The Hartree SCF Method

The simplest kind of ab initio calculation is a Hartree-Fock (HF) calculation. Modem molecular HF calculations grew out of calculations first performed on atoms by Hartree1 in 1928 [3]. The problem that Hartree addressed arises from the fact that for any atom (or molecule) with more than one electron an exact analytic solution of the Schrodinger equation (Section 4.3.2) is not possible, because of the electron-electron repulsion term(s). Thus for the helium atom the Schrodinger equation (cf. Section 4.3.4, Eqs. 4.36 and 4.37) is, in SI units 

Here m is the mass (kg) of the electron, e is the charge (coulombs, positive) of the proton (= minus the charge on the electron), the variables ru r2, and r12 are the distances (m) of electrons 1 and 2 from the nucleus, and from each other, Z = 2 is the number of protons in the nucleus, and 0 is something called the permitivity of empty space the factor 4rc 0 is needed to make SI units consistent. The force (N) between charges qi and q2 separated by r is r2, so the potential energy (J) 

Hamiltonians can be written much more simply by using atomic units. Let s take Planck s constant, the electron mass, the proton charge, and the permitivity of space as the building blocks of a system of units in which h/2n, m, e, and 47i 0 are numerically equal to 1 (i.e. h = 2%, m = 1, e = 1, and e0 = 1/4tt the numerical values of physical constants are always dependent on our system of units). These 

Hartree s method was to write a plausible approximate polyelectronic wavefunc-tion (a guess ) for an atom as the product of one-electron wavefunctions  

This function is called a Hartree product. Here 11 0 is a function of the coordinates of all the electrons in the atom, is a function of the coordinates of 

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These defects of the Hartree SCF method were corrected by Fock (Section 4.3.4) and by Slater2 in 1930 [8], and Slater devised a simple way to construct a total wavefunction from one-electron functions (i.e. orbitals) such that will be antisymmetric to electron switching. Hartree s iterative, average-field approach supplemented with electron spin and antisymmetry leads to the Hartree-Fock equations. 

The Hartree SCF Method. Because of the interelectronic repulsion terms the Schrodinger equation for an atom is not separable. Recalling the perturbation treatment of helium (Section 9.3), we can obtain a zeroth-order wave function by neglecting these repulsions. The Schrddinger equation would then separate into n one-electron hydrogenlike equations. The zeroth-order wave function would be a product of n hydrogenlike (one-electron) orbitals ... 

The Hartree SCF method approximates the atomic wave function as a product of one-electron spatial orbitals [Eq. (11.2)] and finds the best possible forms for the orbitals by an iterative calculation in which each electron is assumed to move in the field jffo-duced by the nucleus and a hypothetical charge cloud due to the other electrons. 

True or false (a) Hie spin multiplicity of every term of an atom with an odd number of electrons must be an even number. (b)The spin multiplicity of every term of an atom with an even number of electrons must be an odd number. (c)Hie spin multiplicity of a term is always equal to the number of levels of that term, (d) In the Hartree SCF method, the energy of an atom equals the sum of the orbital energies of the electrons, (e) Hie Hartree-Fock method k capable of giving the exact nonrelativistic energy of a many-electron atom. 

In the late 1920s and early 1930s, a team led by Hartree [9] formulated a self-consistent-field iterative numerical process to treat atoms. In 1930, Fock [10] noted that the Hartree-SCF method needed a correction due to electron exchange and the combined method was known as the Hartree-Fock SCF method. It was not until 1951 that a molecular form of the LCAO-SCF method was derived by Roothaan [11] as given in Appendix B but we can give a brief outline here. The Roothaan method allows the LCAO to be used for more than one atomic center and so the path was open to treat molecules Now, all we have to do is to carry out the integral for the expectation value of the energy as... 

Not all iterative semi-empirical or ab initio calculations converge for all cases. For SCF calculations of electronic structure, systems with a small energy gap between the highest occupied orbital and the lowest unoccupied orbital may not converge or may converge slowly. (They are generally poorly described by the Hartree-Fock method.)... 

In most work reported so far, the solute is treated by the Hartree-Fock method (i.e., Ho is expressed as a Fock operator), in which each electron moves in the self-consistent field (SCF) of the others. The term SCRF, which should refer to the treatment of the reaction field, is used by some workers to refer to a combination of the SCRF nonlinear Schrodinger equation (34) and SCF method to solve it, but in the future, as correlated treatments of the solute becomes more common, it will be necessary to more clearly distinguish the SCRF and SCF approximations. The SCRF method, with or without the additional SCF approximation, was first proposed by Rinaldi and Rivail [87, 88], Yomosa [89, 90], and Tapia and Goscinski [91], A highly recommended review of the foundations of the field was given by Tapia [71],... 

