Local mode model

It should be realized, though, that either model can be used for levels for all values of v for such stretching vibrations but the normal mode model is more practically useful at low v and the local mode model more useful at higher values of v. 

Starting from the normal mode approximation, one can introduce anharmonicity in different ways. Anharmonic perturbation theory [206] and local mode models [204] may be useful in some cases, where anharmonic effects are small or mostly diagonal. Vibrational self-consistent-field and configuration-interaction treatments [207, 208] can also be powerful and offer a hierarchy of approximation levels. Even more rigorous multidimensional treatments include variational calculations [209], diffusion quantum Monte Carlo, and time-dependent Hartree approaches [210]. 

Henry s group is also involved in theoretical studies to determine sources of local mode overtone intensity. These investigators have developed a very successful approach that uses their harmonically coupled anharmonic oscillator local mode model to obtain the vibrational wavefunctions, and ab initio calculations to obtain the dipole moment functions. The researchers have applied these calculations to relatively large molecules with different types of X-H oscillator. Recently they have compared intensities from their simple model to intensities from sophisticated variational calculations for the small molecules H20 and H2CO. For example, for H2CO they generated a dipole moment function in terms of all six vibrational degrees of freedom.244 This comparison has allowed them to determine the quality of basis set needed to calculate dipole moment... 

As described in connection with the University of Manitoba, Bryan Henry moved to Guelph in 1987 and has continued with his experimental studies of overtone spectra involving X—H stretching and the analysis of the results in terms of the local mode model.284 Indeed, most of his research on the intensities of overtone spectra was completed at Guelph. 

In the present paper, we show that it is possible to calculate both vibrational and electronic transitions of H2SO4 with an accuracy that is useful in atmospheric simulations. We calculate the absorption cross sections from the infrared to the vacuum UV region. In Section 2 we describe the vibrational local mode model used to calculate OH-stretching and SOH-bending vibrational transitions as well as their combinations and overtones [42-44]. This model provides frequencies and intensities of the dominant vibrational transitions from the infrared to the visible region. In Section 3 we present vertical excitation energies and oscillator strengths of the electronic transitions calculated with coupled cluster response theory. These coupled cluster calculations provide us with an accurate estimate of the lowest... 

In the present paper we calculate frequencies and intensities of OH-stretching and SOH-bending vibrational transitions as well as their combinations and overtones— the dominant vibrational transitions from 1000 cm to 20000 cm . The vibrational calculation is based on the harmonically coupled anharmonic oscillator (HCAO) local mode model [42-44] combined with ab initio calculated dipole moment functions [50]. This local mode method has been successful in the calculation of OH- and CH-stretching overtone spectra [19,51,52], The local mode parameters, frequency and anharmonicity, are obtained either from the observed experimental transitions or calculated ab initio [53-56]. 

Our local mode model includes the OH-stretching and SOH-bending modes in the vibrational Hamiltonian [58]. The zeroth order Hamiltonian includes the two vibrational modes as uncoupled Morse oscillators according to... 

VIBRATIONAL SPECTROSCOPY USING TUNABLE LASERS, Robin S. McDowell INFRARED AND RAMAN VIBRATIONAL OPTICAL ACTIVITY, L. A. Nafie RAMAN MICROPROBE SPECTROSCOPIC ANALYSIS, John J. Blaha THE LOCAL MODE MODEL, Bryan R. Henry... 

For the CH3F, the possibility of a model atom approximation will be also discussed. The C-F group will be considered as a model atom. Stretching vibrational frequencies calculated in this work of up to 4 quanta of excitation will be compared with the predictions of the local-mode model for the CH3 group only [26] and with the resonant perturbed spectrum of the molecule observed by Law [26]. 

Figure 2. Predictions such as stick spectra of the six-parameter local and normal mode model (solid lines) and of the three-parameter local-mode model (Ref. [26]) for CH3 group (broken lines) compared with the spectra of vapor CH3F observed by Law [26] in the first v = 2 (a), second v = 3 (b) and third v = 4 (c) C-H stretching overtone regions. Numbers in the first rows are N given in Table II, in nearest row - wavenumbers/cm 1.

In Section 9.4.12.4 the simplest possible local mode HlqCAL, expressed in terms of four independently adjustable parameters (the Morse De and a parameters, and two 1 1 kinetic and potential energy coupling parameters, Grr and km,), is transformed to the simplest possible normal mode H )oRMAL, which is also expressed in terms of four independent parameters. However, the interrelationships between parameters, based on the 1 1 coupled local Morse oscillator model, result in only 3 independent fit parameters. This paradox is resolved when one realizes that the 4 parameter local-Morse model generates the Darling-Dennison 2 2 coupling term in the normal mode model. However, the full effects of this (A ssaa/16hc)[(at + as)2(a+ + aa)2] coupling term are not taken into account in the local mode model. 

The transformation between the standard spectroscopic normal mode model, expressed in terms of 6 independently adjustable fit parameters, to an equivalent 6 parameter local mode model, is considered here. The standard H oRMal model is... 

