formic acid assignment

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• AMEISENSAURE
• AMINIC ACID
• COLLO-BUEGLATT
• COLLO-DIDAX
• FORMIC ACID
• FORMIC ACID (85-95% IN AQUEOUS SOLUTION)
• FORMYLIC ACID
• HYDROGEN CARBOXYLIC ACID
• METHANOIC ACID
• METHANOIC ACID MONOMER

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I. INTRODUCTION

Ii. experimental method, iii. computational method, iv. results and discussion, c. torsional analysis, v. summary and outlook, acknowledgments, synchrotron-based infrared spectroscopy of formic acid: confirmation of the reassignment of fermi-coupled 8 μ m states.

Present address: Department of Chemistry, Emory University, Atlanta, Georgia, 30322, USA.

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Killian Hull , Tyler Wells , Brant E. Billinghurst , Hayley Bunn , Paul L. Raston; Synchrotron-based infrared spectroscopy of formic acid: Confirmation of the reassignment of Fermi-coupled 8 μ m states. AIP Advances 1 January 2019; 9 (1): 015021. https://doi.org/10.1063/1.5063010

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The empirically derived assignment of the strongly interacting 5 1 and 9 2 vibrational states of trans -HCOOH has recently been reassigned on the basis of anharmonic frequency calculations, and this, in turn, affects the assignment of many higher energy states. Here, we investigate the high-resolution synchrotron-based torsional spectrum of trans -HCOOH, and find experimental confirmation that the proposed reassignment is indeed correct, i.e., that 9 2 is in fact lower in energy than 5 1 . This is largely based on examining the intensity ratio of transitions with the same rotational quantum numbers between the 9 2 -9 1 and 5 1 -9 1 hot bands, which indicates that the 5 1 [9 2 ] state has ∼31% 9 2 [5 1 ] character. We also examined the torsional spectrum of trans -HCOOD, and find that the intensity ratios are consistent with 9 2 instead being higher in energy than 6 1 (which is analogous to 5 1 in trans -HCOOD), as previously determined from higher energy spectra.

Formic acid, the simplest carboxylic acid, is an important atmospheric species that is known to be responsible for a large fraction of the acidity of typical precipitation. 1 Significant biogenic sources are from plants and soil, bacterial and human metabolism, and the venom of ants and bees. 2 Its biological importance and simplicity makes it a sought after astrochemical species, and it has been detected in several interstellar molecular clouds (Sagittarius B2 3,4 and others 5 ), and very recently in a protoplanetary disk. 6 In those studies it was the trans rotamer (eclipsed OH and CH bonds 7 ) that was detected, which lies 3.90±0.09 kcal/mol (1365±30 cm -1 ) lower in energy than the cis rotamer. 8 Recently, the cis rotamer has been detected for the first time in interstellar space, in the far-UV illuminated edge of the Orion Bar. 9  

While both rotamers of gas phase formic acid have been investigated by infrared spectroscopy, it is the trans rotamer that has been the most thoroughly studied (e.g., see Refs. 10–14 ). This is on account of the large energy difference between the two, which results in a very low population of formic acid at room temperature in the cis torsional configuration (population ≈ 0.1 %). Bonner and Hofstadter were the first to report the infrared spectrum of gas phase formic acid, where they noted an intense band around 658 cm -1 that they assigned to monomeric formic acid. 10 This was assigned to the OCO bending vibration (ν 7 ) several years later, 15 and then was re-observed with higher resolution at 636 cm -1 where it was reassigned to both OCO bending and (correctly) OH torsion (ν 9 ) 16 (but refined to just OH torsion some two years later 12 ). The first high resolution gas phase infrared investigation was by Deroche in 1979, 17 where they performed a thorough analysis of the Coriolis interacting 7 1 and 9 1 states of trans- HCOOH. It wasn’t until much more recently (2006) that an infrared band of the cis rotamer has been observed in the gas phase for the first time (ν 9 fundamental), 18 which was made possible by the large difference between the energy levels in the cis and the trans potential wells. This results in a large separation between the torsional fundamentals, by an amount that is coincidentally about the same as that for vinyl alcohol. 19  

The interest in investigating the infrared spectra of formic acid has sparked a great deal of theoretical interest over the years; from early “experimental” force constant calculations (e.g., Refs. 14 , 20–22 , to more computationally demanding ab initio calculations (e.g., Refs. 23–27 ). Accurately predicting the moderately high 1365 cm -1 difference between the ground states of the two rotamers has been a subject of interest, and in part results from the partial double bond character of the C-O bond about which rotation occurs. 28,29 Modern multi-dimensional ab initio calculations have managed to provide differences that are in good agreement with experiment, 8 although they are unanimously somewhat larger (1412 cm -1 and 1415 cm -1 ). 26,27 One of the most important recent theoretical developments involve anharmonic calculation of overtone and combination band states, where reassignment of the experimentally determined 30 5 1 and 9 2 state assignments was made. 25,31 Even more recently, a large number of additional vibrational bands were reassigned that involve the Fermi coupled 5 1 and 9 2 states of trans- HCOOH. 26,27 These calculations, which use high quality ab initio data and very different vibrational methods, suggest that most assignments involving these bands in previous spectroscopic investigations should be switched (note that we use these switched labels throughout, unless stated otherwise). Motivated by this major reassignment, we decided to investigate the synchrotron-based infrared spectra of trans- HCOOH and HCOOD in an effort to see if their computational predictions hold true.

The FTIR spectra of HCOOH and HCOOD was measured at the far-infrared beamline of the Canadian Light Source. This beamline is equipped with a Bruker IFS 125HR FTIR spectrometer, and a 2 m long multipass cell that was utilized in this work. The cell alignment was optimized for 36 transits of the synchrotron radiation, giving an optical pathlength of 72 m. We introduced low pressure (4 mTorr - 56 mTorr) samples, and used an elevated cell temperature (∼333 K) so that higher energy torsional states were more populated. The maximum achievable resolution was used (0.00096 cm -1 ), which corresponds to an optical path length difference between the interferometer mirrors of 9.4 m. Interferograms were accumulated over the space of 21-35 hours and then Fourier transformed. The resulting transmittance spectra were then converted to absorbance spectra by dividing by an appropriate reference spectrum. For HCOOH, we optimized the optics for the 400-1250 cm -1 range, and for HCOOD we chose optics that was best suited to cover 330-550 cm -1 . For further details of the far-infrared beamline, please see Ref. 32 .

The torsional potential for formic acid was constructed by performing a relaxed scan of the H-O-C=O dihedral (torsional) coordinate in 15° increments using Gaussian 09. 33 We performed the optimizations at the CCSD(T) level of theory, with correlation consistent double, triple, and quadruple ζ basis sets (cc-pVDZ, cc-pVTZ, cc-pVQZ). The potential energies, V ( ϕ ), were extrapolated to the complete basis set (CBS) limit, as were the dihedral angle dependant internal rotational constants, F ( ϕ ). The F ( ϕ ) values were determined by the Pitzer method, which requires atomic masses and optimized coordinates as inputs, as previously outlined. 34  

We then fit the potential energies and internal rotation constants to the following Fourier expansions:

The resulting V n and F n coefficients were used in the following internal rotation Hamiltonian, which was solved for the eigenvalues using 100 sine-cosine basis functions: H = − d d φ F ( φ ) d d φ + V ( φ ) ⁠ . We used a modified version 35 of a previously used script 36 in the calculations. For further details of this well established method, see Refs. 37–38 .

Figure 1 shows the spectrum of HCOOH in the vicinity of the torsional fundamental band. It is shown along with a simulation (in PGOPHER 39 ) using previously reported parameters 40 to illustrate there is little unaccounted for intensity. Upon close inspection, however, there are hot band lines that are buried in the much stronger fundamental band transitions; their band centers are indicated in Figure 1 by the labels “HB1” and “HB2” (HB = hot band). It is interesting to note that in a pioneering study on formic acid, Miyazawa and Pitzer (incorrectly) assigned HB1 to the torsional fundamental of cis- HCOOH and (correctly) assigned the main peak to the torsional fundamental of trans- HCOOH. 12  

High-resolution experimental (black) and simulated (red) spectra of the ν 9 fundamental of trans- formic acid. The simulation was made using the spectroscopic parameters reported in Ref. 40 , using PGOPHER. 39 The inset shows the principal inertial axes on the CCSD(T)/cc-pVQZ optimized structure. The cell pressure was ∼4 mTorr.

At about twice the energy of the 9 1 state are the 5 1 and 9 2 states, and it is these that must correspond to the upper states in the observed hot bands. While a band at ∼1230 cm -1 was identified and assigned to the ν 5 fundamental over 60 years ago, 13 as far as we are aware, a band involving 9 2 was only first assigned much more recently. 30 In that investigation, Baskakov et al. assigned the 9 2 state based on 1) their analysis of the “9 2 -9 1 ” hot band and 2) its closeness in proximity to the predicted value “expected from the harmonic approximation”. As mentioned above, modern ab initio calculations suggest that the 5 1 and 9 2 state assignments should be switched. 25 In the following we analyze HB1 and HB2 in an effort to find definitive experimental evidence for either assignment; in particular, we address points 1) and 2) mentioned above.

Figure 2 shows the high resolution spectrum of HCOOH in the region of the p Q 1 ( J ) series for both HB1 and HB2. Lines were attributed to specific rovibrational transitions using predictions that were generated from the spectroscopic parameters in a previous analysis [that included 114 diagonal and 105 off diagonal parameters (!)]. 30 We found some discrepancies that were particularly evident at high K a values, and a thorough analysis of the spectra discussed here is ongoing and will be presented elsewhere. As can be seen in Figure 2 , the intensity of the peaks in the lower wavenumber band (HB1) are higher than in the higher wavenumber band (HB2), and this finding is pretty consistent throughout the entire spectrum. We can thus already conclude that the reassignment recently made is correct (i.e., HB1 = 9 2 -9 1 and HB2 = 5 1 -9 1 ), since the lower wavenumber band has a “bright” upper state that mixes with the “dark” upper state of the higher wavenumber band.

High-resolution spectrum of trans- HCOOH in the vicinity of the p Q 1 ( J ) series of the 9 2 -9 1 band (top) and 5 1 -9 1 band (bottom; inverted). The spectra cover the same absorbance range (so the intensities are directly comparable). The value of J is indicated by the numeral, and (apart from J = 14) lies between the corresponding peaks of each band. Assignments were made using the spectroscopic parameters reported in Ref. 30 . The cell pressure was ∼10 mTorr.

More quantitatively, assuming that the unperturbed 5 1 -9 1 band has zero intensity, and there is no rotational dependence on the mixing of states, then in a two state model, the intensity ratio, I A : I B , gives the ratio of the square of the wavefunction coefficients, a 2 : b 2 , where ψ A = a A ψ n 0 - b A ψ i 0 , and ψ B = b B ψ n 0 + a B ψ i 0 . 41 Here, the subscripted n and i correspond to ν’ = 9 2 and 5 1 (respectively), the superscripted 0 indicates the unperturbed wavefunction, and A [ B ] corresponds to the lower [upper] interacting state. Note that the choice of coefficients was made for ease of comparison with theory, and that for the two level system considered in our experimental analysis, a = a A = a B , and b = b A = b B . Now, using the normalization condition ( a 2 + b 2 = 1) and the experimental intensity ratio (ca. 2.2:1), the coefficients are a = 0.83, and b = 0.56. These values compare favourably with the coefficients determined by Tew & Mizukami (| a A | = 0.813, | b A | = 0.516, | b B | = 0.485, and | a B | = 0.836), 26 who used VSCF reference (unperturbed) states, and reasonably well with those from Richter and Carbonnière (| a A | = 0.781, | b A | = 0.447, | b B | = 0.490, and | a B | = 0.624), 27 who used internal modes as reference states.

Naturally, a direct comparison of the wavefunction coefficients between experiment and theory is somewhat hampered by the necessity of assuming a two level system in the experimental analysis. While the recent anharmonic frequency calculations predict significant contributions to the wavefunctions ( ψ A and ψ B ) from other states, their coefficients were smaller than those for 9 2 and 5 1 (so that our comparison is at least qualitatively valid). Alternatively, it is perhaps more valid to simply compare the observed intensity ratio with the calculated ratio of the square of the wavefunction coefficients; the observed value of 2.2 compares well with the calculated values of 2.7 and 2.2 (from Refs. 26 and 27 ).

It is interesting to note that the calculated wavefunction coefficients are also in agreement with expectations based on entering the Fermi interaction matrix element, W ni , and the diagonal (unperturbed) separation between vibrational states, δ , (both from Ref. 30 ) to the following formulae:

We suspect that it was not feasible to use this method in previous attempts to assign bands in the region of the ν 5 fundamental/2ν 9 overtone bands, because the uncoupled 9 2 -0 transition moment is not small compared to that for 5 1 -0.

Figure 3 shows the high-resolution spectrum of trans- HCOOD in the region of the r Q 0 ( J ) series at low J , for the same hot bands we looked at in HCOOH (with ν’ = 9 2 and 6 1 ). In the case of trans- HCOOD the intensities of the bands are flipped, i.e., the higher wavenumber band carries more intensity. This implies that the higher [lower] wavenumber band corresponds to ν’ = 9 2 [6 1 ; note that ν 6 is the COD bend, analogous to the ν 5 COH bend in HCOOH], which is in agreement with a previous analysis. 42 In that analysis, however, a first order Fermi interaction was not considered, which would mean (to first order) that we should not see transitions within the 5 1 -9 1 band (which is obviously not the case).

High-resolution spectrum of trans- HCOOD in the vicinity of the r Q 0 ( J ) series of the 5 1 -9 1 band (top) and 9 2 -9 1 band (bottom; inverted). The spectra cover the same absorbance range (so the intensities are directly comparable). The value of J is indicated by the numeral. Lines were identified using the spectroscopic parameters reported in Refs. 44 and 42 . The cell pressure was ∼56 mTorr.

We can estimate the magnitude of the first order Fermi interaction, which mixes the 9 2 and 6 1 states, and give the 9 2 -6 1 band an appreciable amount of infrared intensity. From comparing lines within both hot bands, the intensity ratio was determined to be 2.1:1, and this results in a = 0.82, and b = 0.57. Using these values we can estimate W ni for trans- HCOOD using:

Thus δ = 12.7 cm -1 and W ni ≈ 18.2 cm -1 . Using reported band positions and δ , we find that the deperturbed 9 2 state is at ∼999 cm -1 . Future work (underway) will involve including a first order Fermi coupling constant in the analysis of the interacting 9 2 and 6 1 states of trans- HCOOD.

