Pure vibrational contribution

The CCSD model gives for static and frequency-dependent hyperpolarizabilities usually results close to the experimental values, provided that the effects of vibrational averaging and the pure vibrational contributions have been accounted for. Zero point vibrational corrections for the static and the electric field induced second harmonic generation (ESHG) hyperpolarizability of methane have recently been calculated by Bishop and Sauer using SCF and MCSCF wavefunctions [51]. 

The techniques and approximations involved in obtaining computationally tractable schemes for the calculation of the linear and nonlinear opAcal properties differ for the three contributions given in Eq. (86), and the different strategies will be presented and reviewed in the different chapters of this book. In the next section, we will briefly describe a few of the approximate methods used to calculate hyperpolarizabilities. Most of these methods will be directed toward the electronic contributions, but some of the approaches will also be able to extract Information about the pure vibrational contributions. 

A comparison with an experimental result for a wavelength of 1064 nm of -19.2 0.9 a.u. [17] is limited by missing data for the pure vibrational contribution at this wavelength. However, the best estimate for the electronic contribution including the aforementioned MP2 value for the ZPV correction obtained in [36] is with —19.81 a.u. in agreement with the experimental value. 

X > refers to the ground state (K = 0) or it refers to an excited state. The electronic property is considered to arise entirely from K 0 intermediate states. Hence, all terms containing one or more intermediate states with K = 0 are considered to be part of what has come to be known as the pure vibrational contribution. 

Pure vibrational contributions (to be discussed in Section 3.2.) were also added to these numbers. D2 as well as H2 was considered. 

The ZPVA contributions for the considered sulfides are very small, but the relativistic contribution has a significant effect on this small quantity. The pure vibrational contribution is small, but not negligible. The relativistic correction has a very significant effect on this contribution of HgS. One may compare the CCSD(T) values for the pure vibrational contribution to Pzzz of HgS, which are —176.6 a.u. (nonrelativistic) and —44.8 a.u. (relativistic). The following sequence is observed (CCSD(T) values with vibrational and relativistic corrections) ... 

Pzzz- The vibrational contribution is quite large. Specifically the pure vibrational correction is 35% of the total property. The ZPVA correction is about an order of magnitude smaller than the pure vibrational contribution. There is a perfect agreement between the CASSCF/NC and the CASSCF/BKPT results. Correlation (CASPT2) has a very significant effect on all contributions (Table 5.3) [20]. 

The NR electronic contribution increases with the atomic number of the metal. At the relativistic level, a maximum value is observed for AgH (Tables 5.5 and 5.6). The relativistic correction has a significant effect on the ZPVA and pure vibrational contributions of AgH, and in particular of AuH. The relaxation and curvature contributions are small (Table 5.5). 

We shall review the electronic and the pure vibrational contributions to the hyperpolarizabilities of pyrrole [50]. The original article involved also dipole moment and polarizabilities. The molecule is placed on the yz plane (Fig. 5.2). The computations have been performed at the Hartree-Fock level, employing the Pol basis set [45]. 

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

We note that the evaluation of CARS requires largely the same quantity as is needed for conventional Raman spectroscopy as well as ROA, namely the polarizability gradient. If the nonresonant contributions are also wanted, then the electronic second hyperpolarizability is also needed, as well as the off-resonant pure vibrational contributions [292]. Ab initio studies of CARS have to date been very limited [292, 293]. 

Secondly, we have not discussed in any detail the effects of nuclear motion. Methods used to calculate these vibrational corrections, for both zero-point vibrational effects and temperature effects, have been described elsewhere in this book. There are, however, other effects that should also be considered. We have not discussed the role of the purely vibrational contributions to molecular (electric) properties (Bishop 1990), which in certain cases can be as large as the electronic contributions (Kirtman et al. 2000). Moreover, in conformationally flexible molecules one has to consider the effects of large nuclear motions. For instance, for a proper comparison with experiment, it may not be sufficient to perform an ab initio calculation for a single molecular structure. In experiment one will always observe the average value of the different thermally accessible isomers (rotamers, conformers), and in order to allow for a direct comparison with these experimental observations, a Boltzmann average of the properties of these isomers must be computed. This is particularly important when the properties of the isomers are very different, possibly even differing in sign (Pecul et al. 2004). 

The quantity (v), an effective atomic charge, is determined by purely vibrational contributions to the dipole moment changes. The elements of Px(v) are not interconnected by die equations resulting from relation (4.19). Still, nine redundancy conditions as defined by Eq. (4.18) are present. The removal of these remaining redundancies can be accomplished by defining the problem in bond Cartesian displacement coordinate space [Eqs. (4.96) and (4.97)]. 

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