Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2B / Section B / 33 / c / iii / Solution

Exact solutions of the Schrödinger equation are unavailable for most interacting or realistic molecular systems. The quantum variational principle turns the search for a ground-state wavefunction into optimization of a computable energy quotient in a tractable trial family. The result is an upper bound: enlarging a nested trial space can only lower or preserve the optimized energy, providing a systematic convergence check. Trial functions can incorporate boundary conditions and known symmetry before calculation. Near an exact normalized ground state, a small orthogonal wavefunction error changes the energy only to second order, which explains why useful energies can result from imperfect trial functions. This does not ensure that every other observable is equally accurate, nor that a restricted trial family captures correlation adequately. Excited states require orthogonality or appropriate symmetry restrictions; unrestricted minimization seeks the ground state, as the preceding calculation demonstrates.