Figure 4 shows the relaxed torsional potential (points) in the vicinity of the trans- formic acid well, along with the fitted Fourier series function (black curve); the corresponding V n coefficients are listed in Table I . The eigenvalues for trans- HCOOD obtained by solving the one-dimensional vibrational Schrödinger equation are also plotted along with the square of the wavefunctions. As can be seen, the torsional states become closer together as one goes up the well, which is due to deviation from harmonicity. This deviation can also be evidenced by comparing the torsional potential with a harmonic potential (green curve in Figure 4 ), which reveals significant deviation even for the 2 nd excited torsional state (i.e., 9 2 ). We note this anharmonicity of the potential in the torsional coordinate is well established, nonetheless, we included the analysis presented here largely to obtain predictions of the (unperturbed) torsional eigenstates for HCOOD (see following).

Ab initio torsional potential for formic acid in the trans well (see Table I for Fourier series coefficients). Energy levels and probability distributions are (only) shown for trans- HCOOD, and were determined using the method of Laane and coworkers. 37,38 Even and odd symmetry states are shown by the red and blue lines, respectively. The green curve is a harmonic (i.e. quadratic) potential that was fit to the three lowest energy points.

Fourier series coefficients from fits to the angular dependence of the torsional potential and internal rotation constants (in cm -1 ).

Table II compares the observed and calculated torsional frequencies, and reveals a fair agreement, with an RMS deviation of 15-16 cm -1 for the trans- formic acid isotopologues considered here. While this is certainly not as good as the RMS deviation in the VCI work of Tew et al. , 26 it allows for us to easily compare unperturbed calculated torsional frequencies with deperturbed experimental ones. Most importantly, the deviation from harmonicity in going up the potential ladder (difference between 9 1 -9 0 and 9 2 -9 1 states), is calculated to be ∼28 cm -1 , which agrees well with the experimental value of ∼33 cm -1 (see Table II ); 30,40 the calculated and experimental values for HCOOD also agree well. 42,44 We thus conclude, in accordance with previous theoretical predictions, that the harmonic approximation is a poor one, even close to the bottom of the trans well for formic acid.

Deperturbed experimental frequencies in comparison to unperturbed theoretical frequencies.

From Ref. 40 .

From Ref. 44 .

Determined using values from Refs. 30 (overtone) and 40 (fundamental).

Determined using values from Refs. 42 (overtone) and 44 (fundamental). Note that the overtone band origin was deperturbed in this work using an estimated first order Fermi coupling constant.

From Ref. 18 .

It is worth noting that the potential provides a useful prediction for the ν 9 fundamental of cis -HCOOD, which has not yet been investigated. While we observed the fundamental of cis- HCOOH with reasonably good signal-to-noise (S/N), which has been previously reported, 18 were unable to locate the fundamental for cis- HCOOD, and this is most likely due to the lower S/N in the HCOOD spectra around 370-380 cm -1 .

Here we have shown experimentally that the theoretically predicted reassignment of the 5 1 and 9 2 states is correct, which overturns a long-standing misassignment in the vibrational states of trans- HCOOH. Future work will involve analyzing the high-resolution torsional spectra, and refining the spectroscopic constants for the formic acid isotopologues, HCOOH and HCOOD.

Research described in this paper was performed at the Canadian Light Source, which is supported by the Canada Foundation for Innovation, Natural Sciences and Engineering Research Council of Canada, the University of Saskatchewan, the Government of Saskatchewan, Western Economic Diversification Canada, the National Research Council Canada, and the Canadian Institutes of Health Research. Acknowledgement is made to the donors of The American Chemical Society Petroleum Research Fund (56406-UN16) and the National Science Foundation (NSF-REU CHE-1461175). The authors are grateful to Isaiah Sumner for technical advice and to Colin Western for implementing special features into a development version of PGOPHER.

Note that in some of the older literature, trans was defined as having eclipsed OH and CO (instead of OH and CH) bonds.

It should be noted that a similar finding was reported earlier, but only briefly mentioned. 45,46

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Green Chemistry

Matching emerging formic acid synthesis processes with application requirements †.

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a Centre for Surface Chemistry and Catalysis: Characterization and Application Team (COK-KAT), KU Leuven, Celestijnenlaan 200F, Box 2461, B-3001 Leuven, Belgium E-mail: [email protected]

Formic acid is gaining interest as a carbon capture and utilization (CCU) product produced electrochemically from CO 2 , water and renewable energy. A common premiss is that concentrated formic acid is the desired product. This originates from the current situation with centralized large-scale industrial production of a concentrated product for reasons of economizing on distribution costs. This premiss is shaping research on green formic acid synthesis, attempting to eliminate water as much as possible from the electrolysis product. Water elimination adds substantially to the overall production cost. Interestingly, the application field of formic acid is shifting to more diluted feedstock. Emerging applications can handle diluted formic acid, and even require dilution. Examples are the use of formic acid as an energy vector, hydrogen gas carrier, syngas storage medium and carbon source for bioprocesses. This review provides an overview of formic acid concentration requirements in emerging applications. The matching of the product specifications from the electrochemical formic acid production process with the requirements of emerging applications is documented.

• This article is part of the themed collection: Green Chemistry Reviews

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Matching emerging formic acid synthesis processes with application requirements

B. Thijs, J. Rongé and J. A. Martens, Green Chem. , 2022,  24 , 2287 DOI: 10.1039/D1GC04791D

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Formic acid

• Formula : CH 2 O 2
• Molecular weight : 46.0254

• IUPAC Standard InChIKey: BDAGIHXWWSANSR-UHFFFAOYSA-N Copy
• CAS Registry Number: 64-18-6

• Other names: Methanoic acid; Aminic acid; Bilorin; Collo-Bueglatt; Collo-Didax; Formisoton; Formylic acid; Hydrogen carboxylic acid; Myrmicyl; HCOOH; Acide formique; Acido formico; Ameisensaeure; Kwas metaniowy; Kyselina mravenci; Mierenzuur; Rcra waste number U123; UN 1779; Formira; Add-F; Amasil
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Gibson, Latimer, et al., 1920 Gibson, G.E. ; Latimer, W.M. ; Parks, G.S. , Entropy changes at low temperatures. I. Formic acid and urea. A test of the third law of thermodynamics , J. Am. Chem. Soc. , 1920, 42, 1533-1542. [ all data ]

Glagoleva and Chervov, 1936 Glagoleva, A.A. ; Chervov, S.I. , Investigation of the heat capacity of formic acid and its aqueous solutions , Zhur. Obshch. Khim. , 1936, 6, 685-690. [ all data ]

Radulescu and Jula, 1934 Radulescu, D. ; Jula, O. , Beiträge zur Bestimmung der Abstufung der Polarität des Aminstickstoffes in den organischen Verbindungen , Z. Phys. Chem. , 1934, B26, 390-393. [ all data ]

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Taylor and Bruton, 1952 Taylor, M.D. ; Bruton, J. , The vapour phase dissociation of some carboxylic acids. II. Formic and propionic acids. , J. Am. Chem. Soc. , 1952, 74, 4151. [ all data ]

Anselme and Teja, 1990 Anselme, M.J. ; Teja, A.S. , The critical properties of rapidly reacting substances , AIChE Symp. Ser. , 1990, 86, 279, 128-32. [ all data ]

Ambrose and Ghiassee, 1987 Ambrose, D. ; Ghiassee, N.B. , Vapor Pressures and Critical Temperatures and Critical Pressures of Some Alkanoic Acids: C1 to C10 , J. Chem. Thermodyn. , 1987, 19, 505. [ all data ]

Majer and Svoboda, 1985 Majer, V. ; Svoboda, V. , Enthalpies of Vaporization of Organic Compounds: A Critical Review and Data Compilation , Blackwell Scientific Publications, Oxford, 1985, 300. [ all data ]

Stephenson and Malanowski, 1987 Stephenson, Richard M. ; Malanowski, Stanislaw , Handbook of the Thermodynamics of Organic Compounds , 1987, https://doi.org/10.1007/978-94-009-3173-2 . [ all data ]

Konicek, Wadsö, et al., 1970 Konicek, Jiri ; Wadsö, Ingemar ; Munch-Petersen, J. ; Ohlson, Ragnar ; Shimizu, Akira , Enthalpies of Vaporization of Organic Compounds. VII. Some Carboxylic Acids. , Acta Chem. Scand. , 1970, 24, 2612-2616, https://doi.org/10.3891/acta.chem.scand.24-2612 . [ all data ]

Stout and Fisher, 1941, 3 Stout, J.W. ; Fisher, Leon H. , The Entropy of Formic Acid. The Heat Capacity from 15 to 300°K. Heats of Fusion and Vaporization , J. Chem. Phys. , 1941, 9, 2, 163, https://doi.org/10.1063/1.1750869 . [ all data ]

Ambrose and Ghiassee, 1987, 2 Ambrose, D. ; Ghiassee, N.B. , Vapour pressures and critical temperatures and critical pressures of some alkanoic acids: C1 to C10 , The Journal of Chemical Thermodynamics , 1987, 19, 5, 505-519, https://doi.org/10.1016/0021-9614(87)90147-9 . [ all data ]

Dreisbach and Shrader, 1949 Dreisbach, R.R. ; Shrader, S.A. , Vapor Pressure--Temperature Data on Some Organic Compounds , Ind. Eng. Chem. , 1949, 41, 12, 2879-2880, https://doi.org/10.1021/ie50480a054 . [ all data ]

Dreisbach and Martin, 1949 Dreisbach, R.R. ; Martin, R.A. , Physical Data on Some Organic Compounds , Ind. Eng. Chem. , 1949, 41, 12, 2875-2878, https://doi.org/10.1021/ie50480a053 . [ all data ]

Campbell and Campbell, 1934 Campbell, Alan Newton ; Campbell, Alexandra Jean Robson , The thermodynamics of binary liquid mixtures : formic acid and water , Trans. Faraday Soc. , 1934, 30, 1109, https://doi.org/10.1039/tf9343001109 . [ all data ]

Coolidge, 1930 Coolidge, Albert Sprague , THE VAPOR PRESSURE AND HEATS OF FUSION AND VAPORIZATION OF FORMIC ACID , J. Am. Chem. Soc. , 1930, 52, 5, 1874-1887, https://doi.org/10.1021/ja01368a018 . [ all data ]

Kahlbaum, 1894 Kahlbaum, G.W.A. , Z. Phys. Chem., Stoechiom. Verwandtschaftsl. , 1894, 13, 14. [ all data ]

Kahlbaum, 1883 Kahlbaum, Georg W.A. , Ueber die Abhängigkeit der Siedetemperatur vom Luftdruck , Ber. Dtsch. Chem. Ges. , 1883, 16, 2, 2476-2484, https://doi.org/10.1002/cber.188301602178 . [ all data ]

Kahlbaum, 1894, 2 Kahlbaum, G.W.A. , Studien uber Dampfspannkraftsmessungen , Z. Phys. Chem. (Leipzig) , 1894, 13, 14-55. [ all data ]

Calis-Van Ginkel, Calis, et al., 1978 Calis-Van Ginkel, C.H.D. ; Calis, G.H.M. ; Timmermans, C.W.M. ; de Kruif, C.G. ; Oonk, H.A.J. , Enthalpies of sublimation and dimerization in the vapour phase of formic, acetic, propanoic, and butanoic acids , The Journal of Chemical Thermodynamics , 1978, 10, 11, 1083-1088, https://doi.org/10.1016/0021-9614(78)90082-4 . [ all data ]

Stull, 1947 Stull, Daniel R. , Vapor Pressure of Pure Substances. Organic and Inorganic Compounds , Ind. Eng. Chem. , 1947, 39, 4, 517-540, https://doi.org/10.1021/ie50448a022 . [ all data ]

Jones, 1960 Jones, A.H. , Sublimation Pressure Data for Organic Compounds. , J. Chem. Eng. Data , 1960, 5, 2, 196-200, https://doi.org/10.1021/je60006a019 . [ all data ]

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On the vibrations of formic acid predicted from first principles

Anna klára kelemen.

Department of Chemistry, University of Zurich, CH-8057 Zurich Switzerland, +41 44 635 42 98

Sandra Luber

Department of Chemistry, University of Zurich, CH-8057 Zurich Switzerland, [email protected] +41 44 635 44 64

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

In this article, we review recent first principles, anharmonic studies on the molecular vibrations of gaseous formic acid in its monomer form.

1. Introduction

Vibrational spectroscopies continue to be a shared interest of both theoretical and experimental approaches in chemistry, with the harmonic approximation 1 as the dominating theoretical description in practice. In contrast, methods that describe vibrations from first principles, obey to the rules of quantum mechanics and go beyond the harmonic approximation are far from routine, despite the apparent anharmonicity of many molecular vibrations. Since a clear hierarchy is not well established for these methods, investigating their performance and characteristics in a joint discussion with regard to the accuracy of the predicted vibrations of a selected molecule may be of interest in addition to the comparison to high-accuracy experimental data. Formic acid, the smallest carboxylic acid, emerges as a good candidate in this respect; it exhibits structural cis – trans isomerism and it forms a hydrogen-bonded dimer at room temperature, giving rise to complex nuclear dynamics which are difficult to model accurately. 2 In a broader perspective, understanding the nuclear dynamics of carboxylic acids could also give insight into the conformation and folding mechanisms of proteins. After a brief discussion of selected concepts employed in the anharmonic and variational computation of molecular spectra, we review experimental results and discuss ab initio calculations of the anharmonic molecular vibrations of the monomer forms of formic acid in a chronological order.

2. Selected concepts of the variational calculation of molecular vibrations

Arguably one of the most difficult aspects of the calculation of molecular vibrations beyond the harmonic approximation is the definition of an appropriate coordinate system and the derivation of the corresponding vibrational Hamiltonian. The separation of the purely vibrational motion of the molecule requires internal coordinates, 1,3 but the conditions which define a unique and an optimal coordinate system which is appropriate for various types of nuclear motion are not known. 4 The Watson Hamiltonian which was derived for non-linear 5 and linear 6 reference configurations and defines the rovibrational motion in terms of rectilinear normal coordinates remains to be one of the most popular choices, since its definition ensures minimal rovibrational coupling and an optimal representation of molecular vibrations with small displacements from the reference geometry. 1,4 It follows, that difficulties are encountered if vibrations corresponding to e.g. torsional motions or significant displacements from the equilibrium structure are computed in this representation. Slow convergence of the computed vibrational eigenstates or inaccurate results in approximate methods are often encountered in these cases. 7 Since the Watson Hamiltonian is invariant under the unitary transformations of the normal modes, attempts for improvement have been made by optimizing 8–10 and localizing 11 the normal modes. The choice of the vibrational Hamiltonian also affects the ease of interpretation of the computed results. Whereas vibrations which are well described by the harmonic approximation are found to be mostly localized on the respective normal coordinate, anharmonic vibrations which are encountered e.g. in the high-energy region, can have contributions along multiple normal coordinates, leading to increasingly difficult assignments. 12 Whereas an ill-defined coordinate system can introduce artificial correlation, a well-choosen one may minimize the correlation between the different degrees of freedom. 13

The main focus of this section is not a to give a comprehensive review of available vibrational methods, for which already vast literature is available, 4,14–19 but rather to give an overview of selected concepts and methods necessary for discussing the literature on formic acid, reviewed below. We also shall not discuss the computation of intensities, but emphasize the importance of the development of e.g. accurate electronic dipole surfaces aiding the interpretation of IR spectra.

