Three body coefficients

Measurements such as these can be conducted to determine the three-body virial coefficients, M 12) and M 21) of collision-induced absorption. To that end, it is useful to measure the variation of yi (and also of yo> Eq. 3.6, where possible) with small amounts of gas 1 mixed with large amounts of the other gas 2, and with small amounts of 2 mixed with 1, to determine the ternary spectral moments M 12 and M 21 separately, with a minumum of interference from the weaker terms. In a mixture of helium and argon, for example, two different three-body coefficients can be determined, those of the He-Ar-Ar and the He-He-Ar complexes. 

Table 6.7. Comparison of the computed third virial coefficients of the induced spectra of various gases and mixtures with the existing measurements. The three-body coefficients were computed with the assumption of pairwise-additive dipole components [48].

In Fig. 11 the influence of four-body terms is shown in fit F4q. In this fit, the three-body coefficients of fit F3 are used, and only the four-body coefficients are adjusted by regression. It can be seen that the four-body contribution is negligible except between 0° and 30°, and there it is small. This is the case for all other cuts that were examined, and it is the result of having obtained a good three-body fit. 

All the remaining parameters are semiempirical Rq and are cutoff radii adjusted to each atomic pairwise and three-body. Coefficients s (n=8,10, — ) are fitted in benchmark calculations depending on functionals combined, while Se is one or an adjusted value less than 1. For dispersion coefficients, C and time-dependent DFT (TDDFT) and recursion relations are used to determine the values for each atonuc pairwise and three-body. The lowest-order Ca is expressed in the Casimer-Polder formula [51],... 

The self-diffusion coefficient calculated for the three body potential is Z) = 1.3 X lO cm /sec. This is to be compared with the experimental value of 2.3 x 10" cmVsec and to the value of 2.25 x 10 cm /sec for the two-body liquid. It could be said that the three-body liquid shows more rigidity in some sense than the two-body liquid. 

Ion-molecule radiative association reactions have been studied in the laboratory using an assortment of trapping and beam techniques.30,31,90 Many more radiative association rate coefficients have been deduced from studies of three-body association reactions plus estimates of the collisional and radiative stabilization rates.91 Radiative association rates have been studied theoretically via an assortment of statistical methods.31,90,96 Some theoretical approaches use the RRKM method to determine complex lifetimes others are based on microscopic reversibility between formation and destruction of the complex. The latter methods can be subdivided according to how rigorously they conserve angular momentum without such conservation the method reduces to a thermal approximation—with rigorous conservation, the term phase space is utilized. 

Our discussion of complex formation in electron-ion recombination, field effects, and three-body recombination has perhaps posed more questions than it has answered. In the case of H3 recombination, the experimental observations suggest but do not prove that complex formation is an important mechanism. Three-body recombination involving complex formation is not likely to have much effect on the total recombination coefficients of diatomic ions, but it may alter the yield of minor product channels. Complex formation may be most prevalent in the case of large polyatomic ions, but there is a serious lack of experimental data and theoretical calculations that can be adduced for or against complex formation. 

The hyperspherical method, from a formal viewpoint, is general and thus can be applied to any N-body Coulomb problem. Our analysis of the three body Coulomb problem exploits considerations on the symmetry of the seven-dimensional rotational group. The matrix elements which have to be calculated to set up the secular equation can be very compactly formulated. All intervals can be written in closed form as matrix elements corresponding to coupling, recoupling or transformation coefficients of hyper-angular momenta algebra. 

Let us make clear now the correspondence between our treatment here and Erdahl s 1978 treatment [4, Sec. 8]. Erdahl works in general Fock space and his operators conserve only the parity of the number of nuclei. He exhibits two families of operators that are polynomials in the annihilation and creation operators containing a three-body and a one-body term. Generic instances of these operators are denoted y and w. The coefficients are real, and Erdahl stresses that this is essential for his treatment. The one-body term is otherwise unrestricted, but the three-body term must satisfy conditions to guarantee that y+y or H +w does not contain a six-body term. For the first family the conditions amount to the three-body term being even under taking the adjoint, and for... 

Here, /u ° and ju are, respectively, the chemical potentials of pure solvent and solvent at a certain concentration of biopolymer V is the molar volume of the solvent Mn=2 y/M/ is the number-averaged molar mass of the biopolymer (sum of products of mole fractions, x, and molar masses, M, over all the polymer constituent chains (/) as determined by the polymer polydispersity) (Tanford, 1961) A2, A3 and A4 are the second, third and fourth virial coefficients, respectively (in weight-scale units of cm mol g ), characterizing the two-body, three-body and four-body interactions amongst the biopolymer molecules/particles, respectively and C is the weight concentration (g ml-1) of the biopolymer. 

