15068
J. Phys. Chem. 1996, 100, 15068-15072
Effective Valence Bond Model Study on Conjugated Hydrocarbons Containing Four-Membered Rings Jing Ma, Shuhua Li, and Yuansheng Jiang* Department of Chemistry, Nanjing UniVersity, Nanjing, 210093, P. R. China ReceiVed: March 15, 1996; In Final Form: June 3, 1996X
By employing the Lanczos method, the effective valence bond model (EVB) is exactly solved for a series of conjugated hydrocarbons with four-membered rings. Several indices calculated from the EVB ground-state energies and wave functions are applied to interrelate with bond lengths, stabilities, and reactivities of these systems. Our results agree well with related experimental observations and deductions of molecular orbital theory. The failure of the classical valence bond theory in these systems is also exposed.
I. Introduction It is well-known that molecular orbital (MO) theory plays a predominant role in understanding some physical and chemical properties of conjugated molecules;1-3 however, chemists also have never lost their confidence in studying these compounds from valence bond (VB) viewpoints.4-10 Recently, it has been demonstrated that exact results of the classical VB (CVB) theory can successfully interpret many ground-state properties of benzenoid hydrocarbons.9,10 These achievements may clarify some doubtfulness on the quantitative capability of the CVB theory, although much qualitative chemical thinking brought out from this theory has been extensively utilized by chemists until now. However, the extension of the CVB theory to nonbenzenoids, especially the four-membered ring systems, is properly desired to be investigated at first owing to the wellknown failure of this theory in cyclobutadiene.11 Kuwajima7 has made a successful attempt toward accounting for the antiaromaticity of systems composed of even-membered systems by using the spin Hamiltonian, in which a ring permutation term is added to the CVB Hamiltonian. Even so, a more powerful effective VB (EVB) model proposed by Malrieu and Maynau,6 which is derived from the Hubbard Hamiltonian by the application of the quasi-degenerate many-body perturbation theory, is expected to bring to light more striking prospects for avoiding the failure of the CVB theory in systems containing 4n-membered rings. Although the second-order EVB model is essentially equivalent to the CVB theory, the inclusion of fourth- and sixth-order corrections makes this model appropriate for more accurate treatments on various conjugated systems. In this paper, we primarily focus on the application of this model to a class of compounds with four-membered rings. Since the sixth-order corrections play a relatively minor role in these compounds,6 we limit our EVB model to fourth-order corrections for simplicity. On the basis of the exact solution of this model, we’ll interpret certain physical and chemical properties, i.e., bond lengths, stability, and reactivity, for more than a dozen of molecules with up to 20 π electrons. A comparison with related experimental facts and those predictions from MO approach will verify the predictive power of the EVB model in treating this class of compounds. In addition, the CVB theory will be proven to break down in these systems with fourmembered rings as in cyclobutadiene.11 The organization of this paper is as follows. In section II, we introduce the EVB Hamiltonian and provide a brief discussion on the computational method. In section III, we X
Abstract published in AdVance ACS Abstracts, August 1, 1996.
