Communication: VSCF/VCI vibrational spectroscopy of H7 O 3 + and H9 O 4 + using high-level, many-body potential energy surface and dipole moment surfaces

Encoding a Many-Body Potential Energy Surface into a Grid-Based Matrix Product Operator

An efficient algorithm for compressing a given many-body potential energy surface (PES) of molecular systems into a grid-based matrix product operator (MPO) is proposed. The PES is once represented by a full-dimensional or truncated many-body expansion form, which is obtained by ab initio calculations at each grid mesh point, and then all terms in the expansion are compressed and merged into a single MPO while maintaining the bond dimension of the MPO as small as possible. It was shown that the ab initio PES of the H2CO was compressed by more than 2 orders of magnitude in the size of the site operators without loss of accuracy. By the use of grid basis, the tensor rank of the site operators of the MPO is reduced from four to three due to the diagonal nature of the position-dependent operators on grid basis, which significantly reduces the computational cost of the tensor contractions required in the real and imaginary time evolution of the matrix product state (MPS) wave functions with the grid-based MPO (Grid-MPO) Hamiltonian. Similar to other grid-based methods, Grid-MPO is easily applicable to any kinds of potentials of molecular systems, such as analytical empirical model potentials expressed by position operators and ab initio potentials, if the values at the grid points are available. Using the Grid-MPO combined with the MPS, we calculated the time correlation function of the Eigen cation H 3 O + ( H 2 O ) 3 to predict the infrared spectrum and compared with the experimental and the previous theoretical studies. The actual scaling with the size of systems was examined for the multidimensional Henon-Heiles Hamiltonian. It was shown that the method is considerably accelerated by the graphic processing unit (GPU) because the sizes of site operators were kept small and all tensors were able to be stored on the VRAM of a GPU.

No Headache for PIPs: A PIP Potential for Aspirin Runs Much Faster and with Similar Precision Than Other Machine-Learned Potentials

Assessments of machine-learning (ML) potentials are an important aspect of the rapid development of this field. We recently reported an assessment of the linear-regression permutationally invariant polynomial (PIP) method for ethanol, using the widely used (revised) rMD17 data set. We demonstrated that the PIP approach outperformed numerous other methods, e.g., ANI, PhysNet, sGDML, and p-KRR, with respect to precision and notably with respect to speed [Houston et al., J. Chem. Phys. 2022, 156, 044120]. Here, we extend this assessment to the 21-atom aspirin molecule, using the rMD17 data set, with a focus on the speed of evaluation. Both energies and forces are used for training, and the precision of several PIPs is examined for both. Normal mode frequencies, the methyl torsional potential, and 1d vibrational energies for an OH stretch are presented. We show that the PIP approach achieves the level of precision obtained from other ML methods, e.g., atom-centered neural network methods, linear regression ACE, and kernel methods, as reported by Kovács et al. in J. Chem. Theory Comput. 2021, 17, 7696-7711. More significantly, we show that the PIP PESs run much faster than all other ML methods, whose timings were evaluated in that paper. We also show that the PIP PES extrapolates well enough to describe several internal motions of aspirin, including an OH stretch.

Near-infrared spectroscopy of H3O+⋯Xn (X = Ar, N2, and CO, n = 1-3)

Near-infrared (NIR) spectra of H3O+⋯Xn (X = Ar, N2, and CO, n = 1-3) in the first overtone region of OH-stretching vibrations (4800-7000 cm-1) were measured. Not only OH-stretching overtones but also several combination bands are major features in this region, and assignments of these observed bands are not obvious at a glance. High-precision anharmonic vibrational simulations based on the discrete variable representation approach were performed. The simulated spectra show good agreement with the observed ones and provide firm assignments of the observed bands, except in the case of X = CO, in which higher order vibrational mode couplings seem significant. This agreement demonstrates that the present system can be a benchmark for high precision anharmonic vibrational computations of NIR spectra. Band broadening in the observed spectra becomes remarkable with an increase of the interaction with the solvent molecule (X). The origin of the band broadening is explored by rare gas tagging experiments and anharmonic vibrational simulations of hot bands.

Correlations between the Structures and Spectra of Protonated Water Clusters

Badger's rule-like correlations between OH stretching frequencies and intensities and the OH bond length are used to develop a spectral mapping procedure for studies of pure and protonated water clusters. This approach utilizes the vibrationally averaged OH bond lengths, which were obtained from diffusion Monte Carlo simulations that were performed using the general potential developed by Yu and Bowman. Good agreement is achieved between the spectra obtained using this approach and previously reported spectra for H+(H2O)n clusters, with n = 3, 4, and 5, as well as their perdeuterated analogues. The analysis of the spectra obtained by this spectral mapping approach supports previous work that assigned the spectrum of H+(H2O)6 to a mixture of Eigen and Zundel-like structures. Analysis of the calculated spectra also suggests a reassignment of the frequency of one of the transitions that involves the OH stretching vibration of the OH bonds in the hydronium core in the Eigen-like structure of H+(H2O)6 from 1917 cm-1 to roughly 2100 cm-1. For D+(D2O)6, comparison of the measured spectrum to those obtained by using the spectral mapping approach suggests that the carrier of the measured spectrum is one or more of the isomers of D+(D2O)6 that contain a four-membered ring and two flanking water molecules. While there are several candidate structures, the two flanking water molecules most likely form a chain that is bound to the hydronium core.

