Block Diagonalization of the Hamiltonian Matrix

We consider a three-dimensional periodic polymer or molecular crystal containing m orbitals in the elementary cell of one or more atoms. For the sake of simplicity the number of elementary cells in the direction of each crystal axis is taken equal to an odd number Ni = N2 = Ni = 27V 4-1. We assume further that there is an interaction between orbitals belonging to different elementary cells. In that case we can describe, in the one-electron approximation, the delocalized crystal orbitals of the polymer with the aid of the linear-combination-of-atomic-orbitals (LCAO) approximation in the form 

1 Hartree-Fock Crystal-Orbital Theory of Periodic Polymers 

The hypermatrix H of dimension m 2N +1) has submatrices Hp, of dimension m consisting of interactions between orbitals belonging to the elementary cells characterized by the lattice vectors R, and R,. The f th element H j. of the matrix Hp, is then given by 

The overlap matrix S can be partitioned similarly into blocks with elements 

In consequence of the three-dimensional translational symmetry of the polymer and of the Bom-von Karman periodic boundary conditions, matrices H and S are cyclic hypermatrices. For the sake of simplicity we show this for the one-dimensional case the generalization to two- and three-dimensional cases is straightforward. In the one-dimensional case, if we take into account the translational symmetry, the hypermatrices H and S have the form 

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In the nonrelativistic case, with a and p strings. Mg = Ms- Because the nonrelativistic operators are spin-independent, the Hamiltonian matrix is blocked by Ms- This block diagonalization of the Hamiltonian matrix does not persist in the relativistic case, and in the absence of any point group symmetry, the N-particle basis extends to all Mg values. 

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