Fourier-invariant expansions

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. 586-588

Section 6.1.1.6.2. Fourier-invariant expansions

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

6.1.1.6.2. Fourier-invariant expansions

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When truncated, an expression for a multipole expansion, p.d.f. or temperature factor must retain those terms essential to the accuracy required of the expansion. Some authors (e.g. Stewart, 1980b) strongly favour classes of truncated expansion that retain symmetry properties appropriate to particular classes of transformation, such as rotation or Fourier inversion. Others, emphasizing simplicity, retain the minimum set of terms required to preserve the accuracy needed in the expansion. In either case, it is desirable for the expansion to converge rapidly, and to have a form related to physical theory.

In principle, the one-particle potential may be expanded in any complete set of functions. Harmonic oscillator functions simplify simultaneous interpretation of the probability distribution in real and reciprocal space because their form does not change under Fourier inversion (Kurki-Suonio, Merisalo & Peltonen, 1979).

If both anharmonicity and anisotropy are small, the p.d.f. may be expressed as a rapidly converging expansion in spherical polar coordinates $[u,\theta,\varphi]$ : $[\rho({\bf u})=\rho_o({\bf u}){N_0\over N}\bigg[1-\sum_{n,l,m,p}a_{nlmp}R_{nl}(Bu)Y_{lmp}(\theta,\varphi)\bigg]\eqno (6.1.1.48)]$ for non-cubic and $[\rho({\bf u})=\rho_o({\bf u}){N_0\over N}\bigg[1-\sum_{n,l,\,j}a_{nlj}R_{nl}(Bu)K_{l\,j}(\theta,\varphi)\bigg]\eqno (6.1.1.49)]$ for cubic site symmetry. The radial term may be written as $[R_{nl}(x)=x^lL^{l+1/2}_{(n-l)/2}(x^2),\eqno (6.1.1.50)]$ where the associated Laguerre polynomial is $[L_k^\alpha(t)=\sum^k_{\nu=0}\bigg(\matrix{k+\alpha\cr k-\alpha\cr}\bigg)\displaystyle{(-t){}^\nu\over\nu!}\eqno (6.1.1.51)]$ with $[\bigg(\matrix{p\cr q\cr}\bigg)={{\Gamma}(p+1)\over[{\Gamma}(q+1){\Gamma}(p-q+1)]}\eqno (6.1.1.52)]$ and the normalizing factor $[N={8\pi^3\over B^3}\bigg[1-\sum_\nu(-)^\nu\displaystyle{(2\nu+1)!\over2^{2\nu}(\nu!)^2}a_{2\nu00+}\bigg].\eqno (6.1.1.53)]$

The real spherical harmonics $[Y_{lmp}(\theta,\varphi)]$ and the cubic harmonics $[K_{lj}(\theta,\varphi)]$ are as defined in Subsection 6.1.1.4. As in the case of multipole expansions, the non-zero coefficients in these expressions are limited by the site symmetry. The restrictions on the temperature factor are identical to those for the generalized scattering factor listed in Tables 6.1.1.7 and 6.1.1.8.

From the Fourier invariance of harmonic oscillator functions, $[\eqalignno{T({\bf S})&={N_0\over N}\exp(-b^2S^2/2)\cr &\quad\times\bigg[1-\sum_{n,l,m,p}a_{nlmp}i^nR_{nl}(bS)Y_{lmp}(\theta_S,\varphi_S)\bigg]& (6.1.1.54)}]$ and $[\eqalignno{T({\bf S})&={N_0\over N}\exp(-b^2S^2/2)\cr &\quad\times\bigg[1-\sum_{n,l,\,j}a_{nlj}i^nR_{nl}(bS)K_{l\,j}(\theta_S,\varphi_S)\bigg]& (6.1.1.55)}]$ for non-cubic and cubic site symmetries, respectively. $[\theta_S]$ and $[\varphi_S]$ are polar coordinates in reciprocal space.

With an appropriate choice of origin, the first-order (110+) and (111 $[\pm]$ ) terms vanish. The isotropic harmonic (200+) and constant (000+) terms have been removed from the summation. If coordinate axes are chosen coincident with the principal axes for the harmonic approximation, (221 $[\pm]$ ) and (222−) vanish. (220+) indicates the prolateness and (222+) the non-axiality in the harmonic approximation (Kurki-Suonio, 1977). Terms with $[n\ge2]$ describe the anharmonicity.

