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

International Tables for Crystallography (2006). Vol. B, ch. 1.2, pp. 14-15   | 1 | 2 |

Section 1.2.7. Beyond the spherical-atom description: the atom-centred spherical harmonic expansion

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

1.2.7. Beyond the spherical-atom description: the atom-centred spherical harmonic expansion

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1.2.7.1. Direct-space description of aspherical atoms

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Even though the spherical-atom approximation is often adequate, atoms in a crystal are in a non-spherical environment; therefore, an accurate description of the atomic electron density requires non-spherical density functions. In general, such density functions can be written in terms of the three polar coordinates r, θ and ϕ. Under the assumption that the radial and angular parts can be separated, one obtains for the density function: [\Phi (r,\theta,\varphi) = R(r)\Theta (\theta,\varphi). \eqno(1.2.7.1)]

The angular functions Θ are based on the spherical harmonic functions [Y_{lm}] defined by [Y_{lm} (\theta, \varphi) = (-1)^{m} \left[\left({2l + 1 \over 4\pi}\right) {(l - |m|)! \over (l + |m|)!}\right]^{1/2} P_{l}^{m} (\cos \theta) \exp (im \varphi), \eqno(1.2.7.2a)] with [-l \leq m \leq l], where [P_{l}^{m} (\cos \theta)] are the associated Legendre polynomials (see Arfken, 1970[link]). [\eqalign{P_{l}^{m} (x) &= (1 - x^{2})^{|m|/2} {\hbox{d}^{|m|}P_{l}(x) \over \hbox{d}x^{|m|}},\cr P_{l} (x) &= {1 \over l!2^{l}} {\hbox{d}^{l} \over \hbox{d}x^{l}} \left[(x^{2} - 1)^{l}\right].}]

The real spherical harmonic functions [y_{lmp}], [0 \leq m \leq l], [p = + \hbox{ or } -] are obtained as a linear combination of [Y_{lm}]: [\eqalignno{y_{lm+} (\theta, \psi) &= \left[{(2l + 1)(l - |m|)! \over 2\pi (1 + \delta_{m0}) (l + | m |)!}\right]^{1/2} P_{l}^{m} (\cos \theta) \cos m\varphi\cr &= N_{lm} P_{l}^{m} (\cos \theta)\cos m\varphi\cr & = (-1)^{m} (Y_{lm} + Y_{l, \, -m}) &(1.2.7.2b)}] and [\eqalignno{y_{lm-} (\theta, \psi) &= N_{lm} P_{l}^{m} (\cos \theta)\sin m\varphi\cr &= (-1)^{m} (Y_{lm} - Y_{l, \, -m})/2i.& (1.2.7.2c)\cr}] The normalization constants [N_{lm}] are defined by the conditions [{\textstyle\int} y_{lmp}^{2} \hbox{ d}\Omega = 1, \eqno(1.2.7.3a)] which are appropriate for normalization of wavefunctions. An alternative definition is used for charge-density basis functions: [{\textstyle\int} |d_{lmp}| \hbox{ d}\Omega = 2 \hbox{ for } l \;\gt\; 0 \hbox{ and } \textstyle\int |d_{lmp}| \hbox{ d}\Omega = 1 \hbox{ for } l = 0. \eqno(1.2.7.3b)] The functions [y_{lmp}] and [d_{lmp}] differ only in the normalization constants. For the spherically symmetric function [d_{00}], a population parameter equal to one corresponds to the function being populated by one electron. For the non-spherical functions with [l \;\gt\; 0], a population parameter equal to one implies that one electron has shifted from the negative to the positive lobes of the function.

The functions [y_{lmp}] and [d_{lmp}] can be expressed in Cartesian coordinates, such that [y_{lmp} = M_{lm} c_{lmp} \eqno(1.2.7.4a)] and [d_{lmp} = L_{lm} c_{lmp}, \eqno(1.2.7.4b)] where the [c_{lmp}] are Cartesian functions. The relations between the various definitions of the real spherical harmonic functions are summarized by [Scheme scheme1] in which the direction of the arrows and the corresponding conversion factors [X_{lm}] define expressions of the type (1.2.7.4)[link] [link]. The expressions for [c_{lmp}] with [l \leq 4] are listed in Table 1.2.7.1[link], together with the normalization factors [M_{lm}] and [L_{lm}].

Table 1.2.7.1| top | pdf |
Real spherical harmonic functions (x, y, z are direction cosines)

