Hyperpolarizabilities formulas

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... 

Many ab initio packages use the two key equations given above in order to calculate the polarizabilities and hyperpolarizabilities. If analytical gradients are available, as they are for many levels of theory, then the quantities are calculated from the first or second derivative (with respect to the electric field), as appropriate. If analytical formulae do not exist, then numerical methods are used. 

The generality of a simple power series ansatz and an open-ended formulation of the dispersion formulas facilitate an alternative approach to the calculation of dispersion curves for hyperpolarizabilities complementary to the point-wise calculation of the frequency-dependent property. In particular, if dispersion curves are needed over a wide range of frequencies and for several optical proccesses, the calculation of the dispersion coefficients can provide a cost-efficient alternative to repeated calculations for different optical proccesses and different frequencies. The open-ended formulation allows to investigate the convergence of the dispersion expansion and to reduce the truncation error to what is considered tolerable. 

To obtain for 71 and jk compact dispersion formulas similar as Eq. (79) for 7, these hyperpolarizability components must be written as sums of tensor components which are irreducibel with respect to the permu-tational symmetry of the operator indices and frequency arguments ... 

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,... 

In a more conventional CNDO/2 calculation. Hush and Williams calculated the parallel and perpendicular components of a for a series of first-row diatomics, and also calculated a for all first-row atoms using a similar formula. In view of the errors in their formulae as published, it is not clear whether their results are correct or not, although our own experience is that CNDO/2 gives in general very low values for a. Hush and Williams have also extended their CNDO/2 calculations to hyperpolarizabilities (elements only) for several linear molecules together with HjO, NHj, and CHj. The results are unimpressive where comparison with experiment is possible. Meyer and Schweig have published an extensive comparison of MINDO/... 

The earlier experimental EFISH studies have often been linked to analysis of the results by the semi-empirical SOS method. An examination of the exact SOS formulae used to interpret the data can provide confirmatory evidence of the convention that has been adopted. The standard SOS procedure, for the electronic contribution to the first hyperpolarizability for frequency doubling, is summarized below. 

The two state formula of eqns (3.10) and (3.11) have frequently been used in attempts to deduce frequency-dependent results from static field calculations, or conversely to deduce the static hyperpolarizability from measurements at optical frequencies. For any convention (C) defining the hyperpolarizability, the two-state model gives. 

Whatever the validity of these formulae and the underlying cavity field theories it is very desirable that, before applying them, the definition and consistency of the macroscopic quantities as measured by different groups should be assessed. Much of the discussion in the literature deals directly with the final values of the hyperpolarizability presented in the various experimental papers. In calculating these values different versions of the above procedures may have been employed and, in particular, different values of the molecular dipole moment inserted into y to extract the y value. It is relatively easy to identify how the microscopic parameter has been obtained and which molecular convention is being used provided the identity of the macroscopic quantity is clearly established. The symbol, F), (and its derivative, (9F i/9H )o) is introduced here to denote provisionally a reported macroscopic nonlinearity before assessing its precise definition. The unprimed symbols are defined in accordance with eqn (4.16). The most troublesome ambiguity in the... 

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... 

Only the EFISH hyperpolarizabilities, /1//(m) and y(co) will be considered here. In principle the most accurate calculation of P//(co) of the coupled-cluster genre is that of Christiansen et al.44 using a CC3 correction. Their results for the electronic contribution, which show surprisingly small dispersion, are represented by the formula,... 

Just as an explicit formula for the linear polarizability is identified from the linear polarization, we are able to retrieve the corresponding formula for the first-order hyperpolarizability from the second-order polarization. We obtain the second-order polarization from Eq. (35) by insertion of the first-order and second-order... 

We must now once more return to the perturbation expansion of the molecular polarization and consider the third-order polarization in Eq. (36) from which we will identify a formula for the second-order hyperpolarizability in analogy with... 

The secular divergences in Eq. (65), i.e. the singularities due to terms in which one or more of the states equal 0), can be removed in a similar manner as we did for the lower-order properties. In fact, explicit nondivergent formulas have been derived for molecular hyperpolarizabilities up to fourth order [5], and for the second-order hyperpolarizability, the resulting formula can be written as (66) 7a/37s(-"<7 l t 2> 3)= E -<7,1,2,3... 

In tills very general formula we are using the notation of Bishop [5], where conventionally (linear polarizability), (first-order hyperpolariz-... 

The explicit formula for the first-order hyperpolarizability thus becomes d (kE,E)... 

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... 

Bishop, D.M. Explicit nondivergent formula.s for atomic and molecular dynamic hyperpolarizabilities. J. Chem. Phys. 100, 6535-6542 (1994)... 

In formula (34), /u, (f) = (0 /Ai(f) 0 ) is the ground state, permanent, electric dipole moment of species f, while ) is the molecular first hyperpolarizability tensor defined by... 

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 ... 

One of the hurdles in this field is the plethora of definitions and abbreviations in the next section I will attempt to tackle this problem. There then follows a review of calculations of non-linear-optical properties on small systems (He, H2, D2), where quantum chemistry has had a considerable success and to the degree that the results can be used to calibrate experimental equipment. The next section deals with the increasing number of papers on ab initio calculations of frequency-dependent first and second hyperpolarizabilities. This is followed by a sketch of the effect that electric fields have on the nuclear, as opposed to the electronic, motions in a molecule and which leads, in turn, to the vibrational hyperpolarizabilities (a detailed review of this subject has already been published [2]). Section 3.3. is a brief look at the dispersion formulas which aid in the comparison of hyperpolarizabilities obtained from different processes. 

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. 

Dynamic Response Functions. - The perturbation series formula or spectral representation of the response functions can be used only in connection with theories that incorporate experimental information relating to the excited states. Semi-empirical quantum chemical methods adapted for calculations of electronic excitation energies provide the basis for attempts at direct implementation of the sum over states (SOS) approach. There are numerous variants using the PPP,50,51 CNDO(S),52-55 INDO(S)56,57 and ZINDO58 levels of approximation. Extensive lists of publications will be found, for example, in references 5 and 34. The method has been much used in surveying the first hyperpolarizabilities of prospective optoelectronically applicable molecules, but is not a realistic starting point for quantitative calculation in un-parametrized calculations. 

---