Here, S (lkji) circ = 1 + (lkji) + (lkji) 2 + (lkji) is the symmetrizer for the circular shift group of four indices. The elements of S (kji) generate the six left cosets of this subgroup, such that S (kji) S (lkji) circ = S (lkji) is the full symmetrizer of the permutation group of four indices. Out of the 10 terms in Eq. (128), terms 8 and 9 are each other's complex conjugate, i.e., only one of them needs to be calculated explicitly. The complete 4th derivative of the free energy can be written as (passing back from simplified to full notation) ∂ 4 F ∂λ i ∂λ j ∂λ k ∂λ l = S (lkji)