J. Phys. Chem. 1996, 100, 12295-12298
12295
A Density Functional Approach to Hardness, Polarizability, and Valency of Molecules in Chemical Reactions Tapan K. Ghanty and Swapan K. Ghosh* HeaVy Water DiVision, Bhabha Atomic Research Centre, Bombay 400 085, India ReceiVed: January 29, 1996; In Final Form: May 3, 1996X
We have studied the variation of hardness, polarizability, and valency of molecules during the course of a chemical reaction. For isomerization reactions, the quantities are calculated through Kohn-Sham version of spin-polarized density functional theory while for other types of reactions, available data on polarizability are used for the study. It is observed that a state of minimum polarizability usually can be associated with higher stability or maximum hardness.
1. Introduction New concepts have been introduced1 in chemistry from time to time for rationalization and prediction of various physicochemical phenomena. Two important concepts that have been highly successful in providing a better understanding of chemical binding and reactivity in molecular systems are the concepts of electronegativity2 (χ) and hardness3 (η). Denoting respectively the first4 and second5 order derivatives of energy (E) with respect to the number of electrons (N), viz.
χ ) -(∂E/∂N)
(1)
η ) 1/2(∂2E/∂N2)
(2)
these two parameters provide measures of response of a system to the change in the number of electrons at fixed external potential V(r) (for example, due to the nuclei). The alternative response function corresponding to the change in the external potential at fixed N defines the polarizability. An interconnection between these two separate response functions corresponding to changes in N and V(r), respectively, has been established6-8 through density functional theory9 (DFT) and has led to quite efficient schemes7,8 for an accurate prediction of polarizability and hardness of atomic systems through a single calculation. It is in fact well-known that the concepts of hardness and softness (reciprocal of hardness) are closely related to the polarizabilities as well as the sizes for atoms and molecules. Thus a hard (soft) species is associated3 with a low (high) value of the polarizability and small (large) size. This interconnection has been further strengthened through studies by Politzer10 and others11,12 demonstrating the existence of a good correlation among these quantities, viz., hardness, polarizability, and sizes not only for atoms and molecules but also for clusters. A lower polarizability is also known to be associated with greater stability of a species. The success of these studies strengthens the motivation for optimism about the possibility of using polarizability as an index not only for a systematic study of individual systems but also for the study of progress of chemical binding and reactions, for which hardness has already been shown13 to be useful. Thus, following the principle of maximum hardness,14 it has been shown15-19 that the change of hardness along a reaction path (hardness profile) passes through a minimum near or at the transition state for inversion, exchange, deformation, and isomerization types of reactions. Some symmetric stretching X
Abstract published in AdVance ACS Abstracts, July 1, 1996.
S0022-3654(96)00276-6 CCC: $12.00
modes of deformation do not however seem to correspond16 to extremum values of hardness or chemical potential at the equilibrium geometry. Similarly, it has been shown15-19 that the molecular valency reaches its minimum value at the transition state for isomerization type of reactions and maximum value at the equilibrium configuration for normal modes of vibrations and internal rotations. As far as the parameter polarizability is concerned, however, studies have so far been confined mainly to isolated molecules alone, and only very recently its variation (along with the variation of hardness and electronegativity) with distortion of a particular bond in a molecule (keeping the remaining parts of the molecule unaffected) has been reported.20 It is thus of interest to study systematically the variation of this property of molecules during the course of a chemical reaction. Along the lines of maximum hardness, there is already indication for a principle of maximum molecular valency,16 and a quest for the possibility of existence of extremum polarizability condition in a chemical reaction might prove to be rewarding. For the present studies, we employ density functional theory, which has been well-known not only as a versatile tool for the investigation of electronic structure of atoms, molecules, solids, and structure of inhomogeneous liquids but also for providing conceptual basis to the concepts of electronegativity,21 hardness,22 softness, and frontier orbital theory23 and leading to a generalized electronegativity equalization principle for the study of chemical binding and reactivity.24,25 2. Theory and Computational Method In DFT, the energy of a many-particle system can be expressed as a functional of the single particle density F(r), viz.
