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Instanton theory provides a semiclassical approximation for computing quantum tunnelling effects in complex molecular systems. It is typically applied to proton-transfer reactions for which the Born–Oppenheimer approximation is valid. However, many processes in physics, chemistry and biology, such as electron transfers, are non-adiabatic and are correctly described instead using Fermi’s golden rule. In this work, we discuss how instanton theory can be generalized to treat these reactions in the golden-rule limit. We then extend the theory to treat fourth-order processes such as bridge-mediated electron transfer and apply the method to simulate an electron moving through a model system of three coupled quantum dots. By comparison with benchmark quantum calculations, we demonstrate that the instanton results are much more reliable than alternative approximations based on superexchange-mediated effective coupling or a classical sequential mechanism. This article is part of the theme issue ‘Chemistry without the Born–Oppenheimer approximation’.

Based on the predictions of global 3D hybrid simulations, we present a new transport/acceleration path for escaped O+ ions in the upstream solar wind region resulting from the impact of a particular IMF tangential discontinuity (TD) with negative (positive) IMF Bz on the discontinuity's anti‐sunward (sunward) side. For O+ ions escaping to the duskside magnetosheath and with gyro‐radii larger than the TD thickness, when they encounter the TD, they can first go sunward into the upstream solar wind. They then gyrate clockwise to the pre‐noon side and get accelerated within the solar wind region and circulate back to the dawnside magnetosphere. These ions may be accelerated to well within the ring current energy range depending on the solar wind electric field strength. This new transport/acceleration path can bring some of the escaped ions into the inner magnetosphere, thus providing a new mechanism for generating an O+ ring current population. Plain Language Summary O+ ions in the magnetosphere only come from the Earth's ionosphere. For O+ ions escaping the magnetosphere, scientists have been treating them as being lost. Using simulations that can describe the O+ ion's kinetic dynamics, we find that, due to the impact of a particular solar wind structure, some escaped O+ ions can circulate back to the magnetosphere via a transport path in the upstream solar wind region and some of them can even enter the inner magnetosphere. Additionally, they are also energized by solar wind electric field and thus can contribute to the ring current population. Therefore, this study shows a new journey of escaped O+ ions. Key Points First global 3D hybrid simulations to investigate the fate of O+ ions after escaping the dayside magnetosphere New transport/acceleration path for escaped O+ ions in upstream solar wind region after impact of an IMF tangential discontinuity New transport/acceleration path brings some of escaped O+ ions back to the inner magnetosphere, contributing to O+ ring current pressure

We report a rare quiet‐time observation by NOAA‐18 spacecraft flying nearly along the electron isotropy boundary (IBe). Complemented by data from four nearby meridional IBe crossings of other POES spacecraft, this case study provides a novel quantitative information concerning the regular energy dependence of the loss‐cone filling ratio (Jprec/Jtrap) variations near the IBe foot. Specific strong energy‐dependent difference of loss‐cone filling rates is manifested at different spacecraft and in various events in the same way, which is consistent with the pitch‐angle scattering in the magnetotail field reversal region as confirmed by the numerical simulation. Therefore it provides a landmark to distinguish the field line curvature scattering (FLCS) mechanism from wave‐induced precipitation. In FLCS case, a strongly fluctuating loss cone filling ratio implies that equatorial magnetic field is inherently structured in the magnetotail in the IBe foot region.

Three methods for non-adiabatic dynamics are compared to highlight their capabilities. Multi-configurational time-dependent Hartree is a full gridbased solution to the time-dependent Schrödinger equation, variational multi-configurational Gaussian (vMCG) uses a less flexible but unrestricted Gaussian wavepacket basis, and trajectory surface hopping (TSH) replaces the nuclear wavepacket with a swarm of classical trajectories. Calculations with all methods using a model Hamiltonian were performed. The vMCG and TSH were also then run in a direct dynamics mode, with the potential energy surfaces calculated on-the-fly using quantum chemistry calculations. All dynamics calculations used the Quantics package, with the TSH calculations using a new interface to a surface hopping code. A novel approach to calculate adiabatic populations from grid-based quantum dynamics using a time-dependent discrete variable representation is presented, allowing a proper comparison of methods. This article is part of the theme issue ‘Chemistry without the Born–Oppenheimer approximation’.

Modeling the dynamics of non-bound states in molecules requires an accurate description of how electronic motion affects nuclear motion and vice-versa. The exact factorization (XF) approach offers a unique perspective, in that it provides potentials that act on the nuclear subsystem or electronic subsystem, which contain the effects of the coupling to the other subsystem in an exact way. We briefly review the various applications of the XF idea in different realms, and how features of these potentials aid in the interpretation of two different laser-driven dissociation mechanisms. We present a detailed study of the different ways the coupling terms in recently-developed XF-based mixed quantum-classical approximations are evaluated, where either truly coupled trajectories, or auxiliary trajectories that mimic the coupling are used, and discuss their effect in both a surface-hopping framework as well as the rigorously-derived coupled-trajectory mixed quantum-classical approach.

