Dirac-Hartree-Fock-Roothaan calculations

Free-ion average energies of a nonrelativistic configuration relative to the energy of the Ce ion from Dirac-Hartree-Fock-Roothaan calculations, in comparison with experimental data ... 

Numerical discretization methods pose an interesting consequence for fully numerical Dirac-Hartree-Fock calculations. These grid-based methods are designed to directly calculate only those radial functions on a given set of mesh points that occupy the Slater determinant. It is, however, not possible to directly obtain any excess radial functions that are needed to generate new CSFs as excitations from the Dirac-Hartree-Fock Slater determinant. Hence, one cannot directly start to improve the Dirac-Hartree-Fock results by methods which capture electron correlation effects based on excitations that start from a single Slater determinant as reference function. This is very different from basis-set expansion techniques to be discussed for molecules in the next chapter. The introduction of a one-particle basis set provides so-called virtual spinors automatically in a Dirac-Hartree-Fock-Roothaan calculation, which are not produced by the direct and fully numerical grid-based approaches. 

The optimization with respect to the spinors can be accomplished by obeying the minimax principle, and positronic energy states are allowed to relax in this correlation method (as in Dirac-Hartree-Fock-Roothaan calculations)... 

All exact-decoupling approaches can be related to the modified Dirac equation and we closely follow here the work presented in Refs. [16,647]. Two-component electrons-only Hamiltonians can be obtained from block-diagonalizing the four-component (one-electron) modified Dirac equation in matrix representation. As we have discussed in chapters 8 and 10 for four-component Dirac-Hartree-Fock-Roothaan calculations, basis functions for the small component must fulfill certain constraints as otherwise variational instability and a wrong nonrelativistic limit [547] would result. The correct nonrelativistic limit will be obtained if the kinetic-balance condition,

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All-electron (AE) calculations are certainly the most rigorous way to treat atoms and molecules, however, the computational requirements are sometimes prohibitive, especially for molecular Dirac-Hartree-Fock-Roothaan (DHFR) and subsequent configuration interaction (Cl) calculations. Nevertheless, very accurate all-electron calculations on small systems yield important reference data for the calibration of more approximate computational schemes, e.g. valence-electron (VE) methods, which may be applied to larger systems. 

Energies of the lowest levels of a 4f configuration on Eu and their degeneracies (d) in the O crystal field calculated within the Dirac-Hartree-Fock-Roothaan and complete open-shell configuration interaction scheme ... 

The momentum wave functions in various atomic models are calculated for arbitrary atomic orbitals. The nonrelativistic hydrogenic, the Hartree-Fock, the relativistic hydrogenic, and the Dirac-Fock models are considered. The momentum wave functions are obtained as a Fourier transform of the wave function in the position space. The Hartree-Fock and the Dirac-Fock wave functions in atoms are given in terms of Slater-type orbitals (STO s), i.e. the Hartree-Fock-Roothaan (HFR) method and the relativistic HFR (RHFR) method. All the wave functions in the momentum space can be expressed analytically in terms of hypergeometric functions. 

Other calculations tested using this molecule include two-dimensional, fully numerical solutions of the molecular Dirac equation and LCAO Hartree-Fock-Slater wave functions [6,7] local density approximations to the moment of momentum with Hartree-Fock-Roothaan wave functions [8] and the effect on bond formation in momentum space [9]. Also available are the effects of information theory basis set quality on LCAO-SCF-MO calculations [10,11] density function theory applied to Hartree-Fock wave functions [11] higher-order energies in... 

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