Dirac-Hartree-Fock-Roothaan

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 ... 

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. 

Dirac-Hartree-Fock-Roothaan Matrix Equations... 

In order to determine these unknowns the variational minimax principle of chapter 8 is invoked. For this procedure, we may again start from the energy expression of section 10.2 and differentiate it or directly insert the basis set expansion of Eq. (10.3) into the SCF Eqs. (8.185). These options are depicted in Figure 10.2. The resulting Dirac-Hartree-Fock equations in basis set representation are called Dirac-Hartree-Fock-Roothaan equations according to the work by Roothaan [511] and Hall [512] on the nonrelativistic analog. 

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,

[Pg.533]

The relativistic theory and computation of atomic structures and processes has therefore attained some sort of maturity and the various codes now available are widely used. Those mentioned so far were based on ideas originating from Hartree and his students [28], and have been developed in much the same way as the non-relativistic self-consistent field theory recorded in [28-30]. All these methods rely on the numerical solution, using finite differences, of the coupled differential equations for radial orbital wave-functions of the self-consistent field. This makes them unsuitable for the study of molecules, for which it is preferable to expand the radial amplitudes in a suitably chosen set of analytic functions. This nonrelativistic matrix Hartree-Fock method, as it is often termed, was pioneered by Hall and Lennard-Jones [31], Hall [32,33] and Roothaan [34,35], and it was Roothaan s students, Synek [36] and Kim [37] who were the first to attempt to solve the corresponding matrix Dirac-Hartree-Fock equations. Kim was able to obtain solutions for the ground state of neon in 1967, but at the expense of some numerical instability, and it seemed at the time that the matrix Dirac-Hartree-Fock scheme would not be a serious competitor to the finite difference codes. 

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. 

Figure 10.2 Two routes for the derivation of Dirac-Hartree-Fock equations in basis set representation in the iower right oorner the Roothaan equations (recall the caveats in sections 8.2.3 and 8.7.1 required for the application of the variational principle).

Hence, the relativistic analog of the spin-restriction in nonrelativistic closed-shell Hartree-Fock theory is Kramers-restricted Dirac-Hartree-Fock theory. We should emphasize that our derivation of the Roothaan equation above is the pedestrian way chosen in order to produce this matrix-SCF equation step by step. The most sophisticated formulations are the Kramers-restricted quaternion Dirac-Hartree-Fock implementations [286,318,319]. A basis of Kramers pairs, i.e., one adapted to time-reversal s)mimetry, transforms into another basis under quatemionic unitary transformation [589]. This can be exploited not only for the optimization of Dirac-Hartree-Fock spinors, but also for MCSCF spinors. In a Kramers one-electron basis, an operator O invariant under time reversal possesses a specific block structure. 

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... 

Similar to nonrelativistic Hartree-Fock theory, the Dirac-Roothaan Eqs. (10.61) are solved iteratively until self-consistency is reached. However, because of the properties of the one-electron Dirac Hamiltonian entering the Fock operator, molecular spinors representing unphysical negative-energy states (recall section 5.5) show up in this procedure. As many of these negative-continuum... 

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