A Systematic Approach to the Problems of Random Lattices. I: A Self-Contained First-Order Approximation Taking into Account the Exclusion Effect

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Abstract
A systematic method of approximation for the electronic state of a randomly doped lattice or a vibration spectrum of a disordered lattice is given in the present series of work by means of an investigation of the one-electron Green's function. In the present article, an exact form of the first-order self-energy is, with the help of a diagrammatic consideration, evaluated on rigorously including the “exclusion effect”. The resulting “exact” first-order self-energy agrees with the lowest-order approximant of the “total first-order self-energy which has been previously obtained by the author and Matsubara, and satisfies the same equation derived by Taylor for the case of lattice vibrations. It is also identical with the approximation developed by Onodera and Toyozawa. Thus, one of the objects of the analysis given in the present work is to offer a mathematically correct interpretation of these methods. A systematic way to proceed to higher-order approximations is discussed.

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