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

International Tables for Crystallography (2006). Vol. B. ch. 1.2, p. 18   | 1 | 2 |

Section 1.2.8.1. One-centre orbital products

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

1.2.8.1. One-centre orbital products

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If the atomic basis consists of hydrogenic type s, p, d, f, … orbitals, the basis functions may be written as [\varphi (r, \theta, \varphi) = R_{l} (r) Y_{lm} (\theta, \varphi) \eqno(1.2.8.3a)] or [\varphi (r, \theta, \varphi) = R_{l} (r) y_{lmp} (\theta, \varphi), \eqno(1.2.8.3b)] which gives for corresponding values of the orbital products [\varphi_{\mu} ({\bf r}) \varphi_{\nu} ({\bf r}) = R_{l} (r) R_{l^{'}} (r) Y_{lm} (\theta, \varphi) Y_{l^{'}m^{'}} (\theta, \varphi) \eqno(1.2.8.4a)] and [\varphi_{\mu} ({\bf r}) \varphi_{\nu} ({\bf r}) = R_{l} (r) R_{l^{'}} (r) y_{lmp} (\theta, \varphi) y_{l^{'}m^{'}p^{'}} (\theta, \varphi), \eqno(1.2.8.4b)] respectively, where it has been assumed that the radial function depends only on l.

Because the spherical harmonic functions form a complete set, their products can be expressed as a linear combination of spherical harmonics. The coefficients in this expansion are the ClebschGordan coefficients (Condon & Shortley, 1957[link]), defined by [Y_{lm} (\theta, \varphi) Y_{l^{'}m^{'}} (\theta, \varphi) = {\textstyle\sum\limits_{L}} {\textstyle\sum\limits_{M}} C_{Lll^{'}}^{M mm^{'}} Y_{LM} (\theta, \varphi) \eqno(1.2.8.5a)] or the equivalent definition [C_{Lll^{'}}^{M mm^{'}} = {\textstyle\int\limits_{0}^{\pi}} \sin \theta \;\hbox{d} \theta {\textstyle\int\limits_{0}^{2\pi}}\; \hbox{d}\varphi Y_{LM}^{*} (\theta, \varphi) Y_{lm} (\theta, \varphi) Y_{l^{'}m^{'}} (\theta, \varphi). \eqno(1.2.8.5b)] The [C_{Lll^{'}}^{M mm^{'}}] vanish, unless [L + l + l^{'}] is even, [|l - l^{'} | \;\lt\; L \;\lt\; l + l^{'}] and [M = m + m^{'}].

The corresponding expression for [y_{lmp}] is [y_{lmp} (\theta, \varphi) y_{l^{'}m^{'}p^{'}} (\theta, \varphi) = {\textstyle\sum\limits_{L}} {\textstyle\sum\limits_{M}} C^{'} {\textstyle {\openup-4pt{\matrix{_{M mm'}\hfill\cr _{Lll'}\hfill\cr _{P}\hfill}}}} y_{LMP} (\theta, \varphi), \eqno(1.2.8.5c)] with [M = |m + m^{'}|] and [|m - m^{'}|] for [p = p^{'}], and [M = - |m + m^{'}|] and [- |m - m^{'}|] for [p = - p^{'}] and [P = p \times p^{'}].

Values of C and [C^{'}] for [l \leq 2] are given in Tables 1.2.8.1[link] and 1.2.8.2.[link] They are valid for the functions [Y_{lm}] and [y_{lmp}] with normalization [{\textstyle\int} |Y_{lm} |^{2}\; \hbox{d} \Omega = 1] and [{\textstyle\int} y_{lmp}^{2}\; \hbox{d} \Omega = 1].

