## Publications

392 results found

Deumal M, Ribas-Ariño J, Robb MA, 2024, Using ‘designer’ coherences to control electron transfer in a model bis(hydrazine) radical cation: can we still distinguish between direct and superexchange mechanisms?, *Journal of Physics B: Atomic, Molecular and Optical Physics*, Vol: 57, ISSN: 0953-4075

We have simulated two mechanisms, direct and superexchange, for the electron transfer in a model Bis(hydrazine) Radical Cation, which consists of two hydrazine moieties coupled by a benzene ring. The computations, that are inspired by the attochemistry approach, focus on the electron dynamics arising from a coherent superposition of four cationic states. The electron dynamics, originating from a solution of the time dependent Schrödinger equation within the Ehrenfest method, is coupled to the relaxation of the nuclei. Both direct (ca. 15 fs dynamics) and superexchange (ca. 2 fs dynamics) mechanisms are observed and turn out to lie on a continuum depending on the strength of the coupling of the benzene bridge electron dynamics with the hydrazine chromophore dynamics. This contrasts with the chemical pathway approach where the direct mechanism is completely non-adiabatic via a conical intersection, while the superexchange mechanism involves an intermediate radical with the unpaired electron localized on the benzene ring. Thus, with the attochemistry-inspired electron dynamics approach, one can distinguish direct from superexchange mechanisms depending on the strength of the coupling of two types of electron dynamics: the slow hydrazine dynamics (ca. 15 fs) and the fast benzene linker dynamics (ca. 2 fs). In this model bis(hydrazine) radical cation, only when the intermediate coupler is in an anti-quinoid state, does one see the coupling of the bridge and hydrazine chromophore dynamics.

Danilov D, Jenkins AJ, Bearpark MJ,
et al., 2023, Coherent mixing of singlet and triplet states in acrolein and ketene: a computational strategy for simulating the electron-nuclear dynamics of intersystem crossing., *Journal of Physical Chemistry Letters*, Vol: 14, Pages: 6127-6134, ISSN: 1948-7185

We present a theoretical study of intersystem crossing (ISC) in acrolein and ketene with the Ehrenfest method that can describe a superposition of singlet and triplet states. Our simulations illustrate a new mechanistic effect of ISC, namely, that a superposition of singlets and triplets yields nonadiabatic dynamics characteristic of that superposition rather than the constituent state potential energy surfaces. This effect is particularly significant in ketene, where mixing of singlet and triplet states along the approach to a singlet/singlet conical intersection occurs, with the spin-orbit coupling (SOC) remaining small throughout. In both cases, the effects require many recrossings of the singlet/triplet state crossing seam, consistent with the textbook treatment of ISC.

Robb M, Olivucci M, 2023, Basic Concepts of Electronic Excited States., Comprehensive Computational Chemistry, Editors: Yanez, Manuel and Boyd, Russell

Danilov D, Tran T, Bearpark MJJ,
et al., 2022, How electronic superpositions drive nuclear motion following the creation of a localized hole in the glycine radical cation, *JOURNAL OF CHEMICAL PHYSICS*, Vol: 156, ISSN: 0021-9606

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- Citations: 3

Barillot T, Alexander O, Cooper B,
et al., 2021, Correlation-driven transient hole dynamics resolved in space and time in the isopropanol molecule, *Physical Review X*, Vol: 11, Pages: 1-15, ISSN: 2160-3308

The possibility of suddenly ionized molecules undergoing extremely fast electron hole (or, hole)dynamics prior to significant structural change was first recognized more than 20 years ago andtermed charge migration. The accurate probing of ultrafast electron hole dynamics requires measurements that have both sufficient temporal resolution and can detect the localization of a specifichole within the molecule. We report an investigation of the dynamics of inner valence hole states inisopropanol where we use an x-ray pump/x-ray probe experiment, with site and state-specific probing of a transient hole state localized near the oxygen atom in the molecule, together with an abinitio theoretical treatment. We record the signature of transient hole dynamics and make the firsttentative observation of dynamics driven by frustrated Auger-Meitner transitions. We verify thatthe effective hole lifetime is consistent with our theoretical prediction. This state-specific measurement paves the way to widespread application for observations of transient hole dynamics localizedin space and time in molecules and thus to charge transfer phenomena that are fundamental inchemical and material physics.

