  Recent experimental advances in ultrafast science put different processes
occurring on the electronic timescale below a few femtoseconds in focus. In the
present theoretical work, we demonstrate how the transformation and propagation
of the density matrix in the basis of irreducible spherical tensors can be
conveniently used to study sub-few fs spin-flip dynamics in the core-excited
transition metal compounds. With the help of the Wigner-Eckart theorem, such a
transformation separates the essential dynamical information from the geometric
factors governed by the angular momentum algebra. We show that an additional
reduction can be performed by the physically motivated truncation of the
spherical tensor basis. In particular, depending on the degree of coherence,
the ultrafast dynamics can be considered semi-quantitative in the notably
reduced spherical basis when only total populations of the basis states of the
given spin are of interest. Such truncation should be especially beneficial
when the number of the high-spin basis states is vast, as it substantially
reduces computational costs.
