Phase Space Electronic Structure Theory: From Diatomic Lambda-Doubling to Macroscopic Einstein-de Haas

TL;DR

The paper presents phase space electronic structure theory as a beyond-Born-Oppenheimer framework that parameterizes the electronic Hamiltonian by nuclear position ${\bf X}$ and momentum ${\bf P}$ to conserve total angular momentum and capture nonadiabatic effects such as $Λ$-doubling. By applying simple one- and two-dimensional models and ab initio calculations for NO, the authors show that electron-rotation coupling embedded in the phase-space Hamiltonian yields phase-space potential energy surfaces whose minima reflect rotational electron momentum, enabling near-quantitative reproduction of the $Λ$-splitting with a cost similar to standard BO methods. They extract a coupling parameter $\alpha$ (~$1.8\times10^{-3}$ to $2.2\times10^{-3}$ a.u.) from multiple PS-based routes and demonstrate how the $J$-dependence of the splitting emerges in a 2D model, improving toward quantitative agreement when full SOC is included. The work links microscopic nonadiabatic angular momentum transfer to macroscopic phenomena like the Einstein–de Haas effect and argues for expanding PS methods to vibrations and larger systems, where generalized $\hat{\Gamma}$ operators could uncover new spin-rotation-phonon physics.

Abstract

$Λ$-doubling of diatomic molecules is a subtle microscopic phenomenon that has long attracted the attention of experimental groups, insofar as rotation of molecular $\textit{nuclei}$ induces small energetic changes in the (degenerate) $\textit{electronic}$ state. A direct description of such a phenomenon clearly requires going beyond the Born-Oppenheimer approximation. Here we show that a phase space theory previously developed to capture electronic momentum and model vibrational circular dichroism -- and which we have postulated should also describe the Einstein-de Haas effect, a macroscopic manifestation of angular momentum conservation -- is also able to recover the $Λ$-doubling energy splitting (or $Λ$-splitting) of the NO molecule nearly quantitatively. The key observation is that, by parameterizing the electronic Hamiltonian in terms of both nuclear position ($\mathbf{X}$) and nuclear momentum ($\mathbf{P}$), a phase space method yields potential energy surfaces that explicitly include the electron-rotation coupling and correctly conserve angular momentum (which we show is essential to capture $Λ-$doubling). The data presented in this manuscript offers another small glimpse into the rich physics that one can learn from investigating phase space potential energy surfaces $E_{PS}(\mathbf{X},\mathbf{P})$ as a function of both nuclear position and momentum, all at a computational cost comparable to standard Born-Oppenheimer electronic structure calculations.

Figures (5)

• Figure 1: The potential energy surfaces of the ground and first excited states of NO from the phase space Hamiltonian, plotted versus the canonical nuclear angular momentum $L$. The blue and orange curves correspond to states whose valence $\Pi^*$ molecular orbital are approximately $\frac{1}{\sqrt{2}}\left[\left| P_x\right\rangle \left| \downarrow_x\right\rangle - i\left| P_y\right\rangle \left| \uparrow_x\right\rangle \right]$ and $\frac{1}{\sqrt{2}}\left[-\left|P_x\right\rangle \left| \uparrow_x\right\rangle + i\left|P_y\right\rangle \left| \downarrow_x\right\rangle \right]$, respectively. The nuclear coupling breaks the Kramers' degeneracy of the two PESs at a given finite canonical nuclear rotation angular momentum $L$ and form a double well with minima at $L/\hbar \approx \pm 0.0022$.
• Figure 2: Energy diagram of the low-energy space of NO. Energy splittings are not drawn to scale.
• Figure 3: The $\Lambda$-splitting between the lowest $e$ and $f$ states and the associated $\alpha$ parameter for a nuclear rotation $J=1/2\hbar$ around one perpendicular axis. The results are calculated using the exact FCI solver (blue circle) in the STO-3G basis, the DMRG solver (purple star) which converges to the FCI accuracy in the STO-3G and 6-31G bases, and an approximate constrained Hartree-Fock solver in larger bases, from STO-3G, 6-31G, cc-pVDZ, aug-ccpVDZ, to aug-ccpVTZ (red).
• Figure 4: Energy splitting between the $e$ and $f$ states as a function of total angular momentum $J$ from the 2D model parametrized with $\alpha$ derived from FCI in the STO-3G basis (blue) and from cHF in the aug-ccpVTZ basis (red), both using Method III, in comparison to experiments (black). Dashed lines include only the 1-electron SOC, while solid lines include both 1-electron and 2-electron SOC (with a spin-orbit mean-field approximation neese2005efficientlee2023ab).
• Figure 5: a) Our original view (see Fig. \ref{['fig:well']}) of potential energy surfaces of the ground and first excited states of NO according to a phase space electronic structure Hamiltonian as a function of the canonical nuclear angular momentum. b) The same data now labeled differently: The green and red curves can be understood as the $e$ and $f$ parity total electronic and nuclear states, now understood to functions of total angular momentum. The $e/f$ states are the states that are observed spectroscopically.