Onivucci M, Tran T, Worth G,
et al., 2021, Unlocking the double bond in protonated Schiff bases by coherent superposition of S1 and S2, *Journal of Physical Chemistry Letters*, Vol: 12, Pages: 5639-5643, ISSN: 1948-7185

The primary event occurring during the E-to-Z photoisomerization reaction of retinal protonated Schiff base (rPSB) is single-to-double bond inversion. In this work we examine the nuclear dynamics that occurs when the initial excited state is a superposition of the S1 and S2 electronic excited states that might be created in a laser experiment. The nuclear dynamics is dominated by double bond inversion that is parallel to the derivative coupling vector of S1 and S2. Thus, the molecule behaves as if it were at a conical intersection even if the states are nondegenerate.

Tran T, Worth G, Robb M, 2021, Control of nuclear dynamics in the benzene cation by electronic wavepacket composition, *Communications Chemistry*, Vol: 4, ISSN: 2399-3669

The study of coupled electron-nuclear dynamics driven by coherent superpositions of electronic states is now possible in attosecond science experiments. The objective is to understand the electronic control of chemical reactivity. In this work we report coherent 8-state non-adiabatic electron-nuclear dynamics simulations of the benzene radical cation. The computations were inspired by the extreme ultraviolet (XUV) experimental results in which all 8 electronic states were prepared with significant population. Our objective was to study the nuclear dynamics using various bespoke coherent electronic state superpositions as initial conditions in the Quantum-Ehrenfest method. The original XUV measurements were supported by Multi-configuration time-dependent Hartree (MCTDH) simulations, which suggested a model of successive passage through conical intersections. The present computations support a complementary model where non-adiabatic events are seen far from a conical intersection and are controlled by electron dynamics involving non-adjacent adiabatic states. It proves to be possible to identify two superpositions that can be linked with two possible fragmentation paths.

Tran T, Jenkins A, Worth GA,
et al., 2020, The Quantum-Ehrenfest method with the inclusion of an IR pulse: Application to electron dynamics of the allene radical cation, *Journal of Chemical Physics*, Vol: 153, ISSN: 0021-9606

We describe the implementation of a laser control pulse in the Quantum-Ehrenfest method, a molecular quantum dynamics method that solves the time-dependent Schrödinger equation for both electrons and nuclei. The oscillating electric fielddipole interaction is incorporated directly in the one-electron Hamiltonian of the electronic structure part of the algorithm. We then use the coupled electron-nuclear dynamics of the π-system in allene radical cation (•CH2=C=CH2)+ as a simple model of a pump-control experiment. We start (pump) with a two-state superposition of two cationic states. The resulting electron dynamics corresponds to the rapid oscillation of the unpaired electron between the two terminal methlylenes. This electron dynamics is in turn coupled to the torsional motion of the terminal methylenes. There is a conical intersection at 90° twist where the electron dynamics collapses because the adiabatic states become degenerate. After passing the conical intersection the electron dynamics revives. The IR pulse (control) in our simulations is timed to have its maximum at the conical intersection. Our simulations show that the effect of the (control) pulse is to change the electron dynamics at the conical intersection and, as a consequence, the concomitant nuclear dynamics which is dominated by change of the torsional angle.

Tran T, Segarra-Martí J, Bearpark M,
et al., 2019, Molecular vertical excitation energies studied with first-order RASSCF (RAS[1,1]): balancing covalent and ionic excited states, *Journal of Physical Chemistry A*, Vol: 123, Pages: 5223-5230, ISSN: 1089-5639

RASSCF calculations of vertical excitation energies were carried out on a benchmark set of 19 organic molecules studied by Thiel and co-workers [ J. Chem. Phys. 2008, 128, 134110]. The best results, in comparison with the MS-CASPT2 results of Thiel, were obtained using a RASSCF space that contains at most one hole and one particle in the RAS1 and RAS3 spaces, respectively, which we denote as RAS[1,1]. This subset of configurations recovers mainly the effect of polarization and semi-internal electronic correlation that is only included in CASSCF in an averaged way. Adding all-external correlation by allowing double excitations from RAS1 and RAS2 into RAS3 did not improve the results, and indeed, they were slightly worse. The accuracy of the first-order RASSCF computations is demonstrated to be a function of whether the state of interest can be classified as covalent or ionic in the space of configurations built from orbitals localized onto atomic sites. For covalent states, polarization and semi-internal correlation effects are negligible (RAS[1,1]), while for ionic states, these effects are large (because of inherent diffusiveness of these states compared to the covalent states) and, thus, an acceptable agreement with MS-CASPT2 can be obtained using first-order RASSCF with the extra basis set involving 3p orbitals in most cases. However, for those ionic states that are quasi-degenerate with a Rydberg state or for nonlocal nπ* states, there remains a significant error resulting from all external correlation effects.

