= Solution
For $\psi=\sum_n\alpha_n\phi_n$ and an orthonormal trial set,
$$
\frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}
=\frac{\alpha^\dagger\mathcal H\alpha}{\alpha^\dagger\alpha},
\qquad
\mathcal H_{nm}=\langle\phi_n|H|\phi_m\rangle.
$$
Stationarity subject to $\alpha^\dagger\alpha=1$ gives
$$
\mathcal H\alpha=E\alpha.
$$
The <finite-subspace variational method>, equivalently the <Rayleigh-Ritz variational principle>, says that the minimum quotient is the smallest <eigenvalue> of $\mathcal H$. It is therefore the optimal upper bound on $E_0$ available from this span.
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