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

International Tables for Crystallography (2006). Vol. B, ch. 1.2, pp. 17-18   | 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: coppens@acsu.buffalo.edu

1.2.8. Fourier transform of orbital products

| top | pdf |

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(1.2.8.1)] with [n_{i} = 1\hbox{ 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(1.2.8.2)]

For a multi-Slater determinant wavefunction the electron density is expressed in terms of the occupied natural spin orbitals, leading again to (1.2.8.2)[link] but with non-integer values for the coefficients [n_{i}].

The summation (1.2.8.1)[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.

1.2.8.1. One-centre orbital products

| top | pdf |

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−

1.2.8.2. Two-centre orbital products

| top | pdf |

Fourier transform of the electron density as described by (1.2.8.1)[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.

References

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.








































to end of page
to top of page