Photochemical denitrogenation reactions of bicyclic azoalkanes produce strained bicyclic compounds of interest to synthetic organic chemists. We report a computational study on the mechanism of diazabicyclo[2.2.1]heptenes to address long standing mechanistic questions. Indeed, the mechanism of these reactions has been disputed for over six decades. We employed non-adiabatic molecular dynamics (NAMD) simulations combined with state-of-the-art multireference quantum mechanical calculations to understand the photophysical properties and mechanisms of these diazabicyclo[2.2.1]heptenes. The energetically accessible lowest excitations are n NN (σ CN ) → π* and range from 3.94–3.97 eV. From the >292 trajectories, the reaction proceeds through a dynamically concerted but asynchronous denitrogenation reaction. One σ CN bond breaks along the S 1 surface; the other σ CN breaks after hopping to the S 0 . We identified two clusters of S₁/S₀ surface hopping points from these trajectories. In the first cluster, the methylene bridge is fully inverted relative to the reactant geometry. In the second cluster, the inversion is only partial, with one of the carbon atoms in the methylene bridge inverted relative to the reactant. We identified each cluster's corresponding minimum energy conical intersection (MECI), indicating at least two possible S 1 /S 0 -MECIs. Our dynamics simulations illustrate that inversion begins in the excited state immediately after the first σ CN bond breaks. This inversion is driven by the atomic momenta acquired after the bond breaks. These dynamical effects promote the formation of the inverted housane, thereby explaining the observed selectivities. A minority of trajectories undergo thermal conversion in the ground state, producing the minor retained housane product from inverted housane/diradical.

A novel treatment of non-adiabatic couplings is proposed. The derivation is based on a theorem by Hunter stating that the wave function of the complete system of electrons and nuclei can be written, without approximation, as a Born-Oppenheimer (BO)-type product of a nuclear wave function, X(R), and an electronic one, ΦR(r), which depends parametrically on the nuclear configuration R. From the variational principle, we deduce formally exact equations for ΦR(r) and X(R). The algebraic structure of the exact nuclear equation coincides with the corresponding one in the adiabatic approximation. The electronic equation, however, contains terms not appearing in the adiabatic case, which couple the electronic and the nuclear wave functions and account for the electron-nuclear correlation beyond the BO level. It is proposed that these terms can be incorporated using an optimized local effective potential.

This review discusses two triatomic Hamiltonians and their applications to some non-adiabatic spectroscopic and collision problems. Carter and Handy in 1984 presented the first Hamiltonian in bond lengths–bond angle coordinates, that is here applied for studying the NO2 spectroscopy: vibronic states, internal dynamics, and interaction with the radiation due to the X˜2A′(A1)−A˜2A′(B2) conical intersection. The second Hamiltonian was reported by Tennyson and Sutcliffe in 1983 in Jacobi coordinates and is here employed in the study of the Kr + OH(A2Σ+) electronic quenching due to conical intersection and Renner–Teller interactions among the 12A′, 22A′, and 12A″ electronic species. Within the non-relativistic approximation and the expansion method in diabatic electronic representations, the formalism is exact and allows a unified study of various non-adiabatic interactions between electronic states. The rotation, inversion, and nuclear permutation symmetries are considered for defining rovibronic representations, which are symmetry adapted for ABC and AB2 molecules, and the matrix elements of the Hamiltonians are then computed.

Factoring a wave function into marginal and conditional factors partitions the subsystem kinetic energy into two terms. The first depends solely on the marginal wave function, through its gauge-covariant derivative, while the second depends on the quantum metric of the conditional wave function over the manifold of marginal variables. We derive an identity for the rate of change of the second term. This article is part of the theme issue ‘Chemistry without the Born–Oppenheimer approximation’.

We propose a novel framework to describe geometric phases in quantum systems under non-adiabatic conditions by introducing the concept of a hidden geometric phase. Conventional geometric phases, such as the Berry phase, rely on adiabatic evolution, limiting their applicability in rapidly changing systems. Here, we remove this constraint by reinterpreting the geometric phase as arising from a dynamically evolving reference basis, independent of the external topological features. The hidden phase is revealed through transitionless quantum control techniques, ensuring pure geometric phase accumulation even in non-adiabatic regimes. Our method offers an exact solution to the neutron spin rotation phase in the Bitter–Dubbers experiment, aligning more closely with experimental data without depending on adiabatic approximations. This unexpected result broadens our understanding of the geometric phase observed in neutron spin rotation beyond the adiabatic conditions that are conventionally required.