Full-dimensional quantum calculations of ground-state tunneling splitting of malonaldehyde using an accurate ab initio potential energy surface

A Bond-Based Machine Learning Model for Molecular Polarizabilities and A Priori Raman Spectra

The use of machine learning (ML) algorithms in molecular simulations has become commonplace in recent years. There now exists, for instance, a multitude of ML force field algorithms that have enabled simulations approaching ab initio level accuracy at time scales and system sizes that significantly exceed what is otherwise possible with traditional methods. Far fewer algorithms exist for predicting rotationally equivariant, tensorial properties such as the electric polarizability. Here, we introduce a kernel ridge regression algorithm for machine learning of the polarizability tensor. This algorithm is based on the bond polarizability model and allows prediction of the tensor components at the cost similar to that of scalar quantities. We subsequently show the utility of this algorithm by simulating gas phase Raman spectra of biphenyl and malonaldehyde using classical molecular dynamics simulations of these systems performed with the recently developed MACE-OFF23 potential. The calculated spectra are shown to agree very well with the experiments and thus confirm the expediency of our algorithm as well as the accuracy of the used force field. More generally, this work demonstrates the potential of physics-informed approaches to yield simple yet effective machine learning algorithms for molecular properties.

Nonadiabatic Hydrogen Tunneling Dynamics for Multiple Proton Transfer Processes with Generalized Nuclear-Electronic Orbital Multistate Density Functional Theory

Proton transfer and hydrogen tunneling play key roles in many processes of chemical and biological importance. The generalized nuclear-electronic orbital multistate density functional theory (NEO-MSDFT) method was developed in order to capture hydrogen tunneling effects in systems involving the transfer and tunneling of one or more protons. The generalized NEO-MSDFT method treats the transferring protons quantum mechanically on the same level as the electrons and obtains the delocalized vibronic states associated with hydrogen tunneling by mixing localized NEO-DFT states in a nonorthogonal configuration interaction scheme. Herein, we present the derivation and implementation of analytical gradients for the generalized NEO-MSDFT vibronic state energies and the nonadiabatic coupling vectors between these vibronic states. We use this methodology to perform adiabatic and nonadiabatic dynamics simulations of the double proton transfer reactions in the formic acid dimer and the heterodimer of formamidine and formic acid. The generalized NEO-MSDFT method is shown to capture the strongly coupled synchronous or asynchronous tunneling of the two protons in these processes. Inclusion of vibronically nonadiabatic effects is found to significantly impact the double proton transfer dynamics. This work lays the foundation for a variety of nonadiabatic dynamics simulations of multiple proton transfer systems, such as proton relays and hydrogen-bonding networks.

Tunneling splittings using modified WKB method in Cartesian coordinates: The test case of vinyl radical

Modified WKB theory for calculating tunneling splittings in symmetric multi-well systems in full dimensionality is re-derived using Cartesian coordinates. It is explicitly shown that the theory rests on the wavefunction that is exact for harmonic potentials. The theory was applied to calculate tunneling splittings in vinyl radical and some of its deuterated isotopologues in their vibrational ground states and the low-lying vibrationally excited states and compared to exact variational results. The exact results are reproduced within a factor of 2 in most states. Remarkably, all large enhancements of tunneling splittings relative to the ground state, up to three orders in magnitude in some excited mode combinations, are well reproduced. It is also shown that in the asymmetrically deuterated vinyl radical, the theory correctly predicts the states that are localized in a single well and the delocalized tunneling states. Modified WKB theory on the minimum action path is computationally inexpensive and can also be applied without modification to much larger systems in full dimensionality; the results of this test case serve to give insight into the expected accuracy of the method.

Numerical Accuracy Matters: Applications of Machine Learned Potential Energy Surfaces

The role of numerical accuracy in training and evaluating neural network-based potential energy surfaces is examined for different experimental observables. For observables that require third- and fourth-order derivatives of the potential energy with respect to Cartesian coordinates single-precision arithmetics as is typically used in ML-based approaches is insufficient and leads to roughness of the underlying PES as is explicitly demonstrated. Increasing the numerical accuracy to double-precision gives a smooth PES with higher-order derivatives that are numerically stable and yield meaningful anharmonic frequencies and tunneling splitting as is demonstrated for H2CO and malonaldehyde. For molecular dynamics simulations, which only require first-order derivatives, single-precision arithmetics appears to be sufficient, though.

