The Hartree-Fock Method

In Hartree-Fock theory, we approximate the many-electron wave function with a single Slater determinant 

In (2), H0 represents the nuclear-nuclear repulsion, whereas h contains the one-electron kinetic energy operator as well at the operator for the attraction of one electron from all the nuclei. Finally 

Hartree term E describing the Coulomb interaction of the electron density with itself whereas the last term is the HF-exchange energy E . In practical calculations, the occupied orbitals defining 4 are written as a linear combination of known atomic orbitals (LCAO) also referred to as basis functions 

In the 1950s, the Roothan equation was too complex and computational demanding to be of practical use due to the occurrence of the many two electron repulsion integrals rs g tu . Thus, the number of such integrals increases as M4 with the number of atomic orbitals M. As a consequence, application of the Roothan equation was in the 1950s and 1960s characterized by a number of approximations. 

1 The Wolfberg Helmholz Approximation and the Extended Hiickel Method 

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While the equations of the Hartree-Fock approach can he rigorously derived, we present them post hoc and give a physical description of the approximations leading to them. The Hartree-Fock method introduces an effective one-electron Hamiltonian. as in equation (47) on page 194 ... 

Within the periodic Hartree-Fock approach it is possible to incorporate many of the variants that we have discussed, such as LFHF or RHF. Density functional theory can also be used. I his makes it possible to compare the results obtained from these variants. Whilst density functional theory is more widely used for solid-state applications, there are certain types of problem that are currently more amenable to the Hartree-Fock method. Of particular ii. Icvance here are systems containing unpaired electrons, two recent examples being the clci tronic and magnetic properties of nickel oxide and alkaline earth oxides doped with alkali metal ions (Li in CaO) [Dovesi et al. 2000]. 

The premise behind DFT is that the energy of a molecule can be determined from the electron density instead of a wave function. This theory originated with a theorem by Hoenburg and Kohn that stated this was possible. The original theorem applied only to finding the ground-state electronic energy of a molecule. A practical application of this theory was developed by Kohn and Sham who formulated a method similar in structure to the Hartree-Fock method. 

Not all iterative semi-empirical or ab initio calculations converge for all cases. For SCF calculations of electronic structure, systems with a small energy gap between the highest occupied orbital and the lowest unoccupied orbital may not converge or may converge slowly. (They are generally poorly described by the Hartree-Fock method.)... 

Each cell in the chart defines a model chemistry. The columns correspond to differcni theoretical methods and the rows to different basis sets. The level of correlation increases as you move to the right across any row, with the Hartree-Fock method jI the extreme left (including no correlation), and the Full Configuration Interaction method at the right (which fuUy accounts for electron correlation). In general, computational cost and accuracy increase as you move to the right as well. The relative costs of different model chemistries for various job types is discussed in... 

As we have seen throughout this book, the Hartree-Fock method provides a reasonable model for a wide range of problems and molecular systems. However, Hartree-Fock theory also has limitations. They arise principally from the fact that Hartree-Fock theory does not include a full treatment of the effects of electron correlation the energy contributions arising from electrons interacting with one another. For systems and situations where such effects are important, Hartree-Fock results may not be satisfactory. The theory and methodology underlying electron correlation is discussed in Appendix A. 

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. 

Of course the Hartree-Fock method and the configuration interaction... 

But alas most of what has been described so far concerning density theory applies in theory rather than in practice. The fact that the Thomas-Fermi method is capable of yielding a universal solution for all atoms in the periodic table is a potentially attractive feature but is generally not realized in practice. The attempts to implement the ideas originally due to Thomas and Fermi have not quite materialized. This has meant a return to the need to solve a number of equations separately for each individual atom as one does in the Hartree-Fock method and other ab initio methods using atomic orbitals. 

This analysis shows that a refinement of the Hartree-Fock method by means of a correlation factor g(r12> rlz, r2Z,. . . ) seems to be possible along several lines, but the numerical work involved... 

To ensure this, the-many-body wavefunction can be written as a Slater determinant of one particle wavefunctions - this is the Hartree Fock method. The drawbacks of this method are that it is computationally demanding and does not include the many-body correlation effects. 

Even though recent progress in hardware and software development has made it possible to study quite large molecules, systems of the size considered here do not lend themselves to studies with any ab initio technique. The Hartree-Fock method has the advantage of being size consistent, which Is a necessity for this type of study when results for molecules of vastly different size are to be compared. In addition, the method is technically and economically feasible for these large systems. 

The Hartree-Fock method is modified by mixing some important valence electron configurations with the ground-state one 20>. This is called the OVC optimized valence configurations) method. 

In order to find a good approximate wave function, one uses the Hartree-Fock procedure. Indeed, the main reason the Schrodinger equation is not solvable analytically is the presence of interelectronic repulsion of the form e2/r. — r.. In the absence of this term, the equation for an atom with n electrons could be separated into n hydrogen-like equations. The Hartree-Fock method, also called the Self-Consistent-Field method, regards all electrons except one (called, for instance, electron 1), as forming a cloud of electric charge... 

When the Hartree-Fock method is applied to molecules, molecular orbitals are used instead of atomic orbitals. To construct the molecular orbitals, one widely used approximation is LCAO (linear combinations of atomic orbitals). According to molecular orbital theory, the total wave function of the system is written as a combination of molecular orbitals, spin functions describing electrons in terms of spin j(a) or — j p). 

The Hartree-Fock method adequately describes the ground state of most molecules. However, the exact wave function itself should take into account the fact that electrons repel each other and need breathing space. The electrons should be allowed to make use of energy levels which are normally empty in the ground state to maintain this breathing space. In other words, to add terms describing excited states in the ground state wave function. 

In most work reported so far, the solute is treated by the Hartree-Fock method (i.e., Ho is expressed as a Fock operator), in which each electron moves in the self-consistent field (SCF) of the others. The term SCRF, which should refer to the treatment of the reaction field, is used by some workers to refer to a combination of the SCRF nonlinear Schrodinger equation (34) and SCF method to solve it, but in the future, as correlated treatments of the solute becomes more common, it will be necessary to more clearly distinguish the SCRF and SCF approximations. The SCRF method, with or without the additional SCF approximation, was first proposed by Rinaldi and Rivail [87, 88], Yomosa [89, 90], and Tapia and Goscinski [91], A highly recommended review of the foundations of the field was given by Tapia [71],... 

In the case of the Hartree-Fock method for a closed shell systems, the new Fock matrix is written as... 

The Hartree-Fock method was in any case the method of choice for the first quantitative calculations related to homogeneous catalysis. It was the method, for instance, on a study of the bonding between manganese and hydride in Mn-H, published in 1973 [28]. The first studies on single steps of catalytic cycles in the early 1980 s used the HF method [29]. And it was also the method applied in the first calculation of a full catalytic cycle, which was the hydrogenation of olefins with the Wilkinson catalyst in 1987 [30]. The limitations of the method were nevertheless soon noticed, and already in the late 1980 s, the importance of electron correlation was being recognized [31]. These approaches will be discussed in detail in the next section. 

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