Dipole function

Figure 1.5 Typical behavior of the dipole function M(r) as a function of r (in atomic units). The results shown are realistic for HF (adapted from Zemke et al., 1991 see also Arunan et al., 1992).

The algebraic transition operator of the preceding section corresponds to a dipole function, which in configuration space is a constant... 

For a more precise calculation of intensities of infrared bands it is necessary to take into account the variation of the dipole function with intemuclear distance,... 

Figure 8.1. The electric dipole function versus the intemuclear distance for LiH using the two ionic and one covalent functions, individually. The sign of the moment for Li —H+ has been changed to facilitate plotting and comparison of the magnitndes.

Figure 8.3. The dipole function for the full valence wave function of LiH. It is a little difficult to see on the present scale, but the moment is -0.033 an at an intemuclear distance of 0.2 bohr.

This damping function s time scale parameter x is assumed to characterize the average time between collisions and thus should be inversely proportional to the collision frequency. Its magnitude is also related to the effectiveness with which collisions cause the dipole function to deviate from its unhindered rotational motion (i.e., related to the collision strength). In effect, the exponential damping causes the time correlation function <% I Eq ... 

In Chapter 4 we will see that two types of induced dipole functions are of a special importance the overlap-induced dipole, Eq. 4.2, an exponential with a range Rq O.lcr, and the multipole-induced dipole, Eq. 4.3, which falls off as R N (N = 4, 5,...). For these, the optical range becomes... 

This expression is a useful starting point for a computation of spectral moments and profiles. Equation 2.68 allows the computation of the dipole emission profile if / (t) falls off to zero sufficiently fast for t —> +oo. Equation 2.69, on the other hand, has less stringent conditions on the dipole function itself and is more broadly applicable (dense fluids) when combined with Eq. 2.66. 

It is of interest to compare the half-widths at half-intensity of the spectral functions of the three systems shown in Fig. 3.2. These amount to roughly 140, 80 and 50 cm-1 for He-Ar, Ne-Ar and Ar-Kr, respectively, which are enormous widths if compared to the widths of common Doppler profiles, etc. The observed widths reflect the short lifetimes of collisional complexes. From the theory of Fourier transforms we know that the product of lifetime, At, and bandwidth, A/, is of the order of unity, Eq. 1.5. The duration of the fly-by interaction is given roughly by the range of the induced dipole function, Eq. 4.30 (1/a = 0.73 a.u. for He-Ar), divided by the mean relative speed, Eq. 2.12. We obtain readily ... 

Spectroscopic measurement. Specifically, if the induced dipole moment and interaction potential are known as functions of the intermolecular separation, molecular orientations, vibrational excitations, etc., an absorption spectrum can in principle be computed potential and dipole surface determine the spectra. With some caution, one may also turn this argument around and argue that the knowledge of the spectra and the interaction potential defines an induced dipole function. While direct inversion procedures for the purpose may be possible, none are presently known and the empirical induced dipole models usually assume an analytical function like Eqs. 4.1 and 4.3, or combinations of Eqs. 4.1 through 4.3, with parameters po, J o, <32, etc., to be chosen such that certain measured spectral moments or profiles are reproduced computationally. 

For hydrogen and its isotopes (H2, HD, D2,...) in interaction with rare gas atoms or other hydrogen molecules, very accurate ab initio calculations exist that were shown to be in agreement with the known measurements of binary spectra the calculated dipole functions are as good as, or possibly better than, the best empirical models and will be discussed in the next Section. 

Turning our attention to the induced dipole function, one wonders if the principle of corresponding states might have any validity for the latter. The question arises whether, or to what extent, the induced dipole function may be written as a universal function with a small number of adjustable scaling parameters. If such a function existed, it would clearly be of interest to workers in the field of collision-induced absorption. 

In recent years, a dependable dipole function for He-Ar, last column of Table 4.3, has been obtained [278] which we compare with the universal dipole function mentioned [23], Fig. 4.5. The He-Ar interaction potential is one of the better known functions [13] and suggests Rmj = 6.518 bohr. Both functions were normalized to unity at the separation R = 5 bohr in the figure. The comparison shows that at small separations the logarithmic slope of the most dependable dipole function is roughly one half that of the universal p, and p diverges rapidly from p(R) for R — o. Similar discrepancies have been noted for other rare gas systems (Ne-Ar, Ne-Kr, and Ar-Kr [152]). Even if for these other systems the dipole function is not as well known as it is for He-Ar, it seems safe to say that for the rare gas mixtures mentioned the induced dipole function is definitely not identical with the universal function at the distances characteristic of the spectroscopic interactions the universal dipole function is not consistent with some well established facts and data. We note that the ratio of // (/ ) and the He-Ar potential is indeed reasonably constant over the range of separations considered (not shown in the figure). 

Fig. 4.5. Comparison of the dipole function (p(R)), universal dipole (p (R/Rm,n) with Rmin = 6.518 a.u., and force (-V (R)) for He-Ar. The three functions have been normalized to unity at R = 5 bohr o = 5.821 bohr is the root of the potential.

It is clear that intermolecular force and induced dipole function arise from the same physical mechanisms, electron exchange and dispersion. Since at the time neither intermolecular potentials nor the overlap-induced dipole moments were known very dependably, direct tests of the assumptions of a proportionality of force and dipole moment were not possible. However, since the assumption was both plausible and successful, it was widely accepted, even after it was made clear that for an explanation of... 

