Real-Time Time-Dependent Nuclear-Electronic Orbital Approach: Dynamics beyond the Born-Oppenheimer Approximation

Beyond Electrons: Correlation and Self-Energy in Multicomponent Density Functional Theory

Post-Kohn-Sham methods are used to evaluate the ground-state correlation energy and the orbital self-energy of systems consisting of multiple flavors of different fermions. Starting from multicomponent density functional theory, suitable ways to arrive at the corresponding multicomponent random-phase approximation and the multicomponent Green's functionG W ${GW}$ approximation, including relativistic effects, are outlined. Given the importance of both of this methods in the development of modern Kohn-Sham density functional approximations, this work will provide a foundation to design advanced multicomponent density functional approximations. Additionally, theG W ${GW}$ quasiparticle energies are needed to study light-matter interactions with the Bethe-Salpeter equation.

Nuclear-Electronic Orbital Time-Dependent Configuration Interaction Method

Combining real-time electronic structure with the nuclear-electronic orbital (NEO) method has enabled the simulation of complex nonadiabatic chemical processes. However, accurate descriptions of hydrogen tunneling and double excitations require multiconfigurational treatments. Herein, we develop and implement the real-time NEO time-dependent configuration interaction (NEO-TDCI) approach. Comparison to NEO-full CI calculations of absorption spectra for a molecular system shows that the NEO-TDCI approach can accurately capture the tunneling splitting associated with the electronic ground state as well as vibronic progressions corresponding to double electron-proton excitations associated with excited electronic states. Both of these features are absent from spectra obtained with single reference real-time NEO methods. Our simulations of hydrogen tunneling dynamics illustrate the oscillation of the proton density from one side to the other via a delocalized, bilobal proton wave function. These results indicate that the NEO-TDCI approach is highly suitable for studying hydrogen tunneling and other inherently multiconfigurational systems.

Optimization of Quantum Nuclei Positions with the Adaptive Nuclear-Electronic Orbital Approach

The use of multicomponent methods has become increasingly popular over the last years. Under this framework, nuclei (commonly protons) are treated quantum mechanically on the same footing as the electronic structure problem. Under the use of atomic-centered orbitals, this can lead to some complications as the ideal location of the nuclear basis centers must be optimized. In this contribution, we propose a straightforward approach to determine the position of such centers within the self-consistent cycle of a multicomponent calculation, making use of individual proton charge centroids. We test the method on model systems including the water dimer, a protonated water tetramer, and a porphine system. Comparing to numerical gradient calculations, the adaptive nuclear-electronic orbital (NEO) procedure is able to converge the basis centers to within a few cents of an Ångström and with less than 0.1 kcal/mol differences in absolute energies. This is achieved in one single calculation and with a small added computational effort of up to 80% compared to a regular NEO- self-consistent field run. An example application for the human transketolase proton wire is also provided.

A Many-Body Perspective of Nuclear Quantum Effects in Aqueous Clusters

Nuclear quantum effects play an important role in the structure and thermodynamics of aqueous systems. By performing a many-body expansion with nuclear-electronic orbital (NEO) theory, we show that proton quantization can give rise to significant energetic contributions for many-body interactions spanning several molecules in single-point energy calculations of water clusters. Although zero-point motion produces a large increase in energy at the one-body level, nuclear quantum effects serve to stabilize higher-order molecular interactions. These results are significant because they demonstrate that nuclear quantum effects play a nontrivial role in many-body interactions of aqueous systems. Our approach also provides a pathway for incorporating nuclear quantum effects into water potential energy surfaces. The NEO approach is advantageous for many-body expansion analyses because it includes nuclear quantum effects directly in the energies.

Real-Time Time-Dependent Density Functional Theory for Simulating Nonequilibrium Electron Dynamics

The explicit real-time propagation approach for time-dependent density functional theory (RT-TDDFT) has increasingly become a popular first-principles computational method for modeling various time-dependent electronic properties of complex chemical systems. In this Perspective, we provide a nontechnical discussion of how this first-principles simulation approach has been used to gain novel physical insights into nonequilibrium electron dynamics phenomena in recent years. Following a concise overview of the RT-TDDFT methodology from a practical standpoint, we discuss our recent studies on the electronic stopping of DNA in water and the Floquet topological phase as examples. Our discussion focuses on how RT-TDDFT simulations played a unique role in deriving new scientific understandings. We then discuss existing challenges and some new advances at the frontier of RT-TDDFT method development for studying increasingly complex dynamic phenomena and systems.

Nonadiabatic Derivative Couplings through Multiple Franck-Condon Modes Dictate the Energy Gap Law for Near and Short-Wave Infrared Dye Molecules

Near infrared (NIR, 700-1000 nm) and short-wave infrared (SWIR, 1000-2000 nm) dye molecules exhibit significant nonradiative decay rates from the first singlet excited state to the ground state. While these trends can be empirically explained by a simple energy gap law, detailed mechanisms of nearly universal behavior have remained unsettled for many cases. Theoretical and experimental results for two representative NIR/SWIR dye molecules reported here clarify the key mechanism for the observed energy gap law behavior. It is shown that the first derivative nonadiabatic coupling terms serve as major coupling pathways for nonadiabatic decay processes from the first excited singlet state to the ground state for these NIR and SWIR dye molecules and that vibrational modes other than the highest frequency modes also make significant contributions to the rate. This assessment is corroborated by further theoretical comparison with possible alternative mechanisms of intersystem crossing to triplet states and also by comparison with experimental data for deuterated molecules.

