Predicting Molecular Photochemistry Using Machine-Learning-Enhanced Quantum Dynamics Simulations.

The processes which occur after molecules absorb light underpin an enormous range of fundamental technologies and applications, including photocatalysis to enable new chemical transformations, sunscreens to protect against the harmful effects of UV overexposure, efficient photovoltaics for energy generation from sunlight, and fluorescent probes to image the intricate details of complex biomolecular structures. Reflecting this broad range of applications, an enormously versatile set of experiments are now regularly used to interrogate light-driven chemical dynamics, ranging from the typical ultrafast transient absorption spectroscopy used in many university laboratories to the inspiring central facilities around the world, such as the next-generation of X-ray free-electron lasers.Computer simulations of light-driven molecular and material dynamics are an essential route to analyzing the enormous amount of transient electronic and structural data produced by these experimental sources. However, to date, the direct simulation of molecular photochemistry remains a frontier challenge in computational chemical science, simultaneously demanding the accurate treatment of molecular electronic structure, nuclear dynamics, and the impact of nonadiabatic couplings.To address these important challenges and to enable new computational methods which can be integrated with state-of-the-art experimental capabilities, the past few years have seen a burst of activity in the development of "direct" quantum dynamics methods, merging the machine learning of potential energy surfaces (PESs) and nonadiabatic couplings with accurate quantum propagation schemes such as the multiconfiguration time-dependent Hartree (MCTDH) method. The result of this approach is a new generation of direct quantum dynamics tools in which PESs are generated in tandem with wave function propagation, enabling accurate "on-the-fly" simulations of molecular photochemistry. These simulations offer an alternative route toward gaining quantum dynamics insights, circumventing the challenge of generating ab initio electronic structure data for PES fitting by instead only demanding expensive energy evaluations as and when they are needed.In this Account, we describe the chronological evolution of our own contributions to this field, focusing on describing the algorithmic developments that enable direct MCTDH simulations for complex molecular systems moving on multiple coupled electronic states. Specifically, we highlight active learning strategies for generating PESs during grid-based quantum chemical dynamics simulations, and we discuss the development and impact of novel diabatization schemes to enable direct grid-based simulations of photochemical dynamics; these developments are highlighted in a series of benchmark molecular simulations of systems containing multiple nuclear degrees of freedom moving on multiple coupled electronic states. We hope that the ongoing developments reported here represent a major step forward in tools for modeling excited-state chemistry such as photodissociation, proton and electron transfer, and ultrafast energy dissipation in complex molecular systems.

Analyzing Grid-Based Direct Quantum Molecular Dynamics Using Non-Linear Dimensionality Reduction.

Grid-based schemes for simulating quantum dynamics, such as the multi-configuration time-dependent Hartree (MCTDH) method, provide highly accurate predictions of the coupled nuclear and electronic dynamics in molecular systems. Such approaches provide a multi-dimensional, time-dependent view of the system wavefunction represented on a coordinate grid; in the case of non-adiabatic simulations, additional information about the state populations adds a further layer of complexity. As such, wavepacket motion on potential energy surfaces which couple many nuclear and electronic degrees-of-freedom can be extremely challenging to analyse in order to extract physical insight beyond the usual expectation-value picture. Here, we show that non-linear dimensionality reduction (NLDR) methods, notably diffusion maps, can be adapted to extract information from grid-based wavefunction dynamics simulations, providing insight into key nuclear motions which explain the observed dynamics. This approach is demonstrated for 2-D and 9-D models of proton transfer in salicylaldimine, as well as 8-D and full 12-D simulations of cis-trans isomerization in ethene; these simulations demonstrate how NLDR can provide alternative views of wavefunction dynamics, and also highlight future developments.

Experimental and Computational Analysis of Para-Hydroxy Methylcinnamate following Photoexcitation.

Para-hydroxy methylcinnamate is part of the cinnamate family of molecules. Experimental and computational studies have suggested conflicting non-radiative decay routes after photoexcitation to its S1(ππ*) state. One non-radiative decay route involves intersystem crossing mediated by an optically dark singlet state, whilst the other involves direct intersystem crossing to a triplet state. Furthermore, irrespective of the decay mechanism, the lifetime of the initially populated S1(ππ*) state is yet to be accurately measured. In this study, we use time-resolved ion-yield and photoelectron spectroscopies to precisely determine the S1(ππ*) lifetime for the s-cis conformer of para-hydroxy methylcinnamate, combined with time-dependent density functional theory to determine the major non-radiative decay route. We find the S1(ππ*) state lifetime of s-cis para-hydroxy methylcinnamate to be ∼2.5 picoseconds, and the major non-radiative decay route to follow the [1ππ*→1nπ*→3ππ*→S0] pathway. These results also concur with previous photodynamical studies on structurally similar molecules, such as para-coumaric acid and methylcinnamate.

