Variational approach to search the excitations. What will happen if start from wrong reference state?

Question

By 'wrong reference state' I mean a state which cannot be transformed into desired ones via variational ansatz

$\left|\Psi\left[\mathbf{n}\right]\right\rangle =e^{i\hat{O}\left[\mathbf{n}\right]}\left|ref\right\rangle \;,\; H\left[\mathbf{n}\right]=\left\langle \Psi\left[\mathbf{n}\right]\right|\hat{H}\left|\Psi\left[\mathbf{n}\right]\right\rangle $

where $\mathbf{n}$ is the variational parameter (field).

To be concrete, if we started with the first excitation state in a gapped system, i.e. choosing (making a guess) this first excitation state to be the reference state, then it seems to me that we will never reach the true ground state, as well as the correct excitations. The manifolds are disconnected because of the gap.

Are there any concrete examples for my general question here? Maybe I have not properly described this question. Please let me know if you are interested but don't understand. I will try to clarify.

Answer 1

If the true ground state is not in (or close to) the manifold of states over which the variation is done, the solution found will simply be meaningless.

Thus a good numerical scheme will have an accompanying analysis that the approximation scheme is at least in principle able to approximate the desired solution.