Squeezed Protons and Infrared Plasmonic Resonance Energy Transfer

Unusual nuclear quantum effects may emerge near noble metal nanostructures such as squeezed vibrational states in molecular junctions and plasmonic resonance energy transfer in the infrared domain. Herein, nuclear quantum effects near heavy metals are studied by nuclear-electronic orbital density functional theory (NEO-DFT) with an effective core potential. For a quantum proton sandwiched between a pair of gold tips modeled by two Au6 clusters, NEO-DFT calculations suggest that the quantum proton density can be squeezed as the tip distance decreases. For an HF molecule placed near a one-dimensional Au nanowire composed of up to 34 Au atoms, real-time NEO time-dependent density functional theory (RT-NEO-TDDFT) shows that the infrared plasmonic motion within the Au nanowire may resonantly transfer electronic energy to the HF proton vibrational stretch mode. Overall, these calculations illustrate the advantages of the NEO approach for probing nuclear quantum effects, such as squeezed proton vibrational states and infrared plasmonic resonance energy transfer.

First-Principles Approach for Coupled Quantum Dynamics of Electrons and Protons in Heterogeneous Systems

The coupled quantum dynamics of electrons and protons is ubiquitous in many dynamical processes involving light-matter interaction, such as solar energy conversion in chemical systems and photosynthesis. A first-principles description of such nuclear-electronic quantum dynamics requires not only the time-dependent treatment of nonequilibrium electron dynamics but also that of quantum protons. Quantum mechanical correlation between electrons and protons adds further complexity to such coupled dynamics. Here we extend real-time nuclear-electronic orbital time-dependent density functional theory (RT-NEO-TDDFT) to periodic systems and perform first-principles simulations of coupled quantum dynamics of electrons and protons in complex heterogeneous systems. The process studied is an electronically excited-state intramolecular proton transfer of o-hydroxybenzaldehyde in water and at a silicon (111) semiconductor-molecule interface. These simulations illustrate how environments such as hydrogen-bonding water molecules and an extended material surface impact the dynamical process on the atomistic level. Depending on how the molecule is chemisorbed on the surface, excited-state electron transfer from the molecule to the semiconductor surface can inhibit ultrafast proton transfer within the molecule. This Letter elucidates how heterogeneous environments influence the balance between the quantum mechanical proton transfer and excited electron dynamics. The periodic RT-NEO-TDDFT approach is applicable to a wide range of other photoinduced heterogeneous processes.

Nuclear Quantum Effects Made Accessible: Local Density Fitting in Multicomponent Methods

The simulation of nuclear quantum effects (NQEs) is crucial for an accurate description of systems and processes involving light nuclei, such as hydrogen atoms. Within the last years, the importance of those effects has been highlighted for a vast range of systems with tremendous implications in chemistry, biology, physics, and materials sciences. However, while electronic structure theory methods have become routine tools for quantum chemical investigations, there is still a lack of approaches to address NQEs that are computationally accessible and straightforward to use. To address this, we present the first combination of the nuclear-electronic orbital Hartree-Fock approach with both local and density fitting approximations (LDF-NEO-HF). This results in a low-order scaling approach that enables the inclusion of NQEs for large systems within a fraction of a day and for small to medium size systems in minutes. Moreover, we demonstrate the qualitative accuracy and robustness of our approach to retrieve NQEs for three real-use cases motivated by chemical, biological, and materials science applications.

Assessing Implicit and Explicit Polarizable Solvation Models for Nuclear-Electronic Orbital Systems: Quantum Proton Polarization and Solvation Energetics

Accurate simulations of many chemical processes require the inclusion of both nuclear quantum effects and a solvent environment. The nuclear-electronic orbital (NEO) approach, which treats electrons and select nuclei quantum mechanically on the same level, combined with a polarizable continuum model (PCM) for the solvent environment, addresses this challenge in a computationally practical manner. In this work, the NEO-PCM approach is extended beyond the IEF-PCM (integral equation formalism PCM) and C-PCM (conductor PCM) approaches to the SS(V)PE (surface and simulation of volume polarization for electrostatics) and ddCOSMO (domain decomposed conductor-like screening model) approaches. IEF-PCM, SS(V)PE, C-PCM, and ddCOSMO all exhibit similar solvation energies as well as comparable nuclear polarization within the NEO framework. The calculations show that the nuclear density does not leak out of the molecular cavity because it is much more localized than the electronic density. Finally, the polarization of quantized protons is analyzed in both continuum solvent and explicit solvent environments described by the polarizable MB-pol model, illustrating the impact of specific hydrogen-bonding interactions captured only by explicit solvation. These calculations highlight the relationship among solvation formalism, nuclear polarization, and energetics.

Nuclear-Electronic Orbital Quantum Mechanical/Molecular Mechanical Real-Time Dynamics

Simulating the nuclear-electronic quantum dynamics of large-scale molecular systems in the condensed phase is key for studying biologically and chemically important processes such as proton transfer and proton-coupled electron transfer reactions. Herein, the real-time nuclear-electronic orbital time-dependent density functional theory (RT-NEO-TDDFT) approach is combined with a hybrid quantum mechanical/molecular mechanical (QM/MM) strategy to enable the accurate description of coupled nuclear-electronic quantum dynamics in the presence of heterogeneous environments such as solvent or proteins. The densities of the electrons and quantum protons are propagated in real time, while the other nuclei are propagated classically on the instantaneous electron-proton vibronic surface. This approach is applied to phenol bound to lysozyme, intramolecular proton transfer in malonaldehyde, and nonequilibrium excited-state intramolecular proton transfer in o-hydroxybenzaldehyde. These examples illustrate that the RT-NEO-TDDFT framework, coupled with an atomistic representation of the environment, allows the simulation of condensed-phase systems that exhibit significant nuclear quantum effects.

A Perspective on Sustainable Computational Chemistry Software Development and Integration

The power of quantum chemistry to predict the ground and excited state properties of complex chemical systems has driven the development of computational quantum chemistry software, integrating advances in theory, applied mathematics, and computer science. The emergence of new computational paradigms associated with exascale technologies also poses significant challenges that require a flexible forward strategy to take full advantage of existing and forthcoming computational resources. In this context, the sustainability and interoperability of computational chemistry software development are among the most pressing issues. In this perspective, we discuss software infrastructure needs and investments with an eye to fully utilize exascale resources and provide unique computational tools for next-generation science problems and scientific discoveries.

