A New Molecular-Mechanics Model for Simulations of Hydrogen Fluoride in Chemistry and Biology

Esam A Orabi; José D Faraldo-Gómez

doi:10.1021/acs.jctc.0c00247

. Author manuscript; available in PMC: 2021 Sep 10.

Published in final edited form as: J Chem Theory Comput. 2020 Jul 21;16(8):5105–5126. doi: 10.1021/acs.jctc.0c00247

A New Molecular-Mechanics Model for Simulations of Hydrogen Fluoride in Chemistry and Biology

Esam A Orabi ^1,^*, José D Faraldo-Gómez ^1,^*

PMCID: PMC8431972 NIHMSID: NIHMS1736508 PMID: 32615034

Abstract

Hydrogen fluoride (HF) is the most polar diatomic molecule and one of the simplest molecules capable of hydrogen-bonding. HF deviates from ideality both in the gas phase and in solution, and is thus of great interest from a fundamental standpoint. Pure and aqueous HF solutions are also broadly used in chemical and industrial processes, despite their high toxicity. HF is a stable species also in some biological conditions, because it does not readily dissociate in water unlike other hydrogen halides; yet, little is known about how HF interacts with biomolecules. Here, we set out to develop a molecular-mechanics model to enable computer simulations of HF in chemical and biological applications. This model is based on a comprehensive high-level ab initio quantum chemical investigation of the structure and energetics of the HF monomer and dimer; (HF)_n clusters, for n = 3-7; various clusters of HF and H₂O; and complexes of HF with analogs of all 20 amino-acids and of several commonly occurring lipids, both neutral and ionized. This systematic analysis explains the unique properties of this molecule; for example, that interacting HF molecules favor non-linear geometries despite being diatomic, and that HF is a strong H-bond donor but a poor acceptor. The ab initio data also enables us to calibrate a three-site molecular-mechanics model, with which we investigate the structure and thermodynamic properties of gaseous, liquid, and supercritical HF in a wide range of temperatures and pressures; the solvation structure of HF in water and of H₂O in liquid HF; and the free diffusion of HF across a lipid bilayer, a key process underlying the high cytotoxicity of HF. Despite its inherent simplifications, the model presented significantly improves upon previous efforts to capture the properties of pure and aqueous HF fluids by molecular-mechanics methods, and to our knowledge constitutes the first parameter set calibrated for biomolecular simulations.

Graphical Abstract

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1. Introduction

Hydrogen fluoride (HF) was first prepared in the anhydrous form by Fremy in 1856 by distillation from potassium hydrogen fluoride, KHF₂.¹ It is a colorless substance whose boiling^2-4 and melting^4-8 points are about 293 K and 190 K, respectively. HF is produced in large quantities and utilized broadly.⁹ It is used in the preparation of most fluorine-containing compounds,^9-11 including the hydrofluorocarbons and hydrochlorofluorocarbons employed as refrigerants. HF is also used in the production of inorganic fluorides, in uranium enrichment and aluminum manufacture, and as a catalyst in the production of high-octane gasoline.^9,10 HF is miscible with H₂O in all proportions;¹² vapor and liquid HF-H₂O mixtures are also used widely, for example for etching in semiconductor production and for pickling of stainless steel, respectively.¹³ Notwithstanding their widespread use in chemical and industrial processes, both pure and aqueous HF (also known as hydrofluoric acid) are highly toxic and corrosive. Handling of these substances thus requires rigorous safety precautions. Acute harmful effects include damage to skin and lung, cardiac arrhythmias, and renal failure.¹⁰ Multiple cases of death due to accidental exposure to HF have been reported.^9,10

HF combines the least and the most electronegative nonmetallic atoms; the F-H bond is thus thought of as having a strong (~43%) ionic character.¹⁴ Indeed, HF is the most polar diatomic molecule known: its gas-phase dipole moment is 1.83 D,¹⁵ which is comparable to that of the triatomic water molecule (1.85 D).¹⁶ Such large polarity makes HF the simplest neutral diatomic molecule capable of forming strong hydrogen-bonds. In fact, gaseous HF deviates from the ideal behavior even at low vapor pressures^2,8,17-21 due to aggregation driven by strong intermolecular H-bonding. Given the importance of H-bonding in chemistry and biology, HF and its deuterated form (DF) have been the focus of many experimental^{2,3,6,8,15,17-45} and computational^46-78 investigations.

Aqueous HF solutions are also intriguing. Unlike the other hydrogen halides (HCl, HBr, and HI), which dissociate completely in water, HF is a weak acid with pK_a ~ 3.2.^79,80 In contrast to other weak acids, however, the strength of HF increases with increasing concentration.⁸¹ The HF-H₂O mixture has thus been seen as a good model system to investigate deviations from ideality in binary systems, either through experimental^{7,12,79,80,82-86} or computational^87-94 methods. The weak acidity of HF, especially in diluted solutions, also implies HF is stable in some biological settings.^95,96 However, our understanding of how HF interacts with proteins and other cellular components is very limited, and the molecular basis of the regulatory processes controlling the levels of F⁻ and HF in the cell have been largely unexplored until recently.^95,96

The toxicity of pure and aqueous HF solutions makes computational modeling an appealing complement to experimental investigations. The properties of HF have been studied with classical molecular dynamics (MD) simulations using both non-polarizable^{44,47,50,52,58,69,74,75,97} and polarizable⁵¹ molecular-mechanics (MM) force fields as well as using quantum mechanical (QM) methods.^{48,53,55-57,60,62,63,66} Various HF-H₂O systems have also been studied using ab initio MD simulations.^87,91-93 While some QM-based MD simulations are most accurate in principle, the associated computational cost severely limits the size of the systems that can be investigated, and the diversity of molecular configurations that can be sampled in a given simulation to obtain robust thermodynamic quantities. Thus, MM methods will continue to be a complementary approach for the foreseeable future. Among them, those that explicitly account for atomic polarizabilities are under development and increasingly utilized;^98-107 however, non-polarizable force fields remain by far the most used in simulations of complex biomolecular systems, in combination with ad hoc corrections optimized for interactions where polarizability might be important.^108-114 For HF in particular, the small size of the F atom makes the molecular polarizability much smaller (0.83 Å³)¹¹⁵ than, for example, that of water (1.47 Å³),¹¹⁶ suggesting that polarization effects are sufficiently small to be captured through suitable parameterization of a conventional force field.

