Analytical expressions for the time evolution of spin systems affected by two or more interactions

Analytical expressions for the description of the time evolution of spin systems beyond product-operator formalism (POF) can be obtained if a low-dimensional subspace of the Liouville space has been found in which the time evolution of the spin system takes place completely. This can be achieved using a procedure that consists of repeated application of the commutator of the Hamiltonian with the density operator. This iteration continues as long as the result of such a commutator operation contains a term that is linearly independent of all the operators appearing in the previous commutator operations. The coefficients of the resulting system of commutator relations can be immediately inserted into the generic propagation formulae given in this article if the system contains two, three, or four equations. In cases where the validity conditions of any of these propagation formulae are not met, the coefficients are used as intermediate steps to obtain both the Liouvillian and propagator matrices of the system. Several application examples are given where an analytical equation can be obtained for the description of the time evolution of small spin systems under the influence of two or more interactions. This procedure for finding the Liouvillian matrix is not limited to time-independent interactions. Some examples illustrate the treatment of time-dependent problems using this method.