The problem of calculating singlet states with two non-degenerate singly occupied orthogonality-constrained orbitals is investigated. The generalized Brillouin condition of Levy and Berthier leads to problems of accuracy and convergence and may even yield energies far below a variational upper bound. An alternative condition is formulated requiring the cancellation of energy contributions from the interaction with single excitations between the open shells. In calculations using this condition accuracy and convergence is considerably improved although strict upper bounds to the energy are not obtained. Test calculations for some simple molecules show the utility of the new method.