E[F] ) ∫dr V(r) F(r) + F[F]
(3)
and for a fixed external potential V(r), the energy functional E[F] assumes a minimum value for the true density. On minimization of eq 3 with respect to the density, subject to the normalization constraint ∫dr F(r) ) N, for an N-electron system, one obtains the Euler equation
µ)
δF[F] δE[F] ) V(r) + δF(r) δF(r)
(4)
where µ is the Lagrange multiplier representing the chemical potential of the electron cloud, which has been identified21 with the electronegativity parameter of eq 1 (with a negative sign). © 1996 American Chemical Society
12296 J. Phys. Chem., Vol. 100, No. 30, 1996
Ghanty and Ghosh
The exact form of the universal functional F[F] is however not known, and approximations are therefore employed for practical calculations. We consider here the Kohn-Sham26 scheme for density calculation where the two components of F[F], viz., the noninteracting kinetic energy functional and the classical electrostatic energy are evaluated exactly while the remaining exchange-correlation (XC) energy component is usually obtained within the local spin-density (LSD) approximation9,27 or LSD with some nonlocal correction (e.g., involving density gradients). In the spin-polarized version of this theory, the up- and down-spin electron densities FR(r) and Fβ(r) corresponding to the respective number of electrons NR [)∫dr FR(r)] and Nβ [)∫dr Fβ(r)] are the basic variables, and the kinetic and XC energy density functionals denoted respectively as TS[FR,Fβ] and EXC[FR,Fβ] depend on these two spin components of the total density F(r) [) F(r) + Fβ(r)]. The energy minimization leads to the Kohn-Sham equations26 for the spin orbitals (with i ) 1 to N) given by (atomic units are used throughout) σ [-1/2∇2 + Veff (r,{Fσ})]ψiσ(r) ) iσψiσ(r)
(5)
where Fσ(r) ) ∑iniσFiσ(r) and Fiσ(r) ) |ψiσ(r)|2 with the occupation numbers niσ satisfying ∑iniσ ) Nσ, for σ ) R or β. The spin-dependent Kohn-Sham effective potential is given by σ (r) ) V(r) + ∫ Veff
dr′ F(r′) σ (r;[{Fσ}]) + µXC |r - r′|
(6)
σ σ [ ) δEXC /δFσ(r)] denotes the XC potential (for where µXC example, within the LSD approximation). The total energy can be obtained from the expression
E ) ∑∑iσ σ
i
1
∫∫ 2
dr dr′ F(r) F(r′) |r - r′|
LSD [{Fσ}] + EXC
σ [{Fσ}]Fσ(r) ∑σ ∫dr µXC
(7)
The polarizability can be calculated by solving the KohnSham equations for the atom or molecule in presence of different values of uniform external fields and considering the expansion of the calculated field-dependent dipole moment as
mi(F) ) mi + ∑RijFj + 1/2∑βijkFjFk j
(11)
j,k
where mi, Rij, and βijk denote components of the permanent dipole moment, dipole polarizability, and the first dipole hyperpolarizability, respectively. The polarizability components Rij can easily be obtained from least-squares fits of the calculated dipole moments to a polynomial in the field variable. All the calculations in this work has been done using the Gaussian density functional program deMon28 where the KohnSham molecular orbitals are expanded in a basis of Gaussian type orbitals. The Perdew-Wang-91 exchange correlation potential29 and the IGLO-III orbital basis set30 (except gallium, for which the DZVP basis set was used) have been used. The fine-grid option of deMon has been used for density calculation and the default field step size of 0.0005 au along with a sixthorder polynomial fit of the field-dependent dipole moment has been employed for polarizability calculation. 