Revisiting recently published Raman jet spectra of monomeric formic acid with accurate high order perturbative calculations based on two explicitly correlated coupled-cluster quality potential energy surfaces from the literature, we assign and add 11 new vibrational band centers to the trans-HCOOH database and 53 for its three deuterated isotopologs. Profiting from the synergy between accurate calculations and symmetry information from depolarized Raman spectra, we reassign eight literature IR bands up to 4000 cm−1. Experimental detection of highly excited torsional states (ν9) of trans-HCOOH, such as 4ν9 and ν6 + 2ν9, reveals substantial involvement of the C–O stretch ν6 into the O–H bend/torsion resonance ν5/2ν9, which is part of a larger resonance polyad. Depolarization and isotopic C-D substitution experiments further elucidate the nature of Raman peaks in the vicinity of the O–H stretching fundamental (ν1), which seem to be members of a large set of interacting states that can be identified and described with a polyad quantum number and that gain intensity via resonance mixing with ν1.

Since its first spectroscopic detection in 1938,1 the internal dynamics of formic acid have continuously been motivating experimental2–12 and theoretical13–18 investigations. The number of experimentally determined (ro-)vibrational parameters of formic acid in its ground and vibrationally excited states is enormous (see Refs. 6–9, 19, and 20 and references therein), the experimental data further extending to its 13C,21–2818O,3,29,30 and 2H isotopologs.19,31–37 One of the reasons for this interest results from formic acid having two low-lying conformational isomers, depicted in Fig. 1. They are connected through the large-amplitude O–H torsion, which is responsible for many interesting and challenging dynamical features that we will touch upon in this work. It is well known that this motion is not appropriately treated using rectilinear normal coordinates, as the torsion corresponds to atomic displacements perpendicular to the figure plane in straight line paths. To correctly investigate this degree of freedom and its coupling to the remaining degrees of freedom, one has to treat this motion as a true internal torsional coordinate.38,39 From the perspective of this paper, the large anharmonicity of the torsion causes this vibration to tune in and out of resonance interactions as it is excited to increasingly high vibrational levels. This tuning provides an opportunity to interrogate a wide range of anharmonic couplings by matching theory with experiment. As an example, several experimental12 and computational16,17 studies have added clarity to the long-debated question of the assignment of the O–H in-plane bend ν5 and O–H torsional overtone 2ν9, which are in resonance. The higher IR intensity of the overtone has obscured the assignment of this resonance pair in the past as the more intense band was believed to be the fundamental.8,40 Recently published Raman jet spectra intuitively support the recent proposal,40 i.e., to assign the fundamental to 1306 cm−1 and the overtone to 1220 cm−1, where the band integral of the fundamental is seven times higher compared to the overtone.

FIG. 1.

Conformational isomers of the formic acid monomer.

FIG. 1.

Conformational isomers of the formic acid monomer.

Close modal

This reassignment was triggered by recent high level anharmonic vibrational calculations, namely, vibrational configuration interaction using an internal coordinate path Hamiltonian (ICPH)16 and multi-configuration time-dependent Hartree (MCTDH),17 further providing the community with two full-dimensional CCSD(T)-F12 (explicitly correlated coupled-cluster singles, doubles, and perturbative triples) quality potential energy surfaces of cis- and trans-formic acid. While the reported MCTDH and ICPH results for trans-HCOOH agree well with each other and available experimental reference data, deviations for cis are significant with root-mean-square deviations of 14 cm−1 and 49 cm−1 for one- and two-quantum states, respectively.16,17 Richter and Carbonnière consulted VPT2 energies for the nine fundamentals using both surfaces to rule out significant contributions from both potentials to the above mentioned discrepancies.17 

In this work, we use these two surfaces to make detailed comparisons with experiment for several isotopologs of both isomers. Our comparisons will detail the differences in the harmonic and anharmonic contributions, going beyond VPT2 and fundamentals. To solve for the vibrational eigenstates, we use high order canonical Van Vleck perturbation theory (CVPT) as implemented in curvilinear coordinates. We shall see that this approach provides accurate energies, allows us to describe couplings with relatively small matrices, and provides a test of perturbative methods.

To benchmark both potentials, we can build on a wide range of perturbation-free vibrational reference data for trans-HCOOH extending far beyond fundamentals, and particularly noteworthy is the enormous list of IR bands reported by Freytes et al.7 Comprehensive Raman spectra of formic acid and its three deuterated isotopologs were published four decades ago by Bertie et al.4,41 As their significant work was concerned with the formic acid dimer, the assignment of non-fundamental monomeric bands from their spectra is impaired by overlapping dimer contributions.40 Recently published Raman jet spectra, which build on earlier studies,42 are optimized for monomer signals and as an additional feature of the heatable setup that is used,43 peaks due to cold monomer, clusters, or hot transitions can easily be distinguished.11 These Raman spectra revealed a richness of hot and (weak) combination and overtone bands of trans-formic acid, many of which have not been reported in the previous literature, particularly for the deuterated isotopologs. These data that are, in part, published40,44 could not be fully analyzed and interpreted so far using available methods, such as standard VPT2. Particularly promising is the interpretation of hot bands as they are direct probes of anharmonicities. Naturally, the gas phase spectroscopy of cis-formic acid is more complicated due to the low abundance (one per mill) at room temperature.25,45 The cis database25 was significantly extended in recent years using the heatable Raman jet setup.11,40,46 Our goal is to unify and summarize recent developments regarding the theoretical and experimental vibrational spectrum of trans-formic acid and its deuterated isotopologs below 4000 cm−1.

This paper is structured as follows: We begin with discussing newly reported CVPT results, examining convergence, and comparing fundamental transition energies of both conformers to computational and experimental reference data. The main body of discussion is then centered around the assignment of new trans-formic acid peaks from Raman jet spectra and reassignment of literature IR bands. Finally, the two formic acid potential energy surfaces are compared.

Using the same experimental protocol as before,40 we have recorded new spectra to fill spectral gaps up to 3750 cm−1. For experimental details, we therefore refer to Ref. 40. In short, different isotopologs of formic acid were seeded into helium and expanded at different temperatures through a vertical slit nozzle into an evacuated jet chamber. The expansion was probed with a continuous-wave 532 nm laser, which was operated at 20 W for the new measurements. The scattered light was collected perpendicular to the laser and nozzle flow and focused onto a monochromator, which disperses the photons onto a liquid nitrogen-cooled CCD-camera (1340 × 400 pixels) binning over 400 vertical pixels. To account for the final resolution from the combination of laser and monochromator, we generously assign band center errors of ±2 cm−1. Spectra of a temperature series are intensity-scaled to a trans-formic acid fundamental in each spectral region. The advantage is that hot bands, i.e., transitions from thermally populated vibrational states, can be easily distinguished from the cold monomer and cluster bands as they increase in intensity with rising temperature, whereas the cluster bands decrease (for further reading, see Refs. 11 and 46). Overtone and combination bands of the trans-formic acid monomer can furthermore be distinguished as their intensity remains constant over a scaled temperature series, similar to fundamentals.

The two potential energy surfaces we use in this work are referred to as PES-201616 and PES-2018.17 Both surfaces are full-dimensional and semi-global as they describe the cis and trans conformational space of formic acid. In the following, we briefly summarize technical aspects, and for details and references, see Refs. 16 and 17.

PES-2016 was presented by Tew and Mizukami in 2016. They fitted 17 076 single point energies at the CCSD(T)-F12c/VTZ-F12 level using a method similar to LASSO constrained optimization. Note that points along the O–H torsion as defined by the internal coordinate path were included in the fit. The potential is composed of a zero-order surface using Morse oscillators for the atom–atom distances to ensure proper asymptotic behavior. Perturbations are introduced by a more flexible correction surface that is a sum of distributed multivariate Gaussian functions of combinations of the atom–atom distances. The authors report an accuracy of the fit to 0.25% with a root-mean-square deviation of 9 cm−1 to the ab initio data in the energy range 0 cm−1–15 000 cm−1.

In 2018, Richter and Carbonnière presented a second full-dimensional formic acid potential, which they fitted to 660 single point energies at the CCSD(T)-F12a/aVTZ level using the AGAPES program. The potential is defined in a sum-of-product form using internal valence coordinates that minimize the potential energy coupling, e.g., the torsional motion is mostly localized along one dihedral coordinate. The AGAPES algorithm iteratively converges the potential energy expansion to a predefined cut-off value, in the process adding more ab initio points. The dead branching variant of AGAPES was used and the authors report the fit to be accurate between 0 cm−1 and 13 327 cm−1.

In this work, we solve the vibrational eigenstates and their corresponding eigenvalues using a numerical implementation of high order canonical Van Vleck perturbation theory (CVPT).47–49 Our implementation requires a Hamiltonian expanded in terms of the normal coordinates and their conjugate momenta written in terms of raising and lowering operators.

The two potential surfaces we consider, as described above, are expanded in Taylor series in terms of the stretch, bend, and dihedral angles extensions following the coordinate choice of Richter and Carbonnière.17 One difference is that we use Simons–Parr–Finlan coordinates for the stretches to improve the convergence properties.50 The potential includes up to four-body terms. These contributions include expansion terms up to 8th, 6th, 6th, and 4th for the one- to four-body terms, respectively. A comparison of the torsional potentials and their fits is given in Fig. S13 of the supplementary material. The exact kinetic energy operator51 is a function of the G-matrix elements. These elements are expanded through sixth order in the internal coordinates with the exception of four-body terms that are expanded through fourth order. The potential-like contribution to the kinetic energy51 is expanded through fourth order.

The potential and kinetic contributions to the Hamiltonian are re-expressed in normal coordinates and conjugate momenta. The normal coordinates are curvilinear, since they are written as a linear expansion of the internal coordinates R = LQ, where L is obtained by diagonalization of the F and G matrices following the work of Wilson, Decius, and Cross.52 As a final step, in order to carry out CVPT, the Hamiltonian is expressed in terms of the harmonic oscillator raising and lowering operators expressed in normal form.53 

The CVPT is carried out through a series of transformations that order by order remove off-resonance terms. The details of the calculations have been described previously, so they are not repeated here.53,54 The central point is that the more off-diagonal terms that are removed, the poorer the agreement between subsequent orders of perturbation theory and the smaller the sizes of the matrices that need to be diagonalized. The two extremes are a final effective Hamiltonian that equals the initial Hamiltonian and a Hamiltonian with no coupling terms. There are many criteria that are used in deciding the form of the final Hamiltonian.47,55,56 In this work, we follow an approach that was successful for describing the vibrations of HFCO,54 a molecule with many possible resonances. In that approach, a rather large list of possible resonance terms is generated. Our criteria are based solely on energy mismatch; coupling strength is not considered. If a coupling term couples states with an energy difference less than Wi/cm−1, where i is the order of the coupling terms, then the coupling term is retained. Here, we set {W3, W4, …, W8} = {350, 200, 60, 50, 15, 15} and therefore retain many small coupling terms that could be transformed away, knowing that these terms are readily treated in the ensuing variational calculation of the transformed Hamiltonian.

The resulting transformed Hamiltonian is expressed as a matrix using a product basis of harmonic oscillators. The size of this basis is constrained by the number of quanta in each oscillator. We use the constraint ∑iciniNt, where c = {1, 2, 2, 2, 3, 3, 6, 6} when the modes are ordered in terms of increasing frequency and Nt = 8. These numbers are roughly based on the frequencies and have not been optimized in a systematic way. These values lead to modest-sized matrices of 264 and 213 for trans-formic acid for the two symmetry blocks, respectively. Comparing results to Nt = 12 shows that all states are converged to better than 0.1 cm−1 compared to the smaller bases for the sixth order CVPT results of trans-formic acid. Our approach allows for many possible resonance interactions through the large values of W that still yield representations for which the eigenfunctions of the effective Hamiltonians can be readily obtained.

In order to test the quality of the results, it is important to compare eigenvalues calculated at different orders of perturbation theory. We do this by comparing E(4) results to E(6) results where the order of the perturbative result is in parentheses. If one finds significant discrepancies, we consider the possible terms that are leading to the slow convergence and adjust the values of W accordingly. Atomic masses used in the CVPT calculations are reported in the supplementary material.

The convergence of newly reported CVPT vibrational eigenvalues is quantified with the three quantities {Emax, nmax, Nord}, where Nord is the order of perturbation theory and the other two parameters identify the states included in the comparison set. We set Emax = 4000 cm−1 and nmax = 4, constraining states to have energies below the cut-off energy Emax and ∑ininmax. The state identification is based on the leading coefficient in the expansion.

Convergence with respect to the level of perturbation theory is shown in Fig. 2 for trans- and cis-HCOOH where the energy difference between sixth and second as well as sixth and fourth order states is compared. In comparing two orders of perturbation theory, we compare states that are most similar to each other, and we use their overlap as a measure of this similarity. As an example, the overlap between the fourth order and sixth order states is defined by S=U4TU6, where UnTHnUn is the similarity transformation that diagonalizes the nth order Hamiltonian. At low energies, the diagonal elements Sii are close to one. Whenever eigenvalues switch order between two levels of perturbation theory or if two states are in resonance at one level and not at another, then this information is encoded in the elements of S and can be used to match the states appropriately.

FIG. 2.

Convergence plots for trans- (left) and cis-HCOOH (right) vibrational states relative to their respective ground state energy using canonical Van Vleck perturbation theory (CVPT). States of a′ and a″ symmetry are marked as “+” and “×,” respectively. Convergence is quantified with the quantities {Emax, nmax, Nord}, where Nord is the order of perturbation theory, Emax is the maximum allowed E(6) energy in cm−1, and nmax is the maximum sum of quanta of vibrational excitation for states included in the comparison set. Energy differences between sixth and second as well as sixth and fourth order states are shown for Emax = 4000 and nmax = 4.

FIG. 2.

Convergence plots for trans- (left) and cis-HCOOH (right) vibrational states relative to their respective ground state energy using canonical Van Vleck perturbation theory (CVPT). States of a′ and a″ symmetry are marked as “+” and “×,” respectively. Convergence is quantified with the quantities {Emax, nmax, Nord}, where Nord is the order of perturbation theory, Emax is the maximum allowed E(6) energy in cm−1, and nmax is the maximum sum of quanta of vibrational excitation for states included in the comparison set. Energy differences between sixth and second as well as sixth and fourth order states are shown for Emax = 4000 and nmax = 4.

