Vibrational lineshape

The approach to the evaluation of vibrational spectra described above is based on classical simulations for which quantum corrections are possible. The incorporation of quantum effects directly in simulations of large molecular systems is one of the most challenging areas in theoretical chemistry today. The development of quantum simulation methods is particularly important in the area of molecular spectroscopy for which quantum effects can be important and where the goal is to use simulations to help understand the structural and dynamical origins of changes in spectral lineshapes with environmental variables such as the temperature. The direct evaluation of quantum time- correlation functions for anharmonic systems is extremely difficult. Our initial approach to the evaluation of finite temperature anharmonic effects on vibrational lineshapes is derived from the fact that the moments of the vibrational lineshape spectrum can be expressed as functions of expectation values of positional and momentum operators. These expectation values can be evaluated using extremely efficient quantum Monte-Carlo techniques. The main points are summarized below. 

Given equilibrium quantum expectation values, we can calculate moments of the infra-red vibrational lineshape using a procedure originally outlined by Gordon.The infrared vibrational lineshape is given as the Fourier transform of the dipole moment correlation function ... 

VER rates cannot be predicted from vibrational lineshapes alone [5] except for a few exceptional cases [2, 91]. The most detailed deconstruction of vibrational lineshapes to date is the work of Payer and co-workers [92], who studied W(CO)g in glass-forming liquids including 2-methyl pentane (2MP = 80 K). The triply degenerate... 

As another example, the widths of the vibrational peaks can originate from a number of effects, including a dispersion of the optical phonons in bulk metals or on metal particles, anharmonicity, or the presence of impurities. It has recently been demonstrated that neutron inelastic scattering using isotope dilution methods can be a sensitive probe of the origin of vibrational lineshapes. 

J.W. Gadzuk, A.C. Luntz, On vibrational lineshapes of adsorbed molecules. Surf Sci. 1984,144,429. 

This lineshape analysis also implies tliat electron-transfer rates should be vibrational-state dependent, which has been observed experimentally [44]- Spin-orbit relaxation has also been identified as an important factor in controlling tire identity of botli electron and vibrational-state distributions in radiationless ET reactions. 

Condensed phase vibrational or vibronic lineshapes (vibronic transitions create vibrational excitations of electronic excited states) rarely provide infonnation about VER (see example C3.5.6.4). Experimental measurements of VER need much more than just the vibrational spectmm. The earliest VER measurements in condensed phases were ultrasonic attenuation studies of liquids [15], which provided an overall relaxation time for slowly (>10 ns) relaxing small molecule liquids. 

The vibrational echo experiments yielded exponential decays at all temperatures. The Fourier-transfonn of the echo decay gives the homogeneous lineshape, in this case Lorentzian. The echo decay time constant is AT, where is... 

Therefore, the absorjDtion line is massively inlromogeneously broadened at low temperature. An inliomogeneous lineshape can be used to detennine the static or quasistatic frequency spread of oscillators due to a distribution of environments, but it provides no dynamical infonnation whatsoever [94, 95]. As T is increased to 300 K, the absorjDtion linewidth decreases and increases. At 300 K, the lineshape is nearly homogeneously broadened and dominated by vibrational dephasing, because fast dephasing wipes out effects of inliomogeneous environments, a well known phenomenon tenned motional narrowing [951. 

The Time Dependent Processes Seetion uses time-dependent perturbation theory, eombined with the elassieal eleetrie and magnetie fields that arise due to the interaetion of photons with the nuelei and eleetrons of a moleeule, to derive expressions for the rates of transitions among atomie or moleeular eleetronie, vibrational, and rotational states indueed by photon absorption or emission. Sourees of line broadening and time eorrelation funetion treatments of absorption lineshapes are briefly introdueed. Finally, transitions indueed by eollisions rather than by eleetromagnetie fields are briefly treated to provide an introduetion to the subjeet of theoretieal ehemieal dynamies. 

For purely rotational transitions, the initial and final eleetronie and vibrational states are the same. Moreover, the eleetronie and vibrational states are not summed over in the analog of the above development beeause one is interested in obtaining an expression for a partieular Xiv /ie ==> Xfv Vfe eleetronie-vibrational transition s lineshape. As a result, the... 

Fig. 23. Experimental room temperature Raman spectrum from a sample consisting primarily of bundles or ropes of single-wall nanotubes with diameters near that of the (10,10) nanotube. The excitation laser wavelength is 514.5 nm. The inset shows the lineshape analysis of the vibrational modes near 1580 cm . SWNT refers to singlewall carbon nanotubes [195].

Here, L(v) is a lineshape function that integrates to unity, v is the frequency,/ is the Lamb-Mossbauer factor, and the desired side bands have an area fraction / that is proportional to which hence determines the relative peak heights in a NIS spectrum. More details are provided in Appendix 2 (Part III, 3 of CD-ROM). An equivalent and often more suggestive display of the NIS spectrum is the PVDOS approach, which describes the NIS signal in terms of the partial vibrational density of states ... 

Reynolds S, Oxley DP (1996) Measurement and modeling of vibrational-mode lineshape and linewidth in inelastic electron tunneling spectroscopy. Surf Sci 368 324-329... 

Spectral lineshapes were first expressed in terms of autocorrelation functions by Foley39 and Anderson.40 Van Kranendonk gave an extensive review of this and attempted to compute the dipolar correlation function for vibration-rotation spectra in the semi-classical approximation.2 The general formalism in its present form is due to Kubo.11 Van Hove related the cross section for thermal neutron scattering to a density autocorrelation function.18 Singwi et al.41 have applied this kind of formalism to the shape of Mossbauer lines, and recently Gordon15 has rederived the formula for the infrared bandshapes and has constructed a physical model for rotational diffusion. There also exists an extensive literature in magnetic resonance where time-correlation functions have been used for more than two decades.8... 

Vibrational dephasing provides us with a powerful method to probe the interaction of a chemical bond with the surrounding medium. Over the years, many experimental techniques have been developed to study dephasing of bonds in many molecular systems at various temperatures, pressures, and concentrations [122-124]. One popular experimental technique is the isotropic Raman lineshape. The other methods involve a coherent excitation of the vibration with a laser pulse and monitoring the decay of the phase coherence,... 

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