Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.2, pp. 22-23   | 1 | 2 |

Section 1.2.12. Treatment of anharmonicity

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:

1.2.12. Treatment of anharmonicity

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The probability distribution ([link] [link] is valid in the case of rectilinear harmonic motion. If the deviations from Gaussian shape are not too large, distributions may be used which are expansions with the Gaussian distribution as the leading term. Three such distributions are discussed in the following sections. The Gram–Charlier expansion

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The three-dimensional Gram–Charlier expansion, introduced into thermal-motion treatment by Johnson & Levy (1974)[link], is an expansion of a function in terms of the zero and higher derivatives of a normal distribution (Kendall & Stuart, 1958[link]). If [D_{j}] is the operator [\hbox{d/d}u\hskip 2pt^{j}], [\eqalignno{P({\bf u}) &= [1 - c\hskip 2pt^{j}D_{j} + {1 \over 2!} c\hskip 2pt^{jk}D_{j}D_{k} - {1 \over 3!} c\hskip 2pt^{jkl}D_{j}D_{k}D_{l} + \ldots\cr &\quad + (-1)^{r} {c^{\alpha_{1}} \ldots c^{\alpha_{r}} \over r!} D_{\alpha_{1}} D_{\alpha_{r}}] P_{0}({\bf u}), &(}] where [P_{0}({\bf u})] is the harmonic distribution, [\alpha_{1} = 1, 2] or 3, and the operator [D_{\alpha_{1}} \ldots D_{\alpha_{r}}] is the rth partial derivative [\partial^{r}/(\partial u^{\alpha 1} \ldots \partial u^{\alpha r})]. Summation is again implied over repeated indices.

The differential operators D may be eliminated by the use of three-dimensional Hermite polynomials [H_{\alpha_{1}\ldots \alpha_{2}}] defined, by analogy with the one-dimensional Hermite polynomials, by the expression [{D_{\alpha_{1}}\ldots D_{\alpha_{r}} \exp (- {\textstyle {1 \over 2}} \sigma_{jk}^{-1} u\hskip 2pt^{j}u^{k}) = (-1)^{r} H_{\alpha_{1}\ldots \alpha_{r}} ({\bf u}) \exp (- {\textstyle{1 \over 2}} \sigma_{jk}^{-1} u\hskip 2pt^{j}u^{k}),} \eqno(] which gives [\eqalignno{P({\bf u}) &= \left[1 + {1 \over 3!} c\hskip 2pt^{jkl}H_{jkl}({\bf u}) + {1 \over 4!} c\hskip 2pt^{jklm}H_{jklm} ({\bf u}) + {1 \over 5!} c\hskip 2pt^{jklmn}H_{jklmn} ({\bf u})\right.\cr &\quad\left. + {1 \over 6!} c\hskip 2pt^{jklmnp}H_{jklmnp} ({\bf u}) + \ldots\right] P_{0}({\bf u}), &(}] where the first and second terms have been omitted since they are equivalent to a shift of the mean and a modification of the harmonic term only. The permutations of [j, k, l \ldots] here, and in the following sections, include all combinations which produce different terms.

The coefficients c, defined by ([link] and ([link], are known as the quasimoments of the frequency function [P(\bf u)] (Kutznetsov et al., 1960[link]). They are related in a simple manner to the moments of the function (Kendall & Stuart, 1958[link]) and are invariant to permutation of indices. There are 10, 15, 21 and 28 components of c for orders 3, 4, 5 and 6, respectively. The multivariate Hermite polynomials are functions of the elements of [\sigma_{jk}^{-1}] and of [u^{k}], and are given in Table[link] for orders [\leq 6] (IT IV, 1974[link]; Zucker & Schulz, 1982[link]).