In order to present a joint discussion of variational methods and the literature reviewed below, we introduce the following notation for the vibrational coordinates q = { q 1 ,…, q M }. Once defined, the vibrational Hamiltonian may be separated into a kinetic and potential part, where the form of both the kinetic and potential term depends on the employed coordinate system. The potential term, the so called Potential Energy Surface (PES) Created by potrace 1.16, written by Peter Selinger 2001-2019 ( q ) can be calculated either by direct evaluation of ab initio electronic energies along the vibrational coordinates, or it may be pre-fitted to an appropriate analytic form. Most commonly, either a Taylor expansion 20

where q 0 is a reference geometry, or a many-body expansion 21

is employed, where the Created by potrace 1.16, written by Peter Selinger 2001-2019 m ( q m ) are the so-called one-, Created by potrace 1.16, written by Peter Selinger 2001-2019 mn ( q m , q n ) are the two-, and Created by potrace 1.16, written by Peter Selinger 2001-2019 mn … M ( q m , q n ,…, q M ) are the M -body terms, respectively, each dependent on a subset of coordinates. The sum-of product (SOP) form 22,23

is also commonly employed. Here, the potential is expressed as a sum with expansion coefficients c r of potential terms Created by potrace 1.16, written by Peter Selinger 2001-2019 r , where the latter is expressed as a product of terms dependent only on coordinate Q k , respectively. The Q k s may combine multiple coordinates of q , therefore reducing the computational effort, i.e. f < M and r = [ r 1 ,…, r f ]. We note, that the accuracy of any vibrational structure calculation is limited by the accuracy of the PES, and therefore the electronic structure method employed for its construction.

Beyond the well-known harmonic approximation, for which the solutions can be obtained exactly due to approximating the potential as a second order Taylor expansion in normal coordinates, variational solutions 24 to the vibrational Schrödinger equation require an ansatz . In the Vibrational Self-Consistent-Field (VSCF) method, 18,25 the vibrational states are computed in the product form

where the functions dependent on a single vibrational coordinate are eigenfunctions of the mean-field operator 26 Created by potrace 1.16, written by Peter Selinger 2001-2019 n ̄ ( q m )

where we have assumed that the kinetic energy operator can be written as a sum of independent terms, Created by potrace 1.16, written by Peter Selinger 2001-2019 m ( q m ). n = [ n 1 ,…, n M ] collects the available quantum numbers associated with the single-coordinate functions ϕ n m ( q m ) determined self-consistently, and n ̄ denotes all quantum numbers but the m -th. The vibrational energies are then the expectation value of the full Hamiltonian with the VSCF wavefunction ( eqn (4) ) which can be obtained in a state-specific, for a chosen set of quantum numbers, or in a non state-specific fashion. The VSCF approximation has been improved by the correlation-corrected VSCF approach (CC-VSCF) developed by Gerber and co-workers 18,27,28 also known as second order vibrational Møller–Plesset (VMP2) theory. In this method, the deviation of the exact wave function from the mean-field solution is assumed to be small, such that a second order perturbation correction is applied for a correlation-corrected energy expression. The latter differs in the zeroth order problem from the well-known second order vibrational perturbation (VPT2) theory in which the harmonic approximation is corrected.

Going beyond the product form, a linear combination of configurations

with expansion coefficients C n may be used. A full direct product basis can be employed as a basis in eqn (6) ( e.g. products of weighted classical orthogonal polynomials, 29 harmonic oscillator basis functions 30 ), however, without further approximations, this leads to an ansatz with an exponential scaling with regard to the size of the system treated and therefore a blowup in the number of terms in the expansion. The truncation of the direct product basis according to some condition is therefore common practice. In Vibrational Configuration Interaction (VCI) 26,31,32 the configurations are usually taken from preceding VSCF calculations. The ansatz can be written as

where ψ i ( q ) is the reference configuration and Created by potrace 1.16, written by Peter Selinger 2001-2019 μ generates an excited configuration from the quantum numbers i = [ i 1 ,…, i M ]. If the full excitation space is used, the method is called full VCI (FVCI). Similarly to CI in electronic structure theory, singles, doubles, and higher excitations are generated, and the expansion is truncated to allow for a subset of the available quantum numbers and/or excitations, therefore introducing a bound on the, in principle, infinite sum. The expansion coefficients C n of the normalized wavefunction are determined variationally as eigenvectors of the VCI matrix 〈 ψ n ( q )| Created by potrace 1.16, written by Peter Selinger 2001-2019 ( q )| ψ l ( q )〉. However, the straightforward construction of the untruncated VCI matrix and computation of its eigenvectors is prohibitively expensive and in practice approximations are introduced. 33,34 By partitioning the modes into a set of active and a set of bath modes Mizukami and Tew 35 have introduced the vibrational active space self-consistent field theory (VASSCF), vibrational active space configuration interaction (VASCI) and vibrational active space second order perturbation theory (VASPT2) methods. These methods employ a product of a CI type ansatz for the active modes and a simple product ansatz for the bath modes. Whereas in their VASSCF approach all CI coefficients and all single-coordinate functions are optimized, only a subset of single-coordinate functions are optimized in their VASCI approach.

In the Multi-configurational time-dependent Hartree 22,36 (MCTDH) method a time-dependent ansatz is employed with expansion coefficients C r

where f corresponds to the number of combined coordinates (particles) and the orthonormal single-particle functions ϕ rk ( Q k , t ) are determined as linear combination of primitive basis functions with time-dependent expansion coefficients, determined variationally. In practice an upper bound is introduced for the sum by limiting the number of single-particle functions. The eigenstates can be computed via the improved relaxation or block-improved relaxation method of MCTDH. 23 Since MCTDH uses time-dependent configurations, the configurational space can naturally adapt during the evolution of the wave function and can therefore be compared to the vibrational multiconfigurational self-consistent field method. 37

Generally speaking, the underlying coordinate system defines the level of coupling among the coordinates that has to be captured by the Hamiltonian and the wave function of each vibrational state. The convergence of the Hamiltonian with respect to the convergence of the computed energy levels can be tested by e.g. including higher-order terms of a Taylor-expanded PES or higher-order coupling terms in a many-body expanded PES. A systematic improvement of variational wave functions is possible through e.g. the enlargement of the underlying basis set(s) and by employing multiple configurations. Finally, SCF and dynamic methods have the advantage of adapting the single-coordinate functions to a given computed state. Besides obtaining converged states within the respective approach, the accuracy of the latter can be quantified by comparison to high-accuracy experimental frequencies, or a higher level of theory, if available. It follows, that an inaccurate PES computed at a low-level of electronic structure theory or not supportive of the relevant nuclear positions will not yield good results, as will not a single-configurational approach for a coordinate system which is inappropriate for the computed vibration ( e.g. rectilinear coordinates for vibrations involving torsional motions). In addition, resonances between states cannot be captured by single-configurational approaches such that any variational or perturbative approach relying only on one configuration will fail in this respect. 35 We note, that although multi-configurational approaches can provide the most accurate results, due to their inherent scaling with system-size, they are difficult to apply to larger systems or highly-coupled motions requiring large bases. In this respect, perturbative approaches represent cost-effective alternatives beyond the harmonic approximation.

Once the vibrational states are determined, the interpretation of the computed levels presents the next challenge. In the harmonic approximation the vibrations are decoupled, such that a single quantum number is sufficient for characterizing a molecular vibration, i.e. ν n 1 corresponds to the n -th solution of the harmonic oscillator in the coordinate q 1 . If n = 1 the vibration is called a fundamental and otherwise with n > 1 an overtone vibration. As mentioned at the beginning of this section, it is advantageous to find a representation which allows each vibration to be localized along its coordinate and therefore to be decoupled from the remaining degrees of freedom. In this case, the product form is already sufficient and an unequivocal assignment in terms of a single set of quantum numbers is possible. Then, the harmonic or VSCF approximation may already present a satisfactory description of the vibrational state, and the corresponding VCI wave function is characterized by a single large CI coefficient. In the opposite case, the absence of a prevalent configuration may be regarded as the evidence of the vibrational counterpart of the concept of static correlation in electronic structure theory. As a dynamic method, in MCTDH the single-particle functions naturally adapt to the computed state, such that in principle a compact wave function can be achieved. In certain cases a single-configurational description completely fails, as is the case for resonances: in Fermi resonance, 38 for instance, an energy splitting, ascribed to the interaction of an overtone and a fundamental accidentally close in energy, is observed.

3. Formic acid

Despite its size, formic acid exhibits a rich structural complexity, which has led to an ongoing theoretical and experimental interest in its nuclear dynamics over the years. 30,35,39–57 In its monomer form ( Fig. 1 ), two stable configurations of the C s point group symmetry, cis and trans have been identified via microwave 57–59 and submillimeter spectroscopy 60,61 in the gas phase. With a difference of about 1365 cm −1 in zero point energy, as measured by gas-phase microwave relative intensity measurements, 57 and 1412 cm −1  62 and 1415 cm −1  56 determined ab initio , the trans form is significantly more abundant at room temperature. This relative stability has been assumed to be due to an intramolecular hydrogen bond, 20 but this idea was discarded later on. 20,63 The gas-phase structure of the trans form was determined via electron diffraction. 64 Based on the ground-state rotational constants determined by microwave spectroscopy, approximate equilibrium structures have been derived. 65 The geometries can be connected via an internal rotation of hydrogen of the OH group around the CO axis, 66 where the earliest experiments found a barrier height of 4827 cm −1 . 57 At low temperatures, proton tunneling through the torsional barrier was found to limit the lifetime of cis -HCOOH as measured in Argon matrix by Petterson et al. , 42,67 whereas the deuterium tunneling rate was found to be significanty slower. 68

The first IR absorption and Raman spectra of gaseous formic acid were measured in 1938 69 and 1940 70 by Bonner et al. , and numerous experimental studies determining the vibrational transitions have been performed ever since at an increasingly high resolution. 39–47,50–52,54,71–79 The signals arising due to the presence of the cis and trans monomers or the dimer have been differentiated experimentally by signal intensification via population control with thermal 54 or laser 42 excitation. Most variational theoretical methods consider purely the vibrational motion at zero temperature which can be compared to low temperature or high-resolution ( e.g. FT absorption) rotationally analysed data. Two popular low temperature spectroscopic methods have emerged in this respect. In matrix isolation spectroscopy 80,81 the analyte is condensed with an inert gas, e.g. a noble gas or nitrogen at low temperature, such that the molecules are “trapped” with different local environments in a matrix. This results in a shift of the measured frequencies and a splitting of absorption bands. For high-resolution data, the best agreement with theory is therefore achieved if the matrix-analyte interactions are also modeled in the theoretical description. As opposed to matrix isolation spectroscopy, jet-cooled spectroscopy 76 does not suffer from such effects. In this method the analyte is passed through a nozzle with a neutral carrier gas and expanded into an evacuated jet chamber at low temperatures. Population enhancement has been achieved in a jet-setup via thermal control therefore recording and spectra at different temperatures have been compared. 54

Complementary high-resolution IR and Raman spectra of partially and fully deuterated formic acid have been recorded for all isotopologues of the cis and trans monomer (DCOOH, HCOOD, DCOOD). 41,44,45,50,82–88 Some spectral regions of formic acid are dense and therefore require a careful rotational analysis, hot bands and overlapping dimer signals further complicate the spectrum with the density of states generally increasing in the higher energy region. 54,82

Amongst the many experimental studies performed for trans HCOOH, 50,55,61,83,88,89 we mention the work of Freytes et al. 82 , who measured gas-phase room-temperature Fourier-transform (FT) and intracavity laser absorption spectra of trans -HCOOH. They reported band origins from 626 cm −1 to 13 284 cm −1 and performed a detailed analysis on the rotational structure of the first CH stretching vibrational overtone ν 2 2 and the second OH stretch overtone ν 3 1 of formic acid. Their work built on a previous study of Hurtmans et al. , who presented a rovibrational analysis in the range of the first and third overtone of the OH stretching vibration ν 2 1 and ν 4 1 of trans -HCOOH based on FT spectroscopy and intracavity laser absorption spectroscopy measurements. 83 Room temperature Raman measurements were performed by Bertie et al. 40 in the range from 70 cm −1 to 4000 cm −1 for all isotopologues of the dimer, where the monomer signals could also be identified. Recent work of Nejad, Suhm and Meyer 54 provided gas-phase Raman-jet measurements for all four deuterated isotopologues of formic acid monomer of both the cis and trans configuration, where the spectra were recorded with a nozzle temperature of 160 °C. They detected Fermi resonance doublets for the fundamentals ν 1 and ν 5 of trans -HCOOH, ν 3 and ν 5 for trans -DCOOH, ν 2 and ν 5 for trans -HCOOD, and ν 2 and ν 3 for trans -DCOOD. A summary of available benchmark quality experimental values for the vibrational transitions of trans -HCOOH and deuterated forms up to 4000 cm −1 in the far and mid-IR region with additional Raman-jet measurements has been recently compiled by Nejad and Sibert 70 from gas-phase IR and Raman spectra reported in literature.

Vibrational data for cis -HCOOH and isotopologues is more scarce 48,66,68,90–95 due to the experimental difficulty of populating the cis state. Matrix-isolation spectroscopy measurements in Argon have been performed for cis -HCOOH by narrowband IR pumping of the first OH stretching overtone of trans -HCOOH ν 2 1 by Pettersson et al. 42 Maçôas et al. recorded near- and mid-IR spectra for HCOOH, DCOOH and HCOOD in solid argon matrix at 8 K from 400 cm −1 to 7800 cm −1  48 where the cis population was enhanced by pumping a trans to cis transition by narrowband tunable IR radiation. Raman-jet spectra for cis -HCOOH and isotopologues have been recorded recently by Meyer, Nejad and Suhm 54,87,96 via thermal excitation between 100–190 °C prior to the jet expansion. In Table 1 we summarize transition frequencies of gaseous formic acid monomer from high-resolution experiments, which have been assigned as fundamentals of the cis and trans monomer, respectively. We note, that many of the fundamentals have been found to be affected by rovibrational 97 and resonance interactions, 20 of which the ν 5 / ν 2 9 Fermi resonance pair of trans -HCOOH has been the most prominent due to an early experimental misassignment of the more intense overtone band as the fundamental. 82

Throughout this work, the assignment of the fundamentals in terms of nuclear displacements is discussed with the valence coordinate definition of Richter and Carbonniére 56 ( Fig. 2 ), although some of the studies reviewed below employ different coordinates.

