Dirac-Coulomb energies/results

Abstract. An application of the Rayleigh-Ritz variational method to solving the Dirac-Coulomb equation, although resulted in many successful implementations, is far from being trivial and there are still many unresolved questions. Usually, the variational principle is applied to this equation in the standard, Dirac-Pauli, representation. All observables derived from the Dirac equation are invariant with respect to the choice of the representation (i.e. to a similarity transformation in the four-dimensional spinor space). However, in order to control the behavior of the variational energy, the trial functions are subjected to several conditions, as for example the kinetic balance condition. These conditions are usually representation-dependent. The aim of this work is an analysis of some consequences of this dependence. 

Thble3.6 Bond length Rc (A), vibrational constant coe (cm-1) and binding energy De (eV) of Eka-Au hydride (111)H without (with) counterpoise correction of the basis-set superposition error. All-electron (AE) values based on die Dirac-Coulomb-Hamiltonian (Seth and Schw-erdtfeger 2000) are compared with valence-only results obtained with energy-consistent (EC) (Dolg etal. 2001) and shape-consistent (SC) (Han and Hirao 2000) pseudopotentials (PP). The numbers 19 and 34 in parentheses denote the number of valence electrons for the Eka-Au PP. 

The fine structure was calculated by Pirenne [110] and independently by Berestetski [4J. Minor errors are corrected, and numerical results are given by Ferrell [45]. The approach used by these authors is to write down the Dirac equations for the two particles, and the interaction terms as they are expressed in quantum field theory. The equations can be transformed so that the particle spins appear explicitly. The interaction terms are found to comprise the Coulomb energy, the Breit interaction, and a term analogous to the Fermi expression for... 

However, the most remarkable result of these calculations is the rapidly increasing deviation of the DKH4 energies from the numerical, i.e., exact Dirac-Fock-Coulomb (DEC) results. The DEC values for the first four noble gases presented are taken from the benchmark results for point nuclei by Visscher and Dyall [65]. Eor Eka-Radon we have calculated the DEC energy with Molfdir... 

The nonrelativistic Hartree-Fock theory (abbreviated HF in the equations to follow) formally resembles Dirac-Hartree-Fock (DHF) theory for the Dirac-Coulomb Hamiltonian. Of course, for large c of, say, 10 to 10 a.u. they even yield the same results. For this reason we shall make an explicit comparison of both in this section. The total energy for closed-shell atoms after integration over all angular and spin coordinates is in both cases given by... 

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