Proof of Eq

To prove the validity of Eq. (2.41) wo assiimn that the oxpan.sion of cigon-fuiictioiis ill terms of a complete orthonormal basis exists [see Eq. (2.40)] so that we have 

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Appendix B Antilinear Operators and Their Properties Appendix C Proof of Eqs. (18) and (23)... 

We shall not give a rigorous proof of Eq. (3-33) because this would take us too far afield however, the following argument should make the truth of this formula appear quite plausible. 

We have returned to a former notation, denoting the electrostatic potential by , so that (0) is the potential at the jellium surface and bulk property and the right member a surface property. Another proof of Eq. (32) was given by Vannimenus and Budd.72 They also derived a related important theorem,... 

The proof of Eq. (5.15) is a very interesting one and is verified in the following development. From thermodynamics one can write for every point along the Hugoniot curve... 

A direct proof of Eq. (65) without using the explicit form (64) of 5Sim(E) would be instructive. This aim is achieved by following the procedure of deriving Eq. (51) for TrQBw(E) via Eq. (52). Since SSim is just SBw with its Sr replaced by it follows from Eq. (52) that... 

This formula can easily be deduced from a theory due to P. W. Kasteleyn [4] (1961) which allows the number of 1-factors of any planar graph G with an even number of vertices to be expressed as the value of the Pfaffian PfS = j/det S of some skew-symmetric matrix S connected with G. Elementary proofs of Eq. (2) (not using Kasteleyn s formula) for plane graphs in which every face F is a (4k + 2)-gon (where k depends on F) were also given by D. Cvetkovic, I. Gutman and N. Trinajstic [5] (1972) and H. Sachs [6] (1986). 

The proof of eq.(12) is obtained starting from Weyl identity, by showing that the... 

Equation 6.53 implies that there is a value of r ( = mn) somewhere between the limits of the integral, such that Wm is simply equal numerically to the exponential factor evaluated at mn. [The proof of Eq. 6.53 requires only the continuity of V(r) and the fact that r2 > 0.30] If V(r) exhibits a positive maximum (corresponding to a potential barrier ), then V( mn) is likely to be equal to it.28... 

A direct proof of Eq. (67) for the case of ionic crystals is given in Ref.286. Mass action laws are locally still valid, since then the A -term in Zvjjlj disappears because of... 

The proof of eq. (3) is easy if both, the motion along x and are bound, i.e. if the energy spectrum of H is discrete. We will restrict the following considerations to this case. 

Gerischer et al. predicted that the reduction of the =Ge-OH surface occurs by a transfer of an electron via the conduction band and by a simultaneous injection of a hole into the valence band. Details concerning charge transfer processes are given in Chapter 7. A further proof of Eq. (5.44) is given below (see Eqs. 5.46 to 5.48). A pH-dependence was also found in the case of a hydride surface (curve b in Fig. 5.11), which was explained by a dissociation of the double layer as given by [20]... 

A proof of Eq. (47) was given by Allen (1971). Another (perhaps more accessible) path to Eq. (47) starts with the basic deletion formulas known as the Sherman-Morrison-Woodbury theorem, illustrated by Rao (1973) in an exercise. 

The proof of Eq. (2.9) begins by considering two nonnegative distribution functions, F( ) and Fo(r), of the degrees of freedom, which we denote symbolically as an N-dimensional vector r (r is not necessarily the spatial coordinate). We also symbolically denote the sum over the degrees of freedom by the functional integral /dr. The distribution functions are normalized so that... 

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