lSymbol CAngular function, [c_{lmp}]Normalization for wavefunctions, [M_{lmp}]§Normalization for density functions, [L_{lmp}]
ExpressionNumerical valueExpressionNumerical value
0 00 1 1 [(1/4\pi)^{1/2}] 0.28209 [1/4\pi] 0.07958
1 [\!\matrix{11+\cr 11-\cr 10\hfill\cr}] [\!\matrix{1\cr 1\cr 1\cr}] [\left.\!\matrix{x\cr y\cr z\cr}\right\}] [(3/4\pi)^{1/2}] 0.48860 [1/\pi] 0.31831
2 20 [1/2] [3z^{2} - 1] [(5/16\pi)^{1/2}] 0.31539 [\displaystyle{3\sqrt{3} \over 8\pi}] 0.20675
[\!\matrix{21+\cr 21-\cr 22+\cr 22-\cr}] [\!\matrix{3\cr 3\cr 6\cr 6\cr}] [\left.\!\matrix{xz\cr yz\cr (x^{2} - y^{2})/2\cr xy\cr}\right\}] [(15/4\pi)^{1/2}] 1.09255 [3/4] 0.75
3 30 [1/2] [5z^{3} - 3z] [(7/16\pi)^{1/2}] 0.37318 [\displaystyle{10 \over 13\pi}] 0.24485
[\!\matrix{31+\cr 31-\cr}] [\!\matrix{3/2\cr 3/2\cr}] [\left.\!\matrix{x[5z^{2} - 1]\cr y[5z^{2} - 1]\cr}\right\}] [(21/32\pi)^{1/2}] 0.45705 [\displaystyle\left(\hbox{ar} + {14 \over 5} - {\pi \over 4}\right)^{-1}] †† 0.32033
[\!\matrix{32+\cr 32-\cr}] [\!\matrix{15\cr 15\cr}] [\left.\!\matrix{(x^{2} - y^{2})z\cr 2xyz\cr}\right\}] [(105/16\pi)^{1/2}] 1.44531 1 1
[\!\matrix{33+\cr 33-\cr}] [\!\matrix{15\cr 15\cr}] [\left.\!\matrix{x^{3} - 3xy^{2}\cr -y^{3} + 3x^{2}y\cr}\right\}] [(35/32\pi)^{1/2}] 0.59004 [4/3\pi] 0.42441
4 40 [1/8] [35z^{4} - 30z^{2} + 3] [(9/256\pi)^{1/2}] 0.10579 ‡‡ 0.06942
[\!\matrix{41+\cr 41-\cr}] [\!\matrix{5/2\cr 5/2\cr}] [\left.\!\matrix{x[7z^{3} - 3z]\cr y[7z^{3} - 3z]\cr}\right\}] [(45/32\pi)^{1/2}] 0.66905 [\displaystyle{735 \over 512\sqrt{7} + 196}] 0.47400
[\!\matrix{42+\cr 42-\cr}] [\!\matrix{15/2\cr 15/2\cr}] [\left.\!\matrix{(x^{2} - y^{2})[7z^{2} - 1]\cr 2xy[7z^{2} - 1]\cr}\right\}] [(45/64\pi)^{1/2}] 0.47309 [\displaystyle{105\sqrt{7} \over 4(136 + 28\sqrt{7})}] 0.33059
[\!\matrix{43+\cr 43-\cr}] [\!\matrix{105\cr 105\cr}] [\left.\!\matrix{(x^{3} - 3xy^{2})z\cr (-y^{3} + 3x^{2}y)z\cr}\right\}] [(315/32\pi)^{1/2}] 1.77013 [5/4] 1.25
[\!\matrix{44+\cr 44-\cr}] [\!\matrix{105\cr 105\cr}] [\left.\!\matrix{x^{4} - 6x^{2}y^{2} + y^{4}\cr 4x^{3}y - 4xy^{3}\cr}\right\}] [(315/256\pi)^{1/2}] 0.62584 [15/32] 0.46875
5 50 [1/8] [63z^{5} - 70z^{3} - 15z] [(11/256\pi)^{1/2}] 0.11695 0.07674
[\!\matrix{51+\cr 51-\cr}] [15/8] [\left.\!\matrix{(21z^{4} - 14z^{2} + 1)x\cr (21z^{4} - 14z^{2} + 1)y\cr}\right\}] [(165/256\pi)^{1/2}] 0.45295 0.32298
[\!\matrix{52+\cr 52-\cr}] [105/2] [\left.\!\matrix{(3z^{3} - z) (x^{2} - y^{2})\cr 2xy(3z^{3} - z)\cr}\right\}] [(1155/64\pi)^{1/2}] 2.39677 1.68750
[\!\matrix{53+\cr 53-\cr}] [105/2] [\left.\!\matrix{(9z^{2} - 1) (x^{3} - 3xy^{2})\cr (9z^{2} - 1) (3x^{2}y - y^{3})\cr}\right\}] [(385/512\pi)^{1/2}] 0.48924 0.34515
[\!\matrix{54+\cr 54-\cr}] [945] [\left.\!\matrix{z(x^{4} - 6x^{2}y^{2} + y^{4})\cr z(4x^{3}y - 4xy^{3})\cr}\right\}] [(3465/256\pi)^{1/2}] 2.07566 1.50000
[\!\matrix{55+\cr 55-\cr}] [945] [\left.\!\matrix{x^{5} - 10x^{3}y^{2} + 5xy^{4}\cr 5x^{4} y - 10x^{2}y^{3} + y^{5}\cr}\right\}] [(693/512\pi)^{1/2}] 0.65638 0.50930
6 60 [1/16] [231z^{6} - 315z^{4} + 105z^{2} - 5] [(13/1024\pi)^{1/2}] 0.06357 0.04171