Formic Acid-Ammonia Heterodimer: A New Δ-Machine Learning CCSD(T)-Level Potential Energy Surface Allows Investigation of the Double Proton Transfer

The formic acid-ammonia dimer is an important example of a hydrogen-bonded complex in which a double proton transfer can occur. Its microwave spectrum has recently been reported and rotational constants and quadrupole coupling constants were determined. Calculated estimates of the double-well barrier and the internal barriers to rotation were also reported. Here, we report a full-dimensional potential energy surface (PES) for this complex, using two closely related Δ-machine learning methods to bring it to the CCSD(T) level of accuracy. The PES dissociates smoothly and accurately. Using a 2d quantum model the ground vibrational-state tunneling splitting is estimated to be less than 10-4 cm-1. The dipole moment along the intrinsic reaction coordinate is calculated along with a Mullikan charge analysis and supports the mildly ionic character of the minimum and strongly ionic character at the double-well barrier.

Smolyak Scheme for solving the Schrödinger equation: Application to Malonaldehyde in Full Dimensionality

In 1963 Smolyak introduced an approach to overcome the exponential scaling with respect to the number of variables of the direct product size [S. A. Smolyak Soviet Mathematics Doklady, 4, 240 (1963)]. The main idea is to replace a single large direct product by a sum of selected small direct products. It was first used in quantum dynamics in 2009 by Avila and Carrington [G. Avila and T. Carrington, J. Chem. Phys., 131, 174103 (2009)]. Since then, several calculations have been published by Avila and Carrington and by other groups. In the present study, and to push the limit to larger and more complex systems, this scheme is combined with the use of an on-the-fly calculation of the kinetic energy operator and a Block-Davidson procedure to obtain eigenstates in our home-made Fortran codes, ElVibRot and Tnum-Tana. This was applied to compute the tunneling splitting of malonaldehyde in full dimensionality (21D) using the potential of Mizukami et al. [W. Mizukami, S. Habershon, and D.P. Tew, J. Chem. Phys. 141, 1443-10 (2014)]. Our tunneling splitting calculations, 21.7±0.3 cm-1 and 2.9±0.1 cm-1 , show excellent agreement with the experimental values, 21.6 cm-1 and 2.9 cm-1 for the normal isotopologue and the mono-deuterated one, respectively.

Exact tunneling splittings from symmetrized path integrals

We develop a new simulation technique based on path-integral molecular dynamics for calculating ground-state tunneling splitting patterns from ratios of symmetrized partition functions. In particular, molecular systems are rigorously projected onto their J = 0 rotational state by an "Eckart spring" that connects two adjacent beads in a ring polymer. Using this procedure, the tunneling splitting can be obtained from thermodynamic integration at just one (sufficiently low) temperature. Converged results are formally identical to the values that would have been obtained by solving the full rovibrational Schrödinger equation on a given Born-Oppenheimer potential energy surface. The new approach is showcased with simulations of hydronium and methanol, which are in good agreement with wavefunction-based calculations and experimental measurements. The method will be of particular use for the study of low-barrier methyl rotations and other floppy modes, where instanton theory is not valid.

Proton-Tunneling Dynamics along Low-Barrier Hydrogen Bonds: A Full-Dimensional Instanton Study of 6-Hydroxy-2-formylfulvene

Understanding the dynamics of proton transfer along low-barrier hydrogen bonds remains an outstanding challenge of great fundamental and practical interest, reflecting the central role of quantum effects in reactions of chemical and biological importance. Here, we combine ab initio calculations with the semiclassical ring-polymer instanton method to investigate tunneling processes on the ground electronic state of 6-hydroxy-2-formylfulvene (HFF), a prototypical neutral molecule supporting low-barrier hydrogen-bonding. The results emerging from a full-dimensional ab initio instanton analysis reveal that the tunneling path does not pass through the instantaneous transition-state geometry. Instead, the tunneling process involves a multidimensional reaction coordinate with concerted reorganization of the heavy-atom skeletal framework to substantially reduce the donor-acceptor distance and drive the ensuing intramolecular proton-transfer event. The predicted tunneling-induced splittings for HFF isotopologues are in good agreement with experimental findings, leading to percentage deviations of only 20-40%. Our full-dimensional results allow us to characterize vibrational contributions along the tunneling path, highlighting the intrinsically multidimensional nature of the attendant hydron-migration dynamics.