Close modal

Figure 2 (left panels) shows that vibrational eigenstates of trans-HCOOH on PES-2016 are slightly better converged than on PES-2018. On the whole, the deviation between fourth and sixth order shows that eigenstates are converged within a few cm−1. The convergence of individual states can be checked with data provided in Table S7 of the supplementary material. The mean absolute deviation (maximum absolute deviation) for nmax = 1, 2, 3, 4 states is below 1(2), 1(3), 1(6), and 2(7) on PES-2016 and below 1(5), 1(5), 2(13), and 3(21) on PES-2018, all in units of cm−1. The comparison between sixth and second order is especially insightful as second order Van Vleck perturbation theory is the same as standard VPT257 with resonance treatment (VPT2+K).58 On PES-2016, the performance for fundamentals and binary states is quite satisfactory with mean absolute deviations (maximum absolute deviations) below 1(3) cm−1 and 2(6) cm−1, respectively. The limitations of VPT2+K are revealed beyond two quanta where the maximum absolute deviations to sixth order CVPT amount to 19 cm−1 and 16 cm−1 for three- and four-quantum states, respectively (cf. Table S7).

The CVPT calculated fundamentals, as well as states with multiple quanta of excitation, converge more slowly for cis-formic acid on PES-2016 than PES-2018, which is reflected in Fig. 2 (right panels). Fundamental wavenumbers of cis-HCOOH are converged within 1 cm−1 with respect to the order of perturbation on PES-2018, whereas the two high-frequency X–H stretching fundamentals on PES-2016 differ by 4 cm−1 between fourth and sixth order. As expected, the O–H torsion of formic acid is very sensitive to the relative conformation, as reflected in the large down-shift from 640.73 cm−1 to 493.42 cm−1 for the fundamental. The frequency lowering reduces the involvement of the torsion in strong resonance polyads—groups of nearly degenerate states that are described as zero-order coupled states59—which are characteristic of the trans species (vide infra).14 This decoupling for cis appears to improve the convergence of the remaining higher-frequency low-lying states so that overall CVPT converges faster for cis than trans, as can be observed by comparing the left- and right-hand side panels of Fig. 2.

The nine vibrations of trans-formic acid are schematically shown in Fig. 3. These localized representations are idealized, especially the C–H/D and O–H/D in-plane bending, and C–O stretching vibrations ν4, ν5, and ν6 are mixed group vibrations. In this work, we apply the Herzberg nomenclature of HCOOH also to its three deuterated isotopologs. With a large number of perturbation-free experimental reference data available for fundamentals of both conformers, we can compare PES-2016 and PES-2018 using different anharmonic vibrational methods such as the internal coordinate path Hamiltonian (ICPH),16 multi-configuration time-dependent Hartree (MCTDH),17,18 and high order canonical Van Vleck perturbation theory (CVPT).

FIG. 3.

Schematic drawings of fundamental vibrations of trans-HCOOH. In this work, we apply the Herzberg nomenclature of cis- and trans-HCOOH to all their three deuterated isotopologs.

FIG. 3.

Schematic drawings of fundamental vibrations of trans-HCOOH. In this work, we apply the Herzberg nomenclature of cis- and trans-HCOOH to all their three deuterated isotopologs.

Close modal

The anharmonic fundamentals of HCOOH are listed together with experimental reference data in Table I; deviations between experiment and anharmonic calculations employing both PES are visualized in Fig. 4. These figure results include all deuterated isotopologs. Table I shows that the CVPT6 results for trans-HCOOH compare favorably with ICPH and MCTDH reference data on the same surface as they agree within 2 cm−1. Comparison for cis-HCOOH indicates that the internal coordinate path Hamiltonian (ICPH) eigenvalues are not fully converged. In addition, the CVPT6/PES-2016 calculation overestimates the O–H stretch ν1(cis) transition energy by 16 cm−1. Given that CVPT4 and CVPT6 agree within 4 cm−1, this discrepancy is probably due to the surface. Note that ν1(cis) was not reported in Ref. 16 and is therefore missing in Fig. 4. MCTDH and CVPT6 results for cis compare favorably with deviations below 3 cm−1 except for ν1, which differs more (3631 cm−1 MCTDH, 3636 cm−1 CVPT6). Upon C–H deuteration,18ν1 agreement between MCTDH (3625 cm−1) and CVPT6 (3637 cm−1) worsens where the MCTDH predictions seem somewhat more sensitive to isotopic substitution (−6 cm−1, +1 cm−1 CVPT6) than the experimental data indicate (−2 cm−1). On the whole, Fig. 4 demonstrates that vibrational energies for both conformers tend to be higher than experiment on PES-2016, whereas PES-2018 tends to underestimate the experiment.

TABLE I.

Fundamentals of cis- and trans-HCOOH computed using PES-2016 and PES-2018 together with experimental values (see Ref. 40 and references therein).

FundamentalPES-2016PES-2018
LabelΓICPH16 CVPT6MCTDH17 CVPT6Exp.
cis-HCOOH 
ν1 a′ n.r. 3653 3631 3636 3637 
ν2 a′ 2880 2878 2871 2874 2873 
ν3 a′ 1824 1821 1810 1810 1818 
ν4 a′ 1394 1389 1383 1384  
ν5 a′ 1255 1246 1246 1247  
ν6 a′ 1103 1096 1097 1097 1093 
ν7 a′ 668 657 652 652  
ν8 a″ 1038 1020 1011 1014  
ν9 a″ 492 491 491 491 493.42 
trans-HCOOH 
ν1 a′ 3575 3576 3567 3568 3570.5 
ν2 a′ 2938 2940 2937 2939 2942.06 
ν3 a′ 1783 1783 1774 1773 1776.83 
ν4 a′ 1379 1380 1375 1374 1379.05 
ν5 a′ 1305 1305 1301 1300 1306.2 
ν6 a′ 1108 1108 1106 1106 1104.85 
ν7 a′ 627 627 623 623 626.17 
ν8 a″ 1034 1035 1032 1032 1033.47 
ν9 a″ 638 640 637 637 640.73 
FundamentalPES-2016PES-2018
LabelΓICPH16 CVPT6MCTDH17 CVPT6Exp.
cis-HCOOH 
ν1 a′ n.r. 3653 3631 3636 3637 
ν2 a′ 2880 2878 2871 2874 2873 
ν3 a′ 1824 1821 1810 1810 1818 
ν4 a′ 1394 1389 1383 1384  
ν5 a′ 1255 1246 1246 1247  
ν6 a′ 1103 1096 1097 1097 1093 
ν7 a′ 668 657 652 652  
ν8 a″ 1038 1020 1011 1014  
ν9 a″ 492 491 491 491 493.42 
trans-HCOOH 
ν1 a′ 3575 3576 3567 3568 3570.5 
ν2 a′ 2938 2940 2937 2939 2942.06 
ν3 a′ 1783 1783 1774 1773 1776.83 
ν4 a′ 1379 1380 1375 1374 1379.05 
ν5 a′ 1305 1305 1301 1300 1306.2 
ν6 a′ 1108 1108 1106 1106 1104.85 
ν7 a′ 627 627 623 623 626.17 
ν8 a″ 1034 1035 1032 1032 1033.47 
ν9 a″ 638 640 637 637 640.73 
FIG. 4.

Deviation between experimental and computed formic acid fundamental wavenumbers (δexp=ν̃calcν̃exp). Data for all four H/D isotopologs are aggregated for cis- and trans-formic acid with 16 and 42 data points, respectively. These 42 points comprise 35 trans fundamentals and their resonance partners including 2ν9 (HCOOH), 2ν8 and 2ν9 (DCOOH), ν3 + ν6 and 2ν9 (HCOOD), and ν4 + ν6 and 2ν8 (DCOOD). For the ICPH model, 8 and 10 data points are used, respectively, as ν1 of cis is not reported in Ref. 16 and deuterated data are not available. Each box extends from the lower quartile to the upper quartile of the data, and whiskers extend out to twice the interquartile range. Outliers are marked as crosses (+), and each box is bisected by a line indicating the median.

FIG. 4.

Deviation between experimental and computed formic acid fundamental wavenumbers (δexp=ν̃calcν̃exp). Data for all four H/D isotopologs are aggregated for cis- and trans-formic acid with 16 and 42 data points, respectively. These 42 points comprise 35 trans fundamentals and their resonance partners including 2ν9 (HCOOH), 2ν8 and 2ν9 (DCOOH), ν3 + ν6 and 2ν9 (HCOOD), and ν4 + ν6 and 2ν8 (DCOOD). For the ICPH model, 8 and 10 data points are used, respectively, as ν1 of cis is not reported in Ref. 16 and deuterated data are not available. Each box extends from the lower quartile to the upper quartile of the data, and whiskers extend out to twice the interquartile range. Outliers are marked as crosses (+), and each box is bisected by a line indicating the median.

Close modal

The careful reader will have noticed that in Fig. 4, data for 35 instead of 4 × 9 = 36 trans fundamentals are shown. To this day, the C–H out-of-plane vibration ν8 of HCOOD remains the only fundamental not reported in the gas phase. The anharmonic calculations on PES-2016 and PES-2018 consistently predict ν8 to be slightly shifted down by 1 cm−1–2 cm−1 upon O-D isotopic substitution. Experimental data in a weakly perturbing neon matrix indicate a similar shift in the opposite direction, however.3,10 Both overtones could be observed in this work at 2060 cm−1 (HCOOH) and 2055 cm−1 (HCOOD). The overall weak experimental band integrals and “pure” CI coefficients (>0.99) indicate 2ν8 not to be perturbed (significantly) by resonances, as is the case for DCOOH and DCOOD (cf. Table IV and Fig. S9). In the context of second order perturbation theory,57 

E(n)hc=iωĩni+12+ijxi,jni+12nj+12,
(1)

one can extract an xi,i value if one knows the transition energy of both the fundamental and first overtone of the ith mode. Using the fundamental (1033.47 cm−1 from Ref. 7) and first overtone (2060 cm−1) of trans-HCOOH, one obtains a diagonal anharmonicity constant of 2x8,8 = −7 cm−1. Assuming x8,8 to remain unchanged upon O-D isotopic substitution, the gas phase band center of trans-HCOOD can be estimated to be shifted down by 2 cm−1 relative to trans-HCOOH, in line with anharmonic predictions (Table IV).

While trans-HCOOH experimental vibrational data are available up to 14 000 cm−1 (Ref. 7) and select OH overtone data beyond,64 we restrict our discussion in this work to vibrational levels up to 4000 cm−1. On the basis of the available calculations and symmetry information from depolarized Raman spectra, we assign peaks in the Raman jet spectra reported in Refs. 40 and 44 and this work and review previously assigned IR bands. As some of the reassignments make use of depolarization ratios, we give a brief explanation of depolarization ratios and their relation to the symmetry of a vibration as note in the references.65 For further reading, see Refs. 66 and 67.

In order to motivate these assignments, we have examined a wide range of data. We have compiled available benchmark quality, i.e., perturbation-free, experimental reference data from the literature and show them together with Raman jet band centers in Table II for HCOOH; data for the deuterated isotopologs are shown in  Appendix B (Table IV). Newly assigned bands are marked with an asterisk and assignments that deviate from Refs. 16 and 17 (or the experimental reference if not discussed in either) are indicated by a dagger. (Re)Assignments are based on the combined predictions from ICPH,16 MCTDH,17,18 and CVPT6 calculations and taking into account the expected error progression for each state. We use the deviation between the experiment and calculation for a fundamental as a measure of the error associated with this vibrational degree of freedom per quantum of excitation. In the following, we discuss (re)assignments concerning trans-HCOOH and touch upon deuterated isotopologs whenever necessary for the interpretation of HCOOH bands. Convergence data and CI mixing coefficients are shown for states relevant to the discussion in  Appendix A (Table III). The Raman jet spectra in the full spectral range between 870 cm−1 and 3750 cm−1 are shown in Figs. S1–S12 of the supplementary material. The discussion of (re)assignments is sorted by energy, starting with the low-frequency end of the spectrum. The O–H stretching spectrum between 3510 cm−1 and 3630 cm−1 is discussed separately at the end.

TABLE II.

Vibrational wavenumbers (in cm−1) of trans-HCOOH states with one or two vibrational quanta. Other states are included if they are involved in resonances or assignments of experimental bands. Internal coordinate path Hamiltonian (ICPH),16 multi-configuration time-dependent Hartree (MCTDH),17 and canonical Van Vleck perturbation theory (CVPT) predictions using two different analytical PES16,17 are shown together with perturbation-free experimental data. The asterisk denotes newly assigned bands, and the dagger denotes a reassignment. Tentative assignments are in brackets, not fully converged MCTDH energies are set in parentheses (adopted from Ref. 17), and energy levels obtained from hot band assignments are italicized. Numbers in the state labels refer to the vibrational degree of freedom, which is indexed with the respective quanta of excitation.

StateExp.PES-2016aPES-2018b
LabelΓRa. jetcLit.ICPHCVPT6MCTDHCVPT6
 71 a′ 626 626.17d 627 627 623 623 
 91 a″  640.73d 638 640 637 637 
 81 a″  1033.47e 1034 1035 1032 1032 
 61 a′ 1104 1104.85e 1108 1108 1106 1106 
 92 a′ 1220 1220.83f 1222 1221 1216 1216 
 72 a′  1253.44g 1256 1256 1247 1246 
 7191 a″  1268.69g 1268 1269 1261 1260 
 51 a′ 1306 1306.2h 1305 1305 1301 1300 
 41 a′ 1379 1379.05i 1379 1380 1375 1374 
 7181 a″   1661 1661 1654 1655 
8191 a′ 1673  1672 1675 1668 1668 
† 6171 a′ 1727 1726.40j 1733 1732 1725 1725 
† 6191 a″ 1738 1737.96j 1739 1741 1736 1736 
 31 a′ 1776 1776.83j 1783 1783 1774 1773 
 93 a″  1792.63j 1795 1793 1786 1786 
 7192 a′ 1847 1847.8,h 1843.48 1855 1850 1839 1839 
 5171 a′ 1931 1931.1h 1933 1932 1922 1921 
5191 a″ 1951  1947 1950 1943 1942 
 4171 a′   2006 2006 1998 1997 
 4191 a″   2021 2024 2016 2014 
82 a′ 2060  2063 2066 2058 2060 
 6181 a″  2132h 2139 2139 2134 2135 
 