4.  Ab initio studies on the formic acid monomer

4.1. first ab initio studies.

One of the first ab initio , anharmonic studies on the formic acid monomer was concerned with the overtone series of the OH stretching vibration ν n 1 of trans -HCOOH. 83 Hurtmans et al. recorded rotationally resolved overtone spectra of formic acid with high-resolution gas-phase FT and intracavity laser absorption spectroscopy and performed rovibrational analysis for the first ν 2 1 and third ν 4 1 overtones in the same work. They constructed an effective one-dimensional Hamiltonian dependent on the normal coordinate of the OH stretching mode for the analysis of the OH stretching vibration by relaxing the remaining geometrical parameters along the coordinate. The resulting effective potential was constructed at the MP2/cc-pVTZ level, and the corresponding one-dimensional variational solutions were computed for ν n 1 , n ≤ 4 with an agreement within 50 cm −1 compared to the experimental band origins of the same work. It was found that the excitation of these ν n 1 vibrations affects the HOC angle with increasing n . The induced electric dipole moment was found to be initially almost parallel to the OH bond and progressively tilted towards the C Created by potrace 1.16, written by Peter Selinger 2001-2019 O bond with increasing n . Based on these results Hurtmans et al. proposed a possible proton exchange mechanism promoted by the excitation of the OH overtones.

Subsequently Maçôas et al. 48 performed CC-VSCF calculations to support the assignment of the mid- and near-IR transitions measured by matrix-isolation spectroscopy in the same work. The transitions were determined at two trapping sites in solid argon for cis - and trans -HCOOH. Mid-IR spectra of trans and cis -DCOOH and near-IR spectra of trans -DCOOH and trans -HCOOD were also recorded, where the cis transitions were determined by narrowband IR pumping of the first OH overtone transition ν 2 1 of the trans configuration. For cis -HCOOH about half of the expected fundamentals could be recorded and strong matrix effects, such as splitting of the recorded signals, was observed, making the assignments difficult. The CC-VSCF calculations were performed in the normal mode representation without rovibrational coupling terms and the PES was expressed with pairwise coupling terms, constructed directly at the MP2/6-311++G(2d,2p) level of theory for each pair of normal coordinate gridpoints. They computed all fundamentals and selected overtone and combination bands in the mid- and near-IR region for both trans - and cis -HCOOH and -DCOOH. We have included in Tables 2 and 3 the computed transitions identified as fundamentals of cis -and trans -HCOOH reported in their work. Anharmonicity was found to be strongest for the high-frequency stretching vibrations of trans and cis -HCOOH, i.e. ν 1 and ν 2 were found to have a downshift of up to 200 cm −1 compared to their calculated harmonic counterpart, whereas most fundamentals were found to reproduce the matrix-isolation experiments reasonably well. Exceptions are fundamentals corresponding to torsional and deformation type motions, that is, ν 5 and ν 9 were predicted to be below the experimental values, e.g. for ν 9 around 40 cm −1 and 70 cm −1 for trans -HCOOH and cis -HCOOH, respectively. In the near-IR region, the first overtones ν 2 1 and ν 2 2 of trans -HCOOH and cis -HCOOH were found to be reasonably accurate compared to the measured experimental values. On the other hand, most combination and overtone bands involving ν 5 and ν 6 were inaccurate. In addition to limitations based on e.g. the level of ab initio theory and the order of coupling in the potential, the reason for this was found to be the normal modes associated with the aforementioned fundamentals. We note, that this inaccuracy may be due to the inability of VSCF to reproduce Fermi resonances. The well-known ν 5 / ν 2 9 Fermi resonance was observed experimentally for trans -HCOOH and no analogous resonance was identified for cis -HCOOH. For DCOOH an ν 3 / ν 2 8 Fermi resonance was observed for both the trans and cis form with the involvement of the OH stretching vibration ν 1 .

We also briefly mention the work of Scribano and Benoit, 101 who four years later computed the OH stretching frequency ν 1 of trans -HCOOH with CC-VSCF, VCI and single-to-all (STA) CC-VSCF and STA-VCI. The STA methods were introduced in their work and rely on the idea that the modes could be partitioned into an active and a “spectator” set. In these calculations, the rovibrational coupling terms were neglected and the potential was expressed with up to two-body coupling terms in normal coordinates. The direct evaluation of the potential was performed at the MP2 and CCSD(T) level with TZP and polarization augmented SBK bases. The OH stretch fundamental ν 1 was computed using standard VCI and STA-VCI with only one active degree of freedom. At the CCSD(T)/TZP level of theory a the STA-CC-VSCF and STA-VCI energies were found to be only 8 cm −1 worse compared to the CC-VSCF and VCI calculations for the frequency of the ν 1 fundamental, where only singles and doubles excitations were included in the VCI expansion.

4.2. Advanced calculations with analytic potential energy surfaces

A more accurate description of the nuclear dynamics of formic acid has become available due to the development of analytic ab initio PESs, the availability of which leads to a significant speedup in the potential energy evaluation and therefore allows for the use of higher accuracy vibrational structure methods. The first anharmonic ab initio force-fields of formic acid were developed for trans -HCOOH by Demaison et al. 20 at the MP2/VTZ, MP2/aVTZ and CCSD(T)/VTZ(ae) level of theory. In addition to analytic harmonic terms, the cubic and semidiagonal quartic force constants were determined via displacements along the normal coordinates with the method of finite differences. With this force field they computed anharmonic fundamentals using vibrational perturbation theory for trans -HCOOH. In Table 2 we included transitions obtained with the CCSD(T)/VTZ(ae) force-field. Vibration-rotation interaction constants as obtained by standard perturbation theory were also reported. The ν 7 , ν 9 , and ν 3 fundamentals of trans -HCOOH were found to be affected by strong rovibrational interactions. They carried out an analysis of the resonances in terms of the magnitude of the force constants, where the Fermi resonant states were assumed to be connected via the cubic constants of force constants of type k ijj and Darling-Dennison type resonances 102 via the quartic constant k kkll . In contrast to the experimentally derived force constants, several calculated force constants indicating Fermi resonance ( e.g. k 499 for ν 4 / ν 2 9 , k 677 for ν 6 / ν 2 7 ) differed significantly in magnitude and sign whereas the ν 5 / ν 2 9 Fermi resonance matched qualitatively. The resonance interactions were further analyzed by constructing a 5 × 5 symmetric matrix representing interactions involving the ν 2 9 , ν 2 7 , ν 6 , ν 5 , ν 4 states and computing the corresponding eigenstates.

In 2013, Tew and Mizukami reported their first PES for formic acid, initially fitted for the description of the trans form ref. 35 with 5873 randomly generated configurations within the energy range 0–15 000 cm −1 at the CCSD(T)(F12*)/ccpVDZ-F12 level of theory. With this preliminary surface, they performed VSCF, VMP2, VASCI, and VASPT2 calculations based on the normal-coordinate Watson Hamiltonian for the fundamentals of trans -HCOOH (see Table 2 ). Rovibrational coupling terms were neglected and up to three body coupling terms were included in the many-body expansion. Up to seven single-mode functions were employed for the state-specific VSCF calculations and for the VASCI calculations an active space of four modes was selected, involving the normal modes associated with the ν 1 , ν 5 , ν 7 , and ν 9 fundamentals. The authors note a good performance of VMP2 except for the fundamentals ν 1 and ν 5 , where the former was overestimated by 22 cm −1 compared to the rovibrational-interaction-corrected experimental value. The largest deviations of the VASPT2 calculations were found for the ν 2 and ν 4 fundamentals. Since the isomerisation barrier was reached already at six quanta of excitation of the single-coordinate function along the torsional mode associated with the ν 9 fundamental, only a limited excitation space was available. This led to multiple unconverged states due to the involvement of the torsional mode in multiple states. The authors concluded, that for fully converged values adopting a different coordinate system that can treat both the cis and trans minima would be necessary.

Consequently, two semiglobal PESs capable of describing the cis – trans isomerization have been developed. Tew and Mizukami published a PES in 2016 62 constructed using LASSO-based regression 62 with electronic energies obtained at the CCSD(T)(F12*)/cc-pVTZ-F12 level of theory using 17 076 random structures in the energy range 0–15 000 cm −1 relative to the trans minimum. The structures were generated by random displacements from cis and trans equilibria as well as from the transition state of the OH torsional motion and with points generated in the internal coordinate path (ICP) coordinate system which was constructed for formic acid. The latter corresponds to curvilinear path for the torsional motion and 8 normal coordinate displacements. The analytic potential fit consists of a zeroth order surface, which is a sum of Morse functions of atom–atom separations, and a correction surface, which is a sum of distributed multivariate Gaussian functions which are a combination of atom–atom separations. In the same work, the authors constructed an internal-coordinate path Hamiltonian (ICPH) using their PES, where a curvilinear path was used for the torsional motion connecting the trans and cis rotamers and the remaining mutually orthogonal rectilinear modes were defined such that the rovibrational coupling constant associated with the motion along the path was minimized. Trigonometric functions were used for the expansion of the single-mode functions along the torsional path coordinate, and harmonic oscillator functions were used for the remaining coordinates. State-specific VSCF and VMP2 calculations were carried out with the ICP Hamiltonian of HCOOH, where the potential included up to four-body coupling terms, whereas in the ICPH-VCI calculations up to five-body coupling terms were included with a maximum overall excitation of up to 10 quanta in the VCI expansion. The change of the zero point energy by including, instead of four-body terms, five-body terms in the Hamiltonian was 2 cm −1 and for the fundamentals ν 2 , ν 3 , ν 5 , ν 8 ν 9 it was within 1 cm −1 . For the ν 1 , ν 4 , ν 6 , and ν 7 fundamentals slower convergence with a change of up to 6 cm −1 was observed with respect to the increase in the coupling. The ICPH-VCI vibrational energies with respect to the trans -minimum were reported up to 4750 cm −1 with respect to the global trans minimum and 14 bands in the region 0–4720 cm −1 were reassigned based on the VCI coefficients compared to the assignment of Freytes et al. 82 The cis -zero point energy (ZPE) and cis states were identified from the quantum number of the torsional mode. The authors note the overestimation of the ν 2 6 and ν 3 6 states in their ICPH-VCI calculations compared to experiment. They also discuss the ambiguity of the assignment of some states in the high-energy region and of increasing excitation, further complicated by the high density of states in this region. They observed a coupling between the ν 4 , ν 5 , ν 6 , and ν 7 cis states and concluded that the coupling could be an artifact from the definition of the coordinate system. Due to this coupling and the associated slow convergence only states with up to two quanta of excitations relative to the cis -ZPE were reported. The highest cis -HCOOH fundamental ν 1 was not reported in this work, and the ν 2 4 overtone was identified to be part of a Fermi doublet with ν 2 . We report only the transitions identified as fundamentals in their work in Tables 2 and 3 .

The second semiglobal ab initio PES was developed by Richter and Carbonnière 56 in 2018 who fit 660 single point calculations at the CCSD(T)-F12a/aug-cc-pVTZ level of theory in the energy range 0–6000 cm −1 relative to the trans minimum. The PES was fit in terms of internal valence coordinates 1 ( i.e. bond stretches, angles and torsions) with the points generated by the Adaptive Generation of Adiabatic PES (AGAPES) procedure. 103 The coordinate definition used is depicted in Fig. 2 . The barrier between the two minima along the torsional coordinate τ 2 was found to be 4442 cm −1 , where the barrier height was found to be affected by the bond-length r 1 and the bending angle θ 2 . The barrier of this PES is 20 cm −1 higher compared to the value reported by Tew and Mizukami. 56 For the vibrational Hamiltonian, the exact analytical kinetic energy operator was constructed in the SOP form with the TANA program 104 in polyspherical coordinates, and the PES was expanded as a many-body expansion in valence coordinate displacements. The “dead branching” 105 option of the AGAPES procedure was used, in which only certain coordinate combinations were selected based on a remoteness measure defined by geometric mean values for the potential coupling terms. Out of the 256 possible coordinate combinations in the four-body expansion, 36, 84 and 126 were kept, for two, three and four-body terms, respectively. A maximal number of terms was kept for the coupling between the θ 3 and τ 2 coordinates, associated with the ν 5 and ν 9 fundamentals. Below 3500 cm −1 with respect to the trans ZPE only half the period of the grid of the torsional coordinate was used ( τ 2 ≤ π/2), such that no cis – trans delocalization effects were included in the computation of the trans states. Full-period calculations were performed between 3500 cm −1 –3640 cm −1 . The high-lying fundamentals associated with the CH and OH stretch vibrations ( ν 2 , ν 1 ) were computed with the improved relaxation scheme of MCTDH, and all states below 3100 cm −1 for trans and 2700 cm −1 for cis with respect to their respective ground state were computed with the block Davidson scheme of MCTDH. We have included the transitions identified as fundamentals in Tables 2 and 3 . The MCTDH reference state was found to have only a minor effect on the convergence of the states. All vibrational states below 3796 cm −1 were converged in either the cis or the trans well, as differentiated by the torsional coordinate, and the first delocalized state was found at 3796 cm −1 above trans ZPE, which was labeled as the ν 6 9 overtone of cis -HCOOH. Four states could not be converged to sufficient accuracy, these were the states at (assignments of Richter Carbonnière in brackets) 3086 cm −1 ( ν 4 + ν 3 ), 3116 cm −1 ( ν 2 8 + ν 2 6 ) for trans -HCOOH and at 2357 cm −1 ( ν 9 + ν 7 + ν 5 ), 2479 cm −1 ( ν 2 7 + ν 5 ) for cis -HCOOH. For most states which were previously identified in literature to be involved in Fermi resonances, a single-configurational description was found to be sufficient, except for the well known ν 5 / ν 2 9 Fermi resonance, where a mixing of the reference configurations was observed. The computed states of trans -HCOOH were in good agreement with the values reported by Tew and Mizukami with a root mean square deviation (RMSD) of 11 cm −1 : the RMSD of the fundamentals, two, three and four quanta excited states were 5 cm −1 , 8 cm −1 , 9 cm −1 and 25 cm −1 , respectively with maximal deviations (MAXDs) for ν 3 , ν 3 2 , ν 5 + ν 2 7 and ν 2 7 + ν 2 9 . The RMSDs of the cis states were about 49 cm −1 compared to the levels of Tew and Mizukami, where the RMSDs for fundamentals, two and three quanta excited states were 14 cm −1 , 49 cm −1 and 111 cm −1 , respectively, and the MAXDs were found for the ν 8 , ν 4 + ν 8 and ν 8 + ν 2 9 states. This discrepancy of the cis results compared to the work of Tew and Mizukami was further investigated by the authors. 56 For cis -HCOOH the authors validated their results by performing the AGAPES construction referenced on the cis , instead of the trans geometry: the levels computed with the global PES were found to be consistent with the latter. VPT2 and MCTDH calculations were performed with both the present PES and the 2016 PES of Tew and Mizukami, for which the latter was re-fitted in the SOP form. The VPT2 fundamentals were found to compare well with each other and with the original MCTDH values such that the authors concluded that the disagreement for the cis levels stems from the ICPH-VCI calculations, rather than from the PES. For the cis levels, a mixing of the ν 5 and ν 6 levels was identified.

