Stark and Zeeman Effects on the Protonic Structure of Molecules

The effect of a homogeneous electric field on the protonic coordinates of neutral hydrides is shown to be identical to the effect on the electronic coordinates. For ionized hydrides, the effects on the protonic coordinates are shown to be proportional to (1 —$\frac{N{m}_p}{M}$) where $m_p$ and $M$ are the protonic and total mass, respectively, and $N$ is the degree of ionization. A homogeneous magnetic field affects the protonic coordinates through terms proportional to $\frac{{\mathcal{H}}^2{m}_p}{M}$ and $\frac{N_e{\mathcal{H}}^2{m}_p^2}{M^2}$, where $\mathcal{H}$ is the strength of the field and $N_e$ is the number of electrons in the molecule. The terms of first order in $\mathcal{H}$ are found to be negligible for the protonic coordinates. The effects of both the electric and magnetic fields on the motion of the center of mass are also given. We show that in the presence of a homogeneous magnetic field in the $z$ direction, the center of mass of neutral atoms or molecules will vibrate in the $\mathrm{xy}$ plane with a frequency of $\omega ={\left(\frac{N_e{q}_e{}_{}{}^2{}_{}{}^{}\mathcal{H}_{}^2{}_{}{}^{}}{4M{m}_e{c}^2}\right)}^{\frac{1}{2}}$, where $c$ is the velocity of light.