An animated visualization of orbital angular momentum and spin‐orbit coupling

Abstract

This paper presents an approach toward visualizing a complex orbital based on animation using a time‐dependent phase factor. This makes orbital angular momentum clearly visible, in a way that reflects the nature of the orbital angular momentum wavefunction. Visualization of this quantity is also useful for examining the effects of spin‐orbit coupling (SOC), in which higher orbital angular momentum states are admixed into the orbital; in this case, however, scaling of one phase‐component is needed. The phase orientation of a complex orbital, which is generally not guaranteed by the SCF procedure, must be considered when doing this. The method of visualization presented here may prove useful when analyzing properties where SOC is important, such as magnetic resonance parameters. Animated visualizations are performed, and compared with the method of phase‐colored isosurfaces, first for a model p‐orbital to explain the idea, and then for the singly‐occupied molecular orbitals of two small doublet radicals.

Visualizing Complex-Valued Molecular Orbitals

“…The visualization of molecular orbitals and associated properties has been thus far limited to that of real-valued orbitals due to the lack of implementation (see, however, ref , which we became aware of after submission of this manuscript), even though it is widely known that extension to complex-valued orbitals is conceptionally trivial. This is partly because the software for real-valued orbitals does suffice for traditional electronic structure simulations in which the nonrelativistic time-independent Schrödinger equation is solved for bound states with open boundary conditions, owing to the fact that the Hamiltonian for such systems is real and the phase of the orbitals can be chosen to be either +1 or −1.…”

Equation-of-Motion Coupled-Cluster Theory to Model L-Edge X-ray Absorption and Photoelectron Spectra

“…The Dyson orbitals of the SO-split states are complex-valued. Complex orbitals can be visualized in different ways, from simply representing their real and imaginary components separately to more sophisticated representations that take their phases into account. , To visualize the complex Dyson orbitals, we use here the QSimulate-QM program, which implements the algorithm proposed in ref .…”