2307
J. Phys. Chem. 1995,99, 2307-2311
Potential-Energy Surfaces Concerning Deformation of Benzene Hiroshi Ichikawa* and Hirotaka Kagawa Hoshi College of Pharmacy, Shinagawa, Tokyo 142, Japan Received: August 19, 1994; In Final Form: November 22, 1994@
The potential-energy surfaces of the total, n,and a electrons are obtained concerning deformation of benzene: it is shown that there are countless numbers of reaction coordinates for distortion that either lower or increase the n energy. The minimum-energy path (MEP) was found for the distortion from D6h to D3h and was not one previously investigated. Along such an MEP, the role of the n electrons seems to be reversed; vibrational Le., the n electrons are most stable at the D6h symmetry. The reason for the low frequency of the Bz,, modes is given. The molecular geometry has been distorted to perturb the n-electronic structure in the study of the relationship between the n-electronic structure and the n energy, a component of the total energy. However, this method was shown to be inappropriate, since even if the changes of the geometry and total energy are small, those of partitioned energies can enormously fluctuate due to the energy-relationship given by the molecular virial theorem.
The n electrons in benzene have long been believed, by both organic and theoretical chemists, to play a decisive role in determining the characteristics of benzene: Particular stability with uniform delocalization of the cyclic 6n electron system has been regarded as common knowledge and used to explain the properties of various aromatic compounds.' However, most recent quantum-chemical investigations concluded that the delocalized x system of benzene is not stable in the D6h geometry (Scheme 1). This creates considerable confusion among organic chemists. Homig f i s t pointed out the stable KekulC structure to explain the abnormally low force constant concerning the Bz,, vibrations: the potential energy toward the KekulC structure changes slowly, giving rise to an abnormally low force constant.z In 1961, Berry raised a doubt conceming the stability of the uniformly distributed n electrons. Namely, the n electrons are not the major source of stability for the regular hexagon, and the x electrons might well have a lower energy if the ring approaches cyclohexatriene-like shapes; i.e., bond alternation caused by the distortion might lower the n-electron energy in ben~ene.~ More than twenty years later, this problem was examined by Shaik and Hiberty and their c o - ~ o r k e r swho , ~ investigated the changes of x and 0 energies when the D6h benzene is deformed to D3h. (We call the analysis of components of the total energy 'energy-component analysis'.) The result of this study indicated that electronic delocalization of n electrons is a byproduct of the a-imposed geometric symmetry and not a driving force by itself. This seems to be supported by several other We thought that such a result was dubious, since none of the previous workers considered the role of the third term of the molecular vinal theorem.* The theorem is written as
where (T), E, and R are the expectation value of the kinetic energy, the total energy, and the internuclear distance and where the sum runs over all internuclear distances. If the geometry of the system is optimized concerning all nuclear centers, the third term (which is called the 'virial term') tums out to be null @
Abstract published in Advance ACS Abstracts, February 1, 1995.
SCHEME 1: Geometry Distortion of the D M Benzene
D3h
D6h
D3h
(since aE/aR = 0 holds for any of the nuclear centers) to give the simple relationship, (T) -I- E = 0 or equivalently (7) = -(V)/ 2, where (V) is the expectation value of the potential energy. (The bracket indicates an expectation value.) Therefore, due to the virial term of eq 1, the imbalance between (T) and (V) appears when the geometry of the system is not optimized. The problem emerges when the optimized Dah benzene is distorted to D3h. Then how does the virial term disturb the result? Actually the deviations of the partitioned energies were found to be from lo2to lo3times more than that of the total energy.'.l0 Because the n-electron energy includes only parts of (7) and (V), the role of the n electrons depends on the change of the virial term that is totally dependent on how to select the reaction coordinate when the D6h benzene is deformed to D3h. The dependency is clearly demonstrated by Baird" and very recently by Glendening et al.,'* where the previous conclusion is reversed. The present paper describes the theoretical reason for coordinate dependency and the region of reaction coordinates that lowers or increases the n energy in distortion. It also points out that one may not obtain meaningful conclusions in the study of the relationship between electronic structure and partitioned energies if the molecular geometry is distorted to perturb the electronic structure.