Direct Grid-Based Nonadiabatic Dynamics on Machine-Learned Potential Energy Surfaces: Application to Spin-Forbidden Processes.

We have recently shown how high-accuracy wave function grid-based propagation schemes, such as the multiconfiguration time-dependent Hartree (MCTDH) method, can be combined with machine-learning (ML) descriptions of PESs to yield an "on-the-fly" direct dynamics scheme which circumvents potential energy surface (PES) prefitting. To date, our approach has been demonstrated in the ground-state dynamics and nonadiabatic spin-allowed dynamics of several molecular systems. Expanding on this successful previous work, this Article demonstrates how our ML-based quantum dynamics scheme can be adapted to model nonadiabatic dynamics for spin-forbidden processes such as intersystem crossing (ISC), opening up new possibilities for modeling chemical dynamic phenomena driven by spin-orbit coupling. After describing modifications to diabatization schemes to enable accurate and robust treatment or electronic states of different spin-multiplicity, we demonstrate our methodology in applications to modeling ISC in SO2 and thioformaldehyde, benchmarking our results against previous trajectory- and grid-based calculations. As a relatively efficient tool for modeling spin-forbidden nonadiabatic dynamics without demanding any prefitting of PESs, our overall strategy is a potentially powerful tool for modeling important photochemical systems, such as photoactivated pro-drugs and organometallic catalysts.

A new diabatization scheme for direct quantum dynamics: Procrustes diabatization.

We present a new scheme for diabatizing electronic potential energy surfaces for use within the recently implemented direct-dynamics grid-based class of computational nuclear quantum dynamics methods, called Procrustes diabatization. Calculations on the well-studied molecular systems LiF and the butatriene cation, using both Procrustes diabatization and the previously implemented propagation and projection diabatization schemes, have allowed detailed comparisons to be made, which indicate that the new method combines the best features of the older approaches; it generates smooth surfaces, which cross at the correct molecular geometries, reproduces interstate couplings accurately, and hence allows the correct modeling of non-adiabatic dynamics.

Time-Resolved Photoelectron Spectroscopy Studies of Isoxazole and Oxazole.

The excited state relaxation pathways of isoxazole and oxazole upon excitation with UV-light were investigated by nonadiabatic ab initio dynamics simulations and time-resolved photoelectron spectroscopy. Excitation of the bright ππ*-state of isoxazole predominantly leads to ring-opening dynamics. Both the initially excited ππ*-state and the dissociative πσ*-state offer a combined barrier-free reaction pathway, such that ring-opening, defined as a distance of more than 2 Å between two neighboring atoms, occurs within 45 fs. For oxazole, in contrast, the excited state dynamics is about twice as slow (85 fs) and the quantum yield for ring-opening is lower. This is caused by a small barrier between the ππ*-state and the πσ*-state along the reaction path, which suppresses direct ring-opening. Theoretical findings are consistent with the measured time-resolved photoelectron spectra, confirming the timescales and the quantum yields for the ring-opening channel. The results indicate that a combination of time-resolved photoelectron spectroscopy and excited state dynamics simulations can explain the dominant reaction pathways for this class of molecules. As a general rule, we suggest that the antibonding σ*-orbital located between the oxygen atom and a neighboring atom of a five-membered heterocyclic system provides a driving force for ring-opening reactions, which is modified by the presence and position of additional nitrogen atoms.

Can we use on-the-fly quantum simulations to connect molecular structure and sunscreen action?

The ground-up design of new molecular sunscreens, with improved photostability, absorbance and spectral coverage, stands as a challenge to fundamental chemical science. Correlating sunscreen molecular structure and function requires detailed insight into the relaxation pathways available following photoexcitation; however, the complex coupled electron/nuclear dynamics in these systems stands as a tough challenge to computational chemistry. To address this challenge, we have recently developed efficient and accurate simulation methods to model non-adiabatic dynamics of general molecular systems as a route to correlating photoinduced dynamics and potential sunscreen activity. Our approach, combining the multi-configuration time-dependent Hartree (MCTDH) method with PESs generated using machine-learning, represents a new "on-the-fly" strategy for accurate wavefunction propagation, with the potential to provide a new "black box" strategy for interrogating ultrafast dynamics in general photoinduced energy transport processes. Here, we illustrate our attempts to apply this methodology to study the ultrafast photochemistry of four compounds derived from mycosporine-like amino acids (MAAs), compounds which are believed to act as microbial sunscreens in micro-organisms such as algae. Specifically, we investigate how the choice of active vibrational space and diabatic electronic states in MCTDH strongly influences the predictions of ultrafast dynamic relaxation in representative MAAs. Our results serve to demonstrate that "on-the-fly" quantum dynamics using MCTDH is increasingly viable, but important barriers relating to coordinate choices and diabatisation remain.

