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

The spectroscopic data base for cis-formic acid is considerably extended to make it fit for experimental benchmarking of vibrational calculation tools.


Introduction
Vibrational spectra of small molecules effectively probe the quality of potential energy hypersurface (PES) predictions, when combined with accurate anharmonic calculations. 1 Typically, an intense interplay between theory and experiment initially converges the performance for a set of low quantum number states around the global minimum. To explore the globality of a PES, it is then desirable to add experimental data on a secondary minimum structure. Its quantum states start locally, but evolve into mixed structure states at higher excitation, probing the transition state region as well. For three atoms, HCN/HNC is the paradigmatic example. 2 For four atoms, the simultaneous description of the nearly isoenergetic formaldehyde molecule and H 2 -CO complex is challenging. 3,4 For ve atoms, the cis-trans isomerism of formic acid is arguably one of the most interesting systems, calling for suitable experimental reference data for the higher-energy cis-form. These have been surprisingly scarce until very recently, with a single exception. 5 As the smallest carboxylic acid, the formic acid monomer has been addressed by a plethora of theoretical [6][7][8][9][10][11][12][13][14][15] and experimental studies. 5, The isomerisation from the ground state transform to the higher-energy cis-form has been of particular interest, 5,14,22,41,42 even when looking at processes in the interstellar medium. 56 When it comes to the vibrations of cis-formic acid, matrix isolation has been the method of choice thus far, because the possibility of long irradiation times allows for a signicant formation via laser excitation of the transform. 35,41,42,49,50,57 Since the matrix environment shis the band positions compared to the gas phase, a direct comparison with predicted band positions of modern quantum chemical methods would require a very challenging description of the environment. Accurate theoretical predictions for the isolated cis-isomer thus suffer from a lack of gas phase experimental reference data. Two recent studies where this applies are by Tew and Mizukami 14 from 2016 and by Richter and Carbonnière 15 from 2018.
Due to the fairly large energy difference of 1365 AE 30 cm À1 between both rotamers of formic acid, 22 vibrational gas phase data on the cis-form are rare. The rst gas phase band position of cis-formic acid has been published in 2006 by Baskakov and co-workers, who studied the out-of-plane bending vibration with high resolution FTIR spectroscopy. 5 Only very recently, it was complemented by a second example obtained as a side effect when studying excitonic C]O stretch coupling in jetcooled carboxylic acid dimers. 54 This observation has triggered the present work, which represents a systematic study of all four valence stretching modes of cis-formic acid. It is based on a powerful new Raman scattering approach of thermally populated and rapidly re-cooled molecules. Instead of conserving the conformational excitation by cryogenic matrix trapping, 58 the spectra are rotationally simplied by supersonic expansion. Vibrational and high barrier conformational excitation is largely trapped and can be probed without environmental distortion as a function of initial gas temperature. Back-tunnelling to the trans-form is also not an issue on the time scale of the jet expansion, making it an "easy" experiment. 59 By a 400% increase of perturbation-free cis-formic acid vibrational frequencies aer a decade of stagnation, we provide the rst systematic access to the performance of quantum chemical methods towards this model system. This decreases the likelihood of accidental error compensation between electronic structure, vibrational treatment, and matrix shis for cis-formic acid by orders of magnitude.
The structure of this paper is as follows: we briey illustrate the general approach of how the spectra of cis-formic acid were recorded, followed by a detailed analysis of the spectra and a rst benchmark of vibrational perturbation theory and literature variational data against the experimental data. It is hoped that this progress will trigger further growth in the experimental data base and its use in benchmarking the global PES of formic acid and pentatomic vibrational treatments.