 62 a′ 2197 2196.3h 2205 2204 2200 2199 
† 94 a′ [2298] [2298.6]h 2312 2305 2294 2295 
718191 a′ [2301]  2302 2304 2291 2292 
† 6192 a′ 2336 [2338]h 2358 2338 2332 2331 
 5181 a″   2338 2338 2330 2331 
 617191 a″  [2376]h,k 2369 2368 2357 2356 
3171 a′ 2395  2402 2402 2390 2389 
† 5161 a′ 2400 2400.2h 2406 2405 2398 2396 
 3191/7193 a″   2420 2418 2404 2405 
3191/7193 a″ 2424  2447 2427 2414 2413 
 4181 a″   2415 2416 2408 2407 
 4161 a′   2479 2479 2473 2473 
 52/5192 a′ 2504 2504h 2511 2507 2498 2499 
 52/5192 a′   2608 2608 2598 2595 
 4192 a′  [2600]h 2600 2598 2589 2588 
4151 a′ 2678  2678 2678 2670 2669 
 42 a′ 2746 2745h 2746 2747 2738 2736 
 3181 a″  2803h 2810 2810 2801 2800 
 3161 a′  2876.6h 2886 2884 2875 2874 
 21 a′ 2942 2942.06h 2938 2940 2937 2939 
7194 a′ 2960  3066 2964 2950 2952 
 517181 a″   2965 2965 2951 2952 
3192 a′ 2995  3003 3002 2989 2987 
 418191 a′  [3057]h 3058 3061 3047 3046 
 3151 a′ 3081 3083l 3087 3087 3075 3073 
 416171 a′  [3106.5]h 3105 3103 3093 3092 
 416191 a″   3115 3117 3108 3105 
 3141 a′ 3153 3152.3h 3161 3160 (3086) 3145 
 63 a′  [3275]h 3292 3289 n.r. 3281 
 32 a′ 3533 3538/3533m 3547 3545 3530 3529 
 [11-Res] a′ 3559      
2171 a′ 3567  3566 3568 3558 3561 
 11 a′ 3570 3570.5h 3575 3576 3567 3568 
 2191 a″   3579 3581 n.r. 3575 
 [11-Res] a′ 3609      
† 3182 a′  3826h 3833 3835 n.r. 3821 
 4261 a′   3836 3835 n.r. 3825 
2181 a″ 3958  3952 3954 n.r. 3954 
† 3162 a′  [3963.6]h 3978 3975 n.r. 3963 
StateExp.PES-2016aPES-2018b
LabelΓRa. jetcLit.ICPHCVPT6MCTDHCVPT6
 71 a′ 626 626.17d 627 627 623 623 
 91 a″  640.73d 638 640 637 637 
 81 a″  1033.47e 1034 1035 1032 1032 
 61 a′ 1104 1104.85e 1108 1108 1106 1106 
 92 a′ 1220 1220.83f 1222 1221 1216 1216 
 72 a′  1253.44g 1256 1256 1247 1246 
 7191 a″  1268.69g 1268 1269 1261 1260 
 51 a′ 1306 1306.2h 1305 1305 1301 1300 
 41 a′ 1379 1379.05i 1379 1380 1375 1374 
 7181 a″   1661 1661 1654 1655 
8191 a′ 1673  1672 1675 1668 1668 
† 6171 a′ 1727 1726.40j 1733 1732 1725 1725 
† 6191 a″ 1738 1737.96j 1739 1741 1736 1736 
 31 a′ 1776 1776.83j 1783 1783 1774 1773 
 93 a″  1792.63j 1795 1793 1786 1786 
 7192 a′ 1847 1847.8,h 1843.48 1855 1850 1839 1839 
 5171 a′ 1931 1931.1h 1933 1932 1922 1921 
5191 a″ 1951  1947 1950 1943 1942 
 4171 a′   2006 2006 1998 1997 
 4191 a″   2021 2024 2016 2014 
82 a′ 2060  2063 2066 2058 2060 
 6181 a″  2132h 2139 2139 2134 2135 
 
 62 a′ 2197 2196.3h 2205 2204 2200 2199 
† 94 a′ [2298] [2298.6]h 2312 2305 2294 2295 
718191 a′ [2301]  2302 2304 2291 2292 
† 6192 a′ 2336 [2338]h 2358 2338 2332 2331 
 5181 a″   2338 2338 2330 2331 
 617191 a″  [2376]h,k 2369 2368 2357 2356 
3171 a′ 2395  2402 2402 2390 2389 
† 5161 a′ 2400 2400.2h 2406 2405 2398 2396 
 3191/7193 a″   2420 2418 2404 2405 
3191/7193 a″ 2424  2447 2427 2414 2413 
 4181 a″   2415 2416 2408 2407 
 4161 a′   2479 2479 2473 2473 
 52/5192 a′ 2504 2504h 2511 2507 2498 2499 
 52/5192 a′   2608 2608 2598 2595 
 4192 a′  [2600]h 2600 2598 2589 2588 
4151 a′ 2678  2678 2678 2670 2669 
 42 a′ 2746 2745h 2746 2747 2738 2736 
 3181 a″  2803h 2810 2810 2801 2800 
 3161 a′  2876.6h 2886 2884 2875 2874 
 21 a′ 2942 2942.06h 2938 2940 2937 2939 
7194 a′ 2960  3066 2964 2950 2952 
 517181 a″   2965 2965 2951 2952 
3192 a′ 2995  3003 3002 2989 2987 
 418191 a′  [3057]h 3058 3061 3047 3046 
 3151 a′ 3081 3083l 3087 3087 3075 3073 
 416171 a′  [3106.5]h 3105 3103 3093 3092 
 416191 a″   3115 3117 3108 3105 
 3141 a′ 3153 3152.3h 3161 3160 (3086) 3145 
 63 a′  [3275]h 3292 3289 n.r. 3281 
 32 a′ 3533 3538/3533m 3547 3545 3530 3529 
 [11-Res] a′ 3559      
2171 a′ 3567  3566 3568 3558 3561 
 11 a′ 3570 3570.5h 3575 3576 3567 3568 
 2191 a″   3579 3581 n.r. 3575 
 [11-Res] a′ 3609      
† 3182 a′  3826h 3833 3835 n.r. 3821 
 4261 a′   3836 3835 n.r. 3825 
2181 a″ 3958  3952 3954 n.r. 3954 
† 3162 a′  [3963.6]h 3978 3975 n.r. 3963 
a

Reference 16, taken from the respective supplementary material.

b

Reference 17.

c

Fundamentals and resonance partners from Ref. 40 and Raman peak at 2960 cm−1 from Ref. 44; otherwise, this work.

d

Reference 20.

e

Reference 60.

f

Reference 61.

g

Computed from parameters reported in Ref. 8.

h

Reference 7.

i

Reference 62.

j

Reference 9.

k

Probably not corresponding to a ground state trans-HCOOH transition; see the text for discussion.

l

Reference 42.

m

Reference 63.

The first reassignment in Table II, as indicated by the daggers (†), concerns the bands at 1727 cm−1 and 1738 cm−1. These bands are assigned as the combination bands ν6 + ν7 and ν6 + ν9, respectively. One might expect the latter to be higher in energy since ν7 lies below ν9. However, Perrin et al. analyzed several interacting states in the ν3 region by high-resolution IR spectroscopy and fitted ro-vibrational lines with an effective Hamiltonian. They reported (among others) band centers at 1726.40 cm−1 and 1737.96 cm−1 for ν6 + ν9 and ν6 + ν7, respectively.9 This “inverse” order was attributed to resonance interactions with ν3 shifting ν6 + ν7 above ν6 + ν9; the latter can only interact via Coriolis couplings, while the former can additionally interact via Fermi couplings. In contrast, all four calculations in Table II indicate that the naive assignment based on the order of fundamentals is correct as was recognized by Richter and Carbonnière. They had proposed a reassignment based on improved root-mean-square deviations among both ICPH and MCTDH calculations.17 Moreover, we find that the above postulated Fermi coupling is small, being on the order of 1 cm−1, this providing further evidence that the lower band should be assigned to ν6 + ν7.

Coincidentally, in the Raman jet spectrum of HCOOH, peaks at 1737 cm−1 and (much weaker) at 1726 cm−1 are observed, and both correspond to a′ transitions. The agreement with the band assigned to ν6 + ν7 by Perrin et al., however, is coincidental, as the Raman signal at 1737 cm−1 can be ascribed to ν3 of H13COOH due to the naturally occurring 13C isotope. The isotope shift was measured in the gas phase23 and weakly perturbing neon matrix,3 amounting to −40 cm−1, in perfect agreement with the observed −39 cm−1. The corresponding 13C peak can also be identified in the HCOOD spectrum at 1733 cm−1, shifted by −39 cm−1. Using CVPT6 predictions, we can furthermore identify one band of the ν3/2ν8 doublet for C-deuterated and even ν6 for O-deuterated isotopologs of formic acid-13C. Band centers are tabulated in Table S3 of the supplementary material

For further confirmation that the ro-vibrational lines observed by Perrin et al., that correspond to a band center at 1726 cm−1, should be assigned to ν6 + ν7, we turn to hot bands involving the two energy levels in question. Although we cannot observe ν6 + ν9 directly in the Raman spectrum of HCOOH, we can calculate hot bands of ν6 originating from the two (by far) lowest energy levels ν7 and ν9 using experimental data and compare those results with the observed hot bands in Fig. 5. With the assignment by Perrin et al., we expect one hot band to be blue-shifted (+6.94 cm−1) and one red-shifted (−19.18 cm−1) relative to the ν6 fundamental. In the Raman spectrum, three distinctly separated hot bands can be found, shifted from the fundamental by −3 cm−1, −7 cm−1, and −11 cm−1—all three red-shifted from ν6—where the latter was assigned to ν6 of cis-HCOOH by Meyer and Suhm.11,40 An assignment of ν6 + ν7 and ν6 + ν9 inverse to Perrin et al., however, would yield shifts of −4.62 cm−1 and −7.62 cm−1, which are compatible with the observed hot band structure as was recognized by Meyer44 and which is consistent with all anharmonic predictions. We therefore confirm the reassignment proposed by Richter and Carbonnière.

FIG. 5.

Calculated hot band progressions in the vicinity of the C–O stretch ν6 are compared to the Raman jet spectra of formic acid (acid-in-helium concentration <0.2%–0.4%). Wavenumber ticks are separated by 5 cm−1. To compare the relative hot band intensities, the limits on the y-axis are set to 1/3 of the maximum intensity of ν6 for each isotopolog. The width of the Gaussian functions is set to σ=0.5 to roughly match the experiment. See the text for details on computed intensities.

FIG. 5.

Calculated hot band progressions in the vicinity of the C–O stretch ν6 are compared to the Raman jet spectra of formic acid (acid-in-helium concentration <0.2%–0.4%). Wavenumber ticks are separated by 5 cm−1. To compare the relative hot band intensities, the limits on the y-axis are set to 1/3 of the maximum intensity of ν6 for each isotopolog. The width of the Gaussian functions is set to σ=0.5 to roughly match the experiment. See the text for details on computed intensities.

Close modal

Contrary to HCOOH, the hot band structure around the C–O stretching fundamental ν6 is much more complicated for DCOOH, featuring much less intensity but an overall higher density of peaks and a blue-shifted hot band. In the absence of anharmonic Raman intensities, we simulate the ν6 spectrum using a simple model assigning a linear transition term to the C–O stretch. The experimental and simulated spectra using CVPT6/PES-2016 anharmonic energies are compared for all four H/D isotopologs in Fig. 5. We scale the trans fundamental with 3/20 to reproduce the relative experimental intensities and shift the wavenumber axis to match experiment. For the cis fundamental, the scaled transition oscillator strength of the trans fundamental is multiplied with the appropriate Boltzmann factor and the relative cistrans Raman intensity factor from Ref. 40 (ESI, Table S3).

Considering the crudeness of the model, the simulated spectra predict the relative intensities and band contours between different isotopologs surprisingly well. Note that the choice of width for the Gaussian functions and the underestimated redshift for ν6 + ν7ν7 (simulation −2 cm−1, experiment −4 cm−1) causes the hot band for HCOOH to overlap with the fundamental peak in Fig. 5. Hot bands originating from ν7 and ν9 overlap for O-deuterated formic acid and an accidental resonance between ν6 + ν9 and 3ν9 causes ν6 + ν9ν9 to be shifted above the fundamental for DCOOH. With a refined model including higher (anharmonic) terms in the polarizability tensor, it seems possible to assign ν6(cis) for all three deuterated isotopologs in the Raman jet spectra shown in Fig. 5.

From the above mentioned ro-vibrational analysis of ν3, Perrin et al. obtained band centers for several interacting states by simultaneously fitting the interacting vibrational states ν3, ν6 + ν7, ν6 + ν9, 3ν9, and ν7 + 2ν9 (the latter two originally labeled ν5 + ν9 and ν5 + ν7, respectively).9 We find that our Raman band centers for the ν5 + ν7/ν7 + 2ν9 Fermi doublet at 1931 cm−1 and 1847 cm−1 agree well with band centers reported by Freytes et al. at 1931.1 cm−1 and 1847.8 cm−1 (originally assigned to 3ν7 and ν5 + ν7, respectively), which were also determined from the respective Q branches.7 The band center that Perrin et al. reported at 1843.48 cm−1,9 however, significantly deviates from the high-resolution IR value reported by Freytes et al. This discrepancy may be resolved by updating the state assignments and therefore the form of the off-diagonal coupling elements in the fit Hamiltonian.

So far, the discussion has focused on the assignment of peaks that are observed in the Raman jet spectrum. Looking at Table II, it is surprising that while the two-quantum states ν5 + ν7 (1931 cm−1) and 2ν8 (2060 cm−1) are observed, the state ν4 + ν7, which is predicted to lie between the two, is not. Following the method by Dübal and Quack,59,68 we analyze nearby a′ symmetric states for the CI contribution of the ν3 fundamental to the respective state, assuming that “dark” states gain intensity via resonance mixing with ν3. The CVPT6/PES-2016 calculation predicts contributions of ν3 to seven other states with squared CI coefficients of 0.87% (ν7 + 2ν9), 0.31% (2ν8), 0.30% (ν5 + ν7), 0.16% (ν8 + ν9), 0.09% (ν6 + ν7), 0.02% (3ν7), and 0.01% (ν4 + ν7). This simple model satisfactorily explains the absence of ν4 + ν7, 3ν7, and the very weak intensity of ν6 + ν7. In analogy, it further illustrates the absence of 2ν7, which has a total contribution below 0.1% from the much weaker (relative to ν3) fundamentals ν4, ν5, and ν6, and the general lack of a″ symmetric combination/overtones (cf. Tables II and IV). The (un)observed peak structure of the Raman jet spectrum of trans-HCOOH up to 2150 cm−1 can entirely be understood within this simple model where “bright” fundamentals light up nearby dark states via (weak) resonance mixing.

Segments of the HCOOH and DCOOH Raman jet spectra between 2210 cm−1 and 2450 cm−1 are shown together with CVPT6 predictions on both surfaces in Fig. 6 to aid the following discussion. In this spectral window, four HCOOH bands are reported by Freytes et al. at 2400.2 cm−1, 2376 cm−1, 2338 cm−1, and 2298.6 cm−1. We propose reassignments in all four cases.

FIG. 6.