4.3. Recent developments

Fundamentals, combination bands and overtone transitions up to approximately 3000 cm −1 with respect to the global trans minimum were computed by Aerts et al. 53 in 2020 for the deuterated forms of cis and trans formic acid using the above mentioned PES of Richter and Carbonnière 56 with the block-improved relaxation method of MCTDH. The assignment of the states was performed via visualization of the reduced densities of the computed vibrational wavefunctions. The RMSDs with respect to experimental data of the fundamental transitions were found to be 8 cm −1 , 7 cm −1 and 3 cm −1 for trans -DCOOD, trans -HCOOD and trans -DCOOH, respectively. The calculations were performed by defining modes combining r 1 and θ 3 , θ 1 and θ 2 , and τ 1 and τ 2 , respectively (see Fig. 2 ). Aerts et al. confirmed the Fermi resonance ν 3 / ν 2 8 of trans -DCOOH, already reported by Macoas et al. , as identified by the reduced density functions along the important coordinates.

Vibrational transition energies were computed by Nejad and Sibert 79 up to 4000 cm −1 for HCOOH and isotopologues with sixth order Canonical Van Vleck Perturbation theory (CVPT6) employing both surfaces of Tew and Mizukami and Richter and Carbonnière. 56 In their work, the PESs were represented as a Taylor expansion in terms of stretch, bend, and dihedral angles with the valence coordinate definition of Richter and Carbonnière. 56 Up to four-body coupling terms were used for the PES, and the elements of the exact kinetic energy operator were expanded up to the sixth order in the internal coordinates. The CVPT6 calculations were performed with the Hamiltonian constructed in curvilinear normal coordinates based on only one geometry. As a basis for the matrix representations, a product of harmonic oscillator functions was used where the excitation degree of each oscillator was limited. A slightly faster convergence of the trans -HCOOH states obtained with the 2016 PES of Mizukami and Tew as compared to the 2018 PES of Richter and Carbonnière was observed with regard to the order of perturbation, where fourth and sixth order corrections were found to be especially important for higher lying vibrational transitions. Slower convergence was observed for the cis -HCOOH states on the 2016 compared to the 2018 PES. It was found, that the 2016 surface generally overestimates, whereas the 2018 PES often underestimates the transition energies for all isotopologues as compared to experimental values, where increasing deviations were found for higher lying vibrational states. The CVPT6 fundamentals of trans -HCOOH compared within 2 cm −1 to ICPH-VCI and MCTDH results of the respective surface (see Table 2 ). For cis -HCOOH it was suggested by the authors that the ICPH-VCI results of Tew and Mizukami were not fully converged. The band assignment was performed based on the leading coefficient in the VCI expansion and in comparison to previous calculations and previously assigned IR and Raman bands reported in literature. 11 new vibrational band centers were assigned for trans -HCOOH and 53 for the deuterated isotopologues by the authors with multiple reassignments in comparison to previous assignments from theory or experiment. For trans formic acid strong coupling among the states which share the polyad quantum number N p = n 5 + n 9 /2 was found, with an involvement of the ν 1 fundamental at higher energies.

Most recently, Martín Santa Daría, Avila, and Mátyus performed calculations using the 2016 PES of Tew and Mizukami with a Hamiltonian constructed using the cis – trans torsional coordinate and eight curvilinear normal coordinates defined with respect to an instantaneous reference configuration along the torsional motion. 30 The curvilinear normal coordinates were adapted to the cis – trans isomerization of formic acid by relaxing the internal coordinates such that the potential energy was at minimum along the torsional motion. In principle, the kinetic and potential energy coupling among the coordinates was reduced by employing curvilinear normal coordinates. The kinetic energy operator was constructed numerically with the kinetic and potential energy integrals evaluated over the direct product basis with the GENIUSH (general rovibrational code with numerical, internal-coordinate, user-specified Hamiltonians) code. 106 For the curvilinear normal coordinates harmonic oscillator basis functions and for the torsional degree of freedom a Fourier basis was used. A Smolyak grid was then used for the truncated direct product basis for the evaluation of the multi-dimensional integrals. The authors computed converged vibrational states up to ca. 4700 cm −1 with respect to the trans -ZPE, slightly above the cis – trans isomerization barrier and therefore the highest-energy cis ν 1 fundamental was not computed. The convergence was evaluated by increasing the basis set size. The energy difference between calculations using the same basis set size was performed with rectilinear and relaxed curvilinear normal coordinates and the difference was found to be around 8–10 cm −1 , with an increasing difference in the higher energy region. The assignment of the cis -states was performed in terms of the contribution of the 1D torsional basis function, which in the lower energy ranges was localized in either well. A variational improvement of up to 5–10 cm −1 was reported compared to the CVPT6 results and by 10–40 cm −1 compared to the previously computed ICPH-VCI levels. Beyond 3700 cm −1 , mixed cis – trans states were reported and the “tunneling splittings” associated with these mixed states were computed to be below 1–5 cm −1 , almost within the convergence uncertainty of the calculations. Beyond 3900 cm −1 , non-negligible contributions from delocalized torsional states were observed. We note, that although the results of the authors unequivocally indicate the switching of the ν 5 and ν 6 cis -HCOOH states as opposed to previous calculations, in Table 3 we list the labeling to be consistent with previous results.

Finally, we discuss the recent work of Aerts et al. , who simulated the intramolecular vibrational redistribution (IVR) dynamics of formic acid on the 2018 PES of Richter and Carbonniére with the goal of identifying a possible laser-induced trans – cis isomerization pathway. The dynamical evolution following the excitation of an infrared-active vibration was simulated by the time-evolution of an initial vibrational wave function which was prepared by the excitation of the single particle function of the torsional mode τ 2 associated with the isomerization. The probability of finding the molecule in the trans well, as defined by the value of the torsional coordinate, was followed during the evolution, and local energies were ascribed to each mode. By investigating the time-evolution of the fractional energies defined for each overtone of the local torsional vibration of trans -HCOOH, multiple mode-couplings and resonances were identified and confirmed. The ν 5 / ν 2 9 Fermi resonance was observed as a strong, reversible energy flow between the two fundamentals, also for states sharing the previously mentioned polyad quantum number N p = n 5 + n 9 /2. A coupling between the modes of the ν 9 and ν 4 fundamentals was identified. The probability to find the wavepacket in the initial trans well was found to diminish at ν 6 9 at which part of the wavepacket transfered from trans to cis within 100 fs. The non-periodic probability indicated, that this transition between trans and cis is irreversible.

We have reviewed first principles, anharmonic studies on the molecular vibrations of formic acid in its monomer form and collected all reported computed transitions identified as fundamentals for both cis - and trans -HCOOH. Formic acid has been a widely explored albeit challenging system to both experiment and theory. On one hand, the spectrum is dense and complex in certain spectral ranges and requires a careful rovibrational analysis 83 and experimental differentiation of the signals arising due to its possible geometries. 107 Undistorted, high-resolution experimental data has still not been recorded for all of the available cis transitions predicted by theory. On the other hand, the higher-energy vibrations of the molecule are expected to explore trans and cis geometries or both, 30 a formidable task for vibrational methods. The minimal energy path connecting the two minima was found to be a function of the τ 2 torsional coordinate, 56,57 but possible isomerization pathways involving the excitation of the ν n 1 OH overtones 83 or involving ν 6 and ν 9 were also suggested. 2 Early on, multiple resonances affecting the vibrational levels of formic acid were identified, most notably the ν 5 / ν 2 9 Fermi resonance of trans -HCOOH. 48 Hurtmans et al. reported the involvment of the HOC angle in the ν n 1 excitations, 83 however, the first ab initio studies employed rectilinear normal coordinates which were found to be ill-suited for torsional and deformation vibrations. 48 The first PESs were computed directly along the normal coordinates 48 or as a Taylor 20 or a local many-body expansion, 35 such that vibrations with a local trans character could be predicted. The development of analytic semiglobal PESs capable of reproducing transitions both from the cis and trans minima followed. 56,62 Transitions of cis - and trans -HCOOH were computed with a Hamiltonian constructed with normal coordinates along the curvilinear isomerisation path, and the computed high-energy cis states were found to be coupled significantly. 62 The vibrational transitions were also computed with the dynamical MCTDH method in valence coordinates, 56 and the computed cis -states differed significantly from the previously reported results. A comparison between the PESs was made with transitions computed based on a curvilinear normal coordinate Hamiltonian, where the coordinates were defined based on a single geometry. 79 The transitions of the deuterated species 53,79 and assignments up to 4000 cm −1 with respect to trans -ZPE were also described. 79 A curvilinear normal coordinate Hamiltonian adapted to the cis – trans torsional motion was constructed, 30 and vibrational states slightly above the isomerisation barrier were reported including cis – trans entangled and delocalized states. Small splittings associated with the high torsional isomerisation barrier were reported. The study of Martín Santa Daría et al. reported the lowest energy variational solutions up to now 30 with a good agreement with high-accuracy experimental values for both cis ( Table 3 ) and trans ( Table 2 ) fundamentals.

Finally, we discuss the computed vibrations of formic acid which have been identified as fundamentals across the studies mentioned in this work, as compiled in Tables 2 and 3 . Although such an analysis is not necessarily warranted considering that different PES and Hamiltonians are used across these approaches, it is still worth analyzing the results in terms of the employed methods. The harmonic analysis performed by many of the studies shows the worst agreement to experimental values with an RMSD ranging from approximately 80–125 cm −1 where generally a better agreement is achieved for the lower energy states. Single-configurational approaches follow with an RMSD ranging from approximately 3.4–31 cm −1 . Finally, all multi-configurational methods achieve an RMSD better than 4 cm for the trans -states and better than 8 cm −1 for the cis states, where for the latter conclusions should only be drawn with care due to the few available experimental results.

As a small molecule exhibiting large-amplitude motion, the vibrations of formic acid are comparatively difficult to model, where the computation of converged cis states and highly excited states above the delocalization barrier remains to be challenging due to the strong coupling of the coordinates in this region. We note that these difficulties are illustrative of the challenges faced for larger systems, since the nuclear dynamics of bigger molecules and clusters can become increasingly intricate but multiconfigurational methods become prohibitively expensive due to their scaling with respect to system size.

Conflicts of interest

There are no conflicts to declare.

Supplementary Material

Acknowledgments.

This work is supported by the Swiss National Science Foundation (grant no. 200021_197207).

Notes and references

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• Published: 31 August 2024

Temperature and volumetric effects on structural and dielectric properties of hybrid perovskites

• Andrzej Nowok   ORCID: orcid.org/0000-0002-4833-4259 1 , 2 ,
• Szymon Sobczak   ORCID: orcid.org/0000-0001-8234-2503 3 ,
• Kinga Roszak   ORCID: orcid.org/0000-0002-1370-2111 3 ,
• Anna Z. Szeremeta 4 ,
• Mirosław Mączka   ORCID: orcid.org/0000-0003-2978-1093 5 ,
• Andrzej Katrusiak   ORCID: orcid.org/0000-0002-1439-7278 3 ,
• Sebastian Pawlus 4 ,
• Filip Formalik 6 , 7 ,
• Antonio José Barros dos Santos   ORCID: orcid.org/0000-0003-0739-7699 8 ,
• Waldeci Paraguassu 8 &
• Adam Sieradzki   ORCID: orcid.org/0000-0003-4136-5754 2  

Nature Communications volume  15 , Article number:  7571 ( 2024 ) Cite this article

Metrics details

• Chemical physics
• Phase transitions and critical phenomena

Three-dimensional organic-inorganic perovskites are rapidly evolving materials with diverse applications. This study focuses on their two representatives - acetamidinium manganese(II) formate (AceMn) and formamidinium manganese(II) formate (FMDMn) – subjected to varying temperature and pressure. We show that AceMn undergoes atypical pressure-induced structural transformations at room temperature, increasing the symmetry from ambient-pressure P 2 1 / n phase II to the high-pressure Pbca phase III. In turn, FMDMn in its C 2/ c phase II displays temperature- and pressure-induced ordering of cage cations that proceeds without changing the phase symmetry or energy barriers. The FMD + cations do not order under constant volume across the pressure-temperature plane, despite similar pressure and temperature evolution of the unit-cell parameters. Temperature and pressure affect the cage cations differently, which is particularly pronounced in their relaxation dynamics seen by dielectric spectroscopy. Their motion require a rearrangement of the metal-formate framework, resulting in the energy and volumetric barriers defined by temperature-independent activation energy and activation volume parameters. As this process is phonon-assisted, the relaxation time is strongly temperature-dependent. Consequently, relaxation times do not scale with unit-cell volume nor H-bond lengths in formates, offering the possibility of tuning their electronic properties by external stimuli (like temperature or pressure) even without any structural changes.

Introduction

Three-dimensional hybrid organic-inorganic perovskites (3D HOIPs) have emerged as a promising and fast-developing class of materials with a vast array of application possibilities across diverse scientific and technological domains. Sensing, catalysis, gas storage, magnetism, luminescence, optoelectronics, or photovoltaics exemplify some of their diverse applications 1 , 2 , 3 , 4 , 5 , 6 , 7 . The key to their success lies in a tuneable framework composed of metal cations coordinated by bridging ligands (organic or inorganic), with cavities occupied by rationally chosen cation 4 .

Of particular interest are the HCOO − anions, used as linkers in hybrid formates. Despite their small size, formate ions can accommodate relatively large cations, such as bis (3-ammoniumpropyl)ammonium, N,N′-dimethylethylenediammonium and 1,4-butanediammonium, without transforming the 3D cage architecture into lower-dimensional systems 8 , 9 , 10 . Within these structures, the cage cations may exhibit additional dynamic disorder, with their reorientation or hopping between symmetry-equivalent positions triggered by external stimuli, giving rise to orientational polarization and dielectric relaxation 11 . Furthermore, the HCOO − ions are able to pass a canted antiferromagnetic interaction, leading to a weak-ferromagnetic order (canted antiferromagnetic order) 4 . Finally, formate anions can bind metal ions in three different coordination modes: syn - syn , syn - anti , or anti - anti 4 , 11 . Consequently, formate-bridged hybrid materials exhibit a diverse range of structural, electric, and magnetic properties, encompassing order-disorder phases with ferroic or multiferroic features, and ferroelasticity 12 , 13 , 14 .