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.

A Structure-Based Gaussian Expansion for Quantum Reaction Dynamics in Molecules: Application to Hydrogen Tunneling in Malonaldehyde

We developed an approximate method for quantum reaction dynamics simulations, namely, a structure-based Gaussian (SBG) expansion approach, where SBG bases for the expansion of the wave function Ψ, expressed by a product of single-atom Cartesian Gaussians centered at the positions of respective nuclei, are mainly placed around critical structures on reaction pathways such as on the intrinsic reaction coordinate (IRC) through a transition state. In the present approach, the "pseudo-lattice points" at which SBGs are deployed are selected in a perturbative manner so as to make moderate the expansion length. We first applied the SBG idea to a two-dimensional quadruple-well model and obtained accurate tunneling splitting values between the lowest four states. We then applied it to hydrogen tunneling in malonaldehyde and achieved a tunneling splitting of 27.1 cm^-1 with only 875 SBGs at the MP2/6-31G(d,p) level of theory, in good agreement with 25 cm^-1 by the more elaborate multiconfiguration time-dependent Hartree method. Reasonable results were also obtained for singly and doubly deuterated malonaldehyde. We analyzed the tunneling states by utilizing expansion coefficients of individual SBGs and found that 40-45% of the SBGs in Ψ are nonplanar structures and SBGs away from the IRC contribute a little to hydrogen transfer.

Positioning of grid points for spanning potential energy surfaces-How much effort is really needed?

The positions of grid points for representing a multidimensional potential energy surface (PES) have a non-negligible impact on its accuracy and the associated computational effort for its generation. Six different positioning schemes were studied for PESs represented by n-mode expansions as needed for the accurate calculation of anharmonic vibrational frequencies by means of vibrational configuration interaction theory. A static approach, which has successfully been used in many applications, and five adaptive schemes based on Gaussian process regression have been investigated with respect to the number of necessary grid points and the accuracy of the fundamental modes for a small set of test molecules. A comparison with a related, more sophisticated, and consistent approach by Christiansen et al. is provided. The impact of the positions of the ab initio grid points is discussed for multilevel PESs, for which the computational effort of the individual electronic structure calculations decreases for increasing orders of the n-mode expansion. As a result of that, the ultimate goal is not the maximal reduction of grid points but rather the computational cost, which is not directly related.

MC-QTAIM analysis reveals an exotic bond in coherently quantum superposed malonaldehyde

The proton between the two oxygen atoms of the malonaldehyde molecule experiences an effective double-well potential in which the proton's wavefunction is delocalized between the two wells. Herein we employ a state-of-the-art multi-component quantum theory of atoms in molecules partitioning scheme to obtain the molecular structure, i.e. atoms in molecules and bonding network, from the superposed ab initio wavefunctions of malonaldehyde. In contrast to the familiar clamped-proton portrayal of malonaldehyde, in which the proton forms a hydrogen basin, for the superposed states the hydrogen basin disappears and two novel hybrid oxygen-hydrogen basins appear instead, with an even distribution of the proton population between the two basins. The interaction between the hybrid basins is stabilizing thanks to an unprecedented mechanism. This involves the stabilizing classical Coulomb interaction of the one-proton density in one of the basins with one-electron density in the other basin. This stabilizing mechanism yields a bond foreign to the known bonding modes in chemistry.

