Spectral moment classical

Table 3.7 also lists ternary spectral moments for a few systems other than H2-H2-H2. For the H2-He-He system, the pairwise-additive dipole moments are also known from first principles. The measured spectral moments are substantially greater than the ones calculated with the assumption of pairwise additivity - just as this was seen in pure hydrogen. For the other systems listed in the Table, no ab initio dipole surfaces are known and a comparison with theory must therefore be based on the approximate, classical multipole model. 

A better approach actually exists. There is considerable empirical and theoretical evidence that the various spectral functions, g(v), of binary systems are indeed closely modeled by a combination of the BC and K0 profiles, Eqs. 5.105 and 5.108. These are functionals of reduced spectral moments, Mo, Mi/Mo, and M2/M0, which in the classical limit may be expressed in terms of reduced temperature, to the extent that interaction potentials are describable by reduced potentials. 

We start with the basic relationships ( Ansatz ) of collision-induced spectra (Section 5.1). Next we consider spectral moments and their virial expansions (Section 5.2) two- and three-body moments of low order will be discussed in some detail. An analogous virial expansion of the line shape follows (Section 5.3). Quantum and classical computations of binary line shapes are presented in Sections 5.4 and 5.5, which are followed by a discussion of the symmetry of the spectral profiles (Section 5.6). Many-body effects on line shape are discussed in Sections 5.7 and 5.8, particularly the intercollisional dip. We conclude this Chapter with a brief discussion of model line shapes (Section 5.10). 

The expressions, Eqs. 5.9 through 5.16, are quite general, wave mechanical formulae of the spectral moments, which for small order n are suitable for numerical computations classical approximations may be derived readily from these. 

Classical approximations. While the computation of binary, low-order spectral moments, Eqs. 5.37 and 5.38, poses no special problems, we note... 

This leads to expressions of the zeroth and first spectral moments that may be computed in seconds, even on computers of small capacity. However, few systems of practical interest are actually classical and better approximations are often needed. 

Detailed balance. Classical line shapes are symmetric so that all classical, odd spectral moments M of the spectral function vanish. The odd moments of actual measurements are, however, non-vanishing because measured spectral density profiles satisfy the principle of detailed balance, Eq. 5.73. This problem of classical relationships may be largely alleviated by symmetrizing the measured profile prior to determining the moments, using the inverse Egelstaff procedure (P-4) discussed on p. 254 this generates a close approximation to the classical profile from the measurement and use of classical formulae is then justified. 

Classical moment expressions. Spectral moments expressed in the Heisenberg notation can be immediately interpreted in terms of classical physics. For a discussion of classical moments, we consider the moments Af of the spectral density, J co), which are related to the moments, y , of the absorption coefficient, a(co), according to Eq. 5.8. By combining that equation with Eq. 5.16, we get at once... 

Table 5.1. Comparison of binary spectral moments calculated from classical (C.), semi-classical (S.) and quantum (Q.) calculations, based on line shapes (.LS) and sum formulae (.SF), for He-Ar at 295 K. Moments computed from the classical line shape after desymmetrization procedures P-2 and P-4 (scaled) had been applied are also shown. Computations are based on the ab initio dipole, Table 4.3, and an advanced potential [12].

Induced dipole autocorrelation functions of three-body systems have not yet been computed from first principles. Such work involves the solution of Schrodinger s equation of three interacting atoms. However, classical and semi-classical methods, especially molecular dynamics calculations, exist which offer some insight into three-body dynamics and interactions. Very useful expressions exist for the three-body spectral moments, with the lowest-order Wigner-Kirkwood quantum corrections which were discussed above. 

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). 

Welsh and his associates have pointed out early on that the observed spectral profiles are strikingly asymmetric [422]. Of course, line shapes computed on the basis of a quantum formalism will always have the proper asymmetry so that measurement and theory may be directly compared. Problems may arise, however, if classical profiles are employed for analysis of a measurement, or if classical expressions for computation of spectral moments are used for a comparison with the measurement. 

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. 

Starting from the assumption that the only known information concerning the line shape is a finite number of spectral moments, a quantity called information is computed from the probability for finding a given spectral component at a given frequency this quantity is then minimized. Alternatively, one may maximize the number of configurations of the various spectral components. This process yields an expression for the spectral density as function of frequency which contains a small number of parameters which are then related to the known spectral moments. If classical (i.e., Boltzmann) statistics are employed, information theory predicts a line shape of the form... 

Classical and semiclassical moment expressions. The expressions for the spectral moments can be made classical by substituting the classical distribution function, g(R) = exp (— V(R)/kT), for the quantum expressions. Wigner-Kirkwood corrections are known which account to lowest order for the static quantum corrections, Eq. 5.44 [177, 292]. For the second and higher moments, dynamic quantum corrections must also be made [177]. As was mentioned in the previous Chapter, such semiclassical corrections are useful in supplementing quantum computations of the spectral moments at large separations where the quantum effects are small the computational effort of quantum calculations, which is substantial at large separations, may thus be avoided. 

