Supermatrix method

In a small system, where few integrals if any are zero by virtue of the distance of the interactions, the size of the Supermatrix file is the same as that of the 2-electron file (we ignore for the moment the effects of molecular symmetry), and the Supermatrix method has a clear advantage because each Supermatrix element gives rise to only two references to the G operator ... 

Self-Consistent-Field The Open Shell SCF Case. The major point here is that the advantage of the Supermatrix methods is approximately halved over the closed shell case simply because one requires J and K matrices to be computed individually, necessitating the construction and use of 2 Supermatrices. Our present code based on the integral driven algorithm performs at 10.3 Mflops when processing cases described by Roothaan (22). 

At this point we have developed a method for handling symmetric one-electron operators a useful step, but far from adequate to cover all cases we are interested in, since we have said nothing yet about the much more demanding case of two-electron integrals. Let us consider first building the Fock matrix for a closed-shell SCF calculation. This is perhaps most easily expressed in terms of a P supermatrix with elements... 

However a method based on the single-reference Supermatrix P defined above may devised which performs at 135 Mflops (if coded in CAL) The method requires only that all the density matrices be held in store, simultaneously, halving the store requirement of the natural algorithm. The P Supermatrix should be ordered so that for a given ij pair index all kZ pair indices are available. Wfe are then able to compute G.. through an inner product... 

The matrix p is a combined set of the external electrostatic fields that represent the effects of the QM field on the EFP polarizability tensors and the PCM potential, while w is a combined set of induced dipoles and surface charges. The physical meaning of the supermatrix equation (3) is that the EFP induced dipoles and PCM induced charges are uniquely determined by the external field and potential therefore, the right hand side of Eq. (3) involves only the external field /potential, and the left side involves only the induced EFP dipoles and PCM charges. The interactions among the induced dipoles and charges are implicitly described with the matrix B. The supermatrix Eq. (3) can be solved either with direct inversion or various iterative methods. 

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