A Phase-Space Electronic Hamiltonian for Molecules in a Static Magnetic Field II: Quantum Chemistry Calculations with Gauge Invariant Atomic Orbitals

In a companion paper, we have developed a phase-space electronic structure theory of molecules in magnetic fields, whereby the electronic energy levels arise from diagonalizing a phase-space Hamiltonian $\hat H_{PS}(\bf{X},\bf{\Pi})$ that depends parametrically on nuclear position and momentum. The resulting eigenvalues are translationally invariant; moreover, if the magnetic field is in the $z-$direction, then the eigenvalues are also invariant to rotations around the $z-$direction. However, like all Hamiltonians in a magnetic field, the theory has a gauge degree of freedom (corresponding to the position of the magnetic origin in the vector potential), and requires either $(i)$ formally, a complete set of electronic states or $(ii)$ in practice, gauge invariant atomic orbitals (GIAOs) in order to realize such translational and rotational invariance. Here we describe how to implement a phase-space electronic Hamiltonian using GIAOs within a practical electronic structure package (in our case, Q-Chem). We further show that novel phenomena can be observed with finite $\bf{B}-$fields, including minimum energy structures with $\bf{\Pi}_{min} \ne 0$, indicating non-zero electronic motion in the ground-state.