Generalized scattering factors

Maslen, E. N.; Fox, A. G.; O'Keefe, M. A.

doi:10.1107/97809553602060000600

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 6.1, pp. 565-584

Section 6.1.1.4. Generalized scattering factors

E. N. Maslen,^e ^‡ A. G. Fox^b and M. A. O'Keefe^c

6.1.1.4. Generalized scattering factors

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For bound atoms, it may be necessary to account for the perturbation of the electron density by interaction with other atoms, and to analyse its effect on the scattering.

The generalized scattering factor is obtained from the Fourier transform of a perturbed atomic electron-density function. The exponential factor in the transform may be written as an expansion in terms of Legendre polynomials $[P_l(\cos\theta).]$ ¹ $[\exp(i{\bf S}\cdot{\bf r})=\sum^\infty_{l=0}(2l+1)i^lj_l(Sr)P_l\bigg[\cos\bigg(\displaystyle{{\bf S\cdot r}\over Sr}\bigg)\bigg],]$ where [j_l] is a spherical Bessel function of order l and S = |S|. The addition theorem enables this to be expressed as $[\exp(i{\bf S}\cdot{\bf r})=4\pi\textstyle\sum\limits_{l=0}i^lj_l(Sr)\sum\limits^l_{m=-l}Y_{lm}(\theta_S,\varphi_S)Y_{lm}^*(\theta,\varphi).\eqno (6.1.1.17)]$ The $[Y_{lm}(\theta,\varphi)]$ are spherical (surface) harmonics $[\eqalignno{Y_{lm}(\theta,\varphi)&=\bigg[{(2l+1)(l+m)!\over4\pi(l-m)!}\bigg]^{1/2}{(-)^le^{im\varphi}\over2^ll!(\sin\theta)^m}\cr &\quad\times{{\rm d}^{l-m}\over{\rm d}(\cos\theta)^{l-m}}\,(\sin\theta)^{2l}\cr &=\bigg[{(2l+1)(l-m)!\over4\pi(l+m)!}\bigg]^{1/2}(-)^me^{im\varphi}P_l^m(\cos\theta)\quad m\ge0,\cr & & (6.1.1.18)}]$ where $[P^m_l(\cos\theta)]$ is an associated Legendre polynomial .

With this definition of the spherical harmonics, $[Y_{l-m}=(-)^mY^*_{lm}.\eqno (6.1.1.19)]$

Spherical harmonics with alternative phase conventions can be defined. The relationship between those in common use is given by Normand (1980). With the convention given in (6.1.1.18), the spherical harmonics up to fourth order are $[\eqalign{Y_{0\,0}&=(4\pi){}^{-1/2}\cr Y_{1\,\pm1}&=\mp(3/8\pi){}^{1/2}\sin\theta\, e^{\pm i\varphi}\cr Y_{1\,0}&=(3/4\pi){}^{1/2}\cos\theta\cr Y_{2\,\pm2}&=\bigg({15\over32\pi}\bigg)^{1/2}\sin^2\theta\, e^{\pm2i\varphi}\cr Y_{2\,\pm1}&=\mp\bigg({15\over8\pi}\bigg)^{1/2}\sin\theta\cos\theta\, e^{\pm i\varphi}\cr Y_{2\,0}&=\bigg({5\over16\pi}\bigg)^{1/2}(3\cos^2\theta-1)\cr Y_{3\,\pm3}&=\mp\bigg({35\over64\pi}\bigg)^{1/2}\sin^3\theta\, e^{\pm3i\varphi}\cr Y_{3\,\pm2}&=\bigg({105\over32\pi}\bigg)^{1/2}\cos\theta\sin^2\theta\, e^{\pm2i\varphi}\cr Y_{3\,\pm1}&=\mp\bigg({21\over64\pi}\bigg)^{1/2}\sin\theta(4-5\sin^2\theta)\, e^{\pm i\varphi}\cr Y_{3\,0}&=\bigg({7\over16\pi}\bigg)^{1/2}\cos\theta(2-5\sin^2\theta)\cr Y_{4\,\pm4}&=\bigg({315\over512\pi}\bigg)^{1/2}\sin^4\theta\, e^{\pm4i\varphi}\cr Y_{4\,\pm3}&=\mp\bigg({315\over64\pi}\bigg)^{1/2}\cos\theta\sin^3\theta\, e^{\pm3i\varphi}\cr Y_{4\,\pm2}&=\bigg({45\over128\pi}\bigg)^{1/2}\sin^2\theta(6-7\sin^2\theta)\, e^{\pm2i\varphi}\cr Y_{4\,\pm1}&=\mp\bigg({45\over64\pi}\bigg)^{1/2}\cos\theta\sin\theta(4-7\sin^2\theta)\, e^{\pm i\varphi}\cr Y_{4\,0}&=\bigg({9\over256\pi}\bigg)^{1/2}(3-30\sin^2\theta+35\sin^4\theta). }\eqno(6.1.1.20)]$ The perturbed electron density may be written as a multipole expansion in spherical polar coordinates $[r,\theta,\varphi]$ , each term having the form $[\specialfonts\rho_{lm\pm}(r)=\rho_{lm\pm}(r){\bsf y}(\theta,\varphi),\eqno (6.1.1.21)]$ where $[\specialfonts{\bsf y}]$ is a suitably normalized real function of the polar coordinates. A common choice is the real form of the spherical harmonics $[Y_{lm\pm(\theta,\,\varphi)}=\bigg[{(2l+1)(l-m)!\over2\pi(l+m)!(1+\delta_{0m})}\bigg]^{1/2}P_l^m(\cos\theta)\matrix{\cos m\varphi\cr \sin m\varphi},\eqno (6.1.1.22)]$ where $[m=0,1,2,\ldots]$ .

