The highest frequency hydrogen bond fundamental of formic acid dimer, ν_{24} (B_{u}), is experimentally located at 264 cm^{−1}. FTIR spectra of this in-plane bending mode of (HCOOH)_{2} and band centers of its symmetric D isotopologues (isotopomers) recorded in a supersonic slit jet expansion are presented. Comparison to earlier studies at room temperature reveals the large influence of thermal excitation on the band maximum. Together with three B_{u} combination states involving hydrogen bond fundamentals and with recent progress for the Raman-active modes, this brings into reach an accurate statistical thermodynamics treatment of the dimerization process up to room temperature. We obtain *D*_{0} = 59.5(5) kJ/mol as the best experimental estimate for the dimer dissociation energy at 0 K. Further improvements have to wait for a more consistent determination of the room temperature equilibrium constant.

## I. INTRODUCTION

Accurate experimental determinations of the 0 K dissociation energy *D*_{0} for hydrogen bonded clusters are rare. In the elementary case of HF dimer and its isotopologues, a direct measurement based on predissociation fragments became possible^{1} after a spectroscopic absolute intensity analysis.^{2} Recently, a similar fragment analysis provided the first reliable experimental value for water dimer.^{3} In considerably more complex systems with an aromatic chromophore, UV laser experiments can provide tight lower and upper bounds.^{4–6} For formic acid dimer, the prototype of a double hydrogen bridge, neither of these approaches is feasible. However, there is an increasingly complete spectroscopic data set of the dimer at low temperatures.^{7–10} Thus, one may quantify the dissociation equilibrium constant *K*_{p} near room temperature and combine this with the rovibrational partition function of the dimer from a complete jet-spectroscopic characterization of the low frequency modes^{9} and from rotational constants of the vibrational ground state.^{8,11} This reverses the strategy employed for HF dimer,^{12} where the equilibrium constant *K*_{p} was estimated from *D*_{0} and the partition functions.

Currently, the accuracy of this approach is limited for formic acid dimer around room temperature - both the equilibrium constant for dissociation and the vibrational partition function start to be affected by large amplitude motion, isomers,^{13} and metastable collision states,^{14} whereas larger clusters should not be a problem.^{15}

The present work addresses the vibrational partition function issue by adding key missing elements for its accurate extrapolation to room temperature - the reliable band center determination for a poorly characterized low frequency mode and the observation of further combination bands involving low frequency fundamentals. For this purpose, we present the first structured FTIR jet spectrum below 300 cm^{−1} of any hydrogen bonded complex or possibly even molecule.

## II. EXPERIMENTAL

Formic acid (Sigma-Aldrich, >98%) or one of its isotopologues (Sigma-Aldrich, 95%, 98% D) was mixed with varying amounts of He (Linde, 99.996%) and collected in a 63 l reservoir at a pressure of 0.7 bar. From there, about 10% were dumped within 0.24 s (0.42 s) through 6 high-throughput magnetic valves into a 0.5 l pre-expansion volume. Expansion took place through a 600 mm long and 0.2 mm wide slit into a 23 m^{3} vacuum chamber pumped at 2500 m^{3}/h. The spatial and temporal core part of the expansion was probed through CsI optics by a mildly focussed Bruker IFS 66v FTIR beam modulated by a Si-coated Mylar beam splitter. The attenuation of the far infrared Globar radiation was monitored with a l-He cooled Si-bolometer, equipped with a 800 cm^{−1} cut-on filter and a polyethylene window, at a nominal sample rate of 20 kHz (40 kHz), corresponding to a single scan duration of 0.20 s at 4 cm^{−1} (0.37 s at 1 cm^{−1}) resolution. Recovery of the vacuum took place during a 50 s (65 s) waiting time before the next gas pulse. Details may be found in Ref. 16 and references cited therein.

## III. FTIR SPECTRA AND THEIR ANALYSIS

Figure 1 shows the asymmetric hydrogen bond stretching or in-plane bending fundamental ν_{24} at 264 cm^{−1} (see Table S1 of supplementary material^{17} for a list of modes). It is the lowest frequency B_{u} mode and its Raman-active counterpart ν_{12} is observed at 242 cm^{−1}.^{9} At room temperature (trace a), it is strongly red shaded due to hot bands, as jet cooling (traces b and c) reveals. This explains why its band center had previously been placed between 237 and 268 cm^{−1},^{10,18,19} with a preference for 248 cm^{−1}.^{9,18,19} The band marked C at 270 cm^{−1} is likely due to larger clusters formed in the jet, although a higher resolution study would be desirable. Observed band positions are listed in Table S2 of supplementary material^{17} for all isotopologues and spectra are shown in Fig. 1S of supplementary material.^{17}

Weaker absorptions at 311, 419, and 451 cm^{−1} in the jet spectrum can be assigned to B_{u} combination bands based on previous Raman jet data^{9} for ν_{8}, ν_{9}, ν_{12}, and the present work (ν_{24}) as well as high resolution gas phase spectroscopy (ν_{16}).^{10} They provide important off-diagonal anharmonicity constants for these hydrogen bond modes, which are summarized in Table S3 of supplementary material.^{17} The fact that the ν_{12} + ν_{16} combination band is additive with respect to the two fundamentals gives valuable support to the near-harmonic behaviour and correct wavenumber for the lowest frequency mode. Confirmation of the combination band at 395 cm^{−1} observed previously^{20} is not possible due to a spectral transmission gap. The two combination bands involving ν_{24} show a more pronounced (1.5%) anharmonicity. A very weak feature in the Raman spectrum^{9} at 518 cm^{−1} can be attributed to 2ν_{24}, implying 2% diagonal anharmonicity for this mode.

