Efficient Method for Numerical Calculations of Molecular Vibrational Frequencies by Exploiting Sparseness of Hessian Matrix

Molecular vibrational frequency analysis plays an important role in theoretical and computational chemistry. However, in many cases, the analytical frequencies are unavailable, whereas frequency calculations using conventional numerical methods are very expensive. In this work, we propose an efficient method to numerically calculate the frequencies. Our main strategies are to exploit the sparseness of the Hessian matrix and to construct the N-fold two-variable potential energy surfaces to fit the parabola parameters, which are later used for the construction of Hessian matrices. A set of benchmark calculations is performed for typical molecules of different sizes and complexities using the proposed method. The obtained frequencies are compared to those calculated with the analytical methods and conventional numerical methods. It is shown that the results yielded with the new method are in very good agreement with corresponding accurate values (with a maximum error of ∼20 cm-1), while the required computation resource is largely reduced compared to that required by conventional numerical methods. For medium-sized molecules, the calculational scaling is lowered to O(N1.6) (this work) from that of O(N2) (conventional numerical methods). For even larger molecules, more computational savings can be achieved, and the scaling is estimated to be quasilinear with respect to the molecular size.

Searching low-energy conformers of neutral and protonated di-, tri-, and tetra-glycine using first-principles accuracy assisted by the use of neural network potentials

In the last ten years, combinations of state-of-the-art gas-phase spectroscopies and quantum chemistry calculations have suggested several intuitive trends in the structure of small polypeptides that may not hold true. For example, the preference for the cis form of the peptide bond and multiple protonated sites was proposed by comparing experimental spectra with low-energy minima obtained from limited structural sampling using various density functional theory methods. For understanding the structures of polypeptides, extensive sampling of their configurational space with high-accuracy computational methods is required. In this work, we demonstrated the use of deep-learning neural network potential (DL-NNP) to assist in exploring the structure and energy landscape of di-, tri-, and tetra-glycine with the accuracy of high-level quantum chemistry methods, and low-energy conformers of small polypeptides can be efficiently located. We hope that the structures of these polypeptides we found and our preliminary analysis will stimulate further experimental investigations.

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.

Formic acid stability in different solvents by DFT calculations

CONTEXT

New technologies have been developed toward the use of green energies. The production of formic acid (FA) from carbon dioxide (CO[Formula: see text]) hydrogenation with H[Formula: see text] is a sustainable process for H[Formula: see text] storage. However, the FA adduct stabilization is thermodynamically dependent on the type of solvent and thermodynamic conditions. The results suggest a wide range of dielectric permittivity values between the dimethyl sulfoxide (DMSO) and water solvents to stabilize the FA in the absence of base. The thermodynamics analysis and the infrared and charge density difference results show that the formation of the FA complex with H[Formula: see text]O is temperature dependent and has a major influence on aqueous solvents compared to the FA adduct with amine, in good agreement with the experiment. In these conditions, the stability thermodynamic of the FA molecule may be favorable at non-organic solvents and dielectric permittivity values closer to water. Therefore, a mixture of aqueous solvents with possible ionic composition could be used to increase the thermodynamic stability of H[Formula: see text] storage in CO[Formula: see text] conversion processes.

METHODS

Using the Quantum ESPRESSO package, density functional theory (DFT) calculations were performed with periodic boundary conditions, and the electronic wave functions were expanded in plane waves. For the exchange-correlation functional, we use the vdW-DF functional with the inclusion of van der Waals (vdW) forces. Electron-ion interactions are treated by the projector augmented wave (PAW) method with pseudopotentials available in the PSlibrary repository. The wave functions and the electronic densities were expanded employing accurate cut-off energies of 6.80[Formula: see text]10[Formula: see text] and 5.44[Formula: see text]10[Formula: see text] eV, respectively. The electronic density was computed from the wave functions calculated at the [Formula: see text]-point in the first Brillouin-zone. Each structural optimization was minimized according to the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, with force and energy convergence criteria of 25 meV[Formula: see text]Å[Formula: see text] and 1.36 meV, respectively. The electrostatic solvation effects were performed by the [Formula: see text] package with the Self-Consistent Continuum Solvation (SCCS) approach.

