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NASA NTRS1972

On the Poincare noninvariance of a recent alternative to the Dirac equation.

Explicit construction of the infinitesimal generators of the Poincare group, demonstrating that the algebra of commutators closes only for the case of zero mass. Hence, the so-called Stigma equation proposed by Biedenharn et al. (1971) as an alternative to the Dirac equation (1928) for spin one-half, finite mass particles is not Poincare invariant except when the leptonic mass is zero.

Madan, R. N.↗
NASA NTRS1977

Geometrization of the Dirac theory of the electron

Using the concept of parallel displacement of a half vector, the Dirac equations are generally written in invariant form. The energy tensor is formed and both the macroscopic and quantum mechanic equations of motion are set up. The former have the usual form: divergence of the energy tensor equals the Lorentz force and the latter are essentially identical with those of the geodesic line.

Fock, V.↗
NASA NTRS1983

Statistical mechanics of light elements at high pressure. V Three-dimensional Thomas-Fermi-Dirac theory

A numerical technique for solving the Thomas-Fermi-Dirac (TED) equation in three dimensions, for an array of ions obeying periodic boundary conditions, is presented. The technique is then used to calculate deviations from ideal mixing for an alloy of hydrogen and helium at zero temperature and high presures. Results are compared with alternative models which apply perturbation theory to calculation of the electron distribution, based upon the assumption of weak response of the electron gas to the ions. The TFD theory, which permits strong electron response, always predicts smaller deviations from ideal mixing than would be predicted by perturbation theory. The results indicate that predicted phase separation curves for hydrogen-helium alloys under conditions prevailing in the metallic zones of Jupiter and Saturn are very model dependent.

Macfarlane, J. J.↗
NASA NTRS1984

Statistical mechanics of light elements at high pressure. VI - Liquid-state calculations with Thomas-Fermi-Dirac theory

A model free energy is developed for hydrogen-helium mixtures based on solid-state Thomas-Fermi-Dirac calculations at pressures relevant to the interiors of giant planets. Using a model potential similar to that for a two-component plasma, effective charges for the nuclei (which are in general smaller than the actual charges because of screening effects) are parameterized, being constrained by calculations at a number of densities, compositions, and lattice structures. These model potentials are then used to compute the equilibrium properties of H-He fluids using a charged hard-sphere model. The results find critical temperatures of about 0 K, 500 K, and 1500 K, for pressures of 10, 100, and 1000 Mbar, respectively. These phase separation temperatures are considerably lower (approximately 6,000-10,000 K) than those found from calculations using free electron perturbation theory, and suggest that H-He solutions should be stable against phase separation in the metallic zones of Jupiter and Saturn.

Macfarlane, J. J.↗
NASA NTRS1985

Statistical mechanics of light elements at high pressure. VIII - Thomas-Fermi-Dirac theory for binary mixtures of H with He, C, and O

We present three-dimensional Thomas-Fermi-Dirac calculations of lattice mixing energies of hydrogen with carbon and oxygen atoms, respectively. The results are used to derive effective interatomic potentials for use in liquid-state mixture calculations. We then use the potentials to derive analytic expressions for binary mixture-free energies and to map out the phase diagrams of mixtures of hydrogen with, respectively, helium, carbon, and oxygen, over a pressure range of about 5 to about 10 to the 3rd Mbar. Within this pressure range, all three of the latter elements are found to have unlimited solubility in metallic hydrogen over a temperature range which lies above their pure-element melting temperatures, and which includes likely interior temperatures in the Jovian planets.

Hubbard, W. B.↗
NASA NTRS1990

All-electron molecular Dirac-Hartree-Fock calculations: The group 4 tetrahydrides CH4, SiH4, GeH4, SnH4 and PbH4

A basis-set-expansion Dirac-Hartree-Fock program for molecules is described. Bond lengths and harmonic frequencies are presented for the ground states of the group 4 tetrahydrides, CH4, SiH4, GeH4, SnH4, and PbH4. The results are compared with relativistic effective core potential (RECP) calculations, first-order perturbation theory (PT) calculations and with experimental data. The bond lengths are well predicted by first-order perturbation theory for all molecules, but non of the RECP's considered provides a consistent prediction. Perturbation theory overestimates the relativistic correction to the harmonic frequencies; the RECP calculations underestimate the correction.