Wavepacket dynamical study of H-atom tunneling in catecholate monoanion: the role of intermode couplings and energy flow

We present a study of H-atom tunneling in catecholate monoanion through wavepacket dynamical simulations. In our earlier study of this symmetrical double-well system [Phys. Chem. Chem. Phys., 2022, 24, 10887], a limited number of transition state modes were identified as being important for the tunneling process. These include the imaginary frequency mode Q₁, the CO scissor mode Q₁₀, and the OHO bending mode Q₂₉. In this work, starting from non-stationary initial states prepared with excitations in these modes, we have carried out wavepacket dynamics in two and three dimensional spaces. We analyse the dynamical effects of the intermode couplings, in particular the role of energy flow between the studied modes on H-atom tunneling. We find that while Q₁₀ strongly modulates the donor-acceptor distance, it does not exchange energy with Q₁. However, excitation in Q₂₉ or Q₁ does lead to rapid energy exchange between these modes, which modifies the tunneling rate at early times.

Δ-Machine Learned Potential Energy Surfaces and Force Fields

There has been great progress in developing machine-learned potential energy surfaces (PESs) for molecules and clusters with more than 10 atoms. Unfortunately, this number of atoms generally limits the level of electronic structure theory to less than the "gold standard" CCSD(T) level. Indeed, for the well-known MD17 dataset for molecules with 9-20 atoms, all of the energies and forces were obtained with DFT calculations (PBE). This Perspective is focused on a Δ-machine learning method that we recently proposed and applied to bring DFT-based PESs to close to CCSD(T) accuracy. This is demonstrated for hydronium, N-methylacetamide, acetyl acetone, and ethanol. For 15-atom tropolone, it appears that special approaches (e.g., molecular tailoring, local CCSD(T)) are needed to obtain the CCSD(T) energies. A new aspect of this approach is the extension of Δ-machine learning to force fields. The approach is based on many-body corrections to polarizable force field potentials. This is examined in detail using the TTM2.1 water potential. The corrections make use of our recent CCSD(T) datasets for 2-b, 3-b, and 4-b interactions for water. These datasets were used to develop a new fully ab initio potential for water, termed q-AQUA.

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.

The MD17 datasets from the perspective of datasets for gas-phase “small” molecule potentials

There has been great progress in developing methods for machine-learned potential energy surfaces. There have also been important assessments of these methods by comparing so-called learning curves on datasets of electronic energies and forces, notably the MD17 database. The dataset for each molecule in this database generally consists of tens of thousands of energies and forces obtained from DFT direct dynamics at 500 K. We contrast the datasets from this database for three “small” molecules, ethanol, malonaldehyde, and glycine, with datasets we have generated with specific targets for the potential energy surfaces (PESs) in mind: a rigorous calculation of the zero-point energy and wavefunction, the tunneling splitting in malonaldehyde, and, in the case of glycine, a description of all eight low-lying conformers. We found that the MD17 datasets are too limited for these targets. We also examine recent datasets for several PESs that describe small-molecule but complex chemical reactions. Finally, we introduce a new database, “QM-22,” which contains datasets of molecules ranging from 4 to 15 atoms that extend to high energies and a large span of configurations.

Quantitative molecular simulations

All-atom simulations can provide molecular-level insights into the dynamics of gas-phase, condensed-phase and surface processes. One important requirement is a sufficiently realistic and detailed description of the underlying intermolecular interactions. The present perspective provides an overview of the present status of quantitative atomistic simulations from colleagues' and our own efforts for gas- and solution-phase processes and for the dynamics on surfaces. Particular attention is paid to direct comparison with experiment. An outlook discusses present challenges and future extensions to bring such dynamics simulations even closer to reality.

Multidimensional H-atom tunneling in the catecholate monoanion

We present the catecholate monoanion as a new model system for the study of multidimensional tunneling. It has a symmetrical O-H double-well structure, and the H atom motion between the two wells is coupled to both low and high frequency modes with different strengths. With a view to studying mode-specific tunneling in the catecholate monoanion, we have developed a full (33) dimensional potential energy surface in transition state (TS) normal modes using a Distributed Gaussian Empirical Valence Bond (DGEVB) based approach. We have computed eigenstates in different subspaces using both unrelaxed and relaxed potentials based on the DGEVB model. With unrelaxed potentials, we present results up to 7D subspaces that include the imaginary frequency mode and six modes coupled to it. With relaxed potentials, we focus on the two most important coupling modes. The structures of the ground and vibrationally excited eigenstates are discussed for both approaches and mode-specific tunneling splitting and their trends are presented.

