In a magnetic field anions are known to possess, within the approximation of infinite nuclear mass, an infinite manifold of bound states. This approximation, however, neglects effects due to the coupling of the anionic center-of-mass (c.m.) motion to the electronic motion. Recently these effects have been reported to be crucial for the mere existence and properties of the field-induced anions [Phys. Rev. Lett. 86, 5450 (2001)]. We develop a theoretical approach to study anions in the presence of a magnetic field including the motional c.m. effects. We aim at weak to moderate magnetic-field strengths typical for present-day laboratories. Our approach involves canonical transformations of the original Hamiltonian, implementing the integrals of motion and reasonable adiabatic approximations. As a result we derive a three-dimensional Hamiltonian that can be universally applied to atomic or molecular negative ions in a magnetic field by specifying the properties of the underlying neutral systems. We consider this Hamiltonian to be suitable for rigorous investigations of moving anions in fields. For the classical dynamics a reduction to an effective two-dimensional problem is possible. This can serve as a basis for future studies, for instance, of the autodetachment dynamics of anions due to c.m. effects.
A general quantum theory is presented for unconventional anionic states supported by the presence of an external magnetic field. The theory applies to atomic anions and allows for straightforward extensions to anions formed in magnetic fields by other species, e.g., by clusters or small molecules. A special focus of the theory is on the coupling of the anion's motion across the magnetic field to the motion of the attached electron. Neglecting this coupling, the magnetically induced anionic states are known to constitute an infinite manifold of bound states. In reality, the number of bound anionic states is finite. Typically, the quantized motion of the anion in the field results in sequences of excitations. These might include, depending on properties of the anion and on the magnetic field strengths, a few or a substantial number of states. Explicit results obtained by quantum ab initio calculations are presented and discussed on bound states and radiative transitions for some experimentally relevant atomic anions.
The interaction between the center of mass (c.m.) and internal degrees of freedom plays a crucial role for the properties and behavior of moving negative ions in magnetic fields. A promising theoretical approach, including this interaction to describe field-induced anions, has been developed very recently. The theory is aimed at weak to moderate magnetic field strengths and therefore of immediate relevance to laboratory experiments. In the present work, we apply this approach to study the classical dynamics of the anions in magnetic fields. We focus on features of the bound motion of anions and on the analysis of the anions' autodetachment induced by the c.m. motion.
In a magnetic field, an atom (or molecule) can attach an extra electron to form an unconventional anionic state which has no counterparts in field-free space. Assuming the atom to be infinitely heavy, these magnetically induced anionic states are known to constitute an infinite manifold of bound states. In reality, the species can move and its motion across the magnetic field couples to the motion of the attached electron. We treat this coupling, for the first time, quantum mechanically, and show that it makes the number of bound anionic states finite. Explicit numerical quantum results are presented and discussed.
A thorough unified treatment is given of the quantum-mechanical operators and wave functions for a molecular system (composed of
N moving charged particles) in static uniform electric and magnetic fields
E and
B. The treatment is rigorous within the nonrelativistic approximation. The system may either be neutral or charged. The fields may have arbitrary intensities and orientations. Close correspondence is maintained between the classical and quantum-mechanical treatments. The wave functions are expressed both in the time-independent energy representation and in time-dependent wave packets. Three types of momentum play important roles. For single-particle systems they are the canonical momentum
P, the mechanical momentum $\mathit{\ensuremath{\Pi}}=\mathit{P}\ensuremath{-}(\frac{e}{c})\mathit{A}=M\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{R}$, and the pseudomomentum $\mathcal{K}=\mathit{\ensuremath{\Pi}}\ensuremath{-}(\frac{e}{c})\mathit{R}\ifmmode\times\else\texttimes\fi{}\mathit{B}\ensuremath{-}e\mathit{E}t$. The pseudomomentum is a constant of the motion. Except in the absence of magnetic fields, not all of the components of either $\mathit{\ensuremath{\Pi}}$ or
K commute. This complicates the quantum-mechanical formalism. The components of the pseudomomentum have simple classical interpretations, and in quantum mechanics they are related to the operator which performs a boost to a reference frame moving with constant velocity
v. In this moving frame, the electric field intensity is ${\mathit{E}}^{\ensuremath{'}}=\mathit{E}+(\frac{\mathit{v}}{c})\ifmmode\times\else\texttimes\fi{}\mathit{B}$. Thus, in a frame moving with the drift velocity ${\mathit{v}}_{d}=(\frac{c}{{B}^{2}})\mathit{E}\ifmmode\times\else\texttimes\fi{}\mathit{B}$, the components of the electric field intensity perpendicular to
B vanish. In this paper, the velocity boost operators are used to show the relationships between wave functions expressed in reference frames moving with repsect to each other. The dynamics of the
N-particle systems are simplified by making the Power-Zienau-Woolley transformation (which reduces as a special case to a unitary transformation used by Lamb) and by using center-of-mass and internal coordinates. Generalizing previous works, it is shown how the pseudomomentum is involved in separating these degrees of freedom for
N particles in both
E and
B. For neutral molecules, the Schr\"odinger equation is "pseudoseparated" and the internal degrees of freedom are coupled to the center of mass motion only by the "motional Stark Effect," which involves the constant of motion
K. For ionic systems, only one component of the center of mass is coupled to the internal motion. Quantitative estimates of the weak center-of-mass coupling are made for both neutral and ionic
n=1 and
n=2 quantum states of two-body systems by perturbation expansions in powers of the field strengths. In the usual nonrigorous treatments of systems in magnetic fields, no distinction is made between $\mathit{\ensuremath{\Pi}}$ and
K and both are approximated by $M\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{R}$, where
M is the mass of the molecule and
R is taken to be the classical orbital position of a single particle having the same mass and charge. This is an excellent approximation for ground-state molecules and ions in weak fields. However, the resulting errors can be much larger if the system is in either intense fields or high-lying states (e.g., Rydberg levels). These are the conditions under which the rigorous formalism should be most useful. The determination of the wave functions and energy levels for many-particle ionic systems is complicated due to the coupling with the center of mass. The suggestion is made that it might be useful to boost the reference frame so that it moves with the velocity $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{R}$ of a single classical particle. The Schr\"odinger equation in this frame could serve as a more suitable basis for perturbative or variational treatments.