The formalism presented in a previous paper for the introduction of relativistic effects into nonrelativistic calculations is used to develop an approximation which is correct to order alpha(2) and is similar to the Breit-Pauli approximation. Although it corresponds to a projection of the Dirac Hamiltonian onto the nonrelativistic (Levy-Leblond) states which in principle should guarantee a lower bound, the bound depends on the form of the potential. The physical eigenstates correspond to a local minimum in the exponential parameter space which vanishes at large Z. In an extended basis set an approximation to the hydrogenic ground state can always be identified, but the relativistic correction to the energy is grossly overestimated. In molecular calculations, the restriction of the variational space involving the high-exponent functions by general contraction did not yield improved numerical stability in a variational scheme. The use of a low-order approximation in a quasivariational method is therefore not expected to yield reliable results.