International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 1.2, p. 22

Table 1.2.12.1 

P. Coppensa*

aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260-3000, USA
Correspondence e-mail: coppens@acsu.buffalo.edu

Table 1.2.12.1| top | pdf |
Some Hermite polynomials (Johnson & Levy, 1974[link]; Zucker & Schulz, 1982[link])

H(u) = 1
Hj(u) = wj
Hjk(u) = wjwkpjk
Hjkl(u) = wjwkwl − (wjpkl + wkplj + wlpjk) = wjwkwl3w( jpkl)
Hjklm(u) = wjwkwlwm6w( jwkplm) + 3pj( kplm)
Hjklmn(u) = wjwkwlwmwn10w( lwmwnpjk) + 15w( npjkplm)
Hjklmnp(u) = wjwkwlwmwnwp − 15w( jwkwlwmpjk) + 45w( jwkplmpnp) − 15pj( kplmpnp)
where [w_{j}\equiv p{_{jk}}u^{k} \hbox{ and } p_{jk}] are the elements of [\sigma^{-1}], defined in expression (1.2.10.2)[link] [link]. Indices between brackets indicate that the term is to be averaged over all permutations which produce distinct terms, keeping in mind that [p_{jk} = p_{kj} \hbox{ and } w_{j}w_{k} = w_{k}w_{j}] as illustrated for [H_{jkl}].