Richard E. Stanton

Stability of buckminsterfullerene and related carbon clusters.

Under appropriate collisional conditions the mass spectrum of carbon fragments produced by laser vaporization of graphite is dominated by C/sub 60/ and (to a lesser extent) C/sub 70/ clusters. The discoverers of this phenomenon have noted that the carbon valence requirements can be satisfied in closed, hollow structures. For C/sub 60/ they suggest an icosahedral soccer ball network, which they call buckminsterfullerene and we abbreviate as BF. Experimental support has come from studies with lanthanum-impregnated graphite. The resulting mass spectra show intense C/sub 60/La peaks, but no C/sub n/La/sub 2/ or C/sub n/La/sub 3/ peaks. Subsequent experiments have demonstrated the inertness of C/sub 60/ and, indeed, other large C/sub 2n/ clusters under NO attack. We report here the results of quantum calculations which were prompted by the experiments cited above and other earlier work. Our purpose has been to test the intrinsic stability of BF and related polyhedral species and to compare their stability with that of planar graphite fragments. The latter have the advantage of being strain free, but suffer from dangling valences on their perimeters. We also make comparisons with linear carbon chains.

Hellmann‐Feynman Theorem and Correlation Energies

A proof is offered for the applicability of the Hellmann‐Feynman theorem to Hartree‐Fock wave functions. In conjunction with Sinanoglus theory of correlation energies, this leads to the idea that Hartree‐Fock potential energy surfaces and exact potential energy surfaces are parallel over short distances. Finally, a new method of calculating dissociation energies is presented.

Multiple Solutions to the Hartree‐Fock Problem. I. General Treatment of Two‐Electron Closed‐Shell Systems

The number of fully self‐consistent, closed‐shell solutions to the two‐electron Hartree‐Fock problem is investigated. With N orbital basis functions the Hartree‐Fock wavefunction can be expanded as a superposition of N(N + 1) / 2 configurations. The constraints in the coefficients are analyzed exactly for the case N = 2. The free CI problem corresopnds to finding energy extrema on the surface of a sphere. The Hartree‐Fock state point is constrained to lie on a circle on this sphere. There will be two, three or four SCF solutions depending on the manner in which the circle of constraint crosses valleys and ridges on the sphere. This is shown explicitly by detailed calculations for the He atom. These calculations also reveal instances in which aufbau method of solution necessarily diverges. The general problem can be analyzed under the assumption of zero differential overlap. The Hartree‐Fock matrix is diagonal in this approximation, but the diagonal elements still involve the orbital coefficients. One can ...

Intrinsic convergence in closed‐shell SCF calculations. A general criterion

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).

Multiple Solutions to the Hartree–Fock Problem. II. Molecular Wavefunctions in the Limit of Infinite Internuclear Separation

The restricted Hartree–Fock equation for a closed‐shell molecule is investigated in the limit of infinite internuclear separation. The conventional molecular equation reduces, in the limit, to a set of modified Hartree–Fock equations for the individual fragments. The modification consists of admitting fractional occupation numbers for occupied orbitals. The individual fragment equations are coupled by conditions on the orbital energies and on the fractional “charges.” The self‐consistent field equation, being nonlinear, admits a variety of mathematically acceptable solutions. For lithium hydride a wavefunction corresponding to ionic fragments Li+ and H− is one such limiting solution. Contrary to popular notion, it is not the solution of lowest energy even within the restricted SCF formalism. Limiting solutions are obtained for H2, LiH, and CH4. The separated hydrogen molecule is shown to have an SCF energy which is exactly one‐quarter that of the helium atom. The long‐range interaction of two hydrogen ato...

Mathematical Properties of Frost's Local‐Energy Method

Frosts local‐energy method is analyzed in two different ways. An exact, trigonometric solution is carried out for the two‐dimensional problem. The more general problem is treated by perturbation theory.It is shown that not all solutions of the local‐energy equations correspond to minimum variance. Some correspond to maximum variance, others to a saddle point on the variance hypersurface. The number of minimum variance solutions is equal to the number of basis functions when the latter are well chosen, but may otherwise be smaller.

Variation‐Perturbation Approach to Electron‐Cluster Wavefunctions

Sinanoglus cluster expansion formula for the exact wavefunction of a many‐electron system is investigated by variation‐perturbation techniques. The perturbation operator is the difference between the actual Hamiltonian and a symmetric sum of arbitrary one‐electron Hamiltonians. Each perturbation function is then expanded in accordance with the cluster formula.When the occupied orbitals are chosen to be eigenfunctions of the one‐electron Hamiltonian the variational equations determining different nth‐order clusters are completely independent. If the orbitals are unitarily transformed to achieve greater localization the variational equations for each cluster are coupled.The perturbation governing first‐order two‐electron clusters has the form of a dipole potential. This result does not depend on the nature of the one‐electron Hamiltonian. It is primarily useful, however, in studying the question of interorbital vs intraorbital correlation, i.e., when the one‐electron Hamiltonian is the Hartree—Fock operato...

Canonical orthonormalization and neglect of differential overlap

Abstract Lowdins canonical orthonormalization procedure is non-unique if the overlap matrix has degenerate eigenvalues. A recent attempt by Roby to justify neglect of differential overlap is re-examined taking proper account of this lack of uniqueness. We conclude that his central result, eq. (6) in our paper, is formally correct in the limit of complete basis sets, but Robys presentation lends a misleading interpretation to this equation. Consequently, some conclusions which are claimed to follow are, in fact, not valid. We briefly investigate the Roby approximation for finite basis sets and indicate the strengths and weaknesses when used for practical computations.

Overcomplete multicenter basis sets

Abstract The distribution of eigenvalues of the overlap matrix is analyzed for large and, in the limit, infinite overcomplete multicenter basis sets. It is shown that in the limit the eigenvalues become infinitely degenerate. A numerical example is given and discussed in terms of a semiclassical distribution in phase space.

Criterion for Ordering the Bond Strengths of Neutral and Positively Charged Hydrides

A simple criterion is developed for ordering the bond strengths of isoelectronic hydrides, particularly positive ions with the formula MZ+1Hn+ and the corresponding neutral molecules MZHn. For monohydrides the criterion states that the bond strength of the ion will be greater than that of the neutral molecule whenever the ionization potential of the atom MZ+1 is less than 13.6 eV. When this ionization potential is significantly greater than 13.6 eV the neutral molecule has the stronger bond. For polyhydrides the total dissociation energy of the ion is almost always greater than that of the neutral molecule. Extensive experimental evidence in support of this criterion is presented.