Atoms numerical Hartree-Fock methods

For small highly symmetric systems, like atoms and diatomic molecules, the Hartree-Fock equations may be solved by mapping the orbitals on a set of grid points. These are referred to as numerical Hartree-Fock methods. However, essentially all calculations use a basis set expansion to express the unknown MOs in terms of a set of known functions. Any type of basis function may in principle be used expo ... 

All the early work was concerned with atoms, with Sir William Hartree regarded as the father of the technique. His son, Douglas R. Hartree, published the definitive book, The Calculation of Atomic Structures, in 1957, and in this he derived the atomic HF equations and described numerical algorithms for their solution. Charlotte Froese Fischer was a research student working under the guidance of D. R. Hartree, and she published her own definitive book. The Hartree—Fock Method for Atoms A Numerical Approach in 1977. The Appendix lists a number of freely available atomie structure programs. Most of these can be obtained from the Computer Physics Communications Program Library. 

Fischer, C. F. (1977) The Hartree-Fock Method for Atoms A Numerical Approach, Wiley, New York. 

The radial functions f (r) will be different for different atoms. Only for the hydrogen atom is the exact analytical form of the i2((r) s known. For other atoms the f (r) s will be approximate and their form will depend on the method used to find them. They might be analytical functions (e.g. Slater orbitals) or tabulated sets of numbers (e.g. numerical Hartree-Fock orbitals). 

Schrddinger equation (280) represents a time-dependent scheme which is numerically much less involved than, e.g., the time-dependent Hartree-Fock method. Numerical results obtained with this scheme for atoms in strong laser pulses will be described in Sect. 8. 

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 best-known and widely-quoted tabulation of atomic Dirac-Hartree-Fock energies was published by Desclaux [11], covered elements in the range Z=1 to Z=120 using finite difference methods. A number of computer packages are available to perform MCDHF calculations [19]. Published DHF and Dirac-Fock-Slater (DFS) calculations for atoms are now too numerous to construct a comprehensive catalogue. It is, however, possible to sort the purposes for which these calculations have been performed into general classes. 

The cluster model of HAp/methyl acetate interface was shown in Fig.2 overlap population analysis was applied to this model. Using Monte Carlo method, 300 sampling points were put around each atom in the cluster. Molecular orbitals in the cluster were constructed by a linear combination of atomic orbitals (LCAO). Atomic orbitals used in this model were ls-2p for C, ls-2p for O, Is for H, ls-3d for P and ls-4p for Ca, which were numerically calculated for atomic Hartree-Fock method. Overlap population was evaluated by Mulliken s population analysis. 

The generator coordinate method (GCM), as initially formulated in nuclear physics, is briefly described. Emphasis is then given to mathematical aspects and applications to atomic systems. The hydrogen atom Schrodinger equation with a Gaussian trial function is used as a model for former and new analytical, formal and numerical derivations. The discretization technique for the solution of the Hill-Wheeler equation is presented and the generator coordinate Hartree-Fock method and its applications for atoms, molecules, natural orbitals and universal basis sets are reviewed. A connection between the GCM and density functional theory is commented and some initial applications are presented. 

Originally, Hartree-Fock atomic calculations were done by using numerical methods to solve the Hartree-Fock differential equations (11.12), and the resulting orbitals were given as tables of the radial functions for various values of r. [The Numerov method (Sections 4.4 and 6.9) can be used to solve the radial Hartree-Fock equations for the radial factors in the Hartree-Fock orbiteds the angular factors are spherical harmonics. See D. R. Heu tree, The Calculation of Atomic Structures, Wiley, 1957 C. Froese Fischer, The Hartree-Fock Method for Atoms, Wiley, 1977.]... 

Froese-Fischer C (1977) The Hartree-Fock method for atoms a numerical approach. Wiley-VCH, New York... 

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