[\!\matrix{61+\cr 61-\cr}] [21/8] [\left.\!\matrix{(33z^{5} - 30z^{3} + 5z)x\cr (33z^{5} - 30z^{3} + 5z)y\cr}\right\}] [(273/256\pi)^{1/2}] 0.58262 0.41721
[\!\matrix{62+\cr 62-\cr}] [105/8] [\left.\!\matrix{(33z^{4} - 18z^{2} + 1) (x^{2} - y^{2})\cr 2xy (33z^{4} - 18z^{2} + 1)\cr}\right\}] [(1365/2048\pi)^{1/2}] 0.46060 0.32611
[\!\matrix{63+\cr 63-\cr}] [315/2] [\left.\!\matrix{(11z^{3} - 3z) (x^{3} - 3xy^{2})\cr (11z^{3} - 3z) (3x^{2}y - 3y)\cr}\right\}] [(1365/512\pi)^{1/2}] 0.92121 0.65132
[\!\matrix{64+\cr 64-\cr}] [945/2] [\left.\!\matrix{(11z^{2} - 1) (x^{4} - 6x^{2}y^{2} + y^{4})\cr (11z^{2} - 1) (4x^{3}y - 4xy^{3})\cr}\right\}] [(819/1024\pi)^{1/2}] 0.50457 0.36104
[\!\matrix{65+\cr 65-\cr}] 10395 [\left.\!\matrix{z(x^{5} - 10x^{3}y^{2} + 5xy^{4})\cr z(5x^{4}y - 10x^{2}y^{3} + y^{5})\cr}\right\}] [(9009/512\pi)^{1/2}] 2.36662 1.75000
[\!\matrix{66+\cr 66-\cr}] 10395 [\left.\!\matrix{x^{6} - 15x^{4}y^{2} + 15x^{2}y^{4} - y^{6}\cr 6x^{5}y - 20x^{3}y^{3} + 6xy^{5}\cr}\right\}] [(3003/2048\pi)^{1/2}] 0.68318 0.54687
7 70 [1/16] [429z^{7} - 693z^{5} + 315z^{3} - 35z] [(15/1024\pi)^{1/2}] 0.06828 0.04480
[\!\matrix{71+\cr 71-\cr}] [7/16] [\left.\!\matrix{(429z^{6} - 495z^{4} + 135z^{2} - 5)x\cr (429z^{6} - 495z^{4} + 135z^{2} - 5)y\cr}\right\}] [(105/4096\pi)^{1/2}] 0.09033 0.06488
[\!\matrix{72+\cr 72-\cr}] [63/8] [\left.\!\matrix{(143z^{5} - 110z^{3} + 15z) (x^{2} - y^{2})\cr 2xy(143z^{5} - 110z^{3} + 15z)\cr}\right\}] [(315/2048\pi)^{1/2}] 0.22127 0.15732
[\!\matrix{73+\cr 73-\cr}] [315/8] [\left.\!\matrix{(143z^{4} - 66z^{2} + 3) (x^{3} - 3xy^{2})\cr (143z^{4} - 66z^{2} + 3) (3x^{2}y - y^{3})\cr}\right\}] [(315/4096\pi)^{1/2}] 0.15646 0.11092
[\!\matrix{74+\cr 74-\cr}] [3465/2] [\left.\!\matrix{(13z^{3} - 3z) (x^{4} - 6x^{2}y^{2} + y^{4})\cr (13z^{3} - 3z) (4x^{3}y - 4xy^{3})\cr}\right\}] [(3465/1024\pi)^{1/2}] 1.03783 0.74044
[\!\matrix{75+\cr 75-\cr}] [10395/2] [\left.\!\matrix{(13z^{3} - 1) (x^{5} - 10x^{3}y^{2} + 5xy^{4})\cr (13z^{3} - 1) (5x^{4}y - 10x^{2}y^{3} + y^{5})\cr}\right\}] [(3465/4096\pi)^{1/2}] 0.51892 0.37723
[\!\matrix{76+\cr 76-\cr}] 135135 [\left.\!\matrix{z(x^{6} - 15x^{4}y^{2} + 15x^{2}y^{4} - y^{6})\cr z(6x^{5}y + 20x^{3}y^{3} - 6xy^{5})\cr}\right\}] [(45045/2048\pi)^{1/2}] 2.6460 2.00000
[\!\matrix{77+\cr 77-\cr}] 135135 [\left.\!\matrix{x^{7} - 21x^{5}y^{2} + 35x^{3}y^{4} - 7xy^{6}\cr 7x^{6}y - 35x^{4}y^{3} + 21x^{2}y^{5} - y^{7}\cr}\right\}] [(6435/4096\pi)^{1/2}] 0.70716 0.58205
Common factor such that [C_{lm}c_{lmp} = P_{l}^{m} (\cos \theta)_{\sin m\varphi}^{\cos m\varphi}.]
[x = \sin \theta \cos \varphi], [y = \sin \theta \sin \varphi], [z = \cos \theta].
§As defined by [y_{lmp} = M_{lmp}c_{lmp}] where [c_{lmp}] are Cartesian functions.
Paturle & Coppens (1988)[link], as defined by [d_{lmp} = L_{lmp}c_{lmp}] where [c_{lmp}] are Cartesian functions.
††ar = arctan (2).
‡‡[N_{\rm ang} = \{(14A_{-}^{5} - 14A_{+}^{5} + 20A_{+}^{3} - 20A_{-}^{3} + 6A_{-} - 6A_{+}) 2\pi\}^{-1}] where [A_{\pm} = [(30\pm \sqrt{480})/70]^{1/2}].

