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 a spherical harmonic may be defined as:
Q_{lm}() = Y_{lm}〈θ(), φ()〉  (1) 
where Y_{lm}(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 .
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 [a, b]:
Q_{l} = ^{2}  (3) 
where is defined by:
= 〈Q_{lm}()〉  (2) 
and represents an average of the Y_{lm}(θ, φ) over all 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 infdormation for the presence/absence of a particular local atomic symmetry.
