In the context of some quantum-mechanical problem, there occurs a series development which gives the dependence of an "energy-type" eigenvalue $\varepsilon$ on some parameter $Q$. $\Psi_i$ are vectors in a Hilbert space, that are eigenvectors of a symmetric (real) operator $\hat{H}(Q)$ that depends parametrically on $Q$.
The eigenvalues of $\hat{H} = \hat{H}(Q=0)$ in increasing order with $i$ are $\varepsilon_i^0 \le 0$. With that one can express the dependence of $\varepsilon$ on $Q$ assuming $\Psi_i$ form a complete basis like \begin{eqnarray*} \varepsilon(Q) = & \varepsilon_0^0 + c_1 Q + \frac{1}{2} c_2 Q^2 + \mathcal{O}(Q^3) \\ = & \varepsilon_0^0 + \\ + & <{\Psi_0|\frac{\partial\hat{H}}{\partial Q}|\Psi_0}> Q + \\ + & \frac{1}{2} \Bigg(<{\Psi_0|\frac{\partial^2\hat{H}}{\partial Q^2}|\Psi_0}> + 2 \sum_{i>0}\frac{<{\Psi_0|\frac{\partial\hat{H}}{\partial Q}|\Psi_i}>^2}{\varepsilon_0^0 - \varepsilon_i^0} \Bigg) Q^2\\ + & \mathcal{O}(Q^3). \end{eqnarray*} (this is something called perturbation expansion, but should be assumed valid)
I am interested in the second order term. The first summand in there shall be assumed to be positiv: \begin{equation*} S_1 = <{\Psi_0|\frac{\partial^2\hat{H}}{\partial Q^2}|\Psi_0}>\;\;\; > 0. \end{equation*} The second one \begin{equation} S_2 = 2 \sum_{i>0}\frac{<{\Psi_0|\frac{\partial\hat{H}}{\partial Q}|\Psi_i}^2>}{\varepsilon_0^0 - \varepsilon_i^0} < 0 \end{equation} is negative since \begin{equation*} \varepsilon_i^0 \le \varepsilon_j^0 < 0, \end{equation*} for $ i < j$.
I am interested to see in which cases $$S_1 + S_2 > 0$$ would hold.
In particular it would be interesting to see if when can impose conditions on the terms in $S_2$ s.t. $S_1 + S_2 > 0$ would follow.
Conditions on $S_2$ which are given and should be used are
$<\Psi_0|\frac{\partial\hat{H}}{\partial Q}|\Psi_0>$ = 0.
$|S_2|$ is large in general
$ |\frac{<{\Psi_0|\frac{\partial\hat{H}}{\partial Q}|\Psi_1}>^2}{\varepsilon_0^0 - \varepsilon_1^0}|$ is large.
assumptions on $|{\varepsilon_0^0 - \varepsilon_1^0}|$ can be made
$Q$ belongs to a generator representation of the symmetry group of $\hat{H}$
I fail to see if such assumptions could possibly imply anything in general on the relative size of $S_1$ and $S_2$, but I can imagine that having informations on all of the summands in $S_2$ could have implications on $S_1$ in the way that $\Psi_0$ is a kind of orthogonal complement on the subspace spanned by $\Psi_i$ with $i>0$.
Can anyone help?
If $$\hat{H}(Q) = H + QV +Q^2 W$$ then $S_1 = \langle \Psi_0 | W |\Psi_0\rangle $, so you can satisfy your condition by choice of $W$.
If the perturbation expansion is assumed valid then we might as well assume that an expansion of $\hat{H}(Q)$ in powers of $Q$ is also valid, so your condition just depends on the relative size of the first and second terms.