Vibrational Tunneling Spectra of Molecules with Asymmetric Wells: A Combined Vibrational Configuration Interaction and Instanton Approach

A combined approach that uses the vibrational configuration interaction (VCI) and semiclassical instanton theory was developed to study vibrational tunneling spectra of molecules with multiple wells in full dimensionality. The method can be applied to calculate low-lying vibrational states in the systems with an arbitrary number of minima, which are not necessarily equal in energy or shape. It was tested on a two-dimensional double-well model system and on malonaldehyde, and the calculations reproduced the exact quantum mechanical (QM) results with high accuracy. The method was subsequently applied to calculate the vibrational spectrum of the asymmetrically deuterated malonaldehyde with nondegenerate vibrational frequencies in the two wells. The spectrum is obtained at a cost of single-well VCI calculations used to calculate the local energies. The interactions between states of different wells are computed semiclassically using the instanton theory at a comparatively negligible computational cost. The method is particularly suited to systems in which the wells are separated by large potential barriers and tunneling splittings are small, for example, in some water clusters, when the exact QM methods come at a prohibitive computational cost.

Analytical gradients for nuclear–electronic orbital multistate density functional theory: Geometry optimizations and reaction paths

Hydrogen tunneling plays a critical role in many biologically and chemically important processes. The nuclear–electronic orbital multistate density functional theory (NEO-MSDFT) method was developed to describe hydrogen transfer systems. In this approach, the transferring proton is treated quantum mechanically on the same level as the electrons within multicomponent DFT, and a nonorthogonal configuration interaction scheme is used to produce delocalized vibronic states from localized vibronic states. The NEO-MSDFT method has been shown to provide accurate hydrogen tunneling splittings for fixed molecular systems. Herein, the NEO-MSDFT analytical gradients for both ground and excited vibronic states are derived and implemented. The analytical gradients and semi-numerical Hessians are used to optimize and characterize equilibrium and transition state geometries and to generate minimum energy paths (MEPs), for proton transfer in the deprotonated acetylene dimer and malonaldehyde. The barriers along the resulting MEPs are lower when the transferring proton is quantized because the NEO-MSDFT method inherently includes the zero-point energy of the transferring proton. Analysis of the proton densities along the MEPs illustrates that the proton density can exhibit symmetric or asymmetric bilobal character associated with symmetric or slightly asymmetric double-well potential energy surfaces and hydrogen tunneling. Analysis of the contributions to the intrinsic reaction coordinate reveals that changes in the C–O bond lengths drive proton transfer in malonaldehyde. This work provides the foundation for future reaction path studies and direct nonadiabatic dynamics simulations of a wide range of hydrogen transfer reactions.

Cavity-Modulated Proton Transfer Reactions

Proton transfer is ubiquitous in many fundamental chemical and biological processes, and the ability to modulate and control the proton transfer rate would have a major impact on numerous quantum technological advances. One possibility to modulate the reaction rate of proton transfer processes is given by exploiting the strong light-matter coupling of chemical systems inside optical or nanoplasmonic cavities. In this work, we investigate the proton transfer reactions in the prototype malonaldehyde and Z-3-amino-propenal (aminopropenal) molecules using different quantum electrodynamics methods, in particular, quantum electrodynamics coupled cluster theory and quantum electrodynamical density functional theory. Depending on the cavity mode polarization direction, we show that the optical cavity can increase the reaction energy barrier by 10-20% or decrease the reaction barrier by ∼5%. By using first-principles methods, this work establishes strong light-matter coupling as a viable and practical route to alter and catalyze proton transfer reactions.

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.

A simplistic computational procedure for tunneling splittings caused by proton transfer

In this manuscript, we present an approach for computing tunneling splittings for large amplitude motions. The core of the approach is a solution of an effective one-dimensional Schrödinger equation with an effective mass and an effective potential energy surface composed of electronic and harmonic zero-point vibrational energies of small amplitude motions in the molecule. The method has been shown to work in cases of three model motions: nitrogen inversion in ammonia, single proton transfer in malonaldehyde, and double proton transfer in the formic acid dimer. In the current work, we also investigate the performance of different DFT and post-Hartree–Fock methods for prediction of the proton transfer tunneling splittings, quality of the effective Schrödinger equation parameters upon the isotopic substitution, and possibility of a complete basis set (CBS) extrapolation for the resulting tunneling splittings.

