Analytical Relativistic Self-Consistent Field Theory

Abstract

screened potential were calculated and compared with the result of exact phase-shift calculations, JR CX . Table III lists these results in detail and permits the recognition of the following:(1) The Moliere approximation renders R by Eqs.(15) and (16) with an error of the order £ [Eq. (5) and Table I].(2) The classical approximation renders R c \^s by Eqs. (22) and (23) with an error which is comparable to that of R for scattering angles larger than 10°.(3) The large-angle approximation renders RL.A. by Eqs. (25) and (26) with errors generally larger than °f ^class* (4) The first-order Born approximation gives values, i?Bom, the errors of which exceed those of all the other approximations. Even for the light element Z=29 and for small angles the errors are larger than 10%. ACKNOWLEDGMENTSWe are indebted to J. W. Motz for numerous discussions and for pointing out to us the need for rather simple and yet sufficiently accurate calculations of the elastic electron-scattering cross section.We would also like to acknowledge the valuable assistance of R. L. Scott in preparing the program and performing the computations.The analytical self-consistent field (SCF) theory, based on the relativistic Breit equation generalized for many particles, was developed for closed-shell systems. The relativistic SCF equations, both of the absolute and of the expansion method type, were derived in the four-component spinor representation. The Breit operator was considered in the first-order perturbation theory. The formulas for the relativistic atomic integrals were derived in terms of simple functions. * The work was originated at

Basis set expansion of the dirac operator without variational collapse

“…(1) by its matrix representation ~$ = ( p m c~+ c a p +~) @ =~+ (2) (we use carets to designate operators and bold face italic or underlining for matrices) in analogy what one usually does in nonrelativistic molecular theory is relatively old [14][15][16][17][18]. Applications of this idea have been rare until quite recently.…”

Ab initiocalculation of the enhancement of the electric dipole moment of an electron in the YbF molecule

“…A single Slater determinant of molecular orbitals of the form given in equation (3.1) approximates the molecular wavefunction in the Born-Oppenheimer approximation [41]. The extremum of the expectation value of the Hamiltonian (3.8) with respect to variations in the coefficients X P ,Q Ip yields the relativistic analogue [42] of the Roothaan-Hall selfconsistent-field procedure [43,44], which is used to determine the coefficients X P ,Q Ip . In the atomic case, all basis functions have a single centre: r p → r for all p. The objective of the calculations undertaken for the Yb atom was to describe the J π = 0 + ground state of Yb as accurately as possible within the basis-set Dirac-Fock model limited in size by the computational resources that would be available for the more demanding YbF model that would be taken up subsequently: it is possible to construct basis-set DF models of atoms and ions that are as accurate as numerical models or even more accurate than numerical models [22]; however, the related molecular model then becomes prohibitively large for all but the largest existing computers.…”

Basis-set methods for the Dirac equation

“…Asaad [4], who observed a physically correct variational energy for a Dirac Hamiltonian that was a saddle point instead of a true minimum, appears to have been the first to see variational collapse. Synek [5] was the first to formulate Roothaan's [6] finite-basis-set methods for relativistic self-consistent-field calculations. Kim [7], who discovered that poor variational approximations to the wave function can give energies that are too low, was the first to actually perform such calculations.…”