In this paper we present a non-perturbative approach to the calculation of correlation energies of open-shell systems. The formulation utilizes an Ursell-type expansion about a multi-determinant starting wavefunction. We have proved a theorem which enables us to derive an effective hamiltonian for the system consisting entirely of linked terms. In the symmetrydegenerate case this effective hamiltonian acts within the subspace of a set of symmetry-degenerate functions, and generates the energy eigenvalues of the system. The present theory has been cast in a diagrammatic language which facilitates the analysis of the correlation problem. The workability of the theory has been tested on a 4 7r electron problem, transbutadiene, for which we have calculated the lowest ~r-~r* singlet and triplet energies. The agreement between the results of the present theory and that found from a full C I calculation is excellent. The desirable feature of the theory is that the effective hamiltonian is energy-independent. We have demonstrated the connection of the present theory with open-shell perturbation theories. We have also indicated a method for extending this theory to general open-shell systems.
The question of the conditions under which 1D systems support extended electronic eigenstates is addressed in a very general context. Using real space renormalisation group arguments we discuss the precise criteria for determining the entire spertrum of extended eigenstates and the corresponding eigenfunctions in disordered as well as quasiperiodic systems. For purposes of illustration we calculate a few selected eigenvalues and the corresponding extended eigenfunctions for the quasiperiodic copper-mean chain. So far, for the infinite copper-mean chain, only a single energy has been numerically shown to support an extended eigenstate [ You et al. (1991)] : we show analytically that there is in fact an infinite number of extended eigenstates in this lattice which form fragmented minibands.
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