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

International Tables for Crystallography (2010). Vol. B, ch. 1.2, p. 20

Table 1.2.8.3 

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@buffalo.edu

Table 1.2.8.3| top | pdf |
Products of two real spherical harmonic functions [y_{lmp}] in terms of the density functions [d_{lmp}] defined by equation (1.2.7.3b)[link]

y00 y00 = 1.0000d00
y10 y00 = 0.43301d10
y10 y10 = 0.38490d20 + 1.0d00
y11± y00 = 0.43302d11±
y11± y10 = 0.31831d21±
y11± y11± = 0.31831d22+ − 0.19425d20 + 1.0d00
y11+ y11− = 0.31831d22−
y20 y00 = 0.43033d20
y20 y10 = 0.37762d30 + 0.38730d10
y20 y11± = 0.28864d31± − 0.19365d11±
y20 y20 = 0.36848d40 + 0.27493d20 + 1.0d00
y21± y00 = 0.41094d21±
y21± y10 = 0.33329d31± + 0.33541d11±
y21± y11± = ±0.26691d32+ − 0.21802d30 + 0.33541d10
y21± y11∓ = −0.26691d32−
y21± y20 = 0.31155d41± + 0.13127d21±
y21± y21± = ±0.25791d42+ ± 0.22736d22+ − 0.24565d40 + 0.13747d20 + 1.0d00
y21+ y21− = 0.25790d42− + 0.22736d22−
y22± y00 = 0.41094d22±
y22± y10 = 0.26691d32±
y22± y11± = ± 0.31445d33+ − 0.083323d31+ + 0.33541d11+
y22± y11∓ = 0.31445d33− ± 0.083323d31− ∓ 0.33541d11−
y22± y20 = 0.22335d42± − 0.26254d22±
y22± y21± = ± 0.23873d43+ − 0.089938d41+ + 0.22736d21+
y22± y21∓ = 0.23873d43− ± 0.089938d41− ∓ 0.22736d21−
y22± y22± = ± 0.31831d44+ + 0.061413d40 − 0.27493d20 + 1.0d00
y22+ y22− = 0.31831d44−