A brand-new type of excited-state proton transfer (ESIPT) molecule based on sulfoxide/sulfenic acid tautomerism

The excited-state proton transfer (ESIPT) behavior of organic fluorophores has attracted much attention due to their unique photophysical properties. So far, ESIPT studies have mainly focused on the transfer of hydrogen atoms between N-N, N-O, or O-O. In this work, a brand-new type of ESIPT molecule based on sulfoxide/sulfenic acid tautomerism has been thoroughly investigated. The sulfoxide/sulfenic acid tautomerization process requires one step and two steps in the ground and first excited singlet states, respectively. A range of density functional theory and time-dependent density functional theory methods have been employed to investigate these structures, and the changes in aromaticity may be responsible for obtaining the ESIPT process. This work presents a novel ESIPT process, showcasing molecules that exhibit distinctive properties compared to conventional ESIPT compounds. These findings are expected to expand the horizons of experimental research in ESIPT.

Nuclear-Electronic Orbital Quantum Dynamics of Plasmon-Driven H2 Photodissociation

Leveraging localized surface plasmon resonances of metal nanoparticles to trigger chemical reactions is a promising approach for heterogeneous catalysis. First-principles modeling of such processes is challenging due to the large number of electrons and electronic excited states as well as the significance of nuclear quantum effects when hydrogen is involved. Herein, the nonadiabatic nuclear-electronic quantum dynamics of plasmon-induced H2 photodissociation near an Al13- cluster is simulated with real-time nuclear-electronic orbital time-dependent density functional theory (RT-NEO-TDDFT). This approach propagates the nonequilibrium quantum dynamics of both electrons and protons. The plasmonic oscillations are shown to inject hot electrons into the antibonding orbital of H2, thereby inducing H2 dissociation. The quantum mechanical treatment of the hydrogen nuclei leads to faster H2 photodissociation and slightly larger isotope effects. Analysis of the nonequilibrium electronic density suggests that these findings stem from enhanced excited-state electronic coupling between the plasmonic mode and the H2 antibonding orbital due to proton delocalization or zero-point energy effects. Given the low computational overhead for including nuclear quantum effects with the RT-NEO-TDDFT approach, this work paves the way for simulating nonadiabatic nuclear-electronic quantum dynamics in other plasmonic systems.

Nonadiabatic Nuclear-Electron Dynamics: A Quantum Computing Approach

Coupled quantum electron-nuclear dynamics is often associated with the Born-Huang expansion of the molecular wave function and the appearance of nonadiabatic effects as a perturbation. On the other hand, native multicomponent representations of electrons and nuclei also exist, which do not rely on any a priori approximation. However, their implementation is hampered by prohibitive scaling. Consequently, quantum computers offer a unique opportunity for extending their use to larger systems. Here, we propose a quantum algorithm for simulating the time-evolution of molecular systems and apply it to proton transfer dynamics in malonaldehyde, described as a rigid scaffold. The proposed quantum algorithm can be easily generalized to include the explicit dynamics of the classically described molecular scaffold. We show how entanglement between electronic and nuclear degrees of freedom can persist over long times if electrons do not follow the nuclear displacement adiabatically. The proposed quantum algorithm may become a valid candidate for the study of such phenomena when sufficiently powerful quantum computers become available.

Constrained Nuclear-Electronic Orbital Density Functional Theory with a Dielectric Continuum Solvent Model

Solvent effects are crucial for simulating chemical and biological processes in solutions. The continuum solvation model is widely used for incorporating solvent effects with different levels of theoretical descriptions of solutes. For solutes and solutions containing hydrogen atoms, nuclear quantum effects can also be nonnegligible for reliable simulations. In this work, we couple our recently developed constrained nuclear-electronic orbital density functional theory with a dielectric continuum solvation model to cover nuclear quantum effects and solvent effects simultaneously. This approach is applied to the formate ion, where an anomalous solvatochromic shift in C-H stretch frequency was reported in experiments. By using this new approach to account for nuclear quantum effects and solvent effects, we show that the vibrational frequency of the C-H stretch and the solvatochromic shift are accurately described.

Generalized Nuclear-Electronic Orbital Multistate Density Functional Theory for Multiple Proton Transfer Processes

Proton transfer and hydrogen tunneling play pivotal roles in many chemical and biological processes. The nuclear-electronic orbital multistate density functional theory (NEO-MSDFT) approach was developed to describe hydrogen tunneling systems within the multicomponent NEO framework, where the transferring proton is quantized and treated with molecular orbital techniques on the same level as the electrons. Herein, the NEO-MSDFT framework is generalized to an arbitrary number of quantum protons to allow applications to systems involving the transfer and tunneling of multiple protons. The generalized NEO-MSDFT approach is shown to produce delocalized, bilobal proton densities and accurate tunneling splittings for fixed geometries of the formic acid dimer and asymmetric substituted variants, as well as the porphycene molecule. Investigation of a protonated water chain highlights the applicability of this approach to proton relay systems. This work provides the foundation for nuclear-electronic quantum dynamics simulations of a wide range of multiple proton transfer processes.

Nuclear-Electronic Orbital QM/MM Approach: Geometry Optimizations and Molecular Dynamics

Hybrid quantum mechanical/molecular mechanical (QM/MM) methods allow simulations of chemical reactions in atomistic solvent and heterogeneous environments such as proteins. Herein, the nuclear-electronic orbital (NEO) QM/MM approach is introduced to enable the quantization of specified nuclei, typically protons, in the QM region using a method such as NEO-density functional theory (NEO-DFT). This approach includes proton delocalization, polarization, anharmonicity, and zero-point energy in geometry optimizations and dynamics. Expressions for the energies and analytical gradients associated with the NEO-QM/MM method, as well as the previously developed polarizable continuum model (NEO-PCM), are provided. Geometry optimizations of small organic molecules hydrogen bonded to water in either dielectric continuum solvent or explicit atomistic solvent illustrate that aqueous solvation can strengthen hydrogen-bonding interactions for the systems studied, as indicated by shorter intermolecular distances at the hydrogen-bond interface. We then performed a real-time direct dynamics simulation of a phenol molecule in explicit water using the NEO-QM/MM method. These developments and initial examples provide the foundation for future studies of nuclear-electronic quantum dynamics in complex chemical and biological environments.