Theory (a) Theoretical Evidence of the Influence of the Virial Term in the Distorted Geometry. The total energy ( E ) consists of the kinetic ( r ) and potential energies (V). The relationship, V = -2T, holds in a stationary state. This is reproduced as (V) = -2(T) in the Hartree-Fock theory only when the basis set is complete. Otherwise, the vinal ratio deviates from this simple ratio. Because the T and V operators are of degree -2 and -1 in the coordinates of electrons, by applying a universal scale factor (5) to the wave function, one can obtain the simple virial ratio.1° On the other hand, the virial
0022-3654/95/2099-2307$09.00/0 0 1995 American Chemical Society
Ichikawa and Kagawa
2308 J. Phys. Chem., Vol. 99, No. 8, 1995 term for the unoptimized geometry can also be made zero by applying 5 to the wave function simply because of the degrees - 2 and - 1 for the T and V operators: Through a single 5, the simple virial ratio can be fulfilled for the wave functions of both the incomplete basis set and the unoptimized geometry. Pedersen and Morokuma studied how much the total energy and its components deviate by ~ c a l i n g .The ~ deviations of (q, (v), and E can be expanded as a power series of the scale error Y (=5 - 11,
+ ...) 6 ( 0 = E(-2y + 3 y 2 + ...) d(V) = E(2y - 2y2 + ...) 6 E = E(y2
(2)
(3)
E, = E',
E,' = E:
f
E',
+ E, iE," i-E,-,
+ E," + E: + E,-,
4-?!J
E E"
ET EV E' E," E', EaT Eav Ed 2Ea-x
(4)
Those equations show that unless 5 is unity, partitioned energies deviate in the first-order of the scale error, while the total energy does in the second-order. Namely, even if the change of the total energy is small, those of the partitioned energies can be enormous. This equally takes place when the total energy is partitioned into E, and E, energies in any f o m . (b) Energy Partitioning of the Total Energy. To investigate the role of the x electrons, the total energy must be properly partitioned into the n and 0 portions. We grouped fundamental energy-terms as13
E = E,
TABLE 1: Changes of Energies by Scaling in Benzene" unscaled scaledb AE vinal ratio 1.999 185 2.000 000
(7)
where, ET,E", E.', and EN are the kinetic, one-electron potential, two-electron potential, and internuclear repulsion energies while Ex-, and E,-x are averaged interactions between n and 0 electrons and are the same quantity. The detailed descriptions of the terms in eqs 6 and 7 are shown e1~ewhere.l~ We included EN in the 0 energy as a part of the skeleton of the system. The reason is as follows. It may be a controversial matter how to partition the total energy into x and u energies. The problem concerns the internuclear repulsion energy (EN). In spite of many discussions concerning this bridging organic and quantum chemistries, one must refer to the idea of such n-electron theories as Hiickel molecular orbital (HMO)and Pariser-Parr-Pople (PPP) theories,14 which have been most widely used in organic chemistry. In those theories, n electrons are considered to move in the average field created by u electrons as well as nuclear charges. To reconstruct this idea using fundamental energies, it may be most appropriate to attribute EN to 'u energy'. Hereafter, we call such a u energy 'the skeletal energy' (E;), since it is not an electronic energy any more. The partitioned energies are sensitive to the density matrix;15 the SCF convergence for the density matrix is set to be 1.0 x 10-lo. The calculation has been carried out on an IBM RS/ 6000-550 computer using the GAUSSIAN-90 package,16 to which we added subroutines for energy-partitioning.
