Whole Number Recursion Formulae for High Order Clebsch-Gordan Coupling Coefficients

Clebsch-Gordan (CG) coupling coefficients appear in many wave-based theories of matter and radiation. Here, novel whole number recursion for-mulae are presented for the exact calculation of arbitrarily high order CG coupling coefficients. Using an extended precision integer and rational number arithmetic library, these recursion formulae are shown to be up to four orders of magnitude faster than an exact prime factorisation approach. When coded in the C programming language using floating point " long double " or " double " precision, good numerical precision is obtained for the common case where the total magnetic quantum number, m 3 , is zero for all CG coefficients up to order J = 200, or J = 130, respectively, where the ranges of the principal quantum numbers are restricted to 0 ≤ j 1 ≤ J, 0 ≤ j 2 ≤ j 1 , and (j 1 − j 2) ≤ j 3 ≤ (j 1 + j 2). In this case, the calculation is up to four orders of magnitude faster than the exact prime number factorisation method used here. A C program that demonstrates these calculations is available for download at http://cgc.loria.fr