Invariants of spherical harmonics as atomic order parameters

Invariants formed from bond spherical harmonics allow to obtain quantitative information on the local atomic symmetries in materials. The analysis starts by associating a set of spherical harmonics with every bond linking an atom to its nearest neighbors. For a given bond defined by a vector $$\vec{r}$$ a spherical harmonic may be defined as: $Q_{lm}(\vec{r})\ =\ Y_{lm} \langle \theta (\vec{r}), \psi (\vec{r}) \rangle$ where $$Y_{lm}(\theta, \psi)$$ is the spherical harmonic associated to the bond, $$\theta$$ and $$\psi$$ are the angular components of the spherical coordinates of the bond which Cartesian coordinates are defined by $$\vec{r}$$.

Because the $$Q_{lm}$$ for a given $$l$$ can be scrambled by changing to a rotated coordinate system, it is important to consider rotational invariant combinations, such as [1], [2]: $Q_l\ =\ \left[\frac{4\pi}{2l+1} \sum_{m=-l}^{l} \left| \bar{Q}_{lm} \right|^2 \right]^{1/2}$ where $$\bar{Q}_{lm}$$ is defined by: $\bar{Q}_{lm}\ =\ \langle Q_{lm}( \vec{r} ) \rangle$ and represents an average of the $$Y_{lm}(\theta, \psi)$$ over all $$\vec{r}$$ vectors in the system whether these vectors belong to the same atomic configuration or not. Just as the angular momentum quantum number, $$l$$, is a characteristic quantity of the 'shape' of an atomic orbital, the quantity $$Q_l$$ is a rotationally invariant characteristic value of the shape/symmetry of a given local atomic configuration (if the average is not taken on all bonds of the system but only within a given configuration) or an average of such values for a set of configurations. Thus it is possible to compare $$Q_l$$'s computed for well known crystal structures (e.g. FCC, HFC ...) and some local atomic configurations in a material's model. The results of the comparison gives information for the presence/absence of a particular local atomic symmetry.

Atomes allows to compute the average $$Q_l$$'s for each chemical species as well as the average $$Q_l$$'s for a user specified local atomic coordination.

1. P. Steinhardt, D. R. Nelson, and M. Ronchetti, Phys. Rev. B., vol. 28, no. 2, pp. 784–805, 1983.
2. A. Baranyai et al., Chem. Soc. Faraday Trans. 2, vol. 83, no. 8, pp. 1335–1365, 1987.