Since their intersection is parametrized by k − − 2 coordinates, we may formally state that for k continuous real parameters controlling the perturbed Hamiltonian, the levels (or surfaces) can only cross at a manifold of dimension k − − 2 . However the symmetry of the Hamiltonian has a role to play in the dimensionality. If the original Hamiltonian has asymmetric states, ⟨ ⟨ ψ ψ 1 | P | ψ ψ 2 ⟩ ⟩ ≠ ≠ ⟨ ⟨ ψ ψ 2 | P | ψ ψ 1 ⟩ ⟩ , the off-diagonal terms vanish automatically to ensure hermiticity. This allows us to get rid of the equation W = 0 . Now from similar arguments as posed above, it is straightforward that for an asymmetrical Hamiltonian, the intersection of energy surfaces takes place in a manifold of dimension k − − 1 .