Direct quantum dynamics using variational Gaussian wavepackets and Gaussian process regression.

The method of direct variational quantum nuclear dynamics in a basis of Gaussian wavepackets, combined with the potential energy surfaces fitted on-the-fly using Gaussian process regression, is described together with its implementation. Enabling exact and efficient analytic evaluation of Hamiltonian matrix elements, this approach allows for black-box quantum dynamics of multidimensional anharmonic molecular systems. Example calculations of intra-molecular proton transfer on the electronic ground state of salicylaldimine are provided, and future algorithmic improvements as well as the potential for multiple-state non-adiabatic dynamics are discussed.

Improved on-the-Fly MCTDH Simulations with Many-Body-Potential Tensor Decomposition and Projection Diabatization.

We have recently demonstrated how potential energy surface (PES) interpolation methods, such as kernel ridge regression (KRR), can be combined with accurate wave function time-propagation methods, specifically the multiconfiguration time-dependent Hartree (MCTDH) method, to generate a new "on-the-fly" MCTDH scheme (DD-MCTDH) that does not require the pre-fitting of the PES, which is normally required by MCTDH. Specifically, we have shown how our DD-MCTDH strategy can be used to model non-adiabatic dynamics in a 4-mode/2-state model of pyrazine, with ab initio electronic structure calculations performed directly during propagation, requiring around 100 h of computer wall-time. In this Article, we show how the efficiency and accuracy of DD-MCTDH can be dramatically improved further still by (i) using systematic tensor decompositions of the KRR PES, and (ii) using a novel scheme for diabatization within the framework of configuration interaction (CI) methods which only requires local adiabatic electronic states, rather than non-adiabatic coupling matrix elements. The result of these improvements is that our latest version of DD-MCTDH can perform a 12-mode/2-state simulation of pyrazine, with PES evaluations at CAS level, in just 29-90 h on a standard desktop computer; this work therefore represents an enormous step towards direct quantum dynamics with MCTDH.

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.

Unravelling the Photoprotection Properties of Mycosporine Amino Acid Motifs.

Photoprotection from harmful ultraviolet (UV) radiation exposure is a key problem in modern society. Mycosporine-like amino acids found in fungi, cyanobacteria, macroalgae, phytoplankton, and animals are already presenting a promising form of natural photoprotection in sunscreen formulations. Using time-resolved transient electronic absorption spectroscopy and guided by complementary ab initio calculations, we help to unravel how the core structures of these molecules perform under UV irradiation. Through such detailed insight into the relaxation mechanisms of these ubiquitous molecules, we hope to inspire new thinking in developing next-generation photoprotective molecules.

MCTDH on-the-fly: Efficient grid-based quantum dynamics without pre-computed potential energy surfaces.

We present significant algorithmic improvements to a recently proposed direct quantum dynamics method, based upon combining well established grid-based quantum dynamics approaches and expansions of the potential energy operator in terms of a weighted sum of Gaussian functions. Specifically, using a sum of low-dimensional Gaussian functions to represent the potential energy surface (PES), combined with a secondary fitting of the PES using singular value decomposition, we show how standard grid-based quantum dynamics methods can be dramatically accelerated without loss of accuracy. This is demonstrated by on-the-fly simulations (using both standard grid-based methods and multi-configuration time-dependent Hartree) of both proton transfer on the electronic ground state of salicylaldimine and the non-adiabatic dynamics of pyrazine.

Direct Quantum Dynamics Using Grid-Based Wave Function Propagation and Machine-Learned Potential Energy Surfaces.

We describe a method for performing nuclear quantum dynamics calculations using standard, grid-based algorithms, including the multiconfiguration time-dependent Hartree (MCTDH) method, where the potential energy surface (PES) is calculated "on-the-fly". The method of Gaussian process regression (GPR) is used to construct a global representation of the PES using values of the energy at points distributed in molecular configuration space during the course of the wavepacket propagation. We demonstrate this direct dynamics approach for both an analytical PES function describing 3-dimensional proton transfer dynamics in malonaldehyde and for 2- and 6-dimensional quantum dynamics simulations of proton transfer in salicylaldimine. In the case of salicylaldimine we also perform calculations in which the PES is constructed using Hartree-Fock calculations through an interface to an ab initio electronic structure code. In all cases, the results of the quantum dynamics simulations are in excellent agreement with previous simulations of both systems yet do not require prior fitting of a PES at any stage. Our approach (implemented in a development version of the Quantics package) opens a route to performing accurate quantum dynamics simulations via wave function propagation of many-dimensional molecular systems in a direct and efficient manner.