These harmonics can also be expressed in terms of Cartesian components of a unit vector [q_x,q_y,q_z] .

The normalization in (6.1.1.17) is appropriate to wavefunctions. The physical significance of the normalization for the spherical harmonics depends on the context in which they are utilized. The implications for density functions are not the same as those for wavefunctions. A normalizing condition on the real form of the spherical harmonics that expresses the properties of the functions under integration is $[\textstyle\int|y(\theta,\varphi)|\,{\rm d}(\cos\theta)\,{\rm d}\varphi=2-\delta_{l0}.\eqno (6.1.1.23)]$ We assume the radial function to be constant in sign, and normalized to unity. The scalar function, with l = 0, does not change sign. Integration over the angular coordinates gives the electron content of the scalar function. The multipole terms with l > 0 integrate to zero. Taking the modulus of the angular function, and then integrating, gives twice the electron transfer from the electron-deficient to the electron-enriched volume for that multipole. With this normalization, the angle-dependent factors in the expansion, in terms of the associated Legendre polynomials and in terms of direction cosines, are given in Table 6.1.1.6. For the alternative normalization such that $[\textstyle\int|y_{lmp}|{}^2\,{\rm d}(\cos\theta)\,{\rm d}\varphi=1,]$ the factor multiplying the angle-dependent term is as given in (6.1.1.22).

Table 6.1.1.6| top | pdf |
Angle dependence of multipole functions, normalized as in equation (6.1.1.23); ω = cos $[\theta]$ and S, D, Q, O, H denote scalar, dipole, quadrupole, octupole, and hexadecapole terms, respectively

