Ab Initio Molecular Cavity Quantum Electrodynamics Simulations Using Machine Learning Models

Perturbative analysis of the coherent state transformation in ab initio cavity quantum electrodynamics

Experimental demonstrations of modified chemical structure and reactivity under strong light-matter coupling have spurred theoretical and computational efforts to uncover underlying mechanisms. Ab initio cavity quantum electrodynamics (QED) combines quantum chemistry with cavity QED to investigate these phenomena in detail. Unitary transformations of ab initio cavity QED Hamiltonians have been used to make them more computationally tractable. We analyze one such transformation, the coherent state transformation, using perturbation theory. Applying perturbation theory up to third order for ground state energies and potential energy surfaces of several molecular systems under electronic strong coupling, we show that the coherent state transformation yields better agreement with exact ground state energies. We examine one specific case using perturbation theory up to ninth order and find that coherent state transformation performs better up to fifth order but converges more slowly to the exact ground state energy at higher orders. In addition, we apply perturbation theory up to second order for cavity mode states under bilinear coupling, elucidating how the coherent state transformation accelerates the convergence of the photonic subspace toward the complete basis limit and renders molecular ion energies origin invariant. These findings contribute valuable insights into computational advantages of the coherent state transformation in the context of ab initio cavity quantum electrodynamics methods.

Comparing parameterized and self-consistent approaches to ab initio cavity quantum electrodynamics for electronic strong coupling

Molecules under strong or ultra-strong light-matter coupling present an intriguing route to modify chemical structure, properties, and reactivity. A rigorous theoretical treatment of such systems requires handling matter and photon degrees of freedom on an equal quantum mechanical footing. In the regime of molecular electronic strong or ultra-strong coupling to one or a few molecules, it is desirable to treat the molecular electronic degrees of freedom using the tools of ab initio quantum chemistry, yielding an approach referred to as ab initio cavity quantum electrodynamics (ai-QED), where the photon degrees of freedom are treated at the level of cavity QED. We analyze two complementary approaches to ai-QED: (1) a parameterized ai-QED, a two-step approach where the matter degrees of freedom are computed using existing electronic structure theories, enabling the construction of rigorous ai-QED Hamiltonians in a basis of many-electron eigenstates, and (2) self-consistent ai-QED, a one-step approach where electronic structure methods are generalized to include coupling between electronic and photon degrees of freedom. Although these approaches are equivalent in their exact limits, we identify a disparity between the projection of the two-body dipole self-energy operator that appears in the parameterized approach and its exact counterpart in the self-consistent approach. We provide a theoretical argument that this disparity resolves only under the limit of a complete orbital basis and a complete many-electron basis for the projection. We present numerical results highlighting this disparity and its resolution in a particularly simple molecular system of helium hydride cation, where it is possible to approach these two complete basis limits simultaneously. In this same helium hydride system, we examine and compare the practical issue of the computational cost required to converge each approach toward the complete orbital and many-electron bases limit. Finally, we assess the aspect of photonic convergence for polar and charged species, finding comparable behavior between parameterized and self-consistent approaches.

A simple approach to rotationally invariant machine learning of a vector quantity

Unlike with the energy, which is a scalar property, machine learning (ML) prediction of vector or tensor properties poses the additional challenge of achieving proper invariance (covariance) with respect to molecular rotation. For the energy gradients needed in molecular dynamics (MD), this symmetry is automatically fulfilled when taking analytic derivative of the energy, which is a scalar invariant (using properly invariant molecular descriptors). However, if the properties cannot be obtained by differentiation, other appropriate methods should be applied to retain the covariance. Several approaches have been suggested to properly treat this issue. For nonadiabatic couplings and polarizabilities, for example, it was possible to construct virtual quantities from which the above tensorial properties are obtained by differentiation and thus guarantee the covariance. Another possible solution is to build the rotational equivariance into the design of a neural network employed in the model. Here, we propose a simpler alternative technique, which does not require construction of auxiliary properties or application of special equivariant ML techniques. We suggest a three-step approach, using the molecular tensor of inertia. In the first step, the molecule is rotated using the eigenvectors of this tensor to its principal axes. In the second step, the ML procedure predicts the vector property relative to this orientation, based on a training set where all vector properties were in this same coordinate system. As the third step, it remains to transform the ML estimate of the vector property back to the original orientation. This rotate-predict-rotate (RPR) procedure should thus guarantee proper covariance of a vector property and is trivially extensible also to tensors such as polarizability. The RPR procedure has an advantage that the accurate models can be trained very fast for thousands of molecular configurations, which might be beneficial where many training sets are required (e.g., in active learning). We have implemented the RPR technique, using the MLatom and Newton-X programs for ML and MD, and performed its assessment on the dipole moment along MD trajectories of 1,2-dichloroethane.

