Coupled perturbed Hartree-Fock level

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. 

The algorithm is available in Fortran, within the STOP (Slater Type Orbital Package) set of programs, at the coupled perturbed Hartree-Fock level with the ETOs expanded in Slater type orbitals. 

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 ... 

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. 

Coupled perturbed Hartree-Fock calculations at the SCF level were used to assess the polarizability tensor elements, each of which is defined as... 

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. 

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. ... 

In order to have a more complete picture of the many-body problem for more general or complicated cases that DFT could help to treat, it is necessary to make a correspondence with the use of many-body perturbation theory. Under this wider classification of perturbation theory are included all the methods that treat electron correlation beyond the Hartree-Fock level, including configuration interaction, coupled cluster, etc. This perturbational approach has traditionally been known as second quantization, and its power for some applications can be seen when dealing with problems beyond the standard quantum mechanics. 

A perturbation calculation starting from an approximate wave function (such as T hf) is susceptible to ambiguities and large errors (see [39, p. 18]). In order to obtain reliable results, it is necessary to stabilize the solutions with respect to a certain class of variations. This approach is called stationary perturbation theory its Hartree-Fock level version is called Coupled Hartree-Fock (CHF). 

These compounds have been the subject of several theoretical [7,11,13,20)] and experimental[21] studies. Ward and Elliott [20] measured the dynamic y hyperpolarizability of butadiene and hexatriene in the vapour phase by means of the dc-SHG technique. Waite and Papadopoulos[7,ll] computed static y values, using a Mac Weeny type Coupled Hartree-Fock Perturbation Theory (CHFPT) in the CNDO approximation, and an extended basis set. Kurtz [15] evaluated by means of a finite perturbation technique at the MNDO level [17] and using the AMI [22] and PM3[23] parametrizations, the mean y values of a series of polyenes containing from 2 to 11 unit cells. At the ab initio level, Hurst et al. [13] and Chopra et al. [20] studied basis sets effects on and y. It appeared that diffuse orbitals must be included in the basis set in order to describe correctly the external part of the molecules which is the most sensitive to the electrical perturbation and to ensure the obtention of accurate values of the calculated properties. 

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],... 

However, until today no systematic comparison of methods based on MpUer-Plesset perturbation (MP) and Coupled Cluster theory, the SOPPA or multiconfigurational linear response theory has been presented. The present study is a first attempt to remedy this situation. Calculations of the rotational g factor of HF, H2O, NH3 and CH4 were carried out at the level of Hartree-Fock (SCF) and multiconfigurational Hartree-Fock (MCSCF) linear response theory, the SOPPA and SOPPA(CCSD) [40], MpUer-Plesset perturbation theory to second (MP2), third (MP3) and fourth order without the triples contributions (MP4SDQ) and finally coupled cluster singles and doubles theory. The same basis sets and geometries were employed in all calculations for a given molecule. The results obtained with the different methods are therefore for the first time direct comparable and consistent conclusions about the performance of the different methods can be made. 

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