Dirac-Coulomb equation application

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. 

It should be noted that the two-electron term in this Dirac—Coulomb equation is based on a classical (i.e., nonrelativistic) picture of the interaction, and it is therefore not Lorentz invariant. In many applications this is a problem of minor importance, and results that are in good agreement with experiment often are obtained with the Dirac-Coulomb equation (Eqs. [71]—[74]) or with theory derived from it through further simplifications. For situations calling for a more accurate model, a relativistic picture of the electron—electron interaction is given by the Breit operator ... 

In this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. 

The combination of the Dirac-Kohn-Sham scheme with non-relativis-tic exchange-correlation functionals is sometimes termed the Dirac-Slater approach, since the first implementations for atoms [13] and molecules [14] used the Xa exchange functional. Because of the four-component (Dirac) structure, such methods are sometimes called fully relativistic although the electron interaction is treated without any relativistic corrections, and almost no results of relativistic density functional theory in its narrower sense [7] are included. For valence properties at least, the four-component structure of the effective one-particle equations is much more important than relativistic corrections to the functional itself. This is not really a surprise given the success of the Dirac-Coulomb operator in wave function based relativistic ab initio theory. Therefore a major part of the applications of relativistic density functional theory is done performed non-rela-tivistic functionals. 

We will describe, in some detail, one such modification, an effective Dirac equation (EDE) which was derived in a number of papers [7, 8, 9, 10]. This new equation is more convenient in many applications than the original BS equation, and we will derive some general formulae connected with this equation. The physical idea behind this approach is that in the case of a loosely bound system of two particles of different masses, the heavy particle spends almost all its life not far from its own mass shell. In such case some kind of Dirac equation for the light particle in an external Coulomb field should be an excellent starting point for the perturbation theory expansion. Then it is convenient to choose the free two-particle propagator in the form of the product of the heavy particle mass shell projector A and the free electron propagator... 

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