Pole	Real spherical harmonic	Cartesian representation
S 1	$[{1\over4\pi}P^0_0(\omega)]$	$[{1\over4\pi}]$
D 1	$[{1\over\pi}P^1_1(\omega)\cos\varphi]$	$[{1\over\pi}q_x]$
D 2	$[{1\over\pi}P^1_1(\omega)\sin\varphi]$	$[{1\over\pi}q_y]$
D 3	$[{1\over\pi}P^0_1(\omega)]$	$[{1\over\pi}q_z]$
Q 1	$[{1\over8}P^2_2(\omega)\cos2\varphi]$	$[{3\over8}(q^2_x-q^2_y)]$
Q 2	$[{1\over8}P^2_2(\omega)\sin2\varphi]$	$[{3\over4}q_xq_y]$
Q 3	$[{1\over4}P^1_2(\omega)\cos\varphi]$	$[{3\over4}q_xq_z]$
Q 4	$[{1\over4}P^1_2(\omega)\sin\varphi]$	$[{3\over4}q_yq_z]$
Q 5	$[{3\surd3\over4\pi}P^0_2(\omega)]$	$[{9\surd2\over8\pi}(q^2_z-\textstyle{1\over3})]$
O 1	$[{4\over45\pi}P^3_3(\omega)\cos3\varphi]$	$[{4\over3\pi}(q^2_x-3q^2_y)q_x]$
O 2	$[{4\over45\pi}P^3_3(\omega)\sin3\varphi]$	$[{4\over3\pi}(3q^2_x-q^2_y)q_y]$
O 3	$[\textstyle{1\over15}P^2_3(\omega)\cos2\varphi]$
O 4	$[\textstyle{1\over15}P^2_3(\omega)\sin2\varphi]$
O 5	$[\textstyle{2\over3}\bigg[\tan^{-1}2+\displaystyle{14\over5}-{\pi\over4}\bigg]^{-1}P^1_3(\omega)\cos\varphi]$	$[\bigg[\tan^{-1}2+\displaystyle{14\over5}-{\pi\over4}\bigg]^{-1}(5q^2_z-1)q_x]$
O 6	$[\textstyle{2\over3}\bigg[\tan^{-1}2+\displaystyle{14\over5}-{\pi\over4}\bigg]^{-1}P^1_3(\omega)\sin\varphi]$	$[\bigg[\tan^{-1}2+\displaystyle{14\over5}-{\pi\over4}\bigg]^{-1}(5q^2_z-1)q_y]$
O 7	$[{20\over13\pi}P^0_3(\omega)]$	$[{10\over13\pi}(5q^2_z-3)q_z]$
H 1	$[{\textstyle{1\over224}}P^4_4(\omega)\cos4\varphi]$	$[{\textstyle{105\over224}}(q^4_x-6q^2_xq^2_y+q^4_y)]$
H 2	$[{\textstyle{1\over224}}P^4_4(\omega)\sin4\varphi]$	$[{\textstyle{420\over224}}(q^2_x-q^2_y)q_xq_y]$
H 3	$[{\textstyle{1\over84}}P^3_4(\omega)\cos3\varphi]$	$[{\textstyle{105\over84}}(q^2_x-3q^2_y)q_xq_z]$
H 4	$[{\textstyle{1\over84}}P^3_4(\omega)\sin3\varphi]$	$[{\textstyle{105\over84}}(3q^2_x-q^2_y)q_yq_z]$
H 5	$[\bigg({7\surd7\over272+56\surd7}\bigg)P^2_4(\omega)\cos2\varphi]$	$[{15\over2}\bigg({7\surd7\over272+56\surd7}\bigg)(7q^2_z\!-\!1)(q^2_x\!-\!q^2_y)]$
H 6	$[\bigg({7\surd7\over272+56\surd7}\bigg)P^2_4(\omega)\sin2\varphi]$	$[{15\over2}\bigg({7\surd7\over272+56\surd7}\bigg)(7q^2_z\!-\!1)q_xq_y]$
H 7	$[\bigg({21\surd7\over256+14\surd7}\bigg)P^1_4(\omega)\cos\varphi]$	$[{5\over2}\bigg({21\surd7\over256+14\surd7}\bigg)(7q^2_z-3)q_xq_z]$
H 8	$[\bigg({21\surd7\over256+14\surd7}\bigg)P^1_4(\omega)\sin\varphi]$	$[{5\over2}\bigg({21\surd7\over256+14\surd7}\bigg)(7q^2_z-3)q_xq_z]$
H 9	$[0.55534 P^0_4(\omega)]$	$[\textstyle{5\over8}(0.55534)(7q^4_z-6q^2_z+{3\over5})]$
^† $[H_{\rm cubic}]$	$[{160\over27\surd3\pi}\bigg[{1\over420}P^4_4(\omega)\cos4\varphi+\textstyle{2\over5}P^0_4(\omega)\bigg]]$	$[{160\over27\surd3\pi}(q^4_x+q^4_y+q^4_z-3/5)]$

^† $[H_{\rm cubic}]$ is the fourth-order hexadecapole appropriate to cubic site symmetry.

The site symmetry of the atom restricts multipole terms to those that are invariant under the operations of the relevant point group. The restrictions for the 27 non-cubic crystallographic point groups are given in Table 6.1.1.7.

