Potential expansion

Conserving only the first two terms of the anisotropic potential expansion (7.49) and summing as noted in (7.70) over j, we get... 

Additivity is broken by the second term of (7.73), which corresponds to the second harmonics of the potential expansion... 

Potential expansion of aggregates from hydration reactions... 

Larger particle size ensures that the potential expansion due to hydration is extended over a period of time. Chemical and physical properties of the most widely used proprietary product, Denka CSA, are given in Table 6.10 [74]. Other CSAs include mixtures of C4 ASHj2 and 2 CS (monosulfate and gypsum) and mixtures of Type... 

Fig. 8.20 Convergence of the bcc-fcc d bond energy (full curve) and hcp-fcc d bond energy (dashed curve) with respect to the number of terms in the bond order potential expansion. The left, middle and right panels correspond to keeping terms up to fourth, sixth and eighteenth moments respectively. (From Aoki (1993).)...

The relevant potential here will be the electric potential expansion... 

Fig. 4. The potential expansion of the lowest order self energy. Three terms of this expansion usually are denoted as zero-potential (ZP), one-potential (OP) and many-potential (MP). The notations are the same as in Figures 1, 2...

We now return to the loop-after-loop SESE calculations in [11]. The first two terms of the potential expansion Eq. (2), ZP and OP terms were evaluated in momentum space. For this purpose the Fourier transform was performed for the bound state wave functions n) in coordinate space. The latter were evaluated by the space discretization method. The MP term was calculated entirely in coordinate space. 

An investigation devoted to the evaluation of SESE a) irreducible contribution was accomplished by S. Mallampalli and J. Sapirstein [22]. Using the same covariant renormalization approach with the potential expansion and employing... 

In this section we will calculate the reducible contributions to the graph Fig. 1 a) and the total contributions to the graphs Figs. lb),lc). The general renormalization scheme for these graphs was presented in [30]. This scheme exploits the potential expansion for separating out the divergent terms and is suitable for the application of the PWE approach. Later these results were rederived in [31] by a different method. 

Appendix II Potential expansion of the reducible Green function... 

Here Vr is an appropriate permutational symmetry projection operator for the desired state, T, and YfcM is a product of coupled solid harmonics labeled by the total angular momentum quantum numbers L and M. Permutational symmetry is handled using projection methods in the same manner as described for the potential expansion in the previous section. Again, the reader is referred to the references for details[9,10,12],... 

For consistency with the quadratic approximation used in the potential expansion, the above expression for H has been tmncated at the quadratic term the higher-order and off-diagonal terms in the expansion can be treated as perturbations. Then, the approximate effective Hamiltonian for the reaction dynamics can be written in the form... 

A more refined approach uses the two-centers potential expansion, which considers the interaction between point multipoles located at the centers of the two charge distributions. Since the exchange integral interactions vary as 1/r and the numbers of interactions increase as r, one has to choose spherical shapes for the convergence of the sum to be guaranteed (Ewald sums method)... 

The first thing which strikes one about these widths is their size, some seven orders of magnitude smaller than the rotational predissociation widths discussed above. Thus, it is clear that line broadening of this type will not be observable for these species, and hence cannot be looked on as a possible source of information on the k>0 contributions to the potential expansion (see Equation 7), which are responsible for this type of coupling. On the other hand, as in the case of rotational predissociation, the same type of information about the coupling functions contained in these widths is also found in vibrationally inelastic cross sections, both being sensitive to behaviour at smaller distances than are important for the line shifts in the discrete spectra. 

SFCCCC Calculations of Ar-HD vdW Complex (20). The potential surface for Ar-HD can be obtained from the 3(6,8) potential of Ar-H2 by performing the asymmetric Isotope frame transformation (33) to a coordinate system based on the centre-of-mass of HD. This transformation Introduces Legendre terms of odd order Into the potential expansion, so that the diagonal vlbratlonally averaged Ar-HD potential, for example, has the form... 

Fig. 53. Comparison between exact electrostatic and approximate descriptions of conformational energy for an aziridine-water associate. Solid, line "exact electrostatic potential — expansion truncated after dipole terms - — after quadrupole terms - -----after octopole terms ------after hexadecapole terms. From Ref. B0>...

Consider next the A dependent part originating from the vertex term (Eq. (19)). Making a potential expansion of the bound propagators shows that only the free propagator term leads to divergences and hence needs to be considered. By using similar approximations as for the one-potential part in the wave function term we get... 

The method that helped to avoid this difficulty was first introduced in [10] and applied to the evaluation of SB for the A-sheU electrons in the mercury atom. The general idea of the method is the potential expansion of the SE Feynman graph for the bound electron. This expansion is depicted in Fig.ll. The divergency is concentrated in the first two terms of this expansion. These terms are usually called zero - potential and one - potential terms. The third term, so called many - potential term, is finite but the most difficult for numerical evaluations. To avoid the evaluation of AE the authors of jlO] rearranged the three terms of the expansion... 

The direct potential expansion of the SE was first analyzed in [64]. Recently it was considered in the Pried-Yennie gauge [65] that allows the extension of... 

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