Vibrational Contributions to Molecular Properties

In this chapter we will therefore discuss the contributions from the nuclear wave-function to the molecular properties derived in the previous chapters. However, in doing so we will still make use of the Born-Oppenheimer approximation. In the following, we will use the static polarizability as example and illustrate how these vibrational corrections can be incorporated (Bishop and Cheung, 1980 Bishop et al., 1980). The expression, which we are going to derive, can then easily be transferred to all linear response properties. A detailed description of vibrational corrections to static and frequency-dependent hyperpolarizabilities can be found in the reviews by Bishop (1990 1998). 

In order to incorporate the effects of nuclear motion we have to go back to the Hamiltonian, Eq. (2.1), which includes the kinetic energy operators for the nuclei. The corresponding eigenfunctions are the so-called vibronic wavefunctions ) with energy E J and are characterized by the electronic, k, and vibrational, v, quantum numbers, where v stands throughout the chapter collectively for the vibrational quantum numbers of all vibrational modes of the molecule. The proper approach for the treatment of the nuclear motion effects would be to use these unperturbed vibronic wavefunctions R/f ) instead of the unperturbed electronic wavefunctions k ) hi the derivation of expression for 

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There are two major ways to view tlie vibrational contribution to molecular linear and nonlinear optical properties, i.e. to (hyper)polarizabilities. One of these is from the time-dependent sum-over-states (SOS) perturbation theory (PT) perspective. In the usual SOS-PT expressions [15], based on the adiabatic approximation, the intermediate vibronic states K, k> are of two types. Either the electronic wavefunction... 

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). 

Molecular electric properties exhibit an appreciable dependence on nuclear geometry and knowledge of corresponding property radial functions is essential for the understanding of the role of vibrational and rotational contributions to these properties (see Ref. [22] and references therein). 

In this contribution the concept of instantaneous normal modes is applied to three molecular liquid systems, carbon monoxide at 80 K and carbon disulphide at ambient temperature and two different densities. The systems were chosen in this way because pairs of them show similarities either in structural or in dynamical properties. The systems and their simulation are described in the following section. Subsequently two different types of molecular coordinates are used cis input to normal mode calculations, external, i.e. translational and rotational coordinates, and internal, i.e. vibrational coordinates of strongly infrared active modes, respectively. The normal mode spectra are related quantitatively to molecular properties and to those of liquid structure and dynamics. Finally a synthesis of both calculations is attempted on qualitative grounds aiming at the treatment of vibrational dephcising effects. 

Since their discovery, fullerenes and their derivatives have been the subject of very extensive research. One of the topics investigated intensively are the linear and nonlinear optical (NLO) properties, owing to a variety of possible applications. Here we review some of the recent work of our group in this area, which is concerned with the ab-initio calculation of molecular NLO properties of two different kinds of fullerene derivatives, a) substituted 1,2-dihydro fullerenes and b) fullerenes endohedrally doped with atoms or small molecules. Apart from the purely electronic response, we also focus on the vibrational contributions to the NLO response, that is, to the response of the nuclei to the external electric fields. 

Calculation of Thermodynamic Properties We note that the translational contributions to the thermodynamic properties depend on the mass or molecular weight of the molecule, the rotational contributions on the moments of inertia, the vibrational contributions on the fundamental vibrational frequencies, and the electronic contributions on the energies and statistical weight factors for the electronic states. With the aid of this information, as summarized in Tables 10.1 to 10.3 for a number of molecules, and the thermodynamic relationships summarized in Table 10.4, we can calculate a... 

Table A4.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the various degrees of freedom, including translation, rotation, and vibration (see Section 10.7). For most monatomic gases, only the translational contribution is used. For molecules, the contributions from rotations and vibrations must be included. If unpaired electrons are present in either the atomic or molecular species, so that degenerate electronic energy levels occur, electronic contributions may also be significant see Example 10.2. In molecules where internal rotation is present, such as those containing a methyl group, the internal rotation contribution replaces a vibrational contribution. The internal rotation contributions to the thermodynamic properties are summarized in Table A4.6.

Computational spectrometry, which implies an interaction between quantum chemistry and analysis of molecular spectra to derive accurate information about molecular properties, is needed for the analysis of the pure rotational and vibration-rotational spectra of HeH in four isotopic variants to obtain precise values of equilibrium intemuclear distance and force coefficient. For this purpose, we have calculated the electronic energy, rotational and vibrational g factors, the electric dipolar moment, and adiabatic corrections for both He and H atomic centres for intemuclear distances over a large range 10 °m [0.3, 10]. Based on these results we have generated radial functions for atomic contributions for g g,... 

Recent work improved earlier results and considered the effects of electron correlation and vibrational averaging [278], Especially the effects of intra-atomic correlation, which were seen to be significant for rare-gas pairs, have been studied for H2-He pairs and compared with interatomic electron correlation the contributions due to intra- and interatomic correlation are of opposite sign. Localized SCF orbitals were used again to reduce the basis set superposition error. Special care was taken to assure that the supermolecular wavefunctions separate correctly for R —> oo into a product of correlated H2 wavefunctions, and a correlated as well as polarized He wavefunction. At the Cl level, all atomic and molecular properties (polarizability, quadrupole moment) were found to be in agreement with the accurate values to within 1%. Various extensions of the basis set have resulted in variations of the induced dipole moment of less than 1% [279], Table 4.5 shows the computed dipole components, px, pz, as functions of separation, R, orientation (0°, 90°, 45° relative to the internuclear axis), and three vibrational spacings r, in 10-6 a.u. of dipole strength [279]. 

V is the vibrational frequency in the gas phase, v is the frequency in the solvent of relative permittivity Sr, and C is a constant depending upon the molecular dimensions and electrical properties of the vibrating solute dipole. The electrostatic model leading to Eq. (6-8) assumes that only the electronic contribution to the solvent polarization can follow the vibrational frequencies of the solute ca. 10 " s ). Since molecular dipole relaxations are characterized by much lower frequencies (10 to 10 s ), dipole orientation cannot be involved in the vibrational interaction, and Eq. (6-8) may be written in the following modified form [158, 168] ... 

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