Results and Discussion (a) Influence of the Universal Scale Factor on Partitioned Energies. The energy-component analysis is based on the change in the third term (the vinal term) of eq 1, since it gives an imbalance between (7') and (V). The simple virial ratio, (u>/ (0, is exactly - 2 , when both conditions that the geometry is optimized with respect to any nuclear coordinate and that the
-230.663 035' 204.746 145 230.851 243 -946.375 857 280.115 433 5.887 642 -80.407 300 4.020 514 224.963 601 -865.968 557 211.984 124 64.110 795
6-31 1G. The scale factor
-230.663 757' 204.714 273 230.663 757 -946.102 710 280.060 923 5.883 156 -80.378 861 4.018 273 224.780 601 -865.723 849 211.950 995 64.091 655
-1.9d -83.7 -492.2 717.1 -143.1 -11.8 74.7 -5.9 -480.5 642.5 -87.0 -50.3
(5) was found to be 0.992 282.
In au.
In Id/mol.
TABLE 2: Changes of Energies toward Expansion along AI. in Benzene AE (A$) E EN
ET EV E' ExT E," E,' E,* E," E,' 2E,-,
-230.663 035'(-230.663 757") 204.746 145 (204.714 273) 230.85 1 243 (230.663 757) -946.375 857 (-946.102 710) 280.1 15 433 (280.060 923) 5.887 642 (5.883 156) -80.407 300 (-80.378 861) 4.020 514 (4.018 273) 224.963 601 (224.780 601) -865.968 557 (-865.723 849) 211.984 124 (211.950 995) 64.1 10 795 (64.091 655)
0.w (0,Ode) -327.4 (-327.4) -43.3 (-43.3) 692.6 (692.6) -321.9 (-321.9)
-6.1 (-6.1) 101.6 (101.7) -5.7 (-5.7) -37.2 (-37.2) 591.0 (590.9) -236.3 (-236.2) -79.9 (-80.0)
6-31 1G. A r = 0.001 A. In au. In kJ/mol. e Scale factor (5) = 0.992282.
adopted basis set is complete are satisfied. As discussed in the former section, the simple vinal ratio can be met by applying a universal scale factor, 5, to the wave function of both an incomplete basis set and an unoptimized geometry. (Here, we call the incomplete but so scaled wave function that satisfies the simple virial ratio 'the scaled wave function'.) Since the energy-component analysis is used to analyze the changes of energy components caused only by geometry distortion, the influence of 5 on the unscaled wave function must be examined. We performed optimizations of both scale factor and geometry in benzene. The basis set is determined to use 6-311G,17 since incorporation of polarization functions makes 0-3 separation difficult. The results are shown in Table 1. As stated in the introduction, the total energy does not change much by scaling. However, its components exhibit tremendous amounts of deviation although the change of the scale factor from 1 seems to be small. If those components are gathered into the ,z (Ex)and skeletal energies (E;) according to Eqs 6 and 7, they are figured out to be 31.8 and -33.9 kl/mol, respectively. These deviations are 20 times more than that of the total energy. The wave function with such a usual basis set as 6-311G does not give the simple virial ratio by itself. Besides, obtaining the appropriate scale factor is time-consuming and impractical. Therefore, we must examine whether or not the unscaled wave function given by 6-3 11G is appropriate enough to discuss the deviations of energy components by geometry distortion. To this end, we have calculated the differences of energy components between two geometries. Table 2 shows the results when the C-C bond lengths in the optimized D6h benzene are simply increased by 0.001 A keeping the same symmetry.
