RESULTS OF THE HARTREE-FOCK METHOD

Vgxch has to replace the K operator in the Hartree-Fock equations. In the resulting equation, the Hamiltonian is a function of the density of a or p spin density. The Slater exchange has the correct dependence of the density with power of 1/3, but it seems to overcorrect for some inadequacies in the results of the Hartree-Fock method. 

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) ... 

Application of the Hartree-Fock Method. - Since numerical Hartree-Fock programs dealing with complex numbers are available in many research groups, it seemed natural to apply this scheme also to the scaled Bom-Oppenheimer Hamiltonian (4.15). As a consequence, some numerical results were obtained before the theory was developed, and - as we have emphasized in the Introduction - some features seemed rather astonishing. 

Ground State of CO. The CO molecule has been extensively studied, both experimentally and theoretically. Table 14 compares ground-state 02+) spectroscopic constants calculated by the Hartree-Fock method109 and by the density functional approach99 with experiment.104 In addition to these spectroscopic constants, the polar nature of the molecule provides a further measurable quantity, the dipole moment. Since the intensities of infrared vibration-rotation bands allow the dipole moment to be determined as a function of C-O separation this provides a useful comparison with the results of ab initio calculations. For example, the positive sign obtained from the equilibrium dipole moment by Hartree-Fock calculations was viewed as a reason to question the negative value found experimentally, whereas the current view is that the positive sign is a defect of the Hartree-Fock method. 

A concrete example of the variational principle is provided by the Hartree-Fock approximation. This method asserts that the electrons can be treated independently, and that the -electron wavefimction of the atom or molecule can be written as a Slater determinant made up of orbitals. These orbitals are defined to be those which minimize the expectation value of the energy. Since the general mathematical form of these orbitals is not known (especially in molecules), then the resulting problem is highly nonlinear and formidably difficult to solve. However, as mentioned in subsection (A 1.1.3.2). a common approach is to assume that the orbitals can be written as linear combinations of one-electron basis functions. If the basis functions are fixed, then the optimization problem reduces to that of finding the best set of coefficients for each orbital. This tremendous simplification provided a revolutionary advance for the application of the Hartree-Fock method to molecules, and was originally proposed by Roothaan in 1951. A similar form of the trial function occurs when it is assumed that the exact (as opposed to Hartree-Fock) wavefimction can be written as a linear combination of Slater determinants (see equation (A 1.1.104 ) ). In the conceptually simpler latter case, the objective is to minimize an expression of the form... 

The first target of quantum chemistry was how to solve the SchrOdinger equation for electronic motions in molecules. To address this challenge, the Hartree-Fock method (Hartree 1928) and its variational method (Slater 1928), molecular orbital theory (Hund 1926 Mulliken 1927), and the Slater determinant (Slater 1929) were developed, resulting in the Hartree-Fock method (Fock 1930 Slater 1930), which is accepted as the precursor of quantum chemistry. Soon afterward, the configuration interaction (Cl) method (Condon 1930), M0ller-Plesset perturbation method (Mpller and Plesset 1934), and multiconflgurational SCF method (Frenkel... 

The accuracy of the Hartree-Fock method (lowering of the total energy) has a limit when any further extension of the basis set fails to improve the results. 

A review of thermochemical results from the Hartree-Fock method is... 

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]. 

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. 

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. 

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