Contents 1

Introduction

2

2

3 2.1

An Overview of Density Functional Theory . . . . . . . . . . . . . . . . . 17 2.1.1

Electron Density from Wave Function . . . . . . . . . . . . . . . . 17

2.1.2

Hohenburg-Kohn Theorem . . . . . . . . . . . . . . . . . . . . . . 17

2.1.3

Kohn-Sham Formulation . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.4

Exchange-Correlation: With Dirac Exchange . . . . . . . . . . . . 20

2.1.5

Local Density Approximation . . . . . . . . . . . . . . . . . . . . 21

2.1.6

Generalized Gradient Approximation . . . . . . . . . . . . . . . . 22

2.2

The Broken Symmetry State . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3

Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1

Chapter 1 Introduction

2

Chapter 2

3

CHAPTER 2.

2.1 2.1.1

17

An Overview of Density Functional Theory Electron Density from Wave Function

The motivation for Density Functional Theory1,2 comes from the fact that the ground state property can be can be found from the ground state electron density only. The electron density can be expressed as a function of wave function as Z ρ(r1 ) = Ψ∗ (r1 , r2 , . . . , rn )Ψ(r1 , r2 , . . . , rn )dr2 , . . . drn .

(2.1)

The advantage of DFT approach over wave functional approach is that for an N electron system number of variable is 3 (or 4 if spin is included) whereas for wave functional approach the number of variable is very high 3N (4N for spin included). The earlier work on the development of DFT started from the work of Thomas and Fermi.3 The the total energy was given by the kinetic energy of a uniform electron gas and the classical electrostatic energy between electrons gas and the nuclei. Dirac extended it by adding exchange energy between electrons themselves.4 But, in spite of that, it gave poor results because of large approximation of kinetic energy. It is the work of Kohn, Hohenberg, and Sham from where the modern density functional theory started. Where the Thomas-Fermy approach involved only electron density, Kohn-Shm reintroduced wave function in between density functional and total energy. The role of wave function is only an intermediate one.

2.1.2

Hohenburg-Kohn Theorem

The main theme of Hohenberg-Kohn theorem1 is that the ground state energy of an electronic system is a unique functional of the electron density. The kinetic energy is unique as the density is unique. Thus the potential energy v(r) is unique for a density ρ(r). Let us assume that there exist another potential energy v 0 (r) which corresponds to the same density ρ(r). Let these two potentials give two Hamiltonian operators H and H 0

CHAPTER 2.

18

that correspond to the ground state density ρ(r) but they have different wave functions Ψ and Ψ0 . Thus HΨ = E0 Ψ and H 0 Ψ = E00 Ψ0 . From variational principle, ˆ 0i E0 < hΨ0 |H|Ψ ˆ 0 |Ψ0 i + hΨ0 |H ˆ −H ˆ 0 |Ψ0 i < hΨ0 |H Z 0 < E0 + ρ(r1 )[v(r1 ) − v 0 (r1 )].

(2.2)

Similarly, ˆ 0 |Ψi E00 < hΨ|H ˆ ˆ 0 − H|Ψi ˆ < hΨ|H|Ψi + hΨ|H Z < E0 − ρ(r1 )[v(r1 ) − v 0 (r1 )].

(2.3)

Adding 2.1 and 2.2 leads to E0 + E00 < E00 + E0 , which is a self contradiction. Thus it can be concluded that two different potential v(r) cannot give rise to the same ρ(r). That is, density is a unique functional of the potential. In turn, the ground state energy is a unique functional of particle density. This is the Hohenburg-Kohn Theorem.

2.1.3

Kohn-Sham Formulation

Just like the Hartree-Fock formulation, the Kohn-Sham formulation centers on mapping the fully interacting system with real potential onto a fictitious non-interacting system where the electron moves within an effective “Kohn-Sham” single particle potential vKS (r).2 The advantage of Kohn-Sham method is that the exchange-correlation energy, incorporated in the Hamiltonian, is taken care of within the single determinantal approach. This is unlike the post-Hartree-Fock approaches, and the reason for Kohn-Sham approache to be computationally less expensive. The variation of total ground state energy with the added constraint

R

n(r)dr − N = 0

CHAPTER 2.

19

multiplied by a Lagrange multiplier looks like Z Z δ[F [n(r)] + vext (r)n(r) − µ( n(r)dr − N )] = 0.