Robb M, Jenkins AJ, 2019, The Damped Ehrenfest (D-Eh) method: Application to non-adiabatic reaction paths, *Computational and Theoretical Chemistry*, Vol: 1152, Pages: 53-61, ISSN: 0166-1280

An implementation of the Ehrenfest method with damped velocity is discussed. The method is then applied to study the non-adiabatic reaction paths for two simple chemical systems: the isomerization of the allene radical cation in its excited state and the channel 3 photochemical transformation of benzene to benzvalene. For both systems the initial conditions for the Ehrenfest trajectory were either an adiabatic eigenstate with the geometry close to a conical intersection, or a superposition of eigenstates at the geometry close to a conical intersection. In allene we were able to show that the adiabatic reaction, which passes through a conical intersection, stimulates electron dynamics. In benzene we were able to show the importance of the phase at the conical intersection for the generation of the benzevalene intermediate.

Spinlove KE, Richings GW, Robb MA,
et al., 2018, Curve crossing in a manifold of coupled electronic states: direct quantum dynamics simulations of formamide, *FARADAY DISCUSSIONS*, Vol: 212, Pages: 191-215, ISSN: 1359-6640

Jenkins AJ, Spinlove K, Vacher M,
et al., 2018, The Ehrenfest method with fully quantum nuclear motion (Qu-Eh): application to charge migration in radical cations, *Journal of Chemical Physics*, Vol: 149, ISSN: 0021-9606

An algorithm is described for quantum dynamics where an Ehrenfest potential is combined with fully quantum nuclear motion (Quantum-Ehrenfest, Qu-Eh). The method is related to the single-set variational multi-configuration Gaussian approach (vMCG) but has the advantage that only a single quantum chemistry computation is required at each time step since there is only a single time-dependent potential surface. Also shown is the close relationship to the “exact factorization method.” The quantum Ehrenfest method is compared with vMCG for study of electron dynamics in a modified bismethylene-adamantane cation system. Illustrative examples of electron-nuclear dynamics are presented for a distorted allene system and for HCCI+ where one has a degenerate Π system.

Polyak I, Bearpark MJ, Robb MA, 2018, Application of the unitary group approach (UGA) to evaluate spindensity for Configuration Interaction (CI) calculations in a basisof S$^{2}$ eigenfunctions, *International Journal of Quantum Chemistry*, Vol: 118, ISSN: 0020-7608

We present an implementation of the spin-dependent unitary group approachto calculate spin densities for CI calculations in a basis of spinsymmetry-adapted functions. Using S$^{2}$ eigenfunctions helps toreduce the size of configuration space and is beneficial in studiesof the systems where selection of states of specific spin symmetryis crucial. In order to achieve this, we combine the method to calculate$U(n)$ generator matrix elements developed by Robb and Downward~[\onlinecite{downward_1977}]with the approach of Gould and Battle to calculate $U(2n)$ generatormatrix elements~[\onlinecite{battle_1993}]. We also compareand contrast the spin density formulated in terms of the spin-independentunitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one- and two-electron charge densities.

Polyak I, Jenkins A, Vacher M,
et al., 2018, Charge migration engineered by localisation: electron-nuclear dynamics in polyenes and glycine, *Molecular Physics*, Vol: 116, Pages: 2474-2489, ISSN: 0026-8976

We demonstrate that charge migration can be ‘engineered’ in arbitrary molecular systems if a single localised orbital – that diabatically follows nuclear displacements – is ionised. Specifically, we describe the use of natural bonding orbitals in Complete Active Space Configuration Interaction (CASCI) calculations to form cationic states with localised charge, providing consistently well-defined initial conditions across a zero point energy vibrational ensemble of molecular geometries. In Ehrenfest dynamics simulations following localised ionisation of -electrons in model polyenes (hexatriene and decapentaene) and -electrons in glycine, oscillatory charge migration can be observed for several femtoseconds before dephasing. Including nuclear motion leads to slower dephasing compared to fixed-geometry electron-only dynamics results. For future work, we discuss the possibility of designing laser pulses that would lead to charge migration that is experimentally observable, based on the proposed diabatic orbital approach.