3. Results and Discussions We have considered three types of reactions, viz., exchange, dissociation, and isomerization. For the isomerization reactions, we have calculated the polarizability through the density functional procedure as described in section 2. For the exchange and dissociation reactions, however, the polarizability values are taken from the literature.31 We consider the following exchange reactions for which the values of ∆H32,33 and the polarizability are reported below (the ∆H value for each reaction is in kcal/mol; the polarizability value for each species is in Å3) (1) (2)
CF2Cl2 7.81 2CH3F 2.97 CHF3 3.57 3CH3F 2.97 3CH3Cl 5.35 3CF3Cl 5.72 3(CH3)2 4.47 CH3F 2.97 CH3OH 3.26 3(CH3)2O 5.29 CF3H 3.57 CH3OH 3.26 CH3OH 3.26
+ + +
CF2Cl2 7.81 CH3F 2.97 3CHF3 3.57 CH3F 2.97 CH3Cl 5.35 CF3Cl 5.72 (CH3)2 4.47 HI 5.45 HI 5.45 (CH3)2O 5.29 CH3F 2.97 C2H6 4.47 CH3OH 3.26
) ) )
CF4 3.838 2CH4 2.593 CH4 2.593 3CH4 2.593 3CH4 2.593 3CF4 3.838 3CH4 2.593 CH3I 7.59 CH3I 7.59 3CH4 2.593 CF4 3.838 CH4 2.593 (CH3)2O 5.29
+ + +
CCl4 11.2 CHF3 3.57 3CF4 3.838 CF4 3.838 CCl4 11.2 CCl4 11.2 C(CH3)4 10.20 HF 0.80 H2O 1.45 C(OCH3)4 13 CH4 2.593 C2H5OH 5.07 H2O 1.45
∆H ) -16.3 ∆R1/3 ) -0.165 ∆H ) -31.4 ∆R1/3 ) -0.036 ∆H ) -22.9 ∆R1/3 ) -0.043 ∆H ) -63 ∆R1/3 ) -0.063 ∆H ) -6 ∆R1/3 ) -0.637 ∆H ) -27.1 ∆R1/3 ) -0.219 ∆H ) -13 ∆R1/3 ) -0.299 ∆H ) -12.3 ∆R1/3 ) -0.304 ∆H ) -12.6 ∆R1/3 ) -0.145 ∆H ) -52 ∆R1/3 ) -0.497 ∆H ) -19 ∆R1/3 ) -0.026 ∆H ) -5 ∆R1/3 ) -0.038 ∆H ) -4.4 ∆R1/3 ) -0.091
From the calculated total energies of any atom or molecule and its positive and stable negative ions, one can now calculate the electronegativity and hardness as energy derivatives, which however require multiple calculations involving solution of the Kohn-Sham equations for more than one value of N for a particular species. Conventionally they are obtained from the experimental values of the ionization potential (I) and the electron affinity (A) through the finite difference approximations χ ) (I + A)/2 and η ) (I - A)/2. For simplicity, we calculate these quantities through these finite difference formulas using suitable eigenvalues from Kohn-Sham calculation as measures of I and A. Thus, the chemical potential and hardness parameters are approximated in terms of the eigenvalues corresponding to the highest occupied (HOMO) and lowest unoccupied molecular orbitals (LUMO) as
(3)
µ ) 1/2(LUMO + HOMO)
(8)
(13)
η ) 1/2(LUMO - HOMO)
(9)
In these examples each reaction proceeding from left to right is accompanied with negative ∆H and the average value of the cube root of polarizability is less for the products than the reactants. We have considered here the cube root of polarizability as the quantity of interest, since it is this quantity that has been shown11,12 to correlate well with the hardness as well as the size of various species. Also among the four species involved in each reaction one of the products is found to have minimum polarizability. An analogous observation reported by
The other quantity of interest is the molecular valency VM defined as
VM ) 1/2∑VA
(10)
A
where VA, the valency of the atom A in the molecule, is essentially the diagonal element of the bond-order matrix.
