Hyperpolarizabilities Hartree-Fock theory

Molecular polarizabilities and hyperpolarizabilities are now routinely calculated in many computational packages and reported in publications that are not primarily concerned with these properties. Very often the calculated values are not likely to be of quantitative accuracy when compared with experimental data. One difficulty is that, except in the case of very small molecules, gas phase data is unobtainable and some allowance has to be made for the effect of the molecular environment in a condensed phase. Another is that the accurate determination of the nonlinear response functions requires that electron correlation should be treated accurately and this is not easy to achieve for the molecules that are of greatest interest. Very often the higher-level calculation is confined to zero frequency and the results scaled by using a less complete theory for the frequency dependence. Typically, ab initio studies use coupled-cluster methods for the static values scaled to frequencies where the effects are observable with time-dependent Hartree-Fock theory. Density functional methods require the introduction of specialized functions before they can cope with the hyperpolarizabilities and higher order magnetic effects. 

S.-Y. Liu and C. E. Dykstra, ]. Phys. Chem., 91, 1749 (1987). Multipole Polarizabilities and Hyperpolarizabilities of AH and AjHn Molecules from Derivative Hartree-Fock 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. 

JHC735>. The polarizability and hyperpolarizability of 1,2,3-triazole have been computed by the Hartree-Fock perturbation theory on an extended basis CNDO method <90JPC1755>. 

Three self-consistent schemes for computing the first and second hyperpolarizabilities per unit cell of stereoregular polymers have been proposed at the Hartree Fock level of theory. One is the approach taken by the Erlangen group to deal with the unbound position operator and is summarized in Refs. 175, 176, and 177. It consists of expressing the scalar potential as a sum of two terms. 

The chemistry and physics of ftillerenes have constituted one of the most fast growing research fields during the last decade [90]. A summary of the early results for the second hyperpolarizability can be found in [91, 92]. There are a number of factors that make comparison of these results difficult, for instance the type of optical process, the phase of the samples, and the reference standard [91, 93]. The theoretical results, on the other hand, seem to be more consistent, especially among those from the first-principle calculations, such as ab initio Hartree-Fock and the density functional theory (DFT) methods [14, 89, 94, 95], The recent applications of time-dependent DFT [14, 96] to NLO properties of the fullerenes has improved the situation considerably. 

The response to frequency-dependent external fields may be obtained from Hartree-Fock response theory, yielding dynamical polarizabilities and hyperpolarizabilities. The identification of excitation energies as the poles of the dynamical polarizability tensor may be invoked to calculate excitation energies as well as one-photon and two-photon transition moments from the time development of the ground state [40-42]. 

Table 16 Results for hyperpolarizabilities (in a.u.) for N2- TDHF and MBPT(2) are results from time-dependent Hartree-Fock and perturbation-theory calculations, respectively, whereas CCSD and CCSD(T) are coupled-cluster results. Exp. denotes experimental results, and LDA, GGA, and LB94 are results from time-dependent density-functional calculations with different density functionals. For a description of the quantities, see the text. The results are from ref. 95...

Kobayashi, Sasagane and Yamaguchi" have developed the theory of the time-dependent spin-restricted Hartree-Fock method for application to open shell systems (TDROHF). The expression for the cubic hyperpolarizability is obtained from the quasi-energy derivative (QED) method. The theory is applied to the investigation of the frequency-dependent y susceptibility of the Li, Na, K and N atoms. 

On the other hand, the polarizability a and hyperpolarizabflity )3 related to the electro-optical effect of the SiC clusters are calculated using Hartree-Fock and time-dependent DFT approaches at A = 0.633 fim. The calculations are performed for an isolated cluster and then for the one embedded into polymer matrix. The effects of the surrounding media are taken into account via local field theory using the point-dipole approach. The obtained results are compared with the experimental data published recently (Boucle et al. 2006). The obtained polarizabilities a and hyperpolarizabilities )3(w 0,w) are siunmarized in O Table 18-3. Even with large differences observed on the local field calculations (see O Table 18-2) in the different... 

Basis set effect, at the Hartree-Fock level of theory, on the mean dipole polarizabilities per atom of the ground state structures of aluminum phosphide cluster of the type AI P win n =2-9, and basis set effect on the mean second hyperpolarizability of AI9P9... 

In the second, response theory, approach, the response of the Hartree-Fock ground state is calculated by perturbation theory. First-order perturbation theory in the fluctuation potential gives a method known as the random phase approximation (REA). The RPA linear response gives the dynamic polarizability, the quadratic response gives the first hyperpolarizability etc. One can obtain expressions for the response functions as sums over states formulae, but they are not calculated as such, rather they are calculated from coupled linear equations. RPA is equivalent to TDCPHF. 

In principle, density functional theory calculations should be able to give answers that are more reliable than Hartree-Fock but at similar cost. Static a and can be calculated by finite field methods or by coupled perturbed Kohn-Sham theory (CPKS) and give answers that are broadly comparable with MP2. In 1986 Sennatore and Subbaswamy did some calculations of the dynamic polarizability and second hyperpolarizability of rare gas atoms, but there have been no calculations of frequency dependent polarizabilities or hyperpolarizabilities of molecules until very recently. 

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