Gauge invariance and radiative transition probabilities

The relationship between recommendations of some recent papers on the choice of formula (length, velocity, or acceleration type) for calculating radiative transition probabilities is clarified. The problem of choosing the appropriate formula for use in oscillator strength calculations for many-electron systems has been discussed in a number of recent papers (Starace 1971, 1973, Grant 1974). The key difficulty is that the interaction between electrons and the radiation field is not gauge invariant unless the electron charge distribution satisfies exactly a continuity equation. This has the unfortunate consequence that transition probabilities for electric multipoles (but not magnetic multipoles), evaluated with approximate wavefunctions, depend on the choice of gauge and so are in general not unique. Our earlier papers contain apparently conflicting recommendations for overcoming the difficulty. The paper of Starace (1971) is written in the spirit of earlier work by Sachs and Austern (1951) and Korolev (1968). Let Hmod be a model Hamiltonian for a manyelectron system. Construct a Hamiltonian describing the interaction of this system with the electromagnetic field by making the usual replacement p → p – eA and then demand that the resulting operator be gauge invariant. This method of enforcing gauge invariance leads to a unique interaction operator proportional to the commutator [Hmod, Dl], where Dl is a multipole operator of order l. For electric-dipole transitions, the commutator only yields the dipole velocity or dipole acceleration operators for local potentials, whereas for both local and non-local potentials, it may be easily written in terms of the dipole length formula. This leads Starace to recommend the use of the dipole length form for problems in which Hmod contains a non-local potential, for example Hartree-Fock or ci calculations. On the other hand, the relativistic theory of radiative transitions described by Grant (1974) focuses attention on the (covariant) interaction Hamiltonian. He suggests that one should make a consistent choice of gauge for the (exact) atomic Hamiltonian and the interaction Hamiltonian so that gauge invariance is guaranteed for the combined electron-photon system. The conventional choice of the Coulomb gauge for the atomic Hamiltonian implies use of the dipole velocity operator for the interaction in the non-relativistic limit. Other gauges lead to a linear combination of the dipole velocity and dipole length operators in the non-relativistic limit, the result only being gauge independent if the charge distribution