Experimental
A detailed description of the experimental set-up can be found in previous publications. 54,60 Formic acid (Acros Organics, 98+%) was seeded at 0.2% in helium and expanded through a vertical slit nozzle at 1.0 bar into the evacuated jet chamber (background pressure < 0.1 mbar). The expansion was probed by a 532 nm, 25 W continuous-wave laser from below. The scattered radiation was detected perpendicularly with respect to both the expanding ow and the laser with a monochromator equipped with a charge-coupled device camera. cis-Formic acid was formed in small quantities from the trans-rotamer by heating the nozzle and its feed line to temperatures between 100-190 C.

Computations
The quantum chemical calculations shown in this work were performed with the Gaussian 09 program package (revision E.01) 61 using a pruned ultra ne integration grid (99 590). This grid is ner than the default of Gaussian 09 (ne grid, (75 302)). 61 The employed methods are B3LYP, 62 72 For all methods, an augmented quadruple-zeta (aVQZ) basis set has been chosen. Additional augmented double-(aVDZ) and triple-zeta (aVTZ) calculations were carried out for MP2 and B2PLYP-D3(BJ). Moreover, the production calculations have been performed without the use of symmetry, employing opt ¼ tight convergence.
The assignment of cis-formic acid fundamentals has been supported by scaled, harmonic frequency calculations at the B3LYP-D3(BJ)/aVTZ level, which have proven to yield sufficient agreement in a previous study. 54 For the vibrational benchmark in Section 3.2, anharmonic frequency calculations were performed at all levels listed above using vibrational perturbation theory (VPT2), 73 as implemented in Gaussian 09. 61 VPT2 was used under the default settings where resonances identied in a pre-screening are removed and treated variationally.
Additionally, exploratory VPT2 calculations utilising the C s symmetry as well as a ner integration grid (pruned super ne integration grid (150 974) 61 ) were carried out in selected cases to probe their impact on the results. A brief discussion can be found in Section 3.3.

The stretching vibrations of cis-formic acid
To choose suitable spectral regions for cis-rotamer detection, the band positions and Raman scattering cross-sections have been predicted using B3LYP-D3(BJ)/aVTZ alongside those of the trans-form. The results are displayed in Fig. 1. The vibrations have been labelled according to the Herzberg nomenclature. The cis-formic acid vibrations with the largest scattering crosssections are n 1 , n 2 , n 3 , and n 6 , namely the O-H, the C-H, the C]O, and the C-O stretching vibration. In fact, n 6 is the only stretching vibration with a distinctly larger scattering crosssection compared to trans-formic acid.
The experimental spectra of these four vibrational modes of both rotamers (cF, F) can be found in Fig. 2 alongside harmonic, individually F-scaled B3LYP-D3(BJ)/aVTZ calculations below the spectra. For each spectral region, four spectra with increasing nozzle temperature have been recorded. These temperature series have been intensity-scaled to the trans-monomer band of lowest intensity amongst the four. Consequently, any hot band, i.e., cis-formic acid or a non-isomeric hot band originating from thermally populated low-lying energy levels of trans-formic acid, should increase in intensity with nozzle temperature, whereas any formic acid cluster band will decrease due to thermal dissociation.
The spectra in the O-H stretching region show one band that increases in intensity with temperature at 3637 cm À1 . The band position is in good agreement with the harmonically calculated, n 1 (F)-scaled band position of cF with a deviation of only 5 cm À1 . Either the anharmonicity of F and cF is similar or there is error compensation with the density functional used. Another way of validating this assignment is to compare the intensity ratio of the cF (3637 cm À1 ) and F (3570 cm À1 ) bands with the energy difference of both forms. The harmonically calculated energy difference  15.9 kJ mol À1 (B3LYP-D3(BJ)/aVTZ with zero point energy correction) is just below the error bounds of the only experimental value of 1365 AE 30 cm À1 by W. Hocking. 22 Neglecting differences in the partition function of the two complexes, this corresponds to a population of 1-2% of cis-formic acid at 190 C. Aer correction by the theoretical cross-section ratio, the ratio of the experimental band integrals provides a cis-abundance of 2%, thus reaffirming the cF assignment. The additional bands downshied compared to the O-H stretching vibration of trans-formic acid at 3560 cm À1 and 3566 cm À1 are most likely trans-formic acid combination bands of n 2 with the lowest frequency vibrations n 7 (3560 cm À1 ) and n 9 (3566 cm À1 ), which benet from the large Raman scattering cross-section of the C-H stretching vibration. The former is in good agreement with the predicted values of Tew and Mizukami (3566 cm À1 ) 14 as well as Richter and Carbonnière (3558 cm À1 ) 15 and the latter with a prediction of Tew and Mizukami, 14 who reported (n 2 + n 9 ) in Fermi resonance with (n 3 + 3n 9 ) at 3571 cm À1 and 3579 cm À1 .