Robb MA, 2018, Theoretical Chemistry for Electronic Excited States, Publisher: Royal Society of Chemistry

Jornet-Somoza J, Deumal M, Borge J,
et al., 2018, A definition of the magnetic transition temperature using valence bond theory, *Journal of Physical Chemistry A*, Vol: 122, Pages: 2168-2177, ISSN: 1089-5639

Macroscopic magnetic properties are analyzed using Valence Bond theory. Commonly the critical temperature TC for magnetic systems is associated with a maximum in the energy-based heat capacity Cp(T). Here a more broadly applicable definition of the magnetic transition temperature TC is described using spin moment expectation value (i.e. applying the spin exchange density operator) instead of energy. Namely, the magnetic capacity Cs(T) reflects variation in the spin multiplicity as a function of temperature, which is shown to be related to ∂[χT(T)]/∂T. Magnetic capacity Cs(T) depends on long-range spin interactions that are not relevant in the energy-based heat capacity Cp(T). Differences between Cs(T) and Cp(T) are shown to be due to spin order/disorder within the crystal, that can be monitored via a Valence Bond analysis of the corresponding magnetic wavefunction. Indeed the concept of the Boltzmann spin-alignment order is used to provide information about the spin correlation between magnetic units. As a final illustration, the critical temperature is derived from the magnetic capacity for several molecular magnets presenting different magnetic topolo- gies that have been experimentally studied. A systematic shift between the transition temperatures associated with Cs(T) and Cp(T) is observed. It is demonstrated that this shift can be attributed to the loss of long-range spin correlation. This suggests that the magnetic capacity Cs(T) can be used as a predictive tool for the magnetic topology, and thus for the synthetic chemists.

Robb MA, Jenkins AJ, Vacher MA, 2018, How Nuclear Motion Affects Coherent Electron Dynamics in Molecules, *In Attosecond Molecular Dynamics Ed Marc J. J. Vrakking and Franck Lepine RSC Books*

Robb MA, Vacher MA, Jenkins A, 2018, How Nuclear Motion Affects Coherent Electron Dynamics in Molecules, Attosecond Molecular Dynamics, Editors: Vrakking, Lepine, Publisher: RSC, Pages: 275-307, ISBN: 978-1-78262-995-5

Knowledge about the electron dynamics in molecules is essential for our understanding of chemical and biological processes. Because of their light mass, electrons are expected to move on the attosecond (1 as = 10− 18 s) timescale. The first synthesis of attosecond pulses in 2001 has opened up the possibility of probing electronic motion on its intrinsic timescale. Excitation or ionisation of a molecule with such a short pulse leads to the coherent population of several electronic states, called an electronic wavepacket. The interference between electronic states in such a superposition, alternating between constructive and destructive, leads to oscillating motion of the electron cloud. This purely quantum process relies on the coherence of the electronic wavepacket. A fundamental challenge is to understand to what extent the electronic wavepacket retains its coherence, i.e., how long the oscillations in the electron cloud survive, in the presence of interactions with the nuclei of the molecule. To address this question, we have developed semi-classical and quantum mechanical methods to simulate the dynamics upon ionisation of polyatomic molecules. The chapter contains a review of the theoretical methods we have developed and some applications illustrating new important physical insights about the predicted decoherence process.

Vacher M, Bearpark M, Robb MA,
et al., 2017, Electron dynamics upon ionisation of polyatomic molecules: Coupling to quantum nuclear motion and decoherence, *Physical Review Letters*, Vol: 118, Pages: 1-5, ISSN: 1079-7114

Knowledge about the electronic motion in molecules is essential for our understanding of chemicalreactions and biological processes. The advent of attosecond techniques opens up the possibility toinduce electronic motion, observe it in real time and potentially steer it. A fundamental questionremains the factors influencing electronic decoherence and the role played by nuclear motion in thisprocess. Here, we simulate the dynamics upon ionisation of the polyatomic molecules para-xyleneand modified bismethylene-adamantane, with a quantum mechanical treatment of both electron andnuclear dynamics using the direct dynamics variational multi-configuration Gaussian method. Oursimulations give new important physical insights about the expected decoherence process. We haveshown that the decoherence of electron dynamics happens on the time scale of a few femtoseconds,with the interplay of different mechanisms: thedephasingis responsible for the fast decoherencewhile thenuclear overlap decaymay actually help maintaining it and is responsible for small revivals.

