Inverse Overlap Matrix for Periodic Arrays of Atoms

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Abstract
The calculation of the inverse overlap matrix for an infinite chain of single orbital atoms is reduced to the problem of calculating the roots of a polynomial of degree n constructed from the overlap integrals between n neighbors. The familiar method of diagonalizing the overlap matrix, inverting it, and then transforming back to the original representation is used. The final transformation leads to contour integrals which can be evaluated by the method of residues. This diagonalization method is shown to be useful also for finite chains of atoms with Born von Kárman boundary conditions, for chains of atoms with more than one orbital per atom, and for calculating the inverse root of the overlap matrix of single‐orbital atoms when the only overlap is between nearest neighbors. Application of the method to two‐ and three‐dimensional arrays of atoms leads to contour integrals containing branch points which cannot be reduced to simple analytic expressions. However, it is shown that an iterative procedure can be devised which permits the calculation of 2n terms of the Löwdin expansion for the inverse with only 2n matrix multiplications.