Hartree-Fock theory coupled perturbed

In many calculations beyond the Hartree-Fock level a first step is the transformation of at least some integrals. For the simplest such calculation, second-order perturbation theory, integrals with two indices transformed into the occupied MO basis axe required. Such integrals appear in many situations, including the MO basis formulation of coupled-perturbed Hartree-Fock theory. We can represent the first phase of this transformation as obtaining Coulomb and exchange operators ... 

Quantum Chemistry (1992) which is entirely devoted to molecular non-linear optics. The foundations of time-dependent coupled-perturbed Hartree Fock theory, referred to in Section 3.1.2, are beautifully described in Sections 12.5-12.7 of Roy McWeeny s book [7]. 

The last three terms of this expression involve the derivatives of the molecular orbital coefficients and cannot easily be avoided. They are obtained through coupled perturbed Hartree-Fock theory (CPHF). ... 

This is the basis of the so-called coupled perturbed Hartree-Fock (CPHF) theory, which, although generally traced to Ref. 233, actually is presented in essentially its complete form in Ref. 235. In Eq. (12), the Greek symbol p refers to an atomic basis function. 

Complete Active Space (CAS) 141 configuration interaction singles (CIS) 89, 93-95 coupled cluster (CC) theory 140, 142 coupled perturbed Hartree-Fock (CPHF) 19 coupled perturbed Kohn-Sham (CPKS) 19 CPCM 9... 

The complexity of the paramagnetic term is more apparent now. Not only do we require knowledge of the ground state wavefunction but we also need to know all the excited states and their associated energies, if we are to do the calculation exactly, even in the perturbation theory approximation. Clearly, these requirements cannot be met, and other approximations must be employed. Typically one starts at the coupled perturbed Hartree-Fock level, and for many years this was the sole approach. The inclusion of correlation from the Hartree-Fock base is complex but possible, and some important results have been obtained. The other approach, presently gaining favor, is to use density functional theory, i which includes correlation from the beginning, and then employ perturbation theory to treat the second-order shielding effect. 

Second-order properties are often evaluated using coupled-perturbed Hartree-Fock (CPHF) theory. The CPHF wave function is essentially the first-order perturbed wave function, which, as we saw above, must include the negative-energy states. Thus, in the relativistic case, the CPHF method must include both the positive- and negative-energy states. 

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. 

Neese F. 2001. Prediction of electron paramagnetic resonance g values using coupled perturbed Hartree-Fock and Kohn-Sham theory. J Chem Phys 115 11080-11096. 

ASC = apparent surface charge BEM = boundary-element method CPHF = coupled perturbed Hartree-Fock C/RF = classical reaction field GBA = generalized Bom approximation FDM = finite-difference method FEM = finite-element method MPE = multipole expansion PD = potential derived SOS = sum over states SPT = scaled particle theory. 

Passing now to the analytical calculations of the y " tensor elements in solution, there are several SCRF methods in use we quote here some major examples, referring for more details to the relevant papers. " All the quoted methods use spherical or ellipsoidal cavities, with the exception of the PCM version which can treat cavities of general shape, and work at a QM level ranging from semiempirical to MCSCF methods. A MPE approach is generally used to describe solvent effects, with the exception of PCM again, which uses an ASC method. The evaluation of the y " tensor elements is made either with finite differences, response theory and SOS methods, or with coupled perturbed Hartree-Fock (CPHF) methods. ... 

Ferrero, M., Civalleri, B., Rerat, M., Orlando, R., and Dovesi, R. (2009) The calculation of the static first and second susceptibilities of crystalline urea A comparison of Hartree-Fock and density functional theory results obtained with the periodic coupled perturbed Hartree-Fock/Kohn-Sham scheme. 

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