Fourier transform of orbital products

Coppens, P.

doi:10.1107/97809553602060000550

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

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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. Coppens^a ^*

^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

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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) $[\rho ({\bf r}) = {\textstyle\sum\limits_{i}} n_{i} \chi_{i}^{2} = {\textstyle\sum\limits_{\mu}} {\textstyle\sum\limits_{\nu}} P_{\mu \nu} \varphi_{\mu} ({\bf r}) \varphi_{\nu} ({\bf r}), \eqno(1.2.8.1)]$ 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) but with non-integer values for the coefficients $[n_{i}]$ .

The summation (1.2.8.1) 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

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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 Clebsch–Gordan coefficients (Condon & Shortley, 1957), 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 and 1.2.8.2. 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)

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) and (1.2.7.2c)

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) or (1.2.8.5c), 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) 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) and (1.2.7.3b). 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, while the coefficients in the expressions (1.2.8.6) are listed in Table 1.2.8.3.

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)

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

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Fourier transform of the electron density as described by (1.2.8.1) requires explicit expressions for the two-centre orbital product scattering. Such expressions are described in the literature for both Gaussian (Stewart, 1969b) and Slater-type (Bentley & Stewart, 1973; Avery & Ørmen, 1979) atomic orbitals. The expressions can also be used for Hartree–Fock atomic functions, as expansions in terms of Gaussian- (Stewart, 1969b, 1970; Stewart & Hehre, 1970; Hehre et al., 1970) and Slater-type (Clementi & Roetti, 1974) functions are available for many atoms.

References

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International Tables for Crystallography (2006). Vol. B. ch. 1.2, pp. 17-18