Table 6.1.1.7| top | pdf |
Indices allowed by the site symmetry for the real form of the spherical harmonics $[Y_{lmp(\theta,\varphi)}]$ ; λ, μ and j are integers such that l, m ≥ 0; (−)ⁿ implies p = − for n odd and p = + for n even

Site symmetry	Coordinate axes	Indices
1	Any	All
$[\bar1]$	Any	$[(2\lambda,m,p)]$
2	$[2\parallel x]$	$[(l,m,(-)^{l-m})]$
	$[2\parallel y]$	$[(l,m,(-)^{l})]$
	$[2\parallel z]$	$[(l,2\mu,p)]$
m	$[m\,\bot\, x]$	$[(l,m,(-)^{m})]$
	$[m\,\bot\, y]$
	$[m\,\bot\, z]$
	$[2\parallel x,m\,\bot\,x]$	$[(2\lambda ,m,(-)^{m})]$
	$[2\parallel y,m\,\bot\,y]$	$[(2\lambda,m,+)]$
	$[2\parallel z,m\,\bot\,z]$	$[(2\lambda,2\mu,p)]$
222	$[2\parallel z,2\parallel y]$	$[(l,2\mu,(-)^l)]$
	$[2\parallel x,m\,\bot\,z]$
	$[2\parallel y,m\,\bot\,z]$
	$[2\parallel z,m\,\bot\,y]$	$[(l,2\mu,+)]$
	$[m\,\bot\,z,m\,\bot\,y,m\,\bot\,z]$	$[(2\lambda,2\mu,+)]$
4	$[4\parallel z]$	$[(l,4\mu,p)]$
$[\bar4]$	$[\bar4\parallel z]$
	$[4\parallel z,m\,\bot\,z]$	$[(2\lambda,4\mu,p)]$
422	$[4\parallel z,2\parallel y]$	$[(l,4\mu,(-)^l)]$
	$[4\parallel z,m\,\bot\,y]$	$[(l,4\mu,+)]$
$[\bar42m]$	$[\bar4\parallel z,2\parallel x]$
$[\bar42m]$	$[\bar4\parallel z,m\,\bot\,y]$
	$[4\parallel z,m\,\bot\,z,m\,\bot\,x]$	$[(2\lambda,4\mu,+)]$
3	$[3\parallel z]$	$[(l,3\mu,p)]$
$[\bar3]$	$[\bar3\parallel z]$	$[(2\lambda,3\mu,p)]$
32	$[3\parallel z,2\parallel y]$	$[(l,3\mu,(-)^l)]$
32	$[3\parallel z,2\parallel x]$	$[(l,3\mu,(-)^{l-m})]$
	$[3\parallel z,m\,\bot\,y]$	$[(l,3\mu,+)]$
	$[3\parallel z,m\,\bot\,x]$	$[(l,3\mu,(-)^m)]$
$[\bar3m]$	$[\bar3\parallel z,m\,\bot\,y]$	$[(2\lambda,3\mu,+)]$
$[\bar3m]$	$[\bar3\parallel z,m\,\bot\,x]$	$[(2\lambda,3\mu,(-)^m)]$
6	$[6\parallel z]$	$[(l,6\mu,p)]$
$[\bar6]$	$[\bar6\parallel z]$	$[(m+2j,3\mu,p)]$
	$[6\parallel z,m\,\bot\,z]$	$[(2\lambda,6\mu,p)]$
622	$[6\parallel z,2\parallel y]$	$[(l,6\mu,(-)^l)]$
	$[6\parallel z,m\,\bot\,y]$	$[(l,6\mu,+)]$
$[\bar6m2]$	$[\bar6\parallel z,m\,\bot\,y]$	$[(m+2j,3\mu,+)]$
$[\bar6m2]$	$[\bar6\parallel z,m\,\bot\,x]$	$[(m+2j,3\mu,(-)^l)]$
	$[6\parallel z,m\,\bot\,z,m\,\bot\,y]$	$[(2\lambda,6\mu,+)]$

For the five cubic point groups, the functions allowed are the linear combinations of the $[Y_{lmp}(\theta,\varphi)]$ known as the cubic harmonics $[K_{l\,j}(\theta,\varphi)]$ (Altmann & Cracknell, 1965). These are listed in Table 6.1.1.8. The normalization constant $[N^2_{l\,j}]$ is given by $[N^2_{l\,j}=\int K^2_{l\,j}\,{\rm d}(\cos\theta)\,{\rm d}\varphi.]$ The derivation of Tables 6.1.1.7 and 6.1.1.8 is described by Kurki-Suonio (1977).