Raman jet spectra of HCOOH and DCOOH in helium (<0.3% and <0.4%, respectively) between 2210 cm−1 and 2450 cm−1. For each isotopolog, the spectra of a temperature series have been intensity-scaled to a trans fundamental (for HCOOH not shown) with the lowest intensity among all temperatures. Isotopic impurities are marked with a double dagger (‡). For DCOOH, the 160 °C spectrum is additionally shown further intensity-scaled (×0.05) in gray. Additional depolarized spectra at 190 °C are shown with the incident laser polarization perpendicular (⊥, black, default for all other measurements) and parallel (∥, cyan) with respect to the scattering plane. CVPT6 predictions using PES-2016 (▴) and PES-2018 (■) are shown as gray sticks for states of a′ symmetry between 2230 cm−1 and 2450 cm−1.

FIG. 6.

Raman jet spectra of HCOOH and DCOOH in helium (<0.3% and <0.4%, respectively) between 2210 cm−1 and 2450 cm−1. For each isotopolog, the spectra of a temperature series have been intensity-scaled to a trans fundamental (for HCOOH not shown) with the lowest intensity among all temperatures. Isotopic impurities are marked with a double dagger (‡). For DCOOH, the 160 °C spectrum is additionally shown further intensity-scaled (×0.05) in gray. Additional depolarized spectra at 190 °C are shown with the incident laser polarization perpendicular (⊥, black, default for all other measurements) and parallel (∥, cyan) with respect to the scattering plane. CVPT6 predictions using PES-2016 (▴) and PES-2018 (■) are shown as gray sticks for states of a′ symmetry between 2230 cm−1 and 2450 cm−1.

Close modal

The initial assignment of the band at 2400.2 cm−1 to 2ν5 by Freytes et al. was rejected by Tew and Mizukami as well as Richter and Carbonnière on the grounds that their calculations predict 2ν5 at much higher energies.16,17 Richter and Carbonnière pointed out that taking into account the expected error progression of the two possible candidates ν3 + ν7 and ν5 + ν6, the latter is a better match since the former is expected at slightly lower energies around 2395 cm−1–2396 cm−1. We can confirm this reassignment as we observe two bands in our Raman spectra at 2400 cm−1 and 2395 cm−1, which we assign to ν5 + ν6 and ν3 + ν7, respectively.

By re-visiting early IR studies on formic acid with high quality anharmonic calculations, we are also able to assign ν5 + ν6 for trans-HCOOD. In one of the first spectroscopic investigations of deuterated formic acid monomer, Williams reported two bands of trans-HCOOD at 2178.8 cm−1 and 2142.4 cm−1, which he could not assign.69 Nearly 70 years later, these bands remain unassigned, as far as we know. Using the anharmonic predictions on both PESs (Tables III and IV), the lower energy band can either be assigned to ν5 + ν6 or 2ν7 + ν8 and the higher energy band to ν6 + 2ν9 or 3ν7 + ν9, where ν5 + ν6 and ν6 + 2ν9 are predicted to be strongly coupled on the basis of their respective CI coefficients. The best energy matches for the bands at 2142.4 cm−1 and 2178.8 cm−1, and further confirmed by correction for expected error progression, are ν5 + ν6 and ν6 + 2ν9, respectively. ν5 + ν6 of trans-HCOOD was reported by Marushkevich et al. at 2139.8 cm−1 and 2176.8 cm−1 in neon and 2144.1 cm−1 and 2181.0 cm−1 in argon matrices—the similarity to the gas phase band centers introduces the possibility of interpreting the matrix data as the Fermi pair ν5 + ν6 and ν6 + 2ν9, as opposed to a matrix site effect.70 

The band at 2376 cm−1 was initially assigned by Freytes et al.7 to ν3 + ν7 and tentatively reassigned by Tew and Mizukami to ν6 + ν7 + ν9.16 Since ν3 + ν7 is assigned to 2395 cm−1, this leaves ν6 + ν7 + ν9 as the only plausible assignment considering that the energetically next higher state is predicted to be greater than 2410 cm−1. Even taking into account expected error progressions, the predictions are still 9 cm−1–14 cm−1 below experiment. Based on the remarkable agreement with experiment for states up to three and four quanta, and as we shall see below, even for states involving resonances, this assignment seems unlikely, and we believe that this band does not correspond to a ground state trans-HCOOH transition.

A critical reassignment concerns the two IR bands at 2338 cm−1 and 2298.6 cm−1.7 Freytes et al. assigned the latter tentatively to ν5 + ν6, which we have already reassigned to 2400.2 cm−1. Tew and Mizukami reassigned both bands to ν5 + ν8 (a″) and ν7 + ν8 + ν9 (a′),16 respectively; Richter and Carbonnière noted that in light of their MCTDH results, alternative reassignments to 4ν9 (a′) and ν6 + 2ν9 (a′) seem plausible.17 In the Raman spectrum of trans-HCOOH (Fig. 6, top), we observe three peaks at 2336 cm−1, 2301 cm−1, and 2298 cm−1, all of which correspond to totally symmetric a′ transitions. This yields three or four bands in this spectral region, depending on whether the band Freytes et al. observed at 2338 cm−1 corresponds with the signal we observe at 2336 cm−1—the deviation of 2 cm−1 is just within our experimental resolution of ±2 cm−1. The ICPH calculation cannot explain the a′ band at 2336 cm−1, since the “best match” ν5 + ν8 can be ruled out by symmetry considerations. The energetically next possible candidates are 4ν9 at 2358 cm−1 and ν6 + 2ν9 at 2312 cm−1, which (even after expected error correction) are energetically too far away to plausibly be assigned to the Raman peak at 2336 cm−1. Looking at Table II, it can be seen that the MCTDH predictions using PES-2018 are compatible with the CVPT results on PES-2016; CVPT and MCTDH predict the two aforementioned states 4ν9 and ν6 + 2ν9 to be strongly coupled where the higher energy band is predicted between 2331 cm−1 and 2338 cm−1 and the lower energy band between 2294 cm−1 and 2305 cm−1.

We can gain further insights by turning to the Raman spectrum of DCOOH, as the CVPT calculations indicate that this resonance persists upon C–H deuteration. In the same spectral region, we observe five a′ bands at 2428 cm−1, 2342 cm−1, 2290 cm−1, 2277 cm−1, and 2271 cm−1. The spectrum is shown in Fig. 6 alongside CVPT6 predictions in this spectral region. The former peak can be identified as ν5 + ν6, which is shifted toward higher energies upon C–H deuteration, as expected from the sum of fundamentals. The second last and most intense band is identified as 2ν6. According to the CVPT results, this state has the largest CI overlap with ν2 (cf. Table III) and therefore is expected to gain the most intensity via Fermi resonance mixing. Moreover, from the data in Table IV, we observe that if one uses the difference between the CVPT6 and experimental fundamental values for ν6 to adjust the CVPT6 energy of the overtone, PES-2016 transitions yield 2282.2–2 × (2.4) = 2277.4 cm−1. The similarly calculated value for the PES-2018 is 2272.3 + 2 × (3.1) = 2278.5 cm−1, both being in excellent agreement with experiment. The two signals in the vicinity of 2ν6 can then straightforwardly be assigned to ν4 + ν5 (2271 cm−1) and 4ν9 (2290 cm−1) as only a′ states within ±50 cm−1 (Fig. 6). The calculations allow two possible assignments for the remaining signal at 2342 cm−1; ν3 + ν7 or ν6 + 2ν9. Due to the strong resonance between ν3 and 2ν8 in C-deuterated formic acid, combination bands in the Raman spectrum associated with this polyad are always observed as doublets (cf. Table IV). The absence of a second signal therefore rules out ν3 + ν7 as possible assignment. In this context, we note that the Raman band center for 2ν6 (2277 cm−1) significantly deviates from the high-resolution IR value reported by Tan et al. (2254.24 cm−1).71 A likely explanation is the missing Fermi resonance coupling between ν2 and 2ν6, as Tan et al. included only Coriolis-type coupling in the effective fit Hamiltonian (cf. Ref. 71).

With ν6 + 2ν9 assigned to 2342 cm−1 and 4ν9 to 2290 cm−1 for DCOOH, we are now able to assign the remaining HCOOH Raman signals at 2336 cm−1, 2301 cm−1, and 2298 cm−1. We assign the higher energy band to ν6 + 2ν9 and the two remaining bands to 4ν9 and ν7 + ν8 + ν9—the order depending on whether the band observed by Freytes et al. at 2338 cm−1 corresponds to the Raman band at 2336 cm−1 or is indeed ν5 + ν8. Since ν6 has a very strong IR oscillator strength,72 it seems plausible that combination bands involving ν6 are not completely “dark” states, as 2ν6, ν6 + ν8, and ν5 + ν6 are all observed by Freytes et al. (cf. experimental intensities reported in Table II of Ref. 7). It is therefore more convincing to assign the two IR bands at 2338 cm−1 and 2298.6 cm−1 to ν6 + 2ν9 and 4ν9, respectively. The resulting deviation of −2 cm−1 (2338 cm−1 IR to 2336 cm−1 Raman) in turn corroborates the assignment of the Raman signal at 2298 cm−1 to 4ν9.

A second although experimentally less certain indication for discrepancies between ICPH and CVPT/MCTDH predictions for higher excited torsional trans states comes from a blue-shifted hot band of the ν3 fundamental of trans-HCOOH. We observe four distinct peaks at 1783 cm−1, 1770 cm−1, 1763 cm−1, and 1757 cm−1, which are shifted by +7 cm−1, −6 cm−1, −13 cm−1, and −19 cm−1 relative to the ν3 fundamental at 1776 cm−1 (see Fig. S9 of the supplementary material). Increased intensity between 1770 cm−1 and 1776 cm−1 indicates other hot contributions where no Q branch can be identified due to the spectral congestion. Focusing first to the down-shifted hot bands, Meyer and Suhm assigned the hot band at 1770 cm−1 to ν3 + ν7ν7, which is consistent with our assignment of ν3 + ν7 to 2395 cm−1.11 We assign the weakest band shifted by −19 cm−1 to 2ν3ν3, in agreement with 2ν3 band centers from FTIR jet63 and helium nanodroplet73,74 measurements and anharmonic calculations that predict a large redshift of 17 cm−1–21 cm−1. Meyer and Suhm were not able to assign the blue-shifted hot band since all DFT VPT2 anharmonicity constants x3,j were negative, predicting exclusively redshifts. Upon O-D isotopic substitution (Fig. S9), we no longer observe a blue-shifted hot band of ν3. This absence could be an indication of a Fermi resonance between a dark state (involving O–H motion) and a combination state involving ν3, based on our previous discussion of a blue-shifted hot band of ν6 in the Raman spectrum of DCOOH. The ICPH and CVPT calculations on PES-2016 indeed predict strong mixing between ν3 + ν7 and ν5 + ν6. Since an experimental band center is available for ν5 + ν6, we can directly compute the hot transition ν5 + ν6ν7, which is shifted by −3 cm−1 and can therefore not explain the observed blue-shifted band. The MCTDH calculation predicts strong mixing between ν7 + 3ν9 and ν3 + ν9, as does CVPT6 on both surfaces. CVPT6/PES-2016 predicts ν4 + ν8 to be involved in this resonance most likely due to the higher energy associated with ν3. The CI coefficients do not allow us to unambiguously assign ν3 + ν9 and ν7 + 3ν9, and we therefore refer to them as “higher” and “lower” energy state. We obtain predicted shifts of +(3–5) cm−1 and −(4–7) cm−1 for hot transitions originating from ν9 into the higher and lower energy state, respectively. The blue-shifted prediction is therefore compatible with the experimental data where the lower energy hot band could be assigned to the increased intensity between 1776 cm−1 and 1770 cm−1. This mixing is not predicted by the ICPH calculation, which predicts shifts of +27 cm−1 and 0 cm−1.

The previously discussed assignments concerning 4ν9 and possibly ν7 + 3ν9 illustrate the difficulty of high level variational calculations that explicitly take into account wavefunction delocalization effects associated with the necessity to converge more and higher energy eigenvalues compared to single-reference methods.16,75 In the unified treatment of cis- and trans-formic acid in the ICPH model, cis states correspond to higher (torsional) excitations of the global minimum trans isomer. In that treatment, 4ν9 of trans corresponds to 9 with n = 6, as n = 4, 5 correspond to the ground and torsionally first excited state of cis-HCOOH.16 In light of the not fully converged cis fundamentals discussed in Sec. III B, it appears plausible that the ICPH eigenstates for 4ν9 and ν7 + 3ν9 are not fully converged either—mind that the eigenvalues are above, not below the experimental value—and extension of the vibrational basis set is expected to converge these higher excited torsional ICPH eigenvalues toward experiment.

The assignment of the band at 2504 cm−1 is particularly interesting as it was previously reassigned to the nearly degenerate Fermi doublet 2ν5/ν5 + 2ν9, which is predicted around 2500 cm−1 and 2600 cm−1. In this spectral window, Freytes et al. reported two bands at 2504 cm−1 and 2600 cm−1, respectively.7 In the Raman jet spectra, we observe a very weak signal at 2504 cm−1, which nicely matches the value reported by Freytes et al., nothing around 2600 cm−1, and then again a band at 2678 cm−1 (Figs. S5 and S6 in the supplementary material). With the aid of the anharmonic calculations, the HCOOH peak at 2678 cm−1 can unambiguously be assigned to ν4 + ν5 (Table III). Its resonance partner, ν4 + 2ν9, is also predicted around 2600 cm−1, complicating the assignment of the IR band at 2600 cm−1.

In order to get a better understanding of these couplings, we examine the CVPT6 results in more detail. Inspection of the CVPT6 Hamiltonian in this energetic region reveals an interesting situation of different ν5/2ν9 polyads intersecting that we now explore. The presence of significant mixing between states sharing the polyad quantum number Np = n5 + n9/2, which has previously been analyzed for Np = 1,12,16,17 makes it challenging to identify additional resonant interactions between zero-order states. One way to circumvent this issue is to examine the couplings in the polyad eigenstate representation. To obtain this representation, we subsume into the zero-order CVPT6 effective Hamiltonian all the couplings between states sharing common {n1, …, n4, n6, …, n8, Np} quantum numbers. The Hamiltonian matrix of select states in this representation, which is shown in the supplementary material, is visualized in Fig. 7. The states are labeled with {n4, n6, Np} quantum numbers; the blue points correspond to Np = 2, the red and green points correspond to Np = 1, and the remaining states have Np = 0. The y-axis represents the energy difference between the polyad eigenvalues and either those of the 10 × 10 matrix or the fully coupled CVPT6 eigenvalues. The smaller this distance, the better the polyad representation. States within each set of common quantum numbers, i.e., {1, 0, 1}, {0, 1, 1}, and {0, 0, 2}, are uncoupled due to the choice of the zero-order Hamiltonian.

FIG. 7.

Eigenvalue differences between the O–H bend/torsion polyad Hamiltonian and both the fully coupled 10-state Hamiltonian (squares) and the full CVPT6 Hamiltonian (crosses) plotted against the full CVPT6 energy for trans-HCOOH. The states of the 10-state Hamiltonian satisfy the condition n4 + n6 + Np = 2, where Np = n5 + n9/2 is the O–H bend/torsion polyad.

