Dirac-Hartree-Fock solutions

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. 

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. 

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. 

Dirac s relativistic theory for the motion of electrons in molecules was introduced in the preceding chapters. The appearance of positron solutions and the four-component form of the wave function looks problematic at first sight but in practice it turns out that the real challenge is, like in non-relativistic electronic structure theory, the description of the correlation between the motion of electrons. The mean-field approximation that is made in the Dirac-Hartree-Fock (DHF) approach provides a good first step, but gives bond energies and structures that are often too inaccurate for chemical purposes. 

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. 

Solving the canonical Hartree-Fock equations for a particular molecule directly yields the canonical spinors of that molecule. How the equations can be solved is discussed from a technical point of view in the next section, while their explicit solution procedures are presented in chapters 9 and 10. The terms "Dirac-Hartree-Fock", "Dirac-Fock", and "four-component Hartree-Fock" are used synonymously in research papers. 

The developments of this section show that for energy solutions in the domain of interest to us, the Rayleigh quotient is bounded below, and there is therefore no danger of variational collapse when solving the Dirac equation in a kinetically balanced finite basis. For the Dirac-Hartree-Fock equations, the only addition is the electron-electron interaction, which is positive and therefore will not contribute to a variational collapse. 

The modified two-electron terms contain all the relativistic integrals, which means that the integral work is no different from that in the full solution of the Dirac-Hartree-Fock equations. It would save a lot of work if we could approximate the integrals, in the same way as we did for the Douglas-Kroll-Hess approximation. To do so, we must use the normalized Foldy-Wouthuysen transformation. The DKH approximation neglects the commutator of the transformation with the two-electron Coulomb operator, and in so doing removes all the spin-dependent terms. We must therefore also use a spin-free one-electron Hamiltonian. The approximate Hamiltonian (in terms of operators rather than matrices) is... 

Applications to atoms are in most cases based on the publicly available programs using finite difference methods for integration in the solution of the (multi-configurational) Dirac-Hartree-Fock equations. The problem of introducing electron correlation in this framework is most successfully accomplished by employing complete active space (CAS) and restricted active space (RAS) techniques (see Ref. 84 for a recent application with further references to the literature) or coupled-cluster techniques. ... 

Electron correlation effects can be defined as the difference between results obtained from the exact solution of a Schrodinger equation with a specific Hamiltonian, and the results obtained at the imcorrelated level, e.g., at the Hartree-Fock or Dirac-Hartree-Fock level. Since for all but the simplest problems the exact solution of the Schrodinger equation is not accessible and usually approximate correlated wavefunctions are used instead. Sometimes experimental values are used rather than the results for the exact solution, which is reasonable as long as the Hamiltonian used for the uncorrelated solution includes all important terms, e.g., with regard to relativistic contributions, influence of the environment of the studied system, etc. As for relativistic effects, the magnitude of electron correlation effects depends to some extent on the details of their evaluation [23]. 

The Xa multiple scattering method generates approximate singledeterminant wavefunctions, in which the non-local exchange interaction of the Hartree-Fock method has been replaced by a local term, as in the Thomas-Fermi-Dirac model. The orbitals are solutions of the one-electron differential equation (in atomic units)... 

Heavy atoms exhibit large relativistic effects, often too large to be treated perturba-tively. The Schrodinger equation must be supplanted by an appropriate relativistic wave equation such as Dirac-Coulomb or Dirac-Coulomb-Breit. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for the lighter elements, and may be included in a similar manner. 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

Historically16 it is worthy of note that if one resorts in equation (51) to the TF approximation (18) for tr, then the Euler equation of the Thomas-Fermi-Dirac method results. We shall not go into the solutions of the Thomas-Fermi-Dirac equation in this review, though there has been recent interest in this area. Suffice it to say that in the full form of the Euler equation (51), we are working at the customary Hartree-Fock-Slater level. However, we shall content ourselves, until we come to Section 17 below, with understanding in a more intuitive, but inevitably less detailed, way how the corrections to the TF energy in equation (48) arise. 

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