Double exchange in polynuclear mixed-valence clusters. 1. General solution of the electronic problem

The problem of calculating the electronic energy spectra of mixed-valence clusters with one "extra" electron (dⁿ-dⁿ...dⁿ⁺¹) or hole (dⁿ⁺¹-dⁿ⁺¹...dⁿ) delocalized in the paramagnetic cores of transition metals is solved. Unlike the available particular solutions, which are restricted to small numbers of ions and electrons, the solution proposed in this work is general and is suitable for many-electron systems of arbitrary numbers of nuclei and arbitrary symmetries. The new microscopic approach to the double exchange problem is based on the combination of the sequential ("chain") scheme of spin coupling and angular momentum method. In terms of this approach, an analytical dependence of the matrix elements of the double exchange, Heisenberg exchange, and vibronic interaction on all spin quantum numbers is obtained. The final equations contain only the 6j symbols and are free of the higher-order nj symbols, which obstructed the solution of the double exchange problem in previous works.