Electron Trajectories in Molecular Orbitals

31 March 2020, Version 1
This content is a preprint and has not undergone peer review at the time of posting.

Abstract

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.

Keywords

Bohmian mechanics
quantum hydrodynamics
molecular orbitals
Dirac Equation
Electronic Structure

Supplementary materials

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Electron Trajectories in molecular orbitalsSI
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SD-QHD-inputs
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