Quantum Dynamical Investigation of Dihydrogen-Hydride Exchange in a Transition-Metal Polyhydride Complex

The ongoing shift toward clean, sustainable energy is a primary driving force behind hydrogen fuel research. Safe and effective storage of hydrogen is a major challenge (particularly for mobile applications) and requires a detailed understanding of the atomic level interactions of hydrogen with its host materials. The light mass of hydrogen, however, implies that quantum effects are important, so a quantum dynamical treatment is required to properly account for these effects in computational simulations. As one such example, we describe herein the hydrogen exchange dynamics between a hydride and a dihydrogen ligand in the [FeH(H2)(PH3)4]+ model complex. A global three-dimensional (3D) potential energy surface (PES) was constructed by fitting to and interpolating from a discrete set of grid points computed using density functional theory; exact quantum dynamical calculations were then carried out on the 3D PES using discrete variable representation basis sets. Energy levels and their quantum tunneling splittings were computed up to 3000 cm-1 above the ground state. Within that energy range, all three fundamentals have been identified using wave function plots, as well as the first three overtones of the exchange (reaction coordinate) motion and several of its combination bands. From the tunneling splittings, the Boltzmann-averaged tunneling rates were computed. The Arrhenius plot of the total exchange rate shows a clear transition around 150 K, below which the activation energy is essentially zero and above which it is less than half of the electronic structure barrier. This indicates that exchange rates are governed by quantum tunneling throughout the relevant temperature range with the low-temperature regime dominated by a single quantum (ground) state. This work is the first-ever fully quantum dynamical study to investigate the hydrogen exchange dynamics between hydride and dihydrogen ligands coordinated to a transition-metal complex.

MBX: A many-body energy and force calculator for data-driven many-body simulations

Many-Body eXpansion (MBX) is a C++ library that implements many-body potential energy functions (PEFs) within the "many-body energy" (MB-nrg) formalism. MB-nrg PEFs integrate an underlying polarizable model with explicit machine-learned representations of many-body interactions to achieve chemical accuracy from the gas to the condensed phases. MBX can be employed either as a stand-alone package or as an energy/force engine that can be integrated with generic software for molecular dynamics and Monte Carlo simulations. MBX is parallelized internally using Open Multi-Processing and can utilize Message Passing Interface when available in interfaced molecular simulation software. MBX enables classical and quantum molecular simulations with MB-nrg PEFs, as well as hybrid simulations that combine conventional force fields and MB-nrg PEFs, for diverse systems ranging from small gas-phase clusters to aqueous solutions and molecular fluids to biomolecular systems and metal-organic frameworks.

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.

An Ab Initio Neural Network Potential Energy Surface for the Dimer of Formic Acid and Further Quantum Tunneling Dynamics

We construct a full-dimensional ab initio neural network potential energy surface (PES) for the isomerization system of the formic acid dimer (FAD). This is based upon ab initio calculations using the DLPNO-CCSD(T) approach with the aug-cc-pVTZ basis set, performed at over 14000 symmetry-unique geometries. An accurate fit to the obtained energies is generated using a general neural network fitting procedure combined with the fundamental invariant method, and the overall energy-weighted root-mean-square fitting error is about 6.4 cm^-1. Using this PES, we present a multidimensional quantum dynamics study on tunneling splittings with an efficient theoretical scheme developed by our group. The ground-state tunneling splitting of FAD calculated with a four-mode coupled method is in good agreement with the most recent experimental measurements. The PES can be applied for further dynamics studies. The effectiveness of the present scheme for constructing a high-dimensional PES is demonstrated, and this scheme is expected to be feasible for larger molecular systems.

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.

