A generalized time-dependent pseudopotential is defined. Analogue of the Newtonian equation is derived for stationary states and pseudoforces are physically interpreted. Physical points are also made from the analogous equation of continuity and the Euler equations.

A relative kinetic mass operator is defined bym =c⁻²·(E −eΦ), and it is shown that bt using it in a symmetric form one can correlate the (charge) velocity operatorα in the Dirac theory exactly with the general quantum mechanical momentum —ih∇. Then the net force, defined as the rate of change of the relative momentum with time, is exactly equal to the Lorentz force. The contribution due to the time variation of mass equals the negative of space variation of the scalar potential, the Newtonian force, whereas the time variation of the charge current absorbs the entire vector potential dependence. The analogous Euler equations can be written either in terms of the charge current or in terms of the mass current. For a many particle system one needs the usual net single particle parameters and the consideration of both the direct and exchange contributions of the two particle interaction. These Euler equations yield two different conditions of the stationary state. It is shown that the charge-current condition is necessary but not sufficient, whereas the mass-current condition retains the appropriate scalar potential dependence. These two conditions are compared for the spherically symmetric case. The charge density, charge current and relative mass current are tabulated for atomic spinors. Differences between the quantum and classical forces for the H₂⁺ molecular ion exhibit the inadequacy of ordinary atomic spinor basis in forming molecular spinors.

Pseudopotential is used as a formal operator to write the exact time-dependence of a pseudoexciton and hence that of an initial excitation spatially localized in a crystal. The exponential operator where pseudopotential occurs at the argument is readily evaluated using the property of projection operators. Migration of an initially localized excitation is of considerable experimental importance and can be of conceptual use since it should eventually generate the characteristics of a migrating exciton. From the formal time-dependence of a localized excitation, its spread with time can be calculated with relative ease. In a concurrent discussion, the previous work of Merrifield (1958) on the propagation of excitation is criticized and an error is pointed out. The spread, however, remains wavelike and is not dissipative in the absence of a collisional mechanism.

Conditions for the usual form of one-electron relativistic virial theorem to be satisfied are derived and illustrated. These can be applied to generate quality basic spinors for solving a one-electron Dirac equation.

It is shown that the constrained-component variation generally suggested by Rosicky and Mark is very fundamental, has consistent variational features and reproduces, as a special case, earlier variational results for atomic systems obtained by Drake and Goldman. Numerical merits and demerits of this method are qualitatively assessed.

A set of indigenously developed computer programs for ab-initio Hartree-Fock calculations on both closed- and open-shell molecules have been described. These programs have been written for calculations using GTO basis sets. Integral formulae have been taken from Taketaet al [8]. Structures and functions of the programs have been discussed. These programs have been extensively tested. Molecular integrals over GTO basis sets have been chosen for tests and as numerical examples in this paper. Results of calculations using very accurate minimal bases have been given for methane. Time taken for these computations in a CDC Cyber 180/840 machine has been indicated. Trends in the calculations have also been illustrated by employing 4-gaussian expansions for the STO’s and by varying the basis size for LiH and BH⁺.

Computer programs forab-initio Hartree-Fock and Dirac-Hartree-Fock calculations on closed- and open-shell atoms and molecules have been indigenously developed. Sample results of high quality are given for Li, Be, LiH and Be₂. As a byproduct of these calculations the importance of considering relativistic effects in the investigation of the elusive bound-state structure of Be₂ is clearly indicated.