Interaction of molecules with electromagnetic fields. II. The multipole operators and dynamics of molecules with moving nuclei in electromagnetic fields

Abstract

This paper presents a thorough unified treatment of the electric and magnetic multipole operators and the dynamics of a moving molecular system of electrons and nuclei in the presence of an arbitrary (semiclassical) electromagnetic field. The multipole operators are expressed in terms of r j c , the position of each of the particles j relative to the center of mass r c , the velocities ? j c and ? c , and the spins s j . Two levels of precision of the multipole operators and dynamics are considered: The ’’n o n r e l a t i v i s t i c’’ approximation including all terms which vary as 1/c (where c is the velocity of light) suffices for most practical applications. The multipole moments are determined by the Lorentz force on the molecule. Also, the multipole operators are related to the electric and magnetic polarization operators Pop c and Mop c , respectively, as well as to the effective charge and effective current on the molecule. The Lagrangian is then determined by rearranging the ’’Newtonian’’ equations of motion into the Lagrangian form. In both the Hamiltonian and the Lagrangian, terms involving Pop c and Mop c couple the external fields to the molecular dynamics. The Hamiltonian is also derived in the ’’quantum mechanical fashion’’ by making a Power–Zienau–Woolley type transformation of the usual ’’minimal coupling’’ Hamiltonian. The new coordinates are r c and a set of (N−1) linearly independent combinations of the r j c . In the determination of the electric and magnetic properties of molecules, there are significant advantages in considering moving nuclei and center of mass coordinates rather than assuming clamped nuclei. In order to explain a few very sensitive types of experimental properties, it is necessary to use the s e m i r e l a t i v i s t i c approximation which is accurate through all of the α4 m c 2 or 1/c 2 terms and includes all of the fine‐structural effects with the exception of the Lamb shift. The electric and magnetic multipole moments were derived in terms of the Kracjik and Foldy semirelativistic coordinates and spins. In addition to the usual ’’dipole length’’ term, the semirelativistic electric dipole operator contains spin–orbit and purely relativistic terms. The related s e m i r e l a t i v i s t i c Hamiltonian in the Kracjik–Foldy coordinates is stated but its very complicated derivation will be presented in another paper.