Solution to the problem of variational collapse for the Dirac equation
Robert Nyden Hill, C. Krathauser · Bulletin of the American Physical Society · 1993
The conventional Rayleigh-Ritz variational principle as applied to the Dirac equation is merely a stationary principle. It is not a minimum principle because the Dirac Hamiltonian H{sub Dirac} is not bounded below; it is not a maximum principle because the Dirac Hamiltonian is not bounded above. This is the well known problem of variational collapse, which causes difficulty when basis set methods axe used to construct approximations to the bound states of the Dirac equation. Variational collapse can be avoided by applying the Rayleigh-Ritz method to the operator G = 1/H{sub Dirac}. The problem of calculating matrix elements of 1/H{sub Dirac} is avoided by using a variational trial function {vert_bar}{phi}{r_angle} of the form {vert_bar}{phi}{r_angle} = H{sub Dirac} {vert_bar}{psi}{r_angle} where {vert_bar}{psi}{r_angle} is a linear combination of basis functions. Matrix elements of H{sub Dirac} and H{sub Dirac}{sup 2} are needed. If the Gram (overlap) matrix is also available, upper and lower bounds to bound state energies and bounds on the errors in the wave functions in the L{sup 2} norm can be computed.