The spherical harmonic functions are mutually orthogonal and form a complete set, which, if taken to sufficiently high order, can be used to describe any arbitrary angular function.

The spherical harmonic functions are often referred to as multipoles since each represents the components of the charge distribution [\rho ({\bf r})], which gives non-zero contribution to the integral [\Theta_{lmp} = {\textstyle\int} \rho ({\bf r}) c_{lmp} r^{l}\; \hbox{d}{\bf r}], where [\Theta_{lmp}] is an electrostatic multipole moment. Terms with increasing l are referred to as monopolar [(l = 0)], dipolar [(l = 1)], quadrupolar [(l = 2)], octapolar [(l = 3)], hexadecapolar [(l = 4)], triacontadipolar [(l = 5)] and hexacontatetrapolar [(l = 6)].

Site-symmetry restrictions for the real spherical harmonics as given by Kara & Kurki-Suonio (1981)[link] are summarized in Table 1.2.7.2[link].

Table 1.2.7.2| top | pdf |
Index-picking rules of site-symmetric spherical harmonics (Kara & Kurki-Suonio, 1981[link])

λ, μ and j are integers.

SymmetryChoice of coordinate axesIndices of allowed [y_{lmp}], [d_{lmp}]
1 Any [{\rm All}\;(l, m, \pm)]
[\bar{1}] Any [(2\lambda, m, \pm)]
2 [2\!\!\parallel\!\!z] [(l, 2\mu, \pm)]
m [m\perp z] [(l, l-2j, \pm)]
[2/m] [2\!\!\parallel\!\! z, m\perp z] [(2\lambda, 2\mu, \pm)]
222 [2\!\!\parallel\!\! z, 2\!\!\parallel\!\! y] [(2\lambda, 2\mu, +)], [(2\lambda + 1, 2\mu, -)]
mm2 [2\!\!\parallel\!\! z, m\perp y] [(l, 2\mu, +)]
mmm [m\perp z, m\perp y, m\perp x] [(2\lambda, 2\mu, +)]
4 [4\!\!\parallel\!\! z] [(l, 4\mu, \pm)]
[\bar{4}] [\bar{4}\!\!\parallel\!\! z] [(2\lambda, 4\mu, \pm)], [(2\lambda + 1, 4\mu + 2, \pm)]
[4/m] [4\!\!\parallel\!\! z, m\perp z] [(2\lambda, 4\mu, \pm)]
422 [4\!\!\parallel\!\! z, 2\!\!\parallel\!\! y] [(2\lambda, 4\mu, +)], [(2\lambda + 1, 4\mu, -)]
4mm [4\!\!\parallel\!\! z, m\perp y] [(l, 4\mu, +)]
[\bar{4}]2m [\bar{4}\!\!\parallel\!\! z, 2\!\!\parallel\!\! x] [(2\lambda, 4\mu, +)], [(2\lambda + 1, 4\mu + 2, -)]
  [m\perp y] [(2\lambda, 4\mu, +)], [(2\lambda + 1, 4\mu + 2, +)]
[4/mmm] [4\!\!\parallel\!\! z, m\perp z, m\perp x] [(2\lambda, 4\mu, +)]
3 [3\!\!\parallel\!\! z] [(l, 3\mu, \pm)]
[\bar{3}] [\bar{3}\!\!\parallel\!\! z] [(2\lambda, 3\mu, \pm)]
32 [3\!\!\parallel\!\! z, 2\!\!\parallel\!\! y] [(2\lambda, 3\mu, +), (2\lambda + 1, 3\mu, -)]
  [2\!\!\parallel\!\! x] [(3\mu + 2j, 3\mu, +)],
    [(3\mu + 2j + 1, 3\mu, -)]
3m [3\!\!\parallel\!\! z, m \perp y] [(l, 3\mu, +)]
  [m \perp x] [(l, 6\mu, +), (l, 6\mu + 3, -)]
[\bar{3}m] [\bar{3}\!\!\parallel\!\! z, m \perp y] [(2\lambda, 3\mu, +)]
  [m \perp x] [(2\lambda, 6\mu, +), (2\lambda, 6\mu + 3, -)]
6 [6\!\!\parallel\!\! z] [(l, 6\mu, \pm)]
[\bar{6}] [\bar{6}\!\!\parallel\!\! z] [(2\lambda, 6\mu, \pm), (2\lambda + 1, 6\mu + 3, \pm)]
[6/m] [6\!\!\parallel\!\! z, m \perp z] [(2\lambda, 6\mu, \pm)]
622 [6\!\!\parallel\!\! z, 2\!\!\parallel\!\! y] [(2\lambda, 6\mu, +), (2\lambda + 1, 6\mu, -)]
6mm [6\!\!\parallel\!\! z, m\!\!\parallel\!\! y] [(l, 6\mu, +)]
[\bar{6}m2] [\bar{6}\!\!\parallel\!\! z, m \perp y] [(2\lambda, 6\mu, +), (2\lambda + 1, 6\mu + 3, +)]
  [m \perp x] [(2\lambda, 6\mu, +), (2\lambda + 1, 6\mu + 3, -)]
[6/mmm] [6\!\!\parallel\!\! z, m \perp z, m \perp y] [(2\lambda, 6\mu, +)]

In cubic space groups, the spherical harmonic functions as defined by equations (1.2.7.2)[link] [link] [link] are no longer linearly independent. The appropriate basis set for this symmetry consists of the `Kubic Harmonics' of Von der Lage & Bethe (1947)[link]. Some low-order terms are listed in Table 1.2.7.3.[link] Both wavefunction and density-function normalization factors are specified in Table 1.2.7.3[link].