Unsupervised Machine Learning in the Analysis of Nonadiabatic Molecular Dynamics Simulation

The all-atomic full-dimensional-level simulations of nonadiabatic molecular dynamics (NAMD) in large realistic systems has received high research interest in recent years. However, such NAMD simulations normally generate an enormous amount of time-dependent high-dimensional data, leading to a significant challenge in result analyses. Based on unsupervised machine learning (ML) methods, considerable efforts were devoted to developing novel and easy-to-use analysis tools for the identification of photoinduced reaction channels and the comprehensive understanding of complicated molecular motions in NAMD simulations. Here, we tried to survey recent advances in this field, particularly to focus on how to use unsupervised ML methods to analyze the trajectory-based NAMD simulation results. Our purpose is to offer a comprehensive discussion on several essential components of this analysis protocol, including the selection of ML methods, the construction of molecular descriptors, the establishment of analytical frameworks, their advantages and limitations, and persistent challenges.

Theory and quantum dynamics simulations of exciton-polariton motional narrowing

The motional narrowing effect has been extensively studied for cavity exciton-polariton systems in recent decades both experimentally and theoretically, which is featured by (1) the subaverage behavior and (2) the asymmetric linewidths for the upper polariton and the lower polariton. However, a minimal theoretical model that is clear and adequate to address all these effects as well as the linewidth scaling relations remains missing. In this work, based on the single mode 1D Holstein-Tavis-Cummings (HTC) model, we studied the motional narrowing effect of the polariton linear absorption spectra via both semi-analytic derivations and numerically exact quantum dynamics simulations using the hierarchical equations of motion approach. The results reveal that under collective light-matter coupling between a cavity mode and N molecules, the polariton linewidth scales as 1/N under the slow limit, while scales as 1/N under the fast limit, due to the polaron decoupling effect. Furthermore, by varying the detunings, the polariton linewidths exhibit significant motional narrowing, covering both characters mentioned above. Our analytic linewidth expressions [Eqs. (34) and (35)] agree well with the numerical exact simulations in all the parameter regimes we explored. These results indicate that the physics of motional narrowing is adequately accounted for by the single-mode 1D HTC model. We envision that both the numerical results and the analytic polariton linewidths expression presented in this work will offer great theoretical value for providing a better understanding of the exciton-polariton motional narrowing based on the HTC model.

In Silico Chemical Experiments in the Age of AI: From Quantum Chemistry to Machine Learning and Back

Computational chemistry is an indispensable tool for understanding molecules and predicting chemical properties. However, traditional computational methods face significant challenges due to the difficulty of solving the Schrödinger equations and the increasing computational cost with the size of the molecular system. In response, there has been a surge of interest in leveraging artificial intelligence (AI) and machine learning (ML) techniques to in silico experiments. Integrating AI and ML into computational chemistry increases the scalability and speed of the exploration of chemical space. However, challenges remain, particularly regarding the reproducibility and transferability of ML models. This review highlights the evolution of ML in learning from, complementing, or replacing traditional computational chemistry for energy and property predictions. Starting from models trained entirely on numerical data, a journey set forth toward the ideal model incorporating or learning the physical laws of quantum mechanics. This paper also reviews existing computational methods and ML models and their intertwining, outlines a roadmap for future research, and identifies areas for improvement and innovation. Ultimately, the goal is to develop AI architectures capable of predicting accurate and transferable solutions to the Schrödinger equation, thereby revolutionizing in silico experiments within chemistry and materials science.

Formulation of transition dipole gradients for non-adiabatic dynamics with polaritonic states

A general formulation of the strong coupling between photons confined in a cavity and molecular electronic states is developed for the state-interaction state-average spin-restricted ensemble-referenced Kohn-Sham method. The light-matter interaction is included in the Jaynes-Cummings model, which requires the derivation and implementation of the analytical derivatives of the transition dipole moments between the molecular electronic states. The developed formalism is tested in the simulations of the nonadiabatic dynamics in the polaritonic states resulting from the strong coupling between the cavity photon mode and the ground and excited states of the penta-2,4-dieniminium cation, also known as PSB3. Comparison with the field-free simulations of the excited-state decay dynamics in PSB3 reveals that the light-matter coupling can considerably alter the decay dynamics by increasing the excited state lifetime and hindering photochemically induced torsion about the C=C double bonds of PSB3. The necessity of obtaining analytical transition dipole gradients for the accurate propagation of the dynamics is underlined.

Non-Hermitian molecular dynamics simulations of exciton-polaritons in lossy cavities

The observation that materials can change their properties when placed inside or near an optical resonator has sparked a fervid interest in understanding the effects of strong light-matter coupling on molecular dynamics, and several approaches have been proposed to extend the methods of computational chemistry into this regime. Whereas the majority of these approaches have focused on modeling a single molecule coupled to a single cavity mode, changes to chemistry have so far only been observed experimentally when very many molecules are coupled collectively to multiple modes with short lifetimes. While atomistic simulations of many molecules coupled to multiple cavity modes have been performed with semi-classical molecular dynamics, an explicit description of cavity losses has so far been restricted to simulations in which only a very few molecular degrees of freedom were considered. Here, we have implemented an effective non-Hermitian Hamiltonian to explicitly treat cavity losses in large-scale semi-classical molecular dynamics simulations of organic polaritons and used it to perform both mean-field and surface hopping simulations of polariton relaxation, propagation, and energy transfer.