Exploring Proton-Coupled Electron Transfer at Multiple Scales

The coupling of electron and proton transfer is critical for chemical and biological processes spanning a wide range of length and time scales and often occurring in complex environments. Thus, diverse modeling strategies, including analytical theories, quantum chemistry, molecular dynamics, and kinetic modeling, are essential for a comprehensive understanding of such proton-coupled electron transfer reactions. Each of these computational methods provides one piece of the puzzle, and all these pieces must be viewed together to produce the full picture.

Solvent Induced Proton Polarization within the Nuclear-Electronic Orbital Framework

To explicitly account for nuclear quantum effects and solvent environments in simulations of chemical processes, the nuclear-electronic orbital approach is coupled with a polarizable continuum model (PCM). This NEO-PCM approach is used to explore the influence of solvation on nuclear polarization through applications to a water dimer and a set of protonated water tetramers. Nuclear polarization in these species is analyzed in terms of changes in proton density and oxygen-hydrogen bond length. Solvation is shown to enhance nuclear polarization with increasing dielectric constant. For the water dimer, the internal, hydrogen-bonded proton is shown to polarize more than the external, free proton. Moreover, proton quantization leads to greater solvent polarization through their mutual polarization. These calculations highlight the complex interplay among electronic, nuclear, and solvent polarization in chemical systems.

Electronic Born-Oppenheimer approximation in nuclear-electronic orbital dynamics

Within the nuclear-electronic orbital (NEO) framework, the real-time NEO time-dependent density functional theory (RT-NEO-TDDFT) approach enables the simulation of coupled electronic-nuclear dynamics. In this approach, the electrons and quantum nuclei are propagated in time on the same footing. A relatively small time step is required to propagate the much faster electronic dynamics, thereby prohibiting the simulation of long-time nuclear quantum dynamics. Herein, the electronic Born-Oppenheimer (BO) approximation within the NEO framework is presented. In this approach, the electronic density is quenched to the ground state at each time step, and the real-time nuclear quantum dynamics is propagated on an instantaneous electronic ground state defined by both the classical nuclear geometry and the nonequilibrium quantum nuclear density. Because the electronic dynamics is no longer propagated, this approximation enables the use of an order-of-magnitude larger time step, thus greatly reducing the computational cost. Moreover, invoking the electronic BO approximation also fixes the unphysical asymmetric Rabi splitting observed in previous semiclassical RT-NEO-TDDFT simulations of vibrational polaritons even for small Rabi splitting, instead yielding a stable, symmetric Rabi splitting. For the intramolecular proton transfer in malonaldehyde, both RT-NEO-Ehrenfest dynamics and its BO counterpart can describe proton delocalization during the real-time nuclear quantum dynamics. Thus, the BO RT-NEO approach provides the foundation for a wide range of chemical and biological applications.

Time-dependent nuclear-electronic orbital Hartree-Fock theory in a strong uniform magnetic field

In an ultrastrong magnetic field, with field strength B ≈ B₀ = 2.35 × 10⁵ T, molecular structure and dynamics differ strongly from that observed on the Earth. Within the Born-Oppenheimer (BO) approximation, for example, frequent (near) crossings of electronic energy surfaces are induced by the field, suggesting that nonadiabatic phenomena and processes may play a more important role in this mixed-field regime than in the weak-field regime on Earth. To understand the chemistry in the mixed regime, it therefore becomes important to explore non-BO methods. In this work, the nuclear-electronic orbital (NEO) method is employed to study protonic vibrational excitation energies in the presence of a strong magnetic field. The NEO generalized Hartree-Fock theory and time-dependent Hartree-Fock (TDHF) theory are derived and implemented, accounting for all terms that result as a consequence of the nonperturbative treatment of molecular systems in a magnetic field. The NEO results for HCN and FHF^- with clamped heavy nuclei are compared against the quadratic eigenvalue problem. Each molecule has three semi-classical modes owing to the hydrogen-two precession modes that are degenerate in the absence of a field and one stretching mode. The NEO-TDHF model is found to perform well; in particular, it automatically captures the screening effects of the electrons on the nuclei, which are quantified through the difference in energy of the precession modes.

Hydrogen-Atom Electronic Basis Sets for Multicomponent Quantum Chemistry

Multicomponent methods are a conceptually simple way to include nuclear quantum effects into quantum chemistry calculations. In multicomponent methods, the electronic molecular orbitals are described using the linear combination of atomic orbitals approximation. This requires the selection of a one-particle electronic basis set which, in practice, is commonly a correlation-consistent basis set. In multicomponent method studies, it has been demonstrated that large electronic basis sets are required for quantum hydrogen nuclei to accurately describe electron-nuclear correlation. However, as we show in this study, much of the need for large electronic basis sets is due to the correlation-consistent electronic basis sets not being optimized to describe nuclear properties and electron-nuclear correlation. Herein, we introduce a series of correlation-consistent electronic basis sets for hydrogen atoms called cc-pVnZ-mc with additional basis functions optimized to reproduce multicomponent density functional theory protonic densities. These new electronic basis sets are shown to yield better protonic densities with fewer electronic basis functions than the standard correlation-consistent basis sets and reproduce other protonic properties such as proton affinities and protonic excitation energies, even though they were not optimized for these purposes. The cc-pVnZ-mc basis sets should enable multicomponent many-body calculations on larger systems due to the improved computational efficiency they provide for a given level of accuracy.

