Cumulant expansion

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, p. 588

Section 6.1.1.6.3. Cumulant expansion

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

6.1.1.6.3. Cumulant expansion

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In a cumulant expansion (Johnson & Levy, 1974), the entire series is expressed in exponential form. The cumulant expansion about S = 0 for the generalized temperature factor is $[\eqalignno{T({\bf S})&=\exp\bigg[1+i\kappa^jS_j+{i^2\over2!}\kappa^{jk}S_jS_k+{i^3\over3!}\kappa^{jkl}S_jS_kS_l\cr &\quad+{i^4\over4!}\kappa^{jklm}S_jS_kS_lS_m+\ldots\bigg],& (6.1.1.63)}]$ where the coefficient tensor $[\kappa^{\alpha\beta\ldots\zeta}]$ , a symmetric tensor of order p, is the pth-order cumulant. The inverse Fourier transform is the Edgeworth expansion around the Gaussian p.d.f. Cumulants can be expressed in terms of moments and vice versa. The pth moment $[\mu^{\alpha\beta\ldots\zeta}]$ (if it exists) of a general p.d.f., ρ(x), is a symmetric tensor defined as $[\mu^{\alpha\beta\ldots\zeta}({\bf x})=\textstyle\int\limits^\infty_{-\infty}x^\alpha x^\beta\ldots x^\zeta\rho({\bf x})\,{\rm d}{\bf x}.\eqno (6.1.1.64)]$ The relations between the lower-order moments and cumulants are $[\eqalign{\mu^j&=\kappa^j\cr \mu^{jk}&=\kappa^{jk}+\kappa^j\kappa^k\cr \mu^{jkl}&=\kappa^{jkl}+\kappa^j\kappa^{kl}+\kappa^k\kappa^{lj}+\kappa^l\kappa^{jk}+\kappa^j\kappa^k\kappa^l\cr &=\kappa^{jkl}+3\kappa^{(j}\kappa^{kl)}+\kappa^j\kappa^k\kappa^l\cr \mu^{jklm}&=\kappa^{jklm}+3\kappa^{j(k}\kappa^{lm)}+4\kappa^{(j}\kappa^{klm)}\cr &\quad+6\kappa^{(j}\kappa^k\kappa^{lm)}+\kappa^j\kappa^k\kappa^l\kappa^m} \eqno(6.1.1.65)]$ and, conversely, $[\eqalign{\kappa^j&=\mu^j\cr \kappa^{jk}&=\mu^{jk}-\mu^j\mu^k\cr \kappa^{jkl}&=\mu^{jkl}-3\mu^{(j}\mu^{kl)}+2\mu^j\mu^k\mu^l\cr \kappa^{jklm}&=\mu^{jklm}-3\mu^{j(k}\mu^{lm)}-4\mu^{(j}\mu^{klm)}\cr &\quad+12\mu^{(j}\mu^k\mu ^{lm)}-6\mu^j\mu^k\mu^l\mu^m.} \eqno(6.1.1.66)]$ In the Gram–Charlier and Fourier-invariant expansions, the Fourier-transform relationship between the p.d.f. and the temperature factor to given order can be made exact. Each cumulant $[\mu^{jkl}]$ contributes to all higher-order quasi-moment terms and vice versa. Hence, a given cumulant expansion is to an extent arbitrarily truncated (Kuhs, 1983). Care is required when interpreting the coefficients (Zucker & Schulz, 1982).

On the other hand, the cumulant expansion has the advantage of yielding tractable expressions for the one-particle potential in the quantum regime (Mair, 1980a). In that regime, equation (6.1.1.36) for the one-particle potential is invalid, and the expressions relating V(u) to ρ(u) in the Gram–Charlier and Fourier-invariant expansions are cumbersome (Mair & Wilkins, 1976).

Coefficients obtained by applying least-squares methods to structure-factor equations related to the truncated cumulant expansions do not necessarily yield non-negative p.d.f.'s nor are the linear-term coefficients necessarily faithful representations of the mean. Caution must be exercised in interpreting the results.

All the methods are satisfactory in the case of rapidly converging potential series. The methods are equivalent up to λ² in the van Hove order parameter (Mair, 1980b). Difficulties are encountered with convergence of the series in the case of strong anharmonicity, in which case numerical or alternative analytical models may be necessary. If the anharmonicity is such that the difference between the expansions is significant, it may be preferable to evaluate the Fourier transforms directly, as recommended by Mackenzie & Mair (1985).

References

Johnson, C. K. & Levy, H. A. (1974). Thermal motion of independent atoms. International tables for X-ray crystallography. Vol. IV, pp. 317–319. Birmingham: Kynoch Press.Google Scholar

Kuhs, W. F. (1983). Statistical description of multimodal atomic probability densities. Acta Cryst. A39, 149–158.Google Scholar

Mackenzie, J. K. & Mair, S. L. (1985). Anharmonic temperature factors: the limitations of perturbation-theory expressions. Acta Cryst. A41, 81–85.Google Scholar

Mair, S. L. (1980a). Temperature dependence of the anharmonic Debye–Waller factor. J. Phys. C, 13, 2857–2868.Google Scholar

Mair, S. L. (1980b). The anharmonic Debye–Waller factor in the classical limit. J. Phys. C, 13, 1419–1425.Google Scholar

Mair, S. L. & Wilkins, S. W. (1976). Anharmonic Debye–Waller factor using quantum statistics. J. Phys. C, 9, 1145–1158.Google Scholar

Zucker, U. H. & Schulz, H. (1982). Statistical approaches for the treatment of anharmonic motion in crystals. I. A comparison of the most frequently used formalisms of anharmonic thermal vibrations. Acta Cryst. A38, 563–568.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 6.1, p. 588