The Fock Equation

By this time, we have introduced so many approximations and restrictions on our wave function and energy spectrum that is no longer quite legitimate to call it a Schroedinger equation (Schroedinger s initial paper treated the hydrogen atom only.) We now write 

Note that Jj r ) and A)(ri) are operators that go to make up the Fock operator. They operate on functions. One often sees the notation 

Having the Slater atomic orbitals, the linear combination approximation to molecular orbitals, and the SCF method as applied to the Fock matrix, we are in a position to calculate properties of atoms and molecules ab initio, at the Hartree-Fock level of accuracy. Before doing that, however, we shall continue in the spirit of semiempirical calculations by postponing the ab initio method to Chapter 10 and invoking a rather sophisticated set of approximations and empirical substitutions 

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If we assume that S = I (which is not true in general), the matr ix form of the Fock equation can be written... 

Multiplying from the left by a specific basis function and integrating yields the Roothaan-Hall equations (for a closed shell system). These are the Fock equations in the atomic orbital basis, and all the M equations may be collected in a matrix notation. 

The first set of equations govern the Cj amplitudes and are called the CI- secular equations. The second set determine the LCAO-MO coefficients of the spin-orbitals (f>j and are called the Fock equations. The Fock operator F is given in terms of the one- and two-electron operators in H itself as well as the so-called one- and two-electron density matrices yij and Tyj i which are defined below. These density matrices reflect the averaged occupancies of the various spin orbitals in the CSFs of VP. The resultant expression for F is ... 

The most general version of Hartree-Fock (HF) theory, in which each electron is permitted to have its own spin and spatial wave function, is called unrestricted HF (UHF). Remarkably, when a UHF calculation is performed on most molecules which have an equal number of alpha and beta electrons, the spatial parts of the alpha and beta electrons are identical in pairs. Thus the picture that two electrons occupy the same MO with opposite spins comes naturally from this theory. A significant simplification in the solution of the Fock equations ensues if one imposes this natural outcome as a restriction. The form of HF theory where electrons are forced to occupied MOs in pairs is called restricted HF (RHF), and the resulting wave function is of the RHF type. A cal-... 

Molecular orbitals will be very irregular three-dimensional functions with maxima near the nuclei since the electrons are most likely to be found there and falling off toward zero as the distance from the nuclei increases. There will also be many zeros defining nodal surfaces that separate phase changes. These requirements are satisfied by a linear combination of atom-centered basis functions. The basis functions we choose should describe as closely as possible the correct distribution of electrons in the vicinity of nuclei since, when the electron is close to one atom and far from the others, its distribution will resemble an AO of that atom. And yet they should be simple enough that mathematical operations required in the solution of the Fock equations can actually be carried out efficiently. The first requirement is easily satisfied by choosing hydrogenic AOs as a basis... 

To construct the Fock matrix, one must already know the molecular orbitals ( ) since the electron repulsion integrals require them. For this reason, the Fock equation (A.47) must be solved iteratively. One makes an initial guess at the molecular orbitals and uses this guess to construct an approximate Fock matrix. Solution of the Fock equations will produce a set of MOs from which a better Fock matrix can be constructed. After repeating this operation a number of times, if everything goes well, a point will be reached where the MOs obtained from solution of the Fock equations are the same as were obtained from the previous cycle and used to make up the Fock matrix. When this point is reached, one is said to have reached self-consistency or to have reached a self-consistent field (SCF). In practice, solution of the Fock equations proceeds as follows. First transform the basis set / into an orthonormal set 2 by means of a unitary transformation (a rotation in n dimensions),... 

Solution of the Fock equations requires integrals involving the basis functions, either in pairs or four at a time. Some of these we have already seen. The simplest are the overlap integrals, stored in the form of the overlap matrix S, whose elements are given by equation (A.48) as... 

An essential characteristic of the Fock equations resides in the fact that each individual operator F depends on all the orbitals which are occupied in the system (on account of the explicit inclusion of the interaction terms). Thus, each is given by an equation which depends on all the s. The way out of this difficulty is to choose arbitrarily a starting set of s, calculate the F(v) s, solve the series of equations for a new set of s, and go over the same series of operations again and again until the pth set of < s reproduce the (p— l)th set with good accuracy—hence the name self-consistent given to the procedure. The orbitals obtained in this fashion are, in principle, the best possible orbitals compatible with a determinantal W. 

Given this form of molecular orbitals, the Fock equations yield a system of homogeneous linear equations in the c/s, the Roothaan equations77... 

The SCF-Xa Scattered-wave Method.—An alternative to the LCAO-MO method for molecules is the Xa scattered wave or multiple-scattering Xa method, which was suggested less than 10 years ago114 and now has an extensive literative. The basic theory has been refined in several directions115 and there are several excellent reviews.116-180 The method is based on the scattered-wave method in solids181 and aims to solve the Fock equation,... 

The LCAO assumption and the Fock equations lead to the Roothaan-Hall matrix equation, or system of B homogeneous equations in B unknowns ... 

HF theory allows a set of molecular orbitals, < ) , to be constructed which are solutions to the Fock equations ... 

More explicit molecular orbital methods including the electronic repulsion terms are called advanced molecular orbital methods. In such methods, F matrix elements of the Fock equation are calculated by Eqs. (6.6) and (6.7), and the iterative procedure, or self-consistent field (SCF) procedure must be used. The most convenient advanced molecular orbital method for the polycyclic aromatic compounds is the semiempirical method suggested by Pariser, Parr, and Pople (the PPP method). In this PPP method, the required integral values are empirically determined using the following approximations. 

We do not include in this review semiempirical quantum chemical methods that do not initially solve the Fock equations for a set of molecular orbitals. This is a subject unto itself. Such methods create orbitals by a fixed ansatz, such as localized orbitals formed from hybrid atomic orbitals, avoid matrix multiplication and diagonalization, and can be developed up to third... 

We start our derivation of the Fock equations with the stationary state Schrodinger equation,... 

The Fock equations can most easily be solved by solving the matrix equations... 

Although this scheme is generic and is used to solve the Fock equations (Hartree-Fock SCF), other schemes are possible. The use of supercomputers with fast cpu and slow storage has somewhat changed the strategy. Integrals might not be stored, but rather recalculated when they are required in each cycle at step 5. 

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