J. Phys. Chem., Vol. 99, No. 8, 1995 2309
Deformation of Benzene from &h to D3h In the table, values in parentheses are those for the scaled wave function. If one compares the differences with those in parentheses, one may understand that they are essentially the same. Namely, one can use the unscaled wave function as far as the diflerences of energy components are considered. This has been found to be true in other n systems.'* (b) Three-Dimensional Potential Energy Surfaces. Let one C-C bond length of the D 3 h benzene be R1 and the other be R2. We adopted the orthogonal R1-R2 coordinate system. At the point (ro, ro) the geometry (&) gives the minimum total energy of benzene. We set the origin of the coordinate at this point and placed the grid with an interval of 0.002 8, (see Figure 1). It consists of 15 lines for both R1 and R2. Then we performed the calculation for 225 (=15 x 15) points to obtain the potential energy surfaces of the total energy and its components. The bond lengths of C-H were optimized at each point. The results are shown in Figure 1. Concerning stretching vibrations, there are two independent modes in benzene: (1) the AI, mode that keeps the &h symmetry and (2) the B2,, mode (see Scheme 2). The B2,, mode involves concurrent stretching and shrinking of the C-C bond by the same quantity. Two normal modes, AI, and B2u, are, of course, orthogonal in this coordinate system and are marked in the figure. Figure 1A depicts a three-dimensional map of the relationship between the total energies, R1 and R2. The surfaces of the total energy along both AI, and B2u are concave. This indicates that any geometry distortion from ro increases the total energy. However, the changes along B2,, are small in accord with the fact that the vibrational frequency of this mode was observed to be low. Parts B and C of Figure 1 show the changes of E, and E,' on the same coordinate system. Both n and skeletal energies show sloped flat planes, indicating that the E, term increases while the E,' term decreases toward expansion along AI, and that both terms are invariant along B2,,. In fact, the E, term shows a very slow convex curve while E,' shows a concave curve with an extreme point at coordinate (ro, ro) along B2u; they are too small to emerge in the figures. One may understand that energy change along B2,, is very small if compared to that of AI, and that a small deviation from the pure B2" coordinate causes a large amount of deviation in E, and E/ (the scales in parts B and C of Figure 1 are 500 times larger than that of part A. Concerning the low frequency of the B2,, mode, both n and skeletal energies are invariant against such a deformation. This is, however, against the speculation by Berry, who insisted that the n electrons might well have a low energy if the ring approaches a cyclohexatriene-like shape.3 Here it should be noted that any coordinates on the left side of the B2,, line lower and those on the other side increase E, by distortion (see Figure 1B). Namely, there is a countless number of reaction coordinates that lower or increase the n energy in the D6h D3h distortion of benzene. A reaction coordinate is governed by the total energy. If a reasonable reaction coordinate is determined, then the role of the n energy is automatically identified. However, the coordinate on the Bz,, line seems not to give any significant answer because of insusceptibility. (c) Energy Change along the AI, and Bzu Coordinates. To investigate the energetic behavior more closely, fine changes of E, E,, and E,' against the AI, coordinates (Le., those of the D6h benzene as functions of the C-C bond length (R) around the optimized bond length (ro)) are drawn in Figure 2. The E,' term linearly decreases as R increases while the E, term increases. This simply indicates that the n system tends to shrink whereas the 0 skeleton wants to expand, resulting in
/
R2
B: n energy (E,)
C: skeletal energy (E:)
-191.5
-
Figure 1. Potential energy surfaces of the total energy, En,and E,' around the equilibrium bond length (ro = 1.387 775 A): (A) total energy; (B) n energy (En);(C) skeletal energy (E,'). The distance between adjacent lattices is 0.002 A.
the compromised bond length at ro. Namely, the n system has a tendency to bond more closely than ro against that of the skeleton. Both the n and skeletal energies are not at the minimum or maximum at ro. Besides, a change of 0.001 8, in bond length could cause as much as 49.9 and -49.9 kJ/mol of deviation in both E,' and E, in spite of a negligible change of the total energy (ca. 0.0 kJ/mol). These facts immediately
Ichikawa and Kagawa
2310 J. Phys. Chem., Vol. 99, No. 8, 1995 SCHEME 2: Normal Stretching Mode in Benzene" k
TABLE 3: Adopted Points along the Minimum-Energy Path (MEP)" step R1 R? a
1.387775 1.387775
In
1.388775 1.387393
1.389775 1.387012
1.390775 1.386632
A.
