Recent publications have suggested that the satisfaction of a principle we call kinetic balance can provide variational safety in Dirac calculations. The theoretical foundation for this proposal is examined, first in simple one-electron problems, and then (less rigorously) in SCF calculations. The conclusion is that finite basis calculations using kinetic balance are safe from catastrophic variational collapse, but that the ‘‘bounds’’ provided by these calculations can be in error by an amount of order 1/c4. The bounds are applicable to total SCF energies, but not to SCF orbital energies. The theory is illustrated by a series of one-electron calculations.
Given any two sets of spin orbitals ai and bj , there exist equivalent sets Iii and bj such that their overlap matrix is diagonal, i.e., (Iii I bi) =diiOij. This is the basis of the corresponding orbital transformation of Amos and Hall. Their transformation is shown to have widespread application to quantum chemistry. It leads to a simple generalization of the Slater-Condon rules for the expectation value of an operator between two determinantal wavefunctions when the spin orbitals of one function have no simple orthogonality relationship to those of the other function. In the case of single,determinantal wavefunctions, use of the corresponding orbital transformation and the integral Hellmann-Feynman formula leads to a very simple expression for the energy difference associated with two similar configurations of a molecular system. Extensions to limited configuration interaction expansions are discussed. Given single-determinantal wavefunctions for two related molecular systems, it is shown that the corresponding orbitals are those which are most nearly molecularly invariant in the sense of maximum overlap. A comparison of the Pitzer-Lipscomb wavefunctions for the staggered and eclipsed forms of ethane reveals that six of the nine corresponding orbitals have an overlap of no less than 0.999998 in the two configurations. Use of the corresponding orbital transformation overcomes various computational difficulties encountered with LOwdin's cofactor method for treating the nonorthogonality problem.
The MNDO approximation is employed to compute normal modes of vibration for the proposed c 6 0 isomer known as buckminsterfullerene. Group theoretical invariance theorems are derived to aid in the interpretation of the normal modes. One particularly interesting mode (the sole A, vibration) consists entirely of a rotary oscillation of the pentagonal rings of c 6 0 , with all rings rotating in the same direction.
111trating upon the structural chemistry alone overlooked the lessons to be learned about this vitally important field of inquiry.the National Science Foundation. served. So little is known about factors controlling the transitions involved, however, that it is premature to generalize the present treatment to cover them. What the present treatment does that is of greatest value is to focus attention in cluster research upon the dynamics of transformations. Prior investigations concen-
Acknowledgment.MNDO calculations are carried out on several fullerene structures relevant to recent tandem mass spectroscopic, photophysical studies. Estimates are presented for the enthalpy changes of Stone-Wales isomerization, of the excision of C2, C4, and C6 groups from buckminsterfullerene (BF), and of the excision of C2 from a closely related c 6 2 molecule. The calculations support the general belief that the introduction of pentalene units into a fullerene is destabilizing. The removal of C2 from c 6 2 to form BF is predicted to be strongly endothermic but not nearly to the Same extent as the removal of C2 from BF itself.In the case of BF the calculations require the excision to be a multiphoton process, in agreement with reported interpretations of experimental data. Additional calculations indicate that C30 has at least one stable fullerene structure. This result is contrary to the photophysically based suggestion that C32 represents the lower limit of fullerene stability. The difficulty of forming reasonable geometric starting points for MNDO calculations is overcome by the use of inverse stereographic projection and the concept of a planar dual graph.
It is shown that the intrinsic convergence of the classical SCF algorithm for closed shells is governed by the eigenvalues of a supermatrix Q whose elements involve orbital excitation energies and electron repulsion integrals over occupied and virtual orbitals. In general, the SCF process is intrinsically convergent if all eigenvalues of Q have absolute values less than unity, and intrinsically divergent if one ore more is greater than unity. It is also shown how the symmetry preserving features of the SCF algorithm enable one to avoid certain divergences provided one starts a calculation with symmetry adapted orbitals. In cases of intrinsic convergence it is proved that the ratio of successive energy increments produced by the SCF algorithm is equal to the square of the largest eigenvalue of Q. The implications of the theory are illustrated by calculations on linear H8, on the cyclic planar ion S4N3+, and on model Q matrices filled with random numbers over the interval (−0.5,0.5).
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