Controlling Electronic Coherences and the Curvature Induced by the Derivative Coupling at a Conical Intersection: A Quantum Ehrenfest (QuEh) Protocol for Reaction Path Following Application to "Channel 3" Benzene Photochemistry.

We report a protocol for the implementation of "reaction path following" from a transition state through a conical intersection, including both the path curvature induced by the derivative coupling and the corresponding induced electronic coherences. This protocol focuses on the "central" Gaussian wavepacket (initially unexcited) in the quantum Ehrenfest (QuEh) method. Like the reaction path following, the normal mode corresponding to the imaginary frequency at the transition state is given an initial momentum. The protocol is applied to the "channel 3" radiationless decay of benzene. We also demonstrate that one can enhance the effect of the derivative coupling and the electronic coherence with an IR pulse.

Coherent Mixing of Singlet and Triplet States in Acrolein and Ketene: A Computational Strategy for Simulating the Electron-Nuclear Dynamics of Intersystem Crossing.

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.

How electronic superpositions drive nuclear motion following the creation of a localized hole in the glycine radical cation.

In this work, we have studied the nuclear and electron dynamics in the glycine cation starting from localized hole states using the quantum Ehrenfest method. The nuclear dynamics is controlled both by the initial gradient and by the instantaneous gradient that results from the oscillatory electron dynamics (charge migration). We have used the Fourier transform (FT) of the spin densities to identify the "normal modes" of the electron dynamics. We observe an isomorphic relationship between the electron dynamics normal modes and the nuclear dynamics, seen in the vibrational normal modes. The FT spectra obtained this way show bands that are characteristic of the energy differences between the adiabatic hole states. These bands contain individual peaks that are in one-to-one correspondence with atom pair (+·) ↔ (·+) resonances, which, in turn, stimulate nuclear motion involving the atom pair. With such understanding, we anticipate "designer" coherent superpositions that can drive nuclear motion in a particular direction.

Unlocking the Double Bond in Protonated Schiff Bases by Coherent Superposition of S1 and S2.

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.

Control of nuclear dynamics in the benzene cation by electronic wavepacket composition.

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.

The quantum-Ehrenfest method with the inclusion of an IR pulse: Application to electron dynamics of the allene radical cation.

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 field-dipole 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 the 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 methylenes. 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 the change in the torsional angle.

Molecular Vertical Excitation Energies Studied with First-Order RASSCF (RAS[1,1]): Balancing Covalent and Ionic Excited States.

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.

Curve crossing in a manifold of coupled electronic states: direct quantum dynamics simulations of formamide.

Quantum dynamics simulations are an important tool to evaluate molecular behaviour including the, often key, quantum nature of the system. In this paper we present an algorithm that is able to simulate the time evolution of a molecule after photo-excitation into a manifold of states. The direct dynamics variational multi-configurational Gaussian (DD-vMCG) method circumvents the computational bottleneck problems of traditional grid-based methods by computing the potential energy functions on-the-fly, i.e. only where required. Unlike other commonly used direct dynamics methods, DD-vMCG is fully quantum mechanical. Here, the method is combined with a novel on-the-fly diabatisation scheme to simulate the short-time dynamics of the key molecule formamide and its acid analogue formimidic acid. This is a challenging test system due to the nature and large number of excited states, and eight coupled states are included in the calculations. It is shown that the method is able to provide unbiased information on the product channels open after excitation at different energies and demonstrates the potential to be a practical scheme, limited mainly by the quality of the quantum chemistry used to describe the excited states.

The Ehrenfest method with fully quantum nuclear motion (Qu-Eh): Application to charge migration in radical cations.

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.

A Definition of the Magnetic Transition Temperature Using Valence Bond Theory.

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 the 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 wave function. 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 topologies 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.

Electron Dynamics upon Ionization of Polyatomic Molecules: Coupling to Quantum Nuclear Motion and Decoherence.

Knowledge about the electronic motion in molecules is essential for our understanding of chemical reactions and biological processes. The advent of attosecond techniques opens up the possibility to induce electronic motion, observe it in real time, and potentially steer it. A fundamental question remains the factors influencing electronic decoherence and the role played by nuclear motion in this process. Here, we simulate the dynamics upon ionization of the polyatomic molecules paraxylene and modified bismethylene-adamantane, with a quantum mechanical treatment of both electron and nuclear dynamics using the direct dynamics variational multiconfigurational Gaussian method. Our simulations give new important physical insights about the expected decoherence process. We have shown that the decoherence of electron dynamics happens on the time scale of a few femtoseconds, with the interplay of different mechanisms: the dephasing is responsible for the fast decoherence while the nuclear overlap decay may actually help maintain it and is responsible for small revivals.

