Shannon Entropy of Chemical Changes: SN 2 Displacement Reactions∗ ` 1 HARTMUT L. SCHMIDER,1 DONALD F. WEAVER,1 MINHHUY HÔ, VEDENE H. SMITH, JR.,1 ROBIN P. SAGAR,2 RODOLFO O. ESQUIVEL2 1
Department of Chemistry, Queen’s University, Kingston K7L 3N6, Ontario, Canada Departamento de Química, Universidad Autónoma, Metropolitana, Apartado Postal 55-534, Iztapalapa, 09340 México Distrito Federal, Mexico 2
Received 29 July 1999; accepted 20 August 1999
ABSTRACT: The Shannon entropies along the intrinsic reaction coordinates (IRC) of
two SN 2 reactions were calculated at the RHF/6-31++G∗∗ level. The resulting entropic profiles were compared with the corresponding energy profiles. The Shannon entropy profiles in position and momentum space, as well as their sum, show interesting features about the bond forming and breaking process that are not apparent from the conventional c 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 376–382, 2000 reaction energy profile.
Key words: Shannon information entropy; momentum and charge density analysis; SN 2 reaction; intrinsic reaction coordinates; reaction energy profile
or the analysis of a chemical process, the conventional measurement of choice is the energy of the system. However, properties that are based on the electronic density have also played a prominent role in explaining the physics and chemistry of reactions. Since the operators associated with these properties do not commute with the Hamiltonian, and the system is usually described by a wave function that is variational with respect to the energy, the former show an enhanced sensitivity toward subtle changes in the system. The electronic charge Correspondence to: V. H. Smith, Jr. ∗ Dedicated to Professor Michael C. Zerner on the occasion of his 60th birthday.
International Journal of Quantum Chemistry, Vol. 77, 376–382 (2000) c 2000 John Wiley & Sons, Inc.
density, which is, in principle, accessible through experiment , may be derived from the multidimensional many-particle wave function by reduction and allows an interpretation of the system in terms of three-dimensional direct space. The importance of the electronic density is due additionally to the existence of its intrinsic relationship with the ground-state energy of the system, proved by Hohenberg and Kohn . Common to many different methodologies in computational chemistry is the search for an appropriate description. Starting with the independentparticle or Hartree–Fock (HF) model, this goal takes on the problem of electron correlation, and the correlation energy is employed as a measure of the closeness of an approximate wave function to the exact one. Expectation values are usually assumed to be of essentially the same level of accuracy as
CCC 0020-7608 / 00 / 010376-07
SHANNON ENTROPY OF CHEMICAL CHANGES the energy. While this can be assumed to be true in the case of complete or full configuration interaction (FCI) calculations within a model space defined by a sufficiently large basis set, one needs a proper measure of the level of correlation of such expectation values for various inexact calculations. This is due to the fact that if one takes the system energy as the standard measure, post-Hartree–Fock methods perform nonuniformly with respect to various expectation values. Post-HF methods aim at improving the description of the simultaneous interactions of electrons; from the density perspective this leads almost invariably to a delocalization with respect to the density in the independent-particle model. The Shannon information entropy  of the charge density ρ(r), Z (1) Sρ = − ρ(r) ln[ρ(r)] dr, provides a measure for this delocalization. Analogously, the information entropy of the momentum distribution π(p) is defined as Z (2) Sπ = − π(p) ln[π(p)] dp. From these definitions, one can see that Sρ is maximal for a uniform distribution, e.g., of an unbound system, and is minimal when the uncertainty about the structure of the distribution is minimal, e.g., a delta-like distribution. Sπ , on the other hand, is largest for systems where electrons are of indeterminable speed and is smaller when the system contains more relaxed electrons, i.e., low p. Atomic and diatomic Shannon entropies of electron densities have been studied extensively [4 – 8], and the results show that the total entropy ST = Sρ + Sπ is a good indicator of basis set quality. Furthermore, ST follows an uncertainty-like inequality [4, 9], namely ST = Sρ + Sπ ≥ 3(1 + ln π) ≈ 6.4342
for unity-normalized ρ(r) and π(p). This particularity of ST has been reported in several studies of molecular properties [10 – 13] in which the complementary use of the momentum entropy has been shown to be crucial. In recent work , we have also demonstrated that the Shannon entropy can be used as a means to interpret correlation effects that are not inherent in the conventional energy analysis, namely in terms of global delocalizations of the densities in both spaces. In the present work, we continue this avenue by monitoring the Shannon entropies and the energies of two SN 2 reactions following their intrinsic reaction coordinates