Despite extensive research efforts dedicated to the study of hybrid formates, the realm of dielectric relaxations in these materials still encompasses numerous unexplained phenomena. For instance, recent high-pressure dielectric studies have revealed a small sensitivity of relaxation times (and relaxation peak position) to compression in several 3D hybrid formates 12 , 13 , 14 . In the case of 1,4-butanediammonium zinc formate (DABZn), the isothermal compression up to 1.7 GPa at 302 K exerts a similar effect on relaxation times as a mere 17 K decrease in temperature (to 285 K) 14 . These findings are rather unexpected, considering the mechanism underpinning the relaxation phenomenon in these materials and the significant softness of their ionic crystal structures, evidenced by relatively small Young’s moduli and considerable compressibility 15 , 16 , 17 , 18 . This softness is also evident in a moderate pressure regime (below 1 GPa), where most of the pressure-induced phase transition in such materials was observed 19 , 20 . It should be mentioned in this regard, that the precise role of the dynamics of cage cations in stimuli-triggered structural transformations continues to be an area of intensive studies 13 .

To further explore the relaxation phenomena in hybrid formates, we have reinvestigated the structures and dielectric response of their two model derivatives, acetamidinium manganese(II) formate ([NH 2 –C(CH 3 )–NH 2 ][Mn(HCOO) 3 ], further abbreviated as AceMn) and formamidinium manganese(II) formate ([NH 2 –CH–NH 2 ][Mn(HCOO) 3 ], named as FMDMn). In both compounds, manganese Mn 2+ cations and formate HCOO − anions form a 3D framework counterbalanced by formamidinium (FMD + , [NH 2 –CH–NH 2 ] + ) or acetamidinium (Ace + , [NH 2 –C(CH 3 )–NH 2 ] + ) respectively for FMDMn and AceMn 21 , 22 .

Pressure and temperature are thermodynamic parameters that are frequently used to understand the structure-property relationships in hybrid compounds 20 , 23 , 24 . The compression can also induce various phenomena in these materials, e.g., amorphization, tricritical points, polarization enhancement, and metallization 25 , 26 , 27 , 28 , 29 , 30 , 31 . In this article, we demonstrate several unconventional pressure-induced effects, such as a phase transition increasing the symmetry, or an order-disorder transformation with no crystal symmetry change nor increased energy barrier. We also derive general relationships that link dielectric features with structural changes in these prototypical HOIPs. In this regard, we redefine the activation volume parameter and highlight its critical role in regulating relaxation dynamics under pressure. The research is based on the combination of varied-temperature and high-pressure X-ray diffraction (XRD), Raman, and dielectric spectroscopy studies, and supported by quantum density functional theory (DFT) calculations.

The model compounds FMDMn and AceMn were synthesized as crystalline materials. Detailed crystallographic data under varied-temperature and high-pressure conditions are presented in Tables  1 , 2, and Supplementary Tables  1 – 8 . The X-ray diffraction studies show that the FMDMn above 335 K and at ambient pressure is trigonal (space group R \(\bar{3}\) c ), whereas a high-temperature phase I of AceMn stable above room temperature (~300 K) is orthorhombic (space group Imma ), which corroborates the previous reports (c.f. Tables  1 , 2 and Supplementary Table  2 ) 21 , 22 .

At the ambient conditions ( T  = 296 K, p  = 0.1 MPa), the FMDMn crystals are monoclinic of space group C 2/ c (phase II, see Table  1 ), as reported previously 21 . Under these conditions, the cage cations FMD + are disordered with the site occupation factor (SOF) equal to 0.85 and 0.15. Isothermal compression of phase II reveals no anomalies in the unit-cell dimensions up to 3.63 GPa (Fig.  1a ). The softest of the unit-cell parameters is a , which, together with increasing angle β , is responsible for the diagonal contraction of the perovskite cages (see Fig.  1 a, b). This mechanism involving the tilts of MnO 6 octahedra reduces the void space in the unit cell and gradually eliminates the disorder in phase II, forcing the FMD + cations to stack into one energetically privileged position. This phenomenon is manifested in the pressure evolution of SOF parameters: the larger SOF 1 gradually increases and the smaller SOF 2 decreases on compression, approaching 1 and 0, respectively (Fig.  1c ). Ultimately, the SOF parameter remains constant above 1 GPa, while any fluctuations are within the experimental errors. (Fig.  1c ). This progressive change does not affect the crystal symmetry as FMD + cations assume the position with the strongest H-bonds to the formate ligands. This provides additional support for the framework, righting the crystal direction [010] along which the H-bonds are aligned, and resulting in almost no compressibility along this direction up to 3.63 GPa (the parameter b is only reduced by about 1%). The linear compressibility along [010] direction calculated as β y  = −1/ b ∂ b /∂ p for compression range from 1.09 GPa to 1.95 GPa, is close to zero, with β y  = 0.39(3) TPa −1 (see Supplementary Tables  9 and 10 ). Interestingly, the strain tensor at 3.63 GPa reveals the negative linear compressibility (NLC) of −4.2(5) TPa −1 in the direction close to [102] (see Supplementary Table  9 ). This effect can be explained by the wine-rack mechanism 32 , with the strongest compression along the shortest diagonal of the rhomboidal cages and the elongation along the longest diagonal 33 , 34 , 35 . Above 4 GPa the intensity of the diffraction spots decreases significantly. We carried out high-pressure Raman measurements at room temperature to explain this observation.

a Variation of unit cell parameters a II , b II , c II during room-temperature isothermal compression of FMDMn. Upper inset shows pressure-induced changes in β angle. b Relative changes in unit cell parameters a II , b II , c II observed during the compression of FMDMn at room temperature. The changes were calculated relative to the ambient-pressure values of these parameters ( a II (0.1MPa) , b II 0.1MPa , c II 0.1MPa ). c Changes in SOF parameter of FMDMn as a function of pressure. Error bars represent the standard deviation. The structure of FMDMn at room temperature at pressures ( d ) 0.1 MPa and ( e ) 1.95 MPa plotted along b axis. Mn, O, C, N, and H atoms are marked in purple, red, gray, blue, and white, respectively. Source data are provided as a Source Data file.

Raman spectra collected during room-temperature compression of FMDMn are presented in Supplementary Fig.  1 and the corresponding pressure dependence of Raman wavenumbers together with assignment based on previous studies of formamidinium-based formates is shown in Supplementary Fig.  2 . The values of wavenumber intercepts at zero pressure ( ω 0 ) and pressure coefficients ( α  = d ω /d p ) obtained from fitting of the experimental data with a linear function ω ( p ) =  ω 0  +  αp are listed in Supplementary Table  11 . This table also summarizes the assignment of these bands according to previous studies on formamidinium-based formates 21 , 36 . As presented in Supplementary Fig.  1 , the Raman spectra remain qualitatively the same up to about 0.6 GPa. When the pressure reaches 0.9 GPa, the intensity of the two strongest lattice bands near 160 and 107 cm −1 starts to decrease at the expense of the bands at 175 and 125 cm −1 (values at 0.9 GPa, see Supplementary Fig.  1a ). This behavior continues beyond 0.9 GPa. Strong intensity increase is also observed for the 81, 1377, and 1386 cm −1 bands, which become clearly seen above 0.9 GPa (Supplementary Fig.  1a, b ). Since the X-ray diffraction evidenced no phase transition up to 3.63 GPa, it was noted that the pressure at which Raman spectra change (~0.9 GPa) correlates with complete ordering of FMD + cations near about 1 GPa (Supplementary Fig.  1c ). The analysis of the Raman data show that nearly all modes in the 0–3.5 GPa range exhibit a positive pressure dependence and that the largest pressure coefficients α are observed for the lattice modes, especially those involving Mn 2+ translations (Supplementary Table  11 ). This behavior indicates that the compression involves tilts of MnO 6 octahedra, in agreement with the X-ray diffraction. The negative pressure dependence is observed for the C–N stretching modes near 1150 cm −1 only (see Supplementary Table  11 ). This behavior can be attributed to the elongation of C–N bond and the decrease of the N–C–N bond angles on compression.

When the pressure reaches 4.0 GPa, some bands are significantly broadened (see for instance the ν 3 (HCOO − ) and ν 3 (HCOO − ) bands at 797 and 1368 cm −1 ) while others split e.g. δ (CN) +  τ (NH 2 ) band near 595 cm −1 (Supplementary Figs.  1 and 2b ). Supplementary Fig.  2b also shows that the C–N stretching modes change slopes of wavenumbers vs pressure from strongly negative below 4 GPa to positive or weakly negative above 4 GPa. All these changes involve both HCOO − and FMD + units, indicating that FMDMn experiences a structural phase transition near 4 GPa to a lower symmetry phase associated with distortion of the framework and increased FMD-framework interactions. Consequently, the weakening of the diffraction peaks near 4 GPa can be associated with this phase transition and attributed to the deterioration of the crystal quality induced by the structural transformation.

On further compression, changes in relative intensities of bands and the appearance of new bands indicate the onset of the third pressure-induced phase transition associated with weak structural changes (see Supplementary Figs.  1 and 2 , as well as Supplementary Table  11 ). Supplementary Fig.  3 shows that during the decompression run, the initial phase is retained. It shows that the pressure-induced phase transitions are reversible and supports the conclusion that FMDMn does not amorphize in the studied pressure range.

Compared to FMDMn, the behavior of AceMn is much more complex despite the structural similarity of both compounds at ambient pressure (see Fig.  2a–f ). At the ambient temperature and pressure, the AceMn crystals are monoclinic of space group P 2 1 / n (phase II, see Table  2 and Supplementary Table  2 ), in agreement with previous studies 22 . The monotonic compressibility of phase II involves the NLC with β c  = −9.7(3) TPa −1 at 0.86 GPa (Fig.  2 a, b and Supplementary Table  12 ), originating from the analogous mechanism as that observed in FMDMn. This NLC effect remains a relatively rare phenomenon among three-dimensional HOIPs. However, there are some notable examples of materials that exhibit NLC, such as chiral NH 4 Zn(HCOO) 3 ( β x  = −1.8(8) TPa −1 ) 37 and [FA]Mn(H 2 POO) 3 ( β x  = −7.8(6) TPa −1 ) 38 . In comparison, the NLC of AceMn is larger than that of NH 4 Zn(HCOO) 3 . The significance of large NLC phenomena in HOIPs arouses wide interest as it allows to correlate the elongation of the crystal at specific pressure with various other effects, including changes in optoelectronic properties 39 . Such a correlation is important for developing multimodal sensors that can find applications in a wide range of fields, including materials science, nanotechnology, and other technologies 39 , 40 , 41 , 42 , 43 .

a Variation of unit-cell parameters a , b , c , d [101] during isothermal compression of AceMn at 298 K. Lower indices II and III in these parameters refer to the phase number. b Relative changes (referred to as the STP structure) of the unit-cell parameters a , b , c , d [101] observed while compressing the AceMn crystal at room temperature. c Pressure dependence of the monoclinic angle β II in phase, the angle β II in the lattice of phase III, and the orthorhombic angle β III of phase III. The structure of AceMn in phases I ( d ), II ( e ), and III ( f ) plotted along b axis. Mn, O, C, N, and H atoms are marked in purple, red, gray, blue, and white, respectively. Source data are provided as a Source Data file.

Phase II remains stable up to roughly 1.0 GPa when an abrupt volume drop marks a pressure-induced transition of clearly first-order type to phase III of orthorhombic space group Pbca (Fig.  2c ). At the critical pressure, the molecular volume (i.e. unit-cell volume per structural unit) is abruptly reduced by about 16 Å 3 , indicating that the transition involves the collapse of voids in the crystal structure. The presence and first-order character of the phase transition near 1.0 GPa is consistent with previous high-pressure Raman studies, which revealed sudden changes in the spectra between 0.7 and 1.4 GPa 44 . The structural transformation doubles the unit cell of phase III, as described by the matrix:

Hence, the unit-cell contents double, too (the Z number increases to 2 in phase III, see Supplementary Fig.  4 ). It is an atypical feature of the transition from AceMn phase II to the high-pressure phase III that its symmetry increases. In most cases, the symmetry of molecular crystals is reduced in order to increase the number of degrees of freedom in the structure, so it efficiently reduces the strain associated with the compression. There are only few exceptions where pressure-induced transitions increase symmetry 45 , 46 . In AceMn, the monoclinic angle β II of phase II, as expected, increases its opening from 90.099(5)˚ at 0.1 MPa to 91.81(5)˚ at 0.86 GPa (Fig.  2c ). The transition to phase III at 0.9 GPa abruptly increases angle β II to 94.47(6)˚, but at the same time the unit-cell parameters a II and c II become equal in length (Fig.  2a ). It indicates the transition to the orthorhombic system, where directions [ a II ] and [ c II ] become the diagonals d [101] III and d [10-1] III of the doubled unit cell of phase III (Fig.  2e, f ). Transformation of AceMn from the low-density monoclinic P 2 1 / n phase II to the high-density phase III of orthorhombic Pbca space group also changes material compressibility as evidenced by 2 nd Birch–Murnaghan coefficient changing from 26.12(1) to 20.0(1) GPa −1 , respectively (compare Supplementary Tables  12 – 15 ).

The transformation between phases II and III preserves the ambient-pressure H-bonding pattern as well as the alignment of the Ace + cation, consistently with previous high-pressure Raman data, showing that the phase transition weakly affects τ NH 2 and ρ NH 2 modes near 544 + 575 and 1140 cm −1 , respectively 44 . These relatively strong N–H ⋯ O bonds support the structure of phase III up to about 5.0 GPa, when its amorphization occurs. This observation is consistent with previously reported Raman studies, which revealed a sluggish transition in AceMn starting near 5.3 GPa and completed at 8.5 GPa 44 . The formation of the new phase agrees with the progressing increase of the interaction strength between Ace + and the perovskite framework, as evidenced by strong shifts of the ν s CCN, ν as CCN and ν NCN Raman modes to higher wavenumbers (from 906, 1178, and 1524 cm −1 at 5.3 GPa to 933, 1195 and 1555 cm −1 at 8.5 GPa) 44 . Above 5.0 GPa, the compression breaks the H-bonds and the framework gradually collapses about the cations. Thus the mechanism of the strain compensation in phase III involves the rotations and deformation of formate linkers translating onto the MnO 6 octahedra rotations (described in Glazer notation as a 0 a 0 c − ).

