Dipole Polarizabilities and Hyperpolarizabilities

When Pauling published his famous paper in 1927, it was several decades before the invention of the laser. This was an event which would transform scientific research in the second half of the century, not least of all for the investigation of hyperpolarizabilities. If, in Eq. (10), electric fields associated with light are introduced (i.e. dynamic fields) then the F3 factor becomes F F F where co, are the frequencies of the fields, some of which 

The frequency dependence of the polarizability governs the change of refractive index with the frequency of the light source. In Eq. (11), F becomes Fa and we can write a either as a(o)) or a(-03o Oi) with 0) = 0) and 0)1 = ox The theory for a(0)) was developed early on and there have been many calculations this is the linear effect. Computation of y(co), the nonlinear property, is a fairly recent departure and it is spurred on by the potential that non-linear processes have for commercial exploitation. 

Whereas for atoms, the static hyperpolarizability tensor y(0 0, 0, 0), has only one independent component, for y(-03n 331, 0)2, CO3) there are a maximum number of three (how many depends on the values of 03,) and they are identified by Cartesian axis subscripts on y, in general y s where a, p, y, 8 = x, y, z. 

In these equations infers a sum over the terms obtained by permuting the pairs (—0)a,fia), (cq./ip), etc., is the dipole moment operator along the r axis, and hcom = Em - Eg. A full discussion of the origin of these terms has been given by Bishop[78]. 

The frequency dependence of a(co) is well understood and it is usually expressed by a Cauchy series. For y(co) it is only recently that a thorough description has become available[79]-[83]. In general, however, y( ) is calculated for specific frequencies (co) and individual NLO processes. 

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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 values for the dipoles, polarizabilities, and hyperpolarizabilities of the H2 series were obtained using (a) a 16-term basis with a fourfold symmetry projection for the homonuclear species and (b) a 32-term basis with a twofold symmetry projection for the heteronuclear species. These different expansion lengths were used so that when combined with the symmetry projections the resulting wave functions were of about the same quality, and the properties calculated would be comparable. A crude analysis shows that basis set size for an n particle system must scale as k", where k is a constant. In our previous work [64, 65] we used a 244-term wave function for the five-internal-particle system LiH to obtain experimental quality results. This gives a value of... 

Dickson and Becke59 performed finite-field LDA calculations of the dipole polarizabilities and hyperpolarizabilities of the following compounds H2, N2, 02, CO, HF, H20, NH3, and CH4. These studies have a benchmark character (for dipole polarizabilites and first hyperpolarizabilities). The calculated dipole polarizabilities are systematically overestimated (see Table 2-3). Other studies reveal the similar trend that LDA overestimates the dipole polarizabilities of small organic molecules. 

We will divide the survey into three parts (3.1) static dipole polarizabilities, (3.2) static dipole hyperpolarizabilities, and (3.3) dynamic dipole polarizabilities and hyperpolarizabilities. Within each part there will be sub-sections dealing with the three isoelectronic series He, Ne, and Ar. For (3.2) and (3.3) the hydrogen atom will also be included. 

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... 

The vibrational excursions of a molecule may cause it to have sharply changing electrical properties from state to state. This, of course, is essential for mechanisms of absorption and emission of radiation. How sharp these changes may be is illustrated for HF in Figure 3. The curves show the axial elements of a. A, and P in the vicinity of the equilibrium bond length as a function of the H-F distance. The types of changes that may be found in a polyatomic molecule are illustrated by Figures 4 and 5. They show contours of the dipole polarizability and hyperpolarizability elements over the two stretching coordinates of HCN. Both and P yy have zero contours... 

Atom-Atom Interactions. - The methods applied, usually to interactions in the inert gases, are a natural extension of diatomic molecule calculations. From the interaction potentials observable quantities, especially the virial coefficients can be calculated. Maroulis et al.31 have applied the ab initio finite field method to calculate the interaction polarizability of two xenon atoms. A sequence of new basis sets for Xe, especially designed for interaction studies have been employed. It has been verified that values obtained from a standard DFT method are qualitatively correct in describing the interaction polarizability curves. Haskopoulos et al.32 have applied similar methods to calculate the interaction polarizability of the Kr-Xe pair. The second virial coefficients of neon gas have been computed by Hattig et al.,33 using an accurate CCSD(T) potential for the Ne-Ne van der Waals potential and interaction-induced electric dipole polarizabilities and hyperpolarizabilities also obtained by CCSD calculations. The refractivity, electric-field induced SHG coefficients and the virial coefficients were evaluated. The authors claim that the results are expected to be more reliable than current experimental data. 

Bishop, D.M., Kirtman, B. A peturbation method for calculating vibrational dynamic dipole polarizabilities and hyperpolarizabilities. J. Chem. Phys. 95, 2646-2658 (1991)... 

Calculating Vibrational Dynamic Dipole Polarizabilities and Hyperpolarizabilities. 

Maroulis calculated the interaction-induced dipole polarizability and hyperpolarizability of the He2, Ne2, Ar2 and Kr2 homodiatoms relying on finite-field Moller-Plesset perturbation theory and coupled cluster calculations. Special attention was paid to the design of flexible basis sets, suitable for interaction-induced electric property calculations. Atom-specific, prepared basis sets were used on all atoms. The construction is completed in four steps ... 

The perturbation operator in the calculation of electric dipole moments, electric dipole polarizabilities and hyperpolarizabilities, i.e. the electric dipole moment operator Eq. (4.30), contains the position vector r of the electrons, which implies that the tail of the wavefunction becomes important. However, this is not well described in GTOs as discussed before and it is therefore essential to include additional valence functions with very small exponents C - so-called diffuse basis functions. In the Pople-style basis sets this is done in the 6-31G-I- and 6-31G- -- - basis sets, where in the -h basis set one diffuse function is added only for second- and third-row atoms, while in the - -+ basis set one diffuse function is also added for hydrogen (Clark et al., 1983). In the series of correlation consistent and polarization consistent basis sets one set of diffuse functions of each type present in the basis set is added in the aug-cc-pVXZ (Kendall et at, 1992 Woon and Dunning Jr., 1993, 1994 Balabanov and Peterson, 2005) and aug-pc-n (Jensen, 2002c) version of these basis sets. In the series of correlation consistent basis sets it is also possible to add two or more sets of diffuse functions in the d-aug , t-aug and so forth versions. 

In previous work [122] we presented an extended computational study of the interaction-induced electric properties of the water dimer (H20)2- Our findings strongly suggest that the interaction-induced mean dipole polarizability and hyperpolarizability are nearly additive, as... 

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