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Second derivatives of the exchange-correlation energy

We merely summarize the relevant theoretical results rather than give a derivation. Differentiating Eq. (36) with respect to a second parameter y yields [Pg.196]

These are obtained by differentiating the KS potential matrix. It is convenient first to express the MO coefficients as [Pg.197]

As the dimension of the CPKS linear system is too large to make direct methods of solution feasible, the CPKS equations are solved iteratively, and this is the most expensive phase of the second derivative calculation. Evaluation of the explicit second derivatives of the XC energy (the third term in Eq. (46)) is not insignificant and warrants careful implementation, as discussed in Ref. [68] however, as this step is not dominant, we will not discuss it here. Likewise, the matrix Q may be evaluated in terms of known quantities and needs to be done only once, and so for a discussion of its implementation the reader is directed to Ref. [68]. [Pg.197]

The preceding equations are of exactly the same form as those in HF theory, because no fitting was used. From a practical standpoint, this means that once the general computational machinery for solving Eq. (49) is in place, this framework need not be extensively modified when different definitions of Exc are to be [Pg.197]

In our implementation we have taken a different approach, for which it will be seen that the above sorts of terms can be treated exactly without posing computational problems. Our approach fits naturally within the iterative method of solution of the CPKS equations, in which it is not the individual elements of A which are calculated, but rather the contraction of these with the current approximate solution vectors. The XC contribution to these quantities may be expressed as [Pg.198]


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