Atoms self-consistent field method

Application of the variational self-consistent field method to the Haitiee-Fock equations with a linear combination of atomic orbitals leads to the Roothaan-Hall equation set published contemporaneously and independently by Roothaan and Hall in 1951. For a minimal basis set, there are as many matr ix elements as there are atoms, but there may be many more elements if the basis set is not minimal. 

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... 

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

In order to find a good approximate wave function, one uses the Hartree-Fock procedure. Indeed, the main reason the Schrodinger equation is not solvable analytically is the presence of interelectronic repulsion of the form e2/r. — r.. In the absence of this term, the equation for an atom with n electrons could be separated into n hydrogen-like equations. The Hartree-Fock method, also called the Self-Consistent-Field method, regards all electrons except one (called, for instance, electron 1), as forming a cloud of electric charge... 

Garza, J., Vargas, R. and Vela, A. 1988. Numerical self-consistent-field method to solve the Kohn-Sham equations in confined many-electron atoms. Phys. Rev. E. 58 3949-54. 

Various theoretical methods and approaches have been used to model properties and reactivities of metalloporphyrins. They range from the early use of qualitative molecular orbital diagrams (24,25), linear combination of atomic orbitals to yield molecular orbitals (LCAO-MO) calculations (26-30), molecular mechanics (31,32) and semi-empirical methods (33-35), and self-consistent field method (SCF) calculations (36-43) to the methods commonly used nowadays (molecular dynamic simulations (31,44,45), density functional theory (DFT) (35,46-49), Moller-Plesset perturbation theory ( ) (50-53), configuration interaction (Cl) (35,42,54-56), coupled cluster (CC) (57,58), and CASSCF/CASPT2 (59-63)). 

Thus, the state of each electron in a many-electron atom is conditioned by the Coulomb field of the nucleus and the screening field of the charges of the other electrons. The latter field depends essentially on the states of these electrons, therefore the problem of finding the form of this central field must be coordinated with the determination of the wave functions of these electrons. The most efficient way to achieve this goal is to make use of one of the modifications of the Hartree-Fock self-consistent field method. This problem is discussed in more detail in Chapter 28. 

In the conclusion of this section let us notice that a wealth of data on the applications of the relativistic self-consistent field method to the studies of the hyperfine structure of atomic levels is collected in [149]. Investigations of the hyperfine structure by the methods of perturbation theory are described in monograph [17]. 

This expression excludes self-interaction. There have been a number of attempts to include into the Hartree-Fock equations the main terms of relativistic and correlation effects, however without great success, because the appropriate equations become much more complex. For a large variety of atoms and ions both these effects are fairly small. Therefore, they can be easily accounted for as corrections in the framework of first-order perturbation theory. Having in mind the constantly growing possibilities of computers, the Hartree-Fock self-consistent field method in various... 

Roothaan, C.C.J. and Bagus, RS. (1963). Atomic self-consistent field calculations by the expansion method, Methods Comput. Phys. 2,47-94. 

Many of the principles and techniques for calculations on atoms, described in section 6.2 of this chapter, can be applied to molecules. In atoms the electronic wave function was written as a determinant of one-electron atomic orbitals which contain the electrons these atomic orbitals could be represented by a range of different analytical expressions. We showed how the Hartree-Fock self-consistent-field methods could be applied to calculate the single determinantal best energy, and how configuration interaction calculations of the mixing of different determinantal wave functions could be performed to calculate the correlation energy. We will now see that these technques can be applied to the calculation of molecular wave functions, the atomic orbitals of section 6.2 being replaced by one-electron molecular orbitals, constructed as linear combinations of atomic orbitals (l.c.a.o. method). 

Slater, J. C. (1974b). The self-consistent field method for atoms, molecules and. solids. New York McGraw-Hill. 

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. 