S0022-3654(96)00802-7 CCC: $12.00
employ the ground-state wave functions and energies calculated with the computational method described in section II to correlate with bond lengths, stability, and reactivity for a set of compounds involving four-membered rings. II. Methodology The Effective Valence Bond (EVB) Hamiltonian. By starting from the Hubbard Hamiltonian, Malrieu and Maynau6 have derived the effective valence bond (EVB) Hamiltonian including various order corrections via the quasi-degenerate many-body perturbation theory. The second-order form is essentially the well-known Pauling-Wheland VB model, being equivalent to the Heisenberg model of solid-state theory, as shown in eq 1,12 where Si is the spin operator for site i, J is an
H(2) ) J∑(2SiSj - 1/2)
(1)
i-j
(positive) exchange parameter, and i-j denotes nearest-neighbor sites. For alternant hydrocarbons discussed in this paper, there are no third-order terms.13 To quantitatively evaluate the electronic properties of conjugated systems, especially those containing four-membered rings, it has been emphasized that the fourth-order correction plays an important role and should be taken into account.6 Some computations show that in the absence of squares the fourth-order correction can be written in the spin operator form as follows:13
H(4a) ) J′∑[4bij(4SiSj - 1) - cij(4SiSj - 1)]
(2)
i,j
where
bij )
1
if the ith and jth atoms are nearest neighbors
0
otherwise
and cij is the number of atoms which are bonded by both ith and jth atoms. While in the presence of four-membered rings, another term should be included13
H(4a) ) J′
∑ dijkl{-10[(SiSj)(SkSl) + (SiSl)(SjSk) -
i,j,k,l
(SiSk)(SjSl)] + 1/2[SiSj + SjSl + SiSl + SiSk + SjSk + SlSk] 1 /8} (3) where © 1996 American Chemical Society
Effective Valence Bond Model Study
dijkl )
1
if bijbjkbklbil ) 1
0
otherwise
J. Phys. Chem., Vol. 100, No. 37, 1996 15069 TABLE 1: Experimental and Calculated Bond Lengths (Å) of trans-Stilbene
e
In the above three equations, the values of parameters J and J′ have been given by Malrieu and Maynau6 through a theoretical estimate and empirical fit. Obviously, the EVB model consisting of both the second-order and fourth-order terms keeps the topological essence because it assumes that all bonds are identical. Despite that this model has a more complicated form than the CVB model, it also acts on the space of covalent VB structures, which exponential increases in size with the number of atoms. Even if one solves this EVB model in the subspace in which S (or Sz) is conserved by application of spin symmetry, a powerful algorithm for the diagonalization of large matrices is still desirable. Our recent works10,14 have demonstrated that the Lanczos method is applicable to this problem. We’ll give some comments on this method in the following. The Computational Method. The details of the Lanczos method have been introduced elsewhere.15 The central idea of this method is to transform a general symmetric matrix to a tridiagonal matrix by exerting a specially chosen transformation on basis functions. After an initial state is selected, the recursion procedure will yield a tridiagonal matrix. Due to the special characteristics of produced basis sets,16 a few recursion steps are enough to give a good approximation to the ground state of VB models as accurate as possible. In our previous treatments,10,14 the lowest energy determinants in various Sz subspaces were taken as starting states to obtain the low-lying states in these subspaces. It should be noted that this selection is not unique, the group properties17 of this method also allow one to choose the initial state to be the Kekule´ structure with the maximum number of double bonds in a certain S subspace. In fact, this selection will lead to a faster convergence, so it will be adopted in subsequent calculations. Comparing with other techniques,8,9 the Lanczos method seems simpler and requires less storage and computational time.
CVB
EVB
bond
Pijs
calcd
Pijs
calcd
exptla
a b c d e f g h
0.865 0.508 0.677 0.726 0.711 0.711 0.726 0.677
1.337 1.470 1.407 1.389 1.394 1.394 1.389 1.407
0.875 0.499 0.678 0.727 0.710 0.710 0.727 0.678
1.337 1.471 1.407 1.389 1.395 1.395 1.389 1.407
1.338 1.473 1.406 1.393 1.393 1.391 1.390 1.402
a
Values are from ref 20.
