Second-order interaction

We begin with the most simple Coulomb-Coulomb box graph. It is reducible and the evaluation of the corresponding energy shift requires the use of Eq(152). 

The -matrix element that corresponds to the Feynman graph Fig.7a looks like  

The electron propagator S xiX2) is defined by Eq(122). Inserting these propagators in Eq(165) we have additional double integration over the freqtiency variables and the double summation over the Dirac spectrum m ri2. 

We perform first the integration over the time variables ti, and then 

Here Pf, are the projectors onto the positive (negative) states in the sums over ni, nj. The last term in Eq(166), proportional to 7 , corresponds to the reference state contribution. Now we have to use the formula (152) for the energy shift, adding the contributions of the exchange graphs. Then the reference state contribution in Eq(166) will be totally cancelled by the counterterm ( a 5 a) in Eq(152) and the final result will be  

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This expression may be interpreted in a very similar spirit to tliat given above for one-photon processes. Now there is a second interaction with the electric field and the subsequent evolution is taken to be on a third surface, with Hamiltonian H. In general, there is also a second-order interaction with the electric field through which returns a portion of the excited-state amplitude to surface a, with subsequent evolution on surface a. The Feymnan diagram for this second-order interaction is shown in figure Al.6.9. 

To complete the discussion of the second-order interaction between tunneling centers, we note that the corresponding contribution to the heat capacity in the leading low T term comes from the ripplon-TLS term and scales as 7 +2 where a is the anomalous exponent of the specific law. Within the approximation adopted in this section, a = 0. However, it is easily seen that the magnitude of the interaction-induced specific heat is down from the two-level system value by a factor of 10(a/ ) ([Pg.188]

In the same way it was found through analysis incorporating second-order interaction coefficients that the aldehyde AfT is not conditioned by the CHO effect on the close hydrocarbon groups as was the case with ketonic carbonyl, but by the effect of close groups on aldehydic C-H. 

Muller et al. focused on polybead molecules in the united atom approximation as a test system these are chains formed by spherical methylene beads connected by rigid bonds of length 1.53 A. The angle between successive bonds of a chain is also fixed at 112°. The torsion angles around the chain backbone are restricted to three rotational isomeric states, the trans (t) and gauche states (g+ and g ). The three-fold torsional potential energy function introduced [142] in a study of butane was used to calculate the RIS correlation matrix. Second order interactions , reflected in the so-called pentane effect, which almost excludes the consecutive combination of g+g- states (and vice-versa) are taken into account. In analogy to the polyethylene molecule, a standard RIS-model [143] was used to account for the pentane effect. 

The columns are indexed by the three states of bond i, the rows are indexed by the three states of bond i—1, and the order of indexing is t, g+, g. The statistical weight denoted by cr, which is approximately exp (-250/T), is the first-order weight of a gauche state relative to a trans state. The second-order interaction, which arises from the pentane effect where a bond pair is g gT, is weighted by to, which is approximately exp (-1000/T). 

The predominant interactions are all second-order. Interaction K( t ) -+0(ti ) stabilizes an all-suprafacial concerted cycloaddition,... 

The combination of MQMAS and DOR into a single scheme led to the development of the MQDOR method [204,205]. In MQDOR a full averaging of the second-order interaction is achieved in the ti (MQ evolution under DOR condition) and t2 domains (DOR), yielding truly isotropic spectra in both dimensions, where the positions of the lines are given by (75). 

It has also been shown that the technique referred to as multiple-quantum variable angle spinning (MQVAS) can be used for the purpose of separating the CSA and quadrupolar lineshapes [155]. In this experiment, a judicious choice of the rotor orientation is fjRL = 70.12°, which leads to cancelation of the fourth-rank term of the second-order interaction (Fig. 2). Thus, the refocusing ratio k for the quadrupolar interaction depends on the ratio of the second-rank terms... 