R.B. Gerber and M.A. Ratner, Self-consistent field methods for vibrational excitations in polyatomic molecules, Adv. Chem. Phys., 70 (1988) 97. P. Jungwirth and R.B. Gerber, Quantum dynamics of large polyatomic systems using a classically based separable potential method, J. Chem. Phys., 102 (1995) 6046 Quantum dynamics of many atom systems by classically based separable potential (CSP) method Calculations for T (Ar),2 in full dimensionality, J. Chem. Phys., 102 (1995) 8855. 

Looking at the history of correlation from the fifties to the seventies, one may be led to ask whether correlation has been a scientific fashion or a real problem. Twenty years ago, almost everybody seemed to accept the idea that the simple molecular orbital method (MO) must be completed by configuration interaction (Cl), in order to obtain reliable prediction for the physical properties of atoms and molecules. Ten years ago, electron correlation was considered as the central problem of Quantum Chemistry (7). Nowadays, about 90% of the quantum-mechanical calculations on molecules are performed by the self-consistent-field method (SCF) using more or less extended sets of basis functions, without any consideration of the possible effects of correlation. 

These contradictory results about the sequence of calcium phosphate phase nucleation and growth demonstrate vividly how complex and in many important details not yet understood mechanisms appear to govern biomimetic formation of bone-like hydroxyapatite. In particular, the transformation of OCP to HAp was shown to be crystallographically controlled (Fernandez et al., 2003) because hydroxyapatite and octacalcium phosphate can form an epitaxial interface. A new OCP-HA interface model based on an earlier configuration model (Brown, 1962) and using the minimum interface free-energy optimisation was presented. In this new model a structure is formed that consists of half a unit cell of HAp and one unit cell of OCP whereby [0001]HAp is parallel to [001]OCp and [1210]HAp is parallel to [010]Ocp (Figure 7.66). It was shown by self-consistent field methods that the atoms of this model possess similar environments as in the HAp and OCP unit cells and that, as a result of the differences between HAp and OCP unit cell parameters, this interface displays misfit dislocation-like features. 

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 anomalous dispersion effect is associated with the ejection of photoelectrons from inner shell electrons in an atom. The normal scattering describes the interaction of all the electrons in the atom with the X-ray beam. The radial distribution of the electrons in an atom can be calculated using quantum mechanics, originally by Hartree s self-consistent field method (Hartree 1933). In figure 9.12 this distribution is given for rubidium, which has a K edge at 0.8155 A the mean radius for... 

There is, in principle, nothing which limits the self-consistent field method to any particular form of the exchange-correlation potential, and the procedure outlined above has been used in connection with several approximations for exchange and correlation. Most notable in this respect is SLATER S Xa method [1.4] which has been applied to all atoms in the periodic table, to some molecules, and in the majority of the existing electronic-structure calculations for crystalline solids. 

There are no analytical forms for the radial functions, / ni(r), as solutions of the radial wave equation. Hartree, in 1928, developed the standard solution procedure, the self-consistent field method for the helium atom by using the simple product forms of equation 1.10 to represent the two-electron wave function. Herman and Skillman (4) programmed a very useful approximate form of the Hartree method in the early 1960s for atomic structure calculations on all the atoms in the Periodic Table. An executable version of this program, based on their FORTRAN code, modified to output data for use on a spreadsheet is included with the material on the CDROM as hs.exe. 

This chapter extends the numerical methods of the previous chapters to the case of the simplest many-electron atom, helium. Then, at the end of the chapter, the calculation of the electronic energy of helium is carried out as a first example of the standard modem form of the self-consistent field method, in which the integrals over Gaussian primitives are evaluated exactly. 

The Englishman, Hartree (1,60) the Russian, Fock (2,3) and the American, Slater (5-7), in the early development of modern quantum mechanics, pioneered the calculation of atomic electronic structure. Hartree based his method on the variation principle and this led naturally to the development of the self-consistent field method, which is at the heart of the design of modem molecular orbital programs. 

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