(2.4)

The corresponding Eular Equation would be µ=

δF [n(r)] + vext (r). δn(r)

(2.5)

Removal of one of the major sources of error of Thomas-Fermy Model (TFM), namely, the approximation to kinetic energy, was achieved by Kohn-Sham model by invoking a noninteracting system of electrons. The corresponding wave-function, ΨKS , for this type of system is given by a determinant of single-particle orbitals ψi (ri ), 1 0 ΨKS = √ det[ψ10 (r1 )ψ20 (r2 ) . . . ψN (rN )]. N

(2.6)

The universal functional F [n(r)] written above can be partitioned into three parts. First two parts can be determined almost exactly, and also constitute the major part. The third part, though very small and unknown plays the most important role in DFT. It is known as exchange-correlation energy. F [n(r)] = TS [n(r)] + EH [n(r)] + EXC [n(r)]

(2.7)

TS [n(r)] is the kinetic energy of electron assuming them to be non-interacting gas of density n(r). EH [n(r)] is the electron-electron interaction energy also known as Hartree energy. ZZ n(r)n(r0 ) 1 drdr0 (2.8) EH [n(r)] = 0 2 |r − r | The Eular equation can be rewritten as µ=

δTS [n(r)] + vKS (r), δn(r)

(2.9)

where vKS (r) = vext (r) + vH (r) + vXC (r).

(2.10)

CHAPTER 2.

20

Comparing 2.7, 2.9, and 2.10 the following two equation can be written: ZZ n(r)n(r0 ) Î´EH [n(r)] = drdr0 , vH (r) = 0 Î´n(r) |r âˆ’ r |

(2.11)

Î´EXC [n(r)] . (2.12) Î´n(r) After some more calculation and writing kinetic energy separately an equation very similar vXC (r) =

to Fock equation appears. This is the Kohn-Sham equation. 1 [âˆ’ âˆ‡2 + vKS (r)]Ďˆi (r) = i Ďˆi (r), 2

(2.13)

where i is the Lagrange multipliers corresponding to the orthonormality of the N singleparticle states Ďˆi (r). The density of electron is n(r) =

N X

|Ďˆi (r)|2 .

(2.14)

i=1

The non-interacting kinetic energy TS [n(r)] looks N Z 1X TS [n(r)] = âˆ’ Ďˆiâˆ— (r)âˆ‡2 Ďˆi (r)d(r). 2 i=1

(2.15)

Thus in Kohn-Sham theory, in order to handle the kinetic energy in an exact manner, N equations have to be solved to obtain the set of Lagrange multipliers {i } whereas in Thomas Fermi model only one equation have to be solved to determine the multiplier Âľ. The main aim of Kohn-Sham theory was to make the unknown contribution to the total energy of the non-interacting system as small as possible. Though the contribution of exchange-correlation energy is very small, still it is an important contribution as the binding energy of many systems is about the same size as EXC [n(r)]. So an accurate description of exchange-correlation energy is very much crucial in DFT.

2.1.4

Exchange-Correlation: With Dirac Exchange

As described in the previous section, the accuracy of the exchange-correlation energy is the central importance in DFT. The history of EXC stem from the addition of exchange

CHAPTER 2.

21

energy term to Thomas-Fermi model by Dirac.4 One of the shortcoming of Thomas-Fermi model is the oversimplified description of electron-electron interaction. Shortly after the introduction of TFM model Dirac develop an approximate expression for exchange interaction based on the homogeneous electron gas. The resulting formula is simple and a local functional of density. This is probably the first step toward addition of exchange-correlation energy term to classical energy Hamiltonian of atoms and molecules. 1/3 Z 3 3 Ex [n(r)] = âˆ’ n(r)4/3 dr 4 Ď€ Relation 2.16 is written in terms of exchange energy density xc [n(r)] as, Z LDA Exc [n(r)] = n(r)xc [n(r)]dr, where xc [n(r)] can be given in terms of Seitz radius rs ,5 1/3 1 0.4582 3 9 â‰ˆ âˆ’ . xc [n(r)] = âˆ’ 4 4Ď€ 2 rs rs

(2.16)

(2.17)

(2.18)

Though it was very crude, it was the starting point for adding exchange-correlation potentials.

