Multi-configuration Dirac-Hartree-Fock calculations

1 Multi-configuration Dirac-Hartree-Fock calculations 

As it was mentioned before the general hyperfine Hamiltonian is formulated as a summation of rank-A tensor contractions according to 

In our case of nuclear quadrupole coupling k takes on the value 2. When open shell atoms are treated relativistically a specific atomic state function rPJM) (ASF) has definite angular momentum J, Mj and parity P therefore being eigenfunction to the operators and 

Since the ASF is a linear combination of configuration state functions (CSF) according to 

Bieron et al. [106] applied the MCDHF scheme to the light elements Be and F in order to obtain very accurate A and B values including nuclear recoil eflFects, the Breit interaction and corrections for omitted virtual orbitals via the I extrapolation technique described in [107]. Both constants were calculated for the Pz/2 state of the iBe ion 

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In the most recent version of the energy-consistent pseudopotential approach the reference data is derived from finite-dilference all-electron multi-configuration Dirac-Hartree-Fock calculations based on the Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian. As an example the first parametrization of such a potential,... 

Figure 1. Total nonrelativistic multi-configuration Hartree-Fock energy, relativistic corrections (estimated as the difference between the multi-configuration Dirac-Hartree-Fock and Hartree-Fock energies) and correlation contributions (estimated from correlation energy density functional calculations) for the group 4 elements. The multi-configuration treatments were carried out with the atomic structure code GRASP [78] and correspond to complete active space calculations with the open valence p shell as active space. The nonrelativistic results were obtained by multiplying the velocity of light with a factor of 10 .

Orbital energies e (a.u.) and radial expectation values (r) (a,u.) for the valence shells of Ce and Lu from multi-conflguration Dirac-Hartree-Fock calculations for the average of the 4f 5d 6s and 4f 5d 6s configurations, respectively. The ratio of relativistic and corresponding nonrelativistic values is given in parentheses, Data taken... 

Average-level (AL) multi-configuration Dirac-Hartree-Fock (MCDHF) calculations corresponding to one nonrelativistic configuration (Dolg 1995). Only 4f was considered. 

Apparently, a large number of successful relativistic configuration-interaction (RCI) and multi-reference Dirac-Hartree-Fock (MRDHF) calculations [27] reported over the last two decades are supposedly based on the DBC Hamiltonian. This apparent success seems to contradict the earlier claims of the CD. As shown by Sucher [18,28], in fact the RCI and MRDHF calculations are not based on the DBC Hamiltonian, but on an approximation to a more fundamental Hamiltonian based on QED which does not suffer from the CD. At this point, let us defer further discussion until we review the many-fermion Hamiltonians derived from QED. 

A relativistic Dirac-Hartree-Fock calculation is somewhat more complicated than the corresponding nonrelativistic calculation due to the fact that each wavefunction has a large and a small component. Thus for the n electron problem there are 2n coupled equations in the relativistic calculation rather than n as in the nonrelativistic calculation. There is an even more severe complication however, produced by the fact that each nonrelativistic (nl) orbital corresponds to two relativistic orbitals (n,l,j = 1+ J)and (n,l,j =1 -J) (except of course, if 1= 0). Consequently, what is a one configuration Hartree-Fock (HF) calculation non-relativistically usually corresponds to a multi-configuration Hartree-Fock (MCHF) relativistically. What this implies is that a single configuration Hartree-Fock calculation is usually less likely to give accurate results in the relativistic case than in the nonrelativistic case. 

The purpose of this contribution is to give an overview of the results which center around the atomic density function and the recovery of the periodicity. Since all the calculations are based on atomic density functions, it is appropriate to revisit the construction of these densities in some depth. First a workable definition of the density function is established in the framework of the multi-configuration Hartree-Fock method (MCHF) [2] and the spherical harmonic content of the density function is discussed. A spherical density function is established in a natural way, by using spherical tensor operators. The proposed expression can be evaluated for any multi-configuration state function corresponding to an atom in a particular well-defined state and a recently developed extension of the MCHF code [3] is used for that purpose. Three illustrative examples are given. In the next section relativistic density functions for the relativistic Dirac-Hartree-Fock method [4] are defined. The latter will be used for a thorough analysis of the influence of relativistic effects on electron density functions later on in this paper. 

Results on the investigation of atomic density functions are reviewed. First, ways for calculating the density of atoms in a well-defined state are discussed, with particular attention for the spherical symmetry. It follows that the density function of an arbitrary open shell atom is not a priori spherically symmetric. A workable definition for density functions within the multi-configuration Hartree-Fock framework is established. By evaluating the obtained definition, particular influences on the density function are illustrated. A brief overview of the calculation of density functions within the relativistic Dirac-Hartree-Fock scheme is given as well. 

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