The AI, mode keeps the D6h symmetry, while the Bzumode involves equal amounts of stretching and shrinking at the same time.
-38.4437 - 1 92.2189
-230.66299-
-230.66292
I\
- -38.4438
-
-38.3
-
-230.66300-
2 e
-230.66294-
u P
@
- 5i
-
- 4 "
-
i
P
i
-
d
5 c,
W
-38.4
*
m
-
-
h
i d -
i
i=
-
z
m
-m
4"
- -38.4440
W
VI 3
-192.2192 -230.66303-
x
- -38.4441
-38.5 -230.66304
-230.66302-
,
,
,
0
,
+I
,
,
+2
,
I
+3
I
, +4
7
i
- 192.2193
+5
Figure 3. Changes of E , E,, and E,' as functions of distortion from ro I -38.6 -4
-2
ro
+2
along the Bzu coordinate. The basis set is 6-31 1G.
+4
A) Figure 2. Changes of E , Ex, and E,' as functions of distortion from ro along the A,, coordinate. The basis set is 6-31 1G. Reaction coordinate (difference for ro in
suggest that, according to the way of selecting the reaction coordinate, one may obtain various results. Drawings of E, Ex, and E,' against the B2" coordinate seem to reproduce the results of the previous work^,^^-^ although the definitions for the o and n energies are not the same as ours. As the D 6 h benzene is distorted to D3h, the n energy decreases while the total and o skeletal energies increase. As far as one looks at this result, one may reach the conclusion that the n system of benzene is unstable in a regular hexagon and is more stable in an alternated one. However, one should understand that the energy changes in the analysis are small and that such a conclusion is fragile if one considers the following facts: The Bz,, coordinate involves an equal amount of distortion concerning both bond expansion and bond shrinkage and such a reaction path never gives cyclohexatriene, since the differences of bond length between ro and a single bond and between ro and a double bond are not equal. Besides, forces for bond shnnkage and bond expansion are not equal even at a very small distortion. (d) Energy Changes along the Minimum-Energy Path (MEP). It is most probable that molecular distortion takes place keeping its total energy as low as possible. Thus, we considered the minimum-energy path (MEP) that leads the D6h benzene to a D3h symmetry keeping the minimum total energy, which is shown as the MEP in Figure 1. Any other paths are higher in energy than this. The actual procedure used to carry out this calculation is as follows. Only R1 is determined as ro 0.001, ro 0.002 A, and so on. Then, other intemal coordinates, R2 and the CH bond lengths, are allowed to be optimized at each R1. By this procedure, the calculation is carried out along the
+
e!
m
W
-230.66300-
W
5
5
i=
1
A
r;
-230.66302-
5
W
-u
-192.2190
W
v)
W
5:
-230.66304
1.391775 1.386252
+
E
Deformation of Benzene from D6h to
J. Phys. Chem., Vol. 99, No. 8, 1995 2311
D3h
those along the B2, coordinate (-0.057 and 0.065 hartreelA), although the average of AElAr on the MEP (0.0070 hartree/A) is smaller than that of Bz,, (0.0077 hartreel&. The above results show how easily the role of a partitioned energy is changed according to the way of selecting the reaction coordinate. This is not the result of the adopted basis set but that of a profound quantum mechanical reason, i.e. the molecular virial theorem. (e) W h y Energies Are Invariant along the Bzu Coordinate? The fact that the total as well as partitioned energies are rather invariant along the Bzu coordinate seems to mislead us in the understanding of the role of the x energy in benzene. Here, we consider the question why energies are so along Bz,. The answer is simple: As Figure 2 shows, E,' and En (in fact, any partitioned energies) around ro are almost linear with respect to a small change of R. The Bz,, vibration modes involve an equal amount of expansion and shrinkage of the carbon-carbon bond at the same time. Therefore, the energy changes caused by expansion are almost canceled by those caused by shrinking. We are handling the very small differences from such cancellation. Such a study does not seem to give any meaningful conclusion.