2.1.5

Local Density Approximation

The simplest approximation is that the exchange-correlation energy at any point of the system can be treated locally and same as that of an uniform electron gas of same density. This approximation holds for a slowly varying density. The exchange correlation energy for a density Ď (r) is given as LDA Exc

Z =

Ď (r)xc (Ď )dr.

(2.19)

where xc (Ď ) is the exchange-correlation energy per particle of an uniform electron gas of density Ď . Thus exchange-correlation potential can be given by LDA vxc [Ď (r)] =

LDA Î´Exc Î´xc (Ď ) = xc (Ď ) + Ď (r) Î´Ď (r) Î´Ď

(2.20)

CHAPTER 2.

22

For practical use exchange-correlation energy xc [n(r)] is slitted into exchange and correlation energies xc [n(r)] = x [n(r)] + c [n(r)]. The exchange energy can be given from Dirac functional4 3 xc [Ď (r)] = âˆ’ 4

13 3 Ď (r). 4

(2.21)

Values for c (Ď ) have been determined from Quantum Monte Carlo calculations.6 Thereafter analytic forms for c (Ď ) are created by interpolation.

2.1.6

Generalized Gradient Approximation

In LDA approximation the xc [n(r)] is a function of a local and constant density and does not care about the space-future of the density. It holds good for a system of slowly varying density. But it fails in situations, like in molecules, where the density undergoes a rapid change after certain distance. To improve xc [n(r)] one should include space-future of electron density. Gradient of electron density tells about that future. Thus an improvement is made in GGA (Generalized Gradient Approximation). In it xc [n(r)] is made a function of electron density gradient âˆ‡(Ď ) also. Hence the exchange-correlation energy function in GGA looks like xc = xc [Ď (r), âˆ‡Ď (r)].

(2.22)

Nowadays a lot of work is going on to get a good analytical form of xc .

2.2

The Broken Symmetry State

A very effective way of estimating J with less computational effort is a DFT formalism established by Noodleman.7 Post HF methods are reliable in computing J value, but they take a lot of computational time. If one can apply DFT, it will be very much cost effective. But the problem with DFT is that it is very difficult to get the energy of a pure spin state. Moreover, in an unrestricted formalism, it is not possible to calculate the diradical singlet.

CHAPTER 2.

23

Noodleman proved that any mixed spin state can be approximated by a weighted average of different spin states. Noodleman’s treatment is as follows. If B corresponds to a mixed state and S to pure spin states then according to Noodleman:7 |ψB i =

X

A(S)|ΨS i

(2.23)

S

Let us consider two interacting centers A and B and the related spatially orthogonal magnetic orbitals ai and bi on A and B, respectively. If there is one unpaired electron in each magnetic orbital a1 and b1 , then the BS determinant can read as |ψB i = (N !)−1/2 M −1/2 det[(a1 + cb1 )αa2 α . . . an α(b1 + ca1 )βb2 β . . . bn β]

(2.24)

where M is the normalization factor. The latter can also be expressed as |ψB i = M −1/2 [φ1 + c(φ2 + φ3 ) + c2 φ4 ].

(2.25)

Let us drop φ4 as it depends on c2 . The other contribution can be written as, φ1 : the purely covalent (A-B)-state, φ2 and φ3 : the charge transfer states for A+−B − and A−−B + . Using Lowdin8 and Nesbet9 spin projection technique it is possible to write φµ =

XX S

S MSOφν .

(2.26)

Ms

If φν be a slater determinant constituted by orthonormal orbitals, we get S MSOφν

= AS (S, MS )[φν +

X

χνµ (S, MS )].

(2.27)

µ

Using the orthoganality theorem for irreducible representations, one obtains 0

0

hSMSOφµ |H|SMSOφν i = hφµ |H|SMSOφν iδSS 0 δM M 0 .

(2.28)

CHAPTER 2.