Reformulation of All ONIOM-Type Molecular Fragmentation Approaches and Many-Body Theories Using Graph-Theory-Based Projection Operators: Applications to Dynamics, Molecular Potential Surfaces, Machine Learning, and Quantum Computing

We present a graph-theory-based reformulation of all ONIOM-based molecular fragmentation methods. We discuss applications to (a) accurate post-Hartree-Fock AIMD that can be conducted at DFT cost for medium-sized systems, (b) hybrid DFT condensed-phase studies at the cost of pure density functionals, (c) reduced cost on-the-fly large basis gas-phase AIMD and condensed-phase studies, (d) post-Hartree-Fock-level potential surfaces at DFT cost to obtain quantum nuclear effects, and (e) novel transfer machine learning protocols derived from these measures. Additionally, in previous work, the unifying strategy discussed here has been used to construct new quantum computing algorithms. Thus, we conclude that this reformulation is robust and accurate.

Capturing Weak Interactions in Surface Adsorbate Systems at Coupled Cluster Accuracy: A Graph-Theoretic Molecular Fragmentation Approach Improved through Machine Learning

The accurate and efficient study of the interactions of organic matter with the surface of water is critical to a wide range of applications. For example, environmental studies have found that acidic polyfluorinated alkyl substances, especially perfluorooctanoic acid (PFOA), have spread throughout the environment and bioaccumulate into human populations residing near contaminated watersheds, leading to many systemic maladies. Thus, the study of the interactions of PFOA with water surfaces became important for the mitigation of their activity as pollutants and threats to public health. However, theoretical study of the interactions of such organic adsorbates on the surface of water, and their bulk concerted properties, often necessitates the use of ab initio methods to properly incorporate the long-range electronic properties that govern these extended systems. Notable theoretical treatments of "on-water" reactions thus far have employed hybrid DFT and semilocal DFT, but the interactions involved are weak interactions that may be best described using post-Hartree-Fock theory. Here, we aim to demonstrate the utility of a graph-theoretic approach to molecular fragmentation that accurately captures the critical "weak" interactions while maintaining an efficient ab initio treatment of the long-range periodic interactions that underpin the physics of extended systems. We apply this graph-theoretical treatment to study PFOA on the surface of water as a model system for the study of weak interactions seen in the wide range of surface interactions and reactions. The approach divides a system into a set of vertices, that are then connected through edges, faces, and higher order graph theoretic objects known as simplexes, to represent a collection of locally interacting subsystems. These subsystems are then used to construct ab initio molecular dynamics simulations and for computing multidimensional potential energy surfaces. To further improve the computational efficiency of our graph theoretic fragmentation method, we use a recently developed transfer learning protocol to construct the full system potential energy from a family of neural networks each designed to accurately model the behavior of individual simplexes. We use a unique multidimensional clustering algorithm, based on the k-means clustering methodology, to define our training space for each separate simplex. These models are used to extrapolate the energies for molecular dynamics trajectories at PFOA water interfaces, at less than one-tenth the cost as compared to a regular molecular fragmentation-based dynamics calculation with excellent agreement with couple cluster level of full system potential energies.

Graph-|Q⟩⟨C|: A Quantum Algorithm with Reduced Quantum Circuit Depth for Electronic Structure

The accurate determination of chemical properties is known to have a critical impact on multiple fundamental chemical problems but is deeply hindered by the steep algebraic scaling of electron correlation calculations and the exponential scaling of quantum nuclear dynamics. With the advent of new quantum computing hardware and associated developments in creating new paradigms for quantum software, this avenue has been recognized as perhaps one way to address exponentially complex challenges in quantum chemistry and molecular dynamics. In this paper, we discuss a new approach to drastically reduce the quantum circuit depth (by several orders of magnitude) and help improve the accuracy in the quantum computation of electron correlation energies for large molecular systems. The method is derived from a graph-theoretic approach to molecular fragmentation and enables us to create a family of projection operators that decompose quantum circuits into separate unitary processes. Some of these processes can be treated on quantum hardware and others on classical hardware in a completely asynchronous and parallel fashion. Numerical benchmarks are provided through the computation of unitary coupled-cluster singles and doubles (UCCSD) energies for medium-sized protonated and neutral water clusters using the new quantum algorithms presented here.

Anharmonic Assignment of the Water Octamer Spectrum in the OH Stretch Region

We interface the quasi-classical trajectory approach with an ab initio potential energy surface for water to assign the vibrational spectroscopical features of the OH stretch region of the water octamer cluster, which is considered to be a precursor of ice. An attempt by Li et al. to assign their recent reference experiment involved lower-level calculations based on an ad hoc scaled harmonic approach. Differently from the conclusions of this previous assignment, which invoked the contribution of 5 conformers and a solvated form of the water heptamer in the spectrum, we find out that the spectroscopic features can be related to the 4 conformers of the octamer lying lower in energy.