In this work, we report a systematic, detailed QM characterization of the structure and energetics of (HF)_n (n = 1–7), HF·H₂O, (HF)₂·H₂O, and (H₂O)₂·HF clusters. We also study HF complexes with analogs of the 20 naturally-occurring amino-acids and of commonly-occurring lipids, considering different protonation states for those that are ionizable (namely ethane, acetamide, N-methylacetamide (NMA), methylamine, methylammonium, tetramethylammonium, methylguanidinium, methanethiol, methanethiolate, dimethyl sulfide (DMS), methanol, ethylene glycol, acetic acid, acetate, methyl acetate, methyl hydrogen phosphate, dimethyl phosphate, toluene, 3-methylindole, 4-methylphenol, 4-methylphenolate, 4-methylimidazole, 4-methylimidazolium, and prolinamide). Based on this investigation, we develop a three-site MM model for HF, which we use to investigate the properties of liquid and supercritical HF solutions in a broad range of temperatures and pressures. The MM model is further calibrated against the ab initio data to explore the solvation properties of HF in water and of H₂O in liquid HF, as well as the interactions with amino-acid side chains and a lipid bilayer. The proposed MM model reproduces available experimental data for pure and aqueous HF solutions and illustrates how HF spontaneously permeates lipid membranes. To our knowledge this model is the first of its kind calibrated to study HF in biomolecular systems.

2. Computational Methodology

2.1. Ab initio Quantum Chemical Calculations

Quantum chemical calculations were carried out in the gas phase using the Gaussian 16 software.¹¹⁷ Unless otherwise stated, the geometries of HF complexes were optimized without constraints, and the resulting structures correspond to energy minima (no imaginary frequencies). The structures were first optimized with MP2(full)/6-311++G(d,p) and the resulting structures were re-optimized with MP2(full)/6-311++G(3df,3pd). For HF, (HF)₂ and HF·H₂O, the structures were also optimized with CCSD(T) using the above two basis sets in addition to the aug-cc-pVXZ basis sets (X = D, T, Q, and 5). For the (HF)_n clusters, the following strategy was followed. To obtain the maximum possible number of stable conformers for a given value of n, each cluster was investigated by adding an HF molecule to the various conformers previously optimized for cluster (HF)_n−1, considering multiple relative orientations and different modes of H-bonding. For HF complexes with H₂O as well as with lipid and amino-acid side-chain analogous, multiple initial configurations were similarly explored, differing in the relative orientation between the interacting molecules. Interaction energies were calculated as the energy difference between the complexes and the constituent monomers. Interaction energies were calculated at the same theory levels used for the geometry optimizations, but also with CCSD(T)/6-311++G(3df,3pd) using the geometries calculated with MP2(full)/6-311++G(3df,3pd). Corrections of the interaction energies for basis-set superposition error (BSSE) were derived using the counterpoise (CP) procedure of Boys and Bernardi¹¹⁸ and both uncorrected (E) and corrected (E^CP) values are reported. Interaction energies were not corrected for zero-point vibrational energies.

The surface electrostatic potentials of HF and H₂O were calculated with MP2(full)/6-311++G(3df,3pd) and mapped on the MP2 electron density. Rigid-monomer potential-energy curves (PEC) were calculated as a function of the distance between the F atom of HF and the S atom of DMS, in the range between 2.0 and 10.0 Å, in 0.1 Å increments. Two different conformers were considered for the complex, but the geometry of the fragments and their relative orientations was maintained as in the optimized conformers. These PEC curves were corrected for BSSE and later used to calibrate the model for HF interactions with the neutral sulfur-containing ligands (DMS and methanethiol). The same methodology was used to scan the H─F⋯HF angle in the (HF)₂ dimer between 60° and 180°, in 5° increments at a constant intermolecular separation.

2.2. Development of a Molecular-Mechanics model for HF

Cournoyer and Jorgensen,74 Jedlovszky and Vallauri,50 and Kreitmeir et al.44 have previously reported molecular-mechanics models of HF (referred to hereafter as the CJ, JV, and K models), whereby the HF molecule is represented by 3 sites, each carrying a point charge. As discussed below, we found these models to be inadequate in their description of the geometry of the (HF)₂ dimer and the structure and thermodynamic properties of liquid HF, especially at high temperature. The model developed here is based on the CJ model74 and also consists of 3 sites: two charged sites on H and F atoms and a third non-atomic charged site (X) near F and along the H─F bond. The charge on H and F is +q, and on the third site is −2q, but only the F atom is assigned Lennard-Jones (LJ) parameters. The potential energy function U(r) that describes interactions between HF molecules is given as:

U (r) = \sum_{bonds} K_{r} (r - r_{0})^{2} + \sum_{nonbonded} {ϵ_{F F} [{(\frac{R_{min, F F}}{r_{F F}})}^{12} - 2 {(\frac{R_{min, F F}}{r_{F F}})}^{6}] + \frac{q_{i} q_{j}}{r_{i j}}}

(1)

where r and r₀ refers to the instantaneous and equilibrium F─H distances and K_r is the bond force constant. The second term in Eq. 1 describes nonbonded van der Waals interactions between the F atoms via a (6–12) LJ term and nonbonded electrostatic interactions between sites i and j via a Coulomb term. Unless otherwise noted, the minimum interaction radius R_min,ij and the well-depth ϵ_ij in the LJ potential were derived from individual LJ parameters for atoms i and j using the Lorentz–Berthelot combination rules:

ϵ_{i j} = \sqrt{ϵ_{i} \times ϵ_{j}} and R_{min, i j} = \frac{R_{min, i} + R_{min, j}}{2}

(2)

Parameters r₀, ϵ_F and R_min,F were preserved as in the CJ model. Parameter K_r was however adjusted to reproduce the F-H bond stretching vibration. The charges as well as the position of the non-atomic site (r_FX) were also optimized to reproduce the ab initio geometry of the (HF)₂ dimer and the density (ρ) and enthalpy of vaporization (ΔH_vap) of liquid HF at 296 K and 1.184 atm. Default parameter values for the proposed model are given in Table 1, alongside those for the CJ, JV, and K models.

Table 1.

Parameters of the molecular-mechanics models of HF discussed in this article.

Parameter	This work^a	CJ model^b	JV model^c	K model^d
r_FH (Å)	0.917	0.917	0.973	0.950
K_r(HF) (kcal/mol/Å²)	637.0	–	–	–
*r_FX* (Å)	0.141218	0.166	0.1647	0.1608
*q_F* (e)	0.705	0.725	0.592	0.592
*q_H* (e)	0.705	0.725	0.592	0.592
*q_X* (e)	−1.410	−1.450	−1.184	−1.184
*ε_F* (kcal/mol)	0.15050	0.15050	0.11923	0.11923
*R_min,F*/2 (Å)	1.67470	1.67470	1.58830	1.58830

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Pair-specific LJ parameters for F─O and F─S interactions are included in SI.

Ref. 74.

Ref. 50.

Ref. 44.