Incorporating Nuclear Quantum Effects in Molecular Dynamics with a Constrained Minimized Energy Surface

The accurate incorporation of nuclear quantum effects in large-scale molecular dynamics (MD) simulations remains a significant challenge. Recently, we combined constrained nuclear-electronic orbital (CNEO) theory with classical MD and obtained a new approach (CNEO-MD) that can accurately and efficiently incorporate nuclear quantum effects into classical simulations. In this Letter, we provide the theoretical foundation for CNEO-MD by developing an alternative formulation of the equations of motion for MD. In this new formulation, the expectation values of quantum nuclear positions evolve classically on an effective energy surface that is obtained from a constrained energy minimization procedure when solving for the quantum nuclear wave function, thus enabling the incorporation of nuclear quantum effects in classical MD simulations. For comparison with other existing approaches, we examined a series of model systems and found that this new MD approach is significantly more accurate than the conventional way of performing classical MD and generally outperforms centroid MD and ring-polymer MD in describing vibrations in these model systems.

Nonadiabatic Dynamics of Hydrogen Tunneling with Nuclear-Electronic Orbital Multistate Density Functional Theory

Proton transfer reactions play a critical role in many chemical and biological processes. The development of computationally efficient approaches to describe the quantum dynamics of proton transfer, which often involves hydrogen tunneling, is challenging. Herein, the nuclear-electronic orbital multistate density functional theory (NEO-MSDFT) method is combined with both Ehrenfest and surface hopping nonadiabatic dynamics methods to describe hydrogen tunneling. The NEO-MSDFT method treats the transferring hydrogen nucleus quantum mechanically on the same level as the electrons and incorporates both static and dynamical correlation by mixing localized NEO-DFT solutions with a nonorthogonal configuration interaction scheme. The other nuclei are propagated on the NEO-MSDFT vibronic surfaces during the Ehrenfest or surface hopping dynamics. These methods are applied to proton transfer in malonaldehyde as a prototypical hydrogen tunneling system. The inclusion of vibronically nonadiabatic effects is found to significantly impact the proton transfer time and tunneling dynamics. This approach is applicable to a wide range of other proton transfer reactions.

Simultaneous Optimization of Nuclear-Electronic Orbitals

Accurate modeling of important nuclear quantum effects, such as nuclear delocalization, zero-point energy, and tunneling, as well as non-Born-Oppenheimer effects, requires treatment of both nuclei and electrons quantum mechanically. The nuclear-electronic orbital (NEO) method provides an elegant framework to treat specified nuclei, typically protons, on the same level as the electrons. In conventional electronic structure theory, finding a converged ground state can be a computationally demanding task; converging NEO wavefunctions, due to their coupled electronic and nuclear nature, is even more demanding. Herein, we present an efficient simultaneous optimization method that uses the direct inversion in the iterative subspace method to simultaneously converge wavefunctions for both the electrons and quantum nuclei. Benchmark studies show that the simultaneous optimization method can significantly reduce the computational cost compared to the conventional stepwise method for optimizing NEO wavefunctions for multicomponent systems.

Theoretical perspectives on non-Born-Oppenheimer effects in chemistry

The Born-Oppenheimer approximation, which assumes that the electrons respond instantaneously to the motion of the nuclei, breaks down for a wide range of chemical and biological processes. The rate constants of such nonadiabatic processes can be calculated using analytical theories, and the real-time nonequilibrium dynamics can be described using numerical atomistic simulations. The selection of an approach depends on the desired balance between accuracy and efficiency. The computational expense of generating potential energy surfaces on-the-fly often favours the use of approximate, robust and efficient methods such as trajectory surface hopping for large, complex systems. The development of formally exact non-Born-Oppenheimer methods and the exploration of well-defined approximations to such methods are critical for providing benchmarks and preparing for the next generation of faster computers. Thus, the parallel development of rigorous but computationally expensive methods and more approximate but computationally efficient methods is optimal. This Perspective briefly summarizes the available theoretical and computational non-Born-Oppenheimer methods and presents examples illustrating how analytical theories and nonadiabatic dynamics simulations can elucidate the fundamental principles of chemical and biological processes. These examples also highlight how theoretical calculations are able to guide the interpretation of experimental data and provide experimentally testable predictions for nonadiabatic processes. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

Nuclear-Electronic Orbital Approach to Quantization of Protons in Periodic Electronic Structure Calculations

The nuclear-electronic orbital (NEO) method is a well-established approach for treating nuclei quantum mechanically in molecular systems beyond the usual Born-Oppenheimer approximation. In this work, we present a strategy to implement the NEO method for periodic electronic structure calculations, particularly focused on multicomponent density functional theory (DFT). The NEO-DFT method is implemented in an all-electron electronic structure code, FHI-aims, using a combination of analytical and numerical integration techniques as well as a resolution of the identity scheme to enhance computational efficiency. After validating this implementation, proof-of-concept applications are presented to illustrate the effects of quantized protons on the physical properties of extended systems such as two-dimensional materials and liquid-semiconductor interfaces. Specifically, periodic NEO-DFT calculations are performed for a trans-polyacetylene chain, a hydrogen boride sheet, and a titanium oxide-water interface. The zero-point energy effects of the protons, as well as electron-proton correlation, are shown to noticeably impact the density of states and band structures for these systems. These developments provide a foundation for the application of multicomponent DFT to a wide range of other extended condensed matter systems.

Semiclassical Real-Time Nuclear-Electronic Orbital Dynamics for Molecular Polaritons: Unified Theory of Electronic and Vibrational Strong Couplings

Molecular polaritons have become an emerging platform for remotely controlling molecular properties through strong light-matter interactions. Herein, a semiclassical approach is developed for describing molecular polaritons by self-consistently propagating the real-time dynamics of classical cavity modes and a quantum molecular subsystem described by the nuclear-electronic orbital (NEO) method, where electrons and specified nuclei are treated quantum mechanically on the same level. This semiclassical real-time NEO approach provides a unified description of electronic and vibrational strong couplings and describes the impact of the cavity on coupled nuclear-electronic dynamics while including nuclear quantum effects. For a single o-hydroxybenzaldehyde molecule under electronic strong coupling, this approach shows that the cavity suppression of excited state intramolecular proton transfer is influenced not only by the polaritonic potential energy surface but also by the time scale of the chemical reaction. This work provides the foundation for exploring collective strong coupling in nuclear-electronic quantum dynamical systems within optical cavities.

