Coupled Hartree-Fock perturbation theory

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 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). 

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... 

Coupled Hartree-Fock perturbation theory has been used to calculate the polarizability of a pair of He atoms as a function of R. However, the authors conclude that further work is needed, including electron correlation.100... 

There are also properties for which the magnitude is dependent upon transition intensity and for which accurate results can be obtained only with perturbation theory examples occur in currently much studied areas like NMR spectroscopy (described in Chapter 2), but also involve other properties like magnetic susceptibilities and refractive indices, which are not much studied from an electronic structure point of view (although we would argue that, due to advances in theory, such experimental techniques are ripe for further exploration). Within a Hartree-Fock approach the perturbation of a molecule by electric or magnetic fields can be calculated at a number of levels of theory. Coupled Hartree-Fock perturbation theory (Lipscomb, 1966 Ditchfield, 1974), which arrives at a self-... 

Now the expression (19) is an uncoupled formulation of the polarizability. We can replace it by a polarizability derived from coupled Hartree-Fock perturbation theory, which is more accurate, because it takes account of the reorganisation of the electron distribution in a self-consistent manner. Better still would be to evaluate the monomer polarizability by a method that takes account of electron correlation as well . But whatever the level of calculation, we can once again perform a much better calculation of the monomer property than is possible for the dimer. In this way we arrive at a description of the induction energy that is far more accurate than we can obtain through either intermolecular perturbation theory, where the perturbation is treated in an uncoupled fashion, or from a supermolecule calculation, where the size of the basis is limited by the need to perform calculations at a large number of points on the potential energy surface. 

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. 

Recently, quantum chemical computational techniques, such as density functional theory (DFT), have been used to study the electrode interface. Other methods ab initio methods based on Hartree-Fock (HF) theory,65 such as Mollcr-PIcsset perturbation theory,66,67 have also been used. However, DFT is much more computationally efficient than HF methods and sufficiently accurate for many applications. Use of highly accurate configuration interaction (Cl) and coupled cluster (CC) methods is prohibited by their immense computational requirements.68 Advances in computing capabilities and the availability of commercial software packages have resulted in widespread application of DFT to catalysis. 

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. 

Alternatively, analytical methods can be used such as coupled Hartree-Fock (CHF), where the perturbed HF equations are solved directly. Using standard perturbation theory one can also develop a sum-over-states (SOS) formalism and write... 

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. 

This leads to the response theory [38,50,51,64,65] or coupled DFT (CDFT) which is the direct analog of the coupled Hartree-Fock (CHF) approach [3,57]. The equations thus obtained are coupled, since the perturbed KS molecular orbitals (MO) are coupled with each other by self-consistently as in the FPT approach. In contrast with FPT, the CDFT equations (18) remain real also in the case of a purely imaginary perturbation because of the lack of dependency on A. The disadvantage is the need to evaluate the linear response of the KS effective potential v cl] analytically. 

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... 

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. 

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). 

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