FIG. 7.

Eigenvalue differences between the O–H bend/torsion polyad Hamiltonian and both the fully coupled 10-state Hamiltonian (squares) and the full CVPT6 Hamiltonian (crosses) plotted against the full CVPT6 energy for trans-HCOOH. The states of the 10-state Hamiltonian satisfy the condition n4 + n6 + Np = 2, where Np = n5 + n9/2 is the O–H bend/torsion polyad.

Close modal

There are three important results of this representation. First, the diagonal elements of the matrix match reasonably well with the eigenvalues of the 10 × 10 matrix, with only two exceptions. The exceptions, indicated by the double headed arrow, result from an isolated resonance between the lowest energy states of the blue and red polyads. The cubic terms responsible for this interaction are the same as those that couple 2v9 to v6 in the original normal mode representation that was discussed earlier. We conclude that the current representation is an excellent basis for describing the coupling. Second, although not shown, we note that the interpolyad couplings are substantially different from those of the original harmonic oscillator representation. Finally, the similarities (±1 cm−1) between CVPT6 eigenvalues of the full and effective 10 × 10 Hamiltonian indicate that there is only minor coupling between the states shown here and all other states. There is one exception: the C–H bending overtone 2ν4 ({2, 0, 0}) is missing contributions from the C–H stretch (ν2). Upon inclusion of ν2 into the effective Hamiltonian, the {2, 0, 0} eigenvalue agrees to within 1 cm−1 with the CVPT6 value.

Beside 2ν4, there are other nearby states that seem to interact with ν2, as reflected in the CI coefficients of CVPT6 states. We now turn our attention to those peaks. In the vicinity of ±300 cm−1 around the ν2 band center, we observe overall five peaks corresponding to cold formic acid monomer (Fig. 8, top). Based on symmetry information (all a′) and the calculated anharmonic energies, the assignments are straightforward (cf. Table II). It is instructive to return to the bright state picture, which offers a more intuitive explanation for the peak structure in the C–H stretching spectrum. The computed CVPT6/PES-2016 eigenvalues plotted alongside the spectrum in Fig. 8 show that all observed trans peaks correspond to eigenstates with significant contributions from ν2. The bright state model further predicts three more Raman signals to be observable, which are anticipated at 2934 cm−1 (2ν7 + ν8 + ν9), 2934 cm−1 (ν6 + ν7 + 2ν9), and 2964 cm−1 (ν7 + 4ν9). These states interact strongly among each other and one can easily recognize their connection to the interacting states ν7 + ν8 + ν9, ν6 + 2ν9, and 4ν9 of a similar polyad, which was discussed in Sec. IV B. Meyer was able to show that two peaks at 2960 cm−1 and 2928 cm−1 are hidden underneath the rotational contour of ν2, which she could completely remove by subtracting two Raman spectra with different incident laser polarizations from each other (Fig. 8, bottom).44 Since ν2 of H13COOH is shifted down by 11.3 cm−1 in the gas phase,23 the latter peak can most likely be assigned to the 13C isotopolog, whereas 2ν7 + ν8 + ν9 and ν6 + ν7 + 2ν9 are probably hidden under nearby hot bands. As the only a′ symmetric state between 2940 cm−1 and 2980 cm−1 (Table III), we assign the Raman signal at 2960 cm−1 to ν7 + 4ν9.

FIG. 8.

Raman jet spectrum of HCOOH in helium (<0.1%) between 2670 cm−1 and 3170 cm−1 with the incident laser polarization perpendicular with respect to the scattering plane (⊥, top) and residual after subtracting 7/6 of the spectrum obtained with polarization parallel to the scattering plane (⊥−7/6∥, bottom). No peaks are observed in omitted intermediate wavenumber intervals (cf. full spectral range in the supplementary material). The spectra are intensity-scaled to ν2 of trans-HCOOH with the lowest intensity among all temperatures. Clusters are marked with an asterisk, and non-isomeric and isomeric, i.e., cis, hot bands are labeled “h” and “c,” respectively. Isotopic impurities are marked with a double dagger (‡). CVPT6/PES-2016 predictions are shown as gray lines, scaled by the squared contributions from ν2 to the respective eigenstate, and states with zero overlap are therefore not visible. The spectra were partly shown in Fig. 4.8 of Ref. 44 and are kindly provided by the author.

FIG. 8.

Raman jet spectrum of HCOOH in helium (<0.1%) between 2670 cm−1 and 3170 cm−1 with the incident laser polarization perpendicular with respect to the scattering plane (⊥, top) and residual after subtracting 7/6 of the spectrum obtained with polarization parallel to the scattering plane (⊥−7/6∥, bottom). No peaks are observed in omitted intermediate wavenumber intervals (cf. full spectral range in the supplementary material). The spectra are intensity-scaled to ν2 of trans-HCOOH with the lowest intensity among all temperatures. Clusters are marked with an asterisk, and non-isomeric and isomeric, i.e., cis, hot bands are labeled “h” and “c,” respectively. Isotopic impurities are marked with a double dagger (‡). CVPT6/PES-2016 predictions are shown as gray lines, scaled by the squared contributions from ν2 to the respective eigenstate, and states with zero overlap are therefore not visible. The spectra were partly shown in Fig. 4.8 of Ref. 44 and are kindly provided by the author.

Close modal

There is evidence in the Raman jet spectrum of DCOOH for a similar combination band involving 4ν9. At 3352 cm−1, 3404 cm−1, and 3421 cm−1, three peaks are observed, corresponding to cold monomer transitions (Fig. S2). Depolarized measurements reveal the former to be a totally symmetric transition. The overall weak intensity of the latter two does not allow any conclusions; however, it seems plausible to assume the same as we have no evidence of any observable a″ symmetric combination or overtone peak in the Raman jet spectrum of any isotopolog. The CVPT6 calculations allow two different assignments of the signal at 3352 cm−1; ν2 + ν6 or ν3 + ν4 + ν7. The former assignment seems more plausible, as ν2 + ν6 is predicted to significantly mix with ν6 + 4ν9 and 3ν6, which are predicted at 3352 cm−1, 3408 cm−1, and 3424 cm−1, respectively, accounting for the other two weak peaks observed at 3404 cm−1 and 3421 cm−1.

In light of the newly assigned bands, such as 4ν9, ν6 + 2ν9, and ν5 + 2ν9/2ν5 paired with mixing coefficients from CVPT calculations, a consistent picture emerges, which indicates that, while the interaction between ν5 and 2ν9 remains strong, additional coupling to ν6 becomes pronounced above 2200 cm−1 and beyond. The available experimental data for torsional states of trans-formic acid are graphically summarized in Fig. 9. The clumps of nearly degenerate states that are observed in this figure, and that are most evident for trans-DCOOD, are conveniently described as nearly degenerate zero-order coupled states known as polyads.

FIG. 9.

Energy levels (relative to the ground state) of the O–H/D torsion 9 up to and including n = 4 for all four H/D isotopologs of trans-formic acid computed using CVPT6 on PES-2016. As previously discussed (Fig. 7), states above 2200 cm−1 are strongly coupled via the O–H torsion. Pairs of energy levels where the CI coefficients do not allow for a one-to-one mapping are highlighted in green. Arrows mark transitions that have been observed in gas phase and jet studies by Raman and IR spectroscopy (this work and compiled from the literature4,7,9,34,40,41,61), discriminating between cold transitions (blue), i.e., from the vibrational ground state, and hot transitions from vibrationally excited states (red).

FIG. 9.

Energy levels (relative to the ground state) of the O–H/D torsion 9 up to and including n = 4 for all four H/D isotopologs of trans-formic acid computed using CVPT6 on PES-2016. As previously discussed (Fig. 7), states above 2200 cm−1 are strongly coupled via the O–H torsion. Pairs of energy levels where the CI coefficients do not allow for a one-to-one mapping are highlighted in green. Arrows mark transitions that have been observed in gas phase and jet studies by Raman and IR spectroscopy (this work and compiled from the literature4,7,9,34,40,41,61), discriminating between cold transitions (blue), i.e., from the vibrational ground state, and hot transitions from vibrationally excited states (red).

Close modal

Before separately discussing the O–H stretching spectrum of formic acid, we finally review five IR assignments of trans-HCOOH concerning bands at 3057 cm−1, 3106.5 cm−1, 3275 cm−1, 3826 cm−1, and 3963.6 cm−1 previously reported by Freytes et al.7 

The former two bands are only tentatively assigned by Tew and Mizukami to ν4 + ν8 + ν9 (a″) and ν4 + ν6 + ν7 (a′), respectively.16 In light of all four anharmonic calculations, these assignments are plausible. However, in each case, there is a state of the opposite symmetry that seems equally plausible in terms of energetic matching. Neither band is observed in the Raman jet spectrum of HCOOH, which would otherwise help resolve both assignments with additional depolarization information.

The assignment by Freytes et al. regarding the band at 3275 cm−1 (3ν6) was tentatively adopted by Tew and Mizukami. From the perspective of the anharmonic calculations, this assignment is plausible but not unambiguous. There are several other nearby states with 4–6 quanta of excitation. However, in light of the high IR activity of the C–O stretch ν6 and the apparent variety of observed IR combination/overtone bands that correspond to vibrations associated with C–O stretching motion (cf. Table II), this assignment seems more plausible than any alternative, so we retain it as tentative.

Freytes et al. reported an IR band at 3826 cm−1, which they tentatively assigned to 2ν4 + ν6.7 Tew and Mizukami adopted this tentative assignment, and it was not further discussed by Richter and Carbonnière.16,17 On the basis of the ICPH and CVPT6 calculations, two assignments seem plausible: 2ν4 + ν6 and ν3 + 2ν8. These states are even isoenergetic in one of the calculations (Table II). Taking into account the expected error progression, however, 2ν4 + ν6 is expected at slightly higher energies 3831 cm−1–3835 cm−1, whereas ν3 + 2ν8 is uniformly expected at 3826 cm−1–3827 cm−1—in excellent agreement with the observed 3826 cm−1.

The last reassignment concerns the band observed at 3963.6 cm−1.7 Freytes et al. originally assigned it to ν2 + ν8, which Tew and Mizukami tentatively adopted.16 As before, this assignment can be checked by comparing it to hot bands that involve this state. Meyer and Suhm assigned a hot band in the Raman jet spectrum of HCOOH at 2925 cm−1 to ν2 + ν8ν8 (cf. Fig. 8) based on intensity arguments from the expected Boltzmann population of ν8 and off-diagonal anharmonicity constants from DFT VPT2 calculations.11 Their assignment is a much better match with the available anharmonic calculations. Using the literature value for ν8 (1033.47 cm−1 from Ref. 7), we obtain a band center for ν2 + ν8 at 3958 cm−1, which nicely matches the error-corrected predictions between 3955 cm−1 and 3958 cm−1. Table III shows that for the band at 3963.6 cm−1, two assignments are possible; ν3 + 2ν6 and a member of a set of strongly interacting states, including ν4, ν5, ν7, and ν9. In analogy to the previous discussion regarding 3ν6, we tentatively assign the band to ν3 + 2ν6, as both CO stretches are the two most IR active fundamentals of formic acid.7 

Complex vibrational spectra of carboxylic acids in the O–H stretching range are usually associated with their cyclic dimers that form two nearly unstrained O–H⋯O hydrogen bonds76,77 and not their monomers. However, even for monomeric HCOOH, perturbations by skeletal modes seem to complicate the interpretation of the O–H stretching fundamental (ν1). This complexity lead Hurtmans et al. to postpone the ro-vibrational analysis of the high-resolution gas phase spectrum of ν1, which “is extremely dense.”6 Shortly after, in 2002, Madeja et al. used helium nanodroplets to obtain rotationally resolved spectra of the ν1 band. For trans-HCOOH, they observed perturbations of ν1 and reported three vibrational band centers at 3566.35 cm−1, 3568.63 cm−1, and 3570.66 cm−1. They attributed the presence of the two additional bands as being due to Fermi and Coriolis interactions with ν2 + ν7 and ν2 + ν9.73 Freytes et al. suggested resonance interactions of ν1 as a likely explanation for a higher density of ro-vibrational lines but overall less than the expected intensity in their room temperature IR spectra, noting similarities between the gas phase and helium nanodroplet spectrum.7 Two decades later, the ν1 gas phase high-resolution spectrum of HCOOH remains unanalyzed, to the best of our knowledge.

The Raman scattering spectrum of HCOOH, in constrast to the low-resolution FTIR jet spectrum shown alongside the Raman jet spectrum in Fig. 10, is spectrally less congested and allows the discrimination between different HCOOH bands via their respective Q branches at 3570 cm−1, 3567 cm−1, and 3559 cm−1 with an intensity ratio of 20:13:2 and a fourth much weaker band at 3609 cm−1.40ν1 can be assigned to 3570 cm−1, in agreement with the literature value.7 The excerpt of the CVPT/PES-2016 eigenvalues in Table III shows that the density of states in this spectral region is considerably large, making assignments purely based on the anharmonic predictions very difficult for the other bands. Depolarized spectra [Fig. 10, traces 1(b) and 1(c)] reveal that all three satellites are totally symmetric and C −D substitution [Fig. 10, trace 2(a)] further shows that of these three bands, two persist with only modest shifts of −2 cm−1 (3609 cm−1 HCOOH to 3607 cm−1 DCOOH) and +4 cm−1 (3559 cm−1 HCOOH to 3563 cm−1 DCOOH). The band at 3567 cm−1, which is sensitive to C-D isotopic substitution, can then straightforwardly be assigned to ν2 + ν7. The high intensity of two-thirds of ν1 suggests possible intensity stealing via Fermi resonance interaction, which in any case can only be of modest strength due to the small shift of −3 cm−1 relative to ν1; in the simplistic picture of a two level interaction, the magnitude of the off-diagonal coupling parameter represents the lower limit of the observed shift. Significant intrinsic oscillator strength can be ruled out; otherwise, ν2 + ν7 would be observed in the HCOOD spectrum, which is not the case (cf. Figs. S1 and S2 in the supplementary material). While the CI coefficients in all four calculations do not suggest significant resonance mixing between ν2 + ν7 and ν1, only slight modifications to the O–H stretching harmonic force constant to reproduce the experimental value show that this resonance is very sensitive to the energy of the O–H stretch and we obtain mixing ratios of ∼50%:50% and a shift of −2 cm−1, using CVPT6 on PES-2016.

FIG. 10.

