Any spherical function f can be decomposed into spherical harmonics as:

where angles and define directions in space, N is the order of approximation, Y_l^m are spherical harmonics, and coefficients c_l^m give representation of f with respect to the spherical harmonics basis. Rotation invariant descriptors of f can be obtained as:

where f_l are the frequency components of f,

This representation has the property of being independent of the orientation of a spherical function.

Rotation invariant representations of scalar data

Each physicochemical property of a molecule (P) given on a cubic grid,

where indices i, j, k enumerate grid points, and P is assigned zero if point x_i, y_j, z_k does not belong to the surface of the molecule, is transformed to spherical coordinates (system of spherical coordinates is centered on the geometric center of the molecule):

using:

where:

where B defines the dimensions of the equiangular grid 2B x 2B.

This way the property P(x_i,y_j,z_k) given on the surface of molecules is transformed into a set of spherical functions:

given on spheres with radii r_i between 0 and R_max.

Rotation invariant descriptors of the property P are constructed as matrices:

If the property P changes the sign (as for example in case of electrostatic potentials), decomposition is made separately for its positive and negative values, i.e. the property P(x_i,y_j,z_k) is decomposed into two parts, P⁺(x_i,y_j,z_k) and P^–(x_i,y_j,z_k) and described with two sets of descriptors, Z(P⁺) and Z(P^–).