Hartree-Fock approximation self-consistency

The most simple approach is the Hartree-Fock (HF) self-consistent field (SCF) approximation, in which the electronic wave function is expressed as an antisymmetrized product of one-electron functions. In this way, each electron is assumed to move in the average field of all other electrons. The one-electron functions, or spin orbitals, are taken as a product of a spatial function (molecular orbital) and a spin function. Molecular orbitals are constructed as a linear combination of atomic basis functions. The coefficients of this linear combination are obtained by solving iteratively the Roothaan equations. 

The Hartree-Fock or self-consistent-field approximation is a simplification useful in the treatment of systems containing more than one electron. It is motivated partly by the fact that the results of Hartree-Fock calculations are the most precise that still allow the notion of an orbital, or a state of a single electron. The results of a Hartree-Fock calculation are interpretable in terms of individual probability distributions for each electron, distinguished by characteristic sizes, shapes and symmetry properties. This pictorial analysis of atomic and molecular wave functions makes possible the understanding and prediction of structures, spectra and reactivities. 

Table 1 contains some further information useful to characterize the different contributions to the molecule/surface interaction orientation dependence and the typical strength of the different contributions, and whether or not they can be understood on a purely classical basis. If one wants to calculate molecule/surface interactions by means of quantum-mechanical or quantum-chemical methods, the most important question is whether standard density functional (DPT) or Hartree-Fock theory (self consistent field, SCF) is sufficient for a correct and reliable description. Table 1 shows that all contributions except the Van der Waals interaction can be obtained both by DPT and SCF methods. However, the results might be connected with rather large errors. One famous example is that the dipole moment of the CO molecule has the wrong sign in the SCF approximation, with the consequence that SCF might yield a wrong orientation of CO on an oxide surface (see also below). In such cases, the use of post Hartree-Fock methods or improved functionals is compulsory. 

The standard way of treating conjugated systems using the o-n separation approximation in the 1930s and into the 1960s was the so-called Hiickel methody In this method, the electron-electron interactions are not explicitly considered. Rather, the positions of the nuclei are fixed, and the electrons move in the field of the nuclei. Much of the error that results from neglecting the interactions of the electrons with one another can be circumvented with proper adjustment of empirical parameters. These parameters are also adjusted to allow (approximately) for the interaction from the o system, which is thus taken into account without specific calculations. This method is crude but does often give qualitative results that enable rationalization of many chemical phenomena of interest. It was a powerful and useful tool in its time. A better approximation is the Hartree-Fock or self-consistent field (SCF) method in which the electron-electron interactions are explicitly considered (Self-Consistent Field Method in Chapter 3). The quantum mechanical calculations on the n system in this case are carried out in a... 

In the bibliography, we have tried to concentrate the interest on contributions going beyond the Hartree-Fock approximation, and papers on the self-consistent field method itself have therefore not been included, unless they have also been of value from a more general point of view. However, in our treatment of the correlation effects, the Hartree-Fock scheme represents the natural basic level for study of the further improvements, and it is therefore valuable to make references to this approximation easily available. For atoms, there has been an excellent survey given by Hartree, and, for solid-state, we would like to refer to some recent reviews. For molecules, there does not seem to exist something similar so, in a special list, we have tried to report at least the most important papers on molecular applications of the Hartree-Fock scheme, t... 

In view of the preceding considerations it should be emphasized that it is incorrect to talk about the self-consistent-field molecular orbitals of a molecular system in the Hartree-Fock approximation. The correct point of view is to associate the molecular orbital wavefunction of Eq. (1) with the N-dimen-sional linear Hilbert space spanned by the orbitals t/2,... uN any set of N linearly independent functions in this space can be used as molecular orbitals for forming the antisymmetrized product. 

Of paramount importance in this latter category is the Hartree-Fock approximation. The so-called Hartree-Fock limit represents a well-defined plateau, in terms of its methematical and physical properties, in the hierarchy of approximate solutions to Schrodinger s electronic equation. In addition, the Hartree-Fock solution serves as the starting point for many of the presently employed methods whose ultimate goal is to achieve solutions to equation (5) of chemical accuracy. A discussion of the Hartree-Fock method and its associated concept of a self-consistent field thus provides a natural starting point for the discussion of the calculation of potential surfaces. 

We adopt the same chemisorption model as in our previous work [3], which within the unrestricted Hartree-Fock approximation involves a self-consistent calculation of the electronic charge on the adatom. The basis elements needed for the calculation are the... 