The n 2 region is spectrally more congested due to its low sensitivity to hydrogen bonding. In the spectral windows 2970-2945 cm À1 and 2935-2925 cm À1 , there are several bands that decrease in intensity with temperature, i.e., cluster bands. The broad underlying signal is due to rovibrational O and S branches of n 2 . As opposed to the O-H stretching region, there are two distinct bands increasing in intensity with temperature at 2925 cm À1 and 2873 cm À1 . The latter deviates from the predicted band position of cis-formic acid by 14 cm À1 . The amount of cis-formic acid at 190 C deduced from the integrated intensities of the bands amounts to 1%, which ts the energy difference, as detailed above. Therefore, the band at 2873 cm À1 can be assigned to cF. The second hot band at 2925 cm À1 is shied by À17 cm À1 from the fundamental of F (2942 cm À1 ). For an assignment to F, two things need to be considered: rstly, the shi directly yields the off-diagonal anharmonicity constant x 2i between n 2 and a low-lying energy level v i that is thermally populated. Secondly, the intensity ratio is dependent on the Boltzmann population of that level and yields the excitation energy of the latter. Hence, the assignment can be checked by comparing the experimentally determined anharmonicity constant and intensity with the calculated values for the lowest-lying energy levels of trans-formic acid. From the anharmonicity matrix elements in Table 1 it is apparent, that the hot band originating from n 7 will most likely overlap with the fundamental, whereas the hot band originating from n 9 (and n 6 ) could overlap with a cluster band at 2938 cm À1 causing the highest nozzle temperature spectrum (red) and the lowest nozzle temperature spectrum (black) to have similar intensities. However, due to the spectral congestion in this area, reliable assignments are not feasible. Additional depolarisation measurements to subtract the O and S branches from the sharp Q peak are currently ongoing and will be addressed in detail in a subsequent publication. Here we focus on the straightforward assignment of the 2925 cm À1 band. Its observed shi of À17 cm À1 perfectly matches the calculated anharmonicity constant x 28 . The expected intensity ratio at 190 C of around 4% approaches the observed ratio of 3%, so that it can be assigned to n 2 + n 8 À n 8 . A rst analysis of the n 3 spectral region at nozzle temperatures of 23 C, 110 C, 140 C, and 170 C can be found in a previous publication. 54 Omitting the room temperature measurement in this work is intentional, as it is heavily congested with cluster bands due to the fairly high concentrations and reservoir pressures chosen. These, however, are essential to obtain high monomer signals of F, and especially cF, aer thermal dissociation of the clusters. Due to an improved signalto-noise ratio compared to the previous measurements as well as the somewhat higher upper nozzle temperature available in the present work, we have reanalysed the n 3 region. Briey, the cis-formic acid band can be seen at 1818 cm À1 , which deviates from the calculated, n 3 (F)-scaled band position by 3 cm À1 . The hot band downshied by 7 cm À1 from F can most likely be attributed to n 3 + n 7 À n 7 , with a negligible (1 cm À1 ) deviation of the calculated anharmonicity constant x 37 compared to the experimentally observed value and a reasonable Boltzmann population match (14% from the level energy and 10% from the Raman spectrum). The hot band intensity qualitatively rules out major contributions from higher energy levels such as n 8 . There are two weaker potential hot bands shied from F by +6 cm À1 and À13 cm À1 with intensities of around 1-2% compared to F. An assignment is not possible since the shis do not match the predicted anharmonicity constants (cf. x 39 , x 38 , x 36 , and x 35 in Table 1). As previously seen, the intensity ratio gives only a rough estimate of the energy level and as such, cannot serve as a stand-alone assignment criterion. Overall, this highlights the importance of the n 2 region with its much higher monomer signal due to the large Raman scattering cross-section of the C-H compared to the C]O stretching vibration.