High-dimensional neural network potentials for accurate vibrational frequencies: the formic acid dimer benchmark

In recent years, machine learning potentials (MLP) for atomistic simulations have attracted a lot of attention in chemistry and materials science. Many new approaches have been developed with the primary aim to transfer the accuracy of electronic structure calculations to large condensed systems containing thousands of atoms. In spite of these advances, the reliability of modern MLPs in reproducing the subtle details of the multi-dimensional potential-energy surface is still difficult to assess for such systems. On the other hand, moderately sized systems enabling the application of tools for thorough and systematic quality-control are nowadays rarely investigated. In this work we use benchmark-quality harmonic and anharmonic vibrational frequencies as a sensitive probe for the validation of high-dimensional neural network potentials. For the case of the formic acid dimer, a frequently studied model system for which stringent spectroscopic data became recently available, we show that high-quality frequencies can be obtained from state-of-the-art calculations in excellent agreement with coupled cluster theory and experimental data.

Semiclassical and VSCF/VCI Calculations of the Vibrational Energies of trans- and gauche-Ethanol Using a CCSD(T) Potential Energy Surface

A recent full-dimensional Δ-Machine learning potential energy surface (PES) for ethanol is employed in semiclassical and vibrational self-consistent field (VSCF) and virtual-state configuration interaction (VCI) calculations, using MULTIMODE, to determine the anharmonic vibrational frequencies of vibration for both the trans and gauche conformers of ethanol. Both semiclassical and VSCF/VCI energies agree well with the experimental data. We find significant mixing between the VSCF basis states due to Fermi resonances between bending and stretching modes. The same effects are also accurately described by the full-dimensional semiclassical calculations. These are the first high-level anharmonic calculations using a PES, in particular a "gold-standard" CCSD(T) one.

Data-Driven Many-Body Potential Energy Functions for Generic Molecules: Linear Alkanes as a Proof-of-Concept Application

We present a generalization of the many-body energy (MB-nrg) theoretical/computational framework that enables the development of data-driven potential energy functions (PEFs) for generic covalently bonded molecules, with arbitrary quantum mechanical accuracy. The "nearsightedness of electronic matter" is exploited to define monomers as "natural building blocks" on the basis of their distinct chemical identity. The energy of generic molecules is then expressed as a sum of individual many-body energies of incrementally larger subsystems. The MB-nrg PEFs represent the low-order n-body energies, with n = 1-4, using permutationally invariant polynomials derived from electronic structure data carried out at an arbitrary quantum mechanical level of theory, while all higher-order n-body terms (n > 4) are represented by a classical many-body polarization term. As a proof-of-concept application of the general MB-nrg framework, we present MB-nrg PEFs for linear alkanes. The MB-nrg PEFs are shown to accurately reproduce reference energies, harmonic frequencies, and potential energy scans of alkanes, independently of their length. Since, by construction, the MB-nrg framework introduced here can be applied to generic covalently bonded molecules, we envision future computer simulations of complex molecular systems using data-driven MB-nrg PEFs, with arbitrary quantum mechanical accuracy.

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.

Double proton transfer in hydrated formic acid dimer: Interplay of spatial symmetry and solvent-generated force on reactivity

The double proton transfer (DPT) reaction in the hydrated formic acid dimer (FAD) is investigated at molecular-level detail. For this, a global and reactive machine learned (ML) potential energy surface (PES) is developed to run extensive (more than 100 ns) mixed ML/MM molecular dynamics (MD) simulations in explicit molecular mechanics (MM) solvent at MP2-quality for the solute. Simulations with fixed – as in a conventional empirical force field – and conformationally fluctuating – as available from the ML-based PES – charge models for FAD show a significant impact on the competition between DPT and dissociation of FAD into two formic acid monomers. With increasing temperature the barrier height for DPT in solution changes by about 10% (∼1 kcal mol⁻¹) between 300 K and 600 K. The rate for DPT is largest, ∼1 ns⁻¹, at 350 K and decreases for higher temperatures due to destabilisation and increased probability for dissociation of FAD. The water solvent is found to promote the first proton transfer by exerting a favourable solvent-induced Coulomb force along the O–H⋯O hydrogen bond whereas the second proton transfer is significantly controlled by the O–O separation and other conformational degrees of freedom. Double proton transfer in hydrated FAD is found to involve a subtle interplay and balance between structural and electrostatic factors.