In recent molecular dynamics studies attempts were made to reproduce the shapes of the intercollisional dip from reliable pair dipole models and pair potentials [301], The shape and relative amplitude of the intercollisional dip are known to depend sensitively on the details of the intermolecular interactions, and especially on the dipole function. For a number of very dense ( 1000 amagat) rare gas mixtures spectral profiles were obtained by molecular dynamics simulation that differed significantly from the observed dips. In particular, the computed amplitudes were never of sufficient magnitude. This fact is considered compelling evidence for the presence of irreducible many-body effects, presumably mainly of the induced dipole function. 

Collision-induced absorption takes place by /c-body complexes of atoms, with k = 2,3,... Each of the resulting spectral components may perhaps be expected to show a characteristic variation ( Qk) with gas density q. It is, therefore, of interest to consider virial expansions of spectral moments of binary mixtures of monatomic gases, i.e., an expansion of the observed absorption in terms of powers of gas density [314], Van Kranendonk and associates [401, 403, 314] have argued that the virial expansion of the spectral moments is possible, because the induced dipole moments are short-ranged functions of the intermolecular separations, R, which decrease faster than R 3. We label the two components of a monatomic mixture a and b, and the atoms of species a and b are labeled 1, 2, N and 1, 2, N, respectively. A set of fc-body, irreducible dipole functions U 2, Us,..., Uk, is introduced (as in Eqs. 4.46), according to... 

This expression may be viewed as a sum of two terms. The first one, the integral over the squared derivative of the dipole function, p R), models those contributions which arise from the variation of the dipole strength with the separation R. The second models the contributions due to the variation of direction, p, as the rather natural separation of the kinetic energy operator into a radial and angular part suggests [314]. 

The computation of the pair distribution function, g,0 (R), requires the knowledge of the interaction potential. The expressions for the zeroth and first moments require, furthermore, knowledge of the dipole function, p(R). For any given system (i.e., for a given reduced mass, potential and dipole function), the moments yn are functions of temperature, T. 

Early numerical estimates of ternary moments [402] were based on the empirical exp-4 induced dipole model typical of collision-induced absorption in the fundamental band, which we will consider in Chapter 6, and hard-sphere interaction potentials. While the main conclusions are at least qualitatively supported by more detailed calculations, significant quantitative differences are observed that are related to three improvements that have been possible in recent work [296] improved interaction potentials the quantum corrections of the distribution functions and new, accurate induced dipole functions. The force effect is by no means always positive, nor is it always stronger than the cancellation effect. 

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). 

For any given potential and dipole function, at a fixed temperature, the classical and quantum profiles (and their spectral moments) are uniquely defined. If a desymmetrization procedure applied to the classical profile is to be meaningful, it must result in a close approximation of the quantum profile over the required frequency band, or the procedure is a dangerous one to use. On the other hand, if a procedure can be identified which will approximate the quantum profile closely, one may be able to use classical line shapes (which are inexpensive to compute), even in the far wings of induced spectral lines a computation of quantum line shapes may then be unnecessary. 

The spectral profile, J(term depends on the product of real and imaginary parts of A, 91 A and 3 X, Eq. 5.92, which will be positive at low frequencies if both, the dipole function and the interatomic potential are short range [236]. In other words, the intercollisional absorption profile is negative under such conditions an absorption dip is obtained, in agreement with the observations. 

In other words, all required quantities may be computed from classical mechanics if the interaction potential and dipole function are known. Greater detail may be found elsewhere, see, for example, Hirschfelder, Curtiss and Bird (1964), or Chapman and Cowling (1960). 

The superscripts I- TV in this expression designate the first through fourth derivatives of potential and dipole function with respect to R. 

For the case of induced absorption of H2-He pairs, the vibrational matrix elements of potential and dipole function are well known [151, 294]. The spectral moments Mq- M2 have been computed for the main induction components with the corrections for the vibrational dependences... 

It is, therefore, interesting to point out that in a recent molecular dynamics study, shapes of intercollisional dips of collision-induced absorption were obtained. These line shapes are considered a particularly sensitive probe of intermolecular interactions [301]. Using recent pair potentials and empirical pair dipole functions, for certain rare-gas mixtures spectral profiles were obtained that differ significantly from what is observed... 

The close coupled scheme is described on pp. 306 through 308. Specifically, the intermolecular potential of H2-H2 is given by an expression like Eq. 6.39 [354, 358] the potential matrix elements are computed according to Eq. 6.45ff. The dipole function is given by Eq. 4.18. Vibration, i.e., the dependences on the H2 vibrational quantum numbers vu will be suppressed here so that the formalism describes the rototranslational band only. For like pairs, the angular part of the wavefunction, Eq. 6.42, must be symmetrized, according to Eq. 6.47. 

An explanation was offered by van Kranendonk many years after the experimental discovery. Van Kranendonk argued that anticorrelations exist between the dipoles induced in subsequent collisions [404], Fig. 3.4. If one assumed that the induced dipole function is proportional to the intermolecular force - an assumption that is certainly correct for the directions of the isotropic dipole component and the force, and it was then thought, perhaps even for the dipole strength - an interference is to be expected. The force pulses on individual molecules are correlated in... 

The comparison of Eqs. (1,2,3) shows that the ternary dipole functions B arising from the dipole moment induced in molecules 1,2 are related to the pair dipole functions B, according to... 

The dipole function p, for OH is modeled as a Mecke function (adapted from Ref. 28), for SBV we assume the ubiquitous linear relation n=f e q with effective charge/ e and scaling factor/. Equation (2) can be solved either... 

Figure 6. Series of IR femtosecond/picosecond laser pulses for the sequence of vibrational transitions SBV(u = 0) - SBV(u = 6) - SBV(t> = 1) for laser control of the Cope rearrangement of the model substituted semibullvalene (SBV) shown in Fig. 2 (adapted from Ref. 26). The notations are as in Fig. 3. The electric field is scaled by the scaling factor / of the effective charge associated with the dipole function jt =/ e q.

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