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Jonathan Rawlinson

Publications and source records attributed to Jonathan Rawlinson.

At least 19 recordsLinked to original sources
arXiv2026

A Perspective on Phase Space Electronic Structure Theory : From Its Surface Hopping Origins Through To Its Future Promise

We trace the history of phase space electronic structure theory (PSEST), high- lighting how this powerful approach emerged from fundamental questions in semi- classical surface hopping dynamics and evolved into an alternative to standard Born- Oppenheimer based electronic structure theory (with moving instead of frozen nuclei). Our goal herein is not to recapitulate the mathematical details of phase space electronic structure calculations, and few equations are presented so as to maximize readability. Instead, our goal is to provide intuition for those new to the field both (i) regarding the physics present when solving the Schrodinger equation in a non-inertial frame as well as (ii) regarding why PSEST is a necessary step forward towards understanding chemical problems involving spin (with limited practical alternatives). We further high- light some of the many open questions in this fast developing area, which will hopefully inspire new practitioners in this field. This intuitive perspective lacks many equations and is meant to complement (rather than replace) the more technical review given in Bian et al, Chem. Phys. Rev. 7, 011303 (2026)

physics.chem-ph↗
arXiv2026

Symmetry and Topology in Wavepacket Dynamics near Conical Intersections

Whether topology directly shapes chemical dynamics remains an open question in theoretical chemistry. The issue arises because degeneracies of adiabatic electronic states can generate nontrivial topological structure, and such degeneracies are common in polyatomic molecules. Existing work has largely emphasized static characterizations and dynamical studies of low-energy, highly symmetric models. Here we develop a symmetry-based analysis of nonadiabatic dynamics in two-state conical-intersection models that is predictive without invoking topological invariants. We show that the nodal-line structures associated with dynamics near a conical intersection are robust in highly symmetric settings, but should not in general be expected to persist once the relevant symmetry is broken.

physics.chem-ph↗
arXiv2025

Recovering Exact Vibrational Energies Within a Phase Space Electronic Structure Framework

In recent years, there has been a push to go beyond Born-Oppenheimer theory and build electronic states from a phase space perspective, i.e. parameterize electronic states by both nuclear position(R) and nuclear momentum(P). Previous empirical studies have demonstrated that such approaches can yield improved single-surface observables, including vibrational energies, electronic momenta, and vibrational circular dichroism spectra. That being said, unlike the case of BO theory, there is no unique phase space electronic Hamiltonian, nor any theory for using phase space eigenvectors (as opposed to BO eigenvectors) so as to recover exact quantum vibrational eigenvalues. As such, one might consider such phase space approaches ad hoc. To that end, here we show how to formally extract exact quantum energies from a coupled nuclear-electronic Hamiltonian using perturbation theory on top of a phase space electronic framework. Thus, while we cannot isolate an "optimal" phase space electronic Hamiltonian, this work does justify a phase space electronic structure approach by offering a rigorous framework for correcting the zeroth order phase space electronic states.

physics.chem-ph↗
arXiv2024

A Phase-Space Electronic Hamiltonian for Molecules in a Static Magnetic Field I: Conservation of Total Pseudomomentum and Angular Momentum

We develop a phase-space electronic structure theory of molecules in magnetic fields. For a system of electrons in a magnetic field with vector potential $\bf{A}(\hat{\bf{r}})$, the usual Born-Oppenheimer Hamiltonian is the sum of the nuclear kinetic energy and the electronic Hamiltonian, $\frac{(\bf{P} - q\bf{A}(\bf{X}) )^2}{2M} + \hat{H}_{e}(\bf{X})$ (where $q$ is a nuclear charge). To include the effects of coupled nuclear-electron motion in the presence of magnetic field, we propose that the proper phase-space electronic structure Hamiltonian will be of the form $\frac{(\bf{P} - q^{\textit{eff}}\bf{A}(\bf{X}) - e\hat{\bf{\Gamma}})^2}{2M} + \hat{H}_{e}(\bf{X})$. Here, $q^{\textit{eff}}$ represents the {\em screened} nuclear charges and the $\hat{\bf{\Gamma}}$ term captures the local pseudomomentum of the electrons. This form reproduces exactly the energy levels for a hydrogen atom in a magnetic field; moreover, single-surface dynamics along the eigenstates is guaranteed to conserve both the total pseudomomentum as well as the total angular momentum in the direction of the magnetic field. This Hamiltonian form can be immediately implemented within modern electronic structure packages (where the electronic orbitals will now depend on nuclear position ($\bf{X}$) and nuclear momentum ($\bf{P}$)). One can expect to find novel beyond Born-Oppenheimer magnetic field effects for strong enough fields and/or nonadiabatic systems.

