James D. Talman

Numerical Fourier and Bessel transforms in logarithmic variables

A method is described for the numerical calculation of Fourier transforms in variables that are the logarithms of the original variable and transform variable. The method involves only the application of two successive Fourier transforms and can also be applied to Bessel and spherical Bessel transforms. Numerical examples show that the method gives very accurate results up to large values of the transform variable.

Can optimized effective potentials be determined uniquely

Local (multiplicative) effective exchange potentials obtained from the linear-combination- of-atomic-orbital (LCAO) optimized effective potential (OEP) method are frequently unrealistic in that they tend to exhibit wrong asymptotic behavior (although formally they should have the correct asymptotic behavior) and also assume unphysical rapid oscillations around the nuclei. We give an algebraic proof that, with an infinity of orbitals, the kernel of the OEP integral equation has one and only one singularity associated with a constant and hence the OEP method determines a local exchange potential uniquely, provided that we impose some appropriate boundary condition upon the exchange potential. When the number of orbitals is finite, however, the OEP integral equation is ill-posed in that it has an infinite number of solutions. We circumvent this problem by projecting the equation and the exchange potential upon the function space accessible by the kernel and thereby making the exchange potential unique. The o...

A program to compute variationally optimized effective atomic potentials

Abstract A computer program to generate numerical effective local potentials for the electrons in atoms or ions is described. These potentials are variationally optimized; that is, the single-particle orbitals derived from them give the minimum value of the Hartree-Fock energy that can be obtained from a local central potential. The resulting potentials should be useful to generate realistic basis states for further atomic calculations. It can be shown [R.T. Sharp and G.K. Horton, Phys. Rev. 90 (1955) 316; J.D. Talman and W.F. Shadwick, Phys. Rev. A 14 (1976) 36] that the variational problem gives an integral equation for the potential that is solved numerically and self-consistently with the radial Schrodinger equation for the potential.

Some properties of three-dimensional harmonic oscillator wave functions

Abstract A number of properties of harmonic oscillator wave functions in three dimensions are derived using a generating function technique. The problems treated are the following: the matrix element between harmonic oscillator wave functions of different spring constants, the expansion of the product of harmonic oscillator wave functions in harmonic oscillator wave functions, the transformation between wave functions in spherical and cylindrical symmetry, translational properties of harmonic oscillator wave functions and the calculation of matrix elements of two-particle central potentials. In connection with the last problem, a relatively simple expression for the Talmi bracket is obtained, and an efficient method of calculating two-particle matrix elements is described.

LSFBTR: A subroutine for calculating spherical bessel transforms

On presente un programme qui calcule des transformees de Hankel pour des fonctions de Bessel spheriques

Oscillator Strengths in the Aluminum Sequence

The structures of the 3s23p 2P0 ground state and of a number of excited states for aluminum and ions along the isoelectronic sequence up to Ca VIII have been calculated in the Multiconfiguration Optimized Potential Model. The excited states include the following: 3s23d 2D, 3s24s 2S, 3s3p2 2D, 2S, as well as 3s24p, 3p3 and 3s3p(3P0)3d 2P0. There is strong configuration mixing in the spectroscopic terms but on the basis of our calculations we recommend a different designation from that currently adopted for the 3p3 and 3s3p(3P0)3d 2P0 terms. Oscillator strengths for transitions among these terms have been calculated in the length and velocity formulation with particular emphasis on accurate description in the vicinity of term crossings. Comparison is made with other theory and with experiment where results are available. At present, the experimental data are inadequate to verify the many calculated structures in the isoelectronic gf dependence.

NumSBT: A subroutine for calculating spherical Bessel transforms numerically

A previous subroutine, LSFBTR, for computing numerical spherical Bessel (Hankel) transforms is updated with several improvements and modifications. The procedure is applicable if the input radial function and the output transform are defined on logarithmic meshes and if the input function satisfies reasonable smoothness conditions. Important aspects of the procedure are that it is simply implemented with two successive applications of the fast Fourier transform, and it yields accurate results at very large values of the transform variable. Applications to the evaluation of overlap integrals and the Coulomb potential of multipolar charge distributions are described.

Numerical calculation of nuclear attraction three‐center integrals for arbitrary orbitals

The problem of calculating a nuclear attraction three‐center integral by expanding each of the factors about a common center is studied. An analysis is made of the convergence rate for various choices of the expansion center. The optimum choice for the expansion center from the point of view of the convergence rate is found. Several numerical examples of the convergence are described and results for a number of three‐center integrals are given.

A program to compute variationally optimized relativistic atomic potentials

The computer program described in the previous paper is modified to calculate effective local potentials for electrons in atoms or ions described by the single-particle Dirac equation. The electron-electron interaction is assumed to be the Coulomb potential. The modifications to the theory necessary in the relativistic case are described in detail. The resulting potential should be useful for calculating properties of atoms for large Z values. The results of the non-relativistic calculation of the previous program are used to provide a starting approximation for the relativistic case.

Non-additive three-body interaction energies for H3 (quartet spin state)

The results of an Unsold average energy calculation of the non-additive interaction energy for H3 (quartet spin state) are presented for equilateral triangular configurations. They are discussed in the context of the problems associated with the representation of non-additive energies for the interaction of closed-shell species.