George S. Handler

Asymptotic behavior of atomic Hartree–Fock orbitals

The restricted Hartree–Fock equations are solved at large r for closed shell atomic states. Except for states containing only s orbitals, the Hartree–Fock orbitals obey the formula, φi(r)∼kiYlimi(ϑ,φ) rβ−λ−1 exp(−ζr), where −1/2ζ2=eho, the highest occupied orbital energy, where β= (nuclear charge – number of electrons +1−ζ)/ζ, where λ+1=0 for the highest occupied orbital, where λ=‖ li−lho ‖ if li≠lho, where λ=2 if li=lho≠0 and i≠ho, where λ=2lmin+1 if li=lho, and where lmin is the smallest nonzero l value of the occupied Hartree–Fock orbitals.

Exponential Hamiltonian: Convergence in an iterated product representation for a linear harmonic oscillator

Using an iterated product representation of an exponential Hamiltonian, the density matrix for the potential V(x) = 1/2Kx2 is derived in nth‐order approximation. The density matrix so found is shown to converge to the exact quantum mechanical result in the limit of n → ∞.

Exchange energy in the Thomas‐Fermi model

The concept of exchange energy as it arises in Thomas‐Fermi theory is examined. It is found that the imposition of Hermitian symmetry upon the single particle Thomas‐Fermi density matrix changes the single particle exchange energy density operator in a drastic way.

Convergence of a Product Representation of the Exponential Hamiltonian Operator. An Example from Statistical Theory

Using an iterated product representation of an exponential Hamiltonian, the density matrix for the potential V (x)=ax, a>0, is derived in nth‐order approximation. The density matrix so found is shown to converge to the exact quantum mechanical result in the limit of n→ ∞.

CONTINUOUS REPRESENTATIONS IN THE STATISTICAL THEORY OF ELECTRONIC ENERGIES. THE H2(+) ION.

A version of the statistical theory of electronic energies, arising from a consideration of alternative partitionings of the Hamiltonian, has been applied to the H2+ ion, yielding electronic energies ∼6% greater in magnitude than the exact theoretical values at several values of the internuclear separation. The binding energy at the theoretical equilibrium separation was calculated to be 0.1024 a.u. compared to the experimental value of 0.1026 a.u.

S Limit in Helium. Movement of a System on an Energy Surface in Parameter Space

As part of a variational examination of angular correlation in electronic systems, a careful examination of the movement of a system on an energy surface in parameter space has been made. The results indicate that the surface is highly convoluted, and that minimization techniques must take explicit account of this. An eight‐term result close to the S limit in helium is given.

Eigenfunctions and eigenvalues of the Thomas‐Fermi density matrix. The simple harmonic oscillator

The eigenfunctions and eigenvalues of a Hermitian Thomas‐Fermi density matrix for the one‐dimensional harmonic oscillator are examined in detail. The major defect of Thomas‐Fermi theory is shown to be the departure of the density matrix from N representability. The imposition of indempotency leads to greatly improved particle densities, including very well developed density oscillations.

Statistical Two‐Particle Density Matrix. Some Considerations Relating to the Choice of Basis of Representation

A preliminary examination of the problem of the statistical theory of electronic energies from the view‐point of the 2 matrix is made. For the two‐particle, one‐dimensional harmonic oscillator, a good representation of the Fermi hole is found for the triplet state. The employment of a discrete basis for heliumlike systems leads to problems with the virial theorem.

The self-energy of a free electron gas

Abstract In this note we consider the issue of an electrons self-Coulomb interaction (or self-energy) in a homogeneous free electron gas. We observe that, provided the gas is charge neutral and infinite in extent, the proper self-energy contribution yields a lowering of the total energy and does not modify the shape of the orbitals from which the particle density is constructed in a self-consistent theory, when the orbital states are Bloch-like.