Relativistic Effect and Nonconservative Spin–Orbital Forces in the Radiative Transition of Molecules

Abstract

A treatment is given of a one-electron orbital model for a many-electron molecule, in which each electron is allowed to interact with the over-all orbital and spin magnetic fields, as well as the (Coulomb) electric field of the rest of the electrons and nuclei. It is shown that when redundancy is properly taken care of, by introducing a factor of 12 for the mutual-magnetic vector potential Aij between the electrons, the subsequent reduction of Dirac's equation reproduces all of Darwin's orbit–orbit, spin–own-orbit, spin–other-orbit, and spin–spin interactions, etc., given by the Breit–Pauli approximation. The above treatment is extended to a system of Dirac electrons interacting with the (time-dependent) electromagnetic field of radiation, in which field–field interaction in the form of Aij·Aj (radiation), is also included. After integration over the photon space, effective transition operators for the large-component spinors are obtained. When the non-Hermitian part of the “Dirac Hamiltonian” for the large-component spinors is not neglected as was customarily done, it is shown that the effective transition operators are already complete, in the sense that they already contain the relativistic-correction-of-mass terms, nonconservative force terms, and corrections due to operation on small-component spinors. This is shown from the Hermiticity of the operators as well as from explicit derivations. In explicit derivations, the origin of each of the effective transition operators is traced and tabulated. It is shown that for a spinless electron, or a Dirac electron described by an equation linear in time, and by a four-component spinor, the use of the transition dipole-length operator is justified. However, the straightforward use of the transition dipole-length operator on the large-component spinors, approximated by Breit–Pauli eigenstates, would neglect not only corrections due to small-component spinors but also the “ordinary” spin radiation term due to the direct spin interaction with the magnetic field of radiation. This interaction is shown to come from a nonconservative force acting on the electron, which is not derivable from a vector potential. All other external electromagnetic forces derivable from a vector potential are shown to contribute to radiation in a similar manner, and such contributions are obtainable by analogy to the classical Lagrangian–Hamiltonian formalism for conservative forces. The internal orbital and spin magnetic forces, such as spin–own-orbit interaction, are shown to contribute to radiation through the non-Hermitian part of the “Dirac Hamiltonian.” Although these forces can be represented by “effective” vector potentials, it is emphasized that the latter are not usable as (the external) vector potentials in the relativistic Lagrangian, and in the Dirac equation. The magnitudes of the effective transition operators obtained in our treatment are estimated (Table I, see also Table II) for a constant vector potential of radiation over the molecule, and for a (linearly) position-dependent vector potential. For application to single–triplet transitions, the relative importance of the spin-dependent transition operators, coming from nonconservative forces, non-Hermitian terms, and from corrections for small-component spinors is examined. Explicit expressions of these operators for different propagation and polarization directions are also given (Table III). For example, spin–other-orbit interaction is shown to contribute to Δ S = ± 1 radiative transitions between states of the same parity. The two-electron radiative transition operators, due to the inclusion of the mutual electromagnetic interaction of the electrons, may be of interest in cooperative optical phenomena, double excitations, and in the radiative contribution to van der Waals forces. The inclusion of the radiative transition operators from the non-Hermitian term also offers several alternative two-photon transition possibilities.