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Matrix elements in the analytic tetrahedron method
Author(s) -
Brener N. E.,
Fry J. L.,
Johnson R. A.
Publication year - 2009
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560200850
Subject(s) - tetrahedron , matrix (chemical analysis) , analytic function , mathematics , principal value , analytic element method , mathematical analysis , series (stratigraphy) , physics , geometry , finite element method , paleontology , materials science , biology , composite material , thermodynamics
Abstract The analytic tetrahedron method for evaluating various principal value integrals over a Brillouin zone is generalized to allow for variation of matrix elements or other functions in the numerator of the integrand throughout each tetrahedron. A three‐dimensional Taylor series expansion retaining terms involving the gradient is made for the matrix elements, and the resulting integrals over an arbitrarily shaped tetrahedron are evaluated. Principal value integrals may then be written as a sum of analytic functions. Because of the properties of the matrix element integrals, calculational time is comparable to the time required in the usual analytic tetrahedron method even though the expressions are somewhat more complicated. Various limiting forms of the analytic expressions which are needed for numerical applications are presented.