Band Theory, Valence Bond, and Tight-Binding Calculations

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Abstract
In the theory of the electronic structure of crystals, the fundamental features of the band theory, the valence bond method, and the tight‐binding approximation are reviewed. The band theory is studied on the basis of the Hartree‐Fock scheme, and the Bloch functions are formed by a projection technique. The main methods for calculating Hartree‐Fock functions in a solid are briefly discussed. The advantages and disadvantages of the band theory and the valence bond method are emphasized, and special attention is paid to the correlation error. In connection with the tight‐binding approximation, the importance of the continuum part and of the approximate linear dependencies is stressed. It is shown that a complete orthonormal set of translationally connected atomic orbitals may be constructed as a convenient basis for this approach. The implication of the virial theorem in interpreting the cohesive properties of the ionic crystals is further emphasized. Some recent refinements of band theory are then discussed. It is shown that a large part of the correlation error can be removed by permitting ``different orbitals for different spins.'' This leads to a scheme intermediate between band theory and valence bond method and, by means of a single parameter, one can obtain an essential lowering of the energy curve and the correct asymptotic behavior for separated atoms or constituents. This approach may be generalized to an extension of the Hartree‐Fock scheme, where the total wave function is defined as a projection of a Slater determinant. The band theory can be further refined and connected to the exact solution of the many‐electron Schrödinger equation of the crystal by means of an extension of the self‐consistent‐field scheme, utilizing the so‐called reaction operator here exactly defined by means of a simple partitioning technique. The various types of self‐consistent field theories are finally compared.

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