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New algorithms for calculating 3 n ‐ j symbols
Author(s) -
Roothaan Clemens C. J.
Publication year - 1993
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.560480805
Subject(s) - binomial coefficient , computation , integer (computer science) , mathematics , binomial (polynomial) , gaussian binomial coefficient , binomial theorem , point (geometry) , negative binomial distribution , discrete mathematics , algorithm , combinatorics , computer science , statistics , geometry , poisson distribution , programming language
Compact expressions are presented for the 3 n‐j symbols, where 1 ⩽ n ⩽ 4, which feature sums over products of binomial coefficients, and certain integer triangular coefficients. The triangular coefficients in turn can be expressed as products of binomial coefficients. Thus in the formulas presented for the 3 n‐j symbols, the dependence on numerous factorials, formally as well as computationally, has been completely eliminated. While formulas which incorporate summations over products of binomial coefficients have been known for the 3‐ j and 6‐ j symbols, the introduction of the triangular coefficients, and the application of the binomial/triangular scheme to 3 n‐j symbols with n > 2, provide important new results. The new formulas are simpler, and they permit more efficient computations of the 3 n‐j symbols, both in exact and in floating point format, than most schemes which are currently in use. © 1993 John Wiley & Sons, Inc.