Ab Initio Vibrational Spectroscopy of cis- and trans-Formic Acid from a Global Potential Energy Surface

VSCF/VCI theory based on the Podolsky Hamiltonian

While the vibrational spectra of semi-rigid molecules can be computed on approaches relying on the Watson Hamiltonian, floppy molecules or molecular clusters are better described by Hamiltonians, which are capable of dealing with any curvilinear coordinates. It is the kinetic energy operator (KEO) of these Hamiltonians, which render the correlated calculations relying on them rather costly. Novel implementation of vibrational self-consistent field theory and vibrational configuration interaction theory on the basis of the Podolsky Hamiltonian are reported, in which the inverse of the metric tensor, i.e., the G matrix, is represented by an n-mode expansion expressed in terms of polynomials. An analysis of the importance of the individual terms of the KEO with respect to the truncation orders of the n-mode expansion is provided. Benchmark calculations have been performed for the cis-HOPO and methanimine, H2CNH, molecules and are compared to experimental data and to calculations based on the Watson Hamiltonian and the internal coordinate path Hamiltonian.

Mid-infrared Laser Spectroscopy of Jet-Cooled Formic Acid Trimer: Mode-Dependent Line Broadening in the C-O Stretching Region

Building on recent progress in the vibrational spectroscopy of the formic acid trimer, we present the first high-resolution measurements of the jet-cooled laser absorption spectrum of (HCOOH)3. The spectra of the lowest- and highest-frequency C-O stretching fundamentals are analyzed whereas the third band is not observed, complicated by monomer and dimer absorptions at 1219 cm-1 (8.2 μm). Vibration-rotation parameters are obtained for the band at 1172.31512(68) cm-1 whereas the C-O stretch at 1246.33(5) cm-1 exhibits a significantly larger breadth, allowing only resolution of the coarse PQR structure. Vibrational predissociation can be ruled out, and intramolecular vibrational redistribution mechanisms are discussed, particularly coupling to the concerted proton exchange within the cyclic dimer subunit. Ultimately, the question remains open. The prospects of high-resolution measurements of other trimer bands or isotope substitution experiments, which might assist in revealing the mode-specificity of the underlying broadening mechanisms, are discussed.

On the vibrations of formic acid predicted from first principles

In this article, we review recent first principles, anharmonic studies on the molecular vibrations of gaseous formic acid in its monomer form. Transitions identified as fundamentals for both cis- and trans form reported in these studies are collected and supported by results from high-resolution experiments. Attention is given to the effect of coordinate coupling on the convergence of the computed vibrational states.

Intramolecular Vibrational Redistribution in Formic Acid and its Deuterated Forms

The intramolecular vibrational relaxation dynamics of formic acid and its deuterated isotopologues is simulated on the full-dimensional potential energy surface of Richter and Carbonnière [F. Richter and P. Carbonnière, J. Chem. Phys. 148, 064303 (2018)] using the Heidelberg MCTDH package. We focus on couplings with the torsion vibrational modes close to the trans- cis isomerisation coordinate from the dynamics of artificially excited vibrational mode overtones. The C-O stretch bright vibrational mode is coupled to the out-of-the plane torsion mode in HCOOH, where this coupling could be exploited for laser-induced trans-to- cis isomerisation. Strong isotopic effects are observed: deuteration of the hydroxyl group, i.e., in HCOOD and DCOOD, destroys the C-O stretch to torsion mode coupling whereas in DCOOH, little to no effect is observed.

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.

Adaptive Fitting of Potential Energy Surfaces of Small to Medium-Sized Molecules in Sum-of-Product Form: Application to Vibrational Spectroscopy

A semi-automatic sampling and fitting procedure for generating sum-of-product (Born-Oppenheimer) potential energy surfaces based on a high-dimensional model representation is presented. The adaptive sampling procedure and subsequent fitting relies on energies only and can be used for re-fitting existing analytic potential energy surfaces in sum-of-product form or for direct fits from ab initio computa- tions. The method is tested by fitting ground electronic state potential energy surfaces for small to medium sized semi-rigid molecules, i.e., HFCO, HONO, and HCOOH, based upon ab initio computations at the CCSD(T)-F12/cc-pVTZ-F12 or MP2/aug-cc-pVTZ levels of theory. Vibrational eigenstates are computed using block improved relaxation in the Heidelberg MCTDH package and compared to available experimental and theoretical data. The new potential energy surfaces are compared to the best ones currently available for these molecules, in terms of accuracy, including of resulting vibrational states, required numbers of sampling points, and number of fitting parameters. The present procedure leads to compact expansions and scales well with the number of dimensions for simple potentials such as single or double wells.

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.

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.

