Hyperpolarizability term

The formulae for effecting the conversion have been discussed in section (7.1). Data is not completely available to apply the general formula of Teng and Garito, but the simplified versions can readily be applied to the macroscopic measurements described in the previous section. A number of different values of the dipole moment have been adopted in extracting Pz from y. In all cases other than in refs. 42 and 43 the values of and a form of p are given and it is possible to reconstruct the quantity fiPz- In comparing the interpretations of different studies this product, expressed in units of (au.Debye), has been used. The third order hyperpolarizability term, y of... 

The coeflScients a, P, and y are the second, third, and fourth rank tensors and are referred to as the polarizability, first hyperpolarizability, and second hyperpolarizability, respectively. The hyperpolarizability terms are responsible for the nonlinear response of the molecule to impinging radiation. These coefiBcients are not very large, and the associated nonlinear optical effects are usually studied by taking advantage of the high optical field obtainable with laser beams. 

Revisions of the apparent quadrupole moments of quadrupolar systems based on ab initio results for Zi(w) were undertaken in the last few years for CI2 [163] and C2H2 [161, 171] (see Table 13). Direct comparison of the hyperpolarizability terms b(a) ) calculated ab initio and derived from measurements at various temperatures, such as in the case of CO2, CS2 [172] and the most recent results for N2 [170] turns out to be difficult in all cases due to the very large error bars associated with the experimentally derived results (see Table 13). 

From a theoretical point of view, rare gas atoms are ideal test systems. This is also the case for the BE, as for these systems the BE is entirely due to the hyperpolarizability term / ( )). Table 14 collects our best theoretical results for the hyperpolarizability term Z (w) of all atomic and spherical systems investigated so far together with Are available experimental data in Ate literature. 

The inducing power may be directly interpreted as a hyperpolarizability term of the chelate system, so that it is unlikely to be much affected by the details of the metal-ligand bonding. 

The foregoing formula refer to changes in the hamiltonian of the system. The polarizability and hyperpolarizability terms arise from the changes in the wavefunction induced by the perturbed hamiltonian. 

The linear term involving polarizability a describes the well known linear response such as the low-intensity refraction and absorption. The higher hyperpolarizability terms /3 and y describe the molecular nonlinear optical responses. 

Asymmetric molecules with delocalized pi-electron systems tend to large values of the first hyperpolarizability term, which is responsible for second harmonic generation. By incorporating electron donors and/or acceptors into the pi system, charge transfer interactions are enhanced, yielding increased values of higher order susceptibilities. 

B) THE MICROSCOPIC HYPERPOLARIZABILITY IN TERMS OF THE LINEAR POLARIZABILITY THE KRAMERS-HEISENBERG EQUATION AND PLACZEK LINEAR POLARIZABILITY THEORY OF THE RAMAN EFFECT... 

The next terms in the series, denoted. .. in equation 17.1 above, are called the dipole hyperpolarizabilities. The first one is and this also is a tensor. It has three indices, and the corresponding formula for the induced dipole, equation 17.3, becomes... 

There are in fact an infinite number of hyperpolarizabilities, and one occasionally comes across terms higher than fi. 

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. 

The proportionality constants a and (> are the linear polarizability and the second-order polarizability (or first hyperpolarizability), and x(1) and x<2) are the first- and second-order susceptibility. The quadratic terms (> and x<2) are related by x(2) = (V/(P) and are responsible for second-order nonlinear optical (NLO) effects such as frequency doubling (or second-harmonic generation), frequency mixing, and the electro-optic effect (or Pockels effect). These effects are schematically illustrated in Figure 9.3. In the remainder of this chapter, we will primarily focus on the process of second-harmonic generation (SHG). 

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... 

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