Table 6.1.1.8| top | pdf |
Cubic harmonics $[K_{lj}(\theta,\,\varphi)]$ for cubic site symmetries

$[K_{lj}(\theta,\varphi)]$	$[N_{l^{2}j}]$	Site symmetry
$[K_{lj}(\theta,\varphi)]$	$[N_{l^{2}j}]$	23	m3	432	$[{\bar 4}3m]$	m3m
$[K_0 = Y_{00+} = 1]$	$[4\pi]$	×	×	×	×	×
$[K_3 = Y_{32-}]$	$[\displaystyle{{240\pi}\over{7}}]$	×			×
$[K_4 = Y_{40+} + {{1}\over{168}} \, Y_{44+}]$	$[\displaystyle{{16\pi}\over{21}}]$	×	×	×	×	×
$[K_{6,1} = Y_{60+} - {{1}\over{360}}\,Y_{64+}]$	$[\displaystyle{{32\pi}\over{13}}]$	×	×	×	×	×
$[K_{6,2} = Y_{62+} - {{1}\over{792}}Y_{66+}]$	$[\displaystyle{{512\pi}\over{13}} \cdot {{105}\over{11}}]$	×	×
$[K_7 = Y_{72-} + {{1}\over{1560}}Y_{76-}]$	$[\displaystyle{{256\pi}\over{15}} \cdot {{567}\over{13}}]$	×			×
$[K_8 = Y_{80+} + {{1}\over{5940}}\,\,(Y_{84+} + {{1}\over{672}}Y_{88+})]$	$[\displaystyle{{256\pi}\over{17 \cdot 33}}]$	×	×	×	×	×
$[K_{9,1} = Y_{92-} - {{1}\over{2520}}Y_{96-}]$	$[\displaystyle{{512\pi}\over{19}} \cdot 165]$	×			×
$[K_{9,2} = Y_{94-} - {{1}\over{4080}}Y_{98-}]$	$[\displaystyle{{2048\pi}\over{19}} \cdot {{243 \cdot 5005}\over{17}}]$	×		×
$[K_{10,1} = Y_{10,0+} - {{1}\over{5460}}(Y_{10,4} + {{1}\over{4320}}Y_{10,8+})]$	$[\displaystyle{{512\pi}\over{21}} \cdot {{3}\over{65}}]$	×	×	×	×	×
$[K_{10,2} = Y_{10,2+} + {{1}\over{43680}} (Y_{10,6+} + {{1}\over{456}}Y_{10,10+})]$	$[\displaystyle{{2048\pi}\over{21}} \cdot {{4455}\over{247}}]$	×	×

The generalized scattering factor for a particular multipole involves evaluating the Fourier transform of the density $[\textstyle\int\exp(i{\bf S}\cdot{\bf r})\rho_{lm\pm}(r)Y_{lm\pm}(\theta,\varphi)\,{\rm d}{\bf r}=f_{lm\pm}(S)Y_{lm}(\theta_S,\varphi_S),]$ where the right-hand side is obtained by substituting (6.1.1.17) and integrating over the angular coordinates for the direct-space variables. The term $[f_{lm\pm}(S)=\textstyle\int\limits^\infty_0j_l(Sr)\rho_{lm\pm}(r)r^2\,{\rm d}r\eqno (6.1.1.24)]$ gives the radial variation of the generalized scattering factor.

