Investigation of the Proton-Bound Dimer of Dihydrogen Phosphate and Formate Using Infrared Spectroscopy in Helium Droplets

Understanding the structural and dynamic properties of proton-bound complexes is crucial for elucidating fundamental aspects of chemical reactivity and molecular interactions. In this work, the proton-bound complex between dihydrogen phosphate and formate, and its deuterated counterparts, is investigated using IR action spectroscopy in helium droplets. Contrary to the initial expectation that the stronger phosphoric acid would donate a proton to formate, both experiment and theory show that all exchangeable protons are located in the phosphate moiety. The experimental spectra show good agreement with both scaled harmonic and VPT2 anharmonic calculations, indicating that anharmonic effects are small. Some H-bending modes of the nondeuterated complex are found to be sensitive to the helium environment. In the case of the partially deuterated complexes, the experiments indicate that internal dynamics leads to isomeric interconversion upon IR excitation.


Excitation of a Two-level System in Helium Droplets
The experiment described in the manuscript relies on the evaporation of helium atoms from a droplet after photon absorption.This evaporation is the consequence of vibrational energy, from the absorption of photons, being converted to phonons, leading to evaporation.In the following, we want to estimate the amount of energy that a single chromophore in a helium droplet during a light pulse can convert to phonons.
In a simplified picture, the excitation and relaxation processes can be described by rate equations for a two-level system, which include excitation, stimulated emission and relaxation of vibrational energy to phonons (Figure S1).
where F is laser flux [photons/(cm 2 s)], σ is the absorption cross-section [cm 2 /molecule] and kr is the rate constant for relaxation.

Figure S1.
Excitation process in a two-level system.
We consider a single droplet with a single dopant molecule for which the sum of the populations is equal to one: The number of photons of which the energy is transferred to phonons (leading to helium evaporation) νk during a light pulse with length T can be quantified by the integral of the product of the population of the higher energy level with the rate constant for relaxation over the duration of the light pulse: The light from the FEL comes in macropulses of light that consist of micropulses.The macropulse length is 10 μs and the micropulse spacing is 1 ns.We assume that the micropulses have a gaussian temporal structure with a FWHM width of ~6 ps.Typically, quantum chemistry software reports band integrated IR intensities in km/mol.For the conversion to absorption cross sections in cm 2 /molecule, we assume a width of the absorption band of 1 cm -1 .We further consider a typical intense absorption of 500 km/mol, a spot-size of 0.3 cm 2 and a photon energy of 1500 cm -1 .
The differential equations can then be integrated numerically with explicitly considering the time structure of the light.Figure S1 shows in the bottom part the first 10 ns of a macropulse with a total macropulse energy of 60 mJ and micopulse FWHM widths of ~6 ps.In the top two parts, the excited state population when using the parameters given in the text are shown for two relaxation rate constants.

Figure S2.
Bottom: the first 10 ns of a 10 μs FEL macropulse with total energy of 60 mJ and a micropulse FWHM width of ~6 ps.Middle and top: excited state populations for two different relaxation rate constants.
Equation 4 can then be used to numerically calculate the number of photons which relax after absorption and cause droplet heating.
For the case of a continuous-wave (cw) pulse, the equations above can be solved analytically.We consider a cw pulse with the same length as the macropulse (10 µs) and assume that the populations of the two energy levels, N0 and N1 reach equilibrium very fast, compared to the duration of the excitation pulse so that their time derivatives become zero.
d " d = d !d = 0 ( 5 ) Solving the equations yields the following expression for the excited state population: When the flux of photons [photons/(cm 2 s)] is constant over a pulse with length T, the number of photons transferred to phonon energy is given by: Figure S3 shows the number of photons converted to heat as a function of laser energy, obtained by numerically solving the equations 1-4 (solid lines) and from the approximation (eq.7) for various relaxation rate constants.In polyatomic molecules, kr is expected to be in the order of ps to ns 1 .The other parameters used are given above in the text above.It can be observed that the number of photons transferred to phonons is not only strongly dependent on the FEL macropulse energy, but also dependents strongly on the relaxation rate constant kr.In our experiment, we observe the appearance of bare unsolvated ions after resonant IR absorption.Should this occur from the complete evaporation of the droplet, many photons have to be absorbed.Considering 5 cm -1 as binding energy for a helium atom to a droplet3 we can relate the number of photons transferred to phonons to the fraction of doped droplets that are completely evaporated (Figure S5).The amount of unsolvated ions signal is strongly dependent on the relaxation rate constant kr .Due to the droplet size distribution and the excitation dynamics, the observed signal depends non-linearly on and the absorption cross-section.