Table 1.2.7.3| top | pdf |
`Kubic Harmonic' functions

(a) Coefficients in the expression [K_{lj} = {\textstyle\sum\limits_{mp}} k_{mpj}^{l} y_{lmp}] with normalization [{\textstyle\int_{0}^{\pi}} {\textstyle\int_{0}^{2\pi}} |K_{lj}|^{2} \sin \theta\;\hbox{d}\theta \;\hbox{d}\varphi = 1] (Kara & Kurki-Suonio, 1981[link]).

Even l mp
l j0+2+4+6+8+10+
0 1 1          
4 1 [\textstyle{1 \over 2}\left({7 \over 3}\right)^{1/2}]   [\textstyle{1 \over 2}\left({5 \over 3}\right)^{1/2}]      
0.76376   0.64550      
6 1 [\textstyle{1 \over 2}\left({1 \over 2}\right)^{1/2}]   [\textstyle-{1 \over 2}\left({7 \over 2}\right)^{1/2}]      
0.35355   −0.93541      
6 2   [\textstyle{1 \over 4}11^{1/2}]   [\textstyle- {1 \over 4} 5^{1/2}]    
  0.82916   −0.55902    
8 1 [\textstyle{1 \over 8}33^{1/2}]   [\textstyle{1 \over 4}\left({7 \over 3}\right)^{1/2}]   [\textstyle{1 \over 8}\left({65 \over 3}\right)^{1/2}]  
0.71807   0.38188   0.58184  
10 1 [\textstyle{1 \over 8}\left({65 \over 6}\right)^{1/2}]   [\textstyle- {1 \over 4}\left({11 \over 2}\right)^{1/2}]   [\textstyle- {1 \over 8}\left({187 \over 6}\right)^{1/2}]  
0.41143   −0.58630   −0.69784  
10 2   [\textstyle{1 \over 8}\left({247 \over 6}\right)^{1/2}]   [\textstyle{1 \over 16}\left({19 \over 3}\right)^{1/2}]   [\textstyle{1 \over 16}85^{1/2}]
  0.80202   0.15729   0.57622
l j   2− 4− 6− 8−
3 1   1      
7 1   [\textstyle{1 \over 2}\left({13 \over 6}\right)^{1/2}]   [\textstyle{1 \over 2}\left({11 \over 16}\right)^{1/2}]  
  0.73598   0.41458  
9 1   [\textstyle{1 \over 4}3^{1/2}]   [\textstyle- {1 \over 4} 13^{1/2}]  
  0.43301   −0.90139  
9 2   [\textstyle{1 \over 2}\left({17 \over 6}\right)^{1/2}]   [\textstyle- {1 \over 2}\left({7 \over 6}\right)^{1/2}]  
  0.84163   −0.54006  

(b) Coefficients [k_{mpj}^{l}] and density normalization factors [N_{lj}] in the expression [K_{lj} = N_{lj} {\textstyle\sum\limits_{mp}} k_{mpj}^{l} u_{lmp}] where [u_{lm \pm} =P_{l}^{m} (\cos \theta)^{\cos m\varphi}_{\sin m\varphi}] (Su & Coppens, 1994[link]).

Even l [N_{lj}] mp
l j   0+ 2+ 4+ 6+ 8+ 10+
0 1 [1/4\pi = 0.079577] 1          
4 1 0.43454 1   [+1/168]      
6 1 0.25220 1   [-1/360]      
6 2 0.020833   1   [-1/792]    
8 1 0.56292 1   1/5940   [\textstyle{1 \over 672} \times {1 \over 5940}]  
10 1 0.36490 1   1/5460   [\textstyle{1 \over 4320} \times {1 \over 5460}]  
10 2 0.0095165 1     [1/43680]   [\textstyle- {1 \over 456} \times {1 \over 43680}]
l j     2− 4− 6− 8−
3 1 0.066667   1      
7 1 0.014612   1   [1/1560]  
9 1 0.0059569   1   [1/2520]  
9 2 0.00014800     1   [-1/4080]

(c) Density-normalized Kubic harmonics as linear combinations of density-normalized spherical harmonic functions. Coefficients in the expression [K_{lj} = {\textstyle\sum\limits_{mp}} k^{''l}_{mpj} d_{lmp}]. Density-type normalization is defined as [{\textstyle\int_{0}^{\pi}} {\textstyle\int_{0}^{2\pi}} |K_{lj}| \sin \theta\ \hbox{d} \theta\ \hbox{d} \varphi = 2 - \delta_{l0}].

