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We present a phase-space electronic Hamiltonian ĤPS (parameterized by both nuclear position X and momentum P) that boosts each electron into the moving frame of the nuclei that are closest in real space. The final form for the phase space Hamiltonian does not assume the existence of an atomic orbital basis, and relative to standard Born–Oppenheimer theory, the newly proposed one-electron operators can be expressed directly as functions of electronic and nuclear positions and momentum. We show that (i) quantum–classical dynamics along such a Hamiltonian maintains momentum conservation and that (ii) diagonalizing such a Hamiltonian can recover the electronic momentum and electronic current density reasonably well. In conjunction with other reports in the literature that such a phase-space approach can also recover vibrational circular dichroism spectra, we submit that the present phase-space approach offers a testable and powerful approach to post-BO electronic structure theory. Moreover, the approach is inexpensive and can be immediately applied to simulations of chiral induced spin selectivity experiments (where the transfer of angular momentum between nuclei and electrons is considered critical). 

We demonstrate that working with a correct phase-space electronic Hamiltonian captures electronic inertial effects. In particular, we show that phase space surface hopping dynamics do not suffer (at least to very high order) from non-physical non-adiabatic transitions between electronic eigenstates during the course of pure nuclear translational and rotational motion. This work opens up many new avenues for quantitatively investigating complex phenomena, including angular momentum transfer between chiral phonons and electrons as well as chiral-induced spin selectivity effects. 

Electron transfer through chiral molecules displays a strong spin preference 

We demonstrate that, for systems with spin–orbit coupling and an odd number of electrons, the standard fewest switches surface hopping algorithm does not conserve the total linear or angular momentum. This lack of conservation arises not so much from the hopping direction (which is easily adjusted) but more generally from propagating adiabatic dynamics along surfaces that are not time reversible. We show that one solution to this problem is to run along eigenvalues of phase-space electronic Hamiltonians H(R, P) (i.e., electronic Hamiltonians that depend on both nuclear position and momentum) with an electronic–nuclear coupling Γ · P [see Eq. (25)], and we delineate the conditions that must be satisfied by the operator Γ. The present results should be extremely useful as far as developing new semiclassical approaches that can treat systems where the nuclear, electronic orbital, and electronic spin degrees of freedom altogether are all coupled together, hopefully including systems displaying the chiral-induced spin selectivity effect. 

Within the context of fewest-switch surface hopping (FSSH) dynamics, one often wishes to remove the angular component of the derivative coupling between states J and K. In a previous set of papers, Shu et al. [J. Phys. Chem. Lett. 11, 1135–1140 (2020)] posited one approach for such a removal based on direct projection, while we isolated a second approach by constructing and differentiating a rotationally invariant basis. Unfortunately, neither approach was able to demonstrate a one-electron operatorÔ whose matrix element JÔK was the angular component of the derivative coupling. Here, we show that a one-electron operator can, in fact, be constructed efficiently in a semi-local fashion. The present results yield physical insight into designing new surface hopping algorithms and are of immediate use for FSSH calculations. 

We show that standard Ehrenfest dynamics does not conserve linear and angular momentum when using a basis of truncated adiabatic states. However, we also show that previously proposed effective Ehrenfest equations of motion [M. Amano and K. Takatsuka, “Quantum fluctuation of electronic wave-packet dynamics coupled with classical nuclear motions,” J. Chem. Phys. 122, 084113 (2005) and V. Krishna, “Path integral formulation for quantum nonadiabatic dynamics and the mixed quantum classical limit,” J. Chem. Phys. 126, 134107 (2007)] involving the non-Abelian Berry force do maintain momentum conservation. As a numerical example, we investigate the Kramers doublet of the methoxy radical using generalized Hartree–Fock with spin–orbit coupling and confirm that angular momentum is conserved with the proper equations of motion. Our work makes clear some of the limitations of the Born–Oppenheimer approximation when using ab initio electronic structure theory to treat systems with unpaired electronic spin degrees of freedom, and we demonstrate that Ehrenfest dynamics can offer much improved, qualitatively correct results. 

Modern electronic structure theory is built around the Born–Oppenheimer approximation and the construction of an electronic Hamiltonian Ĥel(X) that depends on the nuclear position X (and not the nuclear momentum P). In this article, using the well-known theory of electron translation (Γ′) and rotational (Γ″) factors to couple electronic transitions to nuclear motion, we construct a practical phase-space electronic Hamiltonian that depends on both nuclear position and momentum, ĤPS(X,P). While classical Born–Oppenheimer dynamics that run along the eigensurfaces of the operator Ĥel(X) can recover many nuclear properties correctly, we present some evidence that motion along the eigensurfaces of ĤPS(X,P) can better capture both nuclear and electronic properties (including the elusive electronic momentum studied by Nafie). Moreover, only the latter (as opposed to the former) conserves the total linear and angular momentum in general.