Impact of the Characteristics of Quantum Chemical Databases on Machine Learning Prediction of Tautomerization Energies

An essential aspect for adequate predictions of chemical properties by machine learning models is the database used for training them. However, studies that analyze how the content and structure of the databases used for training impact the prediction quality are scarce. In this work, we analyze and quantify the relationships learned by a machine learning model (Neural Network) trained on five different reference databases (QM9, PC9, ANI-1E, ANI-1, and ANI-1x) to predict tautomerization energies from molecules in Tautobase. For this, characteristics such as the number of heavy atoms in a molecule, number of atoms of a given element, bond composition, or initial geometry on the quality of the predictions are considered. The results indicate that training on a chemically diverse database is crucial for obtaining good results and also that conformational sampling can partly compensate for limited coverage of chemical diversity. The overall best-performing reference database (ANI-1x) performs on average by 1 kcal/mol better than PC9, which, however, contains about 2 orders of magnitude fewer reference structures. On the other hand, PC9 is chemically more diverse by a factor of ∼5 as quantified by the number of atom-in-molecule-based fragments (amons) it contains compared with the ANI family of databases. A quantitative measure for deficiencies is the Kullback-Leibler divergence between reference and target distributions. It is explicitly demonstrated that when certain types of bonds need to be covered in the target database (Tautobase) but are undersampled in the reference databases, the resulting predictions are poor. Examples of this include the poor performance of all databases analyzed to predict C(sp²)-C(sp²) double bonds close to heteroatoms and azoles containing N-N and N-O bonds. Analysis of the results with a Tree MAP algorithm provides deeper understanding of specific deficiencies in predicting tautomerization energies by the reference datasets due to inadequate coverage of chemical space. Capitalizing on this information can be used to either improve existing databases or generate new databases of sufficient diversity for a range of machine learning (ML) applications in chemistry.

Instanton calculations of tunneling splittings in chiral molecules

We report the ground state tunneling splittings (ΔE_± ) of a number of axially chiral molecules using the ring-polymer instanton (RPI) method (J. Chem. Phys., 2011, 134, 054109). The list includes isotopomers of hydrogen dichalcogenides H₂ X₂ (X = O, S, Se, Te, and Po), hydrogen thioperoxide HSOH and dichlorodisulfane S₂ Cl₂ . Ab initio electronic-structure calculations have been performed on the level of second-order Møller-Plesset perturbation (MP2) theory either with split-valance basis sets or augmented correlation-consistent basis sets on H, O, S, and Cl atoms. Energy-consistent pseudopotential and corresponding triple zeta basis sets of the Stuttgart group are used on Se, Te, and Po atoms. The results are further improved using single point calculations performed at the coupled cluster level with iterative singles and doubles and perturbative triples amplitudes. When available for comparison, our computed values of ΔE_± are found to lie within the same order of magnitude as values reported in the literature, although RPI also provides predictions for H₂ Po₂ and S₂ Cl₂ , which have not previously been directly calculated. Since RPI is a single-shot method which does not require detailed prior knowledge of the optimal tunneling path, it offers an effective way for estimating the tunneling dynamics of more complex chiral molecules, and especially those with small tunneling splittings.

Full-dimensional potential energy surface for acetylacetone and tunneling splittings

New, full-dimensional potential energy surface for acetylacetone allows for description of H-tunneling dynamics and characterization of stationary points.

Nuclear-Electronic Orbital Multistate Density Functional Theory

Hydrogen tunneling is essential for a wide range of chemical and biological processes. The description of hydrogen tunneling with multicomponent quantum chemistry approaches, where the transferring hydrogen nucleus is treated on the same level as the electrons, is challenging due to the importance of both static and dynamical electron-proton correlation. Herein the nuclear-electronic orbital multistate density functional theory (NEO-MSDFT) method is presented as a strategy to include both types of correlation. In this approach, two localized nuclear-electronic wave functions obtained with the NEO-DFT method are combined with a nonorthogonal configurational interaction approach to produce bilobal, delocalized ground and excited vibronic states. By including a correction function, the NEO-MSDFT approach can produce quantitatively accurate hydrogen tunneling splittings for fixed geometries of systems such as malonaldehyde and acetoacetaldehyde. This approach is computationally efficient and can be combined with methods such as vibronic coupling theory to describe tunneling dynamics and to compute vibronic couplings in many types of systems.