Even l mp
l j0+2+4+6+8+10+
0 1 1          
4 1 0.78245   0.57939      
6 1 0.37790   −0.91682      
6 2   0.83848   −0.50000    
l j 2− 4− 6− 8−
3 1 1      
7 1 0.73145   0.63290  

(d) Index rules for cubic symmetries (Kurki-Suonio, 1977[link]; Kara & Kurki-Suonio, 1981[link]).

lj23 [m\bar{3}]432 [\bar{4}3m] [m\bar{3}m]
T [T_{h}] O [T_{d}] [O_{h}]
0 1 × × × × ×
3 1 ×     ×  
4 1 × × × × ×
6 1 × × × × ×
6 2 × ×      
7 1 ×     ×  
8 1 × × × × ×
9 1 ×     ×  
9 2 ×   ×    
10 1 × × × × ×
10 2 × ×      

A related basis set of angular functions has been proposed by Hirshfeld (1977)[link]. They are of the form [\cos^{n} \theta_{k}], where [\theta_{k}] is the angle with a specified set of [(n + 1)(n + 2)/2] polar axes. The Hirshfeld functions are identical to a sum of spherical harmonics with [l = n], [n - 2], [n - 4,\ldots (0, 1)] for [n \;\gt\; 1], as shown elsewhere (Hirshfeld, 1977[link]).

The radial functions [R(r)] can be selected in different manners. Several choices may be made, such as [{R_{l}(r) = {\zeta^{n_{l} + 3} \over (n_{l} + 2)!} r^{n(l)}\exp (-\zeta_{l}r) \qquad \hbox{(Slater type function)},} \eqno(1.2.7.5a)] where the coefficient [n_{l}] may be selected by examination of products of hydrogenic orbitals which give rise to a particular multipole (Hansen & Coppens, 1978[link]). Values for the exponential coefficient [\zeta_{l}] may be taken from energy-optimized coefficients for isolated atoms available in the literature (Clementi & Raimondi, 1963[link]). A standard set has been proposed by Hehre et al. (1969)[link]. In the bonded atom, such values are affected by changes in nuclear screening due to migrations of charge, as described in part by equation (1.2.6.1)[link].

Other alternatives are: [R_{l} (r) = {\alpha^{n + 1} \over n!} r^{n} \exp (-\alpha r^{2})\qquad (\hbox{Gaussian function}) \eqno(1.2.7.5b)] or [{R_{l} (r) = r^{l} L_{n}^{2l + 2} (\gamma r) \exp \left(-{\gamma r \over 2}\right)\quad (\hbox{Laguerre function}),} \eqno(1.2.7.5c)] where L is a Laguerre polynomial of order n and degree [(2l + 2)].

In summary, in the multipole formalism the atomic density is described by [\eqalignno{\rho_{\rm atomic}({\bf r}) &= P_{c} \rho_{\rm core} + P_{\nu} \kappa^{3} \rho_{\rm valence} (\kappa r)\cr &\quad + {\textstyle\sum\limits_{l = 0}^{l_{\max}}} \kappa'^{3} R_{l}(\kappa' r) {\textstyle\sum\limits_{m = 0}^{l}} {\textstyle\sum\limits_{p}} P_{lmp} d_{lmp} ({\bf r}/r), &(1.2.7.6)}] in which the leading terms are those of the kappa formalism [expressions (1.2.6.1)[link], (1.2.6.2)[link]]; the subscript p is either + or −.

The expansion in (1.2.7.6)[link] is frequently truncated at the hexadecapolar [(l = 4)] level. For atoms at positions of high site symmetry the first allowed functions may occur at higher l values. For trigonally bonded atoms in organic molecules the [l = 3] terms are often found to be the most significantly populated deformation functions.

1.2.7.2. Reciprocal-space description of aspherical atoms

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The aspherical-atom form factor is obtained by substitution of (1.2.7.6)[link] in expression (1.2.4.3a)[link]: [f_{j}({\bf S}) = {\textstyle\int} \rho_{j}({\bf r}) \exp (2\pi i{\bf S} \cdot {\bf r})\ \hbox{d}{\bf r}. \eqno(1.2.4.3a)] In order to evaluate the integral, the scattering operator [{\exp (2\pi i{\bf S} \cdot {\bf r})}] must be written as an expansion of products of spherical harmonic functions. In terms of the complex spherical harmonic functions, the appropriate expression is (Weiss & Freeman, 1959[link]; Cohen-Tannoudji et al., 1977[link]) [{\exp (2\pi i{\bf S} \cdot {\bf r}) = 4\pi {\textstyle\sum\limits_{l = 0}^{\infty}}\; {\textstyle\sum\limits_{m = -l}^{l}} i^{l} j_{l} (2\pi Sr) Y_{lm} (\theta, \varphi) Y_{lm}^{*} (\beta, \gamma).} \eqno(1.2.7.7a)]

The Fourier transform of the product of a complex spherical harmonic function with normalization [{\textstyle\int} |Y_{lm}|^{2}\ \hbox{d}\Omega = 1] and an arbitrary radial function [R_{l}(r)] follows from the orthonormality properties of the spherical harmonic functions, and is given by [{{\textstyle\int} Y_{lm} R_{l}(r) \exp (2\pi i {\bf S} \cdot {\bf r})\ \hbox{d}\tau = 4\pi i^{l}{\textstyle\int}j_{l} (2\pi {S}r) R_{l}(r) r^{2}\ \hbox{d}r Y_{lm} (\beta, \gamma),} \eqno(1.2.7.8a)] where [j_{l}] is the lth-order spherical Bessel function (Arfken, 1970[link]), and θ and ϕ, β and γ are the angular coordinates of r and S, respectively.