## IV. VIBRATIONAL PARTITION FUNCTION

The goal is to determine the vibrational partition function of formic acid dimer *z*_{vD}(*T*) as reliably as possible. The analysis focusses on *T* = 296 K, but lower temperatures are even better suited for our approach. The starting point are the 24 anharmonic experimental fundamental transitions, in particular the low frequency ones which are thermally accessible. Our choice, including the value for ν_{24} obtained in the present work, is given in Table S1 of supplementary material.^{17} Infinite harmonic summation over all states building on these fundamental excitations yields *z*_{vD} (296 K) = 42.5, i.e., a vibrational ground state population of less than 2.5%. We estimate the total error in this vibrational partition function due to inaccurate band centers at ±2 or less. The 4 × 10^{5} levels up to ≈3000 cm^{−1} above the vibrational ground state account for about 97% of the 296 K population. We apply further anharmonic corrections using experimental data for some of them from this work and the literature.^{9,10} All other anharmonicities are roughly estimated, assuming 2

^{13,21,22}which start to become populated around 300 K. However, an energy difference of 20 kJ/mol and a 100-fold larger partition function due to a flatter potential energy hypersurface for these singly bound dimers would still not contribute more than 3% to

*z*

_{vD}. We consider the two errors to be comparable in size and conservatively estimate the size of the anharmonic correction at 11% ± 7%. Therefore, the final estimate for

*z*

_{vD}(296 K) is 47(4). For lower temperatures (see Table S5 of supplementary material

^{17}), the predicted vibrational partition function is more accurate.

The limiting values for the ν_{24} band center prior to this work^{9} (248–268 cm^{−1}) lead to a 4% difference in the final partition function, close to the now remaining uncertainty due to *all* 24 vibrations. A first complete set of low resolution gas phase fundamentals became available in the 1980s.^{20,23} Based on this set, the vibrational partition function was 30% higher than the present best value, largely because of the softening of the modes due to thermal excitation. Little was known about the corresponding anharmonic corrections, adding further to the error bar. The current best estimate of 47(4) for the vibrational partition function of formic acid dimer at 296 K is an order of magnitude more certain.

## V. DISSOCIATION ENERGY

The dissociation equilibrium constant *K*_{p} for gaseous formic acid dimers (*D*) into ideally behaving monomers (*M*) at temperature *T* may be written as

where *p*_{M, D} are partial pressures and *p*° = 10^{5} Pa denotes the standard pressure. *K*_{p} is related to the molecular partition functions *z*_{v}, *z*_{r}, *z*_{t} for vibrational, rotational, and translational motion and to the molar dissociation energy *D*_{0} of the dimer at 0 K via

According to this exponential relationship, a moderately accurate knowledge of the monomer and dimer partition functions and of the dissociation equilibrium for a given temperature determines the dissociation energy of formic acid dimer rather accurately.

The value of *z*_{rM} is based on the rigid rotor rotational constants.^{24} For the rotational partition function of the dimer, we assume a symmetry number of 2 and neglect tunnelling splittings due to their small size,^{8,25} which is equivalent to a doubling of the states and the use of the *D*_{2h} symmetry number of 4. High resolution IR spectroscopy^{8} and four-wave mixing experiments^{11} are within less than 0.5%. We start from the IR ground state result of *z*_{rD} (296 K) = 8.94(1) × 10^{4} and round it up to 9.0(1) × 10^{4} to account for rovibrational coupling in the low-lying vibrational states. *z*_{vM} is straightforward^{26} within the infinite harmonic summation limit using experimental (anharmonic) fundamentals (1.12 for 296 K) and the lack of low-frequency modes renders the error of this approximation small, probably well within 1%.

We have analyzed a range of experimental determinations of *K*_{p} and converted them to the present standard pressure and 296 K, using the temperature dependence of the equilibrium constant determined by Δ*H*° (Table S4 of supplementary material^{17}). These equilibrium constants fall in two broad categories - vapor density measurements^{27} and spectroscopic monomer determinations.^{26} The average value for *K*_{p}*p*° from the vapor density results (including photoacoustic resonance,^{28} but excluding the data which need major temperature extrapolation) is about 265(20) Pa, whereas spectroscopy yields significantly higher values above 350 Pa. Combining the two types of data set with a double weight for the single reliable spectroscopic source, one arrives at 300(50) Pa. Having narrowed down the partition function uncertainty in the present work to less than 10%, this *K*_{p} discrepancy is now determining the accuracy of the resulting dissociation energy of formic acid dimer. Using the vapor density consensus for *K*_{p}, the best value for *D*_{0} is 59.8(3) kJ/mol, whereas *D*_{0} = 59.1(4) kJ/mol follows from the spectroscopic monomer determination and its quoted error bar. A more accurate determination of the vibrational partition function would reduce the error bar by less than 0.2 kJ/mol. Our work thus calls for a more accurate determination of the equilibrium constant near or below room temperature.