Isotope Effects in the Zundel-Eigen Isomerization of H+(H2O)6

The isomerization pathway between the energetically low-lying Zundel and Eigen isomers of the protonated water hexamer was investigated using high-level ab initio calculations including a treatment of zero-point corrections. On the basis of these calculations, the Zundel-Eigen isomerization was found to proceed through a stable intermediate isomer, which consists of a four-membered ring with two single acceptor water molecules. The inclusion of vibrational zero-point energy is shown to be important for accurately establishing the relative energies of the three relevant isomers involved in the Zundel-Eigen isomerization. Diffusion Monte Carlo calculations including anharmonic vibrational effects show that all three isomers of H⁺(H₂O)₆ and D⁺(D₂O)₆ have well-defined structures. The energetic ordering of the three isomers changes upon deuteration. The implications of these results for the vibrational spectra of H⁺(H₂O)₆ and D⁺(D₂O)₆ are also discussed.

Graph-Theoretic Molecular Fragmentation for Potential Surfaces Leads Naturally to a Tensor Network Form and Allows Accurate and Efficient Quantum Nuclear Dynamics

Molecular fragmentation methods have revolutionized quantum chemistry. Here, we use a graph-theoretically generated molecular fragmentation method, to obtain accurate and efficient representations for multidimensional potential energy surfaces and the quantum time-evolution operator, which plays a critical role in quantum chemical dynamics. In doing so, we find that the graph-theoretic fragmentation approach naturally reduces the potential portion of the time-evolution operator into a tensor network that contains a stream of coupled lower-dimensional propagation steps to potentially achieve quantum dynamics with reduced complexity. Furthermore, the fragmentation approach used here has previously been shown to allow accurate and efficient computation of post-Hartree-Fock electronic potential energy surfaces, which in many cases has been shown to be at density functional theory cost. Thus, by combining the advantages of molecular fragmentation with the tensor network formalism, the approach yields an on-the-fly quantum dynamics scheme where both the electronic potential calculation and nuclear propagation portion are enormously simplified through a single stroke. The method is demonstrated by computing approximations to the propagator and to potential surfaces for a set of coupled nuclear dimensions within a protonated water wire problem exhibiting the Grotthuss mechanism of proton transport. In all cases, our approach has been shown to reduce the complexity of representing the quantum propagator, and by extension action of the propagator on an initial wavepacket, by several orders, with minimal loss in accuracy.

The coupling of the hydrated proton to its first solvation shell

The Zundel (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{5}{O}_{2}^{+}$$\end{document}H5O2+) and Eigen (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{9}{O}_{4}^{+}$$\end{document}H9O4+) cations play an important role as intermediate structures for proton transfer processes in liquid water. In the gas phase they exhibit radically different infrared (IR) spectra. The question arises: is there a least common denominator structure that explains the IR spectra of both, the Zundel and Eigen cations, and hence of the solvated proton? Full dimensional quantum simulations of these protonated cations demonstrate that two dynamical water molecules and an excess proton constitute this fundamental subunit. Embedded in the static environment of the parent Eigen cation, this subunit reproduces the positions and broadenings of its main excess-proton bands. In isolation, its spectrum reverts to the well-known Zundel ion. Hence, the dynamics of this subunit polarized by an environment suffice to explain the spectral signatures and anharmonic couplings of the solvated proton in its first solvation shell.

The Zundel [H(H₂O)₂]⁺ and Eigen [H(H₂O)₄]⁺ cations exhibit radicallly different infrared spectra and are the limiting dynamical structures involved in proton mobility in liquid water. Here, the authors find through quantum dynamics simulations that two polarized water molecules and a proton suffice to explain the key spectroscopic features connected to proton mobility for both species.

Graph Theoretic Molecular Fragmentation for Multidimensional Potential Energy Surfaces Yield an Adaptive and General Transfer Machine Learning Protocol

Over a series of publications we have introduced a graph-theoretic description for molecular fragmentation. Here, a system is divided into a set of nodes, or vertices, that are then connected through edges, faces, and higher-order simplexes to represent a collection of spatially overlapping and locally interacting subsystems. Each such subsystem is treated at two levels of electronic structure theory, and the result is used to construct many-body expansions that are then embedded within an ONIOM-scheme. These expansions converge rapidly with many-body order (or graphical rank) of subsystems and have been previously used for ab initio molecular dynamics (AIMD) calculations and for computing multidimensional potential energy surfaces. Specifically, in all these cases we have shown that CCSD and MP2 level AIMD trajectories and potential surfaces may be obtained at density functional theory cost. The approach has been demonstrated for gas-phase studies, for condensed phase electronic structure, and also for basis set extrapolation-based AIMD. Recently, this approach has also been used to derive new quantum-computing algorithms that enormously reduce the quantum circuit depth in a circuit-based computation of correlated electronic structure. In this publication, we introduce (a) a family of neural networks that act in parallel to represent, efficiently, the post-Hartree-Fock electronic structure energy contributions for all simplexes (fragments), and (b) a new k-means-based tessellation strategy to glean training data for high-dimensional molecular spaces and minimize the extent of training needed to construct this family of neural networks. The approach is particularly useful when coupled cluster accuracy is desired and when fragment sizes grow in order to capture nonlocal interactions accurately. The unique multidimensional k-means tessellation/clustering algorithm used to determine our training data for all fragments is shown to be extremely efficient and reduces the needed training to only 10% of data for all fragments to obtain accurate neural networks for each fragment. These fully connected dense neural networks are then used to extrapolate the potential energy surface for all molecular fragments, and these are then combined as per our graph-theoretic procedure to transfer the learning process to a full system energy for the entire AIMD trajectory at less than one-tenth the cost as compared to a regular fragmentation-based AIMD calculation.