Table 1.2.8.1| top | pdf |
Products of complex spherical harmonics as defined by equation (1.2.7.2a)[link]

Y 00 Y 00 = 0.28209479Y 00
Y 10 Y 00 = 0.28209479Y 10
Y 10 Y 10 = 0.25231325Y 20 + 0.28209479Y 00
Y 11 Y 00 = 0.28209479Y 11
Y 11 Y 10 = 0.21850969Y 21
Y 11 Y 11 = 0.30901936Y 22
Y 11 Y 11− = −0.12615663Y 20 + 0.28209479Y 00
Y 20 Y 00 = 0.28209479Y 20
Y 20 Y 10 = 0.24776669Y 30 + 0.25231325Y 10
Y 20 Y 11 = 0.20230066Y 31 − 0.12615663Y 11
Y 20 Y 20 = 0.24179554Y 40 + 0.18022375Y 20 + 0.28209479Y 00
Y 21 Y 00 = 0.28209479Y 21
Y 21 Y 10 = 0.23359668Y 31 + 0.21850969Y 11
Y 21 Y 11 = 0.26116903Y 32
Y 21 Y 11− = −0.14304817Y 30 + 0.21850969Y 10
Y 21 Y 20 = 0.22072812Y 41 + 0.09011188Y 21
Y 21 Y 21 = 0.25489487Y 42 + 0.22072812Y 22
Y 21 Y 21− = −0.16119702Y 40 + 0.09011188Y 20 + 0.28209479Y 00
Y 22 Y 00 = 0.28209479Y 22
Y 22 Y 10 = 0.18467439Y 32
Y 22 Y 11 = 0.31986543Y 33
Y 22 Y 11− = −0.08258890Y 31 + 0.30901936Y 11
Y 22 Y 20 = 0.15607835Y 42 − 0.18022375Y 22
Y 22 Y 21 = 0.23841361Y 43
Y 22 Y 21− = −0.09011188Y 41 + 0.22072812Y 21
Y 22 Y 22 = 0.33716777Y 44
Y 22 Y 22− = 0.04029926Y 40 − 0.18022375Y 20 + 0.28209479Y 00

Table 1.2.8.2| top | pdf |
Products of real spherical harmonics as defined by equations (1.2.7.2b)[link] and (1.2.7.2c)[link]

y 00 y 00 = 0.28209479y 00
y 10 y 00 = 0.28209479y 10
y 10 y 10 = 0.25231325y 20 + 0.28209479y 00
y 11± y 00 = 0.28209479y 11±
y 11± y 10 = 0.21850969y 21±
y 11± y 11± = 0.21850969y 22+ − 0.12615663y 20 + 0.28209479y 00
y 11+ y 11− = 0.21850969y 22−
y 20 y 00 = 0.28209479y 20
y 20 y 10 = 0.24776669y 30 + 0.25231325y 10
y 20 y 11± = 0.20230066y 31± − 0.12615663y 11±
y 20 y 20 = 0.24179554y 40 + 0.18022375y 20 + 0.28209479y 00
y 21± y 00 = 0.28209479y 21±
y 21± y 10 = 0.23359668y 31± + 0.21850969y 11±
y 21± y 11± = ± 0.18467439y 32+ − 0.14304817y 30 + 0.21850969y 10
y 21± y 11∓ = 0.18467469y 32−
y 21± y 20 = 0.22072812y 41± + 0.09011188y 21±
y 21± y 21± = ± 0.18022375y 42+ ± 0.15607835y 22+ − 0.16119702y 40 + 0.09011188y 20 + 0.28209479y 00
y 21+ y 21− = −0.18022375y 42− + 0.15607835y 22−
y 22± y 00 = 0.28209479y 22±
y 22± y 10 = 0.18467439y 32±
y 22± y 11± = ± 0.22617901y 33+ − 0.05839917y 31+ + 0.21850969y 11+
y 22± y 11∓ = 0.22617901y 33− ± 0.05839917y 31− ∓ 0.21850969y 11−
y 22± y 20 = 0.15607835y 42± − 0.18022375y 22±
y 22± y 21± = ± 0.16858388y 43+ − 0.06371872y 41+ + 0.15607835y 21+
y 22± y 21∓ = 0.16858388y 43− ± 0.06371872y 41− ∓ 0.15607835y 21−
y 22± y 22± = ± 0.23841361y 44+ + 0.04029926y 40 − 0.18022375y 20 + 0.28209479y 00
y 22+ y 22− = 0.23841361y 44−