(4) (5) (6) (7) (8) (9) (10) (11) (12)
+ + + + + + + + + +
) ) ) ) ) ) ) ) ) )
+ + + + + + + + + +
A Density Functional Approach to Hardness
J. Phys. Chem., Vol. 100, No. 30, 1996 12297
TABLE 1: Calculated Values of Total Energy, Electronegativity, Hardness, Polarizability, and Valency of Molecules and Ions
a
speciesa
-Etotal (au)
electronegativity χ (eV)
hardness η (eV)
polarizability R (au)
valency VM
HBO HOB HOB-b TS
100.7137 100.6331 100.6383 100.6014
4.8570 3.2087 3.7496 4.8855
3.8672 2.3706 2.0506 1.8607
15.6527 25.4750 25.0742 23.1743
3.2064 2.1991 2.0780 2.5906
HAlO HOAl HOAl-b TS
318.2356 318.2957 318.2955 318.1815
4.2098 3.4610 3.5059 5.1698
1.8714 2.1405 2.0535 0.8816
33.6222 43.0303 43.2406 43.3834
2.7295 1.5458 1.6309 2.3426
HGaO HOGa HOGa-b TS
2000.4223 2000.4733 2000.4730 2000.3623
4.1015 3.8270 3.9275 5.2791
2.0221 1.8464 1.7006 0.8293
31.7584 40.0820 39.0130 40.7457
3.2226 1.9498 1.9923 2.6954
HBS HSB HSB-b TS
423.6429 423.5081 423.5373 423.5250
4.6418 4.6010 4.2557 4.7343
2.6641 2.0156 1.2778 0.8956
30.0907 41.3509 37.8870 37.0058
3.4807 2.7906 2.4573 2.9187
HAlS HSAl HSAl-b TS
641.2211 641.2082 641.2288 641.1620
3.9797 4.0451 3.9091 5.0250
1.9196 2.0449 1.4828 0.6153
49.5222 59.0811 57.4628 59.3694
3.0560 2.1646 2.0521 2.6626
HCN HNC TS
93.4280 93.4042 93.3537
4.8147 4.1246 5.1520
3.9998 3.3780 2.7141
15.1760 16.7190 16.8262
3.9613 3.0982 3.1786
HCO+ HOC+ TS
113.5587 113.4933 113.4414
16.1262 14.2266 16.0587
4.8839 3.2211 2.5770
9.3558 11.1337 11.0890
3.6833 2.9874 3.0337
HCS+ HSC+ HSC+-b TS
436.5238 436.3718 436.4052 436.4047
13.7916 13.9344 13.5910 13.7747
2.8127 2.4655 1.7061 1.4789
20.9109 25.9063 22.2828 22.0389
3.5239 3.4673 2.9701 3.0607
HSiO+ HOSi+ TS
364.9420 365.0415 364.9007
13.2110 12.4440 14.3534
2.3624 2.9852 1.1578
22.1186 22.9385 24.1457
3.2088 2.0595 2.8132
HSiS+ HSSi+ TS
687.9373 687.9620 687.8896
11.8742 12.0647 12.9686
2.1362 1.5811 0.6270
36.8318 38.6685 42.1773
3.2213 2.5073 2.7869
The species HYX-b refers to bent configuration and TS refers to the transition state.
Hati and Datta32 is that in exchange reactions of this type, the average hardness of the products is higher than that of the reactants and the species of largest hardness is found to lie on the product side. Thus the present observation in terms of minimum polarizability (actually R1/3) is essentially an analogue of the existing principle of maximum hardness. It may be noted that due to lack of availability of experimental values of polarizability, the examples considered here have been restricted to exchange reactions involving simple organic molecules. We have also considered a number of dissociation reactions of the type M2 ) 2M, where M represents H, Li, Na, K, Rb, and Cs and also a few reactions of the type MM′ ) M + M′. For convenience, we consider the reverse reactions, viz., M + M′ ) MM′ with negative values of ∆H, and observe that all these reactions are associated with a decrease in the cube root of polarizability, i.e., RM1/3 + RM′1/3 > RMM′1/3. A similar relation is not found to be valid for the hardness of the reactant and product species. It may be noted that recently, complexation reactions involving hard-soft acids and bases have been studied34 using ab initio methods and for these reactions as well, the above type of relations involving hardness does not seem to exist. In view of the recent interest19 in HXY-HYX type (where X ) B, Al, Ga, C, Si, etc., and Y ) O, S, Se, Te, etc.) isomerization reactions, we have studied these reactions in some details. We have calculated the total energy, HOMO-LUMO gap, Mulliken population, dipole moment, Mayer bond order