However, significant stabilization is also contributed by nN—>-szn donor-acceptor interactions, each with estimated second-order interaction energy 49.4 kcal mol-1, as depicted in Fig. 4.52. Each ammine ligand thereby donates about 0.061 e to the zinc cation, primarily to the vacant 4s orbital which acquires about 0.371 e total occupancy. As before, the high formal hypervalency at the metal center is achieved within the limits of the duodectet rule, i.e., without significant involvement of extravalent metal p orbitals. 

Since the shifts produced by the cross-terms between ligand and central ion nuclei are often significant, the strikingly broad ENDOR lines ( 1 MHz) observed for transition metal ions61,62) may be traced back, at least in part, to unresolved splittings due to second order interactions of numerous ligands with the central ion. ENDOR frequencies up to second order for an I = 1 nucleus in the presence of a second ligand nucleus with a spin K = 1 are tabulated in Appendix B, Eqs. (B4). 

K. Murakami, T. Sano and T. Yasunaga, Bull. Chem. Soc. Japan 54, 862 (1981). This is unusual case where there are no temperature-jump relaxations. The interaction of bovine serum albumin with bromophenol blue is accompanied by four relaxations which are attributed to a fast second-order interaction followed by three first-order steps. 

The line widths and shapes to be expected for cubic crystals containing point defects have been derived by Cohen and Reif for both first and second order quadrupole interaction 97). In particular, for point defect concentrations greater than about 0.1 (in terms of probability /, of a lattice site being occupied by a defect) distributed in a random fashion over various possible lattice sites, the second order interaction gives rise to a lopsided central component whose shape is given by (97)... 

From the point of view of the NMR spectrum, it appears that y-AUOj is a highly disordered structure, and one takes as a model for quantitative interpretation the MgO type structure, AljgO, with the aluminum atoms randomly occupying % of the octahedral sites. This is an extremely simple type of lattice for which the necessary lattice sums of 1/R (second order interaction) are known. 

More complex, second order interactions may be imagined, involving more than one natural enemy. For example, consider insects to which tannins are important deterrents and digestion inhibitors. As mentioned above, elevated gut pH appears to be a way of dealing with tannins, since tannin-protein complexes are dissociated or inhibited at alkaline pH (16,32). Indeed, using a model in vitro system in which hemoglobin is employed as a protein substrate, we found that several natural tannins and phenolic extracts do not precipitate this protein when the pH exceeds about... 

Take interaction, for example. Imagine a statistical report that says in this experiment, time and temperature are not significant but the time x temperature interaction is/ Or This experimental design does not permit evaluation of second-order interactions/... 

Unperturbed dimensions and dipole moments of polydialkylsiloxanes are investigated using RIS theory. Polymers are treated as branched molecules in which each silicon atom constitutes a tetrafunctional branch point. All significant first- and second-order interactions are included in the configuration partition function. Higher order interactions not suppressed by second-order interactions are also evaluated and accounted for in the statistical weights used. 

Cyclization of an acyclic polythioethylene chain to form (CHjCHjS)), is studied by Monte-Carlo methods for even x ranging from 4 to 24. Unperturbed acyclic chains are assumed to behave in accord with a RIS treatment which incorporates first- and second-order interactions. The most readily formed macrocycle is that with x = 6, in harmony with results obtained in the POE series. However, cyclization at all x considered is found to be more difficult for polythioethylene than for the equivalent polyoxyethylene. 

Conformational features of meso and racemic diads of PVAc are examined using energy calculations. In contrast to other vinyl chains bearing planar substituents, the g conformation is not prohibited for this polymer. The shifts in the positions of the energy minima from perfect staggering are discussed in terms of the second order interactions. Calculated statistical weight parameters are used to reproduce the experimental data on NMR coupling constants and the characteristic ratios. 

The preceeding treatment is readily applied to longer side chains. Necessary procedures are illustrated by considering a molecule in which the side chain contains three bonds. Some equations are replaced here, oi and C04 denote first- and second-order interactions of the third atom in the side chain with atoms in the main chain. Previous expressions for iU m and U r apply without modification provided each in the definition for t and the fj is replaced by ct, where c - (1 + 2010)4) (1 + 0 +... 

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