physics.chem-ph↗
arXiv2024

A Phase-Space Electronic Hamiltonian for Molecules in a Static Magnetic Field II: Quantum Chemistry Calculations with Gauge Invariant Atomic Orbitals

In a companion paper, we have developed a phase-space electronic structure theory of molecules in magnetic fields, whereby the electronic energy levels arise from diagonalizing a phase-space Hamiltonian $\hat H_{PS}(\bf{X},\bf{\Pi})$ that depends parametrically on nuclear position and momentum. The resulting eigenvalues are translationally invariant; moreover, if the magnetic field is in the $z-$direction, then the eigenvalues are also invariant to rotations around the $z-$direction. However, like all Hamiltonians in a magnetic field, the theory has a gauge degree of freedom (corresponding to the position of the magnetic origin in the vector potential), and requires either $(i)$ formally, a complete set of electronic states or $(ii)$ in practice, gauge invariant atomic orbitals (GIAOs) in order to realize such translational and rotational invariance. Here we describe how to implement a phase-space electronic Hamiltonian using GIAOs within a practical electronic structure package (in our case, Q-Chem). We further show that novel phenomena can be observed with finite $\bf{B}-$fields, including minimum energy structures with $\bf{\Pi}_{min} \ne 0$, indicating non-zero electronic motion in the ground-state.

physics.chem-ph↗
arXiv2024

Eigenvalue crossings in equivariant families of matrices

According to a result of Wigner and von Neumann [1], real symmetric matrices with a doubly degenerate lowest eigenvalue form a submanifold of codimension 2 within the space of all real symmetric matrices. This mathematical result has important consequences for chemistry. First, it implies that degeneracies do not occur within generic one-parameter families of real symmetric matrices - this is the famous non-crossing rule, and is responsible for the phenomenon of avoided crossings in the energy levels of diatomic molecules. Second, it implies that energy levels are expected to cross in polyatomic molecules, with crossings taking place on a submanifold of nuclear configuration space which is codimension 2 - this submanifold is the famous conical intersection seam, of central importance in nonadiabatic chemistry. In this paper we extend the analysis of Wigner and von Neumann to include symmetry. We introduce a symmetry group, and consider parametrised families of matrices which respect an action of that symmetry group on both the parameter space and on the space of matrices. A concrete application is to molecules, for which the relevant symmetry group is generated by permutations of atomic nuclei combined with spatial reflections and rotations. In the presence of this extra symmetry, we find that energy level crossings do not typically occur on codimension 2 submanifolds, and connect our findings with the discovery of confluences of conical intersection seams in the chemical literature. We give a classification of confluences for triatomic molecules and planar molecules, unifying the previous literature on this topic, and predict several new types of confluence.

physics.chem-ph↗
arXiv2024

Can The Mystery of The Born-Oppenheimer Electronic Current Density Be Explained With A Simple Phase Space Electronic Hamiltonian? Yes (And A Lot More Too)

We show that a phase space electronic Hamiltonian $\hat{H}_{PS}(\mathbf{X},\mathbf{P})$, parameterized by both nuclear position $\mathbf{X}$ and momentum $\mathbf{P}$, can recover not just experimental vibrational circular dichroism (VCD) signals, but also a meaningful electronic current density that explains the features of the VCD rotatory strengths. Combined with earlier demonstrations that such Hamiltonians can also recover qualitatively correct electronic momenta with electronic densities that approximately satisfy a continuity equation, the data would suggest that we have isolated a meaningful alternative approach to electronic structure theory, one that entirely avoids Born-Oppenheimer theory and frozen nuclei. While the dynamical implications of such a phase space electronic Hamiltonian are not yet known, we hypothesize that, by offering classical trajectories the conserve the total angular momentum (unlike Born-Oppenheimer theory), this new phase space electronic structure Hamiltonian may well explain some fraction of the chiral-induced spin selectivity effect.

physics.chem-ph↗
arXiv2024

A phase-space view of vibrational energies without the Born-Oppenheimer framework

