Three-Dimensional Rotational Averaging Using Irreducible Sets of Linearly Independent Fundamental Isotropic Cartesian Tensors: A Computational Approach

The theoretical formulation of linear
and nonlinear molecular
spectroscopies
applied to isotropic samples (e.g., liquid or gas solutions) goes
through a fundamental step known as the rotational averaging of Cartesian
tensors. Rotational averaging of Cartesian tensors is a mathematical
procedure from which the expressions for the rotationally invariant
observables (e.g., rates or intensities), associated with a given
spectroscopic process, can be found. In this work, the mathematical/computational
procedure for finding the rotational averages of Cartesian tensors
of any rank n, which is based on the use of the fundamental
isotropic Cartesian tensors (FICTs), is discussed. Moreover, for the
first time, a heuristic computational method for finding a set of
linearly independent FICTs is proposed. The procedure has been tested
for 2 ≤ n ≤ 12, where most of the linear
and nonlinear molecular spectroscopies apply (e.g., one-photon and
multiphoton absorption, emission, electronic circular dichroism, Raman
optical activity, coherent and incoherent mth-harmonic
generation, etc.). Finally, it is shown how this computational procedure
can be extended for n > 12.