All vapor density estimates assume an ideal mixture of (non-polar) dimers and (rather polar) monomers. The polarity of the monomers leads to a long-range influence on binary collisions, even if no dimer is formed along the trajectory. Normally, this would be absorbed into a “physical” negative second virial coefficient of the (monomeric) gas, but this is difficult to separate from “chemical” contributions due to the explicit dimerization.^{29} Spectroscopy detects non-dimer forming collisions as pressure-broadened monomer lines. Therefore, it is expected that a dissociation constant determined by monomer spectroscopy will be larger than a dissociation constant determined by the vapor density, where any contraction relative to the ideal monomeric gas must be blamed on dimers. One can summarize the situation by the following inequality between the measured total pressure *p*_{t}, the ideal gas monomer pressure

*p*

_{D}:

A *K*_{p} method matching our experimental partition function would measure both the monomer and the dimer abundance spectroscopically, ignoring the total pressure. Vapor density measurements underestimate dissociation of dimers into polar monomers, whereas direct spectroscopic determination of

*p*

_{t}. Therefore, the discrepancy in

*K*

_{p}values (Table S4 of supplementary material

^{17}) is a physical one and the truth falls in between. This leads to the best estimate of

*D*

_{0}= 59.5 ± 0.5 kJ/mol.

One may argue that the enthalpy of dissociation at room temperature is a direct observable in the vapor density measurements as a function of temperature. We have extracted the 296 K values (Table S4 of supplementary material^{17}), which typically fall in the 59 kJ/mol range, with the exception of the photoacoustic value of 62 kJ/mol. However, one should not compare these values directly with *D*_{0}, as Fig. 2 shows. Due to the creation of six rotational and translational degrees of freedom upon dissociation, the enthalpy of dissociation initially increases with temperature as long as the hydrogen bond vibrations are not populated. When they become populated, this trend reverses and with increasing anharmonicity influence, the dissociation enthalpy quickly drops below the *D*_{0} value. Our model shows that this is almost the case at 296 K, with an estimated

## VI. CONCLUSIONS

In an experimental approach to the dissociation energy of formic acid dimer at 0 K, we have removed the leading uncertainty about the vibrational partition function by completing the hydrogen bond mode information using far infrared supersonic jet spectroscopy. The last missing hydrogen bond fundamental, which stretches one hydrogen bond while it compresses the other, was located at 264 cm^{−1}. Infrared active combination bands further fix the 296 K vibrational partition function of the dimer at 47(4).

Combined with a critical analysis of the dissociation equilibrium constant at the same temperature, one arrives at a best value for *D*_{0} of 59.5(5) kJ/mol. In simple terms, it makes a difference whether one measures the (partial) pressure of monomeric formic acid exerted on the wall or whether one determines its concentration by spectroscopy. A better method would determine both, the monomer and the dimer concentration, spectroscopically. This would open the way for an improved statistical dimerization analysis.^{30}

Among the carboxylic acid dimers, formic acid dimer is the most weakly bound.^{31} It is challenging to extend the present approach to acetic acid,^{15,31} because the methyl rotors add in a non-trivial way to the partition function and are affected by dimerization. It will also be interesting to uncover zero point energy effects in isotopologues of formic acid dimer.^{12,31}

For direct theoretical comparisons, it would be valuable to extract an experimental *D*_{e} value for formic acid dimer, i.e., to assess the anharmonic zero point energy difference on the order of 5–10 kJ/mol. This requires the full set of accurate experimental fundamentals and their anharmonicities. For the time being, theoretical benchmarking will have to include a calculated zero point energy correction. These calculations can be validated using the growing number of experimental fundamentals and anharmonicity constants available for formic acid dimer. Ultimately, this will lead to an experimentally refined potential energy hypersurface.^{32} The challenge to fix the zero point energy correction within less than 0.5 kJ/mol is that it consists of three contributions of similar size but different sign: The emergence of hydrogen bond modes in the dimer, the lowering of the OH stretching modes in the dimer, and the sum of all other, more subtle effects. However, one can safely assume that the true *D*_{e} value is bracketed by the coupled-cluster singles doubles perturbative triples/aug’-cc-pVTZ results^{33} with (62 kJ/mol) and without (72 kJ/mol) counterpoise correction.

## ACKNOWLEDGMENTS

The present work was supported by the Deutsche Forschungsgemeinschaft (DFG) research training group 782 (www.pcgg.de) as well as by the DFG Grant No. Su 121/2. We thank D. Weiss and U. Schmitt for their contributions to the vibrational state enumeration.

## REFERENCES

*K*

_{p}values from previous vapor density and spectroscopic investigations (S4), and predicted thermodynamic data up to 300 K (S5).