Fock Matrix Space

Due to their spatial localization, it follows that the interaction eneigy of an occupied LMO with any distant virtual LMO will be zero, and so the computational problem becomes reduced to annihilating matrix elements connecting LMOs that are close in space. These LMOs can be easily identified from the molecular connectivity table given the requirement that any allowed LMO spans one or two atoms. The Fock matrix element, Fif takes the form ... 

The construction of the Fock matrix may be simplified in the MCSCF case when the orbital space is partitioned in an inactive, an active, and a secondary space. Siegbahn et al. (1981) have shown that the Fock matrix elements then become... 

W. PoUard and R. Eriesner (1993) Efficient Fock matrix diagonalization by a Krylov-space method. J. Chem. Phys. 99, p. 6742... 

We can now turn to the application of these two simple routines to transform the Hartree-Fock matrix (m x m) to an orthogonal basis. Since the storage R, which was used to form and use the R matrix is now freed, we may use it as work space for an intermediate matrix product and write... 

During the iterative solution of the LCAO equations, at each iteration, the diagonalisation of the current approximation to the self-consistent Hartree-Fock matrix generates such a partition of the total function space, i.e. a current (non-self-consistent) set of occupied orbitals and a set of current victuals a current occupied space and a current virtual space. These current spaces share some of the properties of the final self-consistent spaces in particular the current single-determinant is invariant against linear transformations within the current occupied space. 

Using the concepts from Chapter 16, the variational problem can be considered as a rotation of the coordinate system. In the original function space, the basis functions, the Fock operator depends on all the Mbasis functions, and the corresponding Fock matrix is non-diagonal. By performing a rotation of the coordinate system to the molecular orbitals, however, the matrix can be made diagonal, i.e. in this coordinate system the Fock operator only depends on Nocc functions. 

In constructing the practical Fock matrix two points should be clarified. One of them is the shape and size of the cavity, and the other is the description of the pM- The spatial character of an ideal cavity should be the inclusion of the whole charge distribution of the solutes and the exclusion of empty space where solvent can intrude. A molecular shaped cavity is feasible for fitting such a purpose, but expression of the interaction energy is in a complex form in general. In the case of a regularly shaped cavity, e.g., sphere, ellipsoid, and cylinder, the expression is usually given in analytical form. 

Representing S and F matrices in the Bloch function basis set at every k point of the sampling set. In this basis, the expression of the matrix elements contains a double sum over the direct lattice vectors. For example, a generic element of the Fock matrix represented in the reciprocal space is given by... 

This last equation can be interpreted as a Fourier transform of the Fock matrix from direct to reciprocal space. 

Akin to KM analysis, the RVS analysis decomposes the total INT into chani-cally intuitive components. This analysis utilizes the McWeeney s group function approach, which offers the advantage of diagonaUzation and solving smaller Fock matrix. The function space is separated into the variational and nonvariational space with the flexibility to add or ranove the unoccupied orbitals of the monomers in the functional space. Such a procedure helps to evaluate various conceptual INT terms. 

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