Low-resolution FTIR jet spectrum of HCOOH in helium (0.04%) and Raman jet spectra of HCOOH and DCOOH in helium (<0.2%–0.3% and <0.4%, respectively) between 3510 cm−1 and 3630 cm−1. (a) Raman spectra of a temperature series (100 °C black, 130 °C blue, 160 °C orange, and 190 °C red) are intensity-scaled to ν1 with the lowest intensity among the four nozzle temperatures. (b) Raman spectra where the incident laser polarization is perpendicular (⊥, default in all other measurements) or parallel (∥) with respect to the scattering plane and (c) residual after subtracting 7/6×∥ from ⊥ to remove the rotational contour.67 The FTIR spectrum previously shown in Fig. 4(b) of Ref. 63 was kindly provided by the authors. Due to differing selection rules, the band shape of ν1 is distinctly different in the Raman (dominant Q branch) and IR spectrum (cf. high-resolution IR spectrum of ν1 in Fig. 3 of Ref. 6).

FIG. 10.

Low-resolution FTIR jet spectrum of HCOOH in helium (0.04%) and Raman jet spectra of HCOOH and DCOOH in helium (<0.2%–0.3% and <0.4%, respectively) between 3510 cm−1 and 3630 cm−1. (a) Raman spectra of a temperature series (100 °C black, 130 °C blue, 160 °C orange, and 190 °C red) are intensity-scaled to ν1 with the lowest intensity among the four nozzle temperatures. (b) Raman spectra where the incident laser polarization is perpendicular (⊥, default in all other measurements) or parallel (∥) with respect to the scattering plane and (c) residual after subtracting 7/6×∥ from ⊥ to remove the rotational contour.67 The FTIR spectrum previously shown in Fig. 4(b) of Ref. 63 was kindly provided by the authors. Due to differing selection rules, the band shape of ν1 is distinctly different in the Raman (dominant Q branch) and IR spectrum (cf. high-resolution IR spectrum of ν1 in Fig. 3 of Ref. 6).

Close modal

The currently available calculations do not allow us to unambiguously assign the other two satellites at 3609 cm−1 and 3559 cm−1. Experimentally, we can narrow down the list of possible resonance partners: Isotopic C-D substitution excludes the C–H stretch, in-plane bending, and out-of-plane bending vibrations ν2, ν4, and ν8. The very strong Fermi resonance between the C=O stretch ν3 and 2ν8 in DCOOH also makes ν3 an unlikely candidate. Depolarized spectra shown in Fig. 10 further exclude odd quantum numbers for the torsion 9, as all satellites are totally symmetric. We therefore conclude that these satellites correspond to a member of the interacting states ν5, ν6, 2ν7, and 2ν9, which seem to be strongly coupled in this energetic regime, as can readily be seen by inspection of the CI coefficients reported in Table III and the supplementary material to Ref. 16.

As expected, O–H deuteration detunes these resonance perturbations of ν1 and we only observe one peak for trans-HCOOD and trans-DCOOD, which agree within 1 cm−1 with high-resolution values reported in the literature.37,78 For DCOOD, Goh et al. observed small perturbations of ro-vibrational lines with a high K value that were too weak to be analyzed.78 For HCOOD, A’dawiah et al. observed slight perturbations of ν1, which they ascribed to Coriolis interactions with 3ν7 + ν8.37 By using an effective fit Hamiltonian, they obtained a band center at 2601.13 cm−1 for the perturbing level. Our CVPT calculations disagree with this assignment, as 3ν7 + ν8 is predicted to be significantly higher in energy around 2700 cm−1. The best energy match by far is ν7 + ν8 + 2ν9, which is predicted between 2589 cm−1 and 2604 cm−1 (Tables III and IV).

Using experimental data from the literature and newly assigned in this work, Fig. 4 can be extended beyond fundamentals and their strong resonance partners. Using the same vibrational framework (CVPT), the performance of PES-2016 and PES-2018 is compared for 155 energy levels in Fig. 11, including the three deuterated isotopologs. Similar to the conclusions drawn in Secs. III and IV, it can be seen that PES-2016 almost exclusively overestimates and PES-2018 often underestimates the energy. The experiment is in many cases (halfway) between both, which is also exemplified in Fig. 6. These discrepancies between both PESs seem surprising at first sight, as the underlying ab initio methods are of similar quality.

FIG. 11.

Deviation between 155 experimental and CVPT6 energy levels (δexp=ν̃calcν̃exp) of formic acid against the predicted energy (relative to the vibrational ground state of the trans isotopolog). Data for all four H/D isotopologs are shown in Tables II and IV for trans- and Table S3 (supplementary material) for cis-formic acid. The 2376 cm−1 band of HCOOH (Table II) is omitted.

FIG. 11.

Deviation between 155 experimental and CVPT6 energy levels (δexp=ν̃calcν̃exp) of formic acid against the predicted energy (relative to the vibrational ground state of the trans isotopolog). Data for all four H/D isotopologs are shown in Tables II and IV for trans- and Table S3 (supplementary material) for cis-formic acid. The 2376 cm−1 band of HCOOH (Table II) is omitted.

Close modal

To highlight the role of anharmonic effects, we return to the second order result of Eq. (1) and focus on the xi,i/xi,j values. One can extract these values experimentally using select high-resolution or jet-cooled combination or overtone transitions and compare them to values obtained using the same transition energies calculated with CVPT6. It is more instructive, however, to compare to xi,i/xi,j values obtained with second order perturbation theory. These values are familiar to workers in the field and allow for a global comparison of anharmonic effects with 45 parameters. In contrast, the CVPT6 Hamiltonian has over 4000 terms. Before we make these comparisons, we note that the differences between the two theoretical results arise through resonance effects and higher order perturbative contributions. Even at the second order, with the equivalent treatment of resonances, one will find differences between curvilinear and rectilinear normal mode representations of the xi,i/xi,j values.48 

We report the full x matrices for both PESs computed with CVPT2 in the supplementary material for cis- and trans-HCOOH (Table S5). The values of these terms depend on the resonance terms included in the CVPT transformations, but the same resonant terms are present for both potential surfaces. In comparing these values, we first focus on the C=O stretching degree of freedom ν3, since there are notable differences in the fundamental values (cf. Table I). Comparing the PES-2016 and PES-2018 CVPT6 results, we find 1821 cm−1 and 1810 cm−1, respectively, for the cis isomer and 1783 cm−1 and 1773 cm−1, respectively, for the trans isomer. Inspection of individual anharmonicity constants x3,i, obtained directly with CVPT2, reveals maximum absolute deviations of 2 cm−1 for trans but 7 cm−1 for cis-HCOOH. More generally, the anharmonic terms for trans-formic acid are similar for the two surfaces, and the resulting deviations in anharmonic energy levels can primarily be ascribed to deviations in the underlying harmonic potential. The argument holds when comparisons are made to experimentally extracted63x1,j values for ν1 of trans-HCOOH (see Table S6). For cis-formic acid, the differences between the surfaces in both xi,j values and harmonic values are more pronounced. In several instances, these differences tend to cancel to yield similar fundamental transitions (Table I).

Finally, we turn to the large-amplitude O–H torsion that connects the two conformational isomers. In the energetic regime so far experimentally explored, diagonal anharmonic contributions and resonance couplings are equally well captured by both surfaces. This is elucidated in Fig. 12 where the energetic differences between subsequent energy levels in the trans well are compared. Deviations between higher and lower orders of perturbation theory (CVPT6 vs CVPT2) and anharmonic contributions from both surfaces (CVPT6 on PES-2016 vs PES-2018) appear to become substantial starting with 5ν9. Delocalization effects between the cis and trans well are expected to start with 6ν9.16 The precise energy of 6ν9 should be a sensitive marker for high level calculations due to additional wavefunction mixing with the cis well and anharmonic potential contributions close to the isomerization threshold.

FIG. 12.

Energy differences between subsequent torsional states of trans-HCOOH as a function of the vibrational quantum number n where zero corresponds to the ground state. The VPT2 energies, which are equivalent to CVPT2 without resonances, are computed according to Eq. (1) using the harmonic wavenumber ω̃9 and deperturbed diagonal anharmonicity constant x9,9*; the slope equals 2x9,9*. Delocalization effects between the cis and trans wells are expected to start with n = 6.16 

FIG. 12.

Energy differences between subsequent torsional states of trans-HCOOH as a function of the vibrational quantum number n where zero corresponds to the ground state. The VPT2 energies, which are equivalent to CVPT2 without resonances, are computed according to Eq. (1) using the harmonic wavenumber ω̃9 and deperturbed diagonal anharmonicity constant x9,9*; the slope equals 2x9,9*. Delocalization effects between the cis and trans wells are expected to start with n = 6.16 

Close modal

Utilizing the previously published CCSD(T)-F12 quality potential energy surfaces of formic acid in conjunction with high order canonical Van Vleck perturbation theory, we have revisited the IR and Raman spectra of monomeric trans-formic acid and its three deuterated isotopologs. Overall, we were able to add 11 new vibrational band centers of trans-HCOOH to the database and reassign seven previous IR assignments. We also assigned 53 new vibrational band centers of deuterated trans isotopologs that significantly extend the database, which until now was mostly composed of fundamentals.

Three strategies were pursued in making these assignments. In the absence of anharmonic Raman intensities, the simple model of dark states gaining intensity via resonance mixing with nearby fundamentals proved successful in providing a qualitative explanation for a plethora of (un)observed combination and overtone peaks in the Raman jet spectra of formic acid monomer. Using temperature as a population control, we have identified many hot bands and, in selected cases, used them to critically review other assignments. These bands allow direct insights into the anharmonic contributions to the Hamiltonian. Still many more observed hot bands remain unassigned, and a future direction is to incorporate the calculation of Raman activities into the CVPT6 framework. We have also made use of multiple isotopologs to probe the quality of the potential surface to aid in peak assignments. The approach included identifying bands in the Raman spectra that correspond to the two CO stretching vibrations (ν3 and ν6) of naturally occurring formic acid-13C. The resulting extensive vibrational reference data, particularly highly excited torsional and many resonance coupled states, ultimately will help establish formic acid as a model system to test and evaluate anharmonic vibrational methods for polyatomic molecules.

Comparison between second through sixth order Van Vleck perturbation theory validates the popular approach of second order vibrational perturbation theory with additional resonance treatment for fundamentals and binary combinations/overtones but also shows its limitations for quantitatively predicting highly excited vibrational states beyond two quanta. Errors for three- and four-quantum states were shown to be up to 20 cm−1, even more with the increase in energy and excitation of vibrational quanta. The close experiment-theory interplay revealed increasing deviations of internal coordinate path Hamiltonian eigenvalues to the experiment for highly excited torsional states, further demonstrating the importance of close collaboration between experiment and theory.79 While the question to this discrepancy remains open, several indications point to unconverged torsional eigenvalues.

As we have discussed and shown in Fig. 9, the large-amplitude O–H torsion leads to interesting and challenging dynamical features of trans-formic acid. The data shown in Fig. 9 illustrate how this torsion coupling extends to multiple isotopologs. The strong coupling among states, which share the polyad quantum number Np = n5 + n9/2, is present for many of the isotopologs. Moreover, our analysis of these states (see Fig. 7), which shows a breakdown of the polyad quantum number Np = n5 + n9/2 for Np ≥ 2 due to mixing with the C–O stretch ν6, extends to trans-DCOOH starting with Np = 1.5. The resonance interactions involving the torsion extend to the cis well despite its lower frequency, and they are expected for 6ν9,16 which can be investigated with appropriate methods, such as the internal coordinate path Hamiltonian. This work is intended to help calibrate future endeavors near the isomerization threshold.

Rotationally cold Raman jet spectra of HCOOH and DCOOH allowed us to identify perturbing states via their respective Q branches that interact with the O–H stretching fundamental ν1, leading to a complicated ro-vibrational spectrum.6 We have presented experimental evidence for the involvement of the aforementioned polyad (possibly including the OCO bend) in this resonance with ν1. Next steps in understanding the O–H stretching and torsion dynamics include refinement of the existing potential energy surfaces and analysis of the high-resolution spectrum of ν1, which we strongly encourage. Better understanding of the interaction between O–H stretching and skeletal modes of isolated formic acid is an important stepping stone in elucidating the complex vibrational spectrum of the cyclic formic acid dimer and gaining new physical insight into the dynamics of strongly hydrogen bonded systems. This paves the way for theoretical work revisiting the vibrational spectrum of monomeric formic acid beyond 4000 cm−1, as many of the band centers reported by Freytes et al. remain tentative.7 Further investigation along the lines presented in this work will deepen the understanding of coupling across chemical bonds as resonance mixing between the CH and OH stretches, and short time intramolecular vibration redistribution (IVR) dynamics become significant in the OH overtone spectra of formic acid.13,64

See the supplementary material for a full set of computed CVPT eigenvalues for H/D isotopologs of trans-formic acid, fundamentals of cis-formic acid and trans-formic acid-13C, and Raman jet spectra in the spectral range between 870 cm−1 and 3750 cm−1.

A.N. gratefully acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG; Project No. 389479699/GRK2455), thanks the Fonds der Chemischen Industrie (FCI) for a generous scholarship, and thanks Katharina A. E. Meyer, Martin A. Suhm, and David P. Tew for valuable discussions. E.L.S. gratefully acknowledges support from the NSF via Grant No. CHE-1900095. The authors thank Falk Richter for kindly supplying ab initio harmonic frequencies of trans-HCOOH at the CCSD(T)-F12a/aVTZ level.

The data that support the findings of this study are available within the article (and its supplementary material).

To aid the discussion of trans-formic acid band assignments in Sec. IV, we report selected vibrational eigenstates together with squared coefficients of the two leading contributing basis states in Table III.

TABLE III.

Vibrational energy levels (in cm−1) and two leading squared coefficients (P) for selected states of different trans-formic acid isotopologs (energies relative to the vibrational ground state of the trans isotopolog) obtained from PES-201616 using CVPTn with n = 2, 4, 6. The energy difference Δ(n) = E(n) − E(6) is shown for n = 2, 4. Squared coefficients printed as “0.00” are below 0.5%. Eigenstates are numbered (No.) according to the CVPT6 energy, where “1” corresponds to the vibrational ground state. The full set of eigenvalues is reported in the supplementary material (Tables S7–S10).

