Constraints in the variational principle for stationary states

Constraints in the variational principle for stationary states (VPSS) are classified in accordance with Dirac’s constrained classical mechanics and the time-dependent variational principle (TDVP). All of the VPSS constraints are required to belong to the first-class TDVP as constants of motion to ensure the real-valuedness of the Lagrange multipliers. The VPSS constraints are further classified as either first-class or second-class. The first-class VPSS constraints are constants of variation with symmetry-adapted wave functions. If the representation basis for the constraint operators is incomplete, however, the first-class VPSS constraints lead to broken-symmetry solutions. The nondegenerate energies of \({}^2E'\) at the \(D_{3h}\) geometry in the Jahn–Teller distortion of H\(_3\) are presented as an example of a problem with broken-symmetry. An ad hoc adjustment is suggested by considering the second-class pseudo-VPSS constraints with new adiabatic parameters.