(IRC) . We wish to see if there is a reorganization of information entropy during a chemical reaction based upon which a different profile may be observed. These entropy-based profiles may serve to locate critical stages of the reaction that are specific to the density perspective and otherwise not available via the energy profile. Few studies of entropic changes following the reaction path have been reported. Balakrishnan and Sathyamurthy found that Sρ and Sπ reach a maximum during the time evolution of the He + H+ 2 bimolecular exchange reaction . The SN 2 reaction is one of the most fundamental mechanisms of organic chemistry. Consequently, its literature is extensive. Typically, the mechanism is the nucleophilic attack of an (often anionic) species on the triangular backside of an sp3 -hybridized carbon center. Simultaneously, another species, the leaving group, separates from the molecule, and the remaining residuals on the attacked carbon carry out a movement equivalent to the “flipping of an umbrella.” The energy profile of this type of reaction often consists of a transition state and two minima— the latter associated with adducts of the nucleophile and the starting compound, and the leaving group with the product, respectively. The completely separate species are of course higher in energy than these adducts, i.e., the latter are stable with respect to dissociation.
Results and Discussions Shi and Boyd  studied a variety of SN 2 reactions and examined the change in the charge density by means of topological analysis . We have taken two of their examples: the backside SN 2 reactions between a hydride anion and methane and between a hydride anion and fluoromethane,† H− · · ·H3 C—H → H—CH3 · · ·H−
H− · · ·H3 C—F → H—CH3 · · ·F− .
We studied the entropies of the whole system occurring along the C3v -restricted reaction path. For the construction of the path, we used the same † Note that the hydride anion is not stable within the Hartee– Fock description. However, this fact does not have an appreciable impact on our calculations, since we are only concerned with regions on the potential-energy surface between two local minima, which are stable as a whole. Comparable computations on an MP2 level yield similar results.
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
` ET AL. HÔ basis set as in Ref.  (6-31++G∗∗ /6-31G) to supply a simple comparison. The optimized geometries and energies of the ground states and the transition states obtained for both reactions are virtually identical to the ones reported by Shi and Boyd. We performed IRC calculations  employing the GAMESS program package . For each reaction path, this resulted in 20 geometries that lie evenly distributed in terms of a weighted distance parameter (Eq. (7) of ). We then performed separate single-point calculations in a 6-311++G∗∗ basis set at those geometries to obtain a set of wave functions. Finally, we computed the charge densities in position and momentum space and, subsequently, the unity-normalized entropies from those wave functions, using our own code. All ab initio calculations were performed at the restricted Hartree–Fock (RHF) level. Since only closed-shell species are involved in these reactions, and no homolytic bond breakage occurs, this was deemed sufficient for an adequate description. The integrations of Eqs. (1) and (2) were performed using an automatic threedimensional quadrature scheme [19 – 21] and were set to have an absolute error of less than 10−5 a.u. We have plotted the changes in the geometrical parameters and energy for the first reaction (4) in Figure 1. The x axis shows the step number based on an equal-distance parameter that determines
consecutive geometries on the reaction path. This distance parameter, based on the mass-weighted Cartesian coordinates, is completely linear in the step number, and therefore well suited as an indicator for the smooth progress of the reaction along the coordinate. We have arbitrarily set step 0 for the transition state. Since the reaction coordinate of (4) is symmetric with respect to the transition state, only the data of one half of the reaction are presented. From the energy profile [shown on plot (a)], it may be seen that the rate of change in the energy slows significantly from step 15 onward. The change in the H(nuc)—C bond length [plot (b)], on the other hand, seems to cease at step 6, at which point it almost attains its equilibrium value of ≈1.09 Å. The H(leav)—C bond stretches continuously throughout [also shown in plot (b)]. On closer observation, however, one detects a change in the rate of stretching of the bond after step 5. The equatorial H—C bond lengths [plot (c)] are minimal at the transition state, which has the trigonal bipyramidal (D3h ) symmetry. This presumably is due to the smaller repulsion between the three equatorial H’s and the axial ligands. The H(eq)—C—H(nuc) angles change the fastest at this point, i.e., the slope of curve (d) is largest around the transition state. In Figure 2 the entropies of each point are plotted against the corresponding step number. The posi-
FIGURE 1. Energies and some geometrical parameters of the H− + CH4 reaction [Eq. (4)] along the reaction coordinate from step 0 (transition state) to step 20 (close to ground state).