We show that following the standard mantra of quantum chemistry and diagonalizing the Born-Oppenheimer (BO) Hamiltonian $\hat H_{\rm BO}(\bm R)$ is not the optimal means to construct potential energy surfaces. A better approach is to diagonalize a phase-space electronic Hamiltonian, $\hat H_{\rm PS}(\bm R,\bm P)$, which is parameterized by both nuclear position $\bm R$ and nuclear momentum $\bm P$. The foundation of such a non-perturbative phase-space electronic Hamiltonian can be made rigorous using a partial Wigner transform and the method has exactly the same cost as BO for a semiclassical calculation (and only a slight increase in cost for a quantum nuclear calculation). For a three-particle system, with two heavy particles and one light particle, numerical results show that a phase space electronic Hamiltonian produces not only meaningful electronic momenta (which are completely ignored by BO theory) but also far better vibrational energies. As such, for high level results and/or systems with degeneracies and spin degrees of freedom, we anticipate that future electronic structure and quantum chemistry packages will need to take as input not just the positions of the nuclei but also their momenta.

physics.chem-ph↗
arXiv2024

A Phase Space Approach to Vibrational Circular Dichroism

We show empirically that a phase-space non-Born-Oppenheimer electronic Hamiltonian approach to quantum chemistry (where the electronic Hamiltonian is parameterized by both nuclear position and momentum, (H(R,P)) is both a practical and accurate means to recover vibrational circular dichroism spectra. We further hypothesize that such a phase space approach may lead to very new dynamical physics beyond spectroscopy circular dichroism, with potential implications for understanding chiral induced spin selectivity (CISS), noting that classical phase space approaches conserve the total nuclear plus electronic momentum, whereas classical Born-Oppenheimer approaches do not (they conserve only the nuclear momentum)

physics.chem-ph↗
arXiv2024

Practical Phase-Space Electronic Hamiltonians for Ab Initio Dynamics

Modern electronic structure theory is built around the Born-Oppenheimer approximation and the construction of an electronic Hamiltonian H_{el}(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 (Gamma') and rotational (Gamma'') factors to couple electronic transitions to nuclear motion, we construct a practical phase-space electronic Hamiltonian that depends on both nuclear position and momentum, H_{PS}(X,P). While classical Born-Oppenheimer dynamics that run along the eigensurfaces of the operator H_{el}(X) can recover many nuclear properties correctly, we present some evidence that motion along the eigensurfaces of H_{PS}(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.

physics.chem-ph↗
arXiv2024

A Simple One-Electron Expression for Electron Rotational Factors

Within the context of FSSH dynamics, one often wishes to remove the angular component of the derivative coupling between states $\left|J\right>$ and $\left|K\right>$. In a set of previous papers, Truhlar {\em et al.} posited one approach for such a removal based on direct projection, while we isolated a second approach by constructing and differentiating rotationally invariant basis. Unfortunately, neither approach was able to demonstrate a {\em one-electron operator} $\hat{O}$ whose matrix element $\left $ 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 be of immediate use for FSSH calculations.

physics.comp-ph↗
arXiv2023

Diagonalizing the Born-Oppenheimer Hamiltonian via Moyal Perturbation Theory, Nonadiabatic Corrections and Translational Degrees of Freedom

This article describes a method for calculating higher order or nonadiabatic corrections in Born-Oppenheimer theory and its interaction with the translational degrees of freedom. The method uses the Wigner-Weyl correspondence to map nuclear operators into functions on the classical phase space and the Moyal star product to represent operator multiplication on those functions. The result is a power series in $κ^2$, where $κ=(m/M)^{1/4}$ is the usual Born-Oppenheimer parameter. The lowest order term is the usual Born-Oppenheimer approximation while higher order terms are nonadiabatic corrections. These are needed in calculations of electronic currents, momenta and densities. The method was applied to Born-Oppenheimer theory by Littlejohn and Weigert (1993), in a treatment that notably produced the correction $K_{22}$ to the Born-Oppenheimer Hamiltonian (see {\em infra}). Recently Matyus and Teufel (2019) have applied an improved and more elegant version of the method to Born-Oppenheimer theory, and have calculated the Born-Oppenheimer Hamiltonian for multiple potential energy surfaces to order $κ^6$. One of the shortcomings of earlier methods is that the separation of nuclear and electronic degrees of freedom takes place in the context of the exact symmetries (for an isolated molecule) of translations and rotations, and these need to be a part of the discussion. This article presents an independent derivation of the Moyal expansion in molecular Born-Oppenheimer theory, with special attention to the translational degrees of freedom. We show how electronic currents and momenta can be calculated within the framework of Moyal perturbation theory; we derive the transformation laws of the electronic Hamiltonian, the electronic eigenstates, and the derivative couplings under translations.

physics.chem-ph↗
arXiv2023

Total Angular Momentum Conservation in Ehrenfest Dynamics with a Truncated Basis of Adiabatic States

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[Amano2005,Krishna2007] 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 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.

