Independent-electron models Hartree-Fock

An alternative approach to conventional methods is the density functional theory (DFT). This theory is based on the fact that the ground state energy of a system can be expressed as a functional of the electron density of that system. This theory can be applied to chemical systems through the Kohn-Sham approximation, which is based, as the Hartree-Fock approximation, on an independent electron model. However, the electron correlation is included as a functional of the density. The exact form of this functional is not known, so that several functionals have been developed. 

Chemistry is primarily concerned not with the properties of single molecules but with periodic trends, homologous series and the like. It is, therefore, important that any method which we apply to the problem of molecular electronic structure depends linearly on the number of electrons in the system being studied. Meaningful comparisons of atoms and molecules of different sizes are then possible. This property has been termed size-consistency1-2. Independent electron models, such as the widely used Hartree-Fock approximation, provide a size-consistent theory of atomic and molecular structure. 

The valence structure of argon provides a complete illustration of the application of electron momentum spectroscopy to correlations in the ion. The Hartree—Fock single-electron level diagram of fig. 11.1 illustrates the values of the separation energy e to be expected on the basis of the independent-electron model. The experimental situation is illustrated in fig. 11.2 by the first experiment in the field (Weigold, Hood and Teubner, 1973). The noncoplanar-symmetric differential cross section at 10° is plotted against Eq for =400 eV. There is an ion state at 15.76 eV, as predicted by Hartree—Fock, but there are at least two further states rather than the predicted one. 

The electronic many-body Hamiltonian in equation (1) is treated in the framework of the independent-electron frozen-core model. This means that there is only one active electron, whereas the other electrons are passive (no dynamic conelation is accounted for) and no relaxation occurs. In this model the electron-electron interaction is replaced by an initial-state Hartree-Fock-Slater potential [37]. This treatment is expected to be highly accurate for heavy collision systems at intermediate to high incident energies. The largest uncertainties of the independent-electron model will show up for low-Z few-electron systems, such as H -F H and H + He° or for high multiple-ionization probabilities. 

The simplest independent-electron model results from a complete neglect of electron-electron interactions. This is the bare-nucleus model for which the Hamiltonian h is merely a sum of kinetic energy and nucleus-electron attraction terms. The bare-nucleus model has a number of unique features which make its use in atomic and molecular studies attractive (see, for example. Refs. 7 and 17-22). The most widely used independent-electron model is the Hartree-Fock model. In this model, as is well known, the... 

In independent-electron models of atomic and molecular electronic structure, such as the Hartree-Fock approximation, only fimctions corresponding to the first few values of the angular momentum quantum number / contribute significantly to the energy, or to other expectation values, when multi-centre basis sets are employed. However, in treatments which take account of electron correlation effects, the higher harmonics are known to be important. A considerable amount of data is available on the convergence properties of the harmonic expansion for atoms and the importance of higher-order terms in the harmonic expansion for molecular systems has also been demonstrated. ... 

For Sufficiently large basis sets we reach the so-called Hartree-Fock limit Ejjp, which represents the best energy within the independent electron model of the MO theory. 

M0ller-Plesset second-order perturbation theory [78,162] is the most widely used approach to the electron correlation problem in contemporary ab initio molecular electronic structure studies [163-168], For systems which are well described by a single determinantal reference function, this theory - based on the use of Rayleigh-Schrodinger perturbation theory to describe electron correlation corrections to the Hartree-Fock independent electron model - affords an approach which combines accuracy with computational efficiency. The method, which is often designated mp2 , is based on the lowest order of the many-body perturbation theory expansion to take account of correlation effects. 

Most semi-empirical models are based on the fundamental equations of Hartree-Fock theory. In the following section, we develop these equations for a molecular system composed of A nuclei and N electrons in the stationary state. Assuming that the atomic nuclei are fixed in space (the Born-Oppenheimer approximation), the electronic wavefunction obeys the time-independent Schrodinger equation ... 

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 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. 

The horizontal axis relates the extent to which the motions of electrons in a many-electron system are independent of each other (uncorrelated). At the extreme left are found Hartree-Fock models. 

It is likely that different quantum chemical models will perform differently in each of these situations. Processes which involve net loss or gain of an electron pair are likely to be problematic for Hartree-Fock models, which treat the electrons as essentially independent particles, but less so for density functional models and MP2 models, which attempt to account for electron correlation. Models should fare better for processes in which reactants and products are similar and benefit from cancellation of errors, than those where reactants and products are markedly different. The only exception might be for semi-empirical models, which have been explicitly parameterized to reproduce individual experimental heats of formation, and might not be expected to benefit from error cancellation. 

Orbital interaction theory forms a comprehensive model for examining the structures and kinetic and thermodynamic stabilities of molecules. It is not intended to be, nor can it be, a quantitative model. However, it can function effectively in aiding understanding of the fundamental processes in chemistry, and it can be applied in most instances without the use of a computer. The variation known as perturbative molecular orbital (PMO) theory was originally developed from the point of view of weak interactions [4, 5]. However, the interaction of orbitals is more transparently developed, and the relationship to quantitative MO theories is more easily seen by straightforward solution of the Hiickel (independent electron) equations. From this point of view, the theoretical foundations lie in Hartree-Fock theory, described verbally and pictorially in Chapter 2 [57] and more rigorously in Appendix A. 

In order to improve the theoretical description of a many-body system one has to take into consideration the so-called correlation effects, i.e. to deal with the problem of accounting for the departures from the simple independent particle model, in which the electrons are assumed to move independently of each other in an average field due to the atomic nucleus and the other electrons. Making an additional assumption that this average potential is spherically symmetric we arrive at the central field concept (Hartree-Fock model), which forms the basis of the atomic shell structure and the chemical regularity of the elements. Of course, relativistic effects must also be accounted for as corrections, if they are small, or already at the very beginning starting with the relativistic Hamiltonian and relativistic wave functions. 

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

To account for the magnetic hyperfine interaction in, e.g., the ground state of a lithium atom, it is necessary in the core model to go beyond the usual, restricted Hartree-Fock approximation, which assumes that the spatial parts of one-electron orbitals are independent of mB (the spin-... 

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