InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B, ch. 1.2, pp. 18-19
## Section 1.2.10. The vibrational probability distribution and its Fourier transform in the harmonic approximation |

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

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 , (1.2.10.3*a*) becomes