Solvated Nuclear-Electronic Orbital Structure and Dynamics

Nonadiabatic dynamical processes such as proton-coupled electron transfer and excited state intramolecular proton transfer have been the subject of much research. One of the promising theoretical methods to describe these processes is the nuclear-electronic orbital (NEO) approach. This approach inherently accounts for nuclear quantum effects within quantum chemistry calculations, and it has recently been extended to directly simulate nonadiabatic processes with the development of real-time NEO methods. These processes can also be significantly dependent on the surrounding chemical environment, however, and capturing the effects of the environment is often necessary for analyzing experimentally relevant systems. This work couples the NEO density functional theory and real-time time-dependent density functional theory approaches with solvation through the polarizable continuum model. The effects of this coupling are investigated for ground state properties, solvent-dependent vibrational frequencies, and direct excited state intramolecular proton transfer dynamics.

Quantum Proton Effects from Density Matrix Renormalization Group Calculations

We recently introduced [J. Chem. Phys. 2020, 152, 204103] the nuclear-electronic all-particle density matrix renormalization group (NEAP-DMRG) method to solve the molecular Schrödinger equation, based on a stochastically optimized orbital basis, without invoking the Born-Oppenheimer approximation. In this work, we combine the DMRG method with the nuclear-electronic Hartree-Fock (NEHF-DMRG) approach, treating nuclei and electrons on the same footing. Inter- and intraspecies correlations are described within the DMRG method without truncating the excitation degree of the full configuration interaction wave function. We extend the concept of orbital entanglement and mutual information to nuclear-electronic wave functions and demonstrate that they are reliable metrics to detect strong correlation effects. We apply the NEHF-DMRG method to the HeHHe⁺ molecular ion, to obtain accurate proton densities, ground-state total energies, and vibrational transition frequencies by comparison with state-of-the-art data obtained with grid-based approaches and modern configuration interaction methods. For HCN, we improve on the accuracy of the latter approaches with respect to both the ground-state absolute energy and proton density, which is a major challenge for multireference nuclear-electronic state-of-the-art methods.

Direct Dynamics with Nuclear-Electronic Orbital Density Functional Theory

Direct dynamics simulations of chemical reactions typically require the selection of a method for generating the potential energy surfaces and a method for the dynamical propagation of the nuclei on these surfaces. The nuclear-electronic orbital (NEO) framework avoids this Born-Oppenheimer separation by treating specified nuclei on the same level as the electrons with wave function methods or density functional theory (DFT). The NEO approach is particularly applicable to proton, hydride, and proton-coupled electron transfer reactions, where the transferring proton(s) and all electrons are treated quantum mechanically. In this manner, the zero-point energy, density delocalization, and anharmonicity of the transferring protons are inherently and efficiently included in the energies, optimized geometries, and dynamics.This Account describes how various NEO methods can be used for direct dynamics simulations on electron-proton vibronic surfaces. The strengths and limitations of these approaches are discussed, and illustrative examples are presented. The NEO-DFT method can be used to simulate chemical reactions on the ground state vibronic surface, as illustrated by the application to hydride transfer in C₄H₉⁺. The NEO multistate DFT (NEO-MSDFT) method is useful for simulating ground state reactions in which the proton density becomes bilobal during the dynamics, a characteristic of hydrogen tunneling, as illustrated by proton transfer in malonaldehyde. The NEO time-dependent DFT (NEO-TDDFT) method produces excited electronic, vibrational, and vibronic surfaces. The application of linear-response NEO-TDDFT to H₂ and H₃⁺, as well as the partially and fully deuterated counterparts, shows that this approach produces accurate fundamental vibrational excitation energies when all nuclei and all electrons are treated quantum mechanically. Moreover, when only specified nuclei are treated quantum mechanically, this approach can be used to optimize geometries on excited state vibronic surfaces, as illustrated by photoinduced single and double proton transfer systems, and to conduct adiabatic dynamics on these surfaces. The real-time NEO-TDDFT method provides an alternative approach for simulating nonequilibrium nuclear-electronic dynamics of such systems. These various NEO methods can be combined with nonadiabatic dynamics methods such as Ehrenfest and surface hopping dynamics to include the nonadiabatic effects between the quantum and classical subsystems. The real-time NEO-TDDFT Ehrenfest dynamics simulation of excited state intramolecular proton transfer in o-hydroxybenzaldehyde illustrates the power of this type of combined approach. The field of multicomponent quantum chemistry is in the early stages, and the methods discussed herein provide the foundation for a wide range of promising future directions to be explored. An appealing future direction is the expansion of the real-time NEO-TDDFT method to describe the dynamics of all nuclei and electrons on the same level. Direct dynamics simulations using NEO wave function methods such as equation-of-motion coupled cluster or multiconfigurational approaches are also attractive but computationally expensive options. The further development of NEO direct dynamics methods will enable the simulation of the nuclear-electronic dynamics for a vast array of chemical and biological processes that extend beyond the Born-Oppenheimer approximation.

Multi-configuration electron-nuclear dynamics: An open-shell approach

The multi-configuration electron-nuclear dynamics for open shell systems with a spin-unrestricted formalism is described. The mean fields are evaluated using second-order reduced density matrices for electronic and nuclear degrees of freedom. Applications to light-element diatomics including equilibrium geometries, electronic energies, dipole moments, and absorption spectra are presented. The von Neumann entropies for different spin states of a LiH molecule are compared.

Simulating electronic excitation and dynamics with real-time propagation approach to TDDFT within plane-wave pseudopotential formulation

We give a perspective on simulating electronic excitation and dynamics using the real-time propagation approach to time-dependent density functional theory (RT-TDDFT) in the plane-wave pseudopotential formulation. RT-TDDFT is implemented in various numerical formalisms in recent years, and its practical application often dictates the most appropriate implementation of the theory. We discuss recent developments and challenges, emphasizing numerical aspects of studying real systems. Several applications of RT-TDDFT simulation are discussed to highlight how the approach is used to study interesting electronic excitation and dynamics phenomena in recent years.

Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange-correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear-electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an "open teamware" model and an increasingly modular design.

Nuclear-electronic orbital methods: Foundations and prospects

The incorporation of nuclear quantum effects and non-Born-Oppenheimer behavior into quantum chemistry calculations and molecular dynamics simulations is a longstanding challenge. The nuclear-electronic orbital (NEO) approach treats specified nuclei, typically protons, quantum mechanically on the same level as the electrons with wave function and density functional theory methods. This approach inherently includes nuclear delocalization and zero-point energy in molecular energy calculations, geometry optimizations, reaction paths, and dynamics. It can also provide accurate descriptions of excited electronic, vibrational, and vibronic states as well as nuclear tunneling and nonadiabatic dynamics. Nonequilibrium nuclear-electronic dynamics simulations beyond the Born-Oppenheimer approximation can be used to investigate a wide range of excited state processes. This Perspective provides an overview of the foundational NEO methods and enumerates the prospects for using these methods as building blocks for future developments. The conceptual simplicity and computational efficiency of the NEO approach will enhance its accessibility and applicability to diverse chemical and biological systems.

Analytical Gradients for Nuclear-Electronic Orbital Time-Dependent Density Functional Theory: Excited-State Geometry Optimizations and Adiabatic Excitation Energies

The computational investigation of photochemical processes often entails the calculation of excited-state geometries, energies, and energy gradients. The nuclear-electronic orbital (NEO) approach treats specified nuclei, typically protons, quantum mechanically on the same level as the electrons, thereby including the associated nuclear quantum effects and non-Born-Oppenheimer behavior into quantum chemistry calculations. The multicomponent density functional theory (NEO-DFT) and time-dependent DFT (NEO-TDDFT) methods allow efficient calculations of ground and excited states, respectively. Herein, the analytical gradients are derived and implemented for the NEO-TDDFT method and the associated Tamm-Dancoff approximation (NEO-TDA). The programmable equations for these analytical gradients as well as the NEO-DFT analytical Hessian are provided. The NEO approach includes the anharmonic zero-point energy (ZPE) and density delocalization associated with the quantum protons as well as vibronic mixing in geometry optimizations and energy calculations of ground and excited states. The harmonic ZPE associated with the other nuclei can be computed via the NEO Hessian. This approach is used to compute the 0-0 adiabatic excitation energies for a set of nine small molecules with all protons quantized, exhibiting slight improvement over the conventional electronic approach. Geometry optimizations of two excited-state intramolecular proton-transfer systems, [2,2'-bipyridyl]-3-ol and [2,2'-bipyridyl]-3,3'-diol, are performed with one and two quantized protons, respectively. The NEO calculations for these systems produce electronically excited-state geometries with stronger intramolecular hydrogen bonds and similar relative stabilities compared to conventional electronic methods. This work provides the foundation for nonadiabatic dynamics simulations of fundamental processes such as photoinduced proton transfer and proton-coupled electron transfer.

Multicomponent Unitary Coupled Cluster and Equation-of-Motion for Quantum Computation

The variational quantum eigensolver (VQE) algorithm combined with the unitary coupled cluster (UCC) ansatz has been developed for the quantum computation of molecular energies and wave functions within the Born-Oppenheimer approximation. Herein, this approach is extended to multicomponent systems to enable the quantum mechanical treatment of more than one type of particle, such as electrons and positrons or electrons and nuclei, without invoking the Born-Oppenheimer approximation. Specifically, we introduce the multicomponent unitary coupled cluster (mcUCC) method combined with the VQE algorithm for the calculation of ground-state energies and wave functions as well as the multicomponent equation-of-motion (mcEOM) method for the calculation of excitation energies. These methods are developed within the nuclear-electronic orbital (NEO) framework and are formulated in the qubit basis to enable implementations on quantum computers. Moreover, these methods are used to calculate the ground-state energy and excitation energies of positronium hydride, where both electrons and the positron are treated quantum mechanically, as well as the H₂ molecule, where both electrons and one proton are treated quantum mechanically. These applications validate the implementation and provide benchmark data for future calculations. The errors due to Trotterization of the mcUCC ansatz are also analyzed. This formalism, as well as the accompanying computer code, will serve as the basis for applications to more complex multicomponent systems, such as simulations of photoinduced nonadiabatic molecular processes, on both classical and quantum computers.

First-Principles Simulations of Biological Molecules Subjected to Ionizing Radiation

Ionizing rays cause damage to genomes, proteins, and signaling pathways that normally regulate cell activity, with harmful consequences such as accelerated aging, tumors, and cancers but also with beneficial effects in the context of radiotherapies. While the great pace of research in the twentieth century led to the identification of the molecular mechanisms for chemical lesions on the building blocks of biomacromolecules, the last two decades have brought renewed questions, for example, regarding the formation of clustered damage or the rich chemistry involving the secondary electrons produced by radiolysis. Radiation chemistry is now meeting attosecond science, providing extraordinary opportunities to unravel the very first stages of biological matter radiolysis. This review provides an overview of the recent progress made in this direction, focusing mainly on the atto- to femto- to picosecond timescales. We review promising applications of time-dependent density functional theory in this context.

Excited State Intramolecular Proton Transfer with Nuclear-Electronic Orbital Ehrenfest Dynamics

The recent development of the Ehrenfest dynamics approach in the nuclear-electronic orbital (NEO) framework provides a promising way to simulate coupled nuclear-electronic dynamics. Our previous study showed that the NEO-Ehrenfest approach with a semiclassical traveling proton basis method yields accurate predictions of molecular vibrational frequencies. In this work, we provide a more thorough analysis of the semiclassical traveling proton basis method to elucidate its validity and convergence behavior. We also conduct NEO-Ehrenfest dynamics simulations to study an excited state intramolecular proton transfer process. These simulations reveal that nuclear quantum effects influence the predictions of proton transfer reaction rates and kinetic isotope effects due to the intrinsic delocalized nature of the quantum nuclear wave function. This work illustrates the importance of nuclear quantum effects in coupled nuclear-electronic dynamical processes and shows that the NEO-Ehrenfest approach can be a powerful tool for providing insights and predictions for these processes.