Atomistic Simulations for Reactions and Vibrational Spectroscopy in the Era of Machine Learning─Quo Vadis?

Atomistic simulations using accurate energy functions can provide molecular-level insight into functional motions of molecules in the gas and in the condensed phase. This Perspective delineates the present status of the field from the efforts of others and some of our own work and discusses open questions and future prospects. The combination of physics-based long-range representations using multipolar charge distributions and kernel representations for the bonded interactions is shown to provide realistic models for the exploration of the infrared spectroscopy of molecules in solution. For reactions, empirical models connecting dedicated energy functions for the reactant and product states allow statistically meaningful sampling of conformational space whereas machine-learned energy functions are superior in accuracy. The future combination of physics-based models with machine-learning techniques and integration into all-purpose molecular simulation software provides a unique opportunity to bring such dynamics simulations closer to reality.

Graph-Theory-Based Molecular Fragmentation for Efficient and Accurate Potential Surface Calculations in Multiple Dimensions

We present a multitopology molecular fragmentation approach, based on graph theory, to calculate multidimensional potential energy surfaces in agreement with post-Hartree-Fock levels of theory but at the density functional theory cost. A molecular assembly is coarse-grained into a set of graph-theoretic nodes that are then connected with edges to represent a collection of locally interacting subsystems up to an arbitrary order. Each of the subsystems is treated at two levels of electronic structure theory, the result being used to construct many-body expansions that are embedded within an ONIOM scheme. These expansions converge rapidly with the many-body order (or graphical rank) of subsystems and capture many-body interactions accurately and efficiently. However, multiple graphs, and hence multiple fragmentation topologies, may be defined in molecular configuration space that may arise during conformational sampling or from reactive, bond breaking and bond formation, events. Obtaining the resultant potential surfaces is an exponential scaling proposition, given the number of electronic structure computations needed. We utilize a family of graph-theoretic representations within a variational scheme to obtain multidimensional potential surfaces at a reduced cost. The fast convergence of the graph-theoretic expansion with increasing order of many-body interactions alleviates the exponential scaling cost for computing potential surfaces, with the need to only use molecular fragments that contain a fewer number of quantum nuclear degrees of freedom compared to the full system. This is because the dimensionality of the conformational space sampled by the fragment subsystems is much smaller than the full molecular configurational space. Additionally, we also introduce a multidimensional clustering algorithm, based on physically defined criteria, to reduce the number of energy calculations by orders of magnitude. The molecular systems benchmarked include coupled proton motion in protonated water wires. The potential energy surfaces and multidimensional nuclear eigenstates obtained are shown to be in very good agreement with those from explicit post-Hartree-Fock calculations that become prohibitive as the number of quantum nuclear dimensions grows. The developments here provide a rigorous and efficient alternative to this important chemical physics problem.

Coupled Local-Mode Approach for the Calculation of Vibrational Spectra: Application to Protonated Water Clusters

Spectroscopic studies of protonated water clusters (PWCs) have yielded enormous insights into the fundamental nature of the hydrated proton. Here, we introduce a new coupled local-mode (CLM) approach to calculate PWC OH stretch vibrational spectra. The CLM method combines a sampling of representative configurations from density functional theory (DFT)-based ab initio molecular dynamics (AIMD) simulations with DFT calculations of local-mode vibrational frequencies and couplings. Calculations of inhomogeneous OH stretch vibrational spectra for H⁺(H₂O)₄ and H⁺(H₂O)₂₁ agree well with experiment and higher-level calculations, and decompositions of the calculated spectra in terms of the coupled modes aids in the interpretation of the spectra. This observation is consistent with the idea that capturing anharmonicity and coupling is as important to accuracy as the underlying level of electronic structure theory. The CLM calculations can easily discern the configuration that dominates the experimental measurement for H⁺(H₂O)₅, which can adopt several low-energy conformations.

