Minimum energy conical intersections can be used to rationalize photochemical processes. In this Letter, we examine an algorithm to locate these structures that does not require the evaluation of nonadiabatic coupling vectors, showing that it minimizes the energy on hypersurfaces that envelop the intersection seam. By constraining the states to be separated by a small non-zero energy difference, the algorithm ensures that numerical artifacts and convergence problems of coupled cluster theory at conical intersections are not encountered during the optimization. In this way, we demonstrate for various systems that geometries at minimum energy conical intersections with the ground state are well described by the coupled cluster singles and doubles model, suggesting that coupled cluster theory may, in some cases, provide a good description of relaxation to the ground state in nonadiabatic dynamics simulations.