Hartree-Fock theory coupled

Similar considerations apply to the electric dipole moment the derivatives of the dipole integrals can be easily obtained whilst the derivatives of the density matrix require the use of coupled Hartree-Fock theory (e.g. Gerratt and Mills, 1968). 

In the self-consistent field linear response method [25,46,48] also known as random phase approximation (RPA) [49] or first order polarization propagator approximation [25,46], which is equivalent to the coupled Hartree-Fock theory [50], the reference state is approximated by the Hartree-Fock self-consistent field wavefunction < scf) and the set of operators /i j consists of single excitation and de-excitation operators with respect to orbital rotation operators [51],... 

The matrix equation for the p-h part of p has the familiar form (l-A)X = B, which occurs in coupled Hartree-Fock theory [34]. 

Obviously is evaluated in the spirit of coupled-Hartree-Fock theory. If we label occupied nonrelativistic spin orbitals as e,j,and empty ones as a,6, c..., the only nonvanishing elements of the are Uf and Ua, and the only nonvanishing elements of the Hessean are [90] ... 

VARIATIONAL METHODS. COUPLED HARTREE-FOCK THEORY... 

The first two kinds of terms are called derivative integrals, they are the derivatives of integrals that are well known in molecular structure theory, and they are easy to evaluate. Terms of the third kind pose a problem, and we have to solve a set of equations called the coupled Hartree-Fock equations in order to find them. The coupled Hartree-Fock method is far from new one of the earliest papers is that of Gerratt and Mills. 

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. 

The method of CHF-PT-EB CNDO has been used for several organometallic complexes. This method utilizes a coupled Hartree-Fock (CHF) scheme10 applied through the perturbation theory (PT) of McWeeny,11"13 and an extended basis (EB), complete neglect of differential overlap (CNDO/2) wavefunction. The exponents of the basis set are optimized with respect to experimental polarizabilities and second hyperpolarizabili-ties.14,15 A detailed description of the CNDO/2 method may be found in Ref. 16. 

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 ... 

The basic computational method is that of coupled Hartree-Fock perturbation theory (14). At present we prefer the GIAO implementation mentioned above because of its computational efficiency and ease of use, but we have previously used a common gauge-origin method as implemented in the software SYSMO (15) as well as the random-phase approximation, localized orbital (RPA LORG) approach as implemented in the software RPAC (16). 

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. 

After some sparse papers published by different authors in the 1970s (see e.g. refs. 111-113), in 1987 Tossel et al.114 performed ab initio CHFPT (coupled Hartree-Fock perturbation theory) calculations of Gd and ap/ the diamagnetic and... 

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