The Hartree-Fock Model

In Chapter 4,1 discussed the concept of an idealized dihydrogen molecule where the electrons did not repel each other. After making the Bom-Oppenheimer approximation, we found that the electronic Schrodinger equation separated into two independent equations, one for either electron. These equations are the ones appropriate to the hydrogen molecule ion. 

Once electron repulsion is taken into account, this separation of a many-electron wavefunction into a product of one-electron wavefunctions (orbitals) is no longer possible. This is not a failing of quantum mechanics scientists and engineers reach similar conclusions whenever they have to deal with problems involving three or more mutually interacting particles. We speak of the three-body problem. 

astronomers also suffer from the three-body problem when they try to study the motion of the planets round the sun. They are lucky in that the gravitational force between bodies A and B goes as 

The orbital model is a very attractive one, and it can obviously be used to successfully model atoms, molecules and the solid state because it is now part 

There are several ways in which we can proceed with the derivation of the HF equations. The traditional one is to look for an eigenvalue equation for the HF orbitals 

The basic physical idea of HF theory is a simple one and can be tied in very nicely with our discussion of the electron density given in Chapter 5. We noted the physical significance of the density function pi(r, 5) p (r, s)drdv gives the chance of finding any electron simultaneously in the spin-space volume elements dr and dr, with the other electrons anywhere in space and with either spin, / (r) dr gives the corresponding chance of finding any electron with either spin in the spatial volume element dr. 

In the Hartree-Fock approximation, the electrc ic wave function is approximated by a single configuration of spin orbitals (i.e. by a single Slater determinant or by a single space- and spin-adapted CSF) and the energy is optimized with respect to variations of these spin orbitals. Thus, the wave function may be written in the form 

Since the parameters K occur nonlinearly in the energy expression, an iterative procedure must be invoked to determine the Hartree-Fock state. For a detailed account of Hartree-Fock theory, see Chapter 10. 

Current quantum-mechanical calculations are based on the independent-particle model, where one assumes that the molecular orbitals are either empty or occupied by at most two electrons. This model cannot give a completely correct description of a many-electron system mainly because it treats each of the particles as if it saw the others smeared out in a charge cloud. However, it accounts surprisingly well for many properties, especially those connected with the one-electron density. Consequently, it is worthwhile discussing it in detail. 

From now on, we shall explicitly use spin orbitals, which are derived from orbitals by multiplying each of them by one of the two possible spin functions  

In the frame of the independent particle model, the total wave function can be written as an antisymmetrized product of spin orbitals  

The function 0 is known as a Slater determinant . If one looks for the energy minimum for such a function, one finds that the orbitals have to verify the so-called Hartree-Fock equations I3,i4,i5)  

VnUc its potential energy in the field of the bare nuclei, and 

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It is a well-known fact that the Hartree-Fock model does not describe bond dissociation correctly. For example, the H2 molecule will dissociate to an H+ and an atom rather than two H atoms as the bond length is increased. Other methods will dissociate to the correct products however, the difference in energy between the molecule and its dissociated parts will not be correct. There are several different reasons for these problems size-consistency, size-extensivity, wave function construction, and basis set superposition error. 

In the Hartree-Fock model, where we take account of antisymmetry, it turns out that there is no correlation between the positions of electrons of opposite spin, yet,... 

BSSE also opposes the tendency of the Hartree-Fock model to keep the interacting closed shell fragments too far apart. So, when optimized geometries are considered for the complex, BSSE is found to mimic some of those effects on the electron density distribution which would be induced by the interfragment dispersion contributions. 

As usual, the Hartree-Fock model can be corrected with perturbation theory (e.g., the Mpller-Plesset [MP] method29) and/or variational techniques (e.g., the configuration-interaction [Cl] method30) to account for electron-correlation effects. The electron density p(r) = N f P 2 d3 2... d3r can generally be expressed as... 

The use of the Hartree-Fock model allows the perturbation-theory equations (1.2)-(1.5) to be conveniently recast in terms of underlying orbitals (,), orbital energies (e,), and orbital occupancies (n,). Such orbital perturbation equations will allow us to treat the complex electronic interactions of the actual many-electron system (described by Fock operator F) in terms of a simpler non-interacting system (described by unperturbed Fock operator We shall make use of such one-electron perturbation expressions throughout this book to elucidate the origin of chemical bonding effects within the Hartree-Fock model (which can be further refined with post-HF perturbative procedures, if desired). 

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

B3LYP Model. A Hybrid Density Functional Model which improves on the Local Density Model by accounting explicitly for non-uniformity in electron distributions, and which also incorporates the Exchange Energy from the Hartree-Fock Model. The B3LYP model involves three adjustable parameters. 

The concept of the molecular orbital is, however, not restricted to the Hartree-Fock model. Sets of orbitals can also be constructed for more complex wave functions, which include correlation effects. They can be used to obtain insight into the detailed features of the electron structure. One choice of orbitals are the natural orbitals, which are obtained by diagonalizing the spinless first-order reduced density matrix. The occupation numbers (T ) of the natural orbitals are not restricted to 2, 1, or 0. Instead they fulfill the condition ... 

Many-body calculations which go beyond the Hartree-Fock model can be performed in two ways, i.e. using either a variational or a perturbational procedure. There are a number of variational methods which account for correlation effects superposition-of-configurations (or configuration interaction (Cl)), random phase approximation with exchange, method of incomplete separation of variables, multi-configuration Hartree-Fock (MCHF) approach, etc. However, to date only Cl and MCHF methods and some simple versions of perturbation theory are practically exploited for theoretical studies of many-electron atoms and ions. 

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