Coupled perturbed Hartree-Fock equation

Exact second derivatives methods require the solution of the coupled perturbed Hartree-Fock equations, CPHF [11,34,35]. At the Hartree-Fock level this requires several steps in addition to the usual SCF procedure and the evaluation of the first derivatives. 

We have also examined here the use of approximate solutions of the coupled perturbed Hartree-Fock equations for estimating the Hessian matrix. This Hessian appears to be more accurate than any updated Hessian we have been able to generate during the normal course of an optimization (usually the structure has optimized to within the specified tolerance before the Hessian is very accurate). For semi-empirical methods the use of this approximation in a Newton-like algorithm for minima appears optimal as demonstrated in Table 17. In ab-initio methods searching for minima, the BFGS procedure we describe is the best compromise. 

Once again, the derivative of the density matrix P can be obtained as solution of first-order coupled-perturbed Hartree-Fock equation with derivar tive Fock matrix given by eq. (1.70), exactly as for the nuclear shielding. 

Y. Osamura, Y. Yamaguchi, and H. F. Schaefer, III, Chem. Phys., 103,227 (1986). Second-Order Coupled Perturbed Hartree-Fock Equations for Closed-Shell and Open-Shell Self Consistent Field Wavefunctions. 

The superscripts in Eq. [19] refer to orders in the two perturbations, V and W, respectively. The terms denoted resp refer to the inclusion of higher order terms via the coupled-perturbed Hartree-Fock equations. The term is a cross term involving both exchange and induction. The... 

The analytic determination of dipole moment derivatives by ab initio methods requires derivation of a complete formulation of coupled-perturbed Hartree-Fock equations by nuclear coordinate and electric field variations. 

Mol. Struct. THEOCHEM, 103, 183 (1983). Analytic Second Derivative Techniques for Self-Consistent-Field Wave Functions. A new Approach to the Solution of the coupled Perturbed Hartree-Fock Equations. 

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