VOL. 77, NO. 1
SHANNON ENTROPY OF CHEMICAL CHANGES
FIGURE 2. Entropies of the H− + CH4 reaction [Eq. (4)] along the reaction coordinate for the first 10 steps. The original data points are shifted downward: Sρ (squares, shift 4.6 units), Sπ (triangles, shift 4.0 units), and ST (filled circles, shift 8.6 units).
tion entropy Sρ (squares) decreases as the system moves away from the transition state. This is because the charge density in the methane molecule is as a whole more localized, due to the compact tetrahedral shape (which is more spherical as compared to the trigonal structure of the transition state). The leaving anion has a separate single density sphere. After step 5, however, the methane density does not change significantly anymore, and the total ef-
fect is largely due to the elongation of the density of the CH4 —H− complex. This is a global delocalization effect which raises Sρ . The momentum entropy (triangles) mirrors this behavior in the opposite direction, by showing a maximum at step 5. The total entropy (filled circles) exhibits a more complicated pattern with a minimum at the transition state followed by a shallow hump at step 2, a local minimum at step 4, and an increase thereafter. The changes in the geometries and energy of the reaction (5) of the hydrogen anion with methyl fluoride are plotted in Figure 3. The negative values of the step number and the geometrical distance S belong to the first half (i.e., H− + CH3 F) of the reaction, up to the transition state, where the hydrogen anion closes in on the methyl fluoride molecule. In a similar analysis to the previous reaction, one finds that the asymmetric energy profile [plot (a)] shows significant changes between steps −7 and 20 while the axial bond axes indicate a shorter period, i.e., the changes in the axial bond lengths (plot (b)] occur mainly between steps −7 and 6. Note that the breakage of the F—C bond is much slower than the rate of formation of the C—H bond between steps −7 and 6. As shown below, this difference determines the overall behavior of the entropy. The bond lengths of the equatorial hydrogens continue to increase, albeit at a much slower rate, after step 6 [plot (c)]. The change in the bond angle between
FIGURE 3. Energies and geometrical parameters of the H− + CH3 F reaction [Eq. (5)] along the reaction coordinate.
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
` ET AL. HÔ
FIGURE 4. Entropies of the H− + CH3 F reaction
[Eq. (5)] along the reaction coordinate from step −10 to step +10, where 0 is the transition state. The original data points are shifted downward: Sρ (squares, shift 3.9 units), Sπ (triangles, shift 5.1 units), and ST (filled circles, shift 9.0 units).
