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

International Tables for Crystallography (2006). Vol. B, ch. 1.2, pp. 18-19   | 1 | 2 |

Section 1.2.10. The vibrational probability distribution and its Fourier transform in the harmonic approximation

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@acsu.buffalo.edu

1.2.10. The vibrational probability distribution and its Fourier transform in the harmonic approximation

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For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian, centred at the equilibrium position. For the three-dimensional isotropic harmonic oscillator, the distribution is [P(u) = (2\pi \langle u^{2}\rangle)^{-3/2} \exp \{-| u |^{2} / 2 \langle u^{2}\rangle\}, \eqno(1.2.10.1)] where [\langle u^{2}\rangle] is the mean-square displacement in any direction.

The corresponding trivariate normal distribution to be used for anisotropic harmonic motion is, in tensor notation, [P({\bf u}) = {|{\boldsigma}^{-1}|^{1/2} \over (2\pi)^{3/2}} \exp \{-\textstyle{1 \over 2} {\boldsigma}_{jk}^{-1} (u\hskip 2pt^{j}u^{k})\}. \eqno(1.2.10.2a)] Here σ is the variance–covariance matrix, with covariant components, and [|{\boldsigma}^{-1}|] is the determinant of the inverse of σ. Summation over repeated indices has been assumed. The corresponding equation in matrix notation is [P({\bf u}) = {|{\boldsigma}^{-1}|^{1/2} \over (2\pi)^{3/2}} \exp \{\textstyle-{1 \over 2} ({\bf u})^{T} {\boldsigma}^{-1} ({\bf u})\}, \eqno(1.2.10.2b)] where the superscript T indicates the transpose.

The characteristic function, or Fourier transform, of [P({\bf u})] is [T({\bf H}) = \exp \{-2\pi^{2} \sigma\hskip 2pt^{jk} h_{j}h_{k}\} \eqno(1.2.10.3a)] or [T({\bf H}) = \exp \{-2\pi^{2} {\bf H}^{T} {\bf \boldsigma H}\}. \eqno(1.2.10.3b)] With the change of variable [b\hskip 2pt^{jk} = 2\pi^{2} \sigma\hskip 2pt^{jk}], (1.2.10.3a)[link] becomes [T({\bf H}) = \exp \{-b\hskip 2pt^{jk} h_{j}h_{k}\}.]








































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