Hartree-Fock self-consistent field approximation

Now we want to discuss IR optical spectra of the C60H36 synthesized at high-pressure. Results of this study were published in Bazhenov et al. (2008). There are a lot of publications devoted to theoretical and experimental study of C60H36. We should pay attention on the existing discrepancies in the results of theoretical calculations of the dipole-active spectra C60H36, compare, for example, papers Bini et al. (1998) and Bulusheva et al. (2001). There were used different theoretical models. Semiempirical method of the MNDO type (Dewar and Thiel 1977) was used in (Bini et al. 1998). Ab initio Hartree-Fock self-consistent field approximation was used in (Bulusheva et al. 2001). 

Semiempirical (CNDO, MNDO, ZINDO, AMI, PM3, PM3(tm) and others) methods based on the Hartree-Fock self-consistent field (HF-SCF) model, which treats valence electrons only and contains approximations to simplify (and shorten the time of) calculations. Semiempirical methods are parameterized to fit experimental results, and the PM3(tm) method treats transition metals. Treats systems of up to 200 atoms. 

In the self-consistent field linear response method [25,46,48] also known as random phase approximation (RPA) [49] or first order polarization propagator approximation [25,46], which is equivalent to the coupled Hartree-Fock theory [50], the reference state is approximated by the Hartree-Fock self-consistent field wavefunction < scf) and the set of operators /i j consists of single excitation and de-excitation operators with respect to orbital rotation operators [51],... 

The accurate calculation of these molecular properties requires the use of ab initio methods, which have increased enormously in accuracy and efficiency in the last three decades. Ab initio methods have developed in two directions first, the level of approximation has become increasingly sophisticated and, hence, accurate. The earliest ab initio calculations used the Hartree-Fock/self-consistent field (HF/SCF) methodology, which is the simplest to implement. Subsequently, such methods as Mpller-Plesset perturbation theory, multi-configuration self-consistent field theory (MCSCF) and coupled-cluster (CC) theory have been developed and implemented. Relatively recently, density functional theory (DFT) has become the method of choice since it yields an accuracy much greater than that of HF/SCF while requiring relatively little additional computational effort. 

Ab initio MO computer programmes use the quantum-chemical Hartree-Fock self-consistent-field procedure in Roothaan s LCAO-MO formalism188 and apply Gaussian-type basis functions instead of Slater-type atomic functions. To correct for the deficiencies of Gaussian functions, which are, for s-electrons, curved at the nucleus and fall off too fast with exp( —ar2), at least three different Gaussian functions are needed to approximate one atomic Slater s-function, which has a cusp at the nucleus and falls off with exp(— r). But the evaluation of two-electron repulsion integrals between atomic functions located at one to four different centres is mathematically much simpler for Gaussian functions than for Slater functions. 

Ab initio quantum mechanical (QM) calculations represent approximate efforts to solve the Schrodinger equation, which describes the electronic structure of a molecule based on the Born-Oppenheimer approximation (in which the positions of the nuclei are considered fixed). It is typical for most of the calculations to be carried out at the Hartree—Fock self-consistent field (SCF) level. The major assumption behind the Hartree-Fock method is that each electron experiences the average field of all other electrons. Ab initio molecular orbital methods contain few empirical parameters. Introduction of empiricism results in the various semiempirical techniques (MNDO, AMI, PM3, etc.) that are widely used to study the structure and properties of small molecules. 

The equation is used to describe the behaviour of an atom or molecule in terms of its wave-like (or quantum) nature. By trying to solve the equation the energy levels of the system are calculated. However, the complex nature of multielectron/nuclei systems is simplified using the Born-Oppenheimer approximation. Unfortunately it is not possible to obtain an exact solution of the Schrddinger wave equation except for the simplest case, i.e. hydrogen. Theoretical chemists have therefore established approaches to find approximate solutions to the wave equation. One such approach uses the Hartree-Fock self-consistent field method, although other approaches are possible. Two important classes of calculation are based on ab initio or semi-empirical methods. Ah initio literally means from the beginning . The term is used in computational chemistry to describe computations which are not based upon any experimental data, but based purely on theoretical principles. This is not to say that this approach has no scientific basis - indeed the approach uses mathematical approximations to simplify, for example, a differential equation. In contrast, semi-empirical methods utilize some experimental data to simplify the calculations. As a consequence semi-empirical methods are more rapid than ab initio. 

The general independent particle model described above becomes the Hartree-Fock, self-consistent field (SCF) approximation when [Pg.70]

THE HALL-ROOTHAAN EQUATIONS, THE ORBITAL APPROXIMATION AND THE MODERN HARTREE-FOCK SELF-CONSISTENT FIELD METHOD... 

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]. 

The next step is to make the Hartree-Fock self-consistent field (HF-SCF) approximation as described previously for a multi-electron atom in Section 8.4. The Hartree-Fock approximation results in separation of the electron motions resulting (along with the Pauli principle) in the ordering of the electrons into the molecular orbitals as shown in Figure 9-5 for carbon monoxide. Hence, the many-electron wavefunction i for an N-electron molecule is written in terms of one-electron space wavefunctions,/, and spin functions, a or p, like what was done for complex atoms in Section 8.4. At this stage it is assumed that the N-electron molecule is a closed-shell molecule (all the electrons are paired in the occupied molecular orbitals). How molecules with open shells are represented will be discussed later in this Section. 

In the quantitative development of the structure in the self-consistent field approximation (S.C.F.) using the Hartree-Fock method the energy Ei is made up of three terms, one for the mean kinetic energy of the electron in one for its mean potential energy in the field of the nuclei, and a... 

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. 

The starting point of the creation of the theory of the many-electron atom was the idea of Niels Bohr [1] to consider each electron of an atom as orbiting in a stationary state in the field, created by the charge of the nucleus and the rest of the electrons of an atom. This idea is several years older than quantum mechanics itself. It allows one to construct an approximate wave function of the whole atom with the help of one-electron wave functions. They may be found by accounting for the approximate states of the passive electrons, in other words, the states of all electrons must be consistent. This is the essence of the self-consistent field approximation (Hartree-Fock method), widely used in the theory of many-body systems, particularly of many-electron atoms and ions. There are many methods of accounting more or less accurately for this consistency, usually named by correlation effects, and of obtaining more accurate theoretical data on atomic structure. 

Much more enigmatically, the first-order time-dependent self-consistent field approximation is also widely called the Random Phase Approximation (RPA). This terminology is entrenched and so, although the name time-dependent self-consistent perturbation theory is more descriptive and preferable to both time-dependent Hartree-Fock and RPA all are used more or less interchangeably. The evolution of the concept and phrase random phase approximation is sketched in Appendix 27.A to this chapter. 

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