Cyclic hypermatrices

Since all the matrices A m and B m occurring in equations (82) are, in the case of a linear chain with periodic boundary conditions, cyclic hypermatrices, they can be block-diagonalized with the help of the unitary matrix U [the p,q-th block of U is VM = 1/ (NVl) exp [ 2npq] nx... 

The coefficients are the eigenvector components of the Fock operator of the unperturbed (Est = — 0) polymer). The new matrices A OT are also cyclic hypermatrices and can be block-diagonalized in the same way as before. Therefore, one obtains finally again equation (86) but on the r.h.s. we now have... 

The total Hamiltonian of the system will be the same as in equation (65) and for its periodic part one can use the procedure given in equations (59)-(62). As before one can derive the system of equations (78) with the definitions (79)—(81) for the LCAO case. Finally taking into account that for a periodic polymer all matrices occurring in equation (78) are cyclic hypermatrices, one arrives after block-diagonalization again to equations (86) (in this case without the field ). 

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... 

Returning to the three-dimensional problem, one can show in a similar way that matrices H and S are cyclic hypermatrices in this case too and the unitary matrix U of dimension m 2N + 1) with mXm blocks, namely... 

We now consider a three-dimensional periodic system with (2N -f 1) unit cells and m orbitals within the cell. If we again apply the Bom-von Karman periodic boundary conditions the matrices F , F, and S become cyclic hypermatrices of order m(2N -h 1). Therefore we can again apply to equations (1.97) the unitary transformation described in Section 1.1. Hence... 

It is easy to show that all matrices F > (/ = 1,2,3,4, 5) of order m(2N + 1) X m(2N + 1) are cyclic hypermatrices if one takes into account the translational symmetry of the crystal and introduces periodic boundary conditions. Therefore, one can apply to all the terms in all four equations a unitary transformation which block-diagonalizes all the matrices by multiplying from the left all the equations by and introducing everywhere the unit matrix I = UU+ between matrices F< > and eigenvectors Here we can write... 

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