Cluster expansion methods pair-correlation

It may be noted that at present only the cluster expansion methods are reliable for the investigation of the higher-order limiting laws. Also, the relative quality of different approximation methods is most often judged on the basis of comparison of calculated values for only a small number of thermodynamic coefficients. One compares values for the osmotic coefficient < > or the excess energy B = as determined by different methods, or else (for primitive model calculations) the values of the pair correlation functions of the ions at contact. While it is necessary that all of these coefficients be given with high accuracy, relatively little is known about the accuracy of various approximation methods in the determination of the other measurable coefficients summarized in Section 3. 

T introduces true n-particle correlation, and products like T T etc., arising out of the expansion eq.(3.2 ), generate simultaneous presence of k-particle amd m-particle correlations in a (k+m)-fold excited determinants etc. The truncation of T == T then corresponds to the pair—correlation model of Sinanoglu, while incorporating higher excited states with several disjoint pair excitations induced through the powers T - The amplitudes for T may be called linked or connected clusters for n electrons. The difficulty of a linear variation method such as Cl lies in its inability to realize the cluster expansion structure eq.(3.2),in a simple and practicable manner. 

In 1966, Silverstone and Sinanoglu [77] and Kelly [80] published their extension of previous formal developments and applications of fhe cluster expansion of the wavefunction and MBPT for closed-(sub)shell ground states to analogous formalisms for open-(sub)shell sfafes. In facf, Kelly demonstrated his methods with an impressive calculation of parfs of elec-fron correlation in the oxygen atom, with emphasis on the correlation of pairs of electrons [80]. Among other things, he pointed to the inevitable appearance of terms that correspond to spin-orbital pair excitations such as 2p(-Fl ),2p(0+ 2p(-l+),4/(-F2+, which are called semi-internal by Silverstone and Sinanoglu [77], see below. 

For the calculation of the correlation energy per unit cell in the ground state of a polymer (either conductor or an insulator) one can use any size-consistent method (perturbation theory /18/, coupled cluster expansion /19/, electron pair theories /20/, etc.). In the case of insulators one can Fourier transform the delocalized Bloch orbitals into site semilocalized Wannier functions (WF-s) and perform the excitations between Wannier functions belonging to near lying sites /4/. (For the generation of optimally localized Wannier functions see /21/.) This procedure is, however,... 

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