Comparative studies of IR spectra of deprotonated serine with classical and thermostated ring polymer molecular dynamics simulations

Here we report the vibrational spectra of deprotonated serine calculated from the classical molecular dynamics (MD) simulations and thermostated ring-polymer molecular dynamics (TRPMD) simulation with third-order density-functional tight-binding. In our earlier study [Inakollu and Yu, "A systematic benchmarking of computational vibrational spectroscopy with DFTB3: Normal mode analysis and fast Fourier transform dipole autocorrelation function," J. Comput. Chem. 39, 2067 (2018)] of deprotonated serine, we observed a significant difference in the vibrational spectra with the classical MD simulations compared to the infrared multiple photon dissociation spectra. It was postulated that this is due to neglecting the nuclear quantum effects (NQEs). In this work, NQEs are considered in spectral calculation using the TRPMD simulations. With the help of potential of mean force calculations, the conformational space of deprotonated serine is analyzed and used to understand the difference in the spectra of classical MD and TRPMD simulations at 298.15 and 100 K. The high-frequency vibrational bands in the spectra are characterized using Fourier transform localized vibrational mode (FT-νN AC) and interatomic distance histograms. At room temperature, the quantum effects are less significant, and the free energy profiles in the classical MD and the TRPMD simulations are very similar. However, the hydrogen bond between the hydroxyl-carboxyl bond is slightly stronger in TRPMD simulations. At 100 K, the quantum effects are more prominent, especially in the 2600-3600 cm-1, and the free energy profile slightly differs between the classical MD and TRPMD simulations. Using the FT-νN AC and the interatomic distance histograms, the high-frequency vibrational bands are discussed in detail.

Using Diffusion Monte Carlo Wave Functions to Analyze the Vibrational Spectra of H7O3+ and H9O4. J Phys Chem A 2021;125:7185-7197. [PMID: 34433268 DOI: 10.1021/acs.jpca.1c05025]

An approach for evaluating spectra from ground state probability amplitudes (GSPA) obtained from diffusion Monte Carlo (DMC) simulations is extended to improve the description of excited state energies and allow for coupling among vibrational excited states. This approach is applied to studies of the protonated water trimer and tetramer, and their deuterated analogs. These ions provide models for solvated hydronium, and analysis of these spectra provides insights into spectral signatures of proton transfer in aqueous environments. In this approach, we obtain a separable set of internal coordinates from the DMC ground state probability amplitude. A basis is then developed from products of the DMC ground state wave function and low-order polynomials in these internal coordinates. This approach provides a compact basis in which the Hamiltonian and dipole moment matrix are evaluated and used to obtain the spectrum. The resulting spectra are in good agreement with experiment and in many cases provide comparable agreement to the results obtained using much larger basis sets. In addition, the compact basis allows for interpretation of the spectral features and how they evolve with cluster size and deuteration.

Electron binding energies and Dyson orbitals of OnH2n+1 +,0,- clusters: Double Rydberg anions, Rydberg radicals, and micro-solvated hydronium cations

Ab initio electron propagator methods are employed to predict the vertical electron attachment energies (VEAEs) of OH₃ ⁺(H₂O)_n clusters. The VEAEs decrease with increasing n, and the corresponding Dyson orbitals are diffused over exterior, non-hydrogen bonded protons. Clusters formed from OH₃ ^- double Rydberg anions (DRAs) and stabilized by hydrogen bonding or electrostatic interactions between ions and polar molecules are studied through calculations on OH₃ ^-(H₂O)_n complexes and are compared with more stable H^-(H₂O)_n+1 isomers. Remarkable changes in the geometry of the anionic hydronium-water clusters with respect to their cationic counterparts occur. Rydberg electrons in the uncharged and anionic clusters are held near the exterior protons of the water network. For all values of n, the anion-water complex H^-(H₂O)_n+1 is always the most stable, with large vertical electron detachment energies (VEDEs). OH₃ ^-(H₂O)_n DRA isomers have well separated VEDEs and may be visible in anion photoelectron spectra. Corresponding Dyson orbitals occupy regions beyond the peripheral O-H bonds and differ significantly from those obtained for the VEAEs of the cations.

Weighted-Graph-Theoretic Methods for Many-Body Corrections within ONIOM: Smooth AIMD and the Role of High-Order Many-Body Terms

We present a weighted-graph-theoretic approach to adaptively compute contributions from many-body approximations for smooth and accurate post-Hartree-Fock (pHF) ab initio molecular dynamics (AIMD) of highly fluxional chemical systems. This approach is ONIOM-like, where the full system is treated at a computationally feasible quality of treatment (density functional theory (DFT) for the size of systems considered in this publication), which is then improved through a perturbative correction that captures local many-body interactions up to a certain order within a higher level of theory (post-Hartree-Fock in this publication) described through graph-theoretic techniques. Due to the fluxional and dynamical nature of the systems studied here, these graphical representations evolve during dynamics. As a result, energetic "hops" appear as the graphical representation deforms with the evolution of the chemical and physical properties of the system. In this paper, we introduce dynamically weighted, linear combinations of graphs, where the transition between graphical representations is smoothly achieved by considering a range of neighboring graphical representations at a given instant during dynamics. We compare these trajectories with those obtained from a set of trajectories where the range of local many-body interactions considered is increased, sometimes to the maximum available limit, which yields conservative trajectories as the order of interactions is increased. The weighted-graph approach presents improved dynamics trajectories while only using lower-order many-body interaction terms. The methods are compared by computing dynamical properties through time-correlation functions and structural distribution functions. In all cases, the weighted-graph approach provides accurate results at a lower cost.