physics.chem-ph↗
arXiv2023

Linear and Angular Momentum Conservation in Surface Hopping Methods

We demonstrate that, for systems with spin-orbit coupling and an odd number of electrons, the standard fewest switches surface hopping (FSSH) 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 $Γ\cdot P$ 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 (CISS) effect.

physics.chem-ph↗
arXiv2023

Surface Hopping, Electron Translation Factors, Electron Rotation Factors, Momentum Conservation, and Size Consistency

For a system without spin-orbit coupling, the (i) nuclear plus electronic linear momentum and (ii) nuclear plus orbital electronic angular momentum are good quantum numbers. Thus, when a molecular system undergoes a nonadiabatic transition, there should be no change in the total linear or angular momentum. Now, the standard surface hopping algorithm ignores the electronic momentum and indirectly equates the momentum of the nuclear degrees of freedom to the total momentum. However, even with this simplification, the algorithm still does not conserve either the nuclear linear or the nuclear angular momenta. Here, we show that one way to address these failures is to dress the derivative couplings (i.e. the hopping directions) in two ways: (i) we disallow changes in the nuclear linear momentum by working in a translating basis (which is well known and leads to electron translation factors [ETFs]); and (ii) we disallow changes in the nuclear angular momentum by working in a basis that rotates around the center of mass (which is not well-known and leads to a novel, rotationally removable component of the derivative coupling that we will call electron rotation factors [ERFs] below, cf. Eq. 96). The present findings should be helpful in the short term as far as interpreting surface hopping calculations for singlet systems (without spin) and then developing new surface hopping algorithm in the long term for systems where one cannot ignore the electronic orbital and/or spin angular momentum.

physics.chem-ph↗
arXiv2023

Total Angular Momentum Conservation in Ab Initio Born-Oppenheimer Molecular Dynamics

We prove both analytically and numerically that the total angular momentum of a molecular system undergoing adiabatic Born-Oppenheimer dynamics is conserved only when pseudo-magnetic Berry forces are taken into account. This finding sheds light on the nature of Berry forces for molecular systems with spin-orbit coupling and highlights how ab initio Born-Oppenheimer molecular dynamics simulations can successfully capture the entanglement of spin and nuclear degrees of freedom as modulated by electronic interactions.

physics.chem-ph↗
arXiv2023

Representation and Conservation of Angular Momentum in the Born-Oppenheimer Theory of Polyatomic Molecules

This paper concerns the representation of angular momentum operators in the Born-Oppenheimer theory of polyatomic molecules and the various forms of the associated conservation laws. Topics addressed include the question of whether these conservation laws are exactly equivalent or only to some order of the Born-Oppenheimer parameter $κ=(m/M)^{1/4}$, and what the correlation is between angular momentum quantum numbers in the various representations. These questions are addressed both in problems involving a single potential energy surface, and those with multiple, strongly coupled surfaces; and both in the electrostatic model and those for which fine structure and electron spin are important. The analysis leads to an examination of the transformation laws under rotations of the electronic Hamiltonian; of the basis states, both adiabatic and diabatic, along with their phase conventions; of the potential energy matrix; and of the derivative couplings. These transformation laws are placed in the geometrical context of the structures in the nuclear configuration space that are induced by rotations, which include the rotational orbits or fibers, the surfaces upon which the orientation of the molecule changes but not its shape; and the section, an initial value surface that cuts transversally through the fibers. Finally, it is suggested that the usual Born-Oppenheimer approximation can be replaced by a dressing transformation, that is, a sequence of unitary transformations that block-diagonalize the Hamiltonian. When the dressing transformation is carried out, we find that the angular momentum operator does not change. This is a part of a system of exact equivalences among various representations of angular momentum operators in Born-Oppenheimer theory...

physics.chem-ph↗
arXiv2022

Modeling Spin-Dependent Nonadiabatic Dynamics with Electronic Degeneracy: A Phase-Space Surface-Hopping Method

Nuclear Berry curvature effects emerge from electronic spin degeneracy and canlead to non-trivial spin-dependent (nonadiabatic) nuclear dynamics. However, such effects are completely neglected in all current mixed quantum-classical methods such as fewest switches surface-hopping. In this work, we present a phase-space surface-hopping (PSSH) approach to simulate singlet-triplet intersystem crossing dynamics. We show that with a simple pseudo-diabatic ansatz, a PSSH algorithm can capture the relevant Berry curvature effects and make predictions in agreement with exact quantum dynamics for a simple singlet-triplet model Hamiltonian. Thus, this approach represents an important step towards simulating photochemical and spin processes concomitantly, as relevant to intersystem crossing and spin-lattice relaxation dynamics.

physics.chem-ph↗