24

From the above discussion the energy of the BS state comes as, X 1 X [A1 (S)hφ1 |H|φ1 + χ1µ φ1µ i M S µ X + c2 A2 (S)hφ2 |H|φ2 + χ2µ φ2µ i

hψB |H|ψB i =

µ 2

+ c A3 (S)hφ3 |H|φ3 +

X

χ3µ φ3µ i

(2.29)

µ

+ 2cA1 (S)hφ2 |H|φ1 +

X

χ1µ φ1µ i

µ

+ 2cA1 (S)hφ3 |H|φ1 +

X

χ1µ φ1µ i]

µ

where Ai (S) are the vector coupling coefficients of the φi states which can be expressed in terms os S, n and Smax . Now if n be the total number of magnetic electrons on the two centres, then the resulting state φ1 will be

and

n −n

|SA SB i =

2 2

(2.30)

2 n n n −n S; 0 A1 (S) = c 22 2 2

(2.31)

where c(SA SB S; Ma Mb M ) is a Clebsh-Gordan coupling coefficient.1 With the help of some more constraints and after some more derivation one finally gets the energy of the BS state as7

X

1

n − S(S + 1)

hψB |H|ψB i = < φ1 |H|φ1 > + A1 (S) × ai b j ai b j n2 r12 i,j S X n(n + 1) − S(S + 1) + A1 (S) n2 S

1

Sa¯2¯b

× 2Sa¯¯b ha1 |h|b1 i + ai bj ai bj + (E2 − E1 ) (2.32) r12 2 X

CHAPTER 2.

25

With some similar calculations one get the energy of a pure spin state as

X

1

n − S(S + 1)

× ai b j ai b j hψS |H|ψS i = < φ1 |H|φ1 > + n2 r12 i,j n(n + 1) − S(S + 1) + n2

1

Sa¯2¯b

(E2 − E1 ) (2.33) × 2Sa¯¯b ha1 |h|b1 i + ai bj ai bj + r12 2 From equation 2.32 and 2.33 one can say that the energy of the BS state is a weighted average of the energies of pure spin states and it is also a function of Sa¯¯b , the overlap integral of the magnetic orbitals.

2.3

Methodology

The magnetic exchange interaction between two magnetic sites is normally expressed by the Heisenberg effective Hamiltonian in Eq. (1.1). The eigenfunctions of the Heisenberg Hamiltonian are eigenfunction of S 2 and Sz where S is the total spin angular momentum and directly related to the energy difference between the spin eigenstates. For a diradical, E(S = 1) − E(S = 0) = −2J

(2.34)

One can evaluate J by determining the proper singlet and triplet energy values from a multiconfigurational approach. In literature some MCSCF calculations have been reported on bisverdazyl10 and TME based systems.11 In general, this method faces two major problems. Firstly, it is cumbersome (multiconfigurational) and omits a large amount of correlation corrections (like in CAS [2, 2]). Secondly, a single determinantal wave function gives a poor representation of the ground state singlet, and therefore, an unrestricted approach to yield a proper accounting of this exchange is not feasible. However, Illas et al.12a have

CHAPTER 2.

26

carried out CASSCF calculations for bisverdazyl and discussed the way of effectively introducing dynamical correlation through the CASPT2 method. Within CAS [2, 2], the description of the open-shell singlet is neither single determinantal nor monoconfigurational.12b As broken symmetry method is a reliable alternative way of estimating the energy of pure spin states with much less computational effort, we have used this method for calculating the J values. In the calculations, the spin-polarized, unrestricted density functional method has been used. The BS state is not a pure spin state but an artificial state of mixed ˆ or Sˆ2 . It is an alspin symmetry and lower spatial symmetry. It is not an eigenstate of H most equal mixture of singlet states and triplet states. The coupling constant can be written as J=

(EBS − ET˜ ) 2 1 + Sab

(2.35)

where Sab is the overlap integral between the two magnetically active orbitals a and b. The quantity EBS is the energy of the broken-symmetry solution, and is the energy of triplet state in the unrestricted formalism using the BS orbitals. It is known that there is less spin contamination in the high-spin state in a single-determinantal approach. Hence, ET˜ can be approximated by the triplet state energy which is achieved from a direct computation. The BS state is often found as spin-contaminated. One has to eliminate the effect of spin contamination from the energy of the BS state. To achieve this end, spin-projected methods have been proposed. Equation (2.35) is valid when there is only one pair of magnetically active orbitals. From the same basic methodology, the following three additional spinprojected equations were obtained and these are valid for different general cases: DF T DF T E − E BS T J GN D = , 2 Smax DF T EBS − DF T ET B J = , Smax (Smax + 1) DF T DF T E − E BS T JY = . hS 2 iT − hS 2 iBS

(2.36)

(2.37) (2.38)

CHAPTER 2.

