The time-dependent Schrödinger equation can be rewritten so that its interpretation is no longer probabilistic. Two well-known and related reformulations are Bohmian mechanics and quantum hydrodynamics. In these formulations, quantum particles follow real, deterministic trajectories influenced by a quantum force. Generally, trajectory methods are not applied to electronic structure calculations, since they predict that the electrons in a ground state, real, molecular wavefunction are motionless. However, a spin-dependent momentum can be recovered from the non-relativistic limit of the Dirac equation. Therefore, we developed new, spin-dependent equations of motion for the quantum hydrodynamics of electrons in molecular orbitals. The equations are based on a Lagrange multiplier, which constrains each electron to an isosurface of its molecular orbital, as required by the spin-dependent momentum. Both the momentum and the Lagrange multiplier provide a unique perspective on the properties of electrons in molecules.
Topological BF theory of the quantum hydrodynamics of incompressible polar fluids