second-order correction
The second-order correction is the next refinement to a system's energy under a small static disturbance, beyond the first-order average. Where the first order pretended the state never changed, the second order finally accounts for the way the disturbance bends the state, mixing in small amounts of all the other unperturbed states. Each of those neighbours contributes a term, and we sum over every one of them.
Each contribution has a clear shape: it is the square of how strongly the disturbance connects the state to a neighbour, divided by how far below the neighbour lies in energy. States closer in energy and more strongly coupled matter most. A striking feature follows for the ground state: every other state lies above it, so every contribution lowers its energy. The lowest level is always pushed down at second order — a small but dependable rule.
Second order is where perturbation theory often earns its keep, because it captures effects the first order misses entirely, such as how nearby levels repel one another and how an atom polarizes in a field. The price is a sum over potentially infinitely many states, which can be hard to evaluate exactly. Its honesty, again, is the energy denominator: when two states are degenerate, that denominator vanishes and the naive formula must be replaced by degenerate perturbation theory.
Sum over all other states: coupling squared divided by the energy gap; for the ground state every term lowers the energy.
The infinite sum over intermediate states is exact in principle but rarely tractable. In practice one truncates it, uses sum rules, or recasts it as a solvable equation to avoid summing every level by hand.