Comprehensive data and assignments of nonfundamental modes were reported [7, 37]. Overtones of the v(CH) modes have been measured on liquid and gaseous Ge(CH3)4 up to 6 v (15818 cm ) and have been assigned on the basis of the local-mode model. This gave localmode CH stretching frequencies of 2990.2 + 0.9 cm for the vapor and 2988 + 4 cm for the liquid. The data are compared for M(CH3)4 compounds (M = C, Si, Ge, Sn) and bandwidths are related to theories of vibrational redistribution from highly excited local-mode states [106]. 

Fang, H.L. and Swofford, R.L., Highly excited vibrational states of molecules by thermal lensing spectroscopy and the local mode model. II. Normal, branched, and cycIo-aUcanes, 73(6), 2607-2617, 1980. 

The vinyl group in polybutadienes has been studied more extensively than simple alkenes, and it has a similar spectrum with peaks at about 4484, 4597, and 4660 cm" Bands at 4717 and 4481 cm in polybutadiene have been assigned as second overtones of the symmetric and asymmetric bending vibrations of the CH2 of the uncoupled vinyl group, as interpreted by a local mode model. There are no corresponding fundamental vibrations for these peaks at 1469 and 1572 cm" though ... 

The local mode description with only one bond such as X-H being in motion at a time leads us to consider the polyatomic molecule as a single diatomic molecule M-H where H is hydrogen and M denotes the full rest of the molecule. The main advantage of this description is the dramatic reduction of the number of coupling constants. The anharmonic oscillator potential adopted in the local mode model is the Morse potential, which takes account of the dissociation energy De (53). 

B. R. Henry. The local mode model and overtone spectra a probe of molecular structure and conformation. Acc Chem Res 20 429 35, 1987. 

Finally, a few comments shall be made on the concept of local modes as compared to normal modes [3,33-35], The main idea of the local mode model is to treat a molecule as if it were made up of a set of equivalent diatomic oscillators, and the reason for the local mode behavior at high energy (>8000 cm ) may be understood qualitatively as follows. As the stretching vibrations are excited to high energy levels, the anharmonicity term / vq (Equation (2.9)) tends, in certain cases, to overrule the effect of interbond coupling and the vibrations become uncoupled vibrations and occur as local modes. ... 

De Araujo and Y. Kawano used curve fitting calculations and local mode theory to assign overtone and combination frequencies in the NIR spectra of polyamide 6 (PA 6), poly(vinyl chloride) (PVC), and polychlorotrifluoroethylene (PCTFE). Anharmonicity correction and mechanical frequency were determined from a Birge-Sponer plot. Anharmonicity corrections of 55, 61, and 20 cm were obtained for CH2 NH, and CO stretch modes of PA 6, respectively, and of 60 and 66 cm for CH2 and CH stretch modes of PVC, respectively. The local mode model seemed to be adequate to interpret the origin of the bands of PA 6 and PVC. Anharmonicity corrections of 33, 19, and 16 cm were... 

Shaji S., Eappen S. M Rasheed T. M. A, and Nair K. P. R., NIR vibrational overtone spectra of N-methylaniline, lV,lV-dimethylaniline and N,N-diethylaniline A conformational structural analysis using local mode model, Spectrochim. Acta A Mol Blomol. Spectrosc., 60, 351-355 (2004). 

Figure 2 Relative energies in the local mode model of ground, fundamental, first harmonic and combination states for a system with two degenerate vibrators. A fundamental harmonic wave-number of 3100 cm- and a vibrational anharmonicity of ft>x=60cm- have been assumed. ZPE represents the zero point energy.

Classical trajectory calculations have given some support to the local mode model. Trajectories for the H-C-C H + C=C system are quasiperiodic in an internal coordinate representation. A classical trajectory study of the intramolecular dynamics of highly excited CH bonds in benzene shows that energy remains localized in CH bonds on a picosecond time scale.Deuteration destroys this result, presumably as a result of increased momenta coupling (larger g terms) for CD bonds as compared to CH bonds. Also, the decomposition of model H-Si-Si is found to be more intrinsically non-RRKM than is H-C-C decomposition. This result is consistent with decreased local mode momenta coupling for H-Si-Si. 

R. L. Swofford, M. E, Long, and A. C. Albrecht, C-H vibrational states of benzene, naphthalene, and anthracene in the visible region by thermal leasing spectroscopy and the local mode model, J. Chem, Phys. 65 179 (1976). 

The reduced DMSs by He et al. [63] were used to demonstrate the quality of the local mode model for the intensities of PH3. Towards this end, a number of band intensities were estimated. The model was able to satisfactory describe the vibrational transition moments of PH3 for most of the stretching bands whose intensities had been experimentally observed [63]. The best agreement was found for the total intensity of the (strongest) V1/V3 system ((100 Ai)/(100 E) in the local mode notation). 

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