Coupled perturbed Hartree-Fock procedure

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. 

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. 

This method is valid only in the static field limit (zero frequency), which is a weakness. However, recent advances of a derived procedure (Coupled Perturbed Hartree-Fock) permit the frequency dependence of hyperpolarizabilities to be computed. The FF method mainly uses MNDO (modified neglect of diatomic differential overlap) semi-empirical algorithm and the associated parametrizations of AM-1 and PM-3, which are readily available in the popular MOPAC software package. ... 

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 major advantage of the scheme lies in the facile incorporation of approximate methods, though. If spin-orbit coupling effects are so small that they can be treated perturbatively, one uses the spinfree Hartree-Fock procedure to achieve the exact connection Ms = Mk. With this identification it is then e.g. possible to restrict the Cl space by a cut-off value of AMs or even to run the... 

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. 

There is also a hierarchy of electron correlation procedures. The Hartree-Fock (HF) approximation neglects correlation of electrons with antiparallel spins. Increasing levels of accuracy of electron correlation treatment are achieved by Mpller-Plesset perturbation theory truncated at the second (MP2), third (MP3), or fourth (MP4) order. Further inclusion of electron correlation is achieved by methods such as quadratic configuration interaction with single, double, and (perturbatively calculated) triple excitations [QCISD(T)], and by the analogous coupled cluster theory [CCSD(T)] [8],... 

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. 

For the numerical implementation of these formulae one can either use a perturbation expansion for the functions coupled Hartree-Fock (CHF) equations for each order.124 This procedure seems to be more advantageous for numerical calculations, but the resulting expressions are quite complicated. For this reason we do not reproduce them here, but refer to the original paper.124 It seems to be an acceptable compromise to use the second order CHF equations for the dynamic as and (Js, but use simple perturbation theory for the dynamic ys using as unperturbed wave functions the results of the solutions of the first and second order CHF equations. One should point out, however, that in our calculations124 the second numerical derivatives... 

The question for a more systematic inclusion of electronic correlation brings us back to the realm of molecular quantum chemistry [51,182]. Recall that (see Section 2.11.3) the exact solution (configuration interaction. Cl) is found on the basis of the self-consistent Hartree-Fock wave function, namely by the excitation of the electrons into the virtual, unoccupied molecular orbitals. Unfortunately, the ultimate goal oi full Cl is obtainable for very small systems only, and restricted Cl is size-inconsistent the amount of electron correlation depends on the size of the system (Section 2.11.3). Thus, size-consistent but perturbative approaches (Moller-Plesset theory) are often used, and the simplest practical procedure (of second order, thus dubbed MP2 [129]) already scales with the fifth order of the system s size N, in contrast to Hartree-Fock theory ( N ). The accuracy of these methods may be systematically improved by going up to higher orders but this makes the calculations even more expensive and slow (MP3 N, MP4 N ). Fortunately, restricted Cl can be mathematically rephrased in the form of the so-called coupled clus-... 

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