Transfer Learning to CCSD(T): Accurate Anharmonic Frequencies from Machine Learning Models

The calculation of the anharmonic modes of small- to medium-sized molecules for assigning experimentally measured frequencies to the corresponding type of molecular motions is computationally challenging at sufficiently high levels of quantum chemical theory. Here, a practical and affordable way to calculate coupled-cluster quality anharmonic frequencies using second-order vibrational perturbation theory (VPT2) from machine-learned models is presented. The approach, referenced as "NN + VPT2", uses a high-dimensional neural network (PhysNet) to learn potential energy surfaces (PESs) at different levels of theory from which harmonic and VPT2 frequencies can be efficiently determined. The NN + VPT2 approach is applied to eight small- to medium-sized molecules (H₂CO, trans-HONO, HCOOH, CH₃OH, CH₃CHO, CH₃NO₂, CH₃COOH, and CH₃CONH₂) and frequencies are reported from NN-learned models at the MP2/aug-cc-pVTZ, CCSD(T)/aug-cc-pVTZ, and CCSD(T)-F12/aug-cc-pVTZ-F12 levels of theory. For the largest molecules and at the highest levels of theory, transfer learning (TL) is used to determine the necessary full-dimensional, near-equilibrium PESs. Overall, NN + VPT2 yields anharmonic frequencies to within 20 cm^-1 of experimentally determined frequencies for close to 90% of the modes for the highest quality PES available and to within 10 cm^-1 for more than 60% of the modes. For the MP2 PESs only ∼60% of the NN + VPT2 frequencies were within 20 cm^-1 of the experiment, with outliers up to ∼150 cm^-1, compared to the experiment. It is also demonstrated that the approach allows to provide correct assignments for strongly interacting modes such as the OH bending and the OH torsional modes in formic acid monomer and the CO-stretch and OH-bend mode in acetic acid.

Increasing the weights in the molecular work-out of cis- and trans-formic acid: extension of the vibrational database via deuteration

The higher-energy cis- as well as the global minimum trans-rotamers of the four H/D isotopologues of the formic acid monomer have been examined with Raman jet spectroscopy extending the vibrational gas phase reference database by eleven new cis-band positions for HCOOD, DCOOH, and DCOOD. With these new additions, all O-H/D, C-H/D, and C[double bond, length as m-dash]O stretching as well as the O-D in-plane bending vibrations of these higher-energy rotamers are known in addition to the previously determined C-O stretch and OH torsion of cis-HCOOH. Further, a comparison of the vibrational spectra of all four H/D isotopologues of the globally stable trans-rotamer of formic acid is shown to be very helpful in revealing similarities and differences in these systems, particularly with regard to Fermi resonances. Amongst the most prominent ones is the ν5/2ν9 resonance doublet of trans-HCOOH, for which we provide more insight into a recently suggested label switch of the resonance partners via the comparison of infrared and Raman jet spectra.

Ab initio calculation of rovibrational states for non-degenerate double-well potentials: cis-trans isomerization of HOPO

The rovibrational spectra of metaphosphorous acid, HOPO, and its deuterated isotopologue have been studied by vibrational configuration interaction calculations, relying on the internal coordinate path Hamiltonian and the Watson Hamiltonian. Tunneling effects for the overtones of the torsional mode, which gives rise to the cis-trans isomerization, and its rovibrational transitions have been investigated in detail. Due to strong matrix effects, comparison with experimental data is hindered, and thus, the calculations provide accurate estimates for the fundamental modes of these species.

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.

Anharmonic excited state frequencies of para-difluorobenzene, toluene and catechol using analytic RI-CC2 second derivatives

Analytic second nuclear derivatives for excited electronic state energies have been implemented for the resolution-of-the-identity accelerated CC2, CIS(D_∞) and ADC(2) models. Our efficient implementation with O(N²) memory demands enables the treatment of medium sized molecules with large basis sets and high numerical precision and thereby paves the way for semi-numerical evaluation of the higher-order derivatives required for anharmonic corrections to excited state vibrational frequencies. We compare CC2 harmonic and anharmonic excited state frequencies with experimental values for para-difluorobenzene, toluene and catechol. Basis set problems occur for out-of-plane bending vibrations due to intramolecular basis set superposition error. For non-planar molecules and in plane modes of planar molecules, the agreement between theory and experiment is better than 30 cm^-1 on average and we reassign a number of experimental bands on the basis of the ab initio predictions.

Stretching of cis-formic acid: warm-up and cool-down as molecular work-out

A new technique to rotationally simplify and Raman-probe conformationally and vibrationally excited small molecules is applied to the cis-trans isomerism of formic acid. It quintuples the previously available gas phase vibrational data base on this excited form of a strongly anharmonic planar molecule despite its limited spectral resolution. The newly determined cis-formic acid fundamentals allow for a balanced vibrational benchmark on both rotamers of formic acid. Assuming the adequacy of vibrational perturbation theory, it reveals weaknesses of standard methods for these systems like B3LYP-D3(BJ)/aVQZ VPT2 or PBE0-D3(BJ)/aVQZ VPT2. The functionals ωB97-XD and M06-2X additionally suffer from severe integration grid size and symmetry dependencies. The vibrational benchmark suggests B2PLYP-D3(BJ)/aVQZ VPT2 and MP2/aVQZ VPT2 as partially competitive and in any case efficient alternatives to computationally demanding coupled cluster vibrational configuration interaction calculations. Whether this is due to fortuitous compensation between electronic structure and vibrational perturbation error remains to be explored.