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

International Tables for Crystallography (2010). Vol. B, ch. 1.2, pp. 12-14   | 1 | 2 |

Section 1.2.7.1. Direct-space description of aspherical atoms

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@buffalo.edu

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 nonspherical environment; therefore, an accurate description of the atomic electron density requires nonspherical 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 nonspherical 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)

lSymbolCAngular 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}] [\matrix{1\cr 1\cr 1}] [\left.\matrix{x\cr y\cr z}\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-}] [\matrix{3\cr 3\cr 6\cr 6}] [\left.\matrix{xz\cr yz\cr (x^{2} - y^{2})/2\cr xy}\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-}] [\matrix{3/2\cr 3/2}] [\left.\matrix{x[5z^{2} - 1]\cr y[5z^{2} - 1]}\right\}] [(21/32\pi)^{1/2}] 0.45705 [[{\rm ar} + (14/ 5) - (\pi/4)]^{-1}]†† 0.32033
[\matrix{32+\cr 32-}] [\matrix{15\cr 15}] [\left.\matrix{(x^{2} - y^{2})z\cr 2xyz}\right\}] [(105/16\pi)^{1/2}] 1.44531 1 1
[\matrix{33+\cr 33-}] [\matrix{15\cr 15}] [\left.\matrix{x^{3} - 3xy^{2}\cr -y^{3} + 3x^{2}y}\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-}] [\matrix{5/2\cr 5/2}] [\left.\matrix{x[7z^{3} - 3z]\cr y[7z^{3} - 3z]}\right\}] [(45/32\pi)^{1/2}] 0.66905 [\displaystyle{735 \over 512\sqrt{7} + 196}] 0.47400
[\matrix{42+\cr 42-}] [\matrix{15/2\cr 15/2}] [\left.\matrix{(x^{2} - y^{2})[7z^{2} - 1]\cr 2xy[7z^{2} - 1]}\right\}] [(45/64\pi)^{1/2}] 0.47309 [\displaystyle{105\sqrt{7} \over 4(136 + 28\sqrt{7})}] 0.33059
[\matrix{43+\cr 43-}] [\matrix{105\cr 105}] [\left.\matrix{(x^{3} - 3xy^{2})z\cr (-y^{3} + 3x^{2}y)z}\right\}] [(315/32\pi)^{1/2}] 1.77013 [5/4] 1.25
[\matrix{44+\cr 44-}] [\matrix{105\cr 105}] [\left.\matrix{x^{4} - 6x^{2}y^{2} + y^{4}\cr 4x^{3}y - 4xy^{3}}\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-}] [15/8] [\left.\matrix{(21z^{4} - 14z^{2} + 1)x\cr (21z^{4} - 14z^{2} + 1)y}\right\}] [(165/256\pi)^{1/2}] 0.45295 0.32298
[\matrix{52+\cr 52-}] [105/2] [\left.\matrix{(3z^{3} - z) (x^{2} - y^{2})\cr 2xy(3z^{3} - z)}\right\}] [(1155/64\pi)^{1/2}] 2.39677 1.68750
[\matrix{53+\cr 53-}] [105/2] [\left.\matrix{(9z^{2} - 1) (x^{3} - 3xy^{2})\cr (9z^{2} - 1) (3x^{2}y - y^{3})}\right\}] [(385/512\pi)^{1/2}] 0.48924 0.34515
[\matrix{54+\cr 54-}] 945 [\left.\matrix{z(x^{4} - 6x^{2}y^{2} + y^{4})\cr z(4x^{3}y - 4xy^{3})}\right\}] [(3465/256\pi)^{1/2}] 2.07566 1.50000
[\matrix{55+\cr 55-}] 945 [\left.\matrix{x^{5} - 10x^{3}y^{2} + 5xy^{4}\cr 5x^{4} y - 10x^{2}y^{3} + y^{5}}\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-}] [21/8] [\left.\matrix{(33z^{5} - 30z^{3} + 5z)x\cr (33z^{5} - 30z^{3} + 5z)y}\right\}] [(273/256\pi)^{1/2}] 0.58262 0.41721
[\matrix{62+\cr 62-}] [105/8] [\left.\matrix{(33z^{4} - 18z^{2} + 1) (x^{2} - y^{2})\cr 2xy (33z^{4} - 18z^{2} + 1)}\right\}] [(1365/2048\pi)^{1/2}] 0.46060 0.32611
[\matrix{63+\cr 63-}] [315/2] [\left.\matrix{(11z^{3} - 3z) (x^{3} - 3xy^{2})\cr (11z^{3} - 3z) (3x^{2}y - 3y)}\right\}] [(1365/512\pi)^{1/2}] 0.92121 0.65132
[\matrix{64+\cr 64-}] [945/2] [\left.\matrix{(11z^{2} - 1) (x^{4} - 6x^{2}y^{2} + y^{4})\cr (11z^{2} - 1) (4x^{3}y - 4xy^{3})}\right\}] [(819/1024\pi)^{1/2}] 0.50457 0.36104
[\matrix{65+\cr 65-}] 10395 [\left.\matrix{z(x^{5} - 10x^{3}y^{2} + 5xy^{4})\cr z(5x^{4}y - 10x^{2}y^{3} + y^{5})}\right\}] [(9009/512\pi)^{1/2}] 2.36662 1.75000
[\matrix{66+\cr 66-}] 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}}\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-}] [7/16] [\left.\matrix{(429z^{6} - 495z^{4} + 135z^{2} - 5)x\cr (429z^{6} - 495z^{4} + 135z^{2} - 5)y}\right\}] [(105/4096\pi)^{1/2}] 0.09033 0.06488
[\matrix{72+\cr 72-}] [63/8] [\left.\matrix{(143z^{5} - 110z^{3} + 15z) (x^{2} - y^{2})\cr 2xy(143z^{5} - 110z^{3} + 15z)}\right\}] [(315/2048\pi)^{1/2}] 0.22127 0.15732
[\matrix{73+\cr 73-}] [315/8] [\left.\matrix{(143z^{4} - 66z^{2} + 3) (x^{3} - 3xy^{2})\cr (143z^{4} - 66z^{2} + 3) (3x^{2}y - y^{3})}\right\}] [(315/4096\pi)^{1/2}] 0.15646 0.11092
[\matrix{74+\cr 74-}] [3465/2] [\left.\matrix{(13z^{3} - 3z) (x^{4} - 6x^{2}y^{2} + y^{4})\cr (13z^{3} - 3z) (4x^{3}y - 4xy^{3})}\right\}] [(3465/1024\pi)^{1/2}] 1.03783 0.74044
[\matrix{75+\cr 75-}] [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})}\right\}] [(3465/4096\pi)^{1/2}] 0.51892 0.37723
[\matrix{76+\cr 76-}] 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})}\right\}] [(45045/2048\pi)^{1/2}] 2.6460 2.00000
[\matrix{77+\cr 77-}] 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}}\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 nonzero contribution to the integral [\Theta_{lmp} = {\textstyle\int} \rho ({\bf r}) c_{lmp} r^{\,l}\,{\rm 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\,{\rm d}\theta\,{\rm d}\varphi = 1] (Kara & Kurki-Suonio, 1981[link]).

Even lmp
lj0+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
lj 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 lmp
lj0+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.

References

Arfken, G. (1970). Mathematical Models for Physicists, 2nd ed. New York, London: Academic Press.Google Scholar
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