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Some Hermite polynomials (Johnson & Levy, 1974[link]; Zucker & Schulz, 1982[link])

H(u) = 1
Hj(u) = wj
Hjk(u) = wjwkpjk
Hjkl(u) = wjwkwl − (wjpkl + wkplj + wlpjk) = wjwkwl3w( jpkl)
Hjklm(u) = wjwkwlwm6w( jwkplm) + 3pj( kplm)
Hjklmn(u) = wjwkwlwmwn10w( lwmwnpjk) + 15w( npjkplm)
Hjklmnp(u) = wjwkwlwmwnwp − 15w( jwkwlwmpjk) + 45w( jwkplmpnp) − 15pj( kplmpnp)
where [w_{j}\equiv p{_{jk}}u^{k} \hbox{ and } p_{jk}] are the elements of [\sigma^{-1}], defined in expression ([link] [link]. Indices between brackets indicate that the term is to be averaged over all permutations which produce distinct terms, keeping in mind that [p_{jk} = p_{kj} \hbox{ and } w_{j}w_{k} = w_{k}w_{j}] as illustrated for [H_{jkl}].

The Fourier transform of ([link] is given by [\eqalignno{T({\bf H}) &= \left[1 - {4 \over 3} \pi^{3}ic\hskip 2pt^{jkl}h_{j}h_{k}h_{l} + {2 \over 3} \pi^{4}c\hskip 2pt^{jklm}h_{j}h_{k}h_{l}h_{m}\right.\cr &\quad + {4 \over 15} \pi^{5}ic\hskip 2pt^{jklmn}h_{j}h_{k}h_{l}h_{m}h_{n} &\cr &\quad\left. - {4 \over 45} \pi^{6}c\hskip 2pt^{jklmnp}h_{j}h_{k}h_{l}h_{m}h_{n}h_{p} + \ldots\right] T_{0}({\bf H}), &(}] where [T_{0}({\bf H})] is the harmonic temperature factor. [T({\bf H})] is a power-series expansion about the harmonic temperature factor, with even and odd terms, respectively, real and imaginary. The cumulant expansion

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A second statistical expansion which has been used to describe the atomic probability distribution is that of Edgeworth (Kendall & Stuart, 1958[link]; Johnson, 1969[link]). It expresses the function [P({\bf u})] as [\eqalignno{P({\bf u}) &= \exp \left(\kappa\hskip 2pt^{j}D_{j} + {1 \over 2!} \kappa\hskip 2pt^{jk}D_{j}D_{k} - {1 \over 3!} \kappa\hskip 2pt^{jkl}D_{j}D_{k}D_{l}\right.\cr &\quad\left. + {1 \over 4!} \kappa\hskip 2pt^{jklm}D_{j}D_{k}D_{l}D_{m} - \ldots\right)P_{0}({\bf u}). &(}]

Like the moments μ of a distribution, the cumulants κ are descriptive constants. They are related to each other (in the one-dimensional case) by the identity [{\exp\left\{\kappa_{1}t + {\kappa_{2}t^{2} \over 2!} + \ldots {\kappa_{r}t^{r} \over r!} + \ldots\right\} = 1 + \mu_{1}t + {\mu_{2}t^{2} \over 2!} + \ldots + {\mu_{r}t^{r} \over r!}.} \eqno(] When it is substituted for t, ([link] is the characteristic function, or Fourier transform of [P(t)] (Kendall & Stuart, 1958[link]).

The first two terms in the exponent of ([link] can be omitted if the expansion is around the equilibrium position and the harmonic term is properly described by [P_{0}({\bf u})].