27

These three equations differ in their applicability depending upon the different degree of overlap between the magnetic orbitals. Equation (2.36), (J GN D ), developed by Ginsberg,13 Noodleman et al.,7c,d and Davidson14 (GND), is applicable to the weak overlap limit whereas Eq. (2.37) (J B ) has been proposed by Bencini et al.15 According to Ruiz et al.16 the overlap in Eq. (2.35) is large in binuclear complexes and hence Eq. (2.37) holds. Nevertheless, Illas et al. have explicitly calculated that the overlap is indeed negligibly small. Thus EBS cannot be taken as E(S = 0).12b Equation (2.37) is generally found to be applicable for binuclear transition metal complexes. Equation (2.38), (J Y ), which can be reduced to Eq. (2.36) and Eq. (2.37) in the weak and strong overlap limits respectively, has been derived by Yamaguchi et al.17 Equation (2.38) is equivalent to the proposal by Illas et al. that the BS solution is approximately 50% singlet and 50% triplet.18 In order to avoid a misconception, we put forward a caveat that the strong overlap limit does not hold for practical systems. Therefore, Eq. (2.37) may be overlooked. Borden, Davidson, and Feller19 discussed that the ROHF calculations provide qualitatively correct molecular orbitals but in general fail to produce the correct molecular geometry. They suggested the use of the UHF methodology for a reasonably correct description of triplet (T,S=1) and open-shell singlet, (S, S=0) geometries. We have used spin-polarized unrestricted density functional theory (DFT), more specifically, the UB3LYP method, for both geometry optimization and single point calculation. For diradicals, the ideal BS state corresponds to hS 2 i = 1. Illas et al. found that it is always good to start with correct molecular orbitals for finding the BS solutions.20 Hence, the symmetry breaking in the wave function in the BS calculations has been achieved by using restricted open-shell wave functions for the optimized triplet structure, and then using the corresponding molecular orbitals as an initial guess. All the calculations have been performed by using the Gaussian 03 software.21 Recently, Truhlar et al.22 have proposed the MO6 functional for DFT and found that this functional is about equally as accurate as B3LYP. Range-separated hybrid functionals have

CHAPTER 2.

28

been investigated by Scuseria et al.,23 who have concluded that these functionals provide a significant improvement to B3LYP functional. Nevertheless, much more investigation is required to make the new functionals standard, and therefore, we have opted for the standard hybrids, namely, B3LYP.

CHAPTER 2.

2.4

29

References

1. Hohenberg, P. and Kohn, W. Phys. Rev., 1964, 136, B864. 2. Kohn, W. and Sham, L. J. Phys. Rev., 1965, 140, A1133. 3. (a) Thomas, L. H. Proc. Cambridge Philos. Soc., 1927 23, 542. (b) Fermi, E. Z. Phys., 1928, 48, 73. 4. Dirac, P. A. M. Proc. Cambridge Philos. Soc., 1930, 26, 376. 5. Seitz, F. The modern theory of solids; McGraw-Hill, 1940. 6. Ceperley, D. M. and Alder, B. J. Phys. Rev. Lett., 1980, 45, 566. 7. (a) Noodleman, L. J. Chem. Phys. 1981, 74, 5737. (b) Noodleman, L.; Baerends, E. J. J. Am. Chem. Soc. 1984, 106, 2316. (c) Noodleman, L.; Davidson, E. R. Chem. Phys. 1986, 109, 131. (d) Noodleman, L.; Peng, C. Y.; Case, D. A.; Mouesca, J.-M. Coord. Chem. Rev. 1995, 144, 199. 8. (a) Lowdin, P. O. Phys. Rev. 1955, 97, 1509. (b) Lowdin, P. O. Rev. Mod. Phys. 1962, 34, 80. 9. (a) Nesbet, R. K. Ann. Phys. 1958, 3, 397. (b) Nesbet, R. K. Ann. Phys. 1958, 4, 87. 10. Chung, G.; Lee, D. Chem. Phys. Lett. 2001, 350, 339. 11. (a) Nachtigall, P.; Jordan, K. D. J. Am. Chem. Soc. 1992, 114, 4743. (b) Nachtigall, P.; Jordan, K. D. J. Am. Chem. Soc. 1993, 115, 270. 12. (a) de Graaf, C.; Sousa, C.; Moreira, I. de P. R.; Illas, F. J. Phys. Chem. A 2001, 105, 11371. (b) Moreira, I. de P. R.; Illas, F. Phys. Chem. Chem. Phys. 2006, 8, 1645. 13. Ginsberg, A. P. J. Am. Chem. Soc. 1980, 102, 111. 14. Noodleman, L.; Davidson, E. R. Chem. Phys. 1986, 109, 131. 15. (a) Bencini, A.; Totti, F.; Daul, C. A.; Doclo, K.; Fantucci, P.; Barone, V. Inorg. Chem. 1997, 36, 5022. (b) Bencini, A.; Gatteschi, D.; Totti, F.; Sanz, D. N.; McClevrty, J. A.; Ward, M. D. J. Phys. Chem. A 1998, 102, 10545.

