Polarizability vibrational component

Because an applied field in the y direction Ev can induce a dipole M with a component in the x direction Mx as well as the component in the y direction My, it is necessary that we specify the components of the polarizability tensor by two subscripts (Fig. 3). If the bond A—B of a diatomic molecule stretches during a vibrational mode, Mx and Mv will vary and therefore the corresponding polarizability tensor components will vary. 

More recently, the temperature dependence of the above vibrational component was studied13. Still for the 100 face of NaCl, the contribution of this component to the total surface energy, see Section II.3., was approximately —2 erg/cm2 and to the free surface energy near - 16 erg/cm2, both at 273 °K. No correction for the polarizability of the ions was employed. 

There have been a few recent studies of the corrections due to nuclear motion to the electronic diagonal polarizability (a ) of LiH. Bishop et al. [92] calculated vibrational and rotational contributions to the polarizability. They found for the ground state (v = 0, the state studied here) that the vibrational contribution is 0.923 a.u. Papadopoulos et al. [88] use the perturbation method to find a corrected value of 28.93 a.u. including a vibrational component of 1.7 a.u. Jonsson et al. [91] used cubic response functions to find a corrected value for of 28.26 a.u., including a vibrational contribution of 1.37 a.u. In all cases, the vibrational contribution is approximately 3% of the total polarizability. 

The results for the non-BO diagonal polarizability are shown in Table XIII. Our best—and, as it seems, well-converged—value of a, 29.57 a.u., calculated with a 244-term wave function, is slightly larger than the previously obtained corrected electronic values, 28.93 and 28.26 a.u. [88,91]. It is believed that the non-BO correction to the polarizability will be positive and on the order of less than 1 a.u. [92], but it is not possible to say if the difference between the value obtained in this work and the previous values for polarizability are due to this effect or to other effects, such as the basis set incompleteness in the BO calculations. An effective way of testing this would be to perform BO calculations of the electronic and vibrational components of polarizability using an extended, well-optimized set of explicitly correlated Gaussian functions. This type of calculation is outside of our current research interests and is quite expensive. It may become a possibility in the future. As such, we would like the polarizability value of 29.57 a.u. obtained in this work to serve as a standard for non-BO polarizability of LiH. 

In Table 1 the predicted dipole and quadmpole polarizability tensor components ay and C,y for the vibrational states with quantum number v are given. They were calculated for all vibrational states supported by the potential energy function as expectation values of the polarizability radial functions a(R) and C(R) over the vibrational wavefunction (equation (14)). The latter were obtained from... 

Thus, just as linear polarizabilities are frequency dependent, so are the nonlinear polarizabilities. Perhaps it is not surprising that most organic materials, with almost exclusively electronic nonlinear optical polarization, have similar efficiencies for SHG and the LEO effect. In contrast, inorganic materials, such as lithium niobate, in which there is a substantial vibrational component to the nonlinear polarization, are substantially more efficient for the LEO effect than for SHG. 

Symmetry-lowering effects of the solvent are referred to as the Ham effect (Ham, 1953 Platt, 1962). Thus, in rigid or fluid solutions, symmetry-forbidden vibrational components of the Bju ( Lb) absorption band of benzene appear with increasing intensity as the polarizability and the polarity of the solvent increase. The fact that for pyrene and 2-methylpyrene the direction of the Lb transition moment is inclined on the average 40° and 20° away from the y axis, respectively, has also been ascribed to a symmetrylowering perturbation by the environment, related to the Ham effect (Lang-kilde et al., 1983). The associated intensification of otherwise weak vibronic peaks in the Lb band can be used for an investigation of microenvironments such as micelles. 

In addition to the Raman selection rules described above there are surface selection rules that apply for SERS because the process occurs close to metal surfaces [40—42]. The SERS surface selection rule predicts that the vibrational bands that have contributions from the Raman polarizability tensor component where z is the surface normal, will be most intense with weaker contributions from vibrational bands which have contributions from and o. This is essentially because tlic electric field of the exciting hght is enhanced in the direction of the surface normal (Figure 6.2). The surface selection rule for Raman spectroscopy is more complex than that for infrared spectroscopy. Modes with the bond axis paraUel to... 

The global effect of an applied external field on a molecule involves distortions both in the electronic charge distribution and in the nuclear charge distribution the latter leads to the so-called vibrational contribution to the (hyper)polarizabilities. As said above, the analysis of the vibrational components reveals the presence of the distinct components, the curvar ture related to the effect of the field vibrational motion and including the zero point vibrational correction (ZPV) (see above), and the nuclear relaxation (nr) originates from the shift of the equilibrium geometry induced by the field. 

In this section we report a second extract of the study we have published on the Journal of the American Chemical Society about solvent effects on electronic and vibrational components of linear and nonlinear optical properties of Donor-Acceptor polyenes. In a previous section we have presented the analysis on geometries, here we report the results obtained for the electronic and vibrational (in the double harmonic approximation) static polarizability and hyperpolarizability for the two series of noncentrosym-metric polyenes NH2(CH=CH) R (n=l,2), with R=CHO (series I) and with R=N02 (series II) both in vacuo and in water. 

On the basis of the EM mechanism one can approximately predict the relative enhancements of the different vibrational modes for surface species. Since the component of the enhanced local electric field perpendicular to the surface has the largest enhancement, the vibrational modes with the totally symmetric polarizability tensor component (a ) have the strongest enhancement. The vibrational modes with the nontotally symmetric polarizability tensor elements (Uxz or Uyz) are less enhanced. Vibrational modes with the polarizability tensor... 

The electronic ground-state expectation values in the numerator of the vibrational polarizability are components of the permanent electric dipole moment, Elq. (4.40), and we can therefore write the vibrational polarizability more compactly as... 

