Coupled-perturbed Hartree-Fock analytic second derivatives

Unlike the true propagator, the UCHF approximation is given by a simple closed formula and reqnires only minimum computational effort to evalnate on the fly if the orbitals are available. The nnconpled Hartree-Fock/Kohn-Sham approximation has almost completely vanished from the chemistry literature about 40 years ago when modem derivative techniques became available because of the poor results it produced for second-order properties. Some systematic expositions of analytical derivative methods still use it as a starting point, but it is in our opinion pedagogi-cally inappropriate, as it requires considerable effort to recover the coupled-perturbed Hartree-Fock results which can be derived in a simpler way. UCHF/UCKS is still used in some approximate theories, but we suspect that its only merit is easy computability. According to Geerlings et al. [29], the polarizabilities derived from the uncoupled density response function correlate well with accurate results but can be off by up to a factor of 2, and thus they are only qualitatively useful. Our results in Table 1 confirm this. 

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. 

Analytical second derivatives for closed-shell (or unrestricted Hartree-Fock (UHF)) SCF wavefunctions are used routinely now. The extension to the MCSCF case is relatively new, however. In contrast to the first derivatives, the coupled perturbed SCF equations have to be solved in order to calculate the second and third energy derivatives. The closed-shell case is relatively straightforward, and will be discussed. The multiconfigurational formalism is... 

The second main group of methods analyse the response to successive powers of the perturbation separately using analytical re-arrangements of the perturbed equations. The procedure is typified in the Coupled Perturbed Hartree-Fock39,40,41 method (CPHF). which produces variationally optimized solutions in each order. Since the results represent a solution of the variational Hartree-Fock equations to each order they satisfy the energy derivative equations for the polarisabilities and the (2n+ 1) rule for the derivatives can be used to simplify the calculations. Corrections to the perturbed HF solutions can be made through MP2 or MP4 perturbation theory. 

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