No.ΓE(6)Δ(4)Δ(2)P1StateP2State
trans-HCOOH 
29 a′ 2303.8 −0.2 0.3 0.92 718191 0.03 94 
30 a′ 2304.8 −2.5 −4.5 0.34 94 0.32 6192 
31 a′ 2337.9 0.0 8.0 0.41 94 0.28 6192 
32 a″ 2338.4 0.5 −1.4 0.67 5181 0.30 8192 
33 a′ 2358.6 −0.9 1.4 0.99 6172 0.01 7292 
34 a″ 2368.0 −0.5 −0.6 0.93 617191 0.06 7193 
35 a′ 2402.4 0.1 −2.1 0.65 3171 0.20 5161 
36 a′ 2405.0 0.4 0.2 0.43 5161 0.32 3171 
37 a″ 2415.6 −0.2 0.0 0.52 4181 0.26 3191 
38 a″ 2418.4 0.0 2.1 0.43 4181 0.23 3191 
39 a″ 2427.5 0.1 5.0 0.50 3191 0.38 7193 
⋯ 
42 a′ 2506.9 0.3 −2.0 0.52 52 0.31 5192 
⋯ 
46 a″ 2578.5 1.5 1.2 0.76 517191 0.17 7193 
47 a′ 2597.8 −0.5 −1.4 0.60 4192 0.30 4151 
48 a′ 2607.5 2.4 −0.6 0.42 52 0.41 5192 
49 a′ 2634.4 −0.4 2.9 0.94 4172 0.05 7292 
50 a″ 2652.7 0.2 5.2 0.91 417191 0.07 7193 
51 a′ 2677.6 0.8 5.2 0.57 4151 0.22 4192 
52 a′ 2692.0 −1.7 −1.8 1.00 7182 0.00 3171 
⋯ 
68 a′ 2940.3 −0.1 0.5 0.90 21 0.03 3151 
69 a′ 2964.3 0.8 11.5 0.45 7194 0.23 617192 
70 a″ 2964.9 0.2 −1.3 0.64 517181 0.32 718192 
71 a′ 2983.2 2.1 0.6 0.76 518191 0.18 8193 
⋯ (Between 133 and 147 only a′ states) 
133 a′ 3544.9 0.1 2.7 0.94 32 0.02 315171 
135 a′ 3564.0 −3.8 −2.6 0.31 617292 0.18 7294 
136 a′ 3564.5 −0.7 5.2 0.85 738191 0.05 617292 
137 a′ 3565.2 1.6 1.6 0.66 4162 0.09 5261 
138 a′ 3567.8 0.0 1.6 0.83 2171 0.03 315171 
140 a′ 3576.2 −0.5 −2.8 0.54 11 0.13 4162 
142 a′ 3581.8 1.2 1.3 0.43 11 0.14 5194 
143 a′ 3591.7 1.7 14.2 0.47 7294 0.20 617292 
145 a′ 3610.9 2.0 1.0 0.75 51718191 0.18 718193 
146 a′ 3614.5 −2.6 5.8 0.88 6174 0.03 5261 
147 a′ 3615.0 6.1 10.2 0.30 5261 0.20 5194 
⋯  
182 a″ 3806.9 0.4 7.1 0.67 314191 0.18 417193 
183 a′ 3824.6 −2.5 −0.9 0.60 5174 0.30 7492 
184 a′ 3834.7 −1.6 −2.8 0.84 3182 0.10 4261 
185 a′ 3834.8 2.0 1.3 0.72 4261 0.12 3182 
186 a″ 3840.5 0.7 0.9 0.73 517391 0.16 7393 
⋯  
205 a′ 3934.1 −1.0 5.7 0.50 415172 0.23 417292 
206 a′ 3948.8 −2.5 1.7 0.98 7382 0.01 718292 
207 a″ 3950.7 1.6 9.2 0.61 41517191 0.12 427191 
208 a″ 3954.3 −0.7 −1.7 0.92 2181 0.03 4281 
 
209 a″ 3963.4 −1.9 1.8 0.97 728291 0.01 738192 
210 a′ 3965.3 3.0 −3.0 0.53 4251 0.16 4292 
211 a″ 3967.8 −3.4 −6.1 0.42 61718192 0.24 51617181 
212 a′ 3974.5 −0.2 0.6 0.94 3162 0.02 516271 
213 a′ 3977.3 −0.8 6.5 0.31 4292 0.24 4152 
214 a″ 3983.6 1.9 25.0 0.57 6293 0.21 516291 
trans-DCOOH 
35 a′ 2218.4 0.2 1.0 0.84 21 0.04 62 
36 a″ 2225.9 0.2 1.2 1.00 417191 0.00 7193 
37 a′ 2269.2 −0.2 −1.0 0.68 4151 0.25 4192 
38 a′ 2282.2 0.4 −0.5 0.76 62 0.10 94 
39 a′ 2292.6 0.5 2.6 0.58 94 0.16 62 
40 a′ 2344.5 −1.1 −1.5 0.60 6192 0.27 5161 
41 a′ 2349.9 −0.6 −0.5 0.50 7182 0.45 3171 
42 a″ 2365.3 −0.2 −0.5 0.51 3191 0.42 8291 
43 a′ 2380.1 −0.7 0.9 0.96 6172 0.03 7182 
44 a″ 2381.5 0.2 1.5 0.51 617191 0.34 7193 
45 a′ 2384.0 −0.3 1.0 0.51 3171 0.46 7182 
46 a″ 2397.8 −0.1 0.9 0.34 617191 0.21 8291 
47 a″ 2403.2 −0.2 2.4 0.30 3191 0.28 8291 
48 a′ 2431.1 0.1 −1.2 0.67 5161 0.28 6192 
49 a′ 2454.7 −0.8 0.4 0.67 7292 0.31 5172 
⋯ 
152 a′ 3382.0 −0.3 5.0 0.98 738191 0.01 5173 
153 a′ 3392.4 −0.1 −2.5 0.64 415161 0.19 416192 
154 a′ 3407.6 0.3 4.2 0.48 6194 0.13 63 
155 a″ 3417.1 2.2 15.7 0.70 7195 0.16 517193 
156 a″ 3418.7 −1.3 −1.2 0.59 517281 0.36 728192 
157 a′ 3423.7 −1.5 −5.5 0.61 63 0.14 6194 
158 a′ 3424.1 −0.7 0.7 0.61 417292 0.29 415172 
159 a″ 3433.0 0.3 0.1 0.99 427181 0.00 418291 
160 a′ 3434.9 0.5 2.8 0.62 51718191 0.22 718193 
161 a′ 3437.3 −3.0 −7.0 0.44 3182 0.42 84 
⋯ (Between 186 and 202 only a′ states) 
186 a′ 3542.6 0.6 8.4 0.59 7294 0.17 517292 
189 a′ 3557.9 0.2 −6.3 0.42 5162 0.20 6292 
190 a′ 3563.0 −0.4 2.0 0.34 317192 0.18 315171 
191 a′ 3566.4 2.4 −7.7 0.24 4152 0.23 415192 
192 a′ 3569.9 0.1 −2.0 0.86 11 0.05 4152 
193 a′ 3582.1 −1.6 2.7 0.58 617292 0.28 516172 
194 a′ 3589.9 −1.2 1.9 0.56 3173 0.37 7382 
195 a′ 3597.8 −0.6 −0.1 0.30 718292 0.27 517182 
198 a′ 3607.1 0.0 −1.3 0.76 41618191 0.13 418193 
199 a′ 3615.1 −1.5 −3.7 0.45 5261 0.22 516192 
201 a′ 3620.9 −2.6 3.2 0.97 6174 0.02 7382 
202 a′ 3624.6 −0.6 3.6 0.60 7382 0.38 3173 
trans-HCOOD 
33 a′ 2103.2 −0.6 −0.8 0.72 718191 0.25 5172 
34 a′ 2134.7 0.8 4.5 0.67 7292 0.23 5172 
35 a′ 2145.0 −0.4 −2.5 0.60 5161 0.38 6192 
36 a″ 2150.6 −0.6 0.6 0.94 7281 0.06 7391 
 
37 a′ 2182.3 0.0 0.4 0.58 6192 0.39 5161 
38 a″ 2195.7 −0.5 2.6 0.93 7391 0.06 7281 
39 a″ 2208.5 −0.5 −1.8 0.96 6181 0.03 617191 
⋯ 
62 a″ 2574.6 −0.1 3.4 0.51 517181 0.28 7293 
63 a″ 2604.4 −0.1 1.2 0.63 718192 0.14 517181 
64 a″ 2614.1 0.1 −1.1 0.67 6193 0.29 516191 
65 a′ 2618.3 −1.6 −0.9 0.91 7182 0.08 728191 
66 a″ 2636.1 1.9 7.3 0.55 517291 0.33 7293 
67 a′ 2637.8 −0.1 −0.5 0.99 11 0.01 3151 
No.ΓE(6)Δ(4)Δ(2)P1StateP2State
trans-HCOOH 
29 a′ 2303.8 −0.2 0.3 0.92 718191 0.03 94 
30 a′ 2304.8 −2.5 −4.5 0.34 94 0.32 6192 
31 a′ 2337.9 0.0 8.0 0.41 94 0.28 6192 
32 a″ 2338.4 0.5 −1.4 0.67 5181 0.30 8192 
33 a′ 2358.6 −0.9 1.4 0.99 6172 0.01 7292 
34 a″ 2368.0 −0.5 −0.6 0.93 617191 0.06 7193 
35 a′ 2402.4 0.1 −2.1 0.65 3171 0.20 5161 
36 a′ 2405.0 0.4 0.2 0.43 5161 0.32 3171 
37 a″ 2415.6 −0.2 0.0 0.52 4181 0.26 3191 
38 a″ 2418.4 0.0 2.1 0.43 4181 0.23 3191 
39 a″ 2427.5 0.1 5.0 0.50 3191 0.38 7193 
⋯ 
42 a′ 2506.9 0.3 −2.0 0.52 52 0.31 5192 
⋯ 
46 a″ 2578.5 1.5 1.2 0.76 517191 0.17 7193 
47 a′ 2597.8 −0.5 −1.4 0.60 4192 0.30 4151 
48 a′ 2607.5 2.4 −0.6 0.42 52 0.41 5192 
49 a′ 2634.4 −0.4 2.9 0.94 4172 0.05 7292 
50 a″ 2652.7 0.2 5.2 0.91 417191 0.07 7193 
51 a′ 2677.6 0.8 5.2 0.57 4151 0.22 4192 
52 a′ 2692.0 −1.7 −1.8 1.00 7182 0.00 3171 
⋯ 
68 a′ 2940.3 −0.1 0.5 0.90 21 0.03 3151 
69 a′ 2964.3 0.8 11.5 0.45 7194 0.23 617192 
70 a″ 2964.9 0.2 −1.3 0.64 517181 0.32 718192 
71 a′ 2983.2 2.1 0.6 0.76 518191 0.18 8193 
⋯ (Between 133 and 147 only a′ states) 
133 a′ 3544.9 0.1 2.7 0.94 32 0.02 315171 
135 a′ 3564.0 −3.8 −2.6 0.31 617292 0.18 7294 
136 a′ 3564.5 −0.7 5.2 0.85 738191 0.05 617292 
137 a′ 3565.2 1.6 1.6 0.66 4162 0.09 5261 
138 a′ 3567.8 0.0 1.6 0.83 2171 0.03 315171 
140 a′ 3576.2 −0.5 −2.8 0.54 11 0.13 4162 
142 a′ 3581.8 1.2 1.3 0.43 11 0.14 5194 
143 a′ 3591.7 1.7 14.2 0.47 7294 0.20 617292 
145 a′ 3610.9 2.0 1.0 0.75 51718191 0.18 718193 
146 a′ 3614.5 −2.6 5.8 0.88 6174 0.03 5261 
147 a′ 3615.0 6.1 10.2 0.30 5261 0.20 5194 
⋯  
182 a″ 3806.9 0.4 7.1 0.67 314191 0.18 417193 
183 a′ 3824.6 −2.5 −0.9 0.60 5174 0.30 7492 
184 a′ 3834.7 −1.6 −2.8 0.84 3182 0.10 4261 
185 a′ 3834.8 2.0 1.3 0.72 4261 0.12 3182 
186 a″ 3840.5 0.7 0.9 0.73 517391 0.16 7393 
⋯  
205 a′ 3934.1 −1.0 5.7 0.50 415172 0.23 417292 
206 a′ 3948.8 −2.5 1.7 0.98 7382 0.01 718292 
207 a″ 3950.7 1.6 9.2 0.61 41517191 0.12 427191 
208 a″ 3954.3 −0.7 −1.7 0.92 2181 0.03 4281 
 
209 a″ 3963.4 −1.9 1.8 0.97 728291 0.01 738192 
210 a′ 3965.3 3.0 −3.0 0.53 4251 0.16 4292 
211 a″ 3967.8 −3.4 −6.1 0.42 61718192 0.24 51617181 
212 a′ 3974.5 −0.2 0.6 0.94 3162 0.02 516271 
213 a′ 3977.3 −0.8 6.5 0.31 4292 0.24 4152 
214 a″ 3983.6 1.9 25.0 0.57 6293 0.21 516291 
trans-DCOOH 
35 a′ 2218.4 0.2 1.0 0.84 21 0.04 62 
36 a″ 2225.9 0.2 1.2 1.00 417191 0.00 7193 
37 a′ 2269.2 −0.2 −1.0 0.68 4151 0.25 4192 
38 a′ 2282.2 0.4 −0.5 0.76 62 0.10 94 
39 a′ 2292.6 0.5 2.6 0.58 94 0.16 62 
40 a′ 2344.5 −1.1 −1.5 0.60 6192 0.27 5161 
41 a′ 2349.9 −0.6 −0.5 0.50 7182 0.45 3171 
42 a″ 2365.3 −0.2 −0.5 0.51 3191 0.42 8291 
43 a′ 2380.1 −0.7 0.9 0.96 6172 0.03 7182 
44 a″ 2381.5 0.2 1.5 0.51 617191 0.34 7193 
45 a′ 2384.0 −0.3 1.0 0.51 3171 0.46 7182 
46 a″ 2397.8 −0.1 0.9 0.34 617191 0.21 8291 
47 a″ 2403.2 −0.2 2.4 0.30 3191 0.28 8291 
48 a′ 2431.1 0.1 −1.2 0.67 5161 0.28 6192 
49 a′ 2454.7 −0.8 0.4 0.67 7292 0.31 5172 
⋯ 
152 a′ 3382.0 −0.3 5.0 0.98 738191 0.01 5173 
153 a′ 3392.4 −0.1 −2.5 0.64 415161 0.19 416192 
154 a′ 3407.6 0.3 4.2 0.48 6194 0.13 63 
155 a″ 3417.1 2.2 15.7 0.70 7195 0.16 517193 
156 a″ 3418.7 −1.3 −1.2 0.59 517281 0.36 728192 
157 a′ 3423.7 −1.5 −5.5 0.61 63 0.14 6194 
158 a′ 3424.1 −0.7 0.7 0.61 417292 0.29 415172 
159 a″ 3433.0 0.3 0.1 0.99 427181 0.00 418291 
160 a′ 3434.9 0.5 2.8 0.62 51718191 0.22 718193 
161 a′ 3437.3 −3.0 −7.0 0.44 3182 0.42 84 
⋯ (Between 186 and 202 only a′ states) 
186 a′ 3542.6 0.6 8.4 0.59 7294 0.17 517292 
189 a′ 3557.9 0.2 −6.3 0.42 5162 0.20 6292 
190 a′ 3563.0 −0.4 2.0 0.34 317192 0.18 315171 
191 a′ 3566.4 2.4 −7.7 0.24 4152 0.23 415192 
192 a′ 3569.9 0.1 −2.0 0.86 11 0.05 4152 
193 a′ 3582.1 −1.6 2.7 0.58 617292 0.28 516172 
194 a′ 3589.9 −1.2 1.9 0.56 3173 0.37 7382 
195 a′ 3597.8 −0.6 −0.1 0.30 718292 0.27 517182 
198 a′ 3607.1 0.0 −1.3 0.76 41618191 0.13 418193 
199 a′ 3615.1 −1.5 −3.7 0.45 5261 0.22 516192 
201 a′ 3620.9 −2.6 3.2 0.97 6174 0.02 7382 
202 a′ 3624.6 −0.6 3.6 0.60 7382 0.38 3173 
trans-HCOOD 
33 a′ 2103.2 −0.6 −0.8 0.72 718191 0.25 5172 
34 a′ 2134.7 0.8 4.5 0.67 7292 0.23 5172 
35 a′ 2145.0 −0.4 −2.5 0.60 5161 0.38 6192 
36 a″ 2150.6 −0.6 0.6 0.94 7281 0.06 7391 
 
37 a′ 2182.3 0.0 0.4 0.58 6192 0.39 5161 
38 a″ 2195.7 −0.5 2.6 0.93 7391 0.06 7281 
39 a″ 2208.5 −0.5 −1.8 0.96 6181 0.03 617191 
⋯ 
62 a″ 2574.6 −0.1 3.4 0.51 517181 0.28 7293 
63 a″ 2604.4 −0.1 1.2 0.63 718192 0.14 517181 
64 a″ 2614.1 0.1 −1.1 0.67 6193 0.29 516191 
65 a′ 2618.3 −1.6 −0.9 0.91 7182 0.08 728191 
66 a″ 2636.1 1.9 7.3 0.55 517291 0.33 7293 
67 a′ 2637.8 −0.1 −0.5 0.99 11 0.01 3151 

Experimental band centers from this work and the literature are compactly summarized in Table IV for the three deuterated trans-formic acid isotopologs.