Charge migration in polycyclic norbornadiene cations: Winning the race against decoherence.

The observation of electronic motion remains a key target in the development of the field 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 electron density oscillations. Here we demonstrate electron dynamics in norbornadiene and extended systems where the hole density migrates between two identical chromophores. By studying the effect of nuclear motion and delocalization in these example systems, we present the physical properties that must be considered in candidate molecules in which to observe electron dynamics. Furthermore, we also show a key contribution to nuclear delocalization arises from motion in the branching plane of the cation. For the systems studied, the dephasing time increases with system size while the energy gap between states, and therefore the frequency of the density oscillation, decreases with size (obeying a simple exponential dependence on the inter-chromophore distance). We present a system that balances these two effects and shows several complete oscillations in the spin density before dephasing occurs.

Electron and nuclear dynamics following ionisation of modified bismethylene-adamantane.

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.

Nuclear spatial delocalization silences electron density oscillations in 2-phenyl-ethyl-amine (PEA) and 2-phenylethyl-N,N-dimethylamine (PENNA) cations.

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.

Geometric Rotation of the Nuclear Gradient at a Conical Intersection: Extension to Complex Rotation of Diabatic States.

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.

Polarizable QM/MM Multiconfiguration Self-Consistent Field Approach with State-Specific Corrections: Environment Effects on Cytosine Absorption Spectrum.

We present the formulation and implementation of a polarizable quantum mechanics/molecular mechanics (QM/MM) strategy to describe environment effects in multiconfiguration self-consistent field calculations. The strategy is applied to the calculation of the vertical absorption spectrum of cytosine in water. In our approach, mutual polarization of the solute and the solvent is solved self-consistently at the complete-active-space self-consistent-field (CASSCF) level, and the resulting set of charges and dipoles is used to calculate vertical excitation energies using the complete-active-space second-order perturbative (CASPT2) approach and its multistate (MS-CASPT2) variant. In order to treat multiple excited states, we converge the solvent polarization with respect to the state-averaged density of the solute. In order to obtain the final energies, however, we introduce a state-specific correction, where the solvent polarization is recomputed with the density of each state, and demonstrate that this correction brings the excitation energies closer to the values obtained with state-optimized orbitals. Comparison with PCM and nonpolarizable QM/MM calculations shows the importance of specific solute-solvent interactions and environment polarization in describing experiments. Overall, the calculated excitations for the π → π* states in water show good agreement with the experimental spectrum, whereas the n → π* appear at energies above 6 eV, approximately 1 eV higher than in the gas phase. Beyond solvents, the new method will allow studying the impact of heterogeneous biological environments in multiple excited states, as well as the treatment of multichromophoric systems where charge transfer and exciton states play important roles.

Electron dynamics upon ionization: control of the timescale through chemical substitution and effect of nuclear motion.

Photoionization can generate a non-stationary electronic state, which leads to coupled electron-nuclear dynamics in molecules. In this article, we choose benzene cation as a prototype because vertical ionization of the neutral species leads to a Jahn-Teller degeneracy between ground and first excited states of the cation. Starting with equal populations of ground and first excited states, there is no electron dynamics in this case. However, if we add methyl substituents that break symmetry but do not radically alter the electronic structure, we see charge migration: oscillations in the spin density that we can correlate with particular localized electronic structures, with a period depending on the gap between the states initially populated. We have also investigated the effect of nuclear motion on electron dynamics using a complete active space self-consistent field (CASSCF) implementation of the Ehrenfest method, most previous theoretical studies of electron dynamics having been carried out with fixed nuclei. In toluene cation for instance, simulations where the nuclei are allowed to move show significant differences in the electron dynamics after 3 fs, compared to simulations with fixed nuclei.

Electronic control of initial nuclear dynamics adjacent to a conical intersection.

Photoionization can create a nonstationary electronic state and therefore initiates coupled electron-nuclear dynamics in molecules. Using a CASSCF implementation of the Ehrenfest method, we study the nuclear dynamics following vertical ionization of toluene, starting close to the conical intersection between ground and first excited states of its cation. The results show how the initial nuclear dynamics is controlled by the nonstationary electronic state character. In particular, ionization of this system leading to an equal superposition of the two lowest energy states can initiate nuclear dynamics in an orthogonal direction in the branching space to dynamics on the ground or first excited state potential energy surfaces alone.