Vibrational Coupling in Solvated H3O+: Interplay between Fermi Resonance and Combination Band

Complex vibrational features of solvated hydronium ion, H₃O⁺, in 3 μm enable us to look into the vibrational coupling among O-H stretching modes and other degrees of freedom. Two anharmonic coupling schemes have often been engaged to explain observed spectra: coupling with the OH bending overtone, known as Fermi resonance (FR), has been proposed to account for the splitting of the OH stretch band at ∼3300 cm^-1 in H₃O⁺···Ar₃, but an additional peak in H₃O⁺···(N₂)₃ at the similar frequency region has been assigned to a combination band (CB) with the low-frequency intermolecular stretches. While even stronger vibrational coupling is expected in H₃O⁺···(H₂O)₃, such pronounced peaks are absent. In the present study, vibrational spectra of H₃O⁺···Kr₃ and H₃O⁺···(CO)₃ are measured to complement the existing spectra. Using ab initio anharmonic algorithms, we are able to assign the observed complex spectral features, to resolve seemingly contradictory notions in the interpretations, and to reveal simple pictures of the interplay between FR and CB.

Decoding the 2D IR spectrum of the aqueous proton with high-level VSCF/VCI calculations

The aqueous proton is a common and long-studied species in chemistry, yet there is currently intense interest devoted to understanding its hydration structure and transport dynamics. Typically described in terms of two limiting structures observed in gas-phase clusters, the Zundel H₅O₂ ⁺ and Eigen H₉O₄ ⁺ ions, the aqueous structure is less clear due to the heterogeneity of hydrogen bonding environments and room-temperature structural fluctuations in water. The linear infrared (IR) spectrum, which reports on structural configurations, is challenging to interpret because it appears as a continuum of absorption, and the underlying vibrational modes are strongly anharmonically coupled to each other. Recent two-dimensional IR (2D IR) experiments presented strong evidence for asymmetric Zundel-like motifs in solution, but true structure-spectrum correlations are missing and complicated by the anharmonicity of the system. In this study, we employ high-level vibrational self-consistent field/virtual state configuration interaction calculations to demonstrate that the 2D IR spectrum reports on a broad distribution of geometric configurations of the aqueous proton. We find that the diagonal 2D IR spectrum around 1200 cm^-1 is dominated by the proton stretch vibrations of Zundel-like and intermediate geometries, broadened by the heterogeneity of aqueous configurations. There is a wide distribution of multidimensional potential shapes for the proton stretching vibration with varying degrees of potential asymmetry and confinement. Finally, we find specific cross peak patterns due to aqueous Zundel-like species. These studies provide clarity on highly debated spectral assignments and stringent spectroscopic benchmarks for future simulations.

Analysis of the Proton Transfer Bands in the Infrared Spectra of Linear N2H+···OC and N2D+···OC Complexes Using Electric Field-Driven Classical Trajectories

In this work, we describe ab initio calculations and assignment of infrared (IR) spectra of hydrogen-bonded ion-molecular complexes that involve a fluxional proton: the linear N₂H⁺···OC and N₂D⁺···OC complexes. Given the challenges of describing fluxional proton dynamics and especially its IR activity, we use electric field-driven classical trajectories, i.e., the driven molecular dynamics (DMD) method that was developed by us in recent years and for similar applications, in conjunction with high-level electronic structure theory. Namely, we present a modified and a numerically efficient implementation of DMD specifically for direct (or "on the fly") calculations, which we carry out at the MP2-F12/AVDZ level of theory for the potential energy surface (PES) and MP2/AVDZ for the dipole moment surfaces (DMSs). Detailed analysis of the PES, DMS, and the time-dependence of the first derivative of the DMS, referred to as the driving force, for the highly fluxional vibrations involving H⁺/D⁺ revealed that the strongly non-harmonic PES and non-linear DMS yield remarkably complex vibrational spectra. Interestingly, the classical trajectories reveal a doublet in the proton transfer part of the spectrum with the two peaks at 1800 and 1980 cm^-1. We find that their shared intensity is due to a Fermi-like resonance interaction, within the classical limit, of the H⁺ parallel stretch fundamental and an H⁺ perpendicular bending overtone. This doublet is also observed in the deuterated species at 1360 and 1460 cm^-1.