Simulation of double proton transfer in formic acid dimer by reactive ML potential in explicit molecular mechanics water solvent.

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.

Enabling Rapid and Accurate Construction of CCSD(T)-Level Potential Energy Surface of Large Molecules Using Molecular Tailoring Approach

The construction of a potential energy surface (PES) of even a medium-sized molecule employing correlated theory, such as CCSD(T), is arduous due to the high computational cost involved. The present study reports the possibility of efficiently constructing such a PES of molecules containing up to 15 atoms and 550 basis functions by employing the fragment-based molecular tailoring approach (MTA) on off-the-shelf hardware. The MTA energies at the CCSD(T)/aug-cc-pVTZ level for several geometries of three test molecules, viz., acetylacetone, N-methylacetamide, and tropolone, are reported. These energies are in excellent agreement with their full calculation counterparts with a time advantage factor of 3-5. The energy barrier from the ground to transition state is also accurately captured. Further, we demonstrate the accuracy and efficiency of MTA for estimating the energy gradients at the CCSD(T) level. As a further application of our MTA methodology, the energies of acetylacetone at ∼430 geometries are computed at the CCSD(T)/aug-cc-pVTZ level and used for generating a Δ-machine learning (Δ-ML) PES. This leads to the H-transfer barrier of 3.02 kcal/mol, well in agreement with the benchmarked barrier of 3.19 kcal/mol. The fidelity of this Δ-ML PES is examined by geometry optimization and normal mode frequency calculations of global minima and saddle point geometries. We trust that the present work is a major development for the rapid and accurate construction of PES at the CCSD(T) level for molecules containing up to 20 atoms and 600 basis functions using off-the-shelf hardware.

Conformational Flexibility and Substitution Pattern Lead to Polymorphism of 3-Methyl-2-(phenylamino)benzoic acid

To probe the effect of substitution on the benzoic acid ring on the polymorphism of phenylaminobenzoic acids, five 3-methyl-2-(phenylamino)benzoic acids (MPABAs) were synthesized. A preliminary polymorph screen led to one...

Slow monomer vibrations in formic acid dimer: Stepping up the ladder with FTIR and Raman jet spectroscopy

In an effort to extend the cold gas phase spectroscopic database of the cyclic formic acid dimer (FAD), we present and analyze the jet-cooled vibrational infrared and Raman spectrum of (HCOOH)₂ in the monomer fingerprint region between 600 and 1500 cm^-1. The present study bridges the gap between the intermolecular dimerization-induced and the carbonyl stretching fundamentals that have already been reexamined using jet-cooled or high-resolution spectroscopy. This completes the characterization of the jet-cooled vibrational (HCOOH)₂ spectrum below the complex OH (CH) stretching fundamentals, and we report resonance-induced FAD combination/overtone transitions that will serve as a valuable reference for a theoretical modeling of its vibrational dynamics. As a by-product, several new formic acid trimer fundamentals are identified in the jet spectra and assigned with the help of second-order vibrational perturbation theory (VPT2). The polar formic acid dimer still eludes detection in a supersonic jet, but we are able to estimate an experimental upper-bound of the polar dimer-to-trimer-to-cyclic dimer intensity ratio to about 1:10:100 under typical expansion conditions. Using VPT2 with resonance treatment (VPT2+K), we reinvestigate the notorious ν₂₂ resonance triad. Generally, we find that VPT2, which is, of course, inadequate for modeling the resonance-rich OH stretching spectrum of FAD, is performing very satisfactorily in predicting fundamental and two-quantum state term values for the slower modes below 1500 cm^-1. As these modes are the building blocks for the ultrafast energy dissipation in the OH stretching region, the present work opens the door for its quantitative understanding.