Now, in the Hartree-Fock method (the Roothaan-Hall equations represent one implementation of the Hartree-Fock method) each electron moves in an average field due to all the other electrons (see the discussion in connection with Fig. 53, Section 5.23.2). As the c s are refined the MO wavefunctions improve and so this average field that each electron feels improves (since J and K, although not explicitly calculated (Section 5.2.3.63) improve with the i// s ). When the c s no longer change the field represented by this last set of c s is (practically) the same as that of the previous cycle, i.e. the two fields are consistent with one another, i.e. self-consistent . This Roothaan-Hall-Hartree-Fock iterative process (initial guess, first F, first-cycle c s, second F, second-cycle c s, third F, etc.) is therefore a self-consistent-field procedure or SCF procedure, like the Hartree-Fock procedure... 

The method of calculating wavefunctions and energies that has been described in this chapter applies to closed-shell, ground-state molecules. The Slater determinant we started with (Eq. 5.12) applies to molecules in which the electrons are fed pairwise into the MO s, starting with the lowest-energy MO this is in contrast to free radicals, which have one or more unpaired electrons, or to electronically excited molecules, in which an electron has been promoted to a higher-level MO (e.g. Fig. 5.9, neutral triplet). The Hartree-Fock method outlined here is based on closed-shell Slater determinants and is called the restricted Hartree-Fock method or RHF method restricted means that the electrons of a spin are forced to occupy (restricted to) the same spatial orbitals as those of jl spin inspection of Eq. 5.12 shows that we do not have a set of a spatial orbitals and a set of [l spatial orbitals. If unqualified, a Hartree-Fock (i.e. an SCF) calculation means an RHF calculation. 

Within the Hartree-Fock approximation, calculations on molecules have almost all used the matrix SCF method, in which the HF orbitals are expanded in terms of a finite basis set of functions. Direct numerical solution of the HF equations, routine for atoms, has, however, been thought too difficult, but McCullough has shown that, for diatomic molecules, a partial numerical integration procedure will yield very good results.102 In particular, the Heg results agree well with the usual calculations, and it is claimed that the orbitals are likely to be of more nearly uniform accuracy than in the matrix HF calculations. Extensions to larger molecules should be very interesting so far, published results are available for He, Heg, and LiH. 

In order to solve for the energy and wavefunction within the Hartree-Fock-Roothaan procedure, the AOs must be specified. If the set of AOs is infinite, then the variational principle tells us that we will obtain the lowest possible energy within the HF-SCF method. This is called the HF limit, This is not the actual energy of the molecule recall that the HF method neglects instantaneous electron-electron interactions, otherwise known as electron correlation. 

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. 

As shown in Section 9.1, the A -electron wavefunction is more correctly represented by a Slater determinant of spinorbitals (9.1) rather than a Hartree product of orbitals (9.20), thus accounting automatically for the exclusion principle and the indistinguishability of electrons. The Hartree-Fock method, developed in 1930, is a generalization of the SCF based on Slater determinant wavefimctions. The Hartree-Fock (HF) equations for the spinorbitals (pa have the form... 

The simplest ab initio method used to study van der Waals molecules is the Hartree-Fock method. To the Hartree-Fock interaction energy one should add the dispersion energy, which ranges from some 20% of the dissociation energy for the water dimer to twice the van der Waals well depth for He2, and is thus not negligible. The combined method may be termed SCF + Disp. 

The quantum-mechanical SCF method for obtaining the vibrational energy levels is a direct adaptation of the Hartree approximation for electronic struc-true calculations, which dates back to the early stages of quantum theory. The introduction of the method for vibrational modes is, however, rather recent and is due to Bowman and co-workers,6,7 Carney et al.,8 and Cohen et al.9 The semiclassical version of the SCF, the SC-SCF method, proposed by Gerber and Ratner,10 relies on the characteristically short de Broglie wavelengths typical of vibrational motions (as opposed to electronic ones) to gain some further simplification, but is otherwise based on the same physical considerations as the quantum-mechanical approximation. A brief review of the SCF and SC-SCF methods can be found in Ref. (11). 

Degeneracies of the SCF states are an obvious cause for breakdown of the approximation in the form discussed in the previous sections. We discuss now an extension of the method that applies to such cases, that is, to resonances and near-resonances between SCF modes. Just as the vibrational SCF method is an adaptation of the Hartree approximation from electronic structure calculations, so is the generalization discussed here an application of the configuration interaction (Cl) method, which uses for the wavefunctions a linear combination of the strongly interacting SCF states. Quantum Cl for polyatomic vibrations was introduced by Bowman and co-workers,7-21 the semi-classical version is due to Ratner et al.33... 

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