first-order correction
The first-order correction is the leading change to a system's energy when a small static disturbance is switched on. It is remarkably simple to state: take the disturbance, average it over the original, unperturbed state, and that average is the energy shift. In words, the system mostly stays as it was, and the energy moves by however much the new term contributes on the state it already occupies.
There is a tidy intuition behind this. Because the unperturbed state is the best description we have so far, the cheapest way to estimate the new energy is to evaluate the extra interaction without yet bothering to change the state at all. The wavefunction's own first-order distortion does not feed back into the energy at this leading order — a happy fact that makes first-order energy estimates both easy and surprisingly accurate when the disturbance is genuinely small.
In practice this single average does a lot of work. It tells you the leading splitting of atomic levels in an applied field, the leading shift from a small change in a potential, and the first guess for almost any gentle interaction. When it happens to vanish — for instance by a symmetry that makes the average zero — the real story begins at second order, where the state's distortion finally matters.
The leading energy shift is the disturbance averaged over the unperturbed state — no need to change the state yet.
A zero first-order shift does not mean no effect. It often signals that a symmetry forbids the leading term, and the genuine response shows up only at second order.