Model-based curvilinear density functions

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. 589-590

Section 6.1.1.6.5. Model-based curvilinear density functions

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

6.1.1.6.5. Model-based curvilinear density functions

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For rotational oscillations, which are the curvilinear coordinate analogues of the p.d.f.'s approximating harmonic rectilinear motion, techniques for evaluating the temperature factor are described by Johnson & Levy (1974).

The p.d.f. for an atom in a group of atoms undergoing large-amplitude rotational oscillation (libration) can sometimes be approximated satisfactory by a standard p.d.f. on the circle or on the sphere. The closest analogues of the rectilinear Gaussian p.d.f. are the Brownian-diffusion p.d.f.'s defined on the closed spaces of the circle and the sphere. For statistical analysis, two other p.d.f.'s, the von Mises `circular normal' and the Fisher `spherical normal', are often substituted for the Brownian-diffusion density functions because of their simpler forms.

The p.d.f. for Brownian diffusion on a circle, also called the `wrapped normal' p.d.f. (Feller, 1966; Lévy, 1938), is given by $[\rho(\theta)={1\over(2\pi){}^{1/2}\sigma}\sum^\infty_{n=-\infty}\exp[-(\theta-2n\pi)^2/2\sigma^2],\eqno (6.1.1.76)]$ which may be transformed (Bellman, 1961) into $[\rho(\theta)={1\over2\pi}\sum^\infty_{m=0}(2-\delta_{m0})\exp(-m^2\sigma^2/2)\cos(m\theta).\eqno (6.1.1.77)]$ The von Mises p.d.f. (Gumbel, Greenwood & Durand, 1953; Mardin, 1972; von Mises, 1918) is $[\rho(\theta)={\exp(k_c\cos\theta)\over2\pi I_o(k_c)}={1\over2\pi}\sum^\infty_{m=0}(2-\delta_{m0})\displaystyle{I_m(k_c)\over I_0(k_c)}\cos(m\theta).\eqno (6.1.1.78)]$ [I_m(x)] is the mth-order Bessel function of the first kind with imaginary argument. The parameter $[\sigma^2]$ is the variance; [k_c] is a measure of concentration such that when [k_c] is zero the probability density is uniformly distributed over the circle, and when [k_c] is large the density is concentrated around the modal vector at θ = 0. An approximate relation between $[\sigma^2]$ and [k_c] can be obtained by equating expressions for the centres of mass of the circular Brownian diffusion and von Mises p.d.f.'s (Stephens, 1963), $[\exp(-\sigma^2/2)={I_1(k_c)\over I_0(k_c)}.\eqno (6.1.1.79)]$ For small $[\sigma^2]$ (large [k_c] ), we find that $[\sigma^2\simeq1/k_c.\eqno (6.1.1.80)]$

Equations (6.1.1.76) to (6.1.1.78) can be generalized to describe multimodal density functions with modes (maxima) arranged symmetrically about the circle. The p.d.f. for the s-modal Brownian-diffusion p.d.f. with one of the s modes at θ = θ₀ is $[\eqalignno{\rho(\theta)&={1\over\sqrt{2\pi} s\sigma}\sum^\infty_{m=-\infty}\exp[-(\theta-\theta_0-2\pi m/s)^2/2\sigma^2]\cr &={1\over2\pi}\sum^\infty_{m=0}(2-\delta_{m0})\exp[-(ms\sigma)^2/2]\cos[ms(\theta-\theta_0)]. \cr&&(6.1.1.81)}]$ The two-dimensional Fourier transform (Chidambaram & Brown, 1973) of the last equation in terms of the polar coordinates $[(S,\theta)]$ of the reciprocal-space vector S relative to an origin at the centre of the circle is $[T({\bf S})=\textstyle\sum\limits^\infty_{j=0}(2-\delta_{j 0})i^{js}J_{js}(Sr)\exp[-(js\sigma)^2/2]\cos js\theta_0,\eqno (6.1.1.82)]$ where [J_n(x)] is the Bessel function of the first kind of order n with real argument. Corresponding equations for the von Mises s-modal density function (Atoji, Watanabe & Lipscomb, 1953; King & Lipscomb, 1950; Mardin, 1972) are $[\eqalignno{\rho(\theta)&={1\over2\pi I_o(K_c)}\exp[K_c\cos s(\theta-\theta_0)]\cr &={1\over2\pi}\sum^\infty_{m=0}(2-\delta_{m0})\displaystyle{I_m(K_c)\over I_o(K_c)}\cos ms(\theta-\theta_0)\cr&&(6.1.1.83)}]$ and $[T({\bf S})=\sum^\infty_{j=0}(2-\delta_{j 0})i ^{js}J_{js}(Sr)\displaystyle{I_j(K_c)\over I_0(K_c)}\cos js\theta_0,\eqno (6.1.1.84)]$ where [K_c] , a measure of concentration over 1/sth of the circle about $[\theta_0]$ , is substituted for the [k_c] parameter of the unimodal von Mises density function and [K_c] is related to [k_c] approximately by $[I_1(k_c)/I_0(k_c)=I_s(K_c)/I_0(K_c).\eqno (6.1.1.85)]$

