The time-domain ESEEM calculation incorporates the full density matrix diagonalization from the Mims density matrix treatment of ESEEM. Although the time evolution simulation of a pulsed magnetic resonance experiment is straightforward, we found that the explicit evaluation of the transition frequencies and intensities is computationally advantageous for optimization purposes, relative to methods that calculate the time evolution of the density matrix. Specifically, the following adopted numerical procedure, which we have used previously in ESEEM simulations, is found to be most efficient for constructing the two-pulse and three-pulse modulation in OPTESIM.
The Hamiltonian is first partitioned into sub-matrix representations Hα and Hβ corresponding to α (ms=1/2) and β (ms=-1/2) electron spin manifolds. Hα and Hβ are diagonalized separately with unitary matrices Mα and Mβ to obtain the eigen frequencies να and νβ belonging to the α and β manifolds, respectively. The Hamiltonian, H, is transformed into H’, which corresponds to the coupled representation, as follows:
where 
.
The modulation frequencies and amplitudes are
then calculated for specific types of experiments. For example, for the two-pulse echo envelope
modulation (pulse sequence, Figure 1), the frequencies 
and
amplitudes A are as follows:
The indices i, m, j, and k allow permutation through all elements of M and the eigenvalues.
The echo modulation amplitude 
is formulated as follows:
For the three-pulse echo modulation (pulse
sequence, Figure 1), the frequencies and
amplitudes A are separated into terms representing the α and β manifolds, as
follows:
),
),
The simulated echo modulation amplitude 
is formulated as follows :
For flexibility, users can easily substitute their own time-domain ESEEM calculations into OPTESIM as Matlab functions. Therefore, experimental systems and pulse sequences other than those considered here can be addressed by using the optimization and statistical assessment features of OPTESIM.