Moments theorem

The origins of the Finnis-Sinclair potential [Finnis and Sinclair 1984] lie in the density of states and the moments theorem. Recall that the density of states D(E) (see Section 3.8.5) describes the distribution of electronic states in the system. D(E) gives the number of states between E and E - - 8E. Such a distribution can be described in terms of its moments. The moments are usually defined relative to the energy of the atomic orbital from which the molecular orbitals are formed. The mth moment, fi", is given by ... 

In 1971 Ducastelle and Cyrot-Lackmann proved a very important moments theorem that captures this crossing behaviour of the bond energy curves. Their moments theorem states that if two eigenspectra have moments that are identical up to some level pQ, that is App = 0 for p < p0, then the two bond energy curves must cross at least (p0 — 1) times as a function of electron count. Thus, we have seen in Fig. 4.5 that the triangle and the linear chain curves cross once, since p0 = 2, whereas the square and the linear chain curves cross twice, since p0 = 3. 

In this chapter we will show that the tight binding ( ) description of the covalent bond is able to provide a simple and unifying explanation for the above structural trends and behaviour. We will see that the ideas already introduced in chapter 4 on the structures of small molecules may be taken over to these infinite bulk systems. In particular, we will find that the trends in structural stability across the periodic table or within the structure maps can be linked directly to the topology of the local atomic environment through the moments theorem of Ducastelle and Cyrot-Lackmann (1971). 

In general, this interpolative result is not identical with the rigorous result. Nevertheless, as we shall see in later sections, the information theory result often is in good agreement. Needless to say it would be better to use an optimization procedure which would simultaneously satisfy the moment theorems and give the correct A (0), but we have not been able to devise such a procedure. 

Let the factor multiplying tc1/2/2 be called p. Thus we see that if we assume a functional form for the memory function, then it is possible to determine the parameters in the functional form by using the moment theorems of Eq. (162) and to determine, thereby, the transport coefficients, such as the friction coefficient. Moreover, the time correlation function, i /(t), can also be determined. 

Moment Theorem For the moment of a function, one has the following pair... 

Equation 6.120 is the Fourier transform of the band profile. The statistical moments of the band profile are calculated rather simply making use of the moment theorem of the Fourier transform. The first moment of the peak about the origin is... 

The essential feature is that all higher terms in the expansion of the exponential vanish because of the moment theorem. Successive terms of Z/Sl may be represented by graphs on the lattice. To each neatest neighbor link Ky is associated a continuous line. To each site i must be associated two spin components Su, 5 , (to obtain a nonzero average). These rules mean that the only allowed graphs are closed loops, such shown in Fig. X.3. The loop can never intersect itself. If it did, at one i, this would... 

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