the equatorial atoms and the incoming hydrogen [plot (d)] lags slightly behind the other changes and stabilizes after step 10. The entropic profiles are plotted in Figure 4. The position entropy (squares) has two minima at steps −7 and 7, and a shallow maximum at step −5. The momentum entropy (triangles) has a minimum at step −3 and two maxima at steps −8 and 7. The
total entropy (filled circles) decreases continuously until step 7. The resulting position entropy is the combined effect of the depletion of electron density in the C—F bond, and the migration of electrons to the C—H bond. In the left-hand column of Figure 5, we plot the position density profile along the F· · ·C· · ·H (symmetry) axis at steps −10, 0 (transition state), and 10. The changes in the density show that the entropic effect due to the accumulation of electrons at the C—H bond overwhelms that of the electron depletion at the C—F bond. The overall result consists mainly of the localization of electrons at the newly formed C—H bond, and hence, a decrease in the position entropy (Sρ , see squares in Fig. 4) occurs. The analogous momentum density profiles are plotted in the right-hand column of Fig. 5. Shown is a one-dimensional section along the D3d symmetry axis in momentum space. Note that the point group differs from the C3v position symmetry, since momentum densities exhibit an additional inversion center [22, 23]. While the position density shows more structure (i.e., a lower entropy at the transition state), the momentum density is more delocalized (i.e., it yields a higher entropy). The chemical origin of this behavior requires evaluation. At step −10 (top right), the system consists of a hydrogen anion and the methyl fluoride moiety. Recall that the Dirac–Fourier transformation  establishing the
FIGURE 5. Position and momentum density profiles of the H− + CH3 F reaction along the symmetry axis, for reaction steps −10, 0 (transition state), and 10. The densities at step −10 is shown at the top, step 0 in the middle, and step 10 at the bottom. Position densities are to the left, with the symmetry axis being the Z coordinate and the C atom being arbitrarily set at the origin. Momentum densities are to the right, with the symmetry axis being the pz direction.
VOL. 77, NO. 1
SHANNON ENTROPY OF CHEMICAL CHANGES relationship between the wave functions in position and momentum spaces implies an “inverse” relationship between the respective densities. For example, highly localized inner-shell orbitals will cause a broad, flat shape in the momentum density while the diffuse valence orbitals will be sharply peaked around the origin in the momentum space. In step −10, the single peak of the momentum density profile is due to the highly diffuse hydrogen anion. Consequently, Sπ (Fig. 4, triangles) has a low value. The momentum entropy of the transition state is not much larger; although the momentum distribution is more diffuse, it gains structure through the splitting of the central peak into a pair (middle right plot in Fig. 5). On the other hand, at step 10, the system can be thought of as a neonlike ion (F− ) plus the methane molecule, both of which are known [25, 26] to show a pair of peaks in the momentum density profile due to their strong p-orbital contributions. Here these peaks are, however, very flat and the distribution is so spread out that a larger value of the entropy results. Note that while interpretations of information entropies from density profiles are feasible in these simple examples, Sρ and Sπ are global quantities that take into account all structure of the density. Such analyses will be difficult for larger systems without first partitioning the density into smaller subsystems .
momentum space , phase space  and nuclear configuration space  for the definition of such regions that correspond to specific species, has long been a subject of interest in the literature. The results here suggest that information entropies are suitable candidates for similar studies. Work along these lines, as well as on the influence of solvent effects on the information entropies, is currently underway in this laboratory. ACKNOWLEDGMENTS This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERCC) and the Mexican Research Council (CONACyt).