Transfer learned potential energy surfaces: accurate anharmonic vibrational dynamics and dissociation energies for the formic acid monomer and dimer

The vibrational dynamics of the formic acid monomer (FAM) and dimer (FAD) is investigated from machine-learned potential energy surfaces at the MP2 (PES_MP2) and transfer-learned (PES_TL) to the CCSD(T) levels of theory. The normal mode (MAEs of 17.6 and 25.1 cm⁻¹) and second order vibrational perturbation theory (VPT2, MAEs of 6.7 and 17.1 cm⁻¹) frequencies from PES_TL for all modes below 2000 cm⁻¹ for FAM and FAD agree favourably with experiment. For the OH stretch mode the experimental frequencies are overestimated by more than 150 cm⁻¹ for both FAM and FAD from normal mode calculations. Conversely, VPT2 calculations on PES_TL for FAM reproduce the experimental OH frequency to within 22 cm⁻¹. For FAD the VPT2 calculations find the high-frequency OH stretch at 3011 cm⁻¹, compared with an experimentally reported, broad (∼100 cm⁻¹) absorption band with center frequency estimated at ∼3050 cm⁻¹. In agreement with earlier reports, MD simulations at higher temperature shift the position of the OH-stretch in FAM to the red, consistent with improved sampling of the anharmonic regions of the PES. However, for FAD the OH-stretch shifts to the blue and for temperatures higher than 1000 K the dimer partly or fully dissociates using PES_TL. Including zero-point energy corrections from diffusion Monte Carlo simulations for FAM and FAD and corrections due to basis set superposition and completeness errors yields a dissociation energy of D₀ = −14.23 ± 0.08 kcal mol⁻¹ compared with an experimentally determined value of −14.22 ± 0.12 kcal mol⁻¹.

Neural network based PESs are constructed for formic acid monomer and dimer at the MP2 and transfer learned to the CCSD(T) level of theory. The PESs are used to study the vibrational dynamics and dissociation energy of the molecules.

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.

Theoretical Study of Proton Tunneling in the Imidazole-Imidazolium Complex

Proton tunneling in the hydrogen-bonded imidazole–imidazolium complex ion has been studied theoretically. Ab initio CASSCF/6-311++G(d,p) calculations concerning geometry optimization and vibrational frequencies have been carried out for equilibrium and transition state structures of the system. Two-dimensional double-well model potentials were constructed on the basis of ab initio results and used to analyze the proton dynamics in the hydrogen bond and the influence of the excitation of low-frequency hydrogen-bond vibrations on the proton tunneling splittings. The energy of tunneling-split vibrational sublevels of the high-frequency tunneling mode have been calculated for its ground and first excited vibrational state for the series of excitations of the coupled low-frequency intramolecular hydrogen-bond modes. The promoting and suppressing effect of the low-frequency modes on the proton splittings was shown in the ground and first excited vibrational state of the tunneling mode. The vibrational sublevels form the two separate semicontinuous bands between which the absorption transitions may occur. This mechanism explains the experimentally observed splitting and doublet-component broadening of the high-frequency N–H stretching infrared (IR) absorption band.

CC-stretched formic acid: isomerisation, dimerisation, and carboxylic acid complexation