For the Fourier transform of the real spherical harmonic functions, the scattering operator is expressed in terms of the real spherical harmonics: [\eqalignno{\exp (2\pi i{\bf S} \cdot {\bf r}) &= {\sum\limits_{l = 0}^{\infty}} i^{l} j_{l} (2\pi Sr) (2 - \delta_{m0}) (2l + 1) {\sum\limits_{m = 0}^{l}} {(l - m)! \over (l + m)!}\cr &\quad \times P_{l}^{m} (\cos \theta) P_{l}^{m} (\cos \beta) \cos [m(\phi - \gamma)], &(1.2.7.7b)}] which leads to [{\textstyle\int} y_{lmp} (\theta, \varphi) R_{l}(r) \exp (2\pi i{\bf S} \cdot {\bf r})\ \hbox{d}\tau = 4\pi i^{l}\langle j_{l}\rangle y_{lmp} (\beta, \gamma). \eqno(1.2.7.8b)] Since [y_{lmp}] occurs on both sides, the expression is independent of the normalization selected. Therefore, for the Fourier transform of the density functions [d_{lmp}] [{\textstyle\int} d_{lmp} (\theta, \varphi) R_{l}(r) \exp (2\pi i{\bf S} \cdot {\bf r})\ \hbox{d}\tau = 4\pi i^{l}\langle j_{l}\rangle d_{lmp} (\beta, \gamma). \eqno(1.2.7.8c)]

In (1.2.7.8b)[link] and (1.2.7.8c)[link], [\langle j_{l}\rangle], the Fourier–Bessel transform, is the radial integral defined as [\langle j_{l}\rangle = {\textstyle\int} j_{l}(2\pi Sr) R_{l}(r) r^{2}\ \hbox{d}r \eqno(1.2.7.9)] of which [\langle j_{0}\rangle] in expression (1.2.4.3)[link] is a special case. The functions [\langle j_{l}\rangle] for Hartree–Fock valence shells of the atoms are tabulated in scattering-factor tables (IT IV, 1974[link]). Expressions for the evaluation of [\langle j_{l}\rangle] using the radial function (1.2.7.5a[link][link]c[link]) have been given by Stewart (1980)[link] and in closed form for (1.2.7.5a)[link] by Avery & Watson (1977)[link] and Su & Coppens (1990)[link]. The closed-form expressions are listed in Table 1.2.7.4[link].

Table 1.2.7.4| top | pdf |
Closed-form expressions for Fourier transform of Slater-type functions (Avery & Watson, 1977[link]; Su & Coppens, 1990[link])

[\langle j_{k}\rangle \equiv {\textstyle\int_{0}^{\infty}} r^{N} \exp(-Zr)j_{k}(Kr)\;\hbox{d}r, K = 4\pi \sin \theta/\lambda.]