Figure S5
. Evaporation fraction as a function of FEL macropulse energy considering the excitation of a 500 km/mol transition, a spot-size of 0.3 cm 2 and a photon energy of 1500 cm -1 for different relaxation rate constants.The helium nanodroplets are assumed to follow a log-normal droplet-size distribution with mode= 5.0E+04 and mean=7.1E+04.The solid lines represent the output of the model, which solves the differential equations numerically.The dotted line represents the output of the cwapproximation.

Scaling factor
To select an adequate scaling factor for the harmonic vibrational frequencies, the ratio of the anharmonic (VPT2) to harmonic frequencies are calculated and averaged within the measured region.When performing this for all deuteration levels, a scaling factor of 0.975 is obtained.All calculations are performed at the B3LYP-D3(BJ)/Jul-cc-pV(T+d)Z level of theory. [

Figure S3 .
Figure S3.Number of photons converted to heat as a function of FEL macropulse energy for different constant rate constants.The solid lines are obtained from numerically solving eq.1-4 and the dotted lines from the analytical model (eq.7).

Figure S4 .
Figure S4.Probability Density Function (a) and Cumulative Distribution function (b) according to a log-normal distribution whit mode= 5.0E+04 and mean=7.1E+04,which reassemble the Droplet size-distribution of helium nanodroplets generated in the current experiment.

Figure S8
Figure S8 Experimental IR action spectrum of the phosphoric acid-formate protonbound dimer [FP-H3] -at high (light green) and low (green) FEL macropulse energy in comparison with the calculated harmonic (0.975 scaled, blue) IR spectra of the three lowest minima structures (a), b),c)) computed at the B3LYP-D3(BJ)/Aug-cc-pV(T+d)Z level of theory.

Figure S9
Figure S9 Experimental IR action spectra of the phosphoric acid-formate protonbound dimer [FP-H3] -at high (light green) and low (green) FEL macropulse energy in comparison with the calculated harmonic (0.975 scaled, yellow) IR spectra of the three lowest minima structures (a), b),c)) computed at the WB97-XD/Jul-cc-pV(T+d)Z level of theory.

Figure S10
Figure S10 Experimental IR action spectra of the phosphoric acid-formate protonbound dimer [FP-H3] -at high (light green) and low (green) FEL macropulse energy in comparison with the calculated harmonic (0.975 scaled, red) IR spectra of the three lowest minima structures (a), b),c)) computed at the MP2/Jul-cc-pV(T+d)Z level of theory.

Figure S12
Figure S12The theoretical vibrational bands calculated for [FP-H3] -at the B3LYP-D3(BJ)/Jul-cc-pV(T+d)Z level of theory and scaled by a factor of 0.975 (black line).In the calculations, the mass of one hydrogen atom (green highlight) is artificially increased by 1, 5, 10, and 50\%.The bands that change the most are highlighted in green.The calculations are compared to the high FEL macropulse energy experimental spectrum (top trace, light green filling).

Figure S14
Figure S14Experimental IR action spectra of the [FP-H2D] -complex at high (filled light cyan) and low (dark cyan) FEL macropulse energy in comparison with the calculated IR spectra of structure a) computed at the B3LYP-D3(BJ)/Jul-cc-pV(T+d)Z level of theory.The calculated spectrum in the anharmonic approximation is shown in dashed lines.The spectrum comprises of bands stemming from fundamental modes (gray), combination bands (yellow), and overtones (red).

Figure S15
Figure S15Experimental IR action spectra of the [FP-H2D] -complex at high (filled light pink) and low (magenta) FEL macropulse energy in comparison with the calculated IR spectra of structure a) computed at the B3LYP-D3(BJ)/Jul-cc-pV(T+d)Z level of theory.The calculated spectrum in the anharmonic approximation is shown in dashed lines.The spectrum comprises of bands stemming from fundamental modes (gray), combination bands (yellow), and overtones (red).

Table TS1 .
Comparison of relative energies of [FP-H3 -] at different levels of theory without considering zero-point vibrational energy (ZPE).

Table TS2 .
Comparison of relative energies of [FP-H3 -] at different levels of theory taking into account zero-point vibrational energy (ZPE)