Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.2, pp. 15-17   | 1 | 2 |

Section 1.2.8. Fourier transform of 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:

1.2.8. Fourier transform of orbital products

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If the wavefunction is written as a sum over normalized Slater determinants, each representing an antisymmetrized combination of occupied molecular orbitals [\chi_{i}] expressed as linear combinations of atomic orbitals [\varphi_{\nu}], i.e. [\chi_{i} = {\textstyle\sum\limits_{\nu}} \hbox{c}_{i\nu} \varphi_{\nu}], the electron density is given by (Stewart, 1969a[link]) [\rho ({\bf r}) = {\textstyle\sum\limits_{i}} n_{i} \chi_{i}^{2} = {\textstyle\sum\limits_{\mu}} {\textstyle\sum\limits_{\nu}} P_{\mu \nu} \varphi_{\mu} ({\bf r}) \varphi_{\nu} ({\bf r}), \eqno(]with ni = 1 or 2. The coefficients [P_{\mu \nu}] are the populations of the orbital product density functions [\phi_{\mu} ({\bf r}) \varphi_{\nu} ({\bf r})] and are given by [P_{\mu \nu} = {\textstyle\sum\limits_{i}} n_{i} c_{i\mu} c_{i\nu}. \eqno(]

For a multi-Slater determinant wavefunction the electron density is expressed in terms of the occupied natural spin orbitals, leading again to ([link] but with noninteger values for the coefficients ni.

The summation ([link] consists of one- and two-centre terms for which [\varphi_{\mu}] and [\varphi_{\nu}] are centred on the same or on different nuclei, respectively. The latter represent the overlap density, which is only significant if [\varphi_{\mu} ({\bf r})] and [\varphi_{\nu} ({\bf r})] have an appreciable value in the same region of space. 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(]or [\varphi (r, \theta, \varphi) = R_{l} (r) y_{lmp} (\theta, \varphi), \eqno(]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(]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(]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(]or the equivalent definition [C_{Lll^{\,'}}^{M mm^{'}} = {\textstyle\int\limits_{0}^{\pi}} \sin \theta \,{\rm d} \theta {\textstyle\int\limits_{0}^{2\pi}}\, {\rm d}\varphi Y_{LM}^{*} (\theta, \varphi) Y_{lm} (\theta, \varphi) Y_{l^{\,'}m^{'}} (\theta, \varphi). \eqno(]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(]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[link] and[link] They are valid for the functions [Y_{lm}] and [y_{lmp}] with normalization [{\textstyle\int} |Y_{lm} |^{2}\, {\rm d} \Omega = 1] and [{\textstyle\int} y_{lmp}^{2}\, {\rm d} \Omega = 1].

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Products of complex spherical harmonics as defined by equation ([link]

Y00 Y00 = 0.28209479Y00
Y10 Y00 = 0.28209479Y10
Y10 Y10 = 0.25231325Y20 + 0.28209479Y00
Y11 Y00 = 0.28209479Y11
Y11 Y10 = 0.21850969Y21
Y11 Y11 = 0.30901936Y22
Y11 Y11− = −0.12615663Y20 + 0.28209479Y00
Y20 Y00 = 0.28209479Y20
Y20 Y10 = 0.24776669Y30 + 0.25231325Y10
Y20 Y11 = 0.20230066Y31 − 0.12615663Y11
Y20 Y20 = 0.24179554Y40 + 0.18022375Y20 + 0.28209479Y00
Y21 Y00 = 0.28209479Y21
Y21 Y10 = 0.23359668Y31 + 0.21850969Y11
Y21 Y11 = 0.26116903Y32
Y21 Y11− = −0.14304817Y30 + 0.21850969Y10
Y21 Y20 = 0.22072812Y41 + 0.09011188Y21
Y21 Y21 = 0.25489487Y42 + 0.22072812Y22
Y21 Y21− = −0.16119702Y40 + 0.09011188Y20 + 0.28209479Y00
Y22 Y00 = 0.28209479Y22
Y22 Y10 = 0.18467439Y32
Y22 Y11 = 0.31986543Y33
Y22 Y11− = −0.08258890Y31 + 0.30901936Y11
Y22 Y20 = 0.15607835Y42 − 0.18022375Y22
Y22 Y21 = 0.23841361Y43
Y22 Y21− = −0.09011188Y41 + 0.22072812Y21
Y22 Y22 = 0.33716777Y44
Y22 Y22− = 0.04029926Y40 − 0.18022375Y20 + 0.28209479Y00