In the C-O stretching region, three hot bands can be seen downshied from the fundamental n 6 of trans-formic acid at 1105 cm À1 . The shis amount to À4 cm À1 , À7 cm À1 , and À11 cm À1 with intensities of around 7%, 3%, and 7% of n 6 at 190 C. To assign n 6 of cis-formic acid, the shis are compared to the calculated anharmonicity constants x 6i in Table 1. The predicted anharmonicity constants x 67 and x 68 agree (À3.7 cm À1 ). In addition, x 69 and x 66 are very similar (À5.6 cm À1 and À6.2 cm À1 ). Therefore, it seems likely that the bands at 1101 cm À1 and 1097 cm À1 are a result of overlapping hot bands. The slightly higher intensity of the former is a result of the greater overlap with the fundamental and the lower energy of n 7 and n 8 compared to n 9 and n 6 . The next higher energy level is n 5 with a predicted band position of 1219 cm À1 . A hot band originating from n 5 is expected to be shied by À14.3 cm À1 from n 6 of trans-formic acid (cf. Table 1), which is close to the experimentally observed shi of À11 cm À1 of the third hot band. However, the intensity of that band is with 7% of n 6 distinctly larger than the expected 2% from thermal population at 190 C, especially considering that the observed intensities of all other hot bands are smaller than or equal to the predicted values. Hot bands from higher energy levels can therefore also be ruled out as these should have even lower intensities. The predicted band position of cis-formic acid deviates by À5 cm À1 from the band at 1093 cm À1 , which falls within the accuracy of the n 6 (F)-scaled harmonic B3LYP-D3(BJ)/aVTZ calculations. Additionally, the observed intensity matches the calculated energy difference between both rotamers, considering the four times larger predicted scattering cross-section of n 6 of cis-formic acid compared to the trans-form (cf. Fig. 2). Consequently, the band at 1093 cm À1 can be predominantly assigned to the C-O stretching vibration of cis-formic acid.
The band positions of all stretching vibrations of cis-formic acid as well as that of the out-of-plane O-H bending vibration (n 9 ) determined from high resolution FTIR measurements 5 are summarised in Table 2 in comparison to the values obtained in an argon matrix by Maçôas and co-workers. 42 The argon matrix shis range from +27 to À20 cm À1 or +23 to À21 cm À1 , dependent on the matrix site. This scatter is of a similar order of magnitude as the cis-trans spectral differences themselves, which are also listed in Table 2. It is therefore evident that band positions in a perturbation-free environment are crucial for a direct comparison with theory values such as those of Tew and Mizukami 14 and Richter and Carbonnière. 15

Vibrational benchmark
So far, the band positions of cis-formic acid have been compared to v i (F)-scaled harmonic band positions calculated at the B3LYP-D3(BJ)/aVTZ level, which has shown to be quite valuable in supporting the assignment. The small size of the formic acid monomer and its structural rigidity enable anharmonic vibrational perturbation theory calculations (VPT2), 73 which have proven to be robust for the trans-formic acid monomer at various levels of theory in a study of the trans-formic and -acetic acid monomers and their nitrogen clusters. 53 This is less the case for the trans-acetic acid monomer, where the presence of the oppy methyl groups resulted in instabilities such as a wavenumber increase of the lowest frequency vibration compared to the harmonic case. The newly determined band positions of cis-formic acid thus enable a signicantly extended VPT2 benchmark involving both rotamers, which should not suffer from such methyl torsion instabilities.