Conclusions In summary, through two simple examples of SN 2 reactions, we have shown that the Shannon entropy reveals a great deal of information about the system, from the perspective of geometrical changes to the density. This information is largely transparent in the conventional energetic profile. However, one should not overlook the fact that energy has an irreplaceable role in chemistry and that these Shannon entropies were calculated from wave functions that are derived from energy considerations. For example, the entropy profiles do not seem to be sensitive to the transition state, an energy based species which is eminent in studies of chemical reactions. These results, instead, inspire an investigation of a possible analogous “entropy hypersurface.” The rims and valleys of such a surface could be used to separate different species. Abrupt changes in the information entropies then could signify the formation of new species such as the intermediates and the products as well as the termination of the reactants. The utility of properties in position space ,
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
References 1. Coppens, P. X-Ray Charge Densities and Chemical Bonding; Oxford University Press: New York, 1997. 2. Hohenberg, P.; Kohn, W. Phys Rev B 1964, 136, 864–871. 3. Shannon, C. E. Bell Syst Tech J 1948, 27, 379; reprinted in Claude Elwood Shannon: Collected Papers; Shannon, C. E.; IEEE Press: New York, 1993. 4. Gadre, S. R.; Sears, S. B.; Chakravorty, S. J.; Bendale, R. D. Phys Rev A 1985, 32, 2602–2606. 5. Gadre, S. R.; Bendale, R. D. Curr Sci 1985, 54, 970–977. 6. Allan, N. L.; Cooper, D. L. J Chem Phys 1986, 84, 5594–5605. ` M.; Sagar, R. P.; Smith, Jr., V. H.; Esquivel, R. O. J Phys B 7. Hô, At Mol Opt Phys 1994, 27, 5149–5157. ` M.; Sagar, R. P.; Peréz–Jordá, J.; Smith, Jr., V. H.; Es8. Hô, quivel, R. O. Chem Phys Lett 1994, 219, 15–20. 9. Bialynicki–Birula, I.; Mycielski, J. Commun Math Phys 1975, 44, 129–132. ` M.; Sagar, R. P.; Schmider, H.; Weaver, D. F.; Smith, Jr., 10. Hô, V. H. Int J Quant Chem 1994, 53, 627–633. ` M.; Sagar, R. P.; Weaver, D. F.; Smith, Jr., V. H. Int J Quant 11. Hô, Chem 1995, S29, 109–115. ` M.; Smith, Jr., V. H.; Weaver, D. F.; Gatti, C.; Sagar, R. P.; 12. Hô, Esquivel, R. O. J Chem Phys 1998, 108, 5469–5475. ` M.; Smith, Jr., V. H.; Weaver, D. F.; Sagar, R. P.; Esquivel, 13. Hô, R. O.; Yamamoto, S. J Chem Phys 1998, 109, 10620–10627. 14. Ishida, K.; Morokuma, K.; Komornicki, A. J Chem Phys 1977, 66, 2153–2156. 15. Balakrishnan, N.; Sathyamurthy, N. Chem Phys Lett 1989, 164, 267–269. 16. Shi, Z.; Boyd, R. J. J Am Chem Soc 1989, 111, 1575–1579. 17. Bader, R. F. W. Atoms in Molecules; Clarendon: Oxford, 1994. 18. Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. J.; Koseki, S.; Matsunaga, M.; Nguyen, K. A.; Su, S.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. J Comput Chem 1993, 14, 1347–1363. 19. Pérez-Jordá, J. M.; San-Fabián, E.; Moscardó, F. Comput Phys Commun 1992, 70, 271–284.
` ET AL. HÔ 20. Pérez-Jordá, J. M.; San-Fabián, E. Comput Phys Commun 1993, 77, 46–56.
25. Epstein, I. R.; Tanner, A. C. In Compton Scattering; Williams, B., Ed.; McGraw-Hill: New York, 1977, Chap. 7, p. 209–233.
21. Pérez-Jordá, J. M.; Becke, A. D.; San-Fabián, E. J Chem Phys 1994, 100, 6520–6534.
26. Sagar, R. P.; Esquivel, R. O.; Schmider, H.; Tripathi, A. N.; Smith, Jr., V. H. Phys Rev A 1993, 47, 2625–2627. 27. Tal, Y.; Bader, R. F. W.; Nguyen-Dang, T. T.; Ojha, M.; Anderson, S. G. J Chem Phys 1981, 74, 5162–5167.
22. Löwdin, P.-O. Adv Quantum Chem 1967, 3, 323–381. 23. Kaijser, P.; Smith, Jr., V. H. In Quantum Science: Methods and Structure; Calais, J. L.; Goscinski, O.; Linderberg, J.; Öhrn, Y., Eds.; Plenum: New York, 1976, pp. 417–425. 24. Dirac, P. A. M. The Principles of Quantum Mechanics; Oxford: London, 1958.
28. Kulkarni, S. A.; Gadre, S. R. J Am Chem Soc 1993, 115, 7434– 7438. ` M. J Phys Chem 1996, 100, 17807–17819. 29. Schmider, H.; Hô, 30. Mezey, P. G. Theoret Chim Acta 1981, 58, 309–330.
VOL. 77, NO. 1
This is shannon-entropy paper