Further perturbative results

Further perturbation of the surface charge results in even greater effects. The mutation of either Asp-99 or Glu-156 to a positively charged lysine results in a pAa decrease of 0.6 for His-64, and the pAa of the... 

Equations (7)-(9) are well suited for numerical evaluation with arbitrary functions defining the spatial and temporal distributions of the laser pulse. In addition, the system (7)-(9) is rather convenient for analytical treatment. In particular, one can develop further perturbative expansion of (7)-(8) in terms of the fine structure constant a. In the leading order, this yields nonrelativistic formulas which agree with those formerly derived by us in [19]. For unchirped laser signals (i.e., t) = 0) these reduce further to the result of [31] by expanding all quantities in powers of the laser intensity I(r,t). Extensive numerical tests carried out by us for various forms of the chirp pulse en-... 

The perturbation results initially led us in a systematic way to the form of the more general and accurate "many-electron theory (see below). However, as we have already pointed out, the final many-electron theory in no way necessitates the use of perturbation theory. It can be derived directly from Eq. (20). In fact, perturbation theory can be obtained from the many-electron theory upon further approximations, and for computations often provides a useful first approximation. ... 

The second equivalence in the above equation resulted from the fact that 7 fulfils the CC equation of the first-order with respect to the perturbation 9 K The second term of Eqn. (21), is further decomposed as... 

Results for these CEBEs are presented in Table 1. As can be seen, for the carvone variants I-V the various substitutions have absolutely no effect at the carbonyl C=0 core, and are barely significant at the chiral center that lies between the carbonyl and substituent groups in these molecules. Only upon fluorine substitution at the tail (molecule VI) does the C=0 CEBE shift by one-half of an electronvolt the second F atom substitution adjacent to the C=0 in the difluoro derivative, VII contributes a further 0.6-eV shift. This effect can be rationalized due to the electron-withdrawing power of an F atom. Paradoxically, it is these fluorine-substituted derivatives, VI, VII, that arguably produce b curves most similar to the original carvone conformer, I, yet they are the only ones to produce a perturbation of the ground-state electron density at the C li core. This contributes further evidence to suggest that, at least for the C li... 

Further studies were carried out on the Pd/Mo(l 1 0), Pd/Ru(0001), and Cu/Mo(l 10) systems. The shifts in core-level binding energies indicate that adatoms in a monolayer of Cu or Pd are electronically perturbed with respect to surface atoms of Cu(lOO) or Pd(lOO). By comparing these results with those previously presented in the literature for adlayers of Pd or Cu, a simple theory is developed that explains the nature of electron donor-electron acceptor interactions in metal overlayer formation of surface metal-metal bonds leads to a gain in electrons by the element initially having the larger fraction of empty states in its valence band. This behavior indicates that the electro-negativities of the surface atoms are substantially different from those of the bulk [65]. 

Consider the forward calculation as an example. Earlier in this chapter, we concluded that the most important contribution to a forward free energy calculation comes from the low-x tail of f(x)9 and its poor sampling results in the major systematic error of the calculation. To simplify further the analysis we assume that there is a limit-perturbation x f such that x > Xf is well sampled and x < Xf is never sampled. This is illustrated in Fig. 6.5. 

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