The experimental values that will be employed in the benchmark are the ve band positions of cis-formic acid as well as the band position difference between the cis-and trans- Table 1 Calculated anharmonic (VPT2) band positions (in cm À1 ) of trans-formic acid alongside calculated diagonal (italics) and off-diagonal anharmonicity constants x 2i , x 3i , and x 6i (in cm À1 ) of n 2 , n 3 , and n 6 with all nine fundamentals  The vibrational levels were obtained by using vibrational conguration interaction (VCI) with an internal coordinate path Hamiltonian for the isomerisation path connecting both rotamers. 14 Richter and Carbonnière have constructed a valence coordinate potential energy surface at the CCSD(T)-F12a/aug-cc-pVTZ level and carried out the vibrational energy calculations with the improved relaxation multi-conguration timedependent Hartree (MCTDH) method. 15 Firstly, the performance of VPT2 calculations at various levels of theory will be discussed before these will be compared with the VCI and MCTDH calculations. One should note that for the O-H stretching vibration this comparison can solely be made with the MCTDH calculations, as the theoretical band position of the cis O-H stretching vibration has not been reported by Tew and Mizukami. Due to error compensation, a better agreement between experiment and the tested methods is typically achieved for the shi between the cis-and transrotamers. The absolute band position is predicted correctly in two cases, namely with MP2/aVTZ and MP2/aVQZ for the C-O stretching vibration (n 6 ). All other methods fail to predict the cisformic acid band positions correctly despite generous experimental error bars for the stretching vibrations. An accurate prediction of n 9 (and the respective shi) is evidently unrealistic due to the high accuracy of the high resolution measurements. The lower resolution Raman spectra are seen to be fully adequate to challenge theory on an absolute wavenumber scale. The vibrations where the shi is predicted within the experimental error for most methods are the C]O and C-O stretching vibrations, whereas the largest divergence is observed for the C-H stretching vibration. This is not surprising as the C-H stretching vibration is prone to stretch-bend Fermi resonance, although the VPT2 code employed 61 attempts to include such pronounced resonances. Consequently, part of the discrepancy may be due to a poor vibrational description by VPT2 rather than the electronic structure calculation. The particularly drastic failure for M06-2X is caused by an inversion of the predicted energy sequence for the C-H stretch fundamental and C-H bending overtone of cis-formic acid, which is amplied by Fermi resonance. If the band labels are switched, the agreement increases signicantlythe severe underestimation of the band position of À142 cm À1 (Fig. 3) changes to an overestimation of +10 cm À1 . The band position shi improves from À186 cm À1 (Fig. 3) to À34 cm À1 , compared to the experimental value of À69 cm À1 .
A comparison of the vibrationally averaged, calculated rotational constants for all methods with the experimental values for cis-formic acid obtained by Winnewisser and co-workers 39 is shown in Table 3. Small individual deviations on the order of AE0.5% are observed for B2PLYP-D3(BJ), B3LYP-D3(BJ), and MP2, larger deviations of up to 1-2% for M06-2X, uB97-XD, and PBE0-D3(BJ), and very large deviations for PM3. The average deviation over all three rotational constants (last row in Table 3) supports the overall agreement with the experimental structure. The B3LYP-D3(BJ) structure shows the best agreement with a divergence of À0.1%, directly followed by B2PLYP-D3(BJ) (À0.3%). For MP2, the divergence is slightly larger as all rotational constants are underestimated and thus, do not compensate each other. The same is valid for M06-2X, uB97D, and PBE0-D3(BJ), where all constants are overestimated.
