After the ECI basis is truncated due to strong orthogonality violations, it is convenient to perform spin adaptation considering the set of desired multiplicities. In particular, the ESDs are not necessarily eigenfunctions of the operator of the full-system electronic spin ( Ŝ2 ), and thus not all of them will contribute to the full-system states of a desired multiplicity 2S + 1. However, each ESD has a well-defined spin projection value M S = F m F s . For a non-relativistic full-system Hamiltonian, there is no coupling between two ESDs with different M S , i.e., the Hamiltonian matrix is block-diagonal, with one block for each value of M S . For this reasons, when one calculates the full-system states of spin S, it is sufficient to diagonalize only a single block, corresponding to any M S value in the range -S, -S + 1, . . . , S. The most convenient choice is M S = S, as this block contains the fewest ESDs for a given S. After such a block is diagonalized, only states of the chosen M S are obtained-if states of other M S values are desired, they could be generated by means of the Wigner-Eckart theorem. Taking only ESDs with M S = S into the ECI basis trims the basis substantially and also reduces its scaling with the number of different multiplicities of the provided site states, decreasing computational cost.