Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 4 / 59 / b / iv / Solution

The midpoint-odd perturbation couples the ground state only to even-index states. Direct integration givesInsertion in the second-order expression yieldsIt is negative because all coupled states lie above the ground state. This is the coefficient of ${\lambda }^2$; the physical energy shift includes the additional factor $q^2{\epsilon }^2$.