A comparison of the individual performance of the methods for the determination ofñ i (cF) and Dñ i (cF-F) clearly illustrates that there are few reliable methods. In case of PM3, this is not surprising. It is the only method that fails to predict the energetic order of the vibrations correctly with n 4 and n 6 switched. Other methods with particularly severe deviations from experiment are uB97-XD (cf. n 1 and n 9 ) and M06-2X (n 1 , n 2 , and n 9 ). The large underestimation ofñ 2 (cF) and Dñ 2 (cF-F) of M06-2X is enhanced by a level switch between resonance partners, as discussed above. All other methods predict the correct sequence of fundamental and overtone. Another numerical or fundamental deciency of M06-2X/aVQZ VPT2 is the incorrect sign of the total anharmonicity of n 9 of cis-formic acid, which gives rise to a large overestimation of the anharmonic band position (+163 cm À1 ). In combination with an overestimation of the negative anharmonicity of n 9 of trans-formic acid, this results in a severe overestimation of the shi (+308 cm À1 ) between both rotamers. As such, this data point has been omitted from Fig. 3. The PBE0-D3 calculations match the experimental shis in two cases (n 3 and n 6 ), whereas B3LYP-D3(BJ) predicts the shis correctly in three of the ve cases (n 1 , n 3 and n 6 ). Both exhibit similar deviations with respect to the band positions. Since the rotational constant prediction of B3LYP-D3(BJ) is also more accurate, it is the overall better choice. MP2 is particularly good for the description of the lower frequency modes n 6 and n 9 and overshoots for n 1 and n 2 . For n 3 , an agreement with the shi is reached with the largest basis set aVQZ. It is generally rewarding that basis set sensitive methods tend to move towards the experimental region with increasing basis set size (cf. Fig. 3). Another reliable method is B2PLYP-D3(BJ), which predicts the shis correctly in three cases (n 1 , n 3 and n 6 ) and shows only small deviations for the other two. The band positions are slightly, but consistently underestimated, apart from n 2 , where a small overestimation occurs for the larger basis sets, and n 9 , which is slightly overestimated for all basis set sizes. The band positions and shis obtained from the VCI calculations of Tew and Mizukami 14 show good agreement with experiment. For all stretching vibrations, the band positions are overestimated and n 9 differs solely by À1 cm À1 . With regard to the shis, only one is predicted within the experimental uncertainty (n 3 ), but the shi of n 9 differs solely by about 1 cm À1 . Deviations are generally small and on the same order as for B2PLYP-D3(BJ)/aVQZ VPT2 or MP2/aVQZ VPT2. The agreement of the MCTDH calculations of Richter and Carbonnière 15 with experiment is even slightly better. The band position shis between both rotamers are predicted accurately for all stretching vibrations apart from n 3 , where the value is with 36 cm À1 just outside the experimental condence interval (41 AE 4 cm À1 ). The band position of the C-H stretching vibration is predicted within the experimental accuracy and the n 9 prediction deviates only by 2 cm À1 . However, the latter gas phase value was the only band position of formic acid known in the gas phase before ref. 14 and 15 were published, whereas the other cis-values were true predictions for the isolated molecule.
Another way of visualising the agreement of the theoretical predictions of Tew and Mizukami, Richter and Carbonnière, and results obtained with vibrational perturbation theory (B2PLYP-D3(BJ)/aVQZ VPT2) with experiment is shown in Fig. 4. In these three diagrams, the eight accessible deviations from experiment are plotted in units of experimental condence interval for all four stretching vibrations in the form of octagons. Each axis connecting two vertices of the octagons corresponds to one of the four vibrations. The two directions of each axis display the two experimental observables for each vibration, namely the cis-formic acid band position (c i ) and the band position shi between cis and trans (D i ). The size of the deviation from experiment is encoded in the octagon size. A point on a node with the smallest octagon translates into theoretical agreement within the experimental error bars (AE2 cm À1 for the band position and AE4 cm À1 for the shis). Correspondingly, a point on a node with the nth octagon implies a deviation of that value by up to n condence intervals from experiment. The predicted band position for the C]O stretching vibration of cisformic acid by Tew and Mizukami (1824 cm À1 ) 14 deviates by +6 cm À1 from the Raman jet value of 1818 cm À1 . Considering the experimental condence interval of AE2 cm À1 , the prediction for c 3 lies on the third octagon, or in other words, three nodes away from the origin on the c 3 axis. Note that the origin in these diagrams cannot be met due to the experimental uncertainty. Carbonnière offers a slightly better description of the vibrations scrutinised here. The VPT2 calculations at the B2PLYP-D3(BJ)/ aVQZ level are seen to provide a less expensive alternative. This good performance of the double hybrid functional has recently been illustrated for pyruvic acid by Barone et al. 75 For formic acid, there are some interesting systematic errors, which have consequences when looking at matrix isolation spectroscopy. Supercially and surprisingly, the comparison of VPT2 anharmonic data for trans-formic acid only improves slightly when moving from a matrix to the gas phase. 12 This is largely due to substantial downshis of polar (O-H, C]O) stretching vibrations in an Ar matrix, which mimic the underestimation of these vibrations by the B2PLYP functional in the gas phase. Such good agreements for the wrong reason must be avoided in proper benchmarking. Only the gas phase comparison can provide a realistic picture of the electronic structure performance.