Dyall, Kenneth G.↗
NASA NTRS1990

Polyatomic molecular Dirac-Hartree-Fock calculations with Gaussian basis sets

Numerical methods have been used successfully in atomic Dirac-Hartree-Fock (DHF) calculations for many years. Some DHF calculations using numerical methods have been done on diatomic molecules, but while these serve a useful purpose for calibration, the computational effort in extending this approach to polyatomic molecules is prohibitive. An alternative more in line with traditional quantum chemistry is to use an analytical basis set expansion of the wave function. This approach fell into disrepute in the early 1980's due to problems with variational collapse and intruder states, but has recently been put on firm theoretical foundations. In particular, the problems of variational collapse are well understood, and prescriptions for avoiding the most serious failures have been developed. Consequently, it is now possible to develop reliable molecular programs using basis set methods. This paper describes such a program and reports results of test calculations to demonstrate the convergence and stability of the method.

Dyall, Kenneth G.↗
NASA NTRS1990

Kinetic balance and variational bounds failure in the solution of the Dirac equation in a finite Gaussian basis set

The paper investigates bounds failure in calculations using Gaussian basis sets for the solution of the one-electron Dirac equation for the 2p1/2 state of Hg(79+). It is shown that bounds failure indicates inadequacies in the basis set, both in terms of the exponent range and the number of functions. It is also shown that overrepresentation of the small component space may lead to unphysical results. It is concluded that it is important to use matched large and small component basis sets with an adequate size and exponent range.

Dyall, Kenneth G.↗
NASA NTRS1991

All-electron molecular Dirac-Hartree-Fock calculations - The group IV tetrahydrides CH4, SiH4, GeH4, SnH4, and PbH4

A basis-set-expansion Dirac-Hartree-Fock program for molecules is described. Bond lengths and harmonic frequencies are presented for the ground states of the group 4 tetrahydrides, CH4, SiH4, GeH4, SnH4, and PbH4. The results are compared with relativistic effective core potential (RECP) calculations, first-order perturbation theory (PT) calculations and with experimental data. The bond lengths are well predicted by first-order perturbation theory for all molecules, but none of the RECP's considered provides a consistent prediction. Perturbation theory overestimates the relativistic correction to the harmonic frequencies; the RECP calculations underestimate the correction.

Dyall, Kenneth G.↗
NASA NTRS1992

The Hamiltonian structure of Dirac's equation in tensor form and its Fermi quantization

Currently, there is some interest in studying the tensor forms of the Dirac equation to elucidate the possibility of the constrained tensor fields admitting Fermi quantization. We demonstrate that the bispinor and tensor Hamiltonian systems have equivalent Fermi quantizations. Although the tensor Hamiltonian system is noncanonical, representing the tensor Poisson brackets as commutators for the Heisenberg operators directly leads to Fermi quantization without the use of bispinors.

Reifler, Frank↗
NASA NTRS1991

All-electron molecular Dirac-Hartree-Fock calculations: Properties of the group IV monoxides GeO, SnO and PbO

Dirac-Hartree-Fock calculations have been carried out on the ground states of the group IV monoxides GeO, SnO and PbO. Geometries, dipole moments and infrared data are presented. For comparison, nonrelativistic, first-order perturbation and relativistic effective core potential calculations have also been carried out. Where appropriate the results are compared with the experimental data and previous calculations. Spin-orbit effects are of great importance for PbO, where first-order perturbation theory including only the mass-velocity and Darwin terms is inadequate to predict the relativistic corrections to the properties. The relativistic effective core potential results show a larger deviation from the all-electron values than for the hydrides, and confirm the conclusions drawn on the basis of the hydride calculations.

Dyall, Kenneth G.↗
NASA NTRS1992

All-electron molecular Dirac-Hartree-Fock calculations - Properties of the XH4 and XH2 molecules and the reaction energy XH4 yields XH2 + H2, X = Si, Ge, Sn, Pb

Relativistic corrections to a number of properties of the Group IV hydrides are calculated using the Dirac-Hartree-Fock method. The use of first-order perturbation theory is sufficient to obtain relativistic corrections for Ge, but the effects of spin-orbit interaction and other higher-order effects begin to show for Sn and become important for Pb. The energy of the reaction XH4 yields XH2 + H2 (X = Si, Ge, Sn, and Pb) is also calculated. The results are compared with relativistic effective core potential calculations, first-order perturbation theory calculations, and limited experimental data.

Dyall, Kenneth G.↗