Hartree-Fock method independent particle model

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. 

Most chemists picture the electronic structure of atoms or molecules by invoking orbitals. The orbital concept has its basis in Hartree-Fock theory, which determines the best wavefunction I ) under the approximation that each electron experiences only the average field of the other electrons. This is also called the one-electron, or independent particle model. While the Hartree-Fock method gives very useful results in many situations, it is not always quantitatively or even qualitatively correct. When this approximation fails, it becomes necessary to include the effects of electron correlation one must model the instantaneous electron-electron repulsions present in the molecular Hamiltonian. 

One of the main reasons for the good results obtained with the Hartree-Fock SCF method in electronic structure calculations for atoms and molecules is that the electrons keep away from each other due to the Pauli exclusion principle. This reduces the correlation between them, and provides a basis for the validity of the independent-particle model. The question arises as to the mechanisms that account for the validity of the SCF approximation in the vibrational case, which are obviously quite unrelated to the Pauli principle. 

Yet, even the hnite dimensional standard VB approach runs into a number of difficulties, such as the nightmare of the inner shells [11], neglect of overlap integrals, and the so-called V catastrophe (see, e.g. Ref. [12]). For this very reason, sometime during the second World War, VB theory was eclipsed by the computationally much more amenable molecular orbital (MO) method, relying on the independent-particle model (IPM), which reduces the V-electron problem to effectively a one-electron, though highly non-linear, one. A very important conceptual advance was achieved by the exploitation of the variation principle, which led to the formulation of Hartree-Fock (HF) equations... 

DF theory has the simplicity of an independent-particle model, yet it can be applied successfully to those systems-such as transition metal complexes - where non-dynamical electron correlation is of primary importance. DF-based methods are, in general, very easy to use, no matter how sophisticated the functional employed to describe the electron correlation. Also, more sophisticated functionals do not increase the computational requirements significantly, as opposed to post Hartree-Fock ab initio calculations. The application of approximate density functional theory has been reviewed by Ziegler and others [5]. 

Abstract. The paper by Kohn and Sham (KS) is important for at least two reasons. First, it is the basis for practical methods for density functional calculations. Second, it has endowed chemistry and physics with an independent particle model with very appealing features. As expressed in the title of the KS paper, correlation effects are included at the level of one-electron equations, the practical advantages of which have often been stressed. An implication that has been less widely recognized is that the KS molecular orbital model is physically well-founded and has certain advantages over the Hartree-Fock model. It provides an excellent basis for molecular orbital theoretical interpretation and prediction in chemistry. 

The parameters of the model matrix elements used here are adjusted to fit Hartree-Fock energies. Inner-shell electrons are well described by means of the independent particle model within which the Hartree-Fock method yields accurate one-electron (MO) energies. At low incident velocities, where the MO concept is applicable, the vacancy transfer probabilities are sensitively dependent on the one-electron energies involved. 

As in the Hartree-Fock molecular orbital theory, which is based on the independent particle model, the above Hartree product method also lacks enough correlation among the orbitals, and thereby the resultant accuracy is limited. To overcome the drawback, one can take account of the interaction among possible configurations (or the Hartree products) as in the configuration interaction method and multiconfiguration SCF methods in electronic structure theory. The multiconfigulational time-dependent Hartree... 

In the above discussion we have been concerned with the exact electronic Hamiltonian, energies and wave functions of a supersystem consisting of an array of well-separated subsystems. We now turn our attention to the description afforded by some independent particle model, in which the electrons move in some mean field. The most commonly used approximation of this type is the Hartree-Fock model, but the discussion presented in this section is not restricted to this particular method. In particular, we write the total electronic Hamiltonian operator in the form... 

Unfortunately, the determination of exact solutions of the SchrOdinger equation is intractable for almost all systems of practical interest. On the other hand, independent particle models are not sufficiently accurate for most studies of molecular structure. In particular, the Hartree-Fock model, which is the best independent particle model in the variational sense, does not support sufficient accuracy for many applications. Some account of electron correlation effects has to be included in the theoretical apparatus which underpins practical computational methods. Although the energy associated with electron correlation is a small fraction of the total energy of an atom or molecule, it is of the same order as most energies of chemical interest. However, such theories may not be true many-body theories. They may contain terms which scale non-linearly with electron number and are therefore unphysical and should be discarded. Any theory which contains such unphysical terms is not acceptable as a true many-body method. Either the theory is abandoned or corrections, such as that of Davidson [7] which is used in limited configuration interaction studies, are made in an attempt to restore linear scaling. 

The independent particle model in the theory of many-particle systems is studied by means of the self-consistent-field (SCF) idea. After a review of the characteristic features of the Hartree and Hartree-Fock schemes, the extension of the SCF method developed by Bmeckner is further refined by introducing the exact reaction operator containing aU correlation effects. This operator is here simply defined by means of the partitioning technique, and, if the SCF potentials are derived from this operator, one obtains a formahsm which is completely... 

In all cases where the question concerning the relative stabilities of equidistant versus bond alternating structures arises (polyyne [20,21, polyacetylene 22-27, polymethineimine 28,29 ) the latter are more stable within the framework of the restricted Hartree Fock approximation. For polyyne and polyacetylene this issue is in accord with the well known concept of a Peierls distortion jsoj. The occurence of Hartree Fock instabilities (see e.g. refs. 31,32 ) in the case of the equidistant, metallic structures of polyyne (cumulene) and all-trans polyacetylene points, however, to the need for improved methods going beyond the independent particle model. First efforts in this direction 27 show that at the level of second order Moller-Plesset perturbation theory the alternant configuration of polyacetylene is still preferred energetically although as expected the energy difference to the equidistant structures becomes smaller. 

The central field approximation and the simplifications which result from it allow one to construct a highly successful quantum-mechanical model for the AT-electron atom, by using Hartree s principle of the self-consistent field (SCF). In this method, one equation is obtained for each radial function, and the system is solved iteratively until convergence is obtained, which leaves the total energy stationary with respect to variations of all the functions (the variational principle ). The Hartree-Fock equations for an AT-electron system are equivalent to several one electron radial Schrodinger equations (see equation (2.2)), with terms which make the solution for one orbital dependent on all the others. In essence, the full AT-electron problem is approximated by a smaller number of coupled one-electron problems. This scheme is sometimes (somewhat inappropriately) referred to as a one-electron model in fact, the Hartree-Fock equations are a genuine AT-electron theory, but describe an independent particle system. 

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