With regard to the previous assignment of hot bands of trans-formic acid, the coupling constants to levels with signicant thermal population at 190 C predicted with B3LYP-D3(BJ)/ aVTZ (see Table 1) are in good agreement with those at the B2PLYP-D3(BJ)/aVQZ VPT2 level. The largest discrepancy amounts to 0.6 cm À1 (x 36 ), which is below the spectral resolution of the Raman experiment.   Fig. 3. The color shade shows whether the experimental observable is overestimated (+), underestimated (À), or met.

Instabilities of DFT functionals
As previously mentioned in Section 2.2, all production calculations have been carried out without the use of symmetry using the pruned ultra ne integration grid of Gaussian 09. 61 To explore the inuence of symmetry and grid size, additional calculations have been performed exploiting the C s symmetry and a ner integration grid (super ne integration grid, (150 974)). 61 For the following analysis, the ve vibrations discussed in this work have been considered for both rotamers, i.e., 10 values. All density functional theory methods show deviations for anharmonic frequency (VPT2) calculations with and without the use of symmetry when the integration grid size is kept constant, whereas the results obtained with PM3 and MP2 have a negligible (#0.2 cm À1 ) dependence on symmetry. The size of the deviation depends largely on the density functional theory method used as well as on the vibration. The most sensitive vibrations of the fundamentals discussed in this work are the O-H stretching (n 1 ) and out-of-plane bending vibration (n 9 ), while the smallest deviations are observed for the C]O (n 3 ) and C-O stretching vibrations (n 6 ). For B3LYP-D3(BJ), B2PLYP-D3(BJ), and PBE0-D3(BJ), these deviations are below AE10 cm À1 , with mean absolute deviations of 2.5 cm À1 , 1.7 cm À1 , and 2.1 cm À1 for the ultra ne integration grid, respectively. Particularly severe divergence is observed for uB97-XD and M06-2X with discrepancies of up to À96 cm À1 and 133 cm À1 , respectively. The mean absolute deviations for these methods are as large as 30.2 cm À1 (uB97-XD) and 59.5 cm À1 (M06-2X). These can be reduced by using the ner integration grid (super ne integration grid). This is illustrated in Fig. 5, where the mean absolute deviation of the band positions using C 1 and C s symmetry is plotted for both grid sizes (black and blue squares). This decrease in divergence, however, occurs at the expense of distinctly higher computational costs. In case of uB97-XD and M06-2X, this leads to an mean absolute deviation of 2.8 cm À1 and 48.0 cm À1 . The large value for M06-2X is caused by outliers where the deviation between calculations with C s and C 1 symmetry is enhanced by using the ner grid (n 9 (cF, F) and n 6 (cF)).
