We will here limit our discussion to the optimization and characterization of two-state ConIns. These are points in nuclear coordinate space where two adiabatic electronic states are degenerate, and where that degeneracy is lifted at first order in a two-dimensional coor-dinate sub-space is termed the branching space. These points of degeneracy are not isolated, but form continuously connected seams of degeneracy in the remaining N -2 coordinate subspace, where N is the number of internal degrees of freedom. One of the key approaches for characterizing this seam space is the optimization of minimum energy nuclear configurations on the seam. That is, minimum energy conical intersections (MECIs).