state Pijs within the CVB theory has been established to account for bond lengths of benzenoid hydrocarbons. Similarly, we also resort to this index calculated in the EVB model for measuring the strength of the i-j bond and correlating with the i-j bond length in parallel with the CVB model. Using the EVB groundstate wave functions, we have calculated the ground-state Pijs on various bonds for a series of benzocyclobutadienoid hydrocarbons. To predict the C-C bond lengths in these molecules from the viewpoint of the EVB model, we must first establish an appropriate ground-state singlet probability-bond length formula as done in the MO approach1 and the CVB model.10 Considering that the characteristic phenomenon of short-longbond alternation is found in cyclobutadiene derivatives,19 nearly planar trans-stilbene with considerable variation of bond lengths from 1.338 to 1.473 Å is suitable for deducing this relation. In this way, the ground-state Pijs calculated in both CVB and EVB models and experimental bond lengths for this molecule are shown in Table 1 which are fitted to induce two well-behaved linear formulas as follows:
In the CVB model, III. Results and Discussions Using the EVB model and computational method described above, we have calculated the EVB ground-state energies for a series of molecules containing four-membered rings with up to 20 π electrons. In the following we will interrelate these results with certain physical and chemical properties of these compounds. Bond Lengths. The interrelations of bond lengths and theoretical bond orders in benzenoid hydrocarbons are well established within the framework of both MO1 and CVB theories.10,18 However, for other conjugated systems especially with four-membered rings, the bond order-bond length correlation is assumed to be trivial due to the claim that the CVB theory may be unsuitable in these systems.11 Also, it was recognized that the Pauling bond order-bond length correlation in the empirical version of the CVB theory suffers from the drawback that unlikely bond lengths from 0.839 to 1.964 Å are predicted by unusual large negative and positive Pauling bond orders in these systems.4d Here we try to find a well-behaved bond order-bond length relation in a series of benzocyclobutadienoid hydrocarbons based on EVB calculations and compare the predicted bond lengths with experimental values together with those from MO theory. By defining the index Pijs,18 i.e., the probability of finding a singlet arrangement between atoms i and j, a good linear relationship between bond length and the corresponding ground-
dij (Å) ) 1.661 - 0.375Pijs
(4)
Whereas in the EVB model, dij (Å) ) 1.649 - 0.357Pijs
(5)
Where dij in both formulas denotes the length of i-j bond. Inversely, all bond lengths of stilbene calculated from eqs 4 and 5 are also collected in Table 1 for comparison. Clearly, there exist good linear relationships between the bond length and the corresponding ground-state Pijs with similar small average deviations (0.002 Å referring to experimental values. It may be anticipated from Table 1 that the ground-state Pijs index obtained within these two VB models can be utilized to predict bond lengths for various conjugated systems with reasonable accuracy. Nevertheless, our subsequent discussions will indicate that only Pijs values derived from the EVB calculations are applicable to the cyclobutadiene derivatives we consider here. Accordingly, calculated ground-state Pijs and bond lengths based on both CVB and EVB models, together with experimental (or corresponding MO) values for several benzocyclobutadienoid hydrocarbons, are listed in Table 2. First, one can notice that the agreement between experimental and calculated bond lengths from the EVB theory is satisfactory. The average deviations are (0.017 and (0.020 Å for biphenylene and
15070 J. Phys. Chem., Vol. 100, No. 37, 1996
Ma et al.
TABLE 2: Bond Lengths in Benzocyclobutadienoid Hydrocarbons (Å) CVB compd biphenylenea
angular[3]phenyleneb
[3]phenylenec
naphthocyclobutadiened
anthracyclobutadiened
EVB
bond
Pijs
calcd
Pijs
calcd
exptl (MO)
a b c d e a b c d e f g h i j k l a b c d e f g a b c d e f g h i a b c d e f g h i j k l