Spinlove KE, Vacher M, Bearpark M,
et al., 2017, Using quantum dynamics simulations to follow the competition between charge migration and charge transfer in polyatomic molecules, *CHEMICAL PHYSICS*, Vol: 482, Pages: 52-63, ISSN: 0301-0104

Orr-Ewing AJ, Verlet JRR, Penfold TJ,
et al., 2016, Electronic and non-adiabatic dynamics: general discussion, *Faraday Discussions*, Vol: 194, Pages: 209-257, ISSN: 1359-6640

Robb MA, 2016, Charge migration in polycyclic norbornadiene cations: winning the race against decoherence, *Journal of Chemical Physics*, Vol: 145, ISSN: 1089-7690

The observation of electronic motion remains a key target in the development of the eld of attoscience.However, systems in which long-lived oscillatory charge migration may be observed must be selected carefully,particularly because it has been shown that nuclear spatial delocalization leads to a loss of coherent electrondensity oscillations. Here we demonstrate electron dynamics in norbornadiene and extended systems wherethe hole density migrates between two identical chromophores. By studying the e ect of nuclear motionand delocalization in these example systems, we present the physical properties that must be considered incandidate molecules in which to observe electron dynamics. Furthermore, we also show a key contribution tonuclear delocalization arises from motion in the branching plane of the cation. For the systems studied, thedephasing time increases with system size while the energy gap between states, and therefore the frequency ofthe density oscillation, decreases with size (obeying a simple exponential dependence on the inter-chromophoredistance). We present a system that balances these two e ects and shows several complete oscillations in thespin density before dephasing occurs.

Vacher M, Bearpark MJ, Robb MA, 2016, Direct methods for non-adiabatic dynamics: connecting the single-set variational multi-confguration Gaussian (vMCG) and Ehrenfest perspectives, *Theoretical Chemistry Accounts*, Vol: 135, ISSN: 1432-881X

In this article, we outline the current state-of-theart“on-the-fly” methods for non-adiabatic dynamics, highlightingthe similarities and differences between them. Wederive the equations of motion for both the Ehrenfest andvariational multi-configuration Gaussian (vMCG) methodsfrom the Dirac–Frenkel variational principle. We explorethe connections between these two methods by presentingan alternative derivation of the vMCG method, which givesthe Ehrenfest equations of motion when taking the appropriatelimits.

Vacher M, Albertani FEA, Jenkins AJ,
et al., 2016, Electron and nuclear dynamics following ionisation of modified bismethylene-adamantane, *Faraday Discussions*, Vol: 194, Pages: 95-115, ISSN: 1364-5498

We have simulated the coupled electron and nuclear dynamics using the Ehrenfest method upon valence ionisation of modified bismethylene-adamantane (BMA) molecules where there is an electron transfer between the two π bonds. We have shown that the nuclear motion significantly affects the electron dynamics after a few fs when the electronic states involved are close in energy. We have also demonstrated how the non-stationary electronic wave packet determines the nuclear motion, more precisely the asymmetric stretching of the two π bonds, illustrating “charge-directed reactivity”. Taking into account the nuclear wave packet width results in the dephasing of electron dynamics with a half-life of 8 fs; this eventually leads to the equal delocalisation of the hole density over the two methylene groups and thus symmetric bond lengths.

Jenkins AJ, Vacher M, Bearpark MJ,
et al., 2016, Nuclear spatial delocalization silences electron density oscillations in 2-phenyl-ethyl-amine (PEA) and 2-phenylethyl-N,N-dimethylamine (PENNA) cations (vol 144, 104110, 2016), *JOURNAL OF CHEMICAL PHYSICS*, Vol: 144, ISSN: 0021-9606

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Jenkins AJ, Vacher M, Bearpark MJ,
et al., 2016, Nuclear spatial delocalization silences electron density oscillations in 2-phenyl-ethyl-amine (PEA) and 2-phenylethyl-N,N-dimethylamine (PENNA) cations, *Journal of Chemical Physics*, Vol: 144, ISSN: 1089-7690

We simulate electron dynamics following ionization in 2-phenyl-ethyl-amine and 2-phenylethyl-N,N-dimethylamine as examples of systems where 3 coupled cationic states are involved. We study two nuclear effects on electron dynamics: (i) coupled electron-nuclear motion and (ii) nuclear spatial delocalization as a result of the zero-point energy in the neutral molecule. Within the Ehrenfest approximation, our calculations show that the coherent electron dynamics in these molecules is not lost as a result of coupled electron-nuclear motion. In contrast, as a result of nuclear spatial delocalization, dephasing of the oscillations occurs on a time scale of only a few fs, long before any significant nuclear motion can occur. The results have been rationalized using a semi-quantitative model based upon the gradients of the potential energy surfaces.

