Intermolecular force and energy operators. III

Abstract

Semirelativistic force and energy operators for the interaction of molecules (with moving nuclei) are derived for intermediate range separations. These operators are accurate through all of the (α4mc2 or 1/c2) fine-structural terms (but not the Lamb shift). The molecules interact by means of the internal electromagnetic fields which result from their charges, multipoles, and motions. Thus, the force on molecule a due to b is the generalized Lorentz force: QaE(i)(ra;b)+ (1/c)Ia×B(i)(ra;b), where Qa and Ia are the effective charge and current on a, while E(1)(ra;b) and B(i) (ra;b) are the internal fields of b at ra. It is shown that the interaction Hamiltonian Hab is minus the Lagrangian of the cross internal fields of a and b and can be expressed as Hab = QaQb[1/rab]− [1/c2]{Ia ⋅ Ib[1/rab]+ IaIb:[rabrab/rab3]}, where rab = ra−rb is the distance between the centers-of-mass. Both Qa and Ia are expressed in terms of the electric and magnetic multipole moments. To obtain full semirelavistic precision, the electric multipoles are expressed in terms of the Kracjik–Foldy coordinates. Thus, the electric dipole moment contains spin–orbit and purely relativistic terms. The usual nonrelativistic van der Waals energy operator corresponds to setting Ia = 0. The semirelativistic corrections are usually very small but still observable. For arrays of molecules, the interaction Hamiltonians are pairwise additive and long ranged. Therefore, the semirelativistic terms may be important in condensed media such as electrolytes or biologicals. This research is a byproduct of two previous papers on the interaction of molecules with electromagnetic fields.