0.672 0.675 0.620 0.580 0.616 0.635 0.763 0.667 0.764 0.633 0.602 0.574 0.797 0.603 0.596 0.653 0.678 0.675 0.755 0.645 0.573 0.631 0.657 0.679 0.802 0.641 0.653 0.623 0.687 0.606 0.781 0.644 0.590 0.644 0.778 0.607 0.686 0.648 0.669 0.662 0.637 0.803 0.617 0.572 0.584
1.409 1.408 1.428 1.443 1.430 1.423 1.375 1.411 1.374 1.424 1.435 1.446 1.362 1.435 1.437 1.416 1.407 1.428 1.378 1.419 1.446 1.424 1.415 1.406 1.360 1.421 1.416 1.427 1.403 1.434 1.368 1.419 1.440 1.419 1.369 1.433 1.404 1.418 1.410 1.413 1.422 1.360 1.430 1.446 1.442
0.676 0.739 0.744 0.585 0.499 0.732 0.686 0.730 0.686 0.733 0.496 0.770 0.645 0.763 0.497 0.592 0.542 0.738 0.675 0.743 0.511 0.595 0.576 0.709 0.831 0.538 0.824 0.466 0.558 0.659 0.742 0.683 0.631 0.651 0.773 0.619 0.648 0.710 0.551 0.828 0.533 0.829 0.455 0.588 0.603
1.408 1.385 1.383 1.440 1.471 1.388 1.404 1.388 1.404 1.387 1.472 1.374 1.419 1.377 1.471 1.438 1.455 1.385 1.408 1.384 1.466 1.436 1.443 1.396 1.352 1.457 1.355 1.483 1.450 1.414 1.384 1.405 1.424 1.416 1.373 1.428 1.418 1.395 1.452 1.353 1.459 1.353 1.486 1.439 1.434
1.423 (1.407) 1.385 (1.389) 1.372 (1.389) 1.426 (1.411) 1.514 (1.478) 1.369 (1.394) 1.404 (1.404) 1.370 (1.393) 1.400 (1.404) 1.365 (1.394) 1.503 (1.472) 1.345 (1.385) 1.446 (1.417) 1.348 (1.384) 1.505 (1.472) 1.413 (1.415) 1.449 (1.428) 1.397 (1.389) 1.436 (1.408) 1.359 (1.390) 1.513 (1.478) 1.407 (1.408) 1.397 (1.415) 1.385 (1.400) (1.351) (1.477) (1.367) (1.440) (1.440) (1.413) (1.386) (1.409) (1.404) (1.422) (1.375) (1.428) (1.414) (1.391) (1.447) (1.364) (1.474) (1.352) (1.448) (1.421) (1.412)
a Experimental bond lengths from: Fawcett, J. K.; Trotter, J. Acta Crystallogr. 1966, 20, 87. PPP-MO values from ref 21. b Experimental bond lengths from: Diercks, R.; Vollhardt, K. P. C. Angew. Chem., Int. Ed. Engl. 1986, 25, 266-267. “Iterated” HMO values from ref 22a. c X-ray structure of its tetrasilylated derivative from ref 19. “Iterated” HMO values from ref 22a. d PPP-MO values from ref 21.
angular [3]phenylene, respectively. In fact, the underestimate of the bridge bond in four-membered ring is a major source of average error in our predictions. This situation may arise from the approximations of our EVB model: one is the equal bond length assumption and another involves the neglect of nonnegligible rehybridization of the σ framework in cyclobutadienofused benzenes.23 As expected, all the predicted bridge bonds in the central cyclobutadiene nucleus are of single-bond character so as to minimize the antiaromatic feature of corresponding four-membered ring. In addition, the degree of double-bond fixation in the central benzene moiety in angular [3]phenylene is evaluated to be larger than in [3]phenylene, conforming with experimental observations.19 Second, for naphthocyclobutadiene and anthracyclobutadiene without available experimental bond lengths, our results from the EVB model are very close to corresponding PPP-MO values with average variances of less than (0.012 Å.21 The sole relatively large difference between our results and MO calcula-
tions seems to appear in the common bond shared by sixmembered and four-membered cycles. The experimental lengths of this class of bonds in biphenylene and angular [3]phenylene tend to suggest that the real values may lie between those predicted by these two different theoretical methods. Through comparison of the EVB and MO results described above, we gain an insight that the EVB model can satisfactorily explain bond lengths for these benzocyclobutadienoid hydrocarbons. However, one can also find from Table 2 that the correlation between those predictions from the CVB theory and experimental values (or MO theory) is discouraging. For example, the relative magnitude of bond lengths in fourmembered ring of biphenylene is predicted to be opposite to the experimental fact, and the average error between calculated and experimental values of (0.039 Å for this molecule is also too large even for qualitative purpose. One may want to know why the CVB and EVB theories behave so differently for these systems with cyclobutadiene moieties. As described in section II, the CVB Hamiltonian is limited to the tight-binding interaction, whereas the EVB Hamiltonian adds nonadjacent effect and important four-cyclic contributions on the basis of the