The Fourier transform of ([link] is, by analogy with the left-hand part of ([link] (with t replaced by [2\pi ih]), [\eqalignno{T({\bf H}) &= \exp \left[{(2\pi i)^{3} \over 3!} \kappa\hskip 2pt^{jkl}h_{j}h_{k}h_{l} + {(2\pi i)^{4} \over 4!} \kappa\hskip 2pt^{jklm}h_{j}h_{k}h_{l}h_{m} + \ldots \right] T_{0}({\bf H})\cr &= \exp \left[- {4 \over 3} \pi^{3}i \kappa\hskip 2pt^{jkl}h_{j}h_{k}h_{l} + {2 \over 3} \pi^{4} \kappa\hskip 2pt^{jklm}h_{j}h_{k}h_{l}h_{m} + \ldots \right] T_{0}({\bf H}),\cr &&(}] where the first two terms have been omitted. Expression ([link] is similar to ([link] except that the entire series is in the exponent. Following Schwarzenbach (1986)[link], ([link] can be developed in a Taylor series, which gives [\eqalignno{T({\bf H}) &= \left\{1 + {(2\pi i)^{3} \over 3!} \kappa\hskip 2pt^{jkl}h_{j}h_{k}h_{l} + {(2\pi i)^{4} \over 4!} \kappa\hskip 2pt^{jklm}h_{j}h_{k}h_{l}h_{m} + \ldots\right.\cr &\quad + {(2\pi i)^{6} \over 6!} \left[\vphantom{{\textstyle\sum\limits_{1}^{2}}}\kappa\hskip 2pt^{jklmp} + {6! \over 2!(3!)^{2}} \kappa\hskip 2pt^{jkl}\kappa^{mnp}\right] h_{j}h_{k}h_{l}h_{m}h_{n}h_{p}\cr &\quad + \left.\vphantom{{\textstyle\sum\limits_{1}^{2}}}\hbox{higher-order terms}\right\} T_{0}({\bf H}). &(}]

This formulation, which is sometimes called the Edgeworth approximation (Zucker & Schulz, 1982[link]), clearly shows the relation to the Gram–Charlier expansion ([link], and corresponds to the probability distribution [analogous to ([link]] [\eqalignno{P({\bf u}) &= P_{0}({\bf u}) \left\{1 + {1 \over 3!} \kappa\hskip 2pt^{jkl}H_{jkl}({\bf u}) + {1 \over 4!} \kappa\hskip 2pt^{jklm}H_{jklm}({\bf u}) + \ldots\right.\cr &{\hbox to 7.25pt{}} + {1 \over 6!} \left[\vphantom{{\textstyle\sum\limits_{1}^{2}}}\kappa\hskip 2pt^{jklmnp} + 10 \kappa\hskip 2pt^{jkl} \kappa^{mnp} \vphantom{{\textstyle\sum\limits_{1}^{2}}}\right] H_{jklmnp}\cr &\quad \left.+ \hbox{ higher-order terms}\vphantom{{\textstyle\sum\limits_{1}^{2}}}\right\}. &(}]

The relation between the cumulants [\kappa\hskip 2pt^{jkl}] and the quasimoments [c\hskip 2pt^{jkl}] are apparent from comparison of ([link] and ([link]: [\eqalignno{c\hskip 2pt^{jkl} &= \kappa\hskip 2pt^{jkl}\cr c\hskip 2pt^{jklm} &= \kappa\hskip 2pt^{jklm}\cr c\hskip 2pt^{jklmn} &= \kappa\hskip 2pt^{jklmn}\cr c\hskip 2pt^{jklmnp} &= \kappa\hskip 2pt^{jklmnp} + 10\kappa\hskip 2pt^{jkl} \kappa^{mnp}. &(}]

The sixth- and higher-order cumulants and quasimoments differ. Thus the third-order cumulant [\kappa\hskip 2pt^{jkl}] contributes not only to the coefficient of [H_{jkl}], but also to higher-order terms of the probability distribution function. This is also the case for cumulants of higher orders. It implies that for a finite truncation of ([link], the probability distribution cannot be represented by a finite number of terms. This is a serious difficulty when a probability distribution is to be derived from an experimental temperature factor of the cumulant type. The one-particle potential (OPP) model

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When an atom is considered as an independent oscillator vibrating in a potential well [V({\bf u})], its distribution may be described by Boltzmann statistics. [P({\bf u}) = N \exp \{- V({\bf u})/kT\}, \eqno(] with N, the normalization constant, defined by [{\textstyle\int} P({\bf u})\ \hbox{d}{\bf u} = 1]. The classical expression ([link] is valid in the high-temperature limit for which [kT \gg {\it V}({\bf u})].