The cis-trans-isomerism of the propiolic acid monomer (HC[triple bond, length as m-dash]C-COOH) is examined with linear Raman jet spectroscopy, yielding the first environment-free vibrational band centres of a higher-energy cis-rotamer beyond formic acid (HCOOH) in addition to all fundamentals and a large number of hot and combination/overtone bands of the trans-conformer. Two near-isoenergetic trans-fundamentals of different symmetry (CC[double bond, length as m-dash]O bend and OH torsion) prove to be a sensitive benchmarking target, as their energetic order is susceptible to the choice of electronic structure method, basis set size, and inclusion of vibrational anharmonicity. For the infrared- and Raman-active C[double bond, length as m-dash]O stretching fundamentals of the cyclic (C_2h) trans-propiolic acid dimer, resonance couplings are found that in part extend to the C_s-symmetric heterodimer of trans-propiolic and trans-formic acid. Exploratory vibrational perturbation theory (VPT2) calculations show that all perturbing states involve displacements of the OH moieties located on the doubly hydrogen bonded ring. The comparison of the infrared spectra of the propiolic acid dimer and its heterodimer with formic acid to that of several other carboxylic acid dimers from the literature reveals a notable similarity regarding a non-fundamental dimer band around 1800 cm^-1, which in most cases is so far unassigned. VPT2 calculations and a simple harmonic model suggest an assignment to a combination vibration of the symmetric and antisymmetric OH torsion.

Breaking the Coupled Cluster Barrier for Machine-Learned Potentials of Large Molecules: The Case of 15-Atom Acetylacetone

Machine-learned potential energy surfaces (PESs) for molecules with more than 10 atoms are typically forced to use lower-level electronic structure methods such as density functional theory (DFT) and second-order Møller-Plesset perturbation theory (MP2). While these are efficient and realistic, they fall short of the accuracy of the "gold standard" coupled-cluster method, especially with respect to reaction and isomerization barriers. We report a major step forward in applying a Δ-machine learning method to the challenging case of acetylacetone, whose MP2 barrier height for H-atom transfer is low by roughly 1.1 kcal/mol relative to the benchmark CCSD(T) barrier of 3.2 kcal/mol. From a database of 2151 local CCSD(T) energies and training with as few as 430 energies, we obtain a new PES with a barrier of 3.5 kcal/mol in agreement with the LCCSD(T) barrier of 3.5 kcal/mol and close to the benchmark value. Tunneling splittings due to H-atom transfer are calculated using this new PES, providing improved estimates over previous ones obtained using an MP2-based PES.

Fingerprint region of the formic acid dimer: variational vibrational computations in curvilinear coordinates

Curvilinear kinetic energy models are developed for variational nuclear motion computations including the inter- and the low-frequency intra-molecular degrees of freedom of the formic acid dimer. The coupling of the inter- and intra-molecular modes is studied by solving the vibrational Schrödinger equation for a series of vibrational models, from two up to ten active vibrational degrees of freedom by selecting various combinations of active modes and constrained coordinate values. Vibrational states, nodal assignment, and infrared vibrational intensity information is computed using the full-dimensional potential energy surface (PES) and electric dipole moment surface developed by Qu and Bowman [Phys. Chem. Chem. Phys., 2016, 18, 24835; J. Chem. Phys., 2018, 148, 241713]. Good results are obtained for several fundamental and combination bands in comparison with jet-cooled vibrational spectroscopy experiments, but the description of the ν₈ and ν₉ fundamental vibrations, which are close in energy and have the same symmetry, appears to be problematic. For further progress in comparison with experiment, the potential energy surface, and in particular, its multi-dimensional couplings representation, requires further improvement.

High level ab initio investigation of the catalytic effect of water on formic acid decomposition and isomerization