The vibrations of acetylene provide an example of the so-called mutual exclusion mle. The mle states that, for a molecule with a centre of inversion, the fundamentals which are active in the Raman spectmm (g vibrations) are inactive in the infrared spectmm whereas those active in the infrared spectmm u vibrations) are inactive in the Raman spectmm that is, the two spectra are mutually exclusive. Flowever, there are some vibrations which are forbidden in both spectra, such as the torsional vibration of ethylene shown in Figure 6.23 in the >2 point group (Table A.32 in Appendix A) is the species of neither a translation nor a component of the polarizability. 

Having assigned symmetry species to each of the six vibrations of formaldehyde shown in Worked example 4.1 in Chapter 4 (pages 90-91) use the appropriate character table to show which are allowed in (a) the infrared specttum and (b) the Raman specttum. In each case state the direction of the transition moment for the infrared-active vibrations and which component of the polarizability is involved for the Raman-active vibrations. 

For a molecule belonging to the D2h point group deduce whether the following vibrational transitions, all from the zero-point level, are allowed in the infrared spectmm and/or Raman spectmm, stating the direction of the transition moment and/or the component of the polarizability involved ... 

Raman intensities of the molecular vibrations as well as of their crystal components have been calculated by means of a bond polarizibility model based on two different intramolecular force fields ([87], the UBFF after Scott et al. [78] and the GVFF after Eysel [83]). Vibrational spectra have also been calculated using velocity autocorrelation functions in MD simulations with respect to the symmetry of intramolecular vibrations [82]. 

The components of nucleic acids have been the subject of continuous DFT stud-ies61 S5,67 69. Jasien and Fitzgerald calculated dipole moments and polarizabilities for a series of molecules of biological interest including nucleic acid bases (adenine, thymine, cytosine, and guanine) and their pairs (adenine-thymine and cytosine-guanine)61. A good correlation between DFT(HL), experimental, and MP2 results was obtained for dipole moments and polarizabilities. More detailed analyses of DFT(SVWN) and DFT(B88/P86) results, which included vibrational frequencies, were reported for isolated bases and their... 

A stress-induced alignment can also be detected in Raman experiments. The sensitivity of a vibration to the polarization of the incident and scattered light in a Raman experiment is determined by the polarizability tensor for the vibration. Even in the absence of polarization information, IR absorption or Raman measurements made in the presence of stress can be used to detect a preferential alignment of a defect by the effect the alignment has on the relative intensities of the stress-split-components of a vibrational band. 

The polarizability tensor of a molecule related the components of the induced dipole moment of the molecule to the components of the electric field doing the inducing. It therefore has 9 components, axx, ctxy, etc., only 6 of which are independent. The theory of the Raman effect shows that a vibrational transition, from the totally symmetric ground state to an excited state of symmetry species F, will he Raman active if at least one of the following direct products contains the totally symmetric representation ... 

This bimodal dynamics of hydration water is intriguing. A model based on dynamic equilibrium between quasi-bound and free water molecules on the surface of biomolecules (or self-assembly) predicts that the orientational relaxation at a macromolecular surface should indeed be biexponential, with a fast time component (few ps) nearly equal to that of the free water while the long time component is equal to the inverse of the rate of bound to free transition [4], In order to gain an in depth understanding of hydration dynamics, we have carried out detailed atomistic molecular dynamics (MD) simulation studies of water dynamics at the surface of an anionic micelle of cesium perfluorooctanoate (CsPFO), a cationic micelle of cetyl trimethy-lainmonium bromide (CTAB), and also at the surface of a small protein (enterotoxin) using classical, non-polarizable force fields. In particular we have studied the hydrogen bond lifetime dynamics, rotational and dielectric relaxation, translational diffusion and vibrational dynamics of the surface water molecules. In this article we discuss the water dynamics at the surface of CsPFO and of enterotoxin. 

H2 quadrupole moment, <72(re) at the fixed equilibrium position, and thus the long-range coefficient of the quadrupole-induced dipole component, Eq. 4.3, is about 5% too small relative to the proper vibrational average, <12 = (v = 0 < 2(r) f = 0) [216, 217, 209], A 5% difference of the dipole moment amounts to a 10% difference of the associated spectral intensities. Furthermore, the effects of electron correlation on this long-range coefficient can be estimated. Correlation increases the He polarizability by 5% but decreases the H2 quadrupole moment by 8% [275], a net change of-3% of the leading induction term B R). 

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]. 

While for an accurate (+1%) treatment of the rototranslational spectra (v = v = 0) the matrix elements (vj (9 v f) of the lower rotational states do not much depend on the rotational transitions (j,f), for the vibrational bands (v > 0), for v f v, relatively strong j,f dependences are usually observed (9 designates the multipole and polarizability operator. Similar j dependences are also obtained for the dipole components Bc that are significant for line shape computations [63]. The accounting for the j dependences is relatively easy because the main effect of the j dependence is on the integrated intensity, but not so much on the shape of the profile. The main effect of neglecting the j dependence in the low-temperature spectra is an excess intensity of the Sj(l) lines. 

The selection rules for Raman vibrational transitions are also readily derived from group theory. Here, the transition probability depends on integrals involving the components of the molecular polarizability matrix a. Since a is symmetric, it has only six independent components axx aw axi axr ayi aMf These six quantities can be shown14 to transform the same way the six functions... 

For a vibration to be Raman active, there must be a change in the pofonzability tensor. We need not go into the details of this bere,24 but merely note that the components of the polarizability tensor transform as the quadratic (unctions of x, y, and s. Therefore, in the character tables we are looking fbr x2, y2, r2, xy, x=, yz, or their combinations such as x2 — >>2. Because the irreducible representation fbr x2 is Al and that for yz is B2, all three vibrations of the water molecule are Raman active as well. 

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