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A direct numerical method for the energy band problem: Preliminary results for Li
Author(s) -
Painter G. S.,
Ellis D. E.
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.560040743
Subject(s) - basis (linear algebra) , convergence (economics) , numerical integration , matrix (chemical analysis) , hamiltonian (control theory) , hamiltonian matrix , mathematics , energy (signal processing) , numerical analysis , computer science , mathematical analysis , physics , eigenvalues and eigenvectors , mathematical optimization , chemistry , geometry , quantum mechanics , symmetric matrix , chromatography , economics , economic growth
A numerical method of calculating energy bands is presented which makes it possible to circumvent the difficulties associated with integration of the relevant matrix elements. The method employs a Diophantine numerical integration procedure, previously applied to molecular systems. Preliminary results indicate that the convergence of crystal energy bands is even more rapid than the convergence obtained in previous molecular calculations. The main advantage of our method is that integration is done directly on matrix elements of the hamiltonian without separately evaluating and storing many basis integrals. No multicenter integrals need be done and no approximations are made. Another advantage of the direct evaluation of matrix elements is that there is no loss of significant figures as experienced in going from basis integrals to the final matrix elements.