Following Dawson (1967)[link] and Willis (1969)[link], the potential function may be expanded in terms of increasing order of products of the contravariant displacement coordinates: [{V = V_{0} + \alpha_{j}u\hskip 2pt^{j} + \beta_{jk}u\hskip 2pt^{j}u^{k} + \gamma_{jkl}u\hskip 2pt^{j}u^{k}u^{l} + \delta_{jklm}u\hskip 2pt^{j}u^{k}u^{l}u^{m} + \ldots .} \eqno(] The equilibrium condition gives [\alpha_{j} = 0]. Substitution into ([link] leads to an expression which may be simplified by the assumption that the leading term is the harmonic component represented by [\beta_{jk}]: [\eqalignno{P({\bf u}) &= N \exp \{- \beta^{'}_{jk}u\hskip 2pt^{j}u^{k}\}\cr &\quad \times \{1 - \gamma^{'}_{jkl}u\hskip 2pt^{j}u^{k}u^{l} - \delta^{'}_{jklm}u\hskip 2pt^{j}u^{k}u^{l}u^{m} - \ldots \}, &(}] in which [\beta^{'} = \beta / kT] etc. and the normalization factor N depends on the level of truncation.

The probability distribution is related to the spherical harmonic expansion. The ten products of the displacement parameters [u\hskip 2pt^{j}u^{k}u^{l}], for example, are linear combinations of the seven octapoles [(l = 3)] and three dipoles [(l = 1)] (Coppens, 1980[link]). The thermal probability distribution and the aspherical atom description can be separated only because the latter is essentially confined to the valence shell, while the former applies to all electrons which follow the nuclear motion in the atomic scattering model.

The Fourier transform of the OPP distribution, in a general coordinate system, is (Johnson, 1970a[link]; Scheringer, 1985a[link]) [\eqalignno{T({\bf H}) &= T_{0} ({\bf H}) \left[1 - {4 \over 3} \pi^{3} i \gamma^{'}_{jkl} G\hskip 2pt^{jkl} ({\bf H}) + {2 \over 3} \pi^{4} \delta^{'}_{jklm} G\hskip 2pt^{jklm} ({\bf H})\right.\cr &\quad\left. + {4 \over 15} \pi^{5} i \varepsilon^{'}_{jklmn} G\hskip 2pt^{jklmn} ({\bf H}) - {4 \over 45} \pi^{6} i \varphi^{'}_{jklmnp} G\hskip 2pt^{jklmnp} ({\bf H}) \ldots\right],\cr& &(}] where [T_{0}] is the harmonic temperature factor and G represents the Hermite polynomials in reciprocal space.

If the OPP temperature factor is expanded in the coordinate system which diagonalizes [\beta_{jk}], simpler expressions are obtained in which the Hermite polynomials are replaced by products of the displacement coordinates [u\hskip 2pt^{j}] (Dawson et al., 1967[link]; Coppens, 1980[link]; Tanaka & Marumo, 1983[link]). Relative merits of the three expansions

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The relative merits of the Gram–Charlier and Edgeworth expansions have been discussed by Zucker & Schulz (1982)[link], Kuhs (1983)[link], and by Scheringer (1985b)[link]. In general, the Gram–Charlier expression is found to be preferable because it gives a better fit in the cases tested, and because its truncation is equivalent in real and reciprocal space. The latter is also true for the one-particle potential model, which is mathematically related to the Gram–Charlier expansion by the interchange of the real- and reciprocal-space expressions. The terms of the OPP model have a specific physical meaning. The model allows prediction of the temperature dependence of the temperature factor (Willis, 1969[link]; Coppens, 1980[link]), provided the potential function itself can be assumed to be temperature independent.

It has recently been shown that the Edgeworth expansion ([link] always has negative regions (Scheringer, 1985b[link]). This implies that it is not a realistic description of a vibrating atom.


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