx ) dx = 2 cosbx + sinbx b b 2x b2 x 2 − 2 sinbx 12. ∫ x 2 ( cosbx ) dx = 2 cosbx + b b3 x 2 x sin 2bx cos 2bx 2 + 13. ∫ x cos bx dx = + 4 4b 8b 2 x3 ⎛ x2 1 ⎞ x cos 2bx 14. ∫ x 2 ( cos 2 bx ) dx = + ⎜ − 3 ⎟ sin 2bx + 6 ⎝ 4b 8b ⎠ 4b 2

(

)

(

)

ebx b ebx 16. ∫ xebx dx = 2 ( bx − 1) b x m ebx m m−1 bx m bx − ∫ x e dx 17. ∫ x e dx = b b n+1 x 18. ∫ x n dx = n +1 15.

bx ∫ e dx =


Appendix

cos m+1 bx 19. ∫ ( sinbx ) ( cos bx ) dx = − (m + 1)b m

sin m+1 bx 20. ∫ ( sin bx ) ( cosbx ) dx = (m + 1)b ⎧ cos m−1 bx sin n+1 bx m − 1 ⎪ + cos m−2 bx ) ( sin n bx ) dx ( ∫ m + n b m + n ( ) ⎪ ⎪ m n 21. ∫ cos bx sin bx dx = ⎨ ⎪ sin n−1 bx cos m+1 bx n − 1 ⎪ − + cos m bx ) ( sin n−2 bx ) dx ( ∫ m+n ( m + n)b ⎪⎩ for all integral values of n & m (positive, negative or zero) m

Table A.12 Definite Integrals D1. D2. D3. D4. D5. D6. D7.

π 4b ∞ 1 π 2 −bx 2 ∫0 x e dx = 4b b ∞ 1 − x2 ∫0 xe dx = 2 ∞ π 2 − x2 ∫0 x e dx = 4 1 ∞ 1⋅ 3⋅ 5…(2n − 1) ⎛ π ⎞ 2 2n −bx 2 ⎜⎝ ⎟⎠ ∫0 x e dx = 2 n+1 b n b ∞ n! 2n+1 −bx 2 ∫0 x e dx = 2b n+1 for b>0 ∞ n! n −bx ∫0 x e dx = b n+1 for b>0

0

e−bx dx = 2

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪⎭


Table A.13 Trigonometric identities

sin α sin β = 1 2 cos(α − β ) − 1 2 cos(α + β ) cos α cos β = 1 2 cos(α − β ) + 1 2 cos(α + β ) sin α cos β = 1 2 sin(α + β ) + 1 2 sin(α − β ) cos α sin β = 1 2 sin(α + β ) − 1 2 sin(α − β ) e±iω t = cos ω t ± i sin ω t eiα + e−iα T6. cos α = 2 iα e − e−iα T7. sin α = 2i T1. T2. T3. T4. T5.

Pogil: Quantum Chemistry & Spectroscopy: A Guided Inquiry  

Pogil: Quantum Chemistry & Spectroscopy: A Guided Inquiry

Pogil: Quantum Chemistry & Spectroscopy: A Guided Inquiry  

Pogil: Quantum Chemistry & Spectroscopy: A Guided Inquiry

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