TABLE IV.

Vibrational wavenumbers (in cm−1) of deuterated trans-formic acid states. Multi-configuration time-dependent Hartree (MCTDH)18 and canonical Van Vleck perturbation theory (CVPT) predictions using two different analytical PES16,17 are shown together with perturbation-free experimental data. The asterisk denotes newly assigned bands, and the dagger denotes a reassignment. Tentative assignments are in brackets, not fully converged MCTDH energies are given in energy ranges (adopted from Ref. 18), and energy levels obtained from hot band assignments are italicized. Numbers in the state labels refer to the vibrational degree of freedom, which is indexed with the respective quanta of excitation.

StateExp.PES-2016aPES-2018b
LabelΓRa. jetcLit.CVPT6MCTDHCVPT6
trans-DCOOH 
 71 a′ 620 620.57d 621 617 617 
 91 a″  631.54d 631 628 628 
 81 a″  873.39e 875 872 873 
 41 a′ 971 970.89e 971 971 971 
 61 a′ 1142 1142.31f 1145 1139 1139 
 92 a′ 1206  1206 1202 1202 
 51 a′ 1299 1297g 1299 1294 1294 
 31/82 a′ 1725 1725.87h 1731 1723 1724 
 31/82 a′ 1762 1762.9h 1766 1761 1762 
6191 a″ 1779  1779 1773 1773 
7192 a′ 1828  1830 1819 1820 
5171 a′ 1919  1921 1910 1910 
5191 a″ 1937  1937 1928 1928 
4161 a′ 2103  2106 2101 2101 
4192 a′ 2174  2175 2170 2170 
 21 a′ 2219 2219.69i 2218 2217 2217 
4151 a′ 2271  2269 2263 2263 
 62 a′ 2277 2254.24i 2282 n.r. 2272 
94 a′ 2290  2293 2282 2283 
6192 a′ 2342  2344 n.r. 2337 
5161 a′ 2428  2431 2421 2421 
 
52/5192 a′ 2482  2485 2476 2476 
 52/5192 a′   2593 2582 2581 
3181/83 a″ 2578  2588 2576 2579 
3181/83 a″ 2641  2647 2641 2643 
3141/4182 a′ 2695  2701 2692 2695 
3141/4182 a′ 2731  2736 2730 2733 
3161/6182 a′ 2860  2870 n.r. 2858 
3161/6182 a′ 2898  2906 n.r. 2896 
2161 a′ 3352  3352 n.r. 3347 
6194 a′ 3404  3408 n.r. 3398 
63 a′ 3421  3424 n.r. 3409 
 [11-Res] a′ 3563     
 11 a′ 3569 3566g 3579 3568 3572 
 [11-Res] a′ 3607     
trans-HCOOD 
 91 a″  508.13j 508 505 505 
 71 a′ 558 558.27j 559 556 556 
 51 a′ 972 972.85k 973 969 969 
 92 a′ 1010 1011.68k 1011 1006 1006 
 81 a″ 1031l  1032 1029 1031 
 61 a′ 1176 1177.09m 1180 1179 1179 
 41 a′ 1365 1366.48n 1366 1362 1362 
93 a″ 1447  1448 1441 1441 
5191 a″ 1512  1513 1507 1506 
5171 a′ 1527  1529 n.r. 1522 
8191 a′ 1539  1542 1535 1536 
 6191 a″ 1679 1680.96o 1683 1681 1681 
 6171 a′ 1730 1732.08,o 1735.81p 1734 1730 1730 
 31 a′ 1772 1772.12o 1779 1771 1771 
94 a′ 1897  1899 1890 1889 
52 a′ 1953  1955 1946 1946 
82 a′ 2055  2061 2054 2057 
5161 a′  2142.4q 2145 2141 2141 
6192 a′  2178.8q 2182 2178 2178 
3171 a′ 2327  2334 2322 2323 
62 a′ 2341  2348 2345 2345 
† 718192 a″  2601.13r 2604 2589 2591 
 7381 a″   2712 n.r. 2699 
 11 a′ 2631 2631.64r 2638 2629 631r 
 42 a′ 2713 2714s 2712 2706 2707 
3151 a′ 2741  2749 n.r. 2737 
3192 a′ 2782  2788 2734–2775 2775 
 21 a′ 2938 2938.2s 2941 2934 2936 
 3161 a′ 2954  2961 2954 2955 
416171 a′ 3092  3096 n.r. 3089 
3141 a′ 3137  3145 3131 3132 
1171 a′ 3184  3191 3180 3182 
 32 a′ 3529 3531/3526t 3540 n.r. 3525 
trans-DCOOD 
 91 a″  492.23u 492 489 489 
 71 a′ 554 554.44u 555 552 552 
 
 81 a″  873.2v 874 872 873 
 51 a′ 945 945.0w 946 944 944 
 92 a′   964 961 961 
 41 a′ 1039 1042w 1040 1035 1035 
 61 a′ 1170 1170.80x 1173 1170 1170 
93 a″ 1414  1413 1407 1407 
5191 a″ 1443  1442 1439 1439 
 31/82 a′ 1725 1725.12y 1730 1722 1723 
6191 a″ 1658  1661 1657 1656 
6171 a′ 1720  1723 1717 1717 
 31/82 a′ 1761 1760.0w 1765 1759 1760 
42 a′ 2073  2074 2076 2065 
5161 a′ 2108  2110 n.r. 2106 
6192 a′ 2126  2129 2123 2123 
 4161 a′ 2194 2195.1w 2196 2191 2191 
 21 a′ 2231 2231.8w 2233 2228 2229 
 62 a′ 2330 2326.2w 2338 2330 2329 
3181/83 a″ 2577  2586 2574 2577 
 11 a′ 2632 2631.87z 2638 2629 2631 
3181/83 a″ 2639  2646 2638 2641 
3151/5182 a′ 2668  2673 n.r. 2665 
3151/5182 a′ 2704  2707 n.r. 2702 
3141/4182 a′ 2761  2768 n.r. 2757 
3141/4182 a′ 2797  2803 2792 2795 
3161/6182 a′ 2888  2898 2887–2925 2888 
3161/6182 a′ 2926  2934 n.r. 2926 
2181 a″ 3096  3099 3092 3094 
1171 a′ 3181  3188 n.r. 3178 
StateExp.PES-2016aPES-2018b
LabelΓRa. jetcLit.CVPT6MCTDHCVPT6
trans-DCOOH 
 71 a′ 620 620.57d 621 617 617 
 91 a″  631.54d 631 628 628 
 81 a″  873.39e 875 872 873 
 41 a′ 971 970.89e 971 971 971 
 61 a′ 1142 1142.31f 1145 1139 1139 
 92 a′ 1206  1206 1202 1202 
 51 a′ 1299 1297g 1299 1294 1294 
 31/82 a′ 1725 1725.87h 1731 1723 1724 
 31/82 a′ 1762 1762.9h 1766 1761 1762 
6191 a″ 1779  1779 1773 1773 
7192 a′ 1828  1830 1819 1820 
5171 a′ 1919  1921 1910 1910 
5191 a″ 1937  1937 1928 1928 
4161 a′ 2103  2106 2101 2101 
4192 a′ 2174  2175 2170 2170 
 21 a′ 2219 2219.69i 2218 2217 2217 
4151 a′ 2271  2269 2263 2263 
 62 a′ 2277 2254.24i 2282 n.r. 2272 
94 a′ 2290  2293 2282 2283 
6192 a′ 2342  2344 n.r. 2337 
5161 a′ 2428  2431 2421 2421 
 
52/5192 a′ 2482  2485 2476 2476 
 52/5192 a′   2593 2582 2581 
3181/83 a″ 2578  2588 2576 2579 
3181/83 a″ 2641  2647 2641 2643 
3141/4182 a′ 2695  2701 2692 2695 
3141/4182 a′ 2731  2736 2730 2733 
3161/6182 a′ 2860  2870 n.r. 2858 
3161/6182 a′ 2898  2906 n.r. 2896 
2161 a′ 3352  3352 n.r. 3347 
6194 a′ 3404  3408 n.r. 3398 
63 a′ 3421  3424 n.r. 3409 
 [11-Res] a′ 3563     
 11 a′ 3569 3566g 3579 3568 3572 
 [11-Res] a′ 3607     
trans-HCOOD 
 91 a″  508.13j 508 505 505 
 71 a′ 558 558.27j 559 556 556 
 51 a′ 972 972.85k 973 969 969 
 92 a′ 1010 1011.68k 1011 1006 1006 
 81 a″ 1031l  1032 1029 1031 
 61 a′ 1176 1177.09m 1180 1179 1179 
 41 a′ 1365 1366.48n 1366 1362 1362 
93 a″ 1447  1448 1441 1441 
5191 a″ 1512  1513 1507 1506 
5171 a′ 1527  1529 n.r. 1522 
8191 a′ 1539  1542 1535 1536 
 6191 a″ 1679 1680.96o 1683 1681 1681 
 6171 a′ 1730 1732.08,o 1735.81p 1734 1730 1730 
 31 a′ 1772 1772.12o 1779 1771 1771 
94 a′ 1897  1899 1890 1889 
52 a′ 1953  1955 1946 1946 
82 a′ 2055  2061 2054 2057 
5161 a′  2142.4q 2145 2141 2141 
6192 a′  2178.8q 2182 2178 2178 
3171 a′ 2327  2334 2322 2323 
62 a′ 2341  2348 2345 2345 
† 718192 a″  2601.13r 2604 2589 2591 
 7381 a″   2712 n.r. 2699 
 11 a′ 2631 2631.64r 2638 2629 631r 
 42 a′ 2713 2714s 2712 2706 2707 
3151 a′ 2741  2749 n.r. 2737 
3192 a′ 2782  2788 2734–2775 2775 
 21 a′ 2938 2938.2s 2941 2934 2936 
 3161 a′ 2954  2961 2954 2955 
416171 a′ 3092  3096 n.r. 3089 
3141 a′ 3137  3145 3131 3132 
1171 a′ 3184  3191 3180 3182 
 32 a′ 3529 3531/3526t 3540 n.r. 3525 
trans-DCOOD 
 91 a″  492.23u 492 489 489 
 71 a′ 554 554.44u 555 552 552 
 
 81 a″  873.2v 874 872 873 
 51 a′ 945 945.0w 946 944 944 
 92 a′   964 961 961 
 41 a′ 1039 1042w 1040 1035 1035 
 61 a′ 1170 1170.80x 1173 1170 1170 
93 a″ 1414  1413 1407 1407 
5191 a″ 1443  1442 1439 1439 
 31/82 a′ 1725 1725.12y 1730 1722 1723 
6191 a″ 1658  1661 1657 1656 
6171 a′ 1720  1723 1717 1717 
 31/82 a′ 1761 1760.0w 1765 1759 1760 
42 a′ 2073  2074 2076 2065 
5161 a′ 2108  2110 n.r. 2106 
6192 a′ 2126  2129 2123 2123 
 4161 a′ 2194 2195.1w 2196 2191 2191 
 21 a′ 2231 2231.8w 2233 2228 2229 
 62 a′ 2330 2326.2w 2338 2330 2329 
3181/83 a″ 2577  2586 2574 2577 
 11 a′ 2632 2631.87z 2638 2629 2631 
3181/83 a″ 2639  2646 2638 2641 
3151/5182 a′ 2668  2673 n.r. 2665 
3151/5182 a′ 2704  2707 n.r. 2702 
3141/4182 a′ 2761  2768 n.r. 2757 
3141/4182 a′ 2797  2803 2792 2795 
3161/6182 a′ 2888  2898 2887–2925 2888 
3161/6182 a′ 2926  2934 n.r. 2926 
2181 a″ 3096  3099 3092 3094 
1171 a′ 3181  3188 n.r. 3178 
a

Reference 16.

b

Reference 18.

c

Fundamentals and resonance partners from Ref. 40.

d

Reference 35.

e

Reference 32.

f

Reference 80.

g

Reference 41.

h

Reference 31.

i

Reference 71; for the interaction between ν2 and 2ν6, only Coriolis coupling was included in the model Hamiltonian.

j

Reference 33.

k

Reference 34.

l

Estimated from overtone transition of trans-HCOOD and the experimental diagonal anharmonicity constant for trans-HCOOH (see Sec. III B for details).

m

Reference 81.

n

Reference 62.

o

Reference 82.

p

Reference 83.

q

Reference 69.

r

Reference 37; the band at 2601.13 cm−1 was originally assigned to 3ν7 + ν8, which is shifted toward 2596.31 cm−1 upon 13C substitution.28 

s

Reference 41.

t

Reference 63.

u

Reference 33.

v

Reference 69.

w

Reference 4.

x

Reference 84.

y

Reference 36.

z

Reference 78.

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Supplementary Material