CVB Hamiltonian. Obviously, our calculations have verified that four-cyclic contributions are crucial in treatments of benzocyclobutadienoid systems. Moreover, it is quite meaningful to point out that, although both EVB and HMO models are based on similar topological assumptions, the EVB model does make reasonable predictions in close consistence with modified HMO22 and PPP-MO models21 in which a bond order-bond length relation is used for self-consistent calculations. After analyzing bond lengths obtained in both VB models, we notice that additional terms not included in the CVB theory seem to lengthen the relatively longer bonds and contract the shorter, which plays the same role as the iteration procedure by using the bond order-bond length relation in “iterated” HMO and PPP-MO approaches. Summarizing the above discussion, we conclude that the EVB model can be anticipated to give reasonable predictions on bond lengths of benzocyclobutadienoid hydrocarbons, while the CVB model confronts serious difficulties, confirming the original speculation that the CVB theory may be inappropriate for conjugated compounds with four-membered cycles. Stability and Reactivity. The stability and reactivity of molecules containing four-membered rings has been the subject of extensive experimental and theoretical studies.7,19,21-23 It is noteworthy that those theoretical methods gave similar predictions consistent with experimental observations in most cases. Nevertheless, one also find that in some cases predictions from simplified VB schemes, such as conjugated circuits (CC) theory5 and Resonance-structure theory,4 are not always in line with those from MO methods.22a Moreover, it is well-known that even the classical VB theory may be unsuitable for compounds with cyclobutadiene moieties, although this view comes to relatively large degree from the fact that cyclobutadiene is incorrectly predicted to be aromatic by this theory. In this section, we will provide the direct numerical verifications on the failure of the CVB theory and again investigate the predictive power of the EVB model for those conjugated systems with four-membered rings. For a selection of benzocyclobutadienoid systems as shown in Figure 1, the total π energies calculated within the framework of the CVB model and the EVB model are listed in Table 3, respectively. For comparison, we also give the total π-binding energies obtained from PPP-MO calculations.21 On the basis of these numerical results, we will discuss the relative stability of isomeric species for simplicity. Surprisingly, the predictions
Effective Valence Bond Model Study
J. Phys. Chem., Vol. 100, No. 37, 1996 15071 TABLE 4: EVB Localization Energy of Bond x and EVB Energy Required to Break the Bridge Bond A(B) on the Cyclobutadiene Nucleusa compd
localization energy (eV)b
compd
breaking energy (eV)b,c
1 2 3 4 5 6 7
-1.9438 -2.0714 -1.9478 -1.9254 -2.1689 -2.1089 -1.8983
8 9 10 11 12 13 14
1.2837 0.8583 (0.8724) 0.9467 0.8925 0.9408 (0.9430) 0.9921 0.8277 (0.8588)
a Numbering scheme is shown in Figure 1. b Calculated according to the definition given in ref 22. c Values in brackets are for bond B.
Figure 1. Selected benzocyclobutadienoid hydrocarbons.
TABLE 3: Total π Energies (eV) Obtained by CVB and EVB Models for the Series of Benzocyclobutadienoid Systems Shown in Figure 1a compd
CVB
EVB
MOb
1 2 3 4 5 6 7 8 9 10 11 12 13 14
-15.3890 -22.9880 -23.1617 -19.7471 -19.9629 -30.5996 -30.8084 -23.1014 -30.8499 -30.7312 -35.2343 -35.3350 -38.3707 -38.6067
-20.2611 -31.5556 -31.4320 -25.6353 -25.8788 -42.7132 -42.5026 -31.0207 -43.0651 -43.1394 -48.9651 -49.0134 -54.3573 -54.2240
9.837 15.635 15.334 12.486 13.073 21.169 20.756 16.277 21.841 21.997 24.448 24.492 27.692 27.385
a In both VB models, the energy of the state in which all spins are parallel is taken as the zero. The corresponding π-binding energies (in eV) from PPP-MO calculations in ref 21 are also given for comparison. b For 4 and 5, values are from standard HMO calculations; while for 11 and 12, values come from the “iterated” HMO calculations in ref 22. In both HMO methods, energies are in units of β, and the Coulomb integral R is taken as zero energy. PPP-MO values in ref 21 are adopted for other molecules.