  N
k 1 2 3 4 5 6 7 8
0 [\displaystyle{1 \over K^{2} + Z^{2}}] [\displaystyle{2Z \over (K^{2} + Z^{2})^{2}}] [\displaystyle{2(3Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{3}}] [\displaystyle{24Z(Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{4}}] [\displaystyle{24(5Z^{2} - 10K^{2}Z^{2} + K^{4}) \over (K^{2} + Z^{2})^{5}}] [\displaystyle{240Z(K^{2} - 3Z^{2}) (3K^{2} - Z^{2}) \over (K^{2} + Z^{2})^{6}}] [\displaystyle{720(7Z^{6} - 35K^{2}Z^{4} + 21K^{4}Z^{2} - K^{6}) \over (K^{2} + Z^{2})^{7}}] [\displaystyle{40320(Z^{7} - 7K^{2}Z^{5} + 7K^{4}Z^{3} - K^{6}Z) \over (K^{2} + Z^{2})^{8}}]
1   [\displaystyle{2K \over (K^{2} + Z^{2})^{2}}] [\displaystyle{8KZ \over (K^{2} + Z^{2})^{3}}] [\displaystyle{8K(5Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{4}}] [\displaystyle{48KZ(5Z^{2} - 3K^{2}) \over (K^{2} + Z^{2})^{5}}] [\displaystyle{48K(35Z^{4} - 42K^{2}Z^{2} + 3K^{4}) \over (K^{2} + Z^{2})^{6}}] [\displaystyle{1920KZ(7Z^{4} - 14K^{2}Z^{2} + 3K^{4}) \over (K^{2} + Z^{2})^{7}}] [\displaystyle{5760K(21Z^{6} - 63K^{2}Z^{4} + 27K^{4}Z^{2} - K^{6}) \over (K^{2} + Z^{2})^{8}}]
2     [\displaystyle{8K^{2} \over (K^{2} + Z^{2})^{3}}] [\displaystyle{48K^{2}Z \over (K^{2} + Z^{2})^{4}}] [\displaystyle{48K^{2}(7Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{5}}] [\displaystyle{384K^{2}Z(7Z^{2} - 3K^{2}) \over (K^{2} + Z^{2})^{6}}] [\displaystyle{1152K^{2}(21Z^{4} - 18K^{2}Z^{2} + K^{4}) \over (K^{2} + Z^{2})^{7}}] [\displaystyle{11520K^{2}Z(21Z^{4} - 30K^{2}Z^{2} + 5K^{4}) \over (K^{2} + Z^{2})^{8}}]
3       [\displaystyle{48K^{3} \over (K^{2} + Z^{2})^{4}}] [\displaystyle{384K^{3}Z \over (K^{2} + Z^{2})^{5}}] [\displaystyle{384K^{3}(9Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{6}}] [\displaystyle{11520K^{3}Z(3Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{7}}] [\displaystyle{11520K^{3}(33Z^{4} - 22K^{2}Z^{2} + K^{4}) \over (K^{2} + Z^{2})^{8}}]
4         [\displaystyle{384K^{4} \over (K^{2} + Z^{2})^{5}}] [\displaystyle{3840K^{4}Z \over (K^{2} + Z^{2})^{6}}] [\displaystyle{3840K^{4}(11Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{7}}] [\displaystyle{46080K^{4}Z(11Z^{2} - 3K^{2}) \over (K^{2} + Z^{2})^{8}}]
5           [\displaystyle{3840K^{5} \over (K^{2} + Z^{2})^{6}}] [\displaystyle{46080K^{5}Z \over (K^{2} + Z^{2})^{7}}] [\displaystyle{40680K^{5}(13Z^{2} - K^{2}) \over (K^{2} + Z^{2})^{8}}]
6             [\displaystyle{46080K^{6} \over (K^{2} + Z^{2})^{7}}] [\displaystyle{645120K^{6}Z \over (K^{2} + Z^{2})^{8}}]
7               [\displaystyle{645120K^{7} \over (K^{2} + Z^{2})^{8}}]

Expressions (1.2.7.8)[link] [link] show that the Fourier transform of a direct-space spherical harmonic function is a reciprocal-space spherical harmonic function with the same l, m, or, in other words, the spherical harmonic functions are Fourier-transform invariant.

The scattering factors [f_{lmp}({\bf S})] of the aspherical density functions [R_{l}(r)d_{lmp}(\theta, \phi)] in the multipole expansion (1.2.7.6)[link] are thus given by [f_{lmp}({\bf S}) = 4\pi i^{l} \langle j_{l}\rangle d_{lmp} (\beta, \gamma). \eqno(1.2.7.8d)]

The reciprocal-space spherical harmonic functions in this expression are identical to the functions given in Table 1.2.7.1[link], except for the replacement of the direction cosines x, y and z by the direction cosines of the scattering vector S.

References

International Tables for X-ray Crystallography (1974). Vol. IV. Birmingham: Kynoch Press. (Present distributor Kluwer Academic Publishers, Dordrecht.)
Arfken, G. (1970). Mathematical models for physicists, 2nd ed. New York, London: Academic Press.
Avery, J. & Watson, K. J. (1977). Generalized X-ray scattering factors. Simple closed-form expressions for the one-centre case with Slater-type orbitals. Acta Cryst. A33, 679–680.
Clementi, E. & Raimondi, D. L. (1963). Atomic screening constants from SCF functions. J. Chem. Phys. 38, 2686–2689.
Cohen-Tannoudji, C., Diu, B. & Laloe, F. (1977). Quantum mechanics. New York: John Wiley and Paris: Hermann.
Hansen, N. K. & Coppens, P. (1978). Testing aspherical atom refinements on small-molecule data sets. Acta Cryst. A34, 909–921.
Hehre, W. J., Stewart, R. F. & Pople, J. A. (1969). Self-consistent molecular orbital methods. I. Use of Gaussian expansions of Slater-type atomic orbitals. J. Chem. Phys. 51, 2657–2664.
Hirshfeld, F. L. (1977). A deformation density refinement program. Isr. J. Chem. 16, 226–229.
Kara, M. & Kurki-Suonio, K. (1981). Symmetrized multipole analysis of orientational distributions. Acta Cryst. A37, 201–210.
Stewart, R. F. (1980). Electron and magnetization densities in molecules and solids, edited by P. J. Becker, pp. 439–442. New York: Plenum.
Su, Z. & Coppens, P. (1990). Closed-form expressions for Fourier– Bessel transforms of Slater-type functions. J. Appl. Cryst. 23, 71–73.
Von der Lage, F. C. & Bethe, H. A. (1947). A method for obtaining electronic functions and eigenvalues in solids with an application to sodium. Phys. Rev. 71, 612–622.
Weiss, R. J. & Freeman, A. J. (1959). X-ray and neutron scattering for electrons in a crystalline field and the determination of outer electron configurations in iron and nickel. J. Phys. Chem. Solids, 10, 147–161.








































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