Multipole hyperpolarizability

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] ... 

The properties available include electrostatic charges, multipoles, polarizabilities, hyperpolarizabilities, and several population analysis schemes. Frequency correction factors can be applied automatically to computed vibrational frequencies. IR intensities may be computed along with frequency calculations. 

Asymptotic formulae. For a discussion of induced dipoles in highly polarizable species, it is often sufficient to consider the so-called classical multipole induction approximation in its simplest form (i.e., neglecting field gradients and hyperpolarizabilities). In such a case, one needs to know only the vibrational matrix elements of the multipole moments,... 

G. Maroulis and A. J. Thakkar, J. Chem. Phys., 88, 7623 (1988). Multipole Moments, Polarizabilities and Hyperpolarizabilities for N2 from Fourth-Order Many-Body Perturbation Theory Calculations. [Erratum, ibid., 89, 6558 (1988). 

For neutral molecules, the dipole polarizabilities and hyperpolarizabilities are invariant to the choice of the moment center. Other multipole polarizabilities may be invariant in certain cases of high molecular symmetry. The changes that may occur in P , P ,. . . upon shifting an evaluation center are determined by the changes in the moments or moment operators. If a particular origin translation leads to... 

DHF/ELP Basis Multipole Polarizabilities and Hyperpolarizabilities of trons-NjHj... 

M is a first-rank polytensor. The usual multipole polarizabilities can be arranged in a second-rank (two index) polytensor, by using the ordering of element labels in M for the rows and columns in For instance, pS or pPj is a dipole(A) quadrupole(xx) polarizability tensor element. Hyperpolarizabilities... 

It would be relatively easy to extend here our computer symbolic calculations to the hyperpolarizability part of the pair polarizability [see Eqs. (5) and (7)]. However, from all our numerical computations done for N2, C02, and CF4, it results that nonlinear part of the pair polarizability has a weak influence on the resulting spectrum (for details, see Refs. 8, 13, and 15-18). Bearing in mind these results in this review, we restrict our discussion to multipolar light scattering mechanisms. Formula (22) allows us to write the following simple symbolic program in Mathematica calculating the analytical form of the autocorrelation function (16) for a selected dipole-arbitrary order multipole induction operator ... 

As in the Rayleigh case, the pair polarizability results as well from nonlinear light scattering mechanisms (induced by hyperpolarizabilities and permanent multipole moments). For tetrahedral molecules nonlinear mechanisms contribute to some correlation functions listed in Table V—only those related to the depolarized spectrum and governed by double rotational transitions (QQ, QO, and 00). The nonlinear origin corrections Atp j7"1 j2 which must be added to the linear origin terms cp 1 °f Table V for a tetrahedral molecule are successively [17]... 

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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