CHAPTER 2.

30

16. Ruiz, E.; Cano. J.; Alvarez, S.; Alemany, P. J. Comput. Chem. 1999, 20, 1391. 17. (a) Yamaguchi, K.; Takahara, Y.; Fueno, T.; Nasu, K. Jpn. J. Appl. Phys. 1987, 26, L1362. (b) Yamaguchi, K.; Jensen, F.; Dorigo, A.; Houk, K. N. Chem. Phys. Lett. 1988, 149, 537. (c) Yamaguchi, K.; Takahara, Y.; Fueno, T.; Houk, K. N. Theo. Chim. Acta 1988, 73, 337. 18. (a) Martin, R. L.; Illas, F. Phys. Rev. Lett. 1997, 79, 1539. (b) Caballol, R.; Castell, O.; Illas, F.; Moreira, I. de P. R.; Malrieu, J. P. J. Phys. Chem. A 1997, 101, 7860. (c) Barone, V.; di Matteo, A.; Mele, F.; Moreira, I. de P. R.; Illas, F. Chem. Phys. Lett. 1999, 302, 240. (d) Illas, F.; Moreira, I. de P. R.; de Graaf, C.; Barone, V. Theor. Chem. Acc. 2000, 104, 265. (e) Illas, F.; Moreira, I. de P. R.; Bofill, J. M.; Filatov, M. Phys. Rev. B 2004, 70, 132414. 19. (a) Borden, W. T.; Davidson, E. R. J. Am. Chem. Soc. 1977, 99, 4587. (b) Borden, W. T.; Davidson, E. R.; Feller, D. Tetrahedron 1982, 38, 737. (c) Feller, D.; Davidson, E. R.; Borden, W. T. Isr. J. Chem. 1983, 23, 105. (d) Kato, S.; Morokuma, K.; Feller, D.; Davidson, E. R.; Borden, W. T. J. Am. Chem. Soc. 1983, 105, 1791. 20. (a) Martin, R. L.; Illas, F. Phys. Rev. Lett. 1997, 79, 1539. (b) Caballol, R.; Castell, O.; Illas, F.; Moreira, I. de P. R.; Malrieu, J. P. J. Phys. Chem. A 1997, 101, 7860. (c) Barone, V.; di Matteo, A.; Mele, F.; Moreira, I. de P. R.; Illas, F. Chem. Phys. Lett. 1999, 302, 240. (d) Illas, F.; Moreira, I. de P. R.; de Graaf, C.; Barone, V. Theor. Chem. Acc. 2000, 104, 265. (e) de Graaf, C.; Sousa, C.; Moreira, I. de P. R.; Illas, F. J. Phys. Chem. A 2001, 105, 11371. (f) Illas, F.; Moreira, I. de P. R.; Bofill, J. M.; Filatov, M. Phys. Rev. B 2004, 70, 132414. 21. M. J. Frisch, et al., GAUSSIAN 03 (Revision E), Gaussian, Inc., Pittsburgh, PA, 2004. 22. Valero, R.; Costa, R.; Moreira, I. de P. R.; Truhlar, D. G.; and Illas, F.; J. Chem. Phys. 2008, 128, 114103. 23. Pablo, R.; Moreira, I. de P. R.; and Illas, F.; Scuseria, G. E.; J. Chem. Phys. 2008, 129, 184110.

CHAPTER 2.

31

1+2=3 RRRR 5+4=9

<1

a11 = b11

a12 = b12

(2.39)

a21 = b21

a22 = b22 + c22

(2.40)