For symmetrical Brownian diffusion on a sphere (Furry, 1957; Lévy, 1938; Mardin, 1972; Perrin, 1928), the p.d.f. in terms of the angular displacement θ from the pole is $[\rho(\theta)=\sum^\infty_{n=0}\displaystyle{2n+1\over4\pi}\exp[-n(n+1)V]P_n(\cos\theta)\sin\theta,\eqno (6.1.1.86)]$ where [P_n(x)] is the nth-order Legendre polynomial. The Fisher (1953) `spherical normal' p.d.f. (Mardin, 1972) is a similar density function given by $[\eqalignno{\rho(\theta)&={k_s\over4\pi\sinh k_s}\exp(k_s\cos\theta)\sin\theta\cr &=\sum^\infty_{n=0}\displaystyle{(2n+1)\over 4\pi}{I_{n+1/2}(k_s)\over I_{1/2}(k_s)}P_n(\cos\theta)\sin\theta.& (6.1.1.87)}]$ The parameters V (variance) and [k_s] are measures of concentration analogous to those for the circle and may be related (Roberts & Ursell, 1960) by an equation analogous to (6.1.1.79), $[\exp(-V/2)=\coth k_s-{1\over k_s}={I_{3/2}(k_s)\over I_{1/2}(k_s)},\eqno (6.1.1.88)]$ the small V approximation being $[V\simeq2/k_s.\eqno (6.1.1.89)]$ Equations (6.1.1.86) and (6.1.1.87) are generalized to place the mode of the density at $[(r,\theta',\varphi')]$ by replacing $[\cos\theta]$ by $[\cos\theta\cos\theta'+\sin\theta\sin\theta'\cos(\varphi-\varphi')]$ and by replacing $[P_n(\cos\theta)]$ by $[\eqalign{&P(\cos\theta)P_n(\cos\theta')+2\sum^n_{m=1}\displaystyle{(n-m)!\over(n-m)!}\cr &\quad\times P^m_n(\cos\theta)P^m_n(\cos\theta')\cos m(\varphi-\varphi').}]$

The three-dimensional Fourier transform of the generalized form of (6.1.1.86) in terms of S in spherical coordinates $[(S,\theta_S,\varphi_S)]$ is $[\eqalignno{T({\bf S})&=\sum^\infty_{q=0}i^q {(2q+1)\over r^2}\exp[-q(q+1)V]\cr &\quad\times\sum^\infty_{s=0}{4\over2p+1}Y_{qs+}(\theta',\varphi')Y_{qs+}(\theta_S,\varphi_S)j_q(Sr),& (6.1.1.90)}]$ where r is the radius of the sphere, and [j_n] is the nth-order spherical Bessel function of the first kind. The real spherical harmonics $[Y_{lmp}]$ are normalized as in (6.1.1.22).

The Fourier transform of the generalized form of (6.1.1.87) is identical to (6.1.1.90) except that the term $[\exp[-q(q+1)V]]$ in (6.1.1.90) is replaced by $[I_{q+1/2}(k_s)/I_{1/2}(k_s).]$

The foregoing equations describe isotropic distributions on a sphere. The p.d.f. for general anisotropic Brownian diffusion (or rotation) on a sphere is not available in a convenient form. However, some of the results of Perrin (1934) and Favro (1960) on rotational Brownian motion are applicable to thermal motion. For example, the centre of mass of a p.d.f. resulting from anisotropic diffusion on a sphere is given by equation (6.8) of Favro (1960). The following equation valid in Cartesian coordinates is obtained if the diffusion tensor D of Favro's equation is replaced by the substitution L = 2D $[\eqalignno{\langle{\bf x}\rangle&=\exp[-\textstyle{1\over2}({\rm tr}({\bf L}){\bf I}-{\bf L})]{\bf r}\cr &={\bf r}-\textstyle{1\over2}[{\rm tr}({\bf L}){\bf I}-{\bf L}]{\bf r}+{1\over8}[{\rm tr}({\bf L}){\bf I}-{\bf L}]^2{\bf r}-\ldots, & (6.1.1.91)}]$ where r is the vector from the centre of the sphere to the mode of the p.d.f. on the sphere and $[\langle{\bf x}\rangle]$ is the vector to the centre of mass. This equation, which is valid for all amplitudes of libration L, can be used to describe the apparent shrinkage effect in molecules undergoing librational motion.

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International Tables for Crystallography (2006). Vol. C. ch. 6.1, pp. 589-590