The density function $[\rho_{lm\pm}(r_a)]$ may be derived from atomic basis functions, which asymptotically have the form of simple exponential functions $[A_nr^n\exp(-\alpha r)]$ . Expansions in terms of Gaussian functions $[B_nr^n\exp(-\beta r^2)]$ or of Laguerre functions $[C_nr^lL_n^{2l+2}\exp(-\gamma r/2)]$ , where L is a Laguerre polynomial of order n and degree [2l+2] , are also convenient for some purposes. [A_n] , [B_n] and [C_n] are normalizing factors, which, when specified as $[\eqalignno{&A_n={\alpha^{l+n+3}\over4\pi(l+n+2)!},\quad B_n={2^{\beta(l+n+3)/2}\over{\Gamma}[(l+n+3)/2]},\cr &\qquad \qquad \quad C_n={(-)^n n!(\gamma/2)^{2l+3}\over4\pi(2l+n+2)!},& (6.1.1.25)}]$ impose the normalization condition (Stewart, 1980a) $[\textstyle\int\limits^\infty_0\rho_{lm}(r_a)r_a^{l+2}\,{\rm d}r_a=1.\eqno (6.1.1.26)]$ With this normalization, the Fourier–Bessel transforms are, for the simple exponential, $[\eqalignno{f_{nl}(\alpha,S)&={S\over(2l+1)!![1+(S/\alpha)^2]^{n+2}}\cr &\quad\times {_2F_1}\bigg[{l-n-1\over2},{l-n\over2};l+{3\over2};-(S/\alpha)^2\bigg]; \cr&&(6.1.1.27)}]$ for the Gaussian function, $[g_{nl}(\beta,S)={S^1\over(2l+1)!!}\exp(-S^2/4\beta)_1F_1\bigg[{l-n\over2};l+{3\over2};{S^2\over4\beta}\bigg];\eqno (6.1.1.28)]$ and, for the Laguerre function, $[h_{nl}(\gamma,S)={(-)^nn!2^nS^l\over[2(l+n)+1]!![1+(2S/\gamma){}^2]{}^{l+2}}P_n^{(l+{3\over2},l+{1\over2})}(t);]$ where the Jacobi polynomial is given by $[\eqalign{P_n^{(a,b)}(x)&=2^{-n}\sum^n_{m=0}\bigg(\matrix{n+a\cr m\cr} \bigg)\bigg(\matrix{n+b\cr n-m\cr} \bigg)(x-1)^{n-m}(x+1)^m\cr &={{\Gamma}(a+n+1)\over n!{\Gamma}(a+b+n+1)}\cr &\quad\times\sum^n_{m=0}\bigg(\matrix{n\cr m\cr}\bigg)\displaystyle{{\Gamma}(a+b+n+m+1)\over2^m{\Gamma}(a+m+1)}(x-1)^m\cr &=\bigg(\matrix{n+a\cr n\cr}\bigg)\,{_2F_1}\bigg(-n,n+a+b+1;a+1;{1-x\over2}\bigg) \cr&\kern145pt\hfill a\ge-1,b\ge a}]$ and $[t={[(2S/\gamma){}^2-1]\over[(2S/\gamma){}^2+1]}.\eqno (6.1.1.29)]$ Further details are given by Stewart (1980a).

In the case of Slater-type orbitals , a simpler form of the radial term may be obtained via the recurrence relations (Avery & Watson, 1977) $[\eqalign{(S^2+\alpha^2) &f_{\mu+1,\nu}+(\mu+\nu)(\mu-\nu-1) f_{\mu-1,\nu}=2\nu \alpha f_{\mu\nu}\cr &S f_{\mu,\nu-1}+(\mu-\nu-1) f_{\mu-1,\nu}=\alpha f_{\mu\nu}.}]$ Thus, for the lower-order Slater-type functions, we obtain the values listed in Table 6.1.1.9.

Table 6.1.1.9| top | pdf |
f_nl(α, S) = ∫^∞₀rⁿ exp(−αr)j_l(Sr) dr

n	1	2	3	4
0	$[\displaystyle{1\over(S^2+\alpha^2)}]$	$[\displaystyle{2\alpha\over(S^2+\alpha^2)^2}]$	$[\displaystyle{2(3\alpha^2-S^2)\over(S^2+\alpha^2)^3}]$	$[\displaystyle{24\alpha(\alpha^2-S^2)\over(S^2+\alpha^2)^4}]$
1		$[\displaystyle{2S\over(S^2+\alpha^2)^2}]$	$[\displaystyle{8S\alpha\over(S^2+\alpha^2)^3}]$	$[\displaystyle{8S(5\alpha^2-S^2)\over(S^2+\alpha^2)^4}]$
2			$[\displaystyle{8S^2\over(S^2+\alpha^2)^3}]$	$[\displaystyle{48S^2\alpha\over(S^2+\alpha^2)^4}]$
3				$[\displaystyle{48S^3\over(S^2+\alpha^2)^4}]$

Atomic wavefunctions, in the form of sets of orbital contributions using Slater-type functions, are tabulated by Clementi & Roetti (1974 ). Basis sets for Gaussian orbitals are described by Veillard (1968), Roos & Siegbahn (1970), Huzinaga (1971), van Duijneveldt (1971), Dunning & Jeffrey-Hay (1977), and by McLean & Chandler (1979, 1980). The application of these basis sets to molecular calculations is reviewed by Ahlrichs & Taylor (1981).

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International Tables for Crystallography (2006). Vol. C. ch. 6.1, pp. 565-584