As discussed below, a caveat of our generic MM model is that it overestimates the gas-phase affinity of HF for H₂O and accordingly, the hydration free energy of HF; however, it underestimates the affinity of HF for neutral S-containing analogs (DMS and methanethiol). To correct for these errors, we identified LJ parameters specifically optimized for the F— O (in H₂O) and F—S interactions. These pair-specific LJ parameters are used in the NBFIX section of the CHARMM parameter file and override the default values deduced with the Lorentz–Berthelot combination rules (Eq. 2). In particular, for HF-H₂O we varied R_min,FO while keeping ϵ_FO at its default value (0.1513 kcal/mol). Agreement between calculated and experimental values⁸² was achieved with R_min,FO = 3.510 Å (see Results). For the HF-DMS pair, we followed a procedure similar to that reported elsewhere,^108,119,120 and adjusted parameters ϵ_FS and R_min,FS for the F and S atoms to improve the agreement between MM and ab initio calculations. The parameters were optimized to minimize the following error function:

χ^{2} = \sum_{k} (E_{k}^{C P} - E_{k}^{M M})^{2}

(3)

where $E_{k}^{C P}$ are the 10 lowest E^CP values in each PEC calculated for the complex (covering a range of 0.9 Å with increments of 0.1 Å in each curve) and $E_{k}^{M M}$ are the corresponding interaction energies from the FF. Note that these 10 points cover shorter and longer intermolecular separations than that of the global minimum structure. The optimized parameters for DMS (ϵ_FS = 1.7 kcal/mol and R_min,FS = 3.25 Å) were found to be transferable to the HF-methanethiol interaction. The complete parameter set for our model HF, including the adjusted pair-specific LJ parameters, are provided in the Supporting Information (SI).

All Molecular Mechanics (MM) calculations were carried out with CHARMM¹²¹ using the CHARMM-compatible TIP3P model for H₂O^122,123 and the CHARMM36 all-atom force field for lipid and amino-acid side-chain analogs¹²⁴, in addition to the HF models included in Table 1. MM interaction energies (E^MM and E^MM,opt) for complexes other than the (HF)₂ dimer were calculated as the difference in energy between the complex and its isolated constituents, with all species constrained to the geometries obtained in QM optimizations. For the (HF)2 dimer, we explicitly tested the ability of the MM models to reproduce its non-linear geometry (see Results), and thus carried out MM energy-minimizations of the structure in each case, using 3000 steps of Adaptive-Basis-Newton-Raphson to an RMS gradient of 10⁻⁶ kcal/mol/Å¹⁰² while keeping the F─H bonds at their reference lengths with the RATTLE/Roll algorithm.¹²⁵

2.3. Molecular Dynamics Simulations and Free-Energy Calculations

Molecular dynamics (MD) simulations of bulk HF were carried out with CHARMM,¹²¹ using a cubic box of 500 molecules with periodic boundaries, in the isothermal-isobaric ensemble (NPT) at various thermodynamic conditions that span the temperature and pressure ranges of 190–973 K and 0.0006–200 MPa, respectively. Note that HF is in the supercooled state at ~190 K (the freezing point)^6-8 and is in the supercritical state at or above 461 K (the critical temperature).²⁷ In each thermodynamic state, the system was simulated for 10 ns using the MM model developed here as well as the CJ, JV, and K models (Table 1). To determine whether the self-diffusion coefficient varies with system size,¹²⁶ simulations of a system of 1000 HF molecules were also carried out at 296 K and 1.184 atm. The solvation structures of HF in liquid H₂O and of H₂O in liquid HF were investigated by simulating one solute molecule in 500 solvent molecules at 298.15 K and 1.0 atm. The energetics and mechanism of HF diffusion across lipid membranes was investigated with MD simulations (5 × 60 ns) of a 1-palmitoyl-2-oleoyl-glycero-3-phosphocholine (POPC) lipid bilayer (2 × 66 molecules) flanked by equimolar aqueous HF solutions (1493 HF + 1493 H₂O) on either side. This simulation system comprises a total of 35,604 atoms and was simulated in a tetragonal box with periodic-boundary conditions at 298.15 K and 1 atm.

Electrostatic interactions were computed using the PME method¹²⁷ with the parameter κ = 0.34 for charge screening and a 1.0 Å grid spacing with sixth-order splines for mesh interpolation. Real-space interactions (LJ and electrostatic) are cut off at 12 Å. The Nosé–Hoover thermostat¹²⁸ and Andersen–Hoover barostat¹²⁹ maintained the temperature and pressure at the preset values with relaxation times of 0.1 and 0.2 ps, respectively. The equations of motion were integrated in 1-fs time steps for the pure and aqueous HF systems, and 2-fs time steps for the lipid system. In all systems, all bonds involving hydrogen atoms were constrained at their reference lengths.

The hydration free energy (ΔG_hyd) of HF in liquid H₂O at 298.15 K and 1.0 atm was calculated using the free-energy perturbation method:¹³⁰

Δ G_{h y d} = Δ G_{elec} + Δ G_{disp} + Δ G_{rep}

(4)

where ΔG_elec is the electrostatic component of the hydration free energy and ΔG_disp and ΔG_rep correspond to the attractive and repulsive components of the Lennard-Jones (LJ) interactions. ΔG_elec and ΔG_disp were computed using thermodynamic integration with λ = 0, 0.1, 0.2, …1 while ΔG_rep was computed with a finer discretization (λ = 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5, …1), using a soft-core scheme¹³⁰ and the weighted-histogram analysis method (WHAM).¹³¹

2.4. Thermodynamic and Transport Properties of HF

The molar enthalpy of vaporization (ΔH_vap) was calculated as^132-134

Δ H_{vap} = R T - Δ u = R T - (〈 u 〉_{1} - 〈 u 〉_{g}) = R T - 〈 u 〉_{1}

(5)

where R is the gas constant, T is the absolute temperature, and ⟨u⟩_l and ⟨u⟩_g are the average potential energy per mole in the liquid and gas phases, respectively. Note that ⟨u⟩_g = 0 for a rigid HF molecule in the ideal state.

The self-diffusion coefficient of liquid HF was obtained from the infinite-time limit of the mean-square displacement of the fluorine atoms^119,134

D = lim_{t \to \infty} \frac{1}{6 t} 〈 \frac{1}{N} \sum_{i = 1}^{N} [r_{F, i} (t) - r_{F, i} (0)]^{2} 〉

(6)

Simulations of 500 and 1000 HF molecules at 296 K and 1.184 atm revealed similar self-diffusivity and thus no correction of D for system-size dependence¹²⁶ was applied.

The dielectric constant (ε) was calculated as^119,132

ε = 1 + \frac{4 π}{3 〈 V 〉 k_{B} T} (〈 M^{2} 〉 - 〈 M 〉^{2})

(7)

where M is the total dipole moment of the simulated molecular system. The convergence of ε was evaluated as a function of the length of the time-window used to average M and M²; converged results were obtained in the last 9 ns of data collection.

