The self-consistent field method

The self-consistent field (SCF) method was introduced in 1928 by Hartree. The goal of this method is similar to that of the variation method in that it seeks to optimize an approximate wave function. It differs from the variation method in two ways First, it deals only with orbital wave functions second, the search is not restricted to one family of functions and is capable of finding the best possible orbital approximation. It is a method that proceeds by successive approximations, or iteration. 

We describe the application of the method to the ground state of the helium atom. If electron 2 were fixed at location r2 the Schrddinger equation would be 

We replace the electron-electron repulsion term in the Hamiltonian of Eq. (19.2-1) by a term containing an average over all positions of electron 2, using the probability density of Eq. (19.2-2)  

The integrodifferential equation is solved by iteration (successive approximations) as follows The first approximation is obtained byreplacing the orbital under the integral by the zero-order function or some other known function, which we denote by l (i])(2). Equation (19.2-3) for the ground state of the helium atom now becomes 

The integral over t2 contains only known functions. It cannot be integrated mathematically, but can be evaluated numerically for various values of ri, giving a table of values for different values of ri. Equation (19.2-4) becomes a differential equation that can be solved numerically, using standard methods of numerical analysis. The solution Vf J )(l) is independent of 0i and and is represented by a table of values of V j l) as a function of ri. 

The idea behind the replacement of the excluded volume interactions by a self-consistent potential was put forward by Edwards6 and subsequently developed by various other physicists.5 7 Concerning the critical exponent v, this method gives the same result as Flory s method (v = 3/( / + 2)). 

It can be assumed that the chain potential is the sum of two terms, a chain term and an interaction term 

The self-consistent field method consists in replacing the true probability law 

In his initial article, Edwards studied the case of a semi-infinite chain starting from a point O (of position vector rQ = 0) and, consequently he admitted that the potential is spherically symmetric around O. He described the method in a manner which differs somewhat from the approach presented here. Actually, we are indebted to Reiss8 for a clear formulation of the method in terms of trial probability. In reality, the calculation of the exponent v made by Reiss does not lead to the value v = 3/5) found by Edwards and Flory but to a larger value (v = 2/3) however, as was shown by Yamakawa,9 the lack of agreement arises from the fact that certain terms were unduly omitted in Reiss s calculation and, finally, the correct calculation made by Yamakawa gives for v the result found by Edwards. 

The self-consistent field method can be summarized in the following simple manner which shows clearly the principle and the deficiencies of this approach. 

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We have said that the Schroedinger equation for molecules cannot be solved exactly. This is because the exact equation is usually not separable into uncoupled equations involving only one space variable. One strategy for circumventing the problem is to make assumptions that pemiit us to write approximate forms of the Schroedinger equation for molecules that are separable. There is then a choice as to how to solve the separated equations. The Huckel method is one possibility. The self-consistent field method (Chapter 8) is another. 

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 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... 

A method of molecular-orbital calculations, also referred to as the self-consistent field method (SCF), to characterize the bonding in unsaturated and aromatic molecules while neglecting electron-electron repulsion. The method has been extended to all valence electrons. 

The Hartree-Fock approach is also called the self-consistent field method. Indeed, the potential of the field in which the electron nl is orbiting is also expressed in terms of the wave functions we are looking for. Therefore the procedure for determination of the radial orbitals must be coordinated with the process of finding the expression for the potential starting with the initial form of the wave function, we find the expression for the potential needed to determine the more accurate wave function. We must continue this process until we reach the desired consistency between these quantities. 

Leading references to the self-consistent field method employed here are cited by Pople, J. A., /. Phys. Chem. 61, 6 (1957). Calculations of the Hiickel and SCF orbitals were executed on an IBM-7090 computer using an entirely automatic program written by Dr. Bessis of the Centre de Mecanique Ondulatoire Appliqufee, Paris, France. 

Such a Slater determinant, as it is often called, would, in fact, be the correct wave function for a system of noninteracting electrons. Electrons, however, do interact in real molecular systems. In order to obtain a more satisfactory representation, the individual orbitals self-consistent field method, whose main features are as follows, (a) One writes the exact total Hamiltonian for the system with explicit inclusion of electron interactions... 

It is of interest to consider the experimental radial electron density distribution in the ions Na+ and Cl- in sodium chloride in relation to corresponding results for the free ions calculated by the self-consistent field method. In Fig. 3 data from the experimental study of Schoknecht... 

The MC SCF method usually takes into account a minimum number of configurations capable of assuring some fundamental requirements, this step being followed by the optimization of the basis functions using the self-consistent-field method. For example, in order to describe accurately the dissociation of ground-state H2 it is only necessary to consider the two-configuration wave function... 

Chaban, I. A., "Calculation of the Effective Parameters of Micro-Inhomogeneous media by the Self-Consistent Field Method," Sov. Phvs. -Acoust.. 1965, PP. 81-86. 

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

The third entry refers to the self-consistent field method, developed by Hartree. Even for the best possible choice of one-electron functions V (r), there remains a considerable error. This is due to failure to include the variable rn in the wave-function. The effect is known as electron correlation. The fourth entry, containing a simple correction for correlation, gives a considerable improvement. Hylleraas in 1929 extended this approach with a variational function of the form... 

J.C, Slater, The Self-Consistent Field Method for Molecules and Solids, McGraw-Hill, New York, Vol. 4, 1974. 

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. 

In brief, the self-consistent field method appears here as completely unrealistic. 

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. 

The formal analysis of the mathematics required incorporating the linear combination of atomic orbitals molecular orbital approximation into the self-consistent field method was a major step in the development of modem Hartree-Fock-Slater theory. Independently, Hall (57) and Roothaan (58) worked out the appropriate equations in 1951. Then, Clement (8,9,63) applied the procedure to calculate the electronic structures of many of the atoms in the Periodic Table using linear combinations of Slater orbitals. Nowadays linear combinations of Gaussian functions are the standard approximations in modem LCAO-MO theory, but the Clement atomic calculations for helium are recognized to be very instructive examples to illustrate the fundamentals of this theory (67-69). 

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