Handy-Schaefer technique

The Cl gradient expression was derived and implemented by Krishnan et al. (1980) and Brooks et al. (1980). The generalization to MRCI is due to Osamura et al. (1981, 1982a,b). The Hessian expression was derived by Jorgensen and Simons(1983) and implemented by Fox et al. (1983). Recently, a more efficient implementation has been reported by Lee et al. (1986). MRCI derivative expressions up to fourth order have been derived by Simons et al. (1984). The introduction of the Handy-Schaefer technique (Handy and Schaefer, 1984) greatly improved the efficiency of Cl derivative calculations. The calculation of Cl derivatives within the Fock-operator formalism has recently been reviewed by Osamura et al. (1987). 

This term may be calculated in the same way as the E(2) contribution to the MRCI Hessian discussed above [Eq. (129)]. The only difference is that we must construct one set of transition densities between CI> and Pll) > for each independent perturbation P(1), and the Fock matrices in Eq. (129) must be calculated from these densities. Furthermore, the term containing k<2) [see Eq. (129)] may be treated by the Handy-Schaefer technique in the same way as the k(2) contribution to E<3> discussed above. This is easily achieved by adding —6Fp to FA + FK in Eq. (144), where Fp are the 31V — 6 Fock matrices calculated from the densities between CI> and P(1)>. 

Finally, this term may be simplified using the Handy-Schaefer technique. We obtain... 

The CC molecular gradient and Hessian expressions were derived by Jorgensen and Simons (1983). Using the Handy-Schaefer technique, Adamowicz et al. (1984) and Bartlett (1986) simplified the expressions for the gradient. The only implementations are the CC molecular gradients reported by Fitzgerald et al. (1985) and by Lee et al. (1987). 

The MP2 gradient expression was derived and implemented by Pople et al. (1979). Second derivative expressions were given by J0rgensen and Simons (1983). Handy et al. (1985, 1986) and Harrison et al. (1986) simplified the gradient and Hessian expressions using the Handy- Schaefer technique and reported implementations of these expressions. 

To calculate the above integrals one must know Q(n> and R(n>. The derivatives of the rotation matrix Q are obtained by solving the appropriate response equations, as discussed in later sections. In some cases the explicit evaluation of Q(n) can be avoided by using the technique of Handy and Schaefer (1984). The expressions for Rin) in terms of the derivatives of the overlap matrix may be obtained by expanding R as... 

However, it is not necessary to solve these equations, as use may be made of an exchange theorem, which can be expressed most simply using the technique of Handy and Schaefer. The coefficients U would multiply elements of the dipole moment integrals

, but these integrals are the right-hand side of a set of simultaneous equations which have already been solved, i.e. 

J. D. Goddard, N. C. Handy, and H. F. Schaefer, Gradient techniques for open-shell restricted Hartree-Fock and multiconfiguration self-consistent-field methods, J. Chem. Phys. 71 1525 (1979). 

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