Formic acid (FA) is a ubiquitous molecule found in the atmosphere, and is relevant to many important processes. The FA molecule generally exists as the trans isomer, which can decompose into H2O and CO (dehydration). It can also exist in the less favorable cis isomer which can decompose into H2 and CO2 (decarboxylation). Our work examines the complexes formed between each isomer of FA with water. We present geometries and vibrational frequencies obtained at the reliable CCSD(T)/aug-cc-pVTZ level of theory for seven FAwater complexes. We utilize the focal point method to determine CCSDT(Q)/CBS plus corrections binding energies of 7.37, 3.36, and 2.02 kcal mol-1 plus 6.07, 3.79, 2.60, and 2.55 kcal mol-1 for the trans-FAwater and cis-FAwater complexes, respectively. Natural bond orbital analysis is used to further decompose the interactions in each complex and gain insight into their relative strengths. Furthermore, we examine the effect that a single water molecule has on the barrier heights to each decomposition pathway by optimizing the transition states and verifying their connectivity with intrinsic reaction coordinate computations as well as utilizing a kinetic model. Water lowers the barrier to dehydration by at most 15.78 kcal mol-1 and the barrier to decarboxylation by up to 15.90 kcal mol-1. Our research also examines for the first time the effect of one water molecule on the interconversion barrier and we find that the barrier from trans to cis is not catalyzed by water due to the strong FA and water interactions. Our results highlight some instances where different binary complexes result in different decomposition pathways and even a case where one binary complex can form the same decomposition products via two distinct mechanisms. Our results provide a reliable benchmark of the FAH2O system as well as provide insight into future studies of similar atmospheric 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.

Efficient Quantum Mechanical Calculations of Mode-Specific Tunneling Splittings upon Fundamental Excitation in the Dimer of Formic Acid

The formic acid dimer (FAD) is an important benchmark system for understanding the double hydrogen transfer process. Most recently, Zhang et al. measured a few tunneling splittings upon fundamental excitation of FAD precisely (Zhang, Y. et al. J. Chem. Phys. 2017, 146, 244306); however, relevant theoretical studies are very limited. Here, we present a multidimensional quantum dynamics study on mode-specific tunneling splittings upon fundamental excitation in FAD with an efficient theoretical scheme developed by our group in which the process-oriented basis function customization strategy is combined with the preconditioned inexact spectral transform method. Various mode-specific tunneling splittings upon fundamental excitation are systematically calculated, and interesting mode-specific excitation effects on tunneling rate are identified. In particular, the calculated tunneling splittings for the ν₂₂ and ν₂₁ states are in good agreement with experiment, and the remarkable mode-specific suppression effects upon excitation should result from that the antisymmetric vibrational modes hinder the concerted double H-transfer. The present work is helpful to acquire a better understanding of the mode-specific excitation effects on tunneling processes.

Plumbing Potentials for Molecules with Up To Tens of Atoms: How to Find Saddle Points and Fix Leaky Holes

Potential energy surfaces (PESs) play an indispensable role in molecular dynamics but are notoriously difficult to flesh out properly in large-dimensional spaces. In particular, the undetected presence of PES holes, i.e., unphysical saddle points beyond which the potential energy drops arbitrarily, can have devastating effects on both classical and quantum dynamics calculations. In this study, the Crystal algorithm is developed as a tool for efficiently and accurately finding PES holes, as well as legitimate saddle points, even in very large-dimensional configuration spaces. The approach is applied to three large-dimensional PESs for molecular systems of current interest: uracil, naphthalene, and formic acid dimer. Low-lying PES holes are discovered and located for the first two systems-including naphthalene, for which no holes were previously suspected, to the best of our knowledge. Likewise, the double-well, double-proton-transfer isomerization saddle point for formic acid dimer is also located.

Fitting potential energy surfaces with fundamental invariant neural network. II. Generating fundamental invariants for molecular systems with up to ten atoms

Symmetry adaptation is crucial in representing a permutationally invariant potential energy surface (PES). Due to the rapid increase in computational time with respect to the molecular size, as well as the reliance on the algebra software, the previous neural network (NN) fitting with inputs of fundamental invariants (FIs) has practical limits. Here, we report an improved and efficient generation scheme of FIs based on the computational invariant theory and parallel program, which can be readily used as the input vector of NNs in fitting high-dimensional PESs with permutation symmetry. The newly developed method significantly reduces the evaluation time of FIs, thereby extending the FI-NN method for constructing highly accurate PESs to larger systems beyond five atoms. Because of the minimum size of invariants used in the inputs of the NN, the NN structure can be very flexible for FI-NN, which leads to small fitting errors. The resulting FI-NN PES is much faster on evaluating than the corresponding permutationally invariant polynomial-NN PES.

