Correlation functions vibrational

Note that in a dense liquid the decay of the vibrational correlation function need not be single exponential as assumed in Eq. (283). It can be biphasic with a Gaussian decay at short times followed by an exponential decay at longer times [123, 124]. Now if the decay of (<2(0 2(0)) is fully Gaussian as in the following form,... 

For dephasing (T2) processes one is concerned with the decay of the vibrational correlation function . In the two-level approximation one may write... 

Several complementary techniques exist for the experimental study of dephasing, and here we will underline only the differences between them. The simplest experimental access to dephasing is by spontaneous Raman scattering. A laser of frequency scattered light at to, around cuj = frequency resolved. The isotropic Raman scattering cross section is directly related to the vibrational correlation function... 

For quantitative comparison of theory and experiment several simple models have been developed which give analytical expressions for the pure dephasing time T2. These models are usually tested against the experimentally observed temperature, density, or concentration dependence of T2. Since these models have been intensively reviewed, we merely recall the results. Hydro-dynamic models have been proposed by Lynden-Bell - and Oxtoby. In the former calculation the vibrational correlation function is replaced by its self-part [see Eq. (96)], yielding... 

Up to this point we have mainly considered homogeneous broadening where the vibrational correlation function C(t) may be reduced to single exponential decay. This limit is not always verified, and it is interesting to examine the experimental consequences to be expected in these cases. 

Figure 6. Test of potentiai separation of Schweizer and Chandier at moderateiy low density (pa = 0.3). Time scales for self part of vibrational correlation function for short- and long-range forces are well separated, but cross term 2 is not negligible. As density increases, cross term progressively cancels out, but not any faster than the force contribution , which also disappears eventually. (From Chesnoy and Weis. ° )...

Fig. 4. Time dipole correlation functions C(t) of water in critical state (left top), in bulk liquid water at 30°C (left center), in a monolayer on fluorophlogopite mica (left bottom), in LTA bonded to the first 4 Na+ ions (right top), in SB A-15 heated to 300°C for 2 hrs (right center), and in fully hydrated SBA-15 (right bottom). The normalized total correlation functions, obtained according to Eq. (9) involve vibrations of the transition dipole of the (v+5) band displayed as rapid oscillations. Rotational correlations including angular perturbations appear as envelopes of the vibrational correlation functions. The inertial rotational motion about the least rotational axis of the water molecule is indicated as a quadratic decay C(t) - (kT/I) t2 at times 0 - 0.05 psec in each C(t) vs. t graph. The graphs on the left are reproduced from ref. 18.

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The vibrational correlation function finally takes the form... 

The object of this chapter is to present an overview of the determination of the reorientational and vibrational correlation functions from infrared and Raman spectroscopic band profiles. I shall concentrate on the objectives of such measurements, on the experimental and computational difficulties and on the possible shortcomings of the rather well-established [1-4] methodology. 

A22. Contributions of Self and Distinct Pair Vibrational Correlation Function to Isotropic and Anisotropic Raman Spectra of the V2(A ) Bands of Pure CD3I and CH3I. 

Since the stochastic Langevin force mimics collisions among solvent molecules and the biomolecule (the solute), the characteristic vibrational frequencies of a molecule in vacuum are dampened. In particular, the low-frequency vibrational modes are overdamped, and various correlation functions are smoothed (see Case [35] for a review and further references). The magnitude of such disturbances with respect to Newtonian behavior depends on 7, as can be seen from Fig. 8 showing computed spectral densities of the protein BPTI for three 7 values. Overall, this effect can certainly alter the dynamics of a system, and it remains to study these consequences in connection with biomolecular dynamics. 

In effect, i is replaced by the vibrationally averaged electronic dipole moment iave,iv for each initial vibrational state that can be involved, and the time correlation function thus becomes ... 

Here, I(co) is the Fourier transform of the above C(t) and AEq f is the adiabatic electronic energy difference (i.e., the energy difference between the v = 0 level in the final electronic state and the v = 0 level in the initial electronic state) for the electronic transition of interest. The above C(t) clearly contains Franck-Condon factors as well as time dependence exp(icOfvjvt + iAEi ft/h) that produces 5-function spikes at each electronic-vibrational transition frequency and rotational time dependence contained in the time correlation function quantity <5ir Eg ii,f(Re) Eg ii,f(Re,t)... 

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole P-avejv (t), the vibrational transition dipole itrans (t) or the electronic transition dipole ii f(Re,t)) and the latter two also contain oscillatory time dependences (i.e., exp(icofv,ivt) or exp(icOfvjvt + iAEi ft/h)) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion. 

If the rotational motion of the molecules is assumed to be entirely unhindered (e.g., by any environment or by collisions with other molecules), it is appropriate to express the time dependence of each of the dipole time correlation functions listed above in terms of a "free rotation" model. For example, when dealing with diatomic molecules, the electronic-vibrational-rotational C(t) appropriate to a specific electronic-vibrational transition becomes ... 

Using the impact approximation presented in Chapter 6, they may easily be found for any rotational band even if rotational-vibrational interaction is nonlinear in J. In 1954 R W. Anderson proved as a theorem [104] that expansion of the spectral wings in inverse powers of frequency is controlled by successive odd derivatives of the correlation function at the origin. In impact approximation the lowest non-zero derivative of this type is the third and therefore asymptotics G/(co) is described by the power expansion [20]... 

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 ... 

Thus the nth vibrational spectral moment is equal to an equilibrium correlation function, the nth derivative of the dipole moment autocorrelation function evaluated at t=0. By using the repeated application of the Heisenberg equation of motion ... 

Without regard for deformational and rotational vibrations of unit vectors e(ij, the qualitative behavior of the time dependence of the correlation function for two-dimensional reorientations is describable by the following relation ... 

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