Dirac-Hartree-Fock equation

The Dirac equation can be readily adapted to the description of one electron in the held of the other electrons (Hartree-Fock theory). This is called a Dirac-Fock or Dirac-Hartree-Fock (DHF) calculation. 

These (see Chapter 2) may be obtained utilizing the relativistic analogue of the Hartree-Fock method, normally called the Dirac-Hartree-Fock method [176-178], The relevant equations may be found in an analogous manner to the non-relativistic case, therefore here we shall present only final results (in a.u. let us recall that X = nlj, X = nl j) ... 

The relativistic or non-relativistic random-phase approximation (RRPA or RPA)t is a generalized self-consistent field procedure which may be derived making the Dirac/Hartree-Fock equations time-dependent. Therefore, the approach is often called time-dependent Dirac/Hartree-Fock. The name random phase comes from the original application of this method to very large systems where it was argued that terms due to interactions between many alternative pairs of excited particles, so-called two-particle-two-hole interactions ((2p-2h) see below) tend to... 

Accounting for relativistic effects in computational organotin studies becomes complicated, because Hartree-Fock (HF), density functional theory (DFT), and post-HF methods such as n-th order Mpller-Plesset perturbation (MPn), coupled cluster (CC), and quadratic configuration interaction (QCI) methods are non-relativistic. Relativistic effects can be incorporated in quantum chemical methods with Dirac-Hartree-Fock theory, which is based on the four-component Dirac equation. " Unformnately the four-component Flamiltonian in the all-electron relativistic Dirac-Fock method makes calculations time consuming, with calculations becoming 100 times more expensive. The four-component Dirac equation can be approximated by a two-component form, as seen in the Douglas-Kroll (DK) Hamiltonian or by the zero-order regular approximation To address the electron cor-... 

An overview of the development of the finite difference Hartree-Fock method is presented. Some examples of it axe given construction of sequences of highly accurate basis sets, generation of exact solutions of diatomic states, Cl with numerical molecular orbitals, Dirac-Hartree-Fock method based on a second-order Dirac equation. 

Any realistic description of molecules containing heavy atoms has to take into account relativistic effects (13,41). Attempts to use the algebraic approach to solve the Dirac-Hartree-Fock (DHF) equations are now well advanced (42-45). The difficulties encountered axe much greater than in the nonrelativistic case since the basis sets used have to be larger and have to fulfil the kinetic balance criterion to guarantee the proper description of the large and small components of the molecular orbitals (46-49). 

For the same reasons as in the nonrelativistic case the availability of a numerical solver of the DHF equations for molecules would be very much desired. One possible way to proceed would be to deal with the DHF method cast in the form of the second-order equations instead of the system of first-order coupled equations and try to solve them by means of techniques used in the FD HF approach. The FD scheme was used by Laaksonen and Grant (50) and Sundholm (51) to solve the Dirac equation. Sundholm used the similar approach to perform Dirac-Hartree-Fock-Slater calculations for LiH, Li2, BH and CH+ systems (52,53). 

The finite difference HF scheme can also be used to solve the Schrodinger equation of a one-electron diatomic system with an arbitrary potential. Thus the approach can be applied, for example, to the construction of exchange-correlation potentials employed by the density functional methods. The eigenvalues of several GaF39+ states have been reported and the Th 79+ system has been used to search for the influence of the finite charge distribution on the potential energy curve. It has been also indicated that the machinery of the finite difference HF method could be used to find exact solutions of the Dirac-Hartree-Fock equations based on a second-order Dirac equation. 

With this ansatz the Dirac-Hartree-Fock equations for the determination of the orbitals can be obtained using the variation method. Their form is... 

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 matrix form of the atomic Dirac-Hartree-Fock (DHF) equations was presented by Kim [37,95], who used a basis set of modified radial Slater-type functions, without the benefit of a balancing presciption for the small component set. A further presentation of the atomic equations was made by Kagawa [96], who generalized Kim s work to open shells and discussed matrix element evaluation. An extension to include the low-ffequency form of the Breit interaction self-consistently in an S-spinor basis was presented by Quiney [97], who demonstrated that this did not produce variational collapse. Our presentation of the DHFB method is based on [97-99]. 

The expression above now contains the response of the wave function to the perturbation Q which can be accomplished by solving the coupled-perturbed Dirac-Hartree-Fock equations (CPDHF) yielding a different set of MO-coefficients for the molecular wave function [138] according... 

The relativistic form of the one-electron Schrodinger equation is the Dirac equation. One can do relativistic Hartree-Fock calculations using the Dirac equation to modify the Fock operator, giving a type of calculation called Dirac-Fock (or Dirac-Hartree-Fock). Likewise, one can use a relativistic form of the Kohn-Sham equations (15.123) to do relativistic density-functional calculations. (Relativistic Xa calculations are called Dirac-Slater or Dirac-Xa calculations.) Because of the complicated structure of the relativistic KS equations, relatively few all-electron fully relativistic KS molecular calculations that go beyond the Dirac-Slater approach have been done. [For relativistic DFT, see E. Engel and R. M. Dreizler, Topics in Current Chemistry, 181,1 (1996).]... 

Early work on the finite basis set problem in relativistic calculations has been reviewed by Kutzelnigg. Spurious unphysical solutions of the Dirac equation or the Dirac-Hartree-Fock equations are observed with too small a kinetic energy, leading to an overestimation of the binding energy. Furthermore, these solutions are found neither to tend to the solutions of the Schrodinger equation in the limit c- co nor to vary systematically with increasing size of basis set. 

---