Calculated Raman intensities

When we compare the calculated Raman intensities for armchair, zigzag and chiral CNTs of similar diameters, we do not see large differences in the lower frequency Raman modes. This is because the lower frequency modes have a long... 

Figures 8 and 9 shows a part of the bending region at low temperature containing the components of Vg (150-160 cm ) and Vs (190-200 cm ). The Vg vibration, IR active in the free molecule, has weak components in the Raman spectrum. According to theoretically calculated Raman intensities, which almost perfectly fit the experimental spectrum, the big component has a very low scattering cross-section [87] and is accidentally degenerate with the b2g component at ca. 188 cm. The IR active components of Vg cause strong absorptions in the IR spectrum even if the crystalline sample used for transmission studies is as thin as 400 pm [107, 109].

The calculated Raman intensities are represented by squares and triangles in Figure 31. The experimental Raman intensities are represented by vertical solid lines. The numerical values of calculated Raman band intensities are compared to the experimental data in Table 8. The calculated Raman intensities are in good agreement with those found experimentally. 

Another source of uncertainty in the calculated bond length changes arises from the uncertainties in measuring the peak intensities from the experimental spectrum. A 10% change in bond length causes about a 20% change in the calculated Raman intensities. The reported fits are only obtainable within... 

Szczesniak and Scheiner considered the effect of these extension effects upon calculated spectroscopic intensities. They concluded that whereas cancellation occurs between the effects on the donor and acceptor molecules of (HF)2, leaving infrared intensities little affected, a great deal of caution must be exercised in calculating Raman intensities where the errors are additive. 

Although this formulation is less intuitive than the Placzek formulation, it has an important advantage, since it offers the possibility of calculating Raman intensities in absolute terms for molecules where the number of... 

Since the symmetry coordinates for Og modes are the same as in /-PA, we can test the transferability of e-ph coupling constants. We account for e-ph coupling in the flg block through F = F - F, with the same J F as in /-PA, and plot the frequencies as functions of rj in Fig. 6.7. The best fit to the experimental Raman frequencies [70,71] of m-PA occurs at t = 0.56. Within Huckel theory, t corresponds to cis)lx(trans). The predicted intensities are again in qualitative agreement with experiment [70,71]. The calculated Raman intensities of pz and... 

Raman-active transition in the spectra of polyenes. Comparison of ab initio calculated Raman intensities for n-alkanes and rrnns-polyenes gives parallel information [91,92]. Note that when the number of carbon atoms in the chains increases, the Raman intensities of saturated and conjugated chains differ remarkably in their behav-... 

Performing a least squares fitting of the set of reop. Calculated Raman intensities and depolarization ratios are compared with experimental observations and the set of reop ensuring the best fit is determined. Usually the best-fit procedure is carried out when Raman intensity data for large series of isotopically substituted molecules are available. 

In this section the predictive power of the bond polarizability model is tested in calculating Raman intensities of propyne by transferring parameters from other molecules. 

The siim-over-states method for calculating the resonant enlrancement begins with an expression for the resonance Raman intensity, /.y, for the transition from initial state to final state /in the ground electronic state, and is given by [14]... 

The intensities are plotted vs. v, the final vibrational quantum number of the transition. The CSP results (which for this property are almost identical with CI-CSP) are compared with experimental results for h in a low-temperature Ar matrix. The agreement is excellent. Also shown is the comparison with gas-phase, isolated I. The solvent effect on the Raman intensities is clearly very large and qualitative. These show that CSP calculations for short timescales can be extremely useful, although for later times the method breaks down, and CTCSP should be used. 

A number of molecular properties can be computed. These include ESR and NMR simulations. Hyperpolarizabilities and Raman intensities are computed using the TDDFT method. The population analysis algorithm breaks down the wave function by molecular fragments. IR intensities can be computed along with frequency calculations. 

In the following sections, we first show the phonon dispersion relation of CNTs, and then the calculated results for the Raman intensity of a CNT are shown as a function of the polarisation direction. We also show the Raman calculation for a finite length of CNT, which is relevant to the intermediate frequency region. The enhancement of the Raman intensity is observed as a function of laser frequency when the laser excitation frequency is close to a frequency of high optical absorption, and this effect is called the resonant Raman effect. The observed Raman spectra of SWCNTs show resonant-Raman effects [5, 8], which will be given in the last section. 