The previously reported Raman spectra indicated that the internal vibrations of the HCOO − ligands and the lattice modes on releasing pressure from 10 GPa to ambient pressure became similar to those observed for the initial ambient‐pressure phase 44 . To investigate it and resolve the nature of this hidden phase, we have quickly compressed a AceMn single crystal to the non-hydrostatic region of Daphne 7575 oil (which freezes above 4.5 GPa) 47 . This quick compression preserves the high crystallinity of the sample due to the effect of over-compressing phase III. Then, by gently heating the sample to 320 K, we recovered the hydrostatic conditions in the diamond anvil cell (DAC), as the heating of Daphne oil is a well-established method for restoring its hydrostatic properties 47 . This procedure allowed us to characterize the crystal structure at 7.35 GPa, and then after decompression at 6.00 and 5.20 GPa. The single-crystal diffraction data collected at these pressures showed that phase III is metastable above 5 GPa. The metastability of phase III was confirmed by a gradual decrease in the intensity of reflections progressing over a period of 2 weeks, due to a slow amorphization process. We have investigated the residual phase III during the amorphization process and the XRD data collected on the sample equilibrated for one week at 5.30 GPa resulted in a unit-cell volume larger than the initial measurement before. The increased unit-cell dimension of residual phase III by ca. 1.3% suggests a mosaic topology of amorphous higher-density regions.

To further elucidate the mechanism behind the structural pressure-induced phase transition in AceMn, we conducted periodic DFT calculations. In this case, we considered only two phases with symmetry P 2 1 / n (phase II) and Pbca (phase III) to explain their stability at various temperature-pressure conditions. The geometry optimization revealed that the orthorhombic phase II is more stable than phase III when excluding temperature and pressure effects. Specifically, the calculated DFT energy of phase III is approximately 0.25 kJ/mol per formula unit lower than that obtained for phase II. As illustrated in Fig.  3a , phase III remains stable until a volume of about 239 Å 3 , beyond which phase II becomes stable. A common tangent line constructed from the volume-energy relationship in Fig.  3a demonstrates the pressure at which the transition from phase III to II occurs:

a Energy vs. volume relation for phases II (orange curve and points) and III (blue curve and points) of AceMn. b Relative stability of structures with P 2 1 / n and Pbca symmetries in AceMn under various pressure conditions without considering temperature effects (enthalpy). Here, \({H}_{{{{\rm{Pbca}}}}}\) and \({H}_{{{{{\rm{P}}}}2}_{1}/{{{\rm{n}}}}}\) denote enthalpies of phases III and II of AceMn, respectively. Areas where \({H}_{{{{\rm{Pbca}}}}} > {H}_{{{{{\rm{P}}}}2}_{1}/{{{\rm{n}}}}}\) and \({H}_{{{{\rm{Pbca}}}}} < {H}_{{{{{\rm{P}}}}2}_{1}/{{{\rm{n}}}}}\) are marked in orange and blue, respectively. The dashed line defines the phase transition conditions and the dotted vertical line marks the tra n sition pressure at 0 K. c Difference in entropy between P 2 1 / n and Pbca symmetries versus temperature, highlighting the significance of entropy effects to the phase transition. Here, \({S}_{{{{\rm{Pbca}}}}}\) and \({S}_{{{{{\rm{P}}}}2}_{1}/{{{\rm{n}}}}}\) denote entropy of phases III and II of AceMn, respectively. d Temperature dependence of Helmholtz free energy, indicating that the monoclinic phase becomes stable at 85 K. Symbols \({F}_{{{{\rm{Pbca}}}}}\) and \({F}_{{{{{\rm{P}}}}2}_{1}/{{{\rm{n}}}}}\) denote Helmholtz free energy for phases III and II of AceMn, respectively. Areas where \({F}_{{Pbca}} > {F}_{{P2}_{1}/n}\) and \({F}_{{Pbca}} < {F}_{{P2}_{1}/n}\) are marked in orange and blue, respectively. The dashed line defines the phase transition conditions and the dotted vertical line marks the transition temperature at 0 GPa. e Gibbs free energy map for phases II and III in AceMn, revealing their mutual stability under various pressure–temperature conditions. Symbols \({G}_{{{{\rm{Pbca}}}}}\) and \({G}_{{{{{\rm{P}}}}2}_{1}/{{{\rm{n}}}}}\) denote Gibbs free energy for phases III and II of AceMn, respectively. The black solid line defines the phase transition conditions.

This line is equivalent to the pressure at which enthalpies of both phases ( \(H=E+{pV}\) ) are the same. The calculated transition pressure is approximately −0.11 GPa (see Fig.  3b ), which contradicts experimental observations and suggests that both temperature and pressure significantly influence the transition.

To incorporate temperature effects, phonon calculations were performed to assess the vibrational partition function and analyze the Helmholtz free energy of both phases. As illustrated in Fig.  3 c, d, phase II becomes more stable around 85 K due to increased entropy arising from the mobility of the counterion in the cavity. The low-frequency regime of the phonon density of states, particularly below 2 THz associated with collective rotation-like vibrations of the ion, is more pronounced in the monoclinic phase (see Supplementary Fig.  7 ). These lower frequencies require less energy to activate the corresponding vibrations of the Ace + . The pore size analysis performed on the ion-free frameworks further confirms this observation and shows that the monoclinic phase features larger cages (by approx. 0.2 Å), providing more freedom for countercation movement at increased temperatures (see Supplementary Fig.  8 ).

Finally, the combined influence of temperature and pressure on the free energy surface is presented in Fig.  3e as a Gibbs free energy map. This landscape should not be treated as a full phase diagram for AceMn as it considers only two phases (II and III) and, thus, can only be used to explain their relative stabilities. Discrepancies in exact values of the phase transition pressure are attributed to the harmonic approximation used in phonon calculations and the potential overestimation of dispersion forces in the Grimme dispersion correction noted in the literature 48 .

In terms of dielectric response, a common feature of FMDMn and AceMn is a single relaxation process, which (according to previous studies) originates from a field-induced motion of FMD + and Ace + cations inside the manganese-formate framework, respectively 12 , 13 , 21 , 22 . This relaxation manifests itself as a bell-shaped peak in the imaginary part of the complex dielectric permittivity, ε “( f ), which amplitude decreases steadily during isothermal compression at 298 K for both compounds (see Fig.  4 a, b and Supplementary Fig.  5 ). Eventually, only residual trace of the relaxation process is detectable for FMDMn when 1 GPa is exceeded (see upper inset in Fig.  4a ). This trend agrees with the pressure-induced stepwise reduction of disordered FMD + cations around 1 GPa, as confirmed by high-pressure diffraction studies. The room-temperature ordering of the cage cations under pressure occurs when the unit-cell volume reaches 824.6(18) Å 3 . In turn, a fully ordered crystal structure of FMDMn at ambient pressure is observed at 110 K when the unit-cell volume is 865.0(7) Å 3 21 . Different volume requirements for this process under various p – T conditions indicate that thermal energy is a critical parameter that controls the cage organic cations ordering in HOIPs. Moreover, a direct correlation between crystal structure and dielectric response of FMDMn proves previous hypotheses that gradual ordering of the organic cage cations due to contraction of the cage-like framework constitutes one of the possible origins of pressure-induced diminishing in relaxation peak amplitude in this 3D HOIPs 12 , 13 .

a Representative dielectric loss ε “( f ) spectra collected between 100 and 500 MPa at 298 K for FMDMn. Experimental data are represented by colored dots, and Havriliak–Negami fitting functions are shown as colored lines. The upper inset illustrates the relaxation peak vanishing during isothermal compression of FMDMn from 100 MPa (green points) to 1500 MPa (navy dots) at 298 K. b Representative ε “( f ) spectra collected during compression at 298 K for AceMn, each coded with different color for clarity. c Comparison of experimentally determined (green circles) and calculated (open blue stars) pressure dependence of relaxation times (log τ max ) for FMDMn at 298 K. The green solid line represents the fit of experimental data with Eq. ( 4 ). d Experimentally determined (red circles) and calculated (open brown stars) pressure dependence of log τ max for AceMn at 298 K. The solid line represents the fit of experimental data with Eq. ( 4 ). Source data are provided as a Source Data file.

Apart from the amplitude-related effect, progressive shifting of the relaxation process towards lower frequencies occurs for FMDMn and AceMn during their isothermal compression. It means that the related relaxation times ( τ max ) extend when the pressure increases since \({\tau }_{\max }=\frac{1}{2\pi {f}_{\max }}\) 49 . In this formula, f max is the frequency of the loss peak maximum. In order to obtain the pressure dependence of τ max at 298 K, we parametrize the dielectric spectra of both compounds with the Havriliak–Negami fit function 50 (see Supplementary Note  1 for more details) and calculate the τ max values based on the fitting parameters τ HN , α , β according to the formula 49 :

As presented in in Fig.  4 c, d, the room-temperature τ max ( p ) dependence exhibits an Arrhenius-like character for both FMDMn and AceMn, that can be parametrized by the equation:

where τ 0.1MPa is the ambient-pressure relaxation time, R is the gas constant, and V a is the activation volume 51 . The determined V a parameter takes the value of 2.6 ± 0.1 and 6.6 ± 0.1 cm 3 mol −1 for FMDMn and AceMn, respectively, being in agreement with previous studies on these compounds 12 , 13 . What is more, both determined τ max ( p ) dependences can be well described by the previously determined equations of state:

(see open points in Fig.  4 c, d) 12 , 13 , 52 . Here, τ 0 is a pre-exponential factor, E a denotes the activation energy, and parameters R , V a do not change the meaning relative to Eq. ( 4 ). Consequently, we use further this formalism to derive general rules that govern ambient- and high-pressure relaxation dynamics in these hybrid perovskites.

Hydrogen bonds are widely recognized as key forces governing the relaxation dynamics in HOIPs. Therefore, we begin our discussion by clarifying the relationship between their length and relaxation times of the reorientating molecular cations.

As exemplified by AceMn, log τ max gradually increases with the shortening (strengthening) of hydrogen bonds (cf. Fig.  5a ). This trend is observed during both isobaric cooling and isothermal compression of this material (see blue and red points, respectively). However, there is a significant disparity between these two thermodynamic paths. The compression up to 860 MPa of the hydrogen bond O6 … H-N2 shortens it from 3.05 Å to 2.94 Å at room temperature and increases in τ max from ~2.46 μs to ~22.7 μs. In contrast, the same change of the hydrogen bonds during isobaric cooling increases τ max up to approximately 100 s. It is evident that relaxation times in AceMn do not scale with the length of hydrogen bonds, indicating that their strength alone cannot be considered the sole determinant of the relaxation dynamics of Ace + cations. A similar observation can also be made for FMDMn.

a Changes in relaxation times (log τ max ) observed in AceMn during room-temperature compression (red squares) and ambient-pressure cooling (blue dots) plotted versus the length of the O6 … H-N2 hydrogen bond. Error bars represent standard deviations. The shaded area defines the difference between these two thermodynamic paths. b , c Variation of unit-cell parameters a , b , c , d [101] during isothermal compression at 298 K (closed symbols) and isobaric changes in temperature under 0.1 MPa (open symbols) for FMDMn and AceMn. Lower indices I, II, and III in these parameters refer to the phase number. d Changes in the β angle for FMDMn (blue symbols) and AceMn (red symbols) observed during isothermal compression at 298 K (closed points) and isobaric changes in temperature under 0.1 MPa (open symbols). Lower indices I, II, and III in the β parameter refer to the phase number. e Dependence of relaxation times on unit-cell volume of FMDMn observed under isobaric conditions of 0.1 MPa (open blue symbols and dotted line) and isothermal high-pressure compression at 298 K (closed blue symbols and solid line). The shaded area indicates volume-related changes. f Dependence of relaxation times on unit-cell volume of AceMn observed under isobaric conditions of 0.1 MPa (open symbols and dotted line) and isothermal high-pressure compression at 298 K (closed symbols and solid line). The shaded area defines volume-related changes. Source data are provided as a Source Data file.

Furthermore, the pressure–temperature evolution of τ max does not correlate molecular volume of FMDMn and AceMn. The X-ray diffraction studies reveal that both FMDMn and AceMn possess relatively soft ionic structures. In AceMn, lowering the temperature from 296 K to 130 K results in a reduction of the unit-cell volume from approximately 950.63 Å 3 to roughly 937.73 Å 3 while compression at room temperature up to 350 MPa reduces the unit-cell volume to 929.82 Å 3 . FMDMn shows a similar behavior, with a reduction of the unit-cell volume from approximately 882.20 Å 3 to 865.03 Å 3 when the temperature is lowered from 300 K to 100 K, while a comparable decrease to 861.91 Å 3 is observed during room-temperature compression to 350 MPa. Notably, in both compounds, the variations of the unit-cell parameters a 0 , b 0 , and c 0 exhibit the same pattern during isobaric cooling (represented by open points) and isothermal compression (represented by closed points). This feature is evident when plotting the a 0 , b 0 , and c 0 , parameters as a function of the unit-cell volume (see Fig.  5 b, c). Only minor differences between these thermodynamic paths are noted in the behavior of the β angle for AceMn (refer to red open and closed points in Fig.  5d ). Consequently, we can conclude that pressure and temperature exert similar effects on the crystal structure in both FMDMn and AceMn. In contrast, the dielectric relaxation proves to be far more sensitive to temperature changes. This becomes evident when plotting τ max , calculated from Eq. ( 5 ) based on the known pressure–temperature, versus the experimentally determined unit-cell volume for FMDMn and AceMn (Fig.  5 e, f). As presented in Fig.  5 e, f, volume contraction has a significantly smaller impact on relaxation times compared to temperature effects for both FMDMn and AceMn. This observation emphasizes the negligible influence of thermal expansion on the energy barrier associated with the cage cations' movements within the cavities. Consequently, these findings provide an explanation for the activation-like Arrhenius behavior observed in the dependence of τ max ( T ) observed in AceMn and FMDMn, with temperature-independent parameter E a . They also emphasize that the volumetric changes and related shifts in the strength of hydrogen bonds or other interactions do not constitute the sole determinant governing the relaxation dynamics in HOIPs.

According to formula (5), τ max scales with the activation volume parameter in AceMn, FMDMn suggesting a similar scaling also applies to other HOIPs. The activation volume is an important parameter in the equation of state as it determines the pressure-induced changes in relaxation times and the corresponding shift of the relaxation peak 51 . It also determines the magnitude of pressure-induced changes in E a . Namely, by combining Eq. ( 5 ) and a classical Arrhenius law, one can derive:

As presented in Fig.  6a , if V a is positive, E a increases with pressure in a linear manner and the slope coefficient of the E a ( p ) dependence is equal to V a :

a Pressure-induced increase in activation energy parameter for FMDMn (blue dots) and AceMn (red stars). The colored lines represent fit curves defined by Eq. ( 6 ). b Pressure dependence of Δlog τ max for FMDMn and AceMn at 298 K. Here, Δlog τ max is calculated as changes in relaxation times log τ max relative to the ambient-pressure value of this parameter. The inset shows activation volume V a at various temperatures for FMDMn and AceMn. c Possible changes in the manganese-formate skeleton induced by reorientation of Ace + cations. Mn, O, C, N, and H atoms are marked in purple, red, gray, blue, and white, respectively. d Schematic illustration of the diversified effect of external pressure on the cage cations in hybrid MOFs due to different compressibility of the organic-inorganic framework. The metal-organic framework is schematically shown as red or green lines. In turn, the H, C, and N atoms of the cage cation are marked in white, black, and blue, respectively. Source data are provided as a Source Data file.

