Induced moment correlation function

Spectroscopic techniques have been applied most successfully to the study of individual atoms and molecules in the traditional spectroscopies. The same techniques can also be applied to investigate intermolecular interactions. Obviously, if the individual molecules of the gas are infrared inactive, induced spectra may be studied most readily, without interference from allowed spectra. While conventional spectroscopy generally emphasizes the measurement of frequency and energy levels, collision-induced spectroscopy aims mainly for the measurement of intensity and line shape to provide information on intermolecular interactions (multipole moments, range of exchange forces), intermolecular dynamics (time correlation functions), and optical bulk properties. 

Fig. 3.51. Logarithmic plot of the normalized induced dipole moment correlation function, C(t), for hydrogen-argon mixtures at 165 K. Measurements at 90 amagat ( ) 450 amagat ( ) and 650 amagat (o). The broken lines at small times represents the portion of C(t) affected by the smoothing of the wings of the spectral profiles. Reproduced with permission by the National Research Council of Canada from [109].

Theoretical models of correlation functions and line shapes have been proposed which satisfy the principle of detailed balance [35, 36, 41, 232]. These profiles, along with a number of extensions that were later added [69, 295, 47, 48], describe the known profiles well. Especially the BC and K0 functions, Eqs. 5.105 and 5.108, model multipole- and overlap-induced lines of the rototranslational bands closely. These three-parameter functions are simple analytical expressions that are readily computable, even on computers of small capacity (pocket calculators) the parameters can be computed from the lowest three spectral moments, see Chapter 5. 

Model correlation functions. Certain model correlation functions have been found that model the intracollisional process fairly closely. These satisfy a number of physical and mathematical requirements and their Fourier transforms provide a simple analytical model of the spectral profile. The model functions depend on the choice of two or three parameters which may be related to the physics (i.e., the spectral moments) of the system. Sears [363, 362] expanded the classical correlation function as a series in powers of time squared, assuming an exponential overlap-induced dipole moment as in Eq. 4.1. The series was truncated at the second term and the parameters of the dipole model were related to the spectral moments [79]. The spectral model profile was obtained by Fourier transform. Levine and Birnbaum [232] developed a classical line shape, assuming straight trajectories and a Gaussian dipole function. The model was successful in reproducing measured He-Ar [232] and other [189, 245] spectra. Moreover, the quantum effect associated with the straight path approximation could also be estimated. We will be interested in such three-parameter model correlation functions below whose Fourier transforms fit measured spectra and the computed quantum profiles closely see Section 5.10. Intracollisional model correlation functions were discussed by Birnbaum et a/., (1982). 

Constant acceleration approximation. An approximation introduced to the time-dependent intermolecular correlation function G, which was commonly referred to as the constant acceleration approximation (CAA), was used to compute the line shapes of collision-induced absorption spectra of rare gas mixtures, but the computed profiles were found to be unsatisfactory [286], It does not give the correct first spectral moment. 

Raman scattering depends on the time correlation function of the many-body polarizability of the liquid, collective dipole moment. In the case of Raman scattering, an external electric field (from a laser) generates an induced collective dipole in the liquid ... 

Spin relaxation in a nucleus is induced by random fluctuations of local magnetic fields. These result from time-dependent modulation of the coupling energy of the resonating nuclear spin with nearby nuclear spins, electron spins, quadrupole moments, etc. Any time-dependent phenomenon able to modulate these couplings can contribute to nuclear relaxation. The distribution of the frequencies contained in these time-dependent phenomena is described by a correlation function, characterized by a parameter Tc, the correlation time. Its reciprocal can be considered as the maximum frequency produced by the fluctuations in the vicinity of the nuclear spin. If more than one process modulates the coupling between the nuclear spin and its surroundings, the reciprocal of the effective correlation time is the sum of the reciprocals of the various contributions... 

The analysis of the dipole moment curves for the motion of the adsorbate perpendicular to the surface provides additional information about the degree of ionicity of a given surface chemical bond. Moreover, the analysis of the dipole moment curves is also related to the interpretation of variations of the surface work function induced by the presence of the adsorbate. However, the response of the surface to the presence of the adsorbate does not permit to extract directly adsorbate charges from the dipole moment curve. A procedure based in the use of frozen densities has been proposed that permits to avoid the effect of the surface polarization on the dipole moment curve. Unfortunately, this method has not been yet extensively used. To close this short discussion about the different procedures commonly used to interpret the chemical bond between chemical species and the surface of a catalyst in terms of net charges we mention the valence bond reading of Hartree-Fock and Configuration Interaction wave functions. This procedure has been used to interpret the electronic correlation effect on the surface chemical bond. ... 

Hyperpolarizabilities can be calculated in a number of different ways. The quantum chemical calculations may be based on a perturbation approach that directly evaluates sum-over-states (SOS) expressions such as Eq. (14), or on differentiation of the energy or induced moments for which (electric field) perturbed wavefunctions and/or electron densities are explicitly calculated. These techniques may be implemented at different levels of approximation ranging from semi-empirical to density functional methods that account for electron correlation through approximations to the exact exchange-correlation functionals to high-level ab initio calculations which systematically include electron correlation effects. 

As in the Rayleigh case, the pair polarizability results as well from nonlinear light scattering mechanisms (induced by hyperpolarizabilities and permanent multipole moments). For tetrahedral molecules nonlinear mechanisms contribute to some correlation functions listed in Table V—only those related to the depolarized spectrum and governed by double rotational transitions (QQ, QO, and 00). The nonlinear origin corrections Atp j7"1 j2 which must be added to the linear origin terms cp 1 °f Table V for a tetrahedral molecule are successively [17]... 

London forces Intermolecular forces resulting from the attraction of correlated temporary dipole moments induced in adjacent molecules, wave function (i/r) The mathematical description of an orbital. The square of the wave function is proportional to the electron density, (p. 39)... 

---