The specific heat capacity at constant pressure (C_p) was calculated from simulations at T = 292, 294, 296, 298, and 300 K and at constant pressure of 1.184 atm. The total energy U_tot and the volume V were averaged over time and C_p was calculated from the linear fit of the plot (U_tot + pV) versus T:¹¹⁹

C_{p} = \frac{1}{N} {(\frac{\partial (〈 U_{t o t} 〉 + p 〈 V 〉)}{\partial T})}_{T = 296 K, p = 1.184 a t m}

(8)

The isothermal compressibility (β_T) measure the relative volume change accompanying a change in pressure:¹³⁴

β_{T} = - \frac{1}{V} {(\frac{\partial V}{\partial P})}_{T}

(9)

β_T of a system of N particles in equilibrium at constant temperature and pressure is related to the volume fluctuations around its average value¹³⁴

β_{T} = \frac{1}{k_{B} T} \frac{〈 Δ V^{2} 〉_{N P T}}{〈 V 〉_{N P T}}

(10)

Where ⟨V⟩_NpT is the average volume ⟨ΔV²⟩_NpT the average volume fluctuations.

3. Results and Discussion

3.1. Molecular Properties of the Gaseous HF Monomer

To begin to understand the similarities and differences between HF vs H₂O, we compared their surface electrostatic potentials, calculated with MP2(full)/6-311++G(3df,3pd) (Figure 1). As expected, on the surface of H₂O the most electronegative region is centered on the more electronegative oxygen atom (V_s,min = −27.9 kcal/mol), while the most electropositive regions are centered on the H atoms (V_s,max = 33.4 kcal/mol); these 3 electrostatic potential peaks thus coincide with the HOH plane. Previously obtained values for the same theory level using the SCF electron density are highly similar (V_s,min = −30.5 kcal/mol and V_s,max = 35.7 kcal/mol).¹³² Compared to H₂O, HF displays a much larger electropositive potential on H (V_s,max = 52.7 kcal/mol) but a much lower electronegative potential on the F atom (V_s,min = −18.7 kcal/mol). HF is thus expected to be a better H-bond donor than H₂O, but a poorer acceptor. This difference is also consistent with the fact that HF is a much stronger acid (pK_a ~ 3.2)^79,80 than H₂O (pK_a = 14). Another key difference, interestingly, is that the surface electronegative potential of HF does not peak at a point along the direction of the H─F bond; instead, the most electronegative region delineates an annulus around the F atom (Figure 1). As will be noted below, this finding explains why the H─F⋯HL angle is smaller than 180° whenever HF acts as an H-bond acceptor from a ligand (L), which might seem counter-intuitive for a diatomic molecule.

Figure 1. — Electrostatic potential maps for H₂O and HF calculated with MP2(full)/6-311++G(3df,3pd) using the MP2 electron density. Electrostatic potentials are mapped on the surface of the MP2 electron density at an isovalue of 0.0004 au. Three orientations are shown alongside the corresponding molecular geometries to help visualize the potential on the O/F (left), H (middle), and for the entire molecule (right). The same data for HF is shown in two different scales (second and third rows, respectively) to facilitate comparison with H2O and to highlight the annulus-like electronegative region on the surface of the F atom. Note that this annulus is consistently observed regardless of the isovalue used in the mapping.

Table 2 reports the F─H bond distance (r_FH), dipole moment (μ), polarizability (α) and bond stretching vibration (υ) calculated with MP2(full) and CCSD(T) using the 6-311++G(d,p) and 6-311++G(3df,3pd) basis sets and with CCSD(T)/aug-cc-pVXZ (X = D, T, Q, and 5). Ab initio values are compared with experimental measurements and the four MM models mentioned above. All ab initio calculations overestimate υ; calculations with the aug-cc-pVDZ and aug-cc-pVTZ overestimate r_FH; and calculations with the small basis set 6-311++G(d,p) overestimate μ but underestimate α. Interestingly, MP2(full)/311++G(3df,3pd) calculations yield μ in best agreement with experiment. The small polarizability resulting from the 6-311++G(d,p) basis set indicates that it would not accurately describe HF-interactions where dispersion or induced dipole forces are important.¹¹⁹ Both our and the CJ⁷⁴ models are intentionally adjusted to overestimate the dipole moment of the HF monomer (by 18 and 12%, respectively); as will be discussed below, this adjustment is meant to implicitly account for inter-molecular polarization effects, and leads to a better dipole moment for the HF dimer and a more realistic dielectric constant for liquid HF.

Table 2.

Molecular properties of the HF monomer.

Method	r_FH (Å)	μ (D)	α (Å³)	υ (cm⁻¹) ^a
MP2(full)/6-311++G(d,p)	0.9163	1.969	0.508	4201
MP2(full)/6-311++G(3df,3pd)	0.9156	1.825	0.725	4187
CCSD(T)/6-311++G(d,p)	0.9162	1.945	0.509	4192
CCSD(T)/6-311++G(3df,3pd)	0.9159	1.811	0.720	4179
CCSD(T)/aug-cc-pVDZ	0.9241	1.799	0.734	4081
CCSD(T)/aug-cc-pVTZ	0.9210	1.797	0.797	4126
CCSD(T)/aug-cc-pVQZ	0.9177	1.798	0.814	4143
CCSD(T)/aug-cc-pV5Z	0.9173	1.800	0.818	4143
CJ model^b	0.917	2.037	–	–
JV model^c	0.973	1.830	–	–
K model^d	0.950	1.787	–	–
Current model	0.917	2.149	–	3962
Expt.	0.917^e	1.826^f	0.83^g	3961^h

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The F─H stretching frequency is not shown for the CJ, JV, and K model as the authors did not report the FH bond force constant.

Ref. 74.

Ref. 50.

Ref. 44.

Ref. 135.

Ref. 15.

Ref. 115.

Refs 32 and 37.

3.2. Properties of the Gaseous (HF)₂ Dimer

Geometry optimizations of the HF dimer revealed two transition state (TS) structures (2fa and 2fb) and a minimum-energy conformer (2fc). Figure 2A shows the geometry, structural parameters, dipole moment, and interaction energies for these three conformers, calculated with either MP2(full)/6-311++G(d,p) or MP2(full)/6-311++G(3df,3pd), which lead to identical conclusions. In the TS structures, the HF molecules are parallel and either displaced relative to each other (2fa), or perfectly aligned (2fb). By contrast, in conformer 2fc the two HF molecules are arranged at an angle. Accordingly, 2fc is less polar than 2fb but more polar than 2fa, due to the antiparallel and parallel alignment of the individual F─H bond dipoles in 2fa and 2fb, respectively. While one might expect conformer 2fb to be the most favorable structure (i.e. most negative E and E^CP), due to the linear alignment of the bond dipoles, the most stable conformer is instead 2fc. This result is explained by the observation that the most electronegative region on the HF surface is an annulus around the F atom (Figure 1). The H-bond distance (F⋯HF) in these structures follows a trend opposite to the interaction energy, i.e. the longest is seen in 2fa and the shortest in 2fc. The large F⋯H and small interaction energy in conformer 2fa is a consequence of poor H-bonding due to a small F─H⋯F angle (110°).