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Products of real spherical harmonics as defined by equations ([link] and ([link]

y00 y00 = 0.28209479y00
y10 y00 = 0.28209479y10
y10 y10 = 0.25231325y20 + 0.28209479y00
y11± y00 = 0.28209479y11±
y11± y10 = 0.21850969y21±
y11± y11± = 0.21850969y22+ − 0.12615663y20 + 0.28209479y00
y11+ y11− = 0.21850969y22−
y20 y00 = 0.28209479y20
y20 y10 = 0.24776669y30 + 0.25231325y10
y20 y11± = 0.20230066y31± − 0.12615663y11±
y20 y20 = 0.24179554y40 + 0.18022375y20 + 0.28209479y00
y21± y00 = 0.28209479y21±
y21± y10 = 0.23359668y31± + 0.21850969y11±
y21± y11± = ± 0.18467439y32+ − 0.14304817y30 + 0.21850969y10
y21± y11∓ = 0.18467469y32−
y21± y20 = 0.22072812y41± + 0.09011188y21±
y21± y21± = ± 0.18022375y42+ ± 0.15607835y22+ − 0.16119702y40 + 0.09011188y20 + 0.28209479y00
y21+ y21− = −0.18022375y42− + 0.15607835y22−
y22± y00 = 0.28209479y22±
y22± y10 = 0.18467439y32±
y22± y11± = ± 0.22617901y33+ − 0.05839917y31+ + 0.21850969y11+
y22± y11∓ = 0.22617901y33− ± 0.05839917y31− ∓ 0.21850969y11−
y22± y20 = 0.15607835y42± − 0.18022375y22±
y22± y21± = ± 0.16858388y43+ − 0.06371872y41+ + 0.15607835y21+
y22± y21∓ = 0.16858388y43− ± 0.06371872y41− ∓ 0.15607835y21−
y22± y22± = ± 0.23841361y44+ + 0.04029926y40 − 0.18022375y20 + 0.28209479y00
y22+ y22− = 0.23841361y44−

By using ([link] or ([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 ([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 ([link] and ([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(]where [R_{LMP} = M_{LMP} \hbox{ (wavefunction)}/L_{LMP}\hbox{ (density function)}]. The normalization constants [M_{lmp}] and [L_{lmp}] are given in Table[link], while the coefficients in the expressions ([link] are listed in Table[link].

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Products of two real spherical harmonic functions [y_{lmp}] in terms of the density functions [d_{lmp}] defined by equation ([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− Two-centre orbital products

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Fourier transform of the electron density as described by ([link] requires explicit expressions for the two-centre orbital product scattering. Such expressions are described in the literature for both Gaussian (Stewart, 1969b[link]) and Slater-type (Bentley & Stewart, 1973[link]; Avery & Ørmen, 1979[link]) atomic orbitals. The expressions can also be used for Hartree–Fock atomic functions, as expansions in terms of Gaussian- (Stewart, 1969b[link], 1970[link]; Stewart & Hehre, 1970[link]; Hehre et al., 1970[link]) and Slater-type (Clementi & Roetti, 1974[link]) functions are available for many atoms.


Avery, J. & Ørmen, P.-J. (1979). Generalized scattering factors and generalized Fourier transforms. Acta Cryst. A35, 849–851.
Bentley, J. & Stewart, R. F. (1973). Two-centre calculations for X-ray scattering. J. Comput. Phys. 11, 127–145.
Clementi, E. & Roetti, C. (1974). Roothaan–Hartree–Fock atomic wavefunctions. At. Data Nucl. Data Tables, 14, 177–478.
Condon, E. V. & Shortley, G. H. (1957). The theory of atomic spectra. London, New York: Cambridge University Press.
Hehre, W. J., Ditchfield, R., Stewart, R. F. & Pople, J. A. (1970). Self-consistent molecular orbital methods. IV. Use of Gaussian expansions of Slater-type orbitals. Extension to second-row molecules. J. Chem. Phys. 52, 2769–2773.
Stewart, R. F. (1969a). Generalized X-ray scattering factors. J. Chem. Phys. 51, 4569–4577.
Stewart, R. F. (1969b). Small Gaussian expansions of atomic orbitals. J. Chem. Phys. 50, 2485–2495.
Stewart, R. F. (1970). Small Gaussian expansions of Slater-type orbitals. J. Chem. Phys. 52, 431–438.
Stewart, R. F. & Hehre, W. J. (1970). Small Gaussian expansions of atomic orbitals: second-row atoms. J. Chem. Phys. 52, 5243–5247.

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