Hartree and Fock

When there is more than one electron, the best Rnps are of the SCF (self-consistent-field) type obtainable by the method of Hartree and Fock. These are not given by analytical expressions, and are usually presented in the form of numerical tables. Moreover, these SCF R t s as tabulated are not always all orthogonal however, it is always jxjssible to find an equivalent set of equally good SCF R,u which are orthogonal.2 For practical computations, as Slater has shown,3 the SCF Rni may be approximated passably well by a finite series similar to that of Eq. (4) but with a different exponential factor in each term ... 

Orbital models are almost invariably the first approximation for the electronic structure of molecules within the Born-Oppenheimer approximation. By considering the motion of each electron in the averaged field of the remaining electrons in the systems in the self-consistent field theory of Hartree and Fock, a model is obtained in which each electron is described by an spin-orbital, a description which is familiar to all chemists. 

The secular equations for the Fock operator will have, of course, the form of the Hartree and Fock-Roothaan equations (cf. Chapter 8, p. 431) ... 

Fig. 11.1 Number of publications that employ DFT each year (from 1980 to September 23,2011) based on search from Web of Science database (http //apps. webofknowledge.com) using density, functional, and theory as the keywords. This is compared with a similar search for keywords Hartree and Fock, using Web of Science database...

Hartree and Fock (Fock 1930 Hartree 1928) formalism uses the single SD form of the total wave function, which is solved under self-consistent field (SCF) approximation. This involves an iterative process in which the orbitals are improved cycle to cycle until the electronic energy reaches a constant minimum and the orbitals no longer change. Upon convergence of the SCF method, the minimum-energy MOs produce an electric field that generates the same orbitals and hence the self-consistency. 

The advent of a rigorous quantum mechanics in the period 1925-26 provided a more deductive approach to electronic configurations at least in principle. But not until methods of approximation had been devised by the likes of Hartree and Fock did it become possible to solve the Schrodinger equation for any particular atom to a reliable level of accuracy. From this time onwards the electronic configurations of atoms could be deduced in an ab initio maimer, a claim that has been disputed by some philosophers of chemistry [Scerri, 2004] but defended by some theoretical chemists and physicists [Schwarz, 2007 2009 Ostrovsky, 2001 Freidrich, 2004]. 

The treatment of the many-body system of nuclei and electrons of which solids and metals consist, makes it necessary to introduce approximations. Traditional approximations in electronic structure theory are those due to Hartree and Fock. We shall here only briefly give the main equations relevant for the discussion of the free-electron gas results (Chapter 9) and the density functional theory (Chapter 10). Consider, for instance, a metal of N atoms and each atom having Z electrons, where the number of atoms is of the order Avogadros number. The number of electrons to be considered for each atom can be lowered from the actual number to the number of valence electrons by introducing effective core potentials. The nonrelativistic Schrddinger equation for the electronic part of the problem is then... 

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. 

In the bibliography, we have tried to concentrate the interest on contributions going beyond the Hartree-Fock approximation, and papers on the self-consistent field method itself have therefore not been included, unless they have also been of value from a more general point of view. However, in our treatment of the correlation effects, the Hartree-Fock scheme represents the natural basic level for study of the further improvements, and it is therefore valuable to make references to this approximation easily available. For atoms, there has been an excellent survey given by Hartree, and, for solid-state, we would like to refer to some recent reviews. For molecules, there does not seem to exist something similar so, in a special list, we have tried to report at least the most important papers on molecular applications of the Hartree-Fock scheme, t... 

Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg,... 

The so-called Hartree-Fock-Slater method is much more widely utilized, and is a hybrid of the Hartree and Thomas-Fermi-Dirac methods. In this method the direct part of the potential is calculated using the Hartree-Fock approach, whereas the exchange part is approximated by some statistical expression of the model of free electrons. The Slater potential is given by ... 

I have dealt at length with the Hartree and the Hartree-Fock models. The father of this field, Sir William Hartree, was concerned with the atomic problem where it is routinely possible to integrate numerically the HF integro-differential equations in order to produce (numerical) wavefunctions that correspond to the Hartree-Fock limit. For molecular applications the LCAO variant of HF theory assumes a dominant role because of the reduced symmetry of the problem. 

This equation cannot be solved exactly. The most often used approximation model to solve this equation is called the self-consistent field (SCF) method, first introduced by D. R. Hartree and V. A. Fock. The physical picture of this method is very similar to our treatment of helium each electron sees an effective nuclear charge contributed by the nuclear charge and the remaining electrons. 

On the other hand, ab initio (meaning from the beginning in Latin) methods use a correct Hamiltonian operator, which includes kinetic energy of the electrons, attractions between electrons and nuclei, and repulsions between electrons and those between nuclei, to calculate all integrals without making use of any experimental data other than the values of the fundamental constants. An example of these methods is the self-consistent field (SCF) method first introduced by D. R. Hartree and V. Fock in the 1920s. This method was briefly described in Chapter 2, in connection with the atomic structure calculations. Before proceeding further, it should be mentioned that ab initio does not mean exact or totally correct. This is because, as we have seen in the SCF treatment, approximations are still made in ab initio methods. 

Cohen AJ, Baerends EJ (2002) Variational density matrix functional calculations for the corrected Hartree and corrected Hartree-Fock functionals, Chem. Phys. Lett, 364 409—419... 

It is not simple to demonstrate that 7/3 is the correct power of Z for real atoms when Z < 100. In fact Foldy showed that up to this value of Z, Hartree and Hartree-Fock data are slightly better fitted with Z12/6. Nevertheless, for a non-relativistic theory there can be no doubt that for really large values of Z the total binding energy must vary as Z7/3. We shall see below, when we deal with... 

An excellent description of the Hartree and Hartree-Fock method is given by Slater, J. C. Quantum theory of atomic structure, Vol. I, II. New York McGraw-Hill 1960. 

When approximate solutions are extended to heavier atoms and to molecules, it is necessary to be content with a rather lower accuracy (e.g, 1 to 2%). The most fruitful approximation method is that of Hartree (1928), later justified and refined by Slater (1930) and Fock (1930). This is suggested by lirst neglecting the electron repulsion and observing that the probability function P (1, 2,. .. N) must then be approximated by the product (1) Pg(2). .. Pj (N), for then the probability of any configuration of the electrons is a product of the probabilities of N independent events. Since P = Pf, this means the many-electron wave function must also be a product... 

A very important conceptual step within the MO framework was achieved by the introduction of the independent particle model (IPM), which reduces the AT-electron problem effectively to a one-electron problem, though a highly nonlinear one. The variation principle based IPM leads to Hartree-Fock (HF) equations [4, 5] (cf. also [6, 7]) that are solved iteratively by generating a suitable self-consistent field (SCF). The numerical solution of these equations for the one-center atomic problems became a reality in the fifties, primarily owing to the earlier efforts by Hartree and Hartree [8]. The fact that this approximation yields well over 99% of the total energy led to the general belief that SCF wave functions are sufficiently accurate for the computation of interesting properties of most chemical systems. However, once the SCF solutions became available for molecular systems, this hope was shattered. 

---