Second-order nondegenerate perturbation theory

For a normalized nondegenerate eigenstate $|n⟩$,For a ground state, every denominator is negative, so the correction is nonpositive.

For a nondegenerate lowest state, every excited-state denominator $E_0-E_k$ is negative. Since each numerator $|V_{k0}{|}^2$ is nonnegative, their sum is nonpositive, and strictly negative if a coupled higher state exists. The statement assumes the perturbative expansion and appropriate sum or spectral integral are defined.