The coefficients M k) describe the (i + k)-body contribution involving i atoms of species 1 and k atoms of 2. At not too high densities, the virial expansion of spectral moments provides a sound basis for the study of the spectroscopic three-body (and possibly higher) effects. We note that theoretically terms like M 30 gj and M g should be included in the expansion, Eq. 3.9. These correspond to homonuclear three-body contributions which, however, were experimentally shown to be insignificant in the rare gases and are omitted, see p. 58 for details. 

It has been known since the early days of collision-induced absorption that spectral moments may be represented in the form of a virial expansion, with the coefficients of the Nth power of density, qn, representing the N-body contributions [402, 400], The coefficients of qn for N = 2 and 3 have been expressed in terms of the induced dipole and interaction potential surfaces. The measurement of the variation of spectral moments with density is, therefore, of interest for the two-body, three-body, etc., induced dipole components. 

For some time it has been known that the spectral moments, which are static properties of the absorption spectra, may be written as a virial expansion in powers of density, q", so that the nth virial coefficient describes the n-body contributions (n = 2, 3. ..) [400]. That dynamical properties like the spectral density, J co), may also be expanded in terms of powers of density has been tacitly assumed by a number of authors who have reported low-density absorption spectra as a sum of two components proportional to q2 and q3, respectively [100, 99, 140]. It has recently been shown by Moraldi (1990) that the spectral components proportional to q2 and q3 may indeed be related to the two- and three-body dynamical processes, provided a condition on time is satisfied [318, 297]. The proof resorts to an extension of the static pair and triplet distribution functions to describe the time evolution of the initial configurations these permit an expansion in terms of powers of density that is analogous to that of the static distribution functions [135],... 

In the framework of the impact approximation of pressure broadening, the shape of an ordinary, allowed line is a Lorentzian. At low gas densities the profile would be sharp. With increasing pressure, the peak decreases linearly with density and the Lorentzian broadens in such a way that the area under the curve remains constant. This is more or less what we see in Fig. 3.36 at low enough density. Above a certain density, the l i(0) line shows an anomalous dispersion shape and finally turns upside down. The asymmetry of the profile increases with increasing density [258, 264, 345]. Besides the Ri(j) lines, we see of course also a purely collision-induced background, which arises from the other induced dipole components which do not interfere with the allowed lines its intensity varies as density squared in the low-density limit. In the Qi(j) lines, the intercollisional dip of absorption is clearly seen at low densities, it may be thought to arise from three-body collisional processes. The spectral moments and the integrated absorption coefficient thus show terms of a linear, quadratic and cubic density dependence,... 

A problem which is even more difficult is to investigate the yield of vibrational energy accompanying three-body combination of atoms. The combination rate coefficients tend to be small compared to the fast (and second order) vibrational relaxation. Thus the stationary concentration of the diatomic product usually corresponds very closely to an ambient Boltzmann distribution. Callear58 observed S2 from S-atom combination, but concluded that the relaxation was too fast for any meaningful quantitative measurements to be made. 

The dipole interaction couples higher sublevel channels too, for which the order of the coefficient matrix on R 2 is larger than for the 2s 2p channel coupling. Numerical values of a and/or (or e () for the decoupled higher sublevel channels are tabulated for some three-body systems [44,53, 66,77-79],... 

The last decade has witnessed an intense interest in the theory of radiative association rate coefficients because of the possible importance of the reactions in the interstellar medium and because of the difficulty of measuring these reactions in the laboratory. Several theories have been proposed these are all directed toward systems of at least three or four atoms and utilize statistical approximations to the exact quantum mechanical treatment. The utility of these treatments can be partially gauged by using them to calculate three body rate coefficients which can be compared with laboratory measurements. In order to explain these theories briefly, it would be helpful to write down equations for the mechanism of association reactions. Consider two species A+ and B that come together with bimolecular rate coefficient kj to form a complex AB+ which can then be stabilized radiatively with rate coefficient kr, be stabilized collisionally with helium with rate coefficient kcoll, or redissociate with rate coefficient k j ... 

The modified thermal and phase space theories reproduce most three body association data equally well, including the inverse temperature dependence of the rate coefficient (Herbst 1981 Adams and Smith 1981), and are capable of reproducing experimental rate coefficients to within an order of magnitude (Bates 1983 Bass, Chesnavich, and Bowers 1979 Herbst 1985b). They should therefore be this accurate for radiative association rate coefficients if kr is treated correctly. 

Other radiative association reactions leading to formic acid, ethanol, and sundry species are discussed in Leung, Herbst, and Huebner (1984). Calculations of radiative association rate coefficients for ion-molecule systems with large numbers of atoms will be necessary to extend gas phase mechanisms to the syntheses of still larger species. Unfortunately, such calculations are often rendered difficult by the lack of suitable thermodynamic, structural, and vibrational data on the product ions which are needed as input into the calculations. A somewhat easier approach is to estimate the radiative association rate coefficient from higher temperature laboratory three-body rates (Smith et al. 1983). Even so, this approach cannot be used for most reactions of interest involving more complex reactants because of a paucity of laboratory measurements. It is clear that more laboratory work will always be needed ... 

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