Hyperpolarizabilities vibrational

Apart from purely electronic effects, an asymmetric nuclear relaxation in the electric field can also contribute to the first hyperpolarizability in processes that are partly induced by a static field, such as the Pockels effect [55, 56], and much attention is currently devoted to the study of the vibrational hyperpolarizability, can be deduced from experimental data in two different ways [57, 58], and a review of the theoretical calculations of p, is given in Refs. [59] and [60]. The numerical value of the static P is often similar to that of static electronic hyperpolarizabilities, and this was rationalized with a two-state valence-bond charge transfer model. Recent ab-initio computational tests have shown, however, that this model is not always adequate and that a direct correlation between static electronic and vibrational hyperpolarizabilities does not exist [61]. 

Keywords nonlinear optical properties, vibrational hyperpolarizabilities, nuclear relaxation hyper-... 

The main problem in the analytical evaluation of vibrational hyperpolarizabilities for medium size and larger molecules is the large number of " -order derivatives with respect to normal modes that must be computed. This number is on the order of (3N-6)" with N being the number of normal modes. The static and infinite optical frequency P" can be computed using the FF procedure and the same is true of pc-zpva assuming that is available. However, in order to calculate or P" and at arbitrary frequencies, the BK analytical expressions are currently... 

The fact that die number of FICs needed to compute any vibrational hyperpolarizability does not depend upon the size of the molecule leads to important computational advantages. For Instance, the calculation of the longitudinal component of the static y" for l,l-dlamino-6,6-diphosphinohexa-l,3,5-triene requires quartic derivatives of the electronic energy with respect to vibrational displacements (i.e. quartic force constants) [34]. Such fourth derivatives may be computed by double numerical differentiation of the analytical Hessian matrix. With normal coordinates it is necessary to compute the Hessian matrix 3660 times, whereas using FICs only 6 Hessian calculations are required. 

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. 

Kirtman et al.149 show how low frequency collective modes contribute to the dynamic vibrational hyperpolarizabilities of different linear chains (polyacetylene, polyyne, polyethylene and polysilane). In another work Champagne et al.150 have calculated the static vibrational yvL/N of polyacetylene and they have found it to be 30% larger than the electronic contribution, while in the case of polyyne yj[/yL is 0.92151 at the extrapolated infinite chain length. For polydiacetylene and polybutatriene both the static electronic and the vibrational contributions to a /N and ctvL/N, and yl/N and yj JN, respectively, were computed. Also in these cases for both chains the vibrational contributions are of the same order as the electronic ones. One should mention, however, that in these cases the yL/N values extrapolated to infinity are about 10% larger than the values obtained for seven or ten units, respectively, in the two chains. Finally for d -trans polysilane154 the static y /N and yvL/N and values were also calculated. Their values have been found also to be of comparable magnitude. 

The vibrational hyperpolarizabilities of push-pull systems in solution... 

Bartkowiak and Misiaszek used the eoupled perturbed Hartree-Fock method and the sum-over-modes formalism to ealeulate the eleetronie and vibrational /i-tensors for 4-nitroaniline, 4-nitro-4 -aminostilbene, 4-amino-4 -nitrobiphenyl and 4-amino-4 -nitrophenylacetylene, all typieal push-pull eonjugated molecules of the kind that have been associated with seeond order optieal non-linearities derived from their large P values. Their ealeulations refer to the gas phase and to chloroform and aqueous solutions, the solvent effects being included through a continuum self-consistent reaction field model. They demonstrate that the solvent effects are much greater for the vibrational hyperpolarizability than for the electronic contribution. 

Luis et a/. have shown that, in evaluating the contribution to the static vibrational hyperpolarizability that comes from the normal co-ordinate derivatives of the zero point energy, it is possible to replace the sum over 3N — 6 normal coordinates by transforming to a set of field induced co-ordinates, when only one term remains. The method has been applied to a typical push-pull polyene, NH2-(C=C)-N02. 

In other words, in both infrared and Raman spectra, vibrational coordinates along these particular directions induce charge redistributions in the electronic cloud that are reflected in the vibrational intensities. Vibrational hyperpolarizabilities, which according to Eqs. (22) and (23) are expressed in terms of vibrational intensities, are obviously greatly affected. 

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