Higher-order classical moments have also been reported. We mention the classical expressions for the translational spectral moments M , with n = 0, 2, 4, and 6, for pairs of linear molecules given in an appendix of [204]. Spectral moments of spherical top molecules have been similarly considered [163, 205], We note that for n > 1, spectral moments show dynamic as well as static quantum correction, which become more important as the order n of the spectral moments is increased. The discussions on pp. 219, and Table 5.1, suggest that, even for the near-classical systems, quantum corrections may be substantial and can rarely be ignored. 

Second moments have also been computed, both from first principles and on the basis of the classical multipole-induction model. These are found to be in close agreement with measurements where these exist. Second moments are of a special interest in connection with modeling of three-parameter line profiles from three spectral moments [52]. In analyses based on classical expressions, the second moment is expressible in terms of the first moment specified above, multiplied by 2kT/h. 

Spectral moments can also be computed from classical expressions with Wigner-Kirkwood quantum corrections [177, 189, 317] of the order lV(H2). For the quadrupole-induced 0223 and 2023 components of H2-H2, at the temperature of 40 K, such results differ from the exact zeroth, first and second moments by -10%, -10%, and +30% respectively. For the leading overlap-induced 0221 and 2021 components, we get similarly +14%, +12%, and -56%. These numbers illustrate the significance of a quantum treatment of the hydrogen pair at low temperatures. At room temperature, the semiclassical and quantum moments of low order differ by a few percent at most. Quantum calculations of higher-order moments differ, however, more strongly from their classical counterparts. 

Very little is known about the irreducible ternary dipole components. An early estimate based on classical electrodynamics, hard-sphere interaction and other simplifying assumptions suggests very small, negative contributions to the zeroth spectral moment [402], namely —0.13 x 10-10 cm-1 amagat-3. 

If both the forward (absorption) and backward (emission) optical transitions are available, their first spectral moments determine the reorganization energies of quantum vibrations, Xy, and of the classical nuclear motions of the donor-acceptor skeleton and the solvent, Xj-i ... 

The two main nuclear modes affecting electronic energies of the donor and acceptor are intramolecular vibrations of the molecular skeleton of the donor-acceptor complex and molecular motions of the solvent. If these two nuclear modes are uncoupled, one can arrive at a set of simple relations between the two spectral moments of absorption and/or emission transitions and the activation parameters of ET. The most transparent representation is achieved when the quantum intramolecular vibrations are represented by a single, effective vibrational mode with the frequency vv (Einstein model).15-17 If both the forward (absorption) and backward (emission) optical transitions are available, their first spectral moments determine the reorganization energies of quantum vibrations, Xv, and of the classical nuclear motions of the donor-acceptor skeleton and the solvent, Xci ... 

We have already observed that the classical spectral moments may be divergent quantities, and then the classical Taylor series (23) becomes meaningless. On the other hand, also if the spectral moments do not diverge, truncation of the series (23) always produces the undesired phenomenon, that is, limT ooRy(T) = oo. 

In this entry only two problems have been presented representation of the power spectral density (PSD) and correlation by means of complex fractional moments and filter equations. Regarding the first problem, it has been shown that both PSD and correlation may be represented by the so-called fractional spectral moments (FSMs). The latter are the extension of the classical spectral moments introduced by Vanmarcke (1972) to complex order. The appealing in using these FSM is that they are able to reconstruct both PSD and correlation. As a result, it may be stated that FSM functiOTi is another equivalent representation of PSD and correlation. Moreover it has been shown that with a limited number of informations (few FSM), the whole PSD and correlation may be restored, including the trend at infinity. Extension to multivariate seismic process has been also provided in both frequency... 

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. 

The zeroth moment (n=0) gives the total intensity and is related to theory by familiar sum formulae (Chapter 5). For nearly classical systems (i.e., massive pairs at high temperature and not too high frequencies), the first moment (n=l) is very small and actually drops to zero in the classical limit as we will see in Chapter 5. The ratio of second and zeroth moment defines some average frequency squared and may be considered a mean spectral width squared. A complete set of moments (n = 0, 1,... 00) may be considered equivalent to the knowledge of the spectral line shape,... 

We note that in a classical formula Planck s constant does not appear. Indeed, the zeroth moment Mo of the spectral density, J (o), does not depend on h, as the combination of Eqs. 5.35 and 5.38 shows. On the other hand, the classical moment y of the absorption profile, a(cu), is proportional to /h because the absorption coefficient a depends on Planck s constant see the discussions of the classical line shape below, p. 246. In a discussion of classical moments it is best to focus on the moments Mn of the spectral density, J co), instead of the moments, yn, of the spectral profile. 

For classical line shape calculations one needs the induced dipole moment as function of time, p(R(t)), averaged over angular momenta and speeds of relative motion. In other words, one solves Newton s equation of motion, or one of its integrals, of the two-particle system. After suitable averaging, one obtains the spectral profile by Fourier transform. 

Thus we see that the first moment of the spectral density multiplied by h is the reorganization energy (i.e., one half of the Stokes shift magnitude), whereas the time dependence of the first moment of p(w) corresponds to the fluorescence Stokes shift. Thus the time dependence of S t) is determined entirely by the spectral density. At high temperature [i.e., when p(w) contains frequencies less than 2kBT], S(t) becomes the classical correlation function [36] used by many previous authors [7-10], This follows from... 

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