Tunneling splittings of vibrationally excited states using general instanton paths

A multidimensional semiclassical method for calculating tunneling splittings in vibrationally excited states of molecules using Cartesian coordinates is developed. It is an extension of the theory by Mil'nikov and Nakamura [J. Chem. Phys. 122, 124311 (2005)] to asymmetric paths that are necessary for calculating tunneling splitting patterns in multi-well systems, such as water clusters. Additionally, new terms are introduced in the description of the semiclassical wavefunction that drastically improves the splitting estimates for certain systems. The method is based on the instanton theory and builds the semiclassical wavefunction of the vibrationally excited states from the ground-state instanton wavefunction along the minimum action path and its harmonic neighborhood. The splittings of excited states are thus obtained at a negligible added numerical effort. The cost is concentrated, as for the ground-state splittings, in the instanton path optimization and the hessian evaluation along the path. The method can thus be applied without modification to many mid-sized molecules in full dimensionality and in combination with on-the-fly evaluation of electronic potentials. The tests were performed on several model potentials and on the water dimer.

Reactive Molecular Dynamics at Constant Pressure via Nonreactive Force Fields: Extending the Empirical Valence Bond Method to the Isothermal-Isobaric Ensemble

The Empirical Valence Bond (EVB) method offers a suitable framework to obtain reactive potentials through the coupling of nonreactive force fields. In this formalism, most of the implemented coupling terms are built using functional forms that depend on spatial coordinates, while parameters are fitted against reference data to model the change of chemistry between the participating nonreactive states. In this work, we demonstrate that the use of such coupling terms precludes the computation of the stress tensor for condensed phase systems and prevents the possibility to carry out EVB molecular dynamics in the isothermal-isobaric (NPT) ensemble. Alternatively, we make use of coupling terms that depend on the energy gaps, defined as the energy differences between the participating nonreactive force fields, and derive a general expression for the EVB stress tensor suitable for computation. Implementation of this new methodology is tested for a model of a single reactive malonaldehyde solvated in nonreactive water. Mass densities and probability distributions for the values of the energy gaps computed in the NPT ensemble reveal a negligible role of the reactive potential in the limit of low concentrated solutions, thus corroborating for the first time the validity of approximations based on the canonical NVT ensemble, customarily adopted for EVB simulations. The presented formalism also aims to contribute to future implementations and extensions of the EVB method to research the limit of highly concentrated solutions.

Instanton theory of tunneling in molecules with asymmetric isotopic substitutions

We consider quantum tunneling in asymmetric double-well systems for which the local minima in the two wells have the same energy, but the frequencies differ slightly. In a molecular context, this situation can arise if the symmetry is broken by isotopic substitutions. We derive a generalization of instanton theory for these asymmetric systems, leading to a semiclassical expression for the tunneling matrix element and hence the energy-level splitting. We benchmark the method using a set of one- and two-dimensional models, for which the results compare favorably with numerically exact quantum calculations. Using the ring-polymer instanton approach, we apply the method to compute the level splittings in various isotopomers of malonaldehyde in full dimensionality and analyze the relative contributions from the zero-point energy difference and tunneling effects.

Efficient Generation of Permutationally Invariant Potential Energy Surfaces for Large Molecules

An efficient method is described for generating a fragmented, permutationally invariant polynomial basis to fit electronic energies and, if available, gradients for large molecules. The method presented rests on the fragmentation of a large molecule into any number of fragments while maintaining the permutational invariance and uniqueness of the polynomials. The new approach improves on a previous one reported by Qu and Bowman by avoiding repetition of polynomials in the fitting basis set and speeding up gradient evaluations while keeping the accuracy of the PES. The method is demonstrated for CH₃–NH–CO–CH₃ (N-methylacetamide) and NH₂–CH₂–COOH (glycine).