In the DFT, as in the Hartree-Fock approach, an effective independent-particle Hamiltonian is arrived at, and the electronic states are solved for self consistency. The many-electron wave function is still written as a Slater determinant. However, the wave functions used to construct the Slater determinant are not the one-electron wave functions of the Hartree-Fock approximation. In the DFT, these wave functions have no individual meaning. They are merely used to construct the total electron-charge density. The difference between the Hartree-Fock and DFT approaches lies in the dependence of the Hamiltonian in DFT on the exchange correlation potential, Vxc[ ](t), a functional derivative of the exchange correlation energy, Exc, that, in turn, is a functional (a function of a function) of the electron density. In DFT, the Schrodinger equation is expressed as ... 

Aspects of the relativistic theory of quantum electrodynamics are first reviewed in the context of the electronic structure theory of atoms and molecules. The finite basis set parametrization of this theory is then discussed, and the formulation of the Dirac-Hartree-Fock-Breit procedure presented with additional detail provided which is specific to the treatment of atoms or molecules. Issues concerned with the implementation of relativistic mean-field methods are outlined, including the computational strategies adopted in the BERTHA code. Extensions of the formalism are presented to include open-shell cases, and the accommodation of some electron correlation effects within the multi-configurational Dirac-Hartree-Fock approximation. We conclude with a survey of representative applications of the relativistic self-consistent field method to be found in the literature. 

The method is an approximate self-consistent-field (SCF) ab initio method, as it contains no empirical parameters. All of the SCF matrix elements depend entirely on the geometry and basis set, which must be orthonormal atomic orbitals. Originally, the impetus for its development was to mimic Hartree-Fock-Roothaan [5] (HFR) calculations especially for large transition metal complexes where full HFR calculations were still impossible (40 years ago). However, as we will show here, the method may be better described as an approximate Kohn-Sham (KS) density functional theory (DFT)... 

We also want to mention that a Dyson-equation approach for propagators like the polarisation and the particle-pcurticle propagator has been formulated and used to derive a self-consistent extension of the RPA, also called cluster-Hartree-Fock approximation, that has been applied in the fields of plasma and nuclear physics [12-14], This formalism, however, has similar problems like Feshbach s theory and does not yield a universal, well-behaved optical potential because the two-particle space has to be restricted in order to make the approach well-defined [14]. 

The solution of Eq. (11.60) is achieved by standard methods once the matrix elements are obtained. Their evaluation involves the calculation of expectation values, which can be found when the solution of Eq. (11.60) is available, and we are, thus, faced with a self-consistency requirement, which is similar to but more complex than the corresponding challenge in the Hartree-Fock approximation. In the next section, the calculation of the matrix elements is addressed. 

A better approximation can be obtained by taking the average repulsion between the electrons into account when determining the orbitals, a procedure known as the Hartree-Fock approximation. If the orbital for one of the electrons were somehow known, the orbital for the second electron could be calculated in the electric field of the nucleus and the first electron, described by its orbital. This argument could just as well be used for the second electron with respect to first electron.The goal is therefore to calculate a set of self-consistent orbitals, and this can be done by iterative methods. 

We note that the global minimization approach to the Hartree-Fock approximation just described is equivalent to the minimization of a set of coupled one-electron subhamiltonians. That is, one could instead minimize (by iteration to self-consistency) the set of one-electron subhamiltonians... 

Consider, as an example, polyethylene if the chains are packed as in the crystal (Fig. 1), there are 5.5 x 10 per m in the plane perpendicular to their length. Hence the tensile fracture stress would be 33 GPa. Variants of this approach, using simple Morse potentials, provide values ranging from about 19 to 36 GPa. Other calculations using Hartree-Fock self-consistent field methods yield values of 66 GPa at 0 K (Crist et al., 1979). This last value seems a bit high and may be due to inaccuracies of the Hartree-Fock approximation at large atomic separations. 

In order to extend these methods to make them feasible for the study dynamical chemical processes in biopolymers, simplifying assumptions are necessary. The most obvious choice is the use of semi-empirical techniques within the Hartree Fock, linear combination of atomic orbitals framework. These methods can achieve speedups on the order of 1000 over typical ab initio calculations using split valence basis sets within the Hartree Fock approximation. Often greater accuracy can be achieved as well because of the parameterization inherent in the semi-empirical approaches. One semi-empirical approach which has proven successful in representing many chemically interesting processes is the AMI and MNDO Hartree Fock Self-Consistent Field methods developed and paramerterized by Dewar and coworkers [46]. These methods have recently been implemented in a mixed quantum/ classical methodology for the study of chemical and biochemical processes by Field et al. [47]. 

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