Invariants formed from bond spherical harmonics allow to obtain quantitative informations 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 $\displaystyle \vec{{r}} $ a spherical harmonic may be defined as:

Qlm($\displaystyle \vec{{r}} $)  =   Ylm〈θ($\displaystyle \vec{{r}} $), φ($\displaystyle \vec{{r}} $)〉 (1)

where Ylm(θ, φ) is the spherical harmonic associated to the bond, θ and φ are the angular components of the spherical coordinates of the bond which cartesian coordinates are defined by $\displaystyle \vec{{r}} $.

Because the Qlm for a given l can be scrambled by changing to a rotated coordinate system, it is important to consider rotational invariant combinations, such as [a, b]:

Ql  = $\displaystyle \left[\vphantom{ \frac{4\pi}{2l+1} \sum_{m=-l}^{l} \vert\bar{Q}_{lm}\vert^{2} }\right.$$\displaystyle {\frac{{4\pi}}{{2l+1}}}$ $\displaystyle \sum_{{m=-l}}^{{l}}$|$\displaystyle \bar{{Q}}_{{lm}}^{}$|2$\displaystyle \left.\vphantom{ \frac{4\pi}{2l+1} \sum_{m=-l}^{l} \vert\bar{Q}_{lm}\vert^{2} }\right]^{{1/2}}_{}$ (3)

where $\displaystyle \bar{{Q}}_{{lm}}^{}$ is defined by:

$\displaystyle \bar{{Q}}_{{lm}}^{}$  = 〈Qlm($\displaystyle \vec{{r}} $)〉 (2)

and represents an average of the Ylm(θ, φ) over all $\displaystyle \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 Ql 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 Ql'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 infdormation for the presence/absence of a particular local atomic symmetry.

a
P.J. Steinhardt, D.R. Nelson and M. Ronchetti.
Phys. Rev. B, 28(2):784-805 (1983).
b
I. Baranyai and Al.
Chem. Soc. Faraday Trans. 2, 83(8):1335-1365 (1987).




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