Concluding Remarks Analysis of the energy components in a chemical phenomenon is a fascinating way to know what happens inside the system. The first approach along this line was demonstrated by Ruedenberg et al. ca. 30 years ago.lg That success prompted a number of authors to pursue similar r e s e a r ~ h . ~However, ,~~ this approach was confronted with a number of difficulties: the values of energy components sensitively fluctuate depending on the molecular geometry, the adopted basis set, and/or the scale factor of the wave function. They were technically unsolvable problems at that time, and energy-component analysis was abandoned for decades. Recent development of computational technologies, however, seems to provide sufficient means to restart energy-component analysis. As a matter of fact, a number of papers have appeared concerning energycomponent analyses for the Jahn-Teller Hund's rule,22 aromaticitie~,~~ c ~ n j u g a t i o n and , ~ ~ the steric effect.25 We strongly recommend energy-component analysis for the problems in organic chemistry. This may lead us to a fundamental understanding of a phenomenon. However, it has a pitfall. The discussion of where and how energy-component analysis can be applied has been stated elsewhere.26 This paper demonstrates how the conclusions obtained by energy-component analysis are fragile if the method is carelessly applied. This paper shows that the x electron energy in benzene is destabilized when the optimized D6h benzene is distorted along the MEP to a D3h structure. This is true as long as the distortion takes place along the MEP. However, we do not insist that the n energy is most stable at the D6h structure only because of this reason. For we are not sure whether or not the actual reaction coordinate is the MEP. What we want to stress is that it is not a right way to distort the molecular geometry in order to perturb the electronic structure. We need a direct method
that can correlate the electronic structure and the total and/or x energies. Concerning this, a new method called 'the constrained Hartree-Fock method' has been presented,'* where the total energy and its components can be obtained as functions of the predetermined electronic structure.
References and Notes (1) For review: (a) In Aromaticity, Pseudo-Aromaticity, Anti-Aromaticify; Bergmann, E. D., Pullmann,B., Eds.;The Israel Academy of Sciences and Humanities: Jerusalem, 1971. (b) Garrat, P. J. Aromaticiiy; John Wiley & Sons: New York, 1986. (c) Robinson, R. Tetrahedron 1958, 3, 323. (2) Honig, D. F. J. Am. Chem. SOC.1950, 72, 5774. (3) Berry, R. S. J. Chem. Phys. 1961, 35, 2253. (4) (a) Shaik, S. S.; Bar, R. Nouv. J. Chim. 1984, 8,411. Shaik, S. S.; Hiberty, P. C. J. Am. Chem. SOC. 1985,107,3089. (b) Hiberty, P. C.; Shaik, S. S.; Lefour, J. M.; Ohanessian, G. J. J. Org. Chem. 1985, 50, 4657. (c) Shaik, S. S.; Hiberty, P. C.; Lefour, J. M.; Ohanessian, G. J. Am. Chem. SOC.1987,109,363. (d) Shaik, S. S.; Hiberty, P. C.; Ohanessian, G.; Lefour, J. M. J. Phys. Chem. 