Vibrational first hyperpolarizability

Vibrational first hyperpolarizability of methane and its fluorinated analogs... 

Vibrational first hyperpolarizabilities. The double harmonic approximation... 

The calculated first hyperpolarizabilities (see Table 2-4) are surprisingly close to the experimental data, which is probably fortuitous because they were calculated without taking into account vibrational effect. These studies demonstrated also that the double-zeta basis set augmented by field-induced polarization functions, although sufficient for calculations of dipole and quadrupole moments of the studied molecules at the Kohn-Sham LDA level, is not sufficient in the case of hyperpolarizabilities. 

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]. 

Quinet, O., Kirtman, B., Champagne, B. Analytical time-dependent Hartree-Fock evaluation of the dynamic zero-point vibrationally averaged (ZPVA) first hyperpolarizability. J. Chem. Phys. 118, 505-513 (2003)... 

Del Zoppo et al.197 have made an ab initio study of a model push-pull polyene and calculated the dependence of the polarizability and first hyperpolarizability on the bond alternation. Vibrational and solvent effects are simulated in the calculation. 

Table 1.8 Vibrational averaged static polarizability (10 esu) and first hyperpolarizability (10 esu) in vacuo and in water at 298K.

A similar analysis can be done on the first hyperpolarizability also here vacuum results are in good accord with previous calculations on conjugated systems, for example for NH2(CHCH)2N02 Kirtman etnd Champagne find a ratio of 2.20 with a RHF/6-31G calculation to be compared with our value of 2.03. Once more, solvent effects lead to large increases in the relative importance of the vibrational contribution with respect to the electronic one multiplying the gas phase value of by factors firom 1.5 (in the first term of series I) to 2 (in the second term of series II). 

Table 5.3 Electronic and vibrational contributions to the parallel components of the polarizability Uzz and first hyperpolarizability Pzzz obtained using the Numerov-Cooley intergration scheme at the CASSCF and CASPT2 levels of theory...

Table 5.16 Analysis of the pure vibrational contribution to the first hyperpolarizability components (a.u.) of pyrrole ...

The electronic and (double harmonic) vibrational contributions to the first hyperpolarizability of three merocyanine dyes have been calculated at different levels of approximation, showing a strong impact of electron correlation on the electronic counterpart . So, the dominant longitudinal p ... 

The first hyperpolarizability of planar and non-planar 4-[A,A-dimethyl-amino]-4 -nitro stilbene has been calculated at the CPHF level to investigate efficient 7i-electron delocalization effects within a global investigation including vibrational spectroscopies. ... 

Calculations of the vibrational contributions to the static polarizability and hyperpolarizability have also been attempted. As far as the EFISH experiment is concerned, which depends on the square of an optical frequency field, it is assumed that there will be no direct contribution to (—2static contribution is comparable with the static electronic contribution to /1(0 0,0). An indirect vibrational effect through the linear polarizability of the solvent molecules is more important. Calculations of the vibrational effects in pNA cannot be carried out reliably even for the static case since the second term in the perturbation theory is much larger then the first and there is no evidence of convergence. 

The popularity of the SOS methods in calculations of non-linear optical properties of molecules is due to the so-called few-states approximations. The sum-over-states formalism defines the response of a system in terms of the spectroscopic parameters, like excitations energies and transition moments between various excited states. Depending on the level of approximation, those states may be electronic or vibronic or electronic-vibrational-rotational ones. Under the assumption that there are few states which contribute more than others, the summation over the whole spectrum of the Hamiltonian can be reduced to those states. In a very special case, one may include only one excited state which is assumed to dominate the molecular response through the given order in perturbation expansion. The first applications of two-level model to calculations of j3 date from late 1970s [93, 94]. The two-states model for first-order hyperpolarizability with only one excited state included can be written as ... 

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. 

There are two ways in which molecular vibrations affect non-linear optical properties. The first, which is well understood, is zero-point-vibrational averaging of the calculated electronic properties. This need not delay us long. The second comes about from the effect that the electromagnetic radiation has on the vibrational motions themselves and this leads to the vibrational polarizabilities and hyperpolarizabilities which are the exact counterparts of the electronic ones which stem from the effect that the radiation has on the electronic motions. This phenomenon is now receiving long overdue attention and will be the main subject of this section. A more extensive review is available elsewhere [2]. 

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