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).

Shifting formic acid dimers into perspective: vibrational scrutiny in helium nanodroplets

A metastable dimer of formic acid has been prepared inside superfluid helium nanodroplets and examined using IR spectroscopy and quantum chemical calculations.

Concerted Pair Motion Due to Double Hydrogen Bonding: The Formic Acid Dimer Case

Formic acid dimer as the prototypical doubly hydrogen-bonded gas-phase species is discussed from the perspective of the three translational and the three rotational degrees of freedom which are lost when two formic acid molecules form a stable complex. The experimental characterisation of these strongly hindered translations and rotations is reviewed, as are attempts to describe the associated fundamental vibrations, their combinations, and their thermal shifts by different electronic structure calculations and vibrational models. A remarkable match is confirmed for the combination of a CCSD(T)-level harmonic treatment and an MP2-level anharmonic VPT2 correction. Qualitatively correct thermal shifts of the vibrational spectra can be obtained from classical molecular dynamics in CCSD(T)-quality force fields. A detailed analysis suggests that this agreement between experiment and composite theoretical treatment is not strongly affected by fortuitous error cancellation but fully converged variational treatments of the six pair or intermolecular modes and their overtones and combinations in this model system would be welcome.

Double Proton Transfer in the Dimer of Formic Acid: An Efficient Quantum Mechanical Scheme

Double proton transfer plays an important role in biology and chemistry, such as with DNA base pairs, proteins and molecular clusters, and direct information about these processes can be obtained from tunneling splittings. Carboxylic acid dimers are prototypes for multiple proton transfer, of which the formic acid dimer is the simplest one. Here, we present efficient quantum dynamics calculations of ground-state and fundamental excitation tunneling splittings in the formic acid dimer and its deuterium isotopologues. These are achieved with a multidimensional scheme developed by us, in which the saddle-point normal coordinates are chosen, the basis functions are customized for the proton transfer process, and the preconditioned inexact spectral transform method is used to solve the resultant eigenvalue problem. Our computational results are in excellent agreement with the most recent experiments (Zhang et al., 2017; Li et al., 2019).

Ping-pong tunneling reactions, part 2: boron and carbon bell-clapper rearrangement

Anthracene can be used as a scaffold for intramolecular SN2 degenerate reactions of the “bell clapper” type, where a central boron atom or its isoelectronic carbocation bonds alternatively towards one or the other lateral Lewis bases at the first and eight anthracene positions. This ping-pong bond-switching reaction possesses a symmetrical double-well potential with low activation barrier and relatively narrow barrier width. Herein we show by computational means the active role played by heavy atom quantum tunneling in this degenerate rearrangement reaction at cryogenic temperatures. At these conditions the thermal “over the barrier” reaction is forbidden, whereas the tunneling effect enhances the rate of reaction up to an experimentally measurable half-life. Kinetic isotope effects and cryogenic NMR spectroscopy can, in principle, experimentally demonstrate the tunneling mechanism.

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.

Teaching vibrational spectra to assign themselves

A new paradigm for assigning vibrational spectra is described. Instead of proceeding from potential to Hamiltonian to eigenvalues/eigenvectors/intensities to spectrum, the new method goes "backwards" directly from spectrum to eigenvectors. The eigenvectors then "assign" the spectrum, in that they identify the basis states that contribute to each eigenvalue. To start, we demonstrate an algorithm that can obtain useful estimates of the eigenvectors connecting a real, symmetric Hamiltonian to its eigenvalues even if the only available information about the Hamiltonian is its diagonal elements. When this algorithm is augmented with information about transition intensities, it can be used to assign a complex vibrational spectrum using only information about (1) eigenvalues (the peak centers of the spectrum) and (2) a harmonic basis set (taken to be the diagonal elements of the Hamiltonian). Examples will be discussed, including application to the vibrationally complex spectral region of the formic acid dimer.