matrix, and polarizability values for each isomer as well as its transition state by the density functional method described in section 2, and the results are reported in Table 1. The input geometries for all the species are taken from the recently reported values in the literature.35 From the results of Table 1 it is clear that, in general the most stable isomer is accompanied by relatively lower value of polarizability, although there are few exceptions. (Typical energy difference is of the order of 0.05 au ≈ 30 kcal/mol, while the corresponding difference in polarizability is typically 10 au.) For some HYX systems, a bent geometry which is reported is more stable than the corresponding linear one and is also found to be associated with lower value of polarizability (except for HOGa). The transition state which is least stable is also often associated with higher polarizability. The corresponding trend of higher hardness for higher stability is also observed in most of the examples considered. Although in general, higher stability indicates higher hardness and lower polarizability, there are exceptions in the trend of both these quantities. The ratio of hardness to polarizability (η/R) is found to be maximum in most of the cases of higher stability with relatively less number of exceptions. Another quantity (R∆) which is the product of polarizability (R) and the polarizability ansiotropy (∆) has been found to be minimum for the energetically most stable isomer except only one (HGaO). From the reported values of chemical potential in Table 1, it is clear that the quantity (-µ) is maximum for the transition state (with a few exception). This
12298 J. Phys. Chem., Vol. 100, No. 30, 1996 can be rationalized in terms of the I ()-HOMO) and A ()LUMO) values of the transition state (TS). The value of HOMO for the TS is only slightly different or comparable to HOMO for the isomers, while LUMO for TS is significantly lower than that of the isomers. As a result the value of η ) (LUMO - HOMO)/2 is small for the TS and -µ ) (LUMO + HOMO)/2 is consequently larger. Thus, η for TS is minimum but χ ()-µ) is maximum. This trend is however not an indicator for relative stability of the two isomers. We have also calculated the molecular valency, and the reported values in Table 1 reveal that it always decreases monotonically from the more stable isomer to the less stable isomer and does not show any extremum for the transition state. For the molecules considered here with the atoms X and Y belonging to the group as indicated above, the HXY species have less polarizability value than the corresponding HYX species irrespective of the energy. This can be rationalized from the bonding features of these two species and the polarizability or hardness of the terminal atoms. For HXY and HYX, one of the terminal atoms is different and is Y and X, respectively. Since atom Y is more hard and hence less polarizable than atom X, one can conclude that the calculated values of the polarizability of HXY and HYX essentially follow the trends of these terminal atoms. Thus, the nature of the terminal atom and also the binding characteristics as well as the geometrical arrangement play crucial role in determining the polarizability of a molecule. Therefore, a model using the parameters polarizability, hardness and molecular valency might be useful in predicting the relative stability of various species. 4. Concluding Remarks The present work has been concerned with a density functional calculation of polarizability for a number of molecular systems. The objective has been to investigate how the energy change during a chemical reaction is related to the change in polarizability or hardness of the molecular species involved. From a study of polarizability in exchange, dissociation, and isomerization types of reactions, it is observed that the condition of minimum polarizability can in general be associated with maximum hardness or energetically more stable situations. Alternatively, a suitable combination of these two parameters can also often lead to better prediction of stability. Acknowledgment. We are extremely grateful to Prof. Dennis R. Salahub for kindly providing us with his deMon density functional program. We also thank Dr. Emil I. Proynov for many helpful correspondence regarding the deMon program. It is a pleasure to thank T. G. Varadarajan and H. K. Sadhukhan for their kind interest and encouragement. References and Notes (1) Pauling, L., The Nature of the Chemical Bond, 3rd ed.; Cornell University Press: Ithaca, NY, 1960. (2) For recent reviews, see: Sen, K. D., Jorgensen, C. K., Eds.; ElectronegatiVity, Structure and Bonding; Springer-Verlag: Berlin, 1987; Vol. 66.