1980.92, 5086. (e) Hiberty, P. C. Top. Curr. Chem. 1990, 153, 27. ( 5 ) Jug, K.; Koster, A. J. Am. Chem. SOC.1990, 112, 6772. (6) Janoschek, R. Theochem 1991, 229, 197. (7) Nakajima, T.; Kataoka, M. Theor. Chim. Acta 1992, 84, 27. (8) Parr, R. G.; Brown, J. E. J. Chem. Phys. 1968, 49,4849. (9) Pedersen, L.; Morokuma, K. J. Chem. Phys. 1967, 46, 3491. (10) Magnoli, D. E.; Murdoch, J. R. Int. J. Quantum Chem. 1982, 22, 1249. (11) Baird, N. C. J. Org. Chem. 1986, 51, 3907. See also: Aihara, J. Bull. Chem. SOC. Jpn. 1990, 63, 1956. (12) Glendening, E. D.; Faust, R.; Streitwieser, A.; Vollhardt, K. P. C.; Weinhold, F. J. Am. Chem. SOC. 1993, 115, 10952. (13) Ichikawa, H.; Ebisawa, Y. J. Am. Chem. SOC. 1985, 107, 1161. Ichikawa, H.; Sameshima, K. Bull. Chem. SOC.Jpn. 1985, 93, 3248. (14) (a) Streitwieser,A. Molecular Orbital Theoryfor Organic Chemists; John Wiley & Sons, Inc.: New York, 1959. (b) Coulson, C. A.; O'Leary, B.; Mallion, R. B. Hiickel Theory for Organic Chemists; Academic Press, Inc.: London, 1978. (c) Pariser, R.; Parr, G. R. J. Chem. Phys. 1953, 21, 466; 1953, 21, 767. (d) Pople, J. A. Trans. Faraday SOC.1953, 49, 1375. (15) Ichikawa, H.; Shigihara, A. Bull. Chem. SOC. Jpn. 1988,61, 1837. (16) Frisch, M. J.; Gordon, M. H.; Schlegel, H. B.; Raghavachari, K.; Binkley, J. S.; Gonzalez, C.; Defree, D. J.; Fox,D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R.; Kahn, L. R.; Stewart, J. J. P.; Nuder, E. M.; Topiol, S.; Pople, J. A. GA USSZAN90; Gaussian, Inc.: Pittsburgh, PA, 1990. (17) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. J. Chem. Phys. 1980, 72, 650. (18) Ichikawa, H.; Kagawa, H. Int. J. Quantum Chem. 1994, 52, 575. (19) (a) Ruedenberg, K. Rev. Mod. Phys. 1962, 34, 326. (b) Feinberg, M. J.; Ruedenberg, K.; Mehler, E. L. In Advances in Quantum Chemistry; Lowdin, P. O., Ed.; Academic Press: New York, 1970; Vol. 5, p 27. (c) Feinberg, M. J.; Ruedenberg, K. J. Chem. Phys. 1971, 54, 1495. (20) (a) Pitzer, R. M.; Lipscomb, W. J. Chem. Phys. 1963, 39, 1995. (b) Bader, R. F. W.; Beddall, P. M.; Peslak, J., Jr. J. Chem. Phys. 1973, 58, 557. (c) Parr, R. G.; Gadre, S. R. J. Chem. Phys. 1980, 72, 3669. (21) Boyd, R. J.; Darvesh, K. V.; Fricker, P. D. J. Chem. Phys. 1991, 94, 6677. Wang, J.; Boyd, R. J. J. Chem. Phys. 1992, 96, 1232. (22) Colpa, J. P.; Brown, R. E. J. Chem. Phys. 1978, 68, 4248. Boyd, R. J. Nature 1984, 320, 480. Reynolds, P. J.; Dupis, M.; Lester, W. A., Jr. J. Chem. Phys. 1985, 82, 1983. (23) Ichikawa, H.; Ebisawa, Y. J. Am. Chem. SOC. 1985, 107, 1161. Ichikawa, H.; Aihara, J.; Dlihne, S. Bull. Chem. SOC. Jpn. 1989, 62, 2798. (24) Ichikawa, H.; Ebisawa, Y.; Sameshima, K. Bull. Chem. SOC.Jpn. 1988, 61, 659. Ichikawa, H.; Sameshima, K. J. Phys. Org. Chem. 1990.3, 587. (25) Tokiwa, H.; Ichikawa, H. Znt. J. Quantum Chem. 1994, 50, 109. (26) Tokiwa, H.; Osamura, Y.; Ichikawa, H. Chem. Phys. 1994, 181, 97.
JP942233G