High-Level VSCF/VCI Calculations Decode the Vibrational Spectrum of the Aqueous Proton

The hydrated excess proton is a common species in aqueous chemistry, which complexes with water in a variety of structures. The infrared spectrum of the aqueous proton is particularly sensitive to this array of structures, which manifests as continuous IR absorption from 1000 to 3000 cm^-1 known as the "proton continuum". Because of the extreme breadth of the continuum and strong anharmonicity of the involved vibrational modes, this spectrum has eluded straightforward interpretation and simulation. Using protonated water hexamer clusters from reactive molecular dynamics trajectories, and focusing on their central H⁺(H₂O)₂ structures' spectral contribution, we reproduce the linear IR spectrum of the aqueous proton with a high-level local monomer quantum method and highly accurate many-body potential energy surface. The accuracy of this approach is first verified in the vibrational spectra of the two isomers of the protonated water hexamer in the gas phase. We then apply this approach to 800 H⁺(H₂O)₆ clusters, also written as [H⁺(H₂O)₂](H₂O)₄, drawn from multistate empirical valence bond simulations of the bulk liquid to calculate the infrared spectrum of the aqueous proton complex. Incorporation of anharmonic effects to the vibrational potential and quantum mechanical treatment of the proton produces a better agreement to the infrared spectrum compared to that of the double-harmonic approximation. We assess the correlation of the proton stretching mode with different atomistic coordinates, finding the best correlation with ⟨R_OH⟩, the expectation value of the proton-oxygen distance R_OH. We also decompose the IR spectrum based on normal mode vibrations and ⟨R_OH⟩ to provide insight on how different frequency regions in the continuum report on different configurations, vibrational modes, and mode couplings.

Hydrated excess protons and their local hydrogen bond transport network as measured by translational, librational, and vibrational frequencies

A clear molecular description of excess hydrated protons and their local hydrogen bond transport network remains elusive. Here, the hydrogen bond network of excess hydrated protons in water bridges was probed by measuring their Raman spectra and comparing them to the spectra of protons in ice and water. The proton vibrational spectrum and the hydrogen bond network translational and librational spectra were recorded. The spectra of the water bridge and water exhibit clear differences, indicating the presence of a structure in water bridges when subjected to an electric field of ∼10⁶ V/m that has not been previously reported. The intermolecular Raman spectrum of the floating water bridge exhibits a hydrogen bond stretching band at 150-250 cm^-1, librational bands within the 300-1000 cm^-1 spectral range, and a large band at 1500-3000 cm^-1, which corresponds to the vibrational signature of excess hydrated protons in the water bridge structure. The excess protons are shown to move predominantly at the air/water interface, and the effect of this distribution is a measurable change in the air/water interfacial tension from ∼80 to ∼32 N/m. Therefore, hydrated protons must have a unique water arrangement that enables them to propagate without sinking into bulk water. This local polarized hydrogen bond network in the interfacial water region is characterized by a translational spectrum similar to that of ice V.

Hydrated Excess Protons in Acetonitrile/Water Mixtures: Solvation Species and Ultrafast Proton Motions

The solvation structure of protons in aqueous media is highly relevant to electric properties and to proton transport in liquids and membranes. At ambient temperature, polar liquids display structural fluctuations on femto- to picosecond time scales with a direct impact on proton solvation. We use two-dimensional infrared (2D-IR) spectroscopy to follow proton dynamics in acetonitrile/water mixtures with the Zundel cation H₅O₂⁺ prepared in neat acetonitrile as a benchmark. The 2D-IR spectra of the proton transfer mode of H₅O₂⁺ demonstrate stochastic large-amplitude motions in the double-minimum proton potential, driven by fluctuating electric fields. In all cases, the excess proton is embedded in a water dimer, forming an H₅O₂⁺ complex as the major solvation species. This observation is rationalized by quantum mechanics/molecular mechanics molecular dynamics simulations including up to four water molecules embedded in acetonitrile. The Zundel motif interacts with its closest water neighbor in an H₇O₃⁺ unit without persistent proton localization.

Quantum approaches to vibrational dynamics and spectroscopy: is ease of interpretation sacrificed as rigor increases?

The subject of this Perspective is quantum approaches, beyond the harmonic approximation, to vibrational dynamics and IR spectroscopy.

Near-Infrared Spectroscopy and Anharmonic Theory of Protonated Water Clusters: Higher Elevations in the Hydrogen Bonding Landscape

Near-infrared spectroscopy measurements are presented for protonated water clusters, H⁺(H₂O) _n, in the size range of n = 1-8. Clusters are produced in a pulsed-discharge supersonic expansion, mass selected, and studied with infrared laser photodissociation spectroscopy in the regions of 3600-4550 and 4850-7350 cm^-1. Although there is some variation with cluster size, the main features of these spectra are a broad absorption near 5300 cm^-1, a sharp doublet near 7200 cm^-1, as well as a structured absorption near 4100 cm^-1 for n ≥ 2. The vibrational patterns measured for the hydronium, Zundel, and Eigen ions are compared to those predicted by different forms of anharmonic theory. Second-order vibrational perturbation theory (VPT2) and a local mode treatment of the OH stretches both capture key aspects of the spectra but suffer understandable deficiencies in the quantitative description of band positions and intensities.

Broadband 2D IR spectroscopy reveals dominant asymmetric H5O2+ proton hydration structures in acid solutions

Given the critical role of the aqueous excess proton in redox chemistry, determining its structure and the mechanism of its transport in water are intense areas of experimental and theoretical research. The ultrafast dynamics of the proton's hydration structure has made it extremely challenging to study experimentally. Using ultrafast broadband two-dimensional infrared spectroscopy, we show that the vibrational spectrum of the aqueous proton is fully consistent with a protonated water complex broadly defined as a Zundel-like H₅O₂⁺ motif. Analysis of the inhomogeneously broadened proton stretch two-dimensional lineshape indicates an intrinsically asymmetric, low-barrier O-H⁺-O potential that exhibits surprisingly persistent distributions in both its asymmetry and O-O distance. This structural characterization has direct implications for the extent of delocalization exhibited by a proton's excess charge and for the possible mechanisms of proton transport in water.