These formulas effectively explain the experimentally confirmed absence of pressure-induced increase of E a in FMDMn despite the electric-field-induced reorientation of the FMD + cations 12 . For this material, Eq. ( 6 ) predicts a small increase of E a at high pressure by 25 meV GPa −1 , resulting in an increment of about 8 meV when applying 300 MPa (or 16 meV when applying 600 MPa), which fits within the experimental uncertainty of 20 meV 12 . In turn, for AceMn, E a increases under pressure by 66 meV GPa −1 (equivalent to 20 meV under 300 MPa pressure or 40 meV under 600 MPa), which was confirmed experimentally 13 . It is thus evident that the pressure-induced changes in E a increase with rising V a value. Similarly, at constant temperature, τ max increases under pressure with rising V a :

This rule is well illustrated by the Δlog τ max ( p ) dependencies for the studied hybrid perovskites FMDMn and AceMn (Fig.  6b ). It can be shown that V a is a temperature-independent material constant if the relaxation dynamics is described by the equation of state given by formula 5 (see inset in Fig.  6b ). This parameter does not depend on the size of reorienting cage cation. Our DFT investigations indicate that the ratio of the Ace + and FMD + cation sizes (defined as the ratio between their van der Waals volumes o \({V}_{{{{{\rm{Ace}}}}}^{+}}^{{{{\rm{vdW}}}}}/{V}_{{{{{\rm{FMD}}}}}^{+}}^{{{{\rm{vdW}}}}}\) ) equals \(1.7\) . In contrast, experiments show that the ratio of the V a parameters for AceMn and FMDMn is equal to \({V}_{{{{\rm{a}}}},{{{\rm{AceMn}}}}}/\,{V}_{{{{\rm{a}}}},{{{\rm{FMDMn}}}}}=2.7\) . Furthermore, the relaxation of larger 1,4-butanediammonium cations (DAB 2+ , [NH 3 -CH 2 -CH 2 -CH 2 -CH 2 -NH 3 ] 2+ ) within the zinc-formate cages is related to much smaller V a compared to FMDMn and AceMn 14 . In order to define the activation volume, we study the exemplary reorientation path of Ace + cations within the P 2 1 / n phase, employing DFT solid-state nudged elastic band calculations.

According to previous studies on FMDMn and AceMn, the low-temperature motion of the cage cations can mirror the mechanism typical for the HT phase 12 , 13 . Consequently, we focus on rotating the Ace + cation around its C–N bond in our simulations. As shown in Fig.  6c , such transformations reshape the organic-inorganic cages and induce variations in their volume. The transient state is not only a higher-energy structure but also features larger cavity sizes. The simulated volumetric barrier for the reorientation of a single Ace + cation within phase II is roughly 7.5 Å 3 (see Supplementary Fig.  9 for more details). This value corresponds reasonably to the experimentally determined V a parameter for AceMn (6.6 ± 0.1 cm 3  mol −1 ), equal to 11 Å 3 per reorienting unit. The discrepancy between theory and experiment may originate from the chosen calculation methodology, which only considers one reorientation possibility out of many possible field-induced motion schemes. Indeed, prior studies on DABZn suggested V a to be a direction-dependent (and, consequently, mechanism-dependent) parameter 14 . Hence, based on the DFT calculations, we define activation volume as a parameter delineating the volumetric barrier for the relaxation of cage cations within the cavities. This definition mirrors that derived for other systems 53 , 54 , 55 , 56 . However, in HOIPs, V a depends on numerous factors, including cage cation flexibility, mechanism of its motion, or rigidity of the entire cavities. As illustrated schematically in Fig.  6d , increased rigidity of the metal-organic framework leads to smaller volume contraction of the cages during isothermal compression, and the reorienting cage cation is less affected by external forces. As a result, the volumetric contribution to the relaxation times is reduced, so τ max changes less with pressure. This decrease in pressure sensitivity, according to Eq. ( 8 ), lowers the V a value. Furthermore, the increased flexibility of the cage cation helps it adapt better to pressure-induced changes in the cavities, enhancing this effect.

Finally, defining V a as a volumetric barrier for the relaxation process also explains the significantly larger influence of temperature on τ max (see Supplementary Note  2 for detailed mathematical formalism). At high temperatures, the metal-organic framework possesses enough thermal energy to adjust and accommodate the reorientations of cage cations even under high pressure. Their structural flexibility decreases significantly with lowering temperature, resulting in a dramatic slowdown of the relaxation dynamics of the cage cation. The non-equivalent pressure and temperature effects, previously reported for methylammonium lead iodide (MAPbI 3 ) 57 , explain well why the FMD + cations do not order under isochronous conditions.

In conclusion, ambient- and high-pressure X-ray diffraction, Raman scattering, and broadband dielectric studies were performed on model formamidinium manganese(II) formate (FMDMn) and acetamidinium manganese(II) formate (AceMn) to expand our knowledge on high-pressure behavior of hybrid formates and derive fundamental principles governing their relaxation dynamics under various pressure regimes. Firstly, the AceMn crystals exhibit a phase transition at 1 GPa from ambient-pressure phase II (space group P 2 1 / n ) to high-pressure orthorhombic phase III (space group Pbca ). Phase III of AceMn can undergo astonishing overcompression above 5 GPa, exceeding 2.3 GPa, coexisting with an amorphous solid and contributing to a mosaic-like topology in this metastable region. Secondly, the compression of FMDMn leads to a reduction of the cage volume followed by the ordering of cage cations, without altering the symmetry nor considerably increasing the activation energy. This mechanism is associated with reorientation of the cage cations and structural adjustments of the framework. Consequently, the relaxation dynamics show two significant contributions (thermal and volumetric), with thermal energy and thermally-activated librations within the crystal mostly affecting the relaxation times. As a result, relaxation times do not scale with unit-cell volume or the strength of hydrogen bonds despite similar pressure and temperature evolution of the unit-cell parameters. This observation is reflected in a small activation volume parameter, redefined for HOIPs as a temperature-independent material constant. This parameter determines the volumetric barrier for relaxation processes and depends on cage cation flexibility, the mechanism of its motion, and the rigidity of the entire cavities. Hence, the results indicate a complex relationship between structure and dielectric response in HOIPs under pressure.

The subject of this research is crystalline compounds FMDMn and AceMn. Both compounds were synthesized as small crystals according to the procedures utilized by us before 21 , 22 . In order to obtain FMDMn, a mixture of cyclobutane-1,1′-dicarboxylic acid (4.2 mmol), MnCl 2 (8.4 mmol), and formamide (50 mL) was heated at 130 °C for 24 h in a Teflon-lined microwave autoclave. Light pink crystals were collected after slow overnight cooling. They were subsequently washed with ethanol (3 × 5 mL) and finally dried at room temperature. In order to synthesize AceMn, acetamidine hydrochloride (40 mmol) was dissolved in methanol (15 mL) and the resulting solution was mixed with formic acid (80 mmol) and trimethylamine (10 mmol). The mixture was placed at the bottom of a plastic vial, which was followed by the careful addition of 10 mL of a methanol solution of manganese(II) perchlorate hydrate (2 mmol). The vial was sealed and left undisturbed for 48 h. The formed crystals of AceMn were subsequently harvested, washed with methanol (3 × 5 mL), and dried at room temperature.

Ambient- and high-pressure broadband dielectric spectroscopy

Ambient-pressure dielectric measurements of FMDMn and AceMn were conducted on polycrystalline and pelletized samples with a diameter of 5 mm. Prior to measurements, parallel planes of the samples were painted with silver paste to ensure optimal electric contact and thoroughly dried in a vacuum oven at 330 K. A Novocontrol Alpha impedance analyzer was utilized for the broadband dielectric investigations. During measurements, an ac current in the frequency range of 10 −1 –10 6  Hz was applied across the samples. For both compounds, dielectric spectra were collected every 2 K between 170 and 320 K under quasi-static conditions. For this purpose, the temperature was stabilized prior to each spectrum collection with nitrogen gas. The temperature was controlled by a Novocontrol Quattro system with a precision exceeding 0.1 K.

The same samples of FMDMn and AceMn were then utilized for high-pressure dielectric measurements. However, in this case, they were glued with a silver paste to the high-pressure capacitor, after which they were subjected to drying in a vacuum oven at 330 K for 24 h. Each so-prepared high-pressure capacitor was subsequently enclosed in a Teflon capsule filled with Julabo Thermal HL90 silicon oil and placed inside an LC20 T-type high-pressure chamber (designed by UNIPRESS for hydrostatic pressure up to 1.8 GPa). The whole high-pressure system was put into a thermostatic mantle and placed on a hydraulic press. High-pressure dielectric measurements were conducted during compression under isothermal conditions of 298 K. During measurements, the temperature was stabilized and controlled by Huber Tango Unistat thermostatic bath with a precision exceeding 0.1 K. Ambient-pressure alike, and dielectric spectra were recorded by a Novocontrol Alpha impedance analyzer. During measurements, an ac current in the frequency range of 10 −1 –10 6  Hz was applied across the samples. Prior to each spectrum collection, pressure was stabilized by at least 15 min to ensure quasi-static conditions and the measurements were performed every 50 MPa up to 1500 MPa for FMDMn and (due to technical reasons) up to 1000 MPa for AceMn.

High-pressure single-crystal X-ray diffraction

The high-pressure experiments were performed with a modified Merrill-Bassett DAC cell 58 with diamond anvils supported on steel disks. The diamond culets were 0.8 mm in diameter. The gasket was made of 0.1 mm thick tungsten foil with sparked-eroded holes of 0.4 to 0.5 mm in diameter. The pressure was calibrated using the ruby fluorescence method 59 , 60 , and Daphne oil 7575 was used as the pressure-transmitting medium. Single-crystal X-ray diffraction measurements were performed using a 4-circle diffractometer equipped with a CCD detector and MoKα X-ray source (λ = 0.71073 Å). The DAC chamber was centered using the gasket-shadowing method 61 . The CrysAlisPro software was used for the data collection and processing. All the structures were solved with direct methods using Shelxs and refined with Shelxl using the Olex2 suite 62 , 63 . Structural drawings were prepared with program Mercury CSD 3.8 64 .

High-pressure Raman scattering

High-pressure Raman spectroscopy was performed using a LabRAM HR Evolution spectrometer (from Horiba), set to a resolution of 2 cm −1 , and equipped with a thermoelectrically cooled CCD detection system. The spectra were collected with a 20× long working distance Olympus objective lens and the excitation was provided by a 514.4 nm argon laser line in backscattering geometry, with a power of 2.0 mW focused onto the sample. A diamond anvil cell (DAC) SS Syntek design (manufactured by NOVARETI LTDA) was employed to achieve high pressures. Mineral oil was used as the pressure-transmitting medium. The sample was loaded into a 100 µm diameter hole drilled in a 200 µm thick stainless-steel gasket using an electric discharge machine from Almax easyLab. The pressure was determined by monitoring the shifts of the ruby R1 and R2 fluorescence lines.

Density functional theory

Density functional theory 65 was used to model and analyze the structure of FMD + and Ace + cations. Their van der Waals volumes in a single-molecule approach by means of B3LYP hybrid density functional 66 , 67 , 68 combined with Pople’s 6-311++G(d,p) basis set 69 . These calculations were carried out using the Gaussian 16C.01 program package 70 .

To study relaxation dynamics and pressure-induced phase transition in AceMn, periodic DFT simulations were conducted using projector-augmented wave (PAW) pseudopotentials within the Vienna Ab initio Simulation Package (VASP, version 5.4.4) 71 , 72 , 73 . The simulations employed the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional 74 , enhanced with D3(BJ) dispersion correction 75 , 76 for a more accurate treatment of van der Waals interactions, and a plane-wave cutoff energy of 500 eV. Our computational models for both phases consisted of a unit cell containing 8 Mn atoms. We utilized a gamma-centered 2 × 2 × 2 k-point grid to ensure accurate Brillouin zone sampling. Considering the presence of unpaired electrons on d-orbitals of Mn atoms, spin polarization was incorporated, adopting a high-spin ferromagnetic configuration with a magnetization value of 5 μ B per Mn atom. To mitigate self-interaction errors, a Hubbard U correction (U = 3.8 eV 77 ) was applied to the d-orbitals of Mn. The convergence criteria for electronic steps and geometry optimization were set to 10 −6  eV and 0.01 eV/Å, respectively. Volume-optimized energy calculations were performed under constant-volume conditions, allowing for variations in lattice parameters (ISIF = 4). Phonon calculations were carried out using Phonopy (version 2.19.1) and the Parlinski–Li–Kawazoe method 78 , 79 , with finite difference single point calculations in a 1 × 1 × 1 supercell. Phonon frequencies were determined using a 14 × 14 × 14 sampling mesh for thermodynamic functions and a 4 × 4 × 4 mesh for the phonon density of states (detailed convergence tests are provided in Supplementary Fig.  10 ). Pore size distribution was analyzed with PoreBlazer software 80 . The minimum energy path for acetamidine rotation was identified using the solid-state nudge elastic band (SS-NEB) method 81 , facilitated by the Transition State Atomistic Simulation Environment (TSASE), an extension of the Atomic Simulation Environment (ASE) 82 .

Data availability

Data that support this research are available in its Supplementary Information files. The source dielectric, Raman, and structural data generated in this study are provided in the Source Data file. All other data are available from the corresponding author upon request. The X-ray crystallographic coordinates for the structures of AceMn and FMDMn reported in this study have been deposited at the Cambridge Crystallographic Data Centre (CCDC), under deposition numbers 2287942-2287953 for FMDMn and 2287917-2287940 for AceMn. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif .  Source data are provided with this paper.

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Acknowledgements

F.F. gratefully acknowledges Polish high-performance computing infrastructure PLGrid (HPC Centers: ACK Cyfronet AGH) for providing computer facilities and support within computational grant no. PLG/2023/016135.

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Conceptualization: A.N., A.K. and A.S.; Sample preparation: M.M.; Experimental investigation: S.S., K.R., A.K., A.N., A.Z.S., A.J.B.S. and W.P.; Theoretical investigation: A.N. and F.F.; Project administration: A.S.; Supervision: A.S., A.K., S.P., and M.M.; Visualization: A.N., S.S. and A.Z.S.; Writing: A.N. and S.S.; Review & editing: all authors.

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Nowok, A., Sobczak, S., Roszak, K. et al. Temperature and volumetric effects on structural and dielectric properties of hybrid perovskites. Nat Commun 15 , 7571 (2024). https://doi.org/10.1038/s41467-024-51396-5

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