HF, Hartree-Fock method

Different ab initio methods can be characterized by their treatment of electron-electron interactions, called electron correlation. The first practical ab initio method was the HF (Hartree-Fock) method, which treats each electron as if it exists in a nniform field made from the total charge and space occnpied by the other electrons. This treatment is only an approximation to the interactions between electrons as point charges in a dynamics system and exclndes the contribution of excited electronic configurations. This neglect of electron correlation can lead to significant error in determining thermochemical properties. 

HF Hartree-Fock method for self-consistent fields ... 

A variation on the HF procedure is the way that orbitals are constructed to reflect paired or unpaired electrons. If the molecule has a singlet spin, then the same orbital spatial function can be used for both the a and P spin electrons in each pair. This is called the restricted Hartree-Fock method (RHF). 

HF (Hartree-Fock) an ah initio method based on averaged electron-electron interactions... 

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. 

The corresponding estimate for the second eigenvalue (2s orbital energy) is —0.1789. These results are in good agreement with the actual HF/STO-3G ( Hartree-Fock method with a variational basis set of three-term Gaussians for each Slater-type orbital 10) eigenvalues eis = —2.3692 and e2s = —0.1801. 

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. 

The various methods used in quantum chemistry make it possible to compute equilibrium intermolecular distances, to describe intermolecular forces and chemical reactions too. The usual way to calculate these properties is based on the independent particle model this is the Hartree-Fock method. The expansion of one-electron wave-functions (molecular orbitals) in practice requires technical work on computers. It was believed for years and years that ab initio computations will become a routine task even for large molecules. In spite of the enormous increase and development in computer technique, however, this expectation has not been fulfilled. The treatment of large, extended molecular systems still needs special theoretical background. In other words, some approximations should be used in the methods which describe the properties of molecules of large size and/or interacting systems. The further approximations are to be chosen carefully this caution is especially important when going beyond the HF level. The inclusion of the electron correlation in the calculations in a convenient way is still one of the most significant tasks of quantum chemistry. 

Although HF theory is useful in its own right for many kinds of investigations, there are some applications for which the neglect of electron correlation or the assumption that the error is constant (and so will cancel) is not warranted. Post-Hartree-Fock methods seek to improve the description of the electron-electron interactions using HF theory as a reference point. Improvements to HF theory can be made in a variety of ways, including the method of configuration interaction (Cl) and by use of many-body perturbation theory (MBPT). It is beyond the scope of this text to treat Cl and MBPT methods in any but the most cursory manner. However, both methods can be introduced from aspects of the theory already discussed. 

An analysis of the ESR spectrum of the dibenzothiophene radical anion46 yields the following hfs (hyperfine splitting) constants (gffuss) 5.16, 4.48, 1.46, and 0.86. The theoretical values based on HMO data for Model A2 are considerably smaller 2.84, 2.48, 1.47, and 0.27, respectively, which led the authors to make spin-density calculations by the Hartree-Fock method. Quite recently, the spin densities have been calculated for Model B (8S = 1, pcs = 0.566),466 and the following constants were obtained 5.03,3.99,0.75, and — 1.23. A study of the ESR spectrum of the radical derived from 2,8-dimethyl-dibenzothiophene permitted the assignment of the lowest hfe constant value to the proton in position 2. In contrast to the dithiins, experimental data for dibenzothiophene radicals are better reproduced by Model B. 

The success of the Hartree-Fock method in describing the electronic structure of most closed-shell molecules has made it natural to analyze the wave function in terms of the molecular orbitals. The concept is simple and has a close relation to experiment through Koopmans theorem. The two fundamental building blocks of Hartree-Fock (HF) theory are the molecular orbital and its occupation number. In closed-shell systems each occupied molecular orbital... 

Thus with a much larger basis, but still using the Hartree-Fock method, the FOOF geometry is about the same and the O3 geometry has become even worse than at the HF/6-31G level ... 

The Hartree-Fock method has the advantage that it can be applied to relatively large molecules, e.g. octanitrocubane, with a reasonably good basis set, such as 6-31G. Since no electronic correlation is taken into account, Hartree-Fock calculations of heats of formation should be attempted only in conjunction with an isodesmic or homodesmic reaction [24]. If suitable ones are chosen, satisfactory results may be obtained [25,26]. On the other hand, an isodesmic reaction produced an HF/6-31G AHf of 166.1 kcal/mole for cubane, while a homodesmic one yielded 133.4 kcal/mole [25]. The experimental value is 148.7 kcal/mole [27], One must also be wary of a fortuitous element the same homodesmic reaction produced an excellent 149.7 kcal/mole with the very small STO-3G basis set [28]. 

Corrections for Improper HF Asymptotic Behaviour.—There are two techniques which may be used to obtain results at what is essentially the Hartree-Fock limit over the complete range of some dissociative co-ordinate in those cases where the single determinants] approximation goes to the incorrect asymptotic limit. These techniques are (i) to describe the system in terms of a linear combination of some minimal number of determinantal wavefunctions (as opposed to just one) 137 and (ii) to employ a single determinantal wavefunction to describe the system but to allow different spatial orbitals for electrons of different spins - the so-called unrestricted Hartree-Fock method. Both methods have been used to determine the potential surfaces for the reaction of an oxygen atom in its 3P and 1Z> states with a hydrogen molecule,138 and we illustrate them through a discussion of this work. 

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