Sanchez-Gonzalez A, Barillot TR, Squibb RJ,
et al., 2015, Auger electron and photoabsorption spectra of glycine in the vicinity of the oxygen K-edge measured with an X-FEL, *JOURNAL OF PHYSICS B-ATOMIC MOLECULAR AND OPTICAL PHYSICS*, Vol: 48, ISSN: 0953-4075

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- Citations: 9

Vacher M, Steinberg L, Jenkins AJ,
et al., 2015, Electron dynamics following photoionization: decoherence due to the nuclear-wave-packet width, *Physical Review A*, Vol: 92, Pages: 1-6, ISSN: 1094-1622

The advent of attosecond techniques opens up the possibility to observe experimentally electron dynamics following ionization of molecules. Theoretical studies of pure electron dynamics at single fixed nuclear geometries in molecules have demonstrated oscillatory charge migration at a well-defined frequency but often neglecting the natural width of the nuclear wave packet. The effect on electron dynamics of the spatial delocalization of the nuclei is an outstanding question. Here, we show how the inherent distribution of nuclear geometries leads to dephasing. Using a simple analytical model, we demonstrate that the conditions for a long-lived electronic coherence are a narrow nuclear wave packet and almost parallel potential-energy surfaces of the states involved. We demonstrate with numerical simulations the decoherence of electron dynamics for two real molecular systems (paraxylene and polycyclic norbornadiene), which exhibit different decoherence time scales. To represent the quantum distribution of geometries of the nuclear wave packet, the Wigner distribution function is used. The electron dynamics decoherence result has significant implications for the interpretation of attosecond spectroscopy experiments since one no longer expects long-lived oscillations.

Santolini V, Malhado JP, Robb MA,
et al., 2015, Photochemical reaction paths of cis-dienes studied with RASSCF: the changing balance between ionic and covalent excited states, *Molecular Physics*, Vol: 113, Pages: 1978-1990, ISSN: 1362-3028

The balanced description of ionic and covalent molecular excited electronic states still presents a challenge for currentelectronic structure methods. In this contribution, we show how the restricted active space self-consistent field (RASSCF)method can be used to address this problem, applied to two dienes in the cis conformation. As with the closely relatedcomplete active space self-consistent field (CASSCF) method, the construction of the orbital active space in the RASSCFmethodology requires the a priori formulation of a physical or theoretical model of the system being studied. In this article,we discuss how the active space can be constructed in a guided and systematic way, using pairs of natural bond orbitalsas correlating partner orbitals (oscillator orbitals) and semi-internal correlation. The resulting balanced description of thecovalent and ionic valence excited states – with the ionic state correctly lower in energy at the Franck–Condon geometry –is suitable to study the photochemistry of these and other molecules.

Robb MA, Meisner J, Vacher M,
et al., 2015, Geometric rotation of the nuclear gradient at a conical intersection: Extension to complex rotation of diabatic states, *Journal of Chemical Theory and Computation*, Vol: 11, Pages: 3115-3122, ISSN: 1549-9618

Nonadiabatic dynamics in the vicinity of conical intersections is of essential importance in photochemistry. It is well known that if the branching space is represented in polar coordinates, then for a geometry represented by angle θ, the corresponding adiabatic states are obtained from the diabatic states with the mixing angle θ/2. In an equivalent way, one can study the relation between the real rotation of diabatic states and the resulting nuclear gradient. In this work, we extend the concept to allow a complex rotation of diabatic states to form a nonstationary superposition of electronic states. Our main result is that this leads to an elliptical transformation of the effective potential energy surfaces; i.e., the magnitude of the initial nuclear gradient changes as well as its direction. We fully explore gradient changes that result from varying both θ and ϕ (the complex rotation angle) as a way of electronically controlling nuclear motion, through Ehrenfest dynamics simulations for benzene cation.

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