Ecuaciones de Bloch Ópticas en Sistemas Complejos con Acoplamiento Intramolecular

Proponemos modificaciones de las ecuaciones de Bloch ópticas convencionales para un sistema molecular, cuando consideramos los efectos de acoplamiento intramolecular. Modelamos la molécula aislada como curvas de energía de Born-Oppenheimer que consisten en dos estados electrónicos cruzados descritos como potenciales armónicos, con los mínimos desplazados en coordenadas nucleares y energía. Consideramos dos estados vibracionales y una perturbación, que puede surgir de una correlación residual electrón-electrón y/o términos de acoplamiento spin-órbita en el Hamiltoniano del sistema, causando la separación de las dos curvas según la regla del cruce evitado. Las ecuaciones extendidas de Bloch ópticas permiten establecer la dinámica de un sistema molecular de estados adiabáticos sujeto a la interacción de un reservorio térmico e interactuando con campos electromagnéticos. En las ecuaciones se observa que haciendo nulo el factor de acoplamiento intramolecular, se recuperan las ecuaciones de Bloch ópticas convencionales para un sistema molecular con potenciales armónicos cuyos mínimos se encuentran exactamente en la misma coordenada nuclear sujeto a la aproximación Born-Oppenheimer.

Nonequilibrium Dynamics of Proton-Coupled Electron Transfer in Proton Wires: Concerted but Asynchronous Mechanisms

The coupling between electrons and protons and the long-range transport of protons play important roles throughout biology. Biomimetic systems derived from benzimidazole-phenol (BIP) constructs have been designed to undergo proton-coupled electron transfer (PCET) upon electrochemical or photochemical oxidation. Moreover, these systems can transport protons along hydrogen-bonded networks or proton wires through multiproton PCET. Herein, the nonequilibrium dynamics of both single and double proton transfer in BIP molecules initiated by oxidation are investigated with first-principles molecular dynamics simulations. Although these processes are concerted in that no thermodynamically stable intermediate is observed, the simulations predict that they are predominantly asynchronous on the ultrafast time scale. For both systems, the first proton transfer typically occurs ∼100 fs after electron transfer. For the double proton transfer system, typically the second proton transfer occurs hundreds of femtoseconds after the initial proton transfer. A machine learning algorithm was used to identify the key molecular vibrational modes essential for proton transfer: a slow, in-plane bending mode that dominates the overall inner-sphere reorganization, the proton donor-acceptor motion that leads to vibrational coherence, and the faster donor-hydrogen stretching mode. The asynchronous double proton transfer mechanism can be understood in terms of a significant mode corresponding to the two anticorrelated proton donor-acceptor motions, typically decreasing only one donor-acceptor distance at a time. Although these PCET processes appear concerted on the time scale of typical electrochemical experiments, attaching these BIP constructs to photosensitizers may enable the detection of the asynchronicity of the electron and multiple proton transfers with ultrafast two-dimensional spectroscopy. Understanding the fundamental PCET mechanisms at this level will guide the design of PCET systems for catalysis and energy conversion processes.

Electron nuclear dynamics of H+ + CO2 (000) → H+ + CO2 (v1v2v3) at ELab = 20.5-30 eV with coherent-states quantum reconstruction procedure

The simplest-level electron nuclear dynamics (SLEND) method with the coherent-states (CSs) quantum reconstruction procedure (CSQRP) is applied to the scattering system H+ + CO2 (000) → H+ + CO2 (v1v2v3) at ELab = 20.5-30 eV. Relevant for astrophysics, atmospheric chemistry and proton cancer therapy, this system undergoes collision-induced vibrational excitations in CO2. SLEND is a time-dependent, variational, direct, and non-adiabatic method that adopts a classical-mechanics description for nuclei and a single-determinantal wavefunction for electrons. The CSQRP employs the canonical CS to reconstruct quantum state-to-state vibrational properties from the SLEND classical nuclear dynamics. Overall, the calculated collision-induced vibrational properties agree well with experimental data. SLEND total differential cross sections (DCSs) agree remarkably well with their experimental counterparts and accurately display rainbow scattering angles structures. SLEND averaged target excitation energies for vibrational + rotational and rotational motions exhibit reasonable and good agreements with experimental data, respectively. These properties show that rotational excitation is low and that the asymmetric stretch normal mode of CO2 is much more excited than the others. SLEND/CSQRP state-to-state vibrational DCSs agree reasonably well with the sparse experimental data for final states v1v2v3 = 000-002, but less satisfactorily for 003. These DCSs also accurately display rainbow scattering angles structures. Finally, SLEND/CSQRP vibrational proton energy loss spectra agree remarkably well with their experimental counterparts for various final vibrational states of CO2, collisions energies and scattering angles. Present results demonstrate the accuracy of SLEND/CSQRP to predict state-to-state vibrational properties in scattering systems with multiple normal modes.

Frequency and Time Domain Nuclear-Electronic Orbital Equation-of-Motion Coupled Cluster Methods: Combination Bands and Electronic-Protonic Double Excitations

The accurate description of excited vibronic states is important for modeling a wide range of photoinduced processes. The nuclear-electronic orbital (NEO) approach, which treats specified protons on the same level as the electrons, can describe excited electronic-protonic states. Herein the multicomponent equation-of-motion coupled cluster with singles and doubles (NEO-EOM-CCSD) method and its time-domain counterpart, TD-NEO-EOM-CCSD, are developed and implemented. The application of these methods to the HCN molecule highlights their capabilities. These methods predict qualitatively reasonable energies and intensities for a combination band corresponding to simultaneous excitation of two vibrational modes, as well as an overtone. These methods also describe states with double excitation character, such as excited electronic-protonic states corresponding to the simultaneous excitation of an electron and a proton. The ability of the NEO-EOM-CCSD method and its time-dependent counterpart to describe combination bands, overtones, and double excitations will enable a wide range of photochemical applications.