On the Dependence of Total Energy on Occupation Numbers

Abstract

Abstract. An equation is developed for describing the pairwise interactions of q electrons, when q is integer or non-integer. This expression iscompared with the usual equation. It is found to give lower total energies, different ionization and transition energies in the Slater Transition State theory, different values for the atomic energies calculated by the Hyper-Hartree-Fock method, but numerically identical results for the Mulliken Electronegativity. APOLOGIAAs a entre remarked, "I am leary of attributing any physical meaning to E(N) no matter how the problem of non-integral occupancies is handled", and so are we in the precise physical and philosophical sense. It may even be that the recent work on the virial partitioning theory and catastrophe theory, ,-4and its applications to molecular systems,' wherein atomic regions of total integral occupances within the molecule are defined: may replace the current use of non-integral occupances in the analysis of molecular wavefunctions. However until this comes to pass, there will always be the need to attempt the accurate counting of non-integral occupancy interactions,' as in the recent and elegant formalism of density functionals," which even now is being extended to redefine electronegativity" as the chemical potential IL, expressed as (aE1an) where n is the occupancy. And it will always be more elegant, efficient and pleasing to produce the numerical results with a more direct algebraic formalism which avoids certain previously required assumptions. We feel that the present discussion gives an alternate interpolation formula which is theoretically sound within the limitations of using non-integer occupations.electrons were extended classical charge distributions, the total number of electron repulsions, including q.l2 self-repulsions, would be q:12.(3)An electron does not repel itself, quantum mechanically, so that the total number of repulsions reduces to (2).When cP' is an Hartree-Fock spin orbital, and q, is unity for all cPh (1) gives the Hartree-Fock energy, the coefficients q, (q, -1)/2 being zero.When cP' are for the Hyper-Hartree-Fock method, (1) gives the average energy of the multiplet states" because in the HHF method'" cP' represents a shell of (21, + 1) degenerate spin orbitals with (ii) and (ij) suitable averages."?" However, when (ii) and (ij) are suitable averages, the q, become non-integer, especially in calculating transition metal atom and ion ground state configurations. Similarly, in Slater's transition state calculations":" of excitation energies and ionization potentials, the q, becomes non-integer.In the current electronegativity theories.P" where the energy of an atom, or orbital, is assumed to be a differentiable quadratic function of the occupancy q, and where the electronegativity X is given by q is also allowed to be non-integer.The problem therefore arises, as to whether (1) is the best expression for E(N) when q is non-integer.

Energy Density Functional Theory of Many-Electron Systems

“…There is an important and somewhat controversial question concerning the meaning of relation (7.7-5). Slater's assumption through which the energy is regarded as a differentiable function of the occupation numbers has been analyzed by Larson and Smith (1971), Harris (1979), and Gopinathan and Whitehead (1980). Slater's assumption through which the energy is regarded as a differentiable function of the occupation numbers has been analyzed by Larson and Smith (1971), Harris (1979), and Gopinathan and Whitehead (1980).…”

Electronegativity in density functional theory: diatomic bond energies and hardness parameters

“…When the total energy, E ( q ) , is expressed within density functional theory (8), and is assumed differentiable (8), where p, is the chemical potential (5).…”

“…The electronegativity is also derivable from the finite difference formula. However, there are two distinct derivations of the electronegativity: one assumes that the dependence of the electronegativity on the charge is linear between A+ and A-; the other assumes a distinct discontinuity at A ' between the A+ 4 A ' and A' + A-lines ( 8 ) . The present derivation uses the finite difference formula and applies it to a discontinuous electronegativity curve.…”

“…The electronegativity variation is that developed by Gopinathan and Whitehead (8) and subsequently discussed by Pam ( 2 ) , in which there is a discontinuity in the energy as a function of charge at the uncharged atom So. This gives rise to two different slopes for q between SO and S+ and between SO and S-.…”

Electronegativity: A Mnemonic Number Code for an Atom

“…Therefore, electronegativity is termed as a function of oxidation number. Zhang has also reported dual parameter equation [12]. In this section, an independent approach to define electronegativity is presented.…”