Coupled perturbed Hartree-Fock procedure CPHF

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. 

As a consequence, field methods, which consist of computing the energy or dipole moment of the system for external electric field of different amplitudes and then evaluating their first, second derivatives with respect to the field amplitude numerically, cannot be applied. Similarly, procedures such as the coupled-perturbed Hartree-Fock (CPHF) or time-dependent Hartree-Fock (TDHF) approaches which determine the first-order response of the density matrix with respect to the perturbation cannot be applied due to the breakdown of periodicity. 

In the following the polarizability and the first and second hyperpolarizabilities for urea calculated at the SCF level in vacuo and in water are reported. Both static and frequency dependent nonlinear properties have been calculated, with the Coupled Perturbed Hartree-Fock (CPHF) and Time Dependent-CPHF procedures that have been described above. The solvent model is the Polarizable Continuum Model (PCM) whereas vibrational averaging of the optical properties along the C-0 stretching coordinate has been obtained by the DiNa package both in vacuo and in solution. 

In the previous equation R is the density matrix derivative with respect to Qi. It can be calculated throu a coupled perturbed Hartree-Fock (CPHF) procedure. Notice that the presence of the term, that is, of the dipole moment matrix derivative in Eq. 7.14, is due... 

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. 

More recently, Caves and Karplus71 have used diagrammatic techniques to investigate Hartree-Fock perturbation theory. They developed a double perturbation expansion in the perturbing field and the difference between the true electron repulsion potential and the Hartree-Fock potential, V. This is compared with a solution of the coupled Hartree-Fock equations. In their interesting analysis they show that the CPHF equations include all terms first order in V and some types of terms up to infinite order. They propose an alternative iteration procedure which sums an additional set of diagrams and thus should give results more accurate than the CPHF scheme. Calculations on Ha and Be confirmed these conclusions. 

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