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Graph theoretical concepts for the unitary group approach to the many‐electron correlation problem
Author(s) -
Shavitt Isaiah
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.560120819
Subject(s) - unitary state , unitary group , matrix representation , formalism (music) , row , mathematics , graph , group (periodic table) , combinatorics , computer science , quantum mechanics , physics , musical , art , database , political science , law , visual arts
The simplified unitary group formalism for electronic states introduced by Paldus has been recast in a form based on a relatively small table of “distinct rows” of the array (or tableau) representatives of the appropriate canonical basis functions. This table can be represented effectively and compactly in the form of a two‐rooted directed graph. Each member (configuration function) of the canonical basis is represented by a walk connecting the head and tail of the graph, and its canonical index (sequence number) can be determined easily and directly from information contained in the distinct row table or the graph. The calculation of the matrix representation of a quantum mechanical operator in the canonical basis can be organized in terms of various subgraphs of the distinct row graph, and in terms of certain “loops” contained in them. A relatively small number of basic subgraphs can be used to generate most of the information needed in the calculation. Certain problems associated with matrix elements of products of generators of the unitary group remain to be solved before an effective formalism for large‐scale electronic structure calculations is obtained, but it is hoped that the concepts introduced here can provide insight which will be helpful in solving these problems.