Using the calculated phonon modes of a SWCNT, the Raman intensities of the modes are calculated within the non-resonant bond polarisation theory, in which empirical bond polarisation parameters are used [18]. The bond parameters that we used in this chapter are an - aj = 0.04 A, aji + 2a = 4.7 A and an - a = 4.0 A, where a and a are the polarisability parameters and their derivatives with respect to bond length, respectively [12]. The Raman intensities for the various Raman-active modes in CNTs are calculated at a phonon temperature of 300K which appears in the formula for the Bose distribution function for phonons. The eigenfunctions for the various vibrational modes are calculated numerically at the T point k=Q). 

Table 1 Harmonic fundamental modes of the three most stable isomers of S4 with infrared and Raman intensities calculated at the B3LYP/6-31G(2df) level of theory [9]. Symmetrical modes (of symmetry A) are shown in italics. For the connectivities of the S4 isomers, see Scheme 1. Experimental wavenumbers are given for comparison assignments according to [9] using experimental data from [17, 76] ...

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]. 

In addition, theoretically calculated dispersion curves and Raman intensities have been reported as well as results of neutron scattering experiments [113, 115]. 

We have used the systems CnH +2 with n = 2,4,...,22, C H +2 with n = 3,5,...,21, and C H +2 with n = 4,6,...,22 to represent pure PA, positively charged solitons, and bipolarons respectively. SCF wavefunctions were calculated with a double-zeta quality basis set (denoted 6-3IG) and optimized geometries for all these systems were determined. In addition for the molecules with n up to 11 or 12 we calculated the vibrational spectrum, including infrared and Raman intensities. 

A group of investigators recently suggested that the density-functional theory (DFT), which calculates IR and Raman spectra, is a useful tool for direct characterization of the structures of diamondoids with increasing complexity [66]. They applied DFT to calculate Raman spectra whose frequencies and relative intensities were shown to be in excellent agreement with the experimental Raman spectra for C26H30, thus providing direct vibrational proof of its existence. 

Pettinger et al. observed a TERS spectrum of monolayer-thick brilliant cresyl blue (BCB) adsorbed on a smooth Au film surface by using a Ag tip, while no STM image of the adsorbed surface was shovm [23]. The Raman intensity increased when the tip was in the tunneling position, meaning that the tip-surface distance was around 1 nm. They calculated the field enhancement factor by the method described by... 

It can be seen from Figures 3.7 and 3.8 that the calculations reproduce very well not only the experimental spectra but also the experimentally observed isotopic shifts indicating a high reliability of the computational method. According to this comparison, definite attribution can be made for even the difficult Raman bands that cannot be assigned based solely on the experimental results. It is, however, necessary to mention at this point that the calculated Raman spectrum provided directly by the ab initio computations correspond to the normal Raman spectrum with the band intensity determined by the polarizability of the correlating vibration. Since the intensity pattern exhibited by the experimentally recorded resonance Raman spectrum is due to the resonance enhancement effect of a particular chromophore, with no consideration of this effect, the calculated intensity pattern may, in many... 

To be precise, this expression employs the so-called double-harmonic approximation, where cubic and higher force constants as well as second and higher dipole derivatives are ignored. This approximation is common to all current implementations of calculating IR and Raman intensities. For details see Amos, 1987. 

Stirling, A., 1996, Raman Intensities from Kohn-Sham Calculations , J. Chem. Phys., 104, 1254. 

The advantage of Raman spectromicroscopy is that very small specimens can be studied while still allowing the determination of the second and fourth moments of the ODF. However, the expressions for the Raman intensities are more complex since the optical effects induced by the microscope objective have to be considered. Although the corrections may be small, they are not necessarily negligible [59]. This problem was first treated by Turrell [59-61] and later by Sourisseau and coworkers [5]. Turrell has mathematically quantified the depolarization of the incident electric field in the focal plane of the objective and the collection efficiency of the scattered light by high numerical aperture objectives. For brevity, only the main results of the calculations will be presented. Readers interested in more details are referred to book chapters and reviews of Turrell or Sourisseau [5,59,61]. The intensity in Raman spectromicroscopy is given by [59-61]... 

The Raman intensities are more difficult to calculate as they involve the derivative of the polarisability along the normal mode. 

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