Hyperpolarizability higher-order

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

Our results indicate that dispersion coefficients obtained from fits of pointwise given frequency-dependent hyperpolarizabilities to low order polynomials can be strongly affected by the inclusion of high-order terms. A and B coefficients derived from a least square fit of experimental frequency-dependent hyperpolarizibility data to a quadratic function in ijf are therefore not strictly comparable to dispersion coefficients calculated by analytical differentiation or from fits to higher-order polynomials. Ab initio calculated dispersion curves should therefore be compared with the original frequency-dependent experimental data. 

We now turn our attention to the higher order polarizabilities (y etc.) in Eq. (9), which were named hyperpolarizabilities by Coulson et a/.[61]. The development of the theory... 

For the hydrogen atom, using the method of Coulson[ll], Bishop and Pipin[62] have calculated y and the next higher-order non-zero hyperpolarizability (Xs) analytically. As well, a number of other mixed dipole-quadrupole hyperpolarizabilities were determined in this paper. The first calculation of y(H) was made by Sewell[63]. 

All applications quoted so far were for the linear response. Very few investigations have dealt with the higher-order response described in Sect. 5.2. The frequency-dependent third-order hyperpolarizabilities of rare-gas atoms were calculated by Senatore and Subbaswamy [86] within the ALDA the calculated values turned out to bee too large by a factor of two, further indicating the need for self-interaction corrected functionals in the calculation of response properties. The effect of adsorbates on second-harmonic generation at simple metal surfaces was invested by Kuchler and Rebentrost [205, 206]. Most recently, the second-order harmonic generation in bulk insulators was calculated within the ALDA [207]. 

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. 

In the deduction of Eq. (77) all the hyperpolarizabilities of order higher than xz,ik were omitted. In particular, in Eq. (77) the term proportional to S 2 responsible for the contribution to a(w, SF, ft) due to the Voight mechanism is absent. Thus, by means of the preceding approximations the contributions to the polarizability that in principle may lead the Langevin-Bom mechanism of birefringence are separated. 

Electron correlation plays a role in electrical response properties and where nondynamical correlation is important for the potential surface, it is likely to be important for electrical properties. It is also the case that correlation tends to be more important for higher-order derivatives. However, a deficient basis can exaggerate the correlation effect. For small, fight molecules that are covalently bonded and near their equilibrium structure, correlation tends to have an effect of 1 5% on the first derivative properties (electrical moments) [92] and around 5 15% on the second derivative properties (polarizabilities) [93 99]. A still greater correlation effect is possible, if not typical, for third derivative properties (hyperpolarizabilities). Ionic bonding can exhibit a sizable correlation effect on hyperpolarizabilities. For instance, the dipole hyperpolarizability p of LiH at equilibrium is about half its size with the neglect of correlation effects [100]. For the many cases in which dynamical correlation is not significant, the nondynamical correlation effect on properties is fairly well determined with MP2. For example, in five small covalent molecules chosen as a test set, the mean deviation of a elements obtained with MP2 from those obtained with a coupled cluster level of treatment was 2% [101]. 

The situation is somewhat different for the convergence with the wavefunction model, i.e. the treatment of electron correlation. As an anisotropic and nonlinear property the first dipole hyperpolarizability is considerably more sensitive to the correlation treatment than linear dipole polarizabilities. Uncorrelated methods like HF-SCF or CCS yield for /3 results which are for small molecules at most qualitatively correct. Also CC2 is for higher-order properties not accurate enough to allow for detailed quantitative studies. Thus the CCSD model is the lowest level which provides a consistent and accurate treatment of dynamic electron correlation effects for frequency-dependent properties. With the CC3 model which also includes the effects of connected triples the electronic structure problem for j8 seems to be solved with an accuracy that surpasses that of the latest experiments (vide infra). 

The comparison of tllS/SOS and CISD/SOS results of calulations of y leads to the opposite conclusions. The double and possibly higher order excitations are mandatory to obtain die second-order hyperpolarizability consistent with experimental data. The total SOS formula for y can be divided into two parts, namely 7 " and The convergence of these terms with respect to the number of electronic states included in summation is presented in Figs. 2 and 3. The first figure show the longitudinal component while the second figure present the average... 

The theoretical and experimental works clearly shows that the environment plays a remarkable role in the considerations of the first- (j8) and the second-order hyperpolarizabilities (y) of the D-tt-A type chromophores [ld-45]. Not only the magnitudes of p and y, but also the signs of these properties can be affected by change of the solvent polarity. On the other hand, the linear polarizability (a) is less affected by the solvent than the higher order polarizabilities, namely p and y. 

Despite extensive research, major problems still remain unsolved in both of the crucial areas required for employing effectively nonlinear materials based on molecules the molecular hyperpolarizability properties (P, 7 and to a very limited extent higher-order responses) are still not either predictable or preparable using any meaningful structure/function understandings. Moreover, utilizing molecules to prepare actual materials with designed nonlinearities. .. remains a very... 

In general, the physical properties of an electron system are defined by referring to a specific perturbation problem and can be classified according to the order of the perturbation effect. For instance, the electric dipole moment is associated with the first-order response to an applied electric field (i.e. the perturbation), the electric polarizability with the second-order response, hyperpolarizabilities with higher-order terms. In addition to dipole moments, there is a number of properties which can be calculated as a first-order perturbation energy and identified with the expectation value... 

The theoretical framework developed above is valid in the electric dipole approximation. In this context, it is assumed that the nonlinear polarization PfL(2 >) is reduced to the electric dipole contribution as given in Eq. (1). This assumption is only valid if the surface susceptibility tensor x (2 > >, a>) is large enough to dwarf the contribution from higher orders of the multipole expansion like the electric quadrupole contribution and is therefore the simplest approximation for the nonlinear polarization. At pure solvent interfaces, this may not be the case, since the nonlinear optical activity of solvent molecules like water, 1,2-dichloroethane (DCE), alcohols, or alkanes is rather low. The magnitude of the molecular hyperpolarizability of water, measured by DC electric field induced second harmonic... 

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