By using (1.2.8.5a)[link] or (1.2.8.5c)[link], the one-centre orbital products are expressed as a sum of spherical harmonic functions. It follows that the one-centre orbital product density basis set is formally equivalent to the multipole description, both in real and in reciprocal space. To obtain the relation between orbital products and the charge-density functions, the right-hand side of (1.2.8.5c)[link] has to be multiplied by the ratio of the normalization constants, as the wavefunctions [y_{lmp}] and charge-density functions [d_{lmp}] are normalized in a different way as described by (1.2.7.3a)[link] and (1.2.7.3b)[link]. Thus [y_{lmp} (\theta, \varphi) y_{l^{'}m^{'}p^{'}} (\theta, \varphi) = {\textstyle\sum\limits_{L}} {\textstyle\sum\limits_{M}} R_{LMP} C^{'} {\let\normalbaselines\relax\openup-4pt{\matrix{_{M mm'}\hfill\cr_{Lll'}\hfill\cr_{P}\hfill}}} d_{LMP}(\theta, \varphi), \eqno(1.2.8.6)] where [R_{LMP} = M_{LMP} \hbox{ (wavefunction)}/L_{LMP}\hbox{ (density function)}]. The normalization constants [M_{lmp}] and [L_{lmp}] are given in Table 1.2.7.1[link], while the coefficients in the expressions (1.2.8.6)[link] are listed in Table 1.2.8.3[link].

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]

y 00 y 00 = 1.0000d 00
y 10 y 00 = 0.43301d 10
y 10 y 10 = 0.38490d 20 + 1.0d 00
y 11± y 00 = 0.43302d 11±
y 11± y 10 = 0.31831d 21±
y 11± y 11± = 0.31831d 22+ − 0.19425d 20 + 1.0d 00
y 11+ y 11− = 0.31831d 22−
y 20 y 00 = 0.43033d 20
y 20 y 10 = 0.37762d 30 + 0.38730d 10
y 20 y 11± = 0.28864d 31± − 0.19365d 11±
y 20 y 20 = 0.36848d 40 + 0.27493d 20 + 1.0d 00
y 21± y 00 = 0.41094d 21±
y 21± y 10 = 0.33329d 31± + 0.33541d 11±
y 21± y 11± = ±0.26691d 32+ − 0.21802d 30 + 0.33541d 10
y 21± y 11∓ = −0.26691d 32−
y 21± y 20 = 0.31155d 41± + 0.13127d 21±
y 21± y 21± = ±0.25791d 42+ ± 0.22736d 22+ − 0.24565d 40 + 0.13747d 20 + 1.0d 00
y 21+ y 21− = 0.25790d 42− + 0.22736d 22−
y 22± y 00 = 0.41094d 22±
y 22± y 10 = 0.26691d 32±
y 22± y 11± = ± 0.31445d 33+ − 0.083323d 31+ + 0.33541d 11+
y 22± y 11∓ = 0.31445d 33− ± 0.083323d 31− ∓ 0.33541d 11−
y 22± y 20 = 0.22335d 42± − 0.26254d 22±
y 22± y 21± = ± 0.23873d 43+ − 0.089938d 41+ + 0.22736d 21+
y 22± y 21∓ = 0.23873d 43− ± 0.089938d 41− ∓ 0.22736d 21−
y 22± y 22± = ± 0.31831d 44+ + 0.061413d 40 − 0.27493d 20 + 1.0d 00
y 22+ y 22− = 0.31831d 44−

References

First citationCondon, E. V. & Shortley, G. H. (1957). The theory of atomic spectra. London, New York: Cambridge University Press.Google Scholar








































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