Rotational-vibrational correlation function

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... 

TIME-CORRELATION FUNCTIONS FOR VIBRATION-ROTATION SPECTRA... 

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]... 

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 ... 

The electron-spin time-correlation functions of Eq. (56) were evaluated numerically by constructing an ensemble of trajectories containing the time dependence of the spin operators and spatial functions, in a manner independent of the validity of the Redfield limit for the rotational modulation of the static ZFS. Before inserting thus obtained electron-spin time-correlation functions into an equation closely related to Eq. (38), Abernathy and Sharp also discussed the effect of distortional/vibrational processes on the electron spin relaxation. They suggested that the electron spin relaxation could be described in terms of simple exponential decay rate constant Ts, expressed as a sum of a rotational and a distortional contribution ... 

The Time Dependent Processes Section uses time-dependent perturbation theory, combined with the classical electric and magnetic fields that arise due to the interaction of photons with the nuclei and electrons of a molecule, to derive expressions for the rates of transitions among atomic or molecular electronic, vibrational, and rotational states induced by photon absorption or emission. Sources of line broadening and time correlation function treatments of absorption lineshapes are briefly introduced. Finally, transitions induced by collisions rather than by electromagnetic fields are briefly treated to provide an introduction to the subject of theoretical chemical dynamics. 

E. Rotational, Translational, and Vibrational Contributions to the Correlation Function... 

The shape of the vibration-rotation bands in infrared absorption and Raman scattering experiments on diatomic molecules dissolved in a host fluid have been used to determine2,15 the autocorrelation functions unit vector pointing along the molecular axis and P2(x) is the Legendre polynomial of index 2. These correlation functions measure the rate of rotational reorientation of the molecule in the host fluid. The observed temperature- and density-dependence of these functions yields a great deal of information about reorientation in solids, liquids, and gases. These correlation functions have been successfully evaluated on the basis of molecular models.15... 

These higher-order correlation functions play a large role in determining many physical properties of polyatomic systems. For example, the vibrational relaxation can, in some cases, be expressed in terms of the rotational kinetic energy autocorrelation function.27... 

The main contributions to the frequency-time correlation function are assumed to be, as in the earlier works [123, 124], from the vibration-rotation coupling and the repulsive and attractive parts of the solvent-solute interactions. In several theories, the (faster) repulsive and the (slower) attractive contributions are assumed to be of widely different time scales and are treated separately. However, this may not be true in real liquids because the solvent dynamic interactions cover a wide range of time scales and there could be a considerable overlap of their contributions. The vibration-rotation coupling contribution takes place in a very short time scale and by neglecting the cross-correlation between this mechanism and the atom-atom forces, they... 

Note that the angular momentum-momentum correlation function related to the vibrational-rotation friction too is highly nonexponential. This again could significantly alter the 2 dependence of the rate. However, for the systems studied here, the contribution of vibration-rotation coupling is negligible. 

Electrons switch between levels characterized by Ms values. Let us examine now an ensemble of n molecules, each with an unpaired electron, in a magnetic field at a given temperature. The bulk system is at constant energy but at the molecular level electrons move, molecules rotate, there are concerted atomic motions (vibrations) within the molecules and, in solution, molecular collisions. Is it possible to have information on these dynamics on a system which is at equilibrium The answer is yes, through the correlation function. The correlation function is a product of the value of any time-dependent property at time zero with the value at time t, summed up to a large number n of particles. It is a function of time. In this case the property can be the Ms value of an unpaired electron and the particles are the molecules. The correlation function has its maximum value at t = 0 since each molecule has one unpaired electron, the product of the... 

When interpreting time-resolved mid-IR spectra, it is beneficial to consider the influence of rotational dynamics on the vibrational spectrum of a heteronuclear diatomic. It was shown more than 30 years ago that the vibrational absorption spectrum of a diatomic is related to its transition dipole correlation function (/z(0) /r(t)> through a Fourier transform (10) ... 

The solution of the simultaneous differential Equations 11 and 12 has already been discussed in detail in Reference 16. Only the four lowest rotational-vibrational energies, as a function of y and D, for all of the isotopic hydrogens have been calculated and tabulated. These are the states which correlate, in the case of the free molecule, with the states... 

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