Matrix Hartree-Fock

The unrestricted L.C.A.O.—S.C.F. method reduces to the restricted method when a and electrons are assigned to spatially identical molecular orbitals. Thus under the INDO method the Hartree-Fock matrix elements for an open-shell system become... 

One can adopt one of two viewpoints either to form the Hamiltonian in another way so that it will transform in the same way as S or to consider that the Hamiltonian matrices (as formed by Equations 6 or 7) are good approximations to the true Hartree-Fock matrix for the unhybridized basis, < in this case, in the transformed basis it must be T HT. We have chosen the latter viewpoint in these calculations. (J. A. Pople (17) has proposed that the off-diagonal elements be constructed as Hy — V2( + Pv)Sij where the fis are empirical energies depending only on the atom and not on the state of hybridization.)... 

Within the SCF HF framework, the energy ej, as a diagonal element of the Hartree-Fock matrix (5.31), can be shown to be equal to the difference between the expectation values of the hamiltonian for the neutral species X (the Hartree-Fock total energy) and for a positive ion X (—i) described by a Slater determinant built on the basis of canonical orbitals identical to those of X (frozen orbital model). This apparently crude description of the X (—i) can be shown to give the lowest energy for the ion. As a result, ei is an approximate estimate of the ionization potential of the i-th electron ... 

It is useful to invoke this matrix in the definition of the electron-repulsion matrices involved in the Hartree-Fock matrix. The matrices J and K which depend on C and do so only through the single matrix R = CC -. 

Orthogonalisation subroutine gmprd, subroutine gtprd. The transformation of the Hartree-Fock matrix from the original basis to an orthogonalised basis involves two matrix multiplications ... 

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. 

During any iteration of the solution of the matrix Hartree-Fock equations, it is computationally convenient to transform the Hartree-Fock matrix into the basis of orbitals defined by the diagonalisation of the previous Hartree-Pock... 

Use current R to form Hartree—Fock matrix in IHFl /... 

How can we attach a physical interpretation to the eigenvalues of the matrix equation if they are to be changed at will The interpretation of the many-shell effective Hartree-Fock matrix is deferred until Chapter 23. 

If we recall the definition of the canonical MOs of the UHF or closed-shell model as those MOs which diagonalise the Hartree-Fock matrix ... 

But wc shall now be dealing with many more matrices we shall need a density matrix for each shell, a Hartree-Fock matrix for each shell etc. 

With these expressions for the Fock matrices, the single Hartree-Fock matrix eqn ( 14.17) is just ... 

The upshot is that all the components of the Hartree-Fock matrix undergo the same transformation when the basis is subject to a linear transformation. This result can be expressed as... 

The effective Hartree-Fock matrix equation for a many-shell system has been derived in Chapter 14 and used in several applications open shells and some MCSCF models. So far, it has been seen simply as the formally correct equation to generate SCF orbitals for these many-sheU structures without any interpretation. In particular, the fact that the effective Hartree-Fock matrix (the McWeenyan ) contains many arbitrary parameters has not been addressed, nor has the practical problem of the actual grounds for the choice of values for these parameters been systematised. In looking at this problem we must bear two points in mind ... 

The inter-shell elements of the Hartree-Fock matrix are simultaneously reduced to zero. 

The most straightforward way to implement the solution of the Kohn Sham equations using all the techniques we have used for the Hartree-Fock case is simply to replace the formation of the exchange part of the Hartree-Fock matrix by a matrix element of some chosen exchange-correlation" potential. 

Add this repulsion matrix to the one-electron Hamiltonian matrix to give a Hartree-Fock matrix defined over the (non-orthogonal) basis orbitals. ... 

Use the orthogonalising matrix in V to generate a Hartree-Fock matrix over an orthogonal basis. 

Hartree-Fock is in quotes here since the matrix generated has the form of the Hartree-Fock matrix but, of course, since the MO coefficients are not yet self-consistent, it is not the Hartree-Fock matrix in this basis. 

The routine presented in the last chapter for the formation of the matrix G(R), the electron repulsion contribution to the Hartree-Fock matrix, was coded to be as direct as possible and to illustrate the possibilities of the use of macros to simplify repetitive coding. No consideration at all was given to questions of speed and efficiency. In particular some attention should be given to ... 

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