= 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.
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