Ghanty and Ghosh (3) Pearson, R. G. Coord. Chem. ReV. 1990, 100, 403. For recent reviews, see: Sen, K. D., Ed. Chemical Hardness. Struct. Bonding 1993, 80. (4) Iczkowski, R. P.; Margrave, J. L. J. Am. Chem. Soc. 1961, 83, 3547. (5) Parr, R. G.; Pearson, R. G. J. Am. Chem. Soc. 1983, 105, 7512. (6) Vela, A.; Gazquez, J. L. J. Am. Chem. Soc. 1990, 112, 1490. (7) Ghanty, T. K.; Ghosh, S. K. J. Am. Chem. Soc. 1994, 116, 8801. (8) Ghanty, T. K.; Ghosh, S. K. J. Phys. Chem. 1994, 98, 9197. (9) Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University Press: New York, 1989. Parr, R. G.; Yang, W. Annu. ReV. Phys. Chem. 1995, 46, 701. (10) Politzer, P. J. Chem. Phys. 1987, 86, 1072. (11) Ghanty, T. K.; Ghosh, S. K. J. Phys. Chem. 1993, 97, 4951. (12) Pal, S.; Chandra, A. K. J. Phys. Chem. 1995, 99, 13865. Hati, S.; Datta, D. J. Phys. Chem. 1994, 98, 10451; 1995, 99, 10742. (13) See, for example: Cardenas-Jiron, G. I.; Toro-Labbe, A. J. Phys. Chem. 1995, 99, 12730 and references therein. For a dynamic generalization, see: Chattaraj, P. K.; Nath, S. Chem. Phys. Lett. 1994, 217, 342. (14) Parr, R. G.; Chattaraj, P. K. J. Am. Chem. Soc. 1991, 113, 1854. Pearson, R. G. Acc. Chem. Res. 1993, 26, 250. (15) Datta, D. J. Phys. Chem. 1992, 96, 2409. (16) Chattaraj, P. K.; Nath, S.; Sannigrahi, A. B. Chem. Phys. Lett. 1993, 212, 223. (17) Chattaraj, P. K.; Nath, S.; Sannigrahi, A. B. J. Phys. Chem. 1994, 98, 9143. (18) Chandra, A. K. J. Mol. Struct. (THEOCHEM) 1994, 312, 297. (19) Kar, T.; Scheiner, S. J. Phys. Chem. 1995, 99, 8121. (20) Roy, R. K.; Chandra, A. K.; Pal, S. J. Phys. Chem. 1994, 98, 10447. Roy, R. K.; Chandra, A. K.; Pal, S. J. Mol. Struct. (THEOCHEM) 1995, 331, 261. (21) Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. J. Chem. Phys. 1978, 68, 3801. (22) Berkowitz, M.; Ghosh, S. K.; Parr, R. G. J. Am. Chem. Soc. 1985, 107, 6811. Ghosh, S. K.; Berkowitz, M. J. Chem. Phys. 1985, 83, 2976. Ghosh, S. K. Chem. Phys. Lett. 1990, 172, 77. (23) Parr, R. G.; Yang, W. J. Am. Chem. Soc. 1984, 106, 4049. Fukui, K. Science 1982, 218, 747. (24) Ghanty, T. K.; Ghosh, S. K. J. Phys. Chem. 1991, 95, 6512. Ghanty, T. K.; Ghosh, S. K. J. Chem. Soc., Chem. Commun. 1992, 1502. Ghanty, T. K.; Ghosh, S. K. Inorg. Chem. 1992, 31, 1951. Ghanty, T. K.; Ghosh, S. K. J. Phys. Chem. 1994, 98, 1840. (25) Ghanty, T. K.; Ghosh, S. K. J. Am. Chem. Soc. 1994, 116, 3943. See also: Ghosh, S. K. Int. J. Quantum Chem. 1994, 49, 239. (26) Kohn, W.; Sham, L. J. Phys. ReV. 1965, 140, A1133. (27) Local Density Approximations in Quantum Chemistry and Solid State Physics; Dahl, J. P., Avery, J., Eds.; Plenum: New York, 1984. (28) St.-Amant, A.; Salahub, D. R. Chem. Phys. Lett. 1990, 169, 387. Salahub, D. R.; Fournier, R.; Mlynarski, P.; Papai, I.; St.-Amant, A.; Ushio, J. In Density Functional Methods in Chemistry; Labanowski, J. K., Andzelm, J. W., Eds.; Springer: New York, 1991. (29) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Phys. ReV. 1992, B46, 6671. (30) Kutzelnigg, W.; Fleischer, U.; Schindler, M. In NMR-Basic Principles and Progress; Springer-Verlag: Heidelberg, 1990; Vol. 23, p 165. (31) For values of polarizabilities of molecules see: (a) Miller, T. M. In CRC Handbook of Chemistry and Physics, 74th ed.; CRC Press: Boca Raton, FL, 1993-1994; pp 10-192. (b) No, K. T.; Cho, K. H.; John, M. S.; Scheraga, H. A. J. Am. Chem. Soc. 1993, 115, 2005. (32) Datta, D. Inorg. Chem. 1992, 31, 2797. Hati, S.; Datta, D. J. Org. Chem. 1992, 57, 6056. (33) Politzer, P.; Seminario, J. M. Trends Phys. Chem. 1992, 3, 175. (34) Chattarj, P. K.; Schleyer, P. v. R. J. Am. Chem. Soc. 1994, 116, 1067. (35) Talaty, E. R.; Huang, Y.; Zandler, M. E. J. Am. Chem. Soc. 1991, 113, 779. Nowek, A.; Leszczynski, J. J. Phys. Chem. 1994, 98, 13210 and references therein.
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