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 where is the mean-square displacement in any direction.

The corresponding trivariate normal distribution to be used for anisotropic harmonic motion is, in tensor notation, Here σ is the variance–covariance matrix, with covariant components, and is the determinant of the inverse of σ. Summation over repeated indices has been assumed. The corresponding equation in matrix notation is where the superscript T indicates the transpose.

The characteristic function, or Fourier transform, of is or With the change of variable , (1.2.10.3a) becomes