The approximations in (6.1.1.48) to (6.1.1.55) fail if the anisotropy, indicated by the size of the (220+) and (222+) terms, or the anharmonicity is large. If the anharmonicity and non-axiality are small, one can invoke Fourier-invariant expansions in cylindrical polar coordinates $[u_r,u_z,\varphi]$ : $[\specialfonts\eqalignno{\rho({\bf u})&=\rho_o({\bf u}){N_0\over N}\cr &\quad\times\bigg[1-\sum_{n_z,n,m,p}b_{n_znmp}H_{n_z}(B_zu_z){\bsf P}_{nm}(B_ru_r)\Phi_{mp}(\varphi)\bigg]\cr&& (6.1.1.56)}]$ and $[\specialfonts\eqalignno{T(S)&={N_0\over N}\exp[-\textstyle{1\over2}(b^2_rS^2_r+b^2_zS^2_z)]\cr &\quad\times\bigg[1-\sum_{n_z,n,m,p}b_{n_znmp}H_{n_z}(b_zS_z){\bsf P}_{nm}(b_rS_r)\Phi_{mp}(\varphi_S)\bigg],\cr&&(6.1.1.57)}]$ where $[S_r,S_z,\varphi_S]$ are cylindrical coordinates for S. $[\specialfonts{\bsf P}_{nm}(x)= x^mL^m_{(n-m)/2}(x^2),\quad\Phi_{m\pm}(\varphi)=\matrix{\cos m\varphi\cr \sin m\varphi\cr} \eqno (6.1.1.58)]$ and $[N={8\pi^3\over B^2_rB_z}\bigg[1-\sum_{\mu\nu}(-)^\nu\displaystyle{(2\mu)!\over\mu!}b_{2\mu2\nu0+}\bigg].\eqno (6.1.1.59)]$ The indices allowed for the site symmetrical basis are as indicated in Table 6.1.1.10.

Table 6.1.1.10| top | pdf |
Indices nmp allowed by the site symmetry for the functions $[H_n(z)\Phi_{mp}(\varphi)]$ ; μ, ν and j are integers such that m, n ≥ 0; (−)ⁿ implies p = − for n odd and p = + for n even

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

Again, the first-order (100+) and (011 $[\pm]$ ) terms vanish with the appropriate choice of origin. For coordinate axes coinciding with the principal axes of the harmonic approximation, (111 $[\pm]$ ) and (022−) vanish. (020+), (200+), and (000+) have been removed from the summation.

Equations (6.1.1.56) and (6.1.1.57) apply accurately to non-cubic symmetries with rotation axes higher than twofold where non-axiality vanishes. Where non-axiality is large, it is preferable to use the Cartesian Fourier invariant expansion $[\eqalignno{\rho({\bf u})&={N_0\over N}\exp\bigg[-1/2\sum_iB^2_iu^2_1\bigg]\cr &\quad\times\bigg[1-\sum_{n_x,n_y,n_z}c_{n_xn_y,n_z}H_{n_x}(B_xu_x)H_{n_y}(B_yu_y)H_{n_z}(B_zu_z)\bigg]\cr && (6.1.1.60)}]$ and $[\eqalignno{T({\bf S})&={N_0\over N}\exp\bigg[-1/2\sum_ib^2_iu^2_1\bigg]\cr &\quad\times\bigg[1-\sum_{n_x,n_y,n_z}c_{n_xn_y,n_z}H_{n_x}(b_xu_x)H_{n_y}(b_yu_y)H_{n_z}(b_zu_z)\bigg], \cr&&(6.1.1.61)}]$ where $[N={8\pi^3\over B_xB_yB_z}\bigg[1-\sum_{\lambda\mu\nu}\displaystyle{(2\lambda)!(2\mu)!(2\nu)!\over\lambda!\mu!\nu!}c_{2\lambda2\mu2\nu}\bigg].\eqno (6.1.1.62)]$ The indices allowed under the site symmetry are listed in Table 6.1.1.11.

Table 6.1.1.11| top | pdf |
Indices n_x, n_y, n_z allowed for the basis functions H_{n_x}(Ax)H_{n_y}(By)H_{n_z}(Cz); λ, μ and ν are non-negative; conditions for other choices of axes are derived by cyclic permutation

Symmetry	Coordinate axes	Allowed indices
1	Any	$[{\rm All}\,\,(n_x,n_y,n_z)]$
$[\bar1]$	Any	$[n_x+n_y+n_z=2\lambda]$
2	$[2\parallel z]$	$[n_x+n_y=2\lambda]$
m	$[m\,\bot\, z]$	$[n_z=2\nu]$
	$[2\parallel z,m\,\bot\,z]$	$[n_x+n_y=2\lambda,n_z=2\nu]$
222	$[2\parallel z,2\parallel y]$	all even or all odd
mm2	$[2\parallel z,m\,\bot\,y]$	$[n_x=2\lambda,n_y=2\mu]$
mmm	$[m\,\bot\,z,m\,\bot\,y,m\,\bot\,z]$	$[n_x=2\lambda,n_y=2\mu,n_z=2\nu]$

The first-order terms vanish with suitable choice of origin. (110), (101), and (011) vanish if the coordinates coincide with the principal axes for the harmonic approximation, and (200), (020), (002), and (000) are removed from the summation. Only anharmonic terms remain.

References

Kurki-Suonio, K. (1977). Electron density mapping in molecules and crystals. IV. Symmetry and its implications. Isr. J. Chem. 16, 115–123.Google Scholar

Kurki-Suonio, K., Merisalo, M. & Peltonen, H. (1979). Site symmetrized Fourier invariant treatment of anharmonic temperature factors. Phys. Scr. 19, 57–63.Google Scholar

Stewart, R. F. (1980b). Multipolar expansions of one-electron densities. Electron and magnetisation densities in molecules and crystals, edited by P. Becker, pp. 405–425. New York: Plenum.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 6.1, pp. 586-588