Energy expression configurational average

Similar expressions are used for (ss r), etc. Except for the integral F°, evaluated from Slater orbitals, semi-empirical values are used for G1 and F2 chosen to give the best fit with atomic spectra. At this level Pople thus expresses the energy of the average state associated with the configuration (2s)m (2p)n as... 

A particular configuration T of a semi-infinite disordered binary alloy Aj,Bi j is characterized by a set of occupation indices vr, where 7 = 1 if the site R is occupied by an atom of the type Q, and Tj = 0 o..ierwise. Configurational averaging of occupation indices ( 7r) = yields the local concentrations c. The energy of such a configuration can be expressed in the form of an effective Hamiltonian of Ising type... 

A simplified closed-shell expression for the Breit interaction can be derived as well [201]. For open-shell atoms, an average of configurations yields a similar total energy expression except for the intra-shell exchange energy [57, p. 164]. 

Denoting the configurational average of G over the cluster configurations by G) = G, a cluster self-energy Is obtained through the expression. 

Since H=K. + V, the canonical ensemble partition fiinction factorizes into ideal gas and excess parts, and as a consequence most averages of interest may be split into corresponding ideal and excess components, which sum to give the total. In MC simulations, we frequently calculate just the excess or configurational parts in this case, y consists just of the atomic coordinates, not the momenta, and the appropriate expressions are obtained from equation b3.3.2 by replacing fby the potential energy V. The ideal gas contributions are usually easily calculated from exact... 

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