An analytic approximation to the singlet order decay rate constant under chemical exchange may be constructed by considering the eigenvalues of the join generator of motion Ĝ. For a two-site exchange model the matrix representation of Ĝ is of dimension dim( Ĝ) = 2 × N 2 H , where N H is the dimension of the underlying Hilbert space. But as discussed in appendix B the singlet order dynamics for the combined ensemble (A+B) are well approximated by the following 4 × 4 matrix sub-block