The Realization of Direct Methods in Quantum Chemistry

In Section 2 we outlined briefly how the ordinary closed-shell SCF problem could be recast as a problem in direct minimization. In this section we consider the problem in a little more detail and also consider its generalization, particularly in respect of incorporating constraints and finding the relevant gradient expressions. 

In McWeeny s3 4 realization of the steepest descent to the method for coefficients, constraints were included in rather an oblique manner. McWeeny concentrated on the fact that in the ordinary closed-shell SCF problem the physical variables were the elements of the R matrix [see equation (6)], and he regarded these as the variables of his problem. To first order the change in energy induced by a charge R-+R + SR is simply  

It is clear that in the absence of constraints the energy function may not possess any minima, so that constraints cannot be completely neglected. McWeeny observed, however, that orthogonality could be preserved to any order by requiring that 

McWeeny then regarded the elements of A as determining the minimization problem and clearly the steepest descent is along a negative multiple (—A, say) of the quantity in square brackets in equation (32). Given that there is a convergent power-series expansion for the inverse in equation (29), it then 

It is easy to see that in general the new R matrix created from equation (33) will not be accurately idempotent (i.e. orthogonality will be lost), and that therefore one will not be able to use this matrix at the next iteration without correcting it for this defect. However, as the minimum is approached the new R matrix will become more and more accurately idempotent. 

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