Deconstructing Prominent Bands in the Terahertz Spectra of H7O3+ and H9O4+: Intermolecular Modes in Eigen Clusters

We report experimental vibrational action spectra (210-2200 cm^-1) and calculated IR spectra, using recent ab initio potential energy and dipole moment surfaces, of H₇O₃⁺ and H₉O₄⁺. We focus on prominent far-IR bands, which postharmonic analyses show, arise from two types of intermolecular motions, i.e., hydrogen bond stretching and terminal water wagging modes, that are similar in both clusters. The good agreement between experiment and theory further validates the accuracy of the potential and dipole moment surfaces, which was used in a recent theoretical study that concluded that infrared photodissociation spectra of the cold clusters correspond to the Eigen isomer. The comparison between theory and experiment adds further confirmation of the need of postharmonic analysis for these clusters.

Spectral signatures of proton delocalization in H+(H2O)n=1−4 ions

Vibrational couplings in protonated water clusters are described by harmonic analysis, vibrational perturbation theory (VPT2) and diffusion Monte Carlo (DMC) approaches.

Anharmonic vibrational spectra from double incremental potential energy and dipole surfaces

Using incremental approaches, size limitations for property surface generations are pushed significantly, enabling accurate large molecule anharmonic vibrational spectra calculations.

IR Spectroscopy of Protonated Acetylacetone and Its Water Clusters: Enol-Keto Tautomers and Ion→Solvent Proton Transfer

Protonated ions of acetylacetone, H⁺(Hacac), and their argon-tagged analogues are produced via a pulsed discharge and cooled in a supersonic expansion. These ions are mass analyzed, selected in a time-of-flight spectrometer, and studied with infrared laser photodissociation spectroscopy using the method of rare-gas atom tagging. Computational studies at the DFT/B3LYP level are employed to elucidate the structures and spectra of these ions, which are expected to exist as either enol- or keto-based tautomers. The protonated acetylacetone ion is found to form a single enol-based isomer. Adding one or two water molecules to this ion, for example, H⁺(Hacac)(H₂O)_1,2, produces primarily enol-based structures, although a small concentration of keto structures also contribute to the spectra. The vibrational patterns resulting from hydrogen bonding in these systems are not well-described by theory. Addition of a third water molecule to form the H⁺(Hacac)(H₂O)₃ ion causes a significant change in the spectroscopy, attributed to proton transfer from the H⁺(Hacac) ion into the water solvent.

Disentangling the Complex Vibrational Spectrum of the Protonated Water Trimer, H+(H2O)3, with Two-Color IR-IR Photodissociation of the Bare Ion and Anharmonic VSCF/VCI Theory

Vibrational spectroscopy of the protonated water trimer provides a stringent constraint on the details of the potential energy surface (PES) and vibrational dynamics governing excess proton motion far from equilibrium. Here we report the linear spectrum of the cold, bare H⁺(H₂O)₃ ion using a two-color, IR-IR photofragmentation technique and follow the evolution of the bands with increasing ion trap temperature. The key low-energy features are insensitive to both D₂ tagging and internal energy. The D₂-tagged D⁺(D₂O)₃ spectrum is reported for the first time, and the isotope dependence of the band pattern is surprisingly complex. These spectra are reproduced by large-scale vibrational configuration interaction calculations based on a new full-dimensional PES, which treat the anharmonic effects arising from large amplitude motion. The results indicate such extensive mode mixing in both isotopologues that one should be cautious about assigning even the strongest features to particular motions, especially for the absorptions that occur close to the intramolecular bending mode of the water molecule.

High-Level Quantum Calculations of the IR Spectra of the Eigen, Zundel, and Ring Isomers of H+(H2O)4 Find a Single Match to Experiment

The protonated water tetramer H⁺(H₂O)₄, often written as the Eigen cluster, H₃O⁺(H₂O)₃, plays a central role in studies of the hydrated proton. The cluster has been investigated spectroscopically both experimentally and theoretically with some differences and controversies. The major issue stems from the existence of higher-energy Zundel isomers of this cluster and the role these isomers might play in the IR spectra. Settling this fundamental issue is one goal of this Communication, where high-level quantum calculations of the IR spectra of the Eigen and three isomeric forms of this cluster are presented. These calculations make use of a many-body representation of the potential and dipole moment surfaces and VSCF/VCI calculations of vibrational eigenstates and the IR spectrum. The calculated spectra for the Eigen H₃O⁺(H₂O)₃ and D₃O⁺(D₂O)₃ isomers compare very well with experiment. The calculated spectra for the cis and trans-Zundel and ring isomers show prominent features that do not match with experiment but which can guide future experiments to search for these interesting and important isomers.