Instanton theory of ground-state tunneling splittings with general paths

We derive a multidimensional instanton theory for calculating ground-state tunneling splittings in Cartesian coordinates for general paths. It is an extension of the method by Mil'nikov and Nakamura [J. Chem. Phys. 115, 6881 (2001)] to include asymmetric paths that are necessary for calculating tunneling splitting patterns in multi-well systems, such as water clusters. The approach avoids multiple expensive matrix diagonalizations to converge the fluctuation prefactor in the ring-polymer instanton (RPI) method, and instead replaces them by an integration of a Riccati differential equation. When combined with the string method for locating instantons, we avoid the need to converge the calculation with respect to the imaginary time period of the semiclassical orbit, thereby reducing the number of convergence parameters of the optimized object to just one: the number of equally spaced system replicas used to represent the instanton path. The entirety of the numerical effort is thus concentrated in optimizing the shape of the path and evaluating hessians along the path, which is a dramatic improvement over RPI. In addition to the standard instanton approximations, we neglect the coupling of vibrational modes to external rotations. The method is tested on the model potential of malonaldehyde and on the water dimer and trimer, giving close agreement with RPI at a much-reduced cost.

Quantum momentum distribution and quantum entanglement in the deep tunneling regime

In this paper, we consider the momentum operator of a quantum particle directed along the displacement of two of its neighbors. A modified open-path path integral molecular dynamics is presented to sample the distribution of this directional momentum distribution, where we derive and use a new estimator for this distribution. Variationally enhanced sampling is used to obtain this distribution for an example molecule, malonaldehyde, in the very low temperature regime where deep tunneling happens. We find no secondary feature in the directional momentum distribution and that its absence is due to quantum entanglement through a further study of the reduced density matrix.

Upper and lower bounds for tunneling splittings in a symmetric double-well potential

Ground state tunneling gaps: solid circles are mean of eigenvalues and lower bound gaps.

Infrared Spectra of Deprotonated Dicarboxylic Acids: IRMPD Spectroscopy and Empirical Valence‐Bond Modeling

Experimental infrared multiple-photon dissociation (IRMPD) spectra recorded for a series of deprotonated dicarboxylic acids, HO₂ (CH₂ )_n CO 2 - (n=2-4), are interpreted using a variety of computational methods. The broad bands centered near 1600 cm^-1 can be reproduced neither by static vibrational calculations based on quantum chemistry nor by a dynamical description of individual structures using the many-body polarizable AMOEBA force field, strongly suggesting that these molecules experience dynamical proton sharing between the two carboxylic ends. To confirm this assumption, AMOEBA was combined with a two-state empirical valence-bond (EVB) model to allow for proton transfer in classical molecular dynamics simulations. Upon suitable parametrization based on ab initio reference data, the EVB-AMOEBA model satisfactorily reproduces the experimental infrared spectra, and the finite temperature dynamics reveals a significant amount of proton sharing in such systems.

Large amplitude motions within molecules trapped in solid parahydrogen

Molecules of the β-diketone and β-dialdehyde families were trapped in solid parahydrogen (pH2) to investigate the vibrational behavior of systems containing an intramolecular hydrogen bond (IHB). In the simplest β-diketone, acetylacetone (AcAc), H transfer related to the IHB is coupled with methyl torsions. In pH2, the study of nuclear spin conversion (NSC) in methyl groups allows the characterisation of the influence of these large amplitude motions on the vibrational modes. The deuteration of the OH group involved in the IHB has important consequences on the vibrational spectrum of the molecule and evidence of NSC in methyl groups is difficult to obtain. In the chlorine derivative (3-chloroacetylacetone), the H-transfer is no longer coupled with methyl torsion, and NSC has undetectable effects on the IR spectrum. A search of H tunnelling splitting in the IR spectra of β-dialdehydes trapped in pH2 was performed. A few modes of 2-chloromalonaldehyde appear as doublets and were assigned to tunnelling levels. The spectroscopic results related to large amplitude motions are detailed and discussed, highlighting puzzling effects.