Cavity Quantum Electrodynamics Complete Active Space Configuration Interaction Theory

Polariton chemistry has attracted great attention as a potential route to modify chemical structure, properties, and reactivity through strong interactions among molecular electronic, vibrational, or rovibrational degrees of freedom. A rigorous theoretical treatment of molecular polaritons requires the treatment of matter and photon degrees of freedom on equal quantum mechanical footing. In the limit of molecular electronic strong or ultrastrong coupling to one or a few molecules, it is desirable to treat the molecular electronic degrees of freedom using the tools of ab initio quantum chemistry, yielding an approach we refer to as ab initio cavity quantum electrodynamics, where the photon degrees of freedom are treated at the level of cavity quantum electrodynamics. Here, we present an approach called Cavity Quantum Electrodynamics Complete Active Space Configuration Interaction theory to provide ground- and excited-state polaritonic surfaces with a balanced description of strong correlation effects among electronic and photonic degrees of freedom. This method provides a platform for ab initio cavity quantum electrodynamics when both strong electron correlation and strong light-matter coupling are important and is an important step toward computational approaches that yield multiple polaritonic potential energy surfaces and couplings that can be leveraged for ab initio molecular dynamics simulations of polariton chemistry.

Machine Learning Framework for Modeling Exciton Polaritons in Molecular Materials

A light-matter hybrid quasiparticle, called a polariton, is formed when molecules are strongly coupled to an optical cavity. Recent experiments have shown that polariton chemistry can manipulate chemical reactions. Polariton chemistry is a collective phenomenon, and its effects increase with the number of molecules in a cavity. However, simulating an ensemble of molecules in the excited state coupled to a cavity mode is theoretically and computationally challenging. Recent advances in machine learning (ML) techniques have shown promising capabilities in modeling ground-state chemical systems. This work presents a general protocol to predict excited-state properties, such as energies, transition dipoles, and nonadiabatic coupling vectors with the hierarchically interacting particle neural network. ML predictions are then applied to compute the potential energy surfaces and electronic spectra of a prototype azomethane molecule in the collective coupling scenario. These computational tools provide a much-needed framework to model and understand many molecules' emerging excited-state polariton chemistry.

Polaritonic Chemistry: Hindering and Easing Ground State Polyenic Isomerization via Breakdown of σ-π Separation

The ground state conformational isomerization in polyenes is a symmetry allowed process. Its low energy barrier is governed by electron density transfer from the formal single bond that is rotated to the nearby formal double bonds. Along the reaction pathway, the transition state is therefore destabilized. The rules of polaritonic chemistry, i.e., chemistry in a nanocavity with reflecting windows, are barely beginning to be laid out. The standing electric field of the nanocavity couples strongly with the molecular wave function and modifies the potential energy curve in unexpected ways. A quantum electrodynamics approach, applied to the torsional degree of freedom of the central bond of butadiene, shows that formation of the polariton mixes the σ-π frameworks thereby stabilizing/destabilizing the planar, reactant-like conformations. The values of the fundamental mode of the cavity field used in the absence of the cavity do not trigger this mechanism.

Theoretical Advances in Polariton Chemistry and Molecular Cavity Quantum Electrodynamics

When molecules are coupled to an optical cavity, new light-matter hybrid states, so-called polaritons, are formed due to quantum light-matter interactions. With the experimental demonstrations of modifying chemical reactivities by forming polaritons under strong light-matter interactions, theorists have been encouraged to develop new methods to simulate these systems and discover new strategies to tune and control reactions. This review summarizes some of these exciting theoretical advances in polariton chemistry, in methods ranging from the fundamental framework to computational techniques and applications spanning from photochemistry to vibrational strong coupling. Even though the theory of quantum light-matter interactions goes back to the midtwentieth century, the gaps in the knowledge of molecular quantum electrodynamics (QED) have only recently been filled. We review recent advances made in resolving gauge ambiguities, the correct form of different QED Hamiltonians under different gauges, and their connections to various quantum optics models. Then, we review recently developed ab initio QED approaches which can accurately describe polariton states in a realistic molecule-cavity hybrid system. We then discuss applications using these method advancements. We review advancements in polariton photochemistry where the cavity is made resonant to electronic transitions to control molecular nonadiabatic excited state dynamics and enable new photochemical reactivities. When the cavity resonance is tuned to the molecular vibrations instead, ground-state chemical reaction modifications have been demonstrated experimentally, though its mechanistic principle remains unclear. We present some recent theoretical progress in resolving this mystery. Finally, we review the recent advances in understanding the collective coupling regime between light and matter, where many molecules can collectively couple to a single cavity mode or many cavity modes. We also lay out the current challenges in theory to explain the observed experimental results. We hope that this review will serve as a useful document for anyone who wants to become familiar with the context of polariton chemistry and molecular cavity QED and thus significantly benefit the entire community.