To further explore the properties of the minimum-energy conformer (2fc), we optimized its geometry with the CCSD(T) method using the two basis sets mentioned above as well as the aug-cc-pVXZ (X = D, T, Q, 5) basis sets. Table 3 shows the F⋯F distance (r_F⋯F), F─H⋯F angle (θF_H─H⋯F), H─F⋯H angle (Θ_H─H⋯H), dipole moment (μ), and interaction energies calculated with various ab initio methods and with the four MM models mentioned above (Table 1), as well as available experimental data. Calculations with the 6-311++G(d,p) basis set underestimate the stability of the dimer and the most expensive CCSD(T)/aug-cc-pV5Z calculation yields the best agreement with experiment. The MM models yield binding energies that overestimate the ab initio values (especially the JV model) but fall in the experimental range of bond energy and enthalpy of dimerization (Table 3). Both the JV and K models favor an incorrect equilibrium structure of the HF dimer with significantly small r_F⋯F, θ_H─F⋯H, and dipole moment. The calculated polarity of the dimer using the CJ model and the model proposed here shows best agreement with experiment and with the high-level ab initio calculations, underscoring the influence of the large dipole moment of the monomer (Table 2). However, our model yields the value for θ_H─F⋯H that is in best agreement with experiment and with CCSD(T)/aug-cc-pV5Z calculations (Table 3). This result is further illustrated in Figure 2B and 2C, which show the change in interaction energy (E^CP or E^MM) and in the dipole moment of the dimer as a function of the θ_H─F⋯H angle, calculated with MP2(full)/6-311++G(3df,3pd) and with the four MM models. As expected, the dipole moment increases with increasing the θ_H─F⋯H angle. Not only does the current model correctly reproduce the equilibrium geometry of the dimer; it also yields the best estimate of the relative stability of its various conformers, which is key for reliable sampling of a multiplicity of energy-accessible configurations in MD simulations.

Table 3.

Molecular properties of the (HF)₂ dimer ^a

Method	r_F⋯F (Å)	θ_H─F⋯H (°)	θ_F─H⋯F (°)	μ (D)	Energy ^b (kcal/mol)
MP2(full)/6-311++G(d,p)	2.79	125	171	3.68	−4.8 [−3.8]
MP2(full)/6-311++G(3df,3pd)	2.73	118	175	3.50	−5.1 [−4.1]
CCSD(T)/6-311++G(d,p)	2.79	124	171	3.62	−4.8 [−3.8]
CCSD(T)/6-311++G(3df,3pd)	2.74	117	174	3.44	−5.1 [−4.1]
CCSD(T)/aug-cc-pVDZ	2.75	113	170	3.28	−4.8 [−4.0]
CCSD(T)/aug-cc-pVTZ	2.74	114	170	3.29	−4.9 [−4.3]
CCSD(T)/aug-cc-pVQZ	2.73	115	170	3.30	−4.8 [−4.5]
CCSD(T)/aug-cc-pV5Z	2.73	115	170		−4.7 [−4.6]
CJ model^c	2.60	103	165	2.83	−5.8
JV model^d	2.34	87	160	2.01	−6.8
K model^e	2.38	92	162	2.14	−6.1
Current model	2.60	115	168	3.37	−6.3
Expt.	2.72±0.03 ^f	121 ±6^f	165±6^f	2.99 ^f	−5,^g −5.8,^h −6±1,ⁱ −6.8±1^j,k −4.6,^l −5.5±0.5^m

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Properties for the MM models were obtained from geometry optimizations while constraining the F─H bonds to their equilibrium values (Table 1).

BSSE-corrected interaction energies are shown in square brackets.

Ref. 74.

Ref. 50.

Ref. 44.

Ref. 33.

^g–k

Enthalpy of dimerization from references 136, 31, 137, 20, 21, respectively.

^l,m

Negative of the dissociation energy (−D_e) of the dimer from references 34 and 138.

3.3. Structure and Energetics of (HF)n (n = 3–7) Gaseous Clusters

Optimized structures of HF clusters varying in geometry and size from trimers to heptamers, obtained with MP2(full)/6-311++G(3df,3pd) calculations, are shown in Figures 3-4. Calculated dipole moments and interaction energies, from both ab initio (E and E^CP) and MM calculations (E^MM) are reported in Table 4. For comparison, H-bond distances obtained from geometry optimization with MP2(full)/6-311++G(d,p) are also shown. (For all gas phase interactions considered in this study, we define H-bonds using cut-off values of 2.7 Å and 100° for the H⋯A distance and D─H⋯A angle, respectively, where A and D designate acceptor and donor atoms.) Interaction energies calculated with MP2(full)/6-311++G(d,p) and CCSD(T)/6-311++G(3df,3pd)//MP2(full)/6-311++G(3df,3pd) are reported in Supplementary Information (Table S1). As could be expected, this analysis shows that the number of stable structures increases with cluster size. For example, we identified 7 conformers for the hexamer (Figure 3) but 15 for the heptamer (Figure 4). Calculations with the small basis set (Table S1) yield BSSE values equal to 20–24% of the interaction energy as compared to 14–20% from calculations with the larger basis sets (Table 4). Interestingly, interaction energies (E and E^CP) calculated with MP2(full) and CCSD(T) methods using the 6-311++G(3df,3pd) basis set differ by less than 1.1 kcal/mol. Note that MP2(full)/6-311++G(d,p) calculations yield longer H-bond distances, which explains the 1–10 kcal/mol weaker interaction energies (less negative E^CP) as compared to MP2(full)/6-311++G(3df,3pd) calculations (Table 4 vs S1). As noted, H-bond strength also decreases as it deviates from the ideal angular geometry. For example, the longest H-bonds in structures 4fb (1.98 Å) and 6fe (2.58 Å) are characterized by F─H⋯F angle = 135° and 107° as compared to angles of 150–180° for the other shorter H-bonds in these structures. Such strained, long H-bonds contribute little to the stability of a given conformer, especially in bulk solutions, where intermolecular interactions are effectively weaker than in the gas phase.^99,109

Figure 3. — Optimized structures and relative stabilities of (HF)_n clusters (n = 3–6). Different configurations for a given cluster size are shown in increasing order of energetic stability, from top left to bottom right, based on MP2(full)/6-311++G(3df,3pd) calculations. Intermolecular H-bonds are indicated by dotted lines, with H-bond distances (in Å) indicated in bold. Atomic coordinates are provided as the SI. H-bond distances derived from geometry optimization with MP2(full)/6-311++G(d,p) are shown in normal font. The latter calculations result in highly similar geometries with slightly larger H-bond distances; the only significant difference is for the right-side ring of **6fe**, which is predicted to have a more cyclic structure.