When just the integration grid is varied and the symmetry is kept xed (either C 1 or C s ), the band positions vary on average between 1-2 cm À1 for B3LYP-D3(BJ), B2PLYP-D3(BJ), and PBE0-D3(BJ). This is on the same order of magnitude as the symmetry effects discussed above. Again, a huge impact of the integration grid size is seen for uB97-XD and M06-2X, where mean absolute deviations of 28.6 cm À1 and up to 63.5 cm À1 are observed (cf. orange and green points in Fig. 5). In both cases, the deviations are larger for the C s symmetry, whereas for the other methods, it is the other way around.
Altogether, these symmetry and integration grid size dependent variations in anharmonic band positions of the fundamentals of cis-and trans-formic acid are on the order of magnitude of the experimental error bars for B3LYP-D3(BJ), B2PLYP-D3(BJ), and PBE0-D3(BJ). Nonetheless, one should keep in mind that individual outliers are slightly larger. Anharmonic frequency calculations with uB97-XD and M06-2X on the other hand, show substantial differences with regard to the symmetry and integration grid chosen, so that these results must be viewed with caution, as has been discussed before. 76,77 For most methods, the best agreement with experiment is achieved with the C s symmetry and the ner integration grid. Since the improvement of the accuracy is below the experimental condence interval for the more reliable DFT methods, if present at all, Fig. 3 and 4 would only change slightly.

Conclusions
Overall, thermal excitation combined with rapid jet quenching and Raman probing as reported in this work provides access to the four stretching vibrations of cis-formic acid in a perturbation-free environment. These reference data points are essential for the validation and comparison of modern quantum chemical methods towards a more global description of this model system. Recent examples are VCI calculations of Tew and Mizukami 14 and MCTDH calculations of Richter and Carbonnière. 15 However, it was also shown that vibrational perturbation theory can be a good compromise between accuracy and computational costs for a reasonably rigid molecule like formic acid, if combined with an adequate method for the electronic structure calculation. In this case, the double hybrid method B2PLYP-D3(BJ)/aVQZ and MP2/aVQZ offer a good compromise between accuracy and cost efficiency, in particular for differences between corresponding cis-and trans-vibrations. A benchmark examining various levels of theory revealed the failure of methods like M06-2X/aVQZ VPT2 or uB97-XD/aVQZ VPT2 to give consistent results, partly due to numerical grid size and symmetry sensitivity. With the single gas phase value from 2006 (ref. 5) available up to a year ago, these conclusions could not have been drawn. A side effect of the thermal population of cis-formic acid is the signicant enhancement of hot bands of trans-formic acid compared to room temperature spectra. The anharmonicity constants that can be deduced from Mean absolute deviations (MAD, in cm À1 ) of anharmonically (VPT2) calculated band positions of the stretching vibrations (n 1 , n 2 , n 3 , and n 6 ) and the O-H out-of-plane bending vibration (n 9 ) of cisand trans-formic acid resulting from the usage of symmetry (C s ) compared to no symmetry (C 1 ) or the increase of the DFT integration grid size (super fine integration grid compared to the ultra fine integration grid), as implemented in Gaussian 09. 61 these can help to validate combination band assignments, which are in some cases still under debate. 15 Finally, further experiments such as deuteration or depolarisation experiments will help to shed more light on various debates surrounding trans-formic acid. A prominent example is the assignment of n 5 and the overtone 2n 9 , where calculations of Tew and Mizukami 14 and Richter and Carbonnière 15 disagree with the experimental, infrared spectroscopic assignments of Freytes and co-workers 37 as well as Raman spectra of Bertie and Michaelian. 27 Additional Raman data recorded in the fashion shown here, i.e., in combination with thermal excitation, show a distinctly higher intensity for the band previously assigned to the overtone of n 9 (1305 cm À1 (ref. 37)) compared to n 5 (1223 cm À1 (ref. 37)), making this a fascinating disagreement of IR and Raman intensity patterns to be resolved. 8,45 Indeed, a very recent IR investigation 55 points into the same direction.

Conflicts of interest
There are no conicts to declare.