Energy Response of Atomic and Molecular Orbitals in Nonuniform Magnetic Fields

The response of atoms and molecules to nonuniform magnetic fields is far less explored than uniform fields. In this work, we analyze the orbital and spin-Zeeman interactions of the electrons with a spatially nonuniform linearly varying magnetic field at the atomic and molecular orbital levels and their role in shaping the net response. Gauge-origin-invariant finite field restricted Hartree-Fock computations are carried out to examine the orbital-Zeeman effect, while the general Hartree-Fock method with a two-component spinor representation of the orbitals helps gauge the combined space-spin response. The nature of degeneracy lifting of the orbitals is found to be distinct from that for uniform fields. Degeneracy lifting of various orbitals by the so-called "diamagnetic" A² term is discussed for the first time to the best of our knowledge. A strong directionality of the response is uncovered and explained. Moreover, the role of the reference point for the gradient of the field, which has hitherto been largely ignored, is investigated in detail. The guiding principles for understanding the energy shifts of the atomic and molecular orbitals with change in the reference point are determined, and the minimum energy position and orientation of the ground states of homo- and heterodiatomic molecules relative to the field applied are discussed.