Figure 4. — Optimized structures of the (HF)₇ cluster, shown in increasing order of energetic stability from top left to bottom right, based on MP2(full)/6-311++G(3df,3pd) calculations. Intermolecular H-bonds are indicated by dotted lines, with H-bond distances (in Å) indicated in bold. Atomic coordinates are provided as SI. H-bond distances derived from geometry optimizations with MP2(full)/6-311++G(d,p) are shown in normal font. The latter calculations result in highly similar geometries with larger H-bond distances; the only significant difference is for the right-side ring of **7fe**, **7ff**, and **7fk**, which is predicted to have a more cyclic structure.

Table 4.

Interaction energies (E, E^CP, E^MM in kcal/mol) and dipole moment (in Debye) for the (HF)_n (n = 3–7) conformers depicted in Figures 3-4.^a

Structure	MP2(full)/6-311++G(3df,3pd)			Model
Structure	Dipole	E	E ^CP	E ^MM ^b
3fa	5.87	−11.9	−9.6	−13.6
3fb	0.00	−16.3	−13.8	−18.8
4fa	8.18	−19.5	−15.9	−21.6
4fb	3.04	−20.7	−17.2	−23.4
4fc	0.00	−31.9	−26.6	−32.3
5fa	10.73	−27.7	−22.7	−29.9
5fb	5.03	−27.9	−23.0	−30.9
5fc	1.64	−28.2	−23.5	−31.8
5fd	2.74	−35.9	−29.5	−36.2
5fe	0.00	−44.7	−37.2	−41.9
6fa	3.09	−33.5	−27.5	−36.8
6fb	13.22	−36.3	−29.8	−38.3
6fc	3.61	−38.9	−31.5	−39.2
6fd	0.00	−39.0	−31.6	−39.6
6fe	2.25	−42.8	−35.2	−43.9
6ff	2.74	−48.6	−40.0	−45.6
6fg	0.00	−54.7	−45.5	−49.5
7fa	3.33	−36.3	−29.6	−40.3
7fb	2.34	−41.0	−32.8	−41.7
7fc	5.97	−40.7	−33.5	−43.9
7fd	15.84	−45.1	−37.1	−46.8
7fe	1.20	−45.9	−37.4	−46.9
7ff	2.72	−46.0	−37.5	−47.4
7fg	2.74	−47.9	−39.2	−48.6
7fh	7.28	−50.3	−41.4	−51.1
7fi	4.12	−51.5	−41.9	−48.4
7fj	1.66	−51.7	−42.1	−48.9
7fk	2.14	−53.5	−44.0	−53.8
7fl	4.49	−55.4	−45.6	−52.7
7fm	4.36	−55.5	−45.6	−52.8
7fn	2.76	−58.7	−48.3	−53.2
7fo	0.04	−64.8	−54.0	−59.1

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Interaction energies calculated with MP2(full)/6-311++G(d,p) and with CCSD(T)/6-311++G(3df,3pd)//MP2(full)/6-311++G(3df,3pd) are reported in Table S1 of the SI.

Calculated with the molecular-mechanics model developed here, for geometries optimized with MP2/6-311++G(3df,3pd).

Among conformers of the same size, zigzag structures are the most polar, as a result of the concerted orientation of individual bond dipoles; however, these zigzag configurations are typically the least stable, as they feature a lower number of H-bonds and weaker dispersion forces. In contrast, cyclic configurations are nonpolar but are the most stable. This result is consistent with the experimental observation, based on electron-diffraction studies, that puckered cyclic hexamers are the main constituent of vapor HF, aside from monomers.²⁹ The cyclic heptamer (7fo) is slightly polar as it is not planar and slightly asymmetric (Figure 4). Inspection of the H-bond distances in structurally similar conformers of different clusters reveals cooperativity in H-bonding interactions. For example, these distances are shorter in 3fa than in 2fc, indicating that upon H-bonding the molecule becomes an even better H-bond donor or acceptor. That is, the charge transfer from an H-bond acceptor to a donor renders the former a better donor (to a third molecule) and the latter a better acceptor. Note that each molecule in a cyclic conformer forms two H-bonds; yet the H-bond distances decrease as the cluster size increases. This effect seems to result from a suboptimal angle, and is particularly clear for the smaller rings. This strain on the H-bond geometry seems to become negligible for the hexamer, as it features H-bond lengths that are similar to those in the heptamer.

As observed for the dimer, the MM model yields interaction energies larger (more negative) than those from MP2(full)/6-311++G(3df,3pd) calculations (Table 4). It should however be noted that the MM interaction energies (E^MM) are calculated on the basis of ab initio geometries. Optimizing the structures at the MM level improves the agreement between E^MM and E^CP. For example, MM geometry optimizations of the global minima for the (HF)_3–7 clusters (3fb, 4fc, 5fe, 6fg, and 7fo) results in E^MM = −17.6, −28.9, −37.9, −46.1, and −54.1 kcal/mol, respectively, which are in better agreement with the E^CP values (−13.8, −26.6, −37.2, −45.5, and −54.0) compared to the E^MM energies evaluated for the ab initio geometries (−18.8, −32.3, −41.9, −49.5, and −59.1 kcal/mol). For consistency, we report interaction energies calculated on the ab initio geometries. While MM-optimized structures are highly similar to the corresponding ab-initio geometries, a few of the abinitio-optimized geometries are not metastable in the MM force field. For example, conformers 5fb and 7fh converge to structures 5fc and 7fo upon MM energy minimization. Interestingly, the deviation between E^MM and E^CP decreases as the cluster size increases (Table 4), indicating that the model reliably describes interactions in bulk HF and in large HF clusters. The non-linear arrangement of any pair of interacting HF molecules in these clusters is again a seemingly surprising feature for a diatomic molecule (Figure 3); as noted above, this geometry is explained by the distribution of the most electronegative potential on the surface of the HF molecule (Figure 1).

3.4. Structure and Energetics of (H₂O)₂, H₂O·HF, (H₂O)₂·HF and (HF)₂·H₂O Clusters

Geometries for (H₂O)₂, H₂O·HF, (H₂O)₂·HF and (HF)₂·H₂O clusters optimized with MP2(full)/6-311++G(3df,3pd) are shown in Figure 5; the corresponding interaction energies (E, E^CP, E^MM, E^MM,opt) are reported in Table 5. For completeness, interaction energies calculated with MP2(full)/6-311++G(d,p) and CCSD(T)/6-311++G(3df,3pd)//MP2(full)/6-311++G(3df,3pd) are reported in Supplementary Information (Table S2).

Table 5.