A Practical Diabatisation Scheme for Use with the Direct-Dynamics Variational Multi-Configuration Gaussian Method.

A method for diabatising multiple electronic states on-the-fly within the direct dynamics variational multi-configuration Gaussian method for calculating quantum nuclear dynamics is presented. The method is based upon the propagation of the adiabatic-diabatic transformation matrix along the paths followed by the Gaussian basis functions that constitute the nuclear wave function, by use of a well-known differential equation relating the matrix and the nonadiabatic vector coupling terms between the electronic states. The implementation of the method is described, and test calculations are presented using the ground and first-excited states of the butatriene cation as an example, allowing comparison to the earlier regularisation diabatisation scheme as well as to full nuclear dynamics on a precomputed potential energy surface. The new scheme is termed propagation diabatisation.

The time-resolved photoelectron spectrum of toluene using a perturbation theory approach.

A theoretical study of the intra-molecular vibrational-energy redistribution of toluene using time-resolved photo-electron spectra calculated using nuclear quantum dynamics and a simple, two-mode model is presented. Calculations have been carried out using the multi-configuration time-dependent Hartree method, using three levels of approximation for the calculation of the spectra. The first is a full quantum dynamics simulation with a discretisation of the continuum wavefunction of the ejected electron, whilst the second uses first-order perturbation theory to calculate the wavefunction of the ion. Both methods rely on the explicit inclusion of both the pump and probe laser pulses. The third method includes only the pump pulse and generates the photo-electron spectrum by projection of the pumped wavepacket onto the ion potential energy surface, followed by evaluation of the Fourier transform of the autocorrelation function of the subsequently propagated wavepacket. The calculations performed have been used to study the periodic population flow between the 6a and 10b16b modes in the S1 excited state, and compared to recent experimental data. We obtain results in excellent agreement with the experiment and note the efficiency of the perturbation method.

Non-resonant dynamic Stark control at a conical intersection: the photodissociation of ammonia.

Control of the photodissociation of ammonia, by the nonresonant dynamic Stark effect, has been studied theoretically by the numerically exact propagation of wavepackets on ab initio potential energy surfaces. An assessment of the feasibility of controlling the proportion of the wavepacket which dissociates to produce ground or electronically excited state NH(2) fragments, mediated by a conical intersection, has been made by use of a simple two-dimensional, two-state model. It was found that modest control was possible for nonresonant pulses applied during and after excitation, and that the control was caused not by alteration of the position or nature of the conical intersection but by modification of the energy surfaces around the Franck-Condon region. This is made possible by the predissociative nature of the mechanism for hydrogen ejection. The control effect is frequency independent but dependent on pulse, i.e., electric field, strength, indicating that it is indeed due to the Stark effect. Analysis of the control is, however, complicated by the presence of vibrational effects which can come into play if the control pulse frequency is not carefully chosen. By systematically varying the excitation energy, it was also found that the capacity for control is only significant at low energies.

Dynamic Stark control: model studies based on the photodissociation of IBr.

The Stark effect is produced when a static field alters molecular states. When the field applied is time dependent, the process is known as the dynamic Stark effect. Of particular interest for the control of molecular dynamics is the Non-Resonant Dynamic Stark Effect (NRDSE), in which the time dependent field is unable to effect a one-photon excitation. The intermediate strength laser pulse instead shapes the potential energy surfaces (PES) and so guides the evolution of the system. A prototype control scheme uses the NRDSE to change the topography of PES in regions where they intersect, thus providing control over photochemistry. Following earlier experimental work, in this paper we study the NRDSE on a new 3 state model of the IBr molecule to gain insight into the mechanism of control at the avoided crossing that governs the branching ratio of the photodissociation.

Implementation of the full explicitly correlated coupled-cluster singles and doubles model CCSD-F12 with optimally reduced auxiliary basis dependence.

An implementation of the full explicitly correlated coupled-cluster singles and doubles model CCSD-F12 using a single Slater-type geminal has been obtained with the aid of automated term generation and evaluation techniques. In contrast to a previously reported computer code [T. Shiozaki et al., J. Chem. Phys. 129, 071101 (2008)], our implementation features a reduced dependence on the auxiliary basis set due to the use of a reformulated evaluation of the so-called Z-intermediate rather than straight forward insertion of an auxiliary basis expansion, which allows an unambiguous comparison to more approximate CCSD-F12 models. First benchmark results for total correlation energies and reaction energies indicate an excellent performance of the much cheaper CCSD(F12) model.