made in terms of the EVB results are in remarkable agreement with those from PPP-MO theory for all molecules we considered, but the CVB theory often fails to assess the order of stability for those isomeric species. For instance, the CVB model predicts that 3 has greater stability than 2, being just the opposite to the conclusions from all other methods. Similarly, for those isomers with 16 π electrons, 6 < 10 < 7 < 9 is judged by this simple VB scheme; however, EVB model correctly predicts the relative stability of this series as 7 < 6 < 9 < 10. Consequently our calculations display the limitation of the CVB theory, which was previously speculated from small 4n systems.7,11 On the other hand, the aromaticity of this class of compounds has been discussed within the CC theory,5b,c and it is of interest to compare their results with ours because in most cases the aromaticity of a molecule parallels its thermal stability.1 We find that the results achieved from the two different theories agree well in those species with one fourmembered ring but diverge in two sets of isomeric species with two cyclobutadiene nuclei (4 and 5, 11 and 12). For example,
the total π energies we obtained predict 12 to be more stable than 11, while the resonance energies from the CC theory suggest that 11 has the stronger aromaticity than 12. Since within the CC theory the total π energy is essentially the sum of the resonance energy and the energy of corresponding bondlocalized reference structure, we cannot carry out a strictly meaningful comparison for these two theories unless the energy of the corresponding reference structure is known. To correlate with available experimental evidence, the reactivity of these benzocyclobutadienoid systems is also worth investigating. For those compounds containing exoteric cyclobutadiene moieties, the bond labeled by X is the most reactive bond; the localization energy of this bond has been taken as a measure of dienophile reactivity in those systems.22b Whereas in the case of polycyclic biphenylene derivatives, it is wellknown that these molecules can readily be hydrogenated with rupture of the four-membered ring; the energy required to break the bridge bond on four-membered rings can be used to analyze the relative reactivity for these compounds.22a Accordingly, we calculate localization energies of bond X in 1-7 and the breaking energies of the bridge bonds in 8-14 referring to ref 22b; the results are tabulated in Table 4. Since the lower localization energy corresponds to the higher reactivity, compounds 3 and 7 are evaluated to be more reactive than their respective isomers 2 and 6, coinciding with the results derived from MO calculations.22 Another aspect that should be noted is that the difference in reactivity may be greater than the difference in stability, as exemplified in 6 and 7. On the other hand, the relative ease of breaking the bridge bonds deduced from Table 4 is basically in accord with the known chemical facts,22b,24 also paralleling the predictions from the modified HMO theory.22b In terms of this index, rupture of the fourmembered ring in biphenylene is verified to be more difficult than in its higher homologous derivatives, and 11 indeed exhibits the higher reactivity than 1221 in accord with EVB results mentioned above that 11 is relatively more unstable than 12. IV. Conclusions We have employed the Lanczos method to solve the EVB model exactly for π systems of a series of benzocyclobutadienoid hydrocarbons with up to 20 carbon atoms. By defining several indices calculated from the exact EVB ground-state energies and wave functions, we have given a detailed discussion on bond lengths, stabilities, and reactivities for these compounds. Encouragingly, our predictions agree well with the available experimental facts and the deductions of MO theory. This agreement is somewhat surprising in view of the simplicity of the EVB model in which only two exchange parameters are used, instead of a large amount of bielectronic integrals in PPPMO calculations. Therefore our work indicates that the EVB model is a promising tool for computing electronic structures
15072 J. Phys. Chem., Vol. 100, No. 37, 1996 of conjugated molecules with four-membered rings. In contrast with the success of the EVB model, our calculations reinforce the conclusion that the classical VB theory is unreliable in predicting the physical and chemical properties for this class of systems. Additionally, it should be emphasized that our bond order-bond length relationship predicts bond lengths in these systems fairly well, avoiding the failure of the empirical version of the CVB theory. Acknowledgment. We thank the two reviewers for their pertinent comments and good suggestions concerning this manuscript. This work was supported by China NSF. References and Notes (1) Streitweiser, A., Jr. Molecular Orbital Theory for Organic Chemists; Wiley: New York, 1961; pp 139-350. (2) Dewar, M. J. S.; Gleicher, G. J. J. Am. Chem. Soc. 1965, 87, 685692. (3) Hess, B. A., Jr.; Schaad, L. J. J. Am. Chem. Soc. 1971, 93, 305310. (4) (a) Simpson, W. T. J. Am. Chem. Soc. 1953, 75, 597-603. (b) Simpson, W. T.; Looney, C. W. J. Am. Chem. Soc. 1954, 76, 6285-6292. (c) Herndon, W. C. J. Am. Chem. Soc. 1973, 95, 2404-2406. (d) Herndon, W. C.; Ellzey, M. L. Jr. J. Am. Chem. Soc. 1974, 96, 6631-6642. (5) (a) Randic, M. Chem. Phys. Lett. 1976, 38, 68-70. (b) Randic, M. Tetrahedron 1977, 33, 1905-1920. (c) Randic, M. J. Am. Chem. Soc. 1977, 99, 444-450. (d) Randic, M.; Trinajstic, N. J. Am. Chem. Soc. 1987, 109, 6923-6926. (e) Nikolic, S.; Randic, M.; Klein, D. J.; Plavsic, D.; Trinajstic, N. J. Mol. Struct. (THEOCHEM) 1989, 198, 223-237. (f) Trinajstic, N.; Schmalz, T. G.; Zivkovic, T. P.; Nikolic, S.; Hite, G. E.; Klein, D. J.; Seitz, W. A. New J. Chem. 1991, 15, 27-31. (6) (a) Malrieu, J.-P.; Maynau, D. J. Am. Chem. Soc. 1982, 104, 30213029. (b) Maynau, D.; Malrieu, J.-P. J. Am. Chem. Soc. 1982, 104, 3029-
Ma et al. 3034. (c) Maynau, D.; Said, M.; Malrieu, J.-P. J. Am. Chem. Soc. 1983, 105, 5244-5252. (d) Said, M.; Maynau, D.; Malrieu, J.-P.; Garcı´a-Bach, M. A. J. Am. Chem. Soc. 1984, 106, 571-579. (e) Said, M.; Maynau, D.; Malrieu, J.-P. J. Am. Chem. Soc. 1984, 106, 580-587. (7) Kuwajima, S. J. Am. Chem. Soc. 1984, 106, 6496-6502. (8) Cioslowski, J. Theor. Chim. Acta 1989, 75, 271-278. (9) Alexander, S. A.; Schmalz, T. G. J. Am. Chem. Soc. 1987, 109, 6933-6937. (10) Li, S.; Jiang, Y. J. Am. Chem. Soc. 1995, 117, 8401-8406. (11) Wheland, G. W. J. Chem. Phys. 1934, 2, 474-481. (12) Klein, D. J. Top. Curr. Chem. 1990, 153, 59-81. (13) Lee, S. J. Chem. Phys. 1989, 90, 2732-2740. (14) Li, S.; Ma, J.; Jiang, Y. Chem. Phys. Lett. 1995, 246, 221-227. (15) Lanczos, C. J. Res. Natl. Bur. Stand. 1950, 45, 255-282. (16) Haydock, R. Solid-State Physics; Ehrenreich, H., Seitz, F., Turnbull, D., Eds.; Academic Press: New York, 1980; pp 216-292. (17) (a) Liu, C. G.; Shao, Y. H.; Jiang, Y. S. Chem. Phys. Lett. 1994, 228, 131-136. (b) Liu, C. G.; Wang, M. M.; Shao, Y. H.; Jiang, Y. S. Phys. Lett. A 1994, 196, 120-124. (18) Maynau, D.; Said, M.; Malrieu, J.-P. J. Am. Chem. Soc. 1983, 105, 5244-5252. (19) Bern’s, B. C.; Hovakeemian, G. H.; Lai, Y. H.; Mestdagh, H.; Vollhardt, K. P. C. J. Am. Chem. Soc. 1985, 107, 5670-5687. (20) Kao, J.; Allinger, N. L. J. Am. Chem. Soc. 1977, 99, 975-986. (21) Dewar, M. J. S.; Gleicher, G. J. Tetrahedron 1965, 21, 18171825. (22) (a) Barron, T. H. K.; Barton, J. W.; Johnson, J. D. Tetrahedron 1966, 22, 2609-2613. (b) Coulson, C. A.; Poole, M. D. Tetrahedron 1964, 20, 1859-1862. (23) Faust, R.; Glendening, E. D.; Streitwieser, A.; Vollhardt, K. P. C. J. Am. Chem. Soc. 1992, 114, 8263-8268. (24) Cava, M. P.; Strucker, J. F. J. Am. Chem. Soc. 1955, 77, 6022-6026.
JP960802+