Interaction energies (in kcal/mol) for the (H₂O)₂ (2w), H₂O·HF (wf), (H₂O)₂·HF (2wf) and (HF)₂·H₂O (_2fw) conformers displayed in Figure 5^a

Structure	MP2(full)/ 6-311++G(3df,3pd)		Model
Structure	E	E ^CP	E ^MM ^b	E ^MM,opt ^c
2w	−5.5	−4.6	−6.1	–
wfa	−1.9	−1.6	−2.8	−2.4
wfb	−2.4	−1.9	−2.5	−2.7
wfc	−3.1	−2.5	−3.7	−3.5
wfd	−9.6	−8.2	−11.6	−10.8
2wfa	−13.5	−11.5	−16.1	−15.3
2wfb	−20.1	−17.6	−20.4	−18.9
2fwa	−7.8	−6.3	−9.6	−9.5
2fwb	−12.8	−11.1	−15.4	−14.3
2fwc	−15.5	−13.0	−17.8	−16.8
2fwd	−20.2	−17.5	−21.7	−20.0

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Interaction energies calculated with MP2(full)/6-311++G(d,p) and with CCSD(T)/6-311++G(3df,3pd)//MP2(full)/6-311++G(3df,3pd) are reported in Table S3 of the SI.

^bc

Calculated with our MM model of HF and TIP3P, using geometries optimized with MP2/6-311++G(3df,3pd), without and with pair-specific LJ parameters for the H₂O-HF interaction.

For the water dimer, the optimization results in an H-bond distance r_O⋯H of 1.93 Å and interaction energies E and E^CP of −5.5 and −4.6 kcal/mol, respectively. That is, the H₂O dimer is slightly more stable than the HF dimer (E = −5.1 and E^CP = −4.1 kcal/mol). For the H₂O·HF pair, the optimizations reveal two TS structures (wfa and wfb) and two stable conformers (wfc and wfd). It is worth noting that conformer wfd is ~3 times more stable than wfc, showing that HF is the better H-bond donor when in complex with H₂O. The shorter r_O⋯H (1.70 Å) and r_O⋯F (2.63 Å) distances in structure wfd compared to r_F⋯H (2.03 Å) and r_O⋯F (2.98 Å) distances in structure wfc are consistent with the higher stability of wfd. These differences are consistent with the properties of the monomers, i.e. the more electropositive potential on the H atom of HF and the larger electronegative potential on the O atom in H₂O (Figure 1). H-bond distances and interaction energies calculated on the minimum-energy wfd using the CCSD(T) method and the 6-311++G(d,p), 6-311++G(3df,3pd), and aug-cc-pVXZ (X = D, T, Q, 5) basis sets are reported in Supplementary Information (Table S3). CCSD(T)/aug-cc-pV5Z calculations yield H-bond distance r_O⋯HF = 1.70 Å and interaction energies E and E^CP = −9.0 and −8.8 kcal/mol, respectively. The relative stability of the homo and heterodimers formed of HF and H₂O molecules reveals a trend in H-bond stability that follows the sequence F─H⋯O >> O─H⋯O ≥ F─H⋯F >> O─H⋯F.

Two and four conformers are obtained for the (H₂O)₂·HF and (HF)₂·H₂O complexes, respectively. While conformers 2wfb, 2fwc, and 2fwd are energy minima, conformers 2wfa, 2fwa, and 2fwb are TS structures. The similar affinity of the HF and H₂O dimers results in comparable stability of the minimum-energy structures for these complexes (2wfb and 2fwd). Interestingly, while comparison of the H-bond distances in the homo (2fc, 2w) and hetero (wfa–wfd) dimers with those in 2wfb, 2fwa, 2fwb, or 2fwd indicates cooperativity, comparison with those in 2wfa or 2fwc suggests anti-cooperativity (shorter H-bonds in the (HF)₂ and H₂O·HF pairs). This effect results from the H₂O molecule acting as a bi H-bond donor or acceptor in structures 2wfa and 2fwc, which weakens the average strength of each H-bond. In other words, when a molecule acts as an H-bond acceptor or donor, charge transfer from or to the other molecule impacts the ability of the former to form additional H-bonds as acceptor or donor.

The TIP3 molecular-mechanics model predicts −6.1 kcal/mol for the ab-initio optimized structure of the water dimer; together with our generic model for HF, they significantly overestimate the stability of the H₂O·HF pair (E^CP vs E^MM, Table 5). However, implementation of the NBFIX parameters optimized to reproduce the hydration free energy of HF yields interaction energies (E^MM,opt) that are in better, though still imperfect, agreement with ab initio values (Table 5). Yet, the corrected model produces excellent agreement with the experimental bond dissociation energy of the pair, D_e = 10.3 ±1.9 kcal/mol.⁸⁴ Compared to the monomer, we observed an elongation of the H─F bond of up to 0.035 Å in the binary HF-H₂O systems investigated. However, it is worth noting that we observed no spontaneous proton transfer from HF to H₂O (Figure 5), in agreement with previous investigations of (H₂O)_n·HF clusters (n = 1–10).^88,89

3.5. Structures and Energies of HF Complexes with Amino Acid and Lipid Analogs

To evaluate and improve our HF model for protein and lipid interactions, we investigated the stability and geometry of HF complexes with model compounds that mimic the amino-acid side chains and the polar and nonpolar functional groups in multiple lipids (e.g. PC, PE, PG, PS, PA, CL and DAG). Optimized geometries and interaction energies calculated with MP2(full)/6-311++G(3df,3pd) for HF complexes with 17 non-aromatic compounds are shown in Figure 6 and Table 6; equivalent data for HF complexes with prolinamide and 6 aromatic compounds are given in Figure 7 and Table 7. For completeness, interaction energies calculated with MP2(full)/6-311++G(d,p) and CCSD(T)/6-311++G(3df,3pd)//MP2(full)/6-311++G(3df,3pd) are reported in the Supplementary Information (Tables S4-S5).

Table 6.

Interaction energies (E, E^CP, E^MM, E^MM,opt in kcal/mol) for the HF complexes with models of non-aromatic amino-acid side chains depicted in Figure 6.^a

Ligand	Conf.	MP2(full)/6-311++G(3df,3pd)		Model ^b
		E	E ^CP	E^MM (E^{MM, opt})
Ethane	5a	−0.4	−0.1	−0.3
	5b	−0.8	−0.4	−0.3
Methanol	5c	−1.0	−0.7	−0.9
	5d	−11.1	−9.5	−11.7
Ethylene glycol	5e	−3.6	−2.8	−4.0
	5f	−12.5	−10.7	−12.7
	5g	−13.5	−11.3	−13.8
Acetamide	5h	−3.4	−2.7	−3.3
	5i	−14.1	−12.3	−13.0
	5j	−16.1	−14.3	−14.5
NMA	5k	−3.4	−2.6	−3.6
	5l	−14.1	−12.2	−13.7
	5m	−14.8	−12.9	−12.4
Acetic acid	5n	−1.2	−0.8	−0.5
	5o	−7.6	−6.1	−6.7
	5p	−11.1	−9.4	−9.5
	5q	−13.6	−11.7	−11.3
Acetate	5r	−9.9	−8.7	−7.0
	5s	−32.5	−30.3	−33.8
	5t	−36.9	−34.2	−35.4
Methyl acetate	5u	−1.1	−0.7	−0.7
	5v	−1.4	−1.0	−0.9
	5w	−8.8	−7.0	−7.9
	5x	−11.1	−9.4	−12.8
	5y	−11.8	−10.1	−11.3
Methanethiol	5z	−1.8	−1.3	−1.3 (−2.4)
	5aa	−7.7	−6.2	−3.0 (−5.0)
Methanethiolate	5ab	−29.0	−26.7	−24.8
DMS	5ac	−1.4	−0.9	−0.9 (−1.3)
	5ad	−9.4	−7.6	−2.8 (−5.0)
Methylamine	5ae	−16.0	−14.2	−15.7
Methylammonium	5af	−7.0	−6.4	−4.5
	5ag	−11.0	−10.4	−9.2
	5ah	−12.0	−11.1	−11.7
Tetramethylammonium	5ai	−6.2	−5.6	−4.4
	5aj	−7.2	−6.4	−6.7
	5ak	−8.3	−7.4	−8.3
Methylguanidinium	5al	−9.2	−8.1	−9.9
	5am	−10.9	−9.9	−11.2
	5an	−10.9	−10.0	−11.4
Methyl hydrogen phosphate	5ao	−21.8	−18.6	−23.1
	5ap	−28.2	−25.6	−30.5
	5aq	−33.2	−30.2	−29.7
Dimethyl phosphate	5ar	−21.0	−18.5	−22.0
	5as	−28.2	−25.7	−28.3

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Interaction energies calculated with MP2(full)/6-311++G(d,p) and CCSD(T)/6-311++G(3df,3pd) are reported in Table S3 of the SI.

Calculated with the additive model for MP2/6-311++G(3df,3pd) geometries using the default Lorentz–Berthelot combination rules (E^MM) and the optimized pair-specific parameters for HF-methanethiol/DMS interactions (E^MM,opt).

Figure 7. — MP2(full)/6-311++G(3df,3pd) optimized structures for HF complexes with toluene, 3-methylindole, 4-methylphenol, 4-methylphenolate, 4-methylimidazole, 4-methylimidazolium, and prolinamide (atomic coordinates are provided in the SI). Intermolecular σ- and π-type H-bonds are indicated by dotted lines, with H-bond distances indicated in Å. Dotted lines in structures 6a/6d/6h represent F⋯C distances while those in 6c/6g/6k/6t are distances between H/F and the ring centroid. The conformers of each system are shown in increasing order of stability (at the theory level employed) from left to right. H-bond distances from geometry optimization with MP2(full)/6-311++G(d,p) are shown in normal font. The smaller basis set predicts longer interfragment distances and smaller interaction energies (Table 6 vs S4).

Table 7.

Interaction energies (E, E^CP, E^MM, E_MM,opt in kcal/mol) for the HF complexes with models of proline and aromatic amino acid side chains depicted in Figure 7.^a

Ligand	Conf.	MP2(full)/ 6-311++G(3df,3pd)		Model ^b
Ligand	Conf.	E	E ^CP	E ^MM
Toluene	6a	−0.8	−0.4	−0.3
	6b	−1.5	−0.9	−0.6
	6c	−6.9	−4.6	−5.1
3-Methylindole	6d	−0.7	−0.3	−0.3
	6e	−1.4	−0.7	−0.7
	6f	−3.7	−2.8	−3.5
	6g	−7.9	−5.3	−4.4
4-Methylphenol	6h	−0.8	−0.4	−0.3
	6i	−1.5	−0.9	−0.7
	6j	−4.2	−3.2	−4.5
	6k	−7.2	−4.7	−4.2
	6l	−9.1	−7.5	−8.8
4-Methylphenolate	6m	−15.5	−13.1	−18.6
	6n	−15.6	−13.1	−12.7
	6o	−38.6	−35.9	−40.2
4-Methylimidazole	6p	−3.8	−3.0	−3.6
	6q	−6.9	−4.8	−4.2
	6r	−16.5	−14.6	−16.8
4-Methylimidazolium	6s	−6.9	−6.2	−7.2
	6t	−7.2	−6.2	−5.5
	6u	−10.0	−8.9	−9.8
	6v	−10.1	−8.9	−9.6
Prolinamide	6w	−2.1	−1.4	−1.0
	6x	−2.1	−1.4	−1.2
	6y	−2.7	−1.9	−2.2

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Interaction energies calculated with MP2(full)/6-311++G(d,p) and CCSD(T)/6-311++G(3df,3pd) are reported in Table S4 of the SI.

Calculated with the additive model for MP2/6-311++G(3df,3pd) geometries using the default Lorentz–Berthelot combination rules.

Due to very weak C─H⋯F H-bonding, complexes of HF with the ethane (5a, 5b) and prolinamide (6w–6y) are the least stable. (Note that for prolinamide, only the interaction of HF with the methylene groups was considered, as interactions with amide groups are represented by acetamide and NMA.) Substituting H in H₂O by the electron-donating methyl group results in a stronger H-bond in methanol and hence in 1.3 kcal/mol larger stability for complex 5d vs wfd. Conversely, disrupting N─H⋯F and O─H⋯F H-bonding upon replacing H by CH₃ results in ~1.5 kcal/mol lower stability for HF interaction with NMA (5m) and methyl acetate (5y) than acetamide (5j) and acetic acid (5q). The 1.8 and 4.5 kcal/mol higher stability of 5g vs 5d and of 5aq vs 5as is due to an extra O─H⋯F H-bond in the HF complexes with ethylene glycol and methyl hydrogen phosphate.

Deprotonation of acetic acid, methanethiol and 4-methylphenol increases their affinity for HF 3–5 fold, reflecting the larger magnitude of ion-dipole interactions in the charged complexes. By contrast, protonation of methylamine and 4-methylimidazole reduces their affinities for HF by 20–65%. This observation is striking as the cationic complexes involve ion-dipole interactions that might be expected to be stronger than the dipole-dipole interactions in their neutral counterparts. This finding is however explained by the fact that HF is a strong H-bond donor but a poor acceptor. The lower stability of the cationic complexes is due to weak N─H⋯F H-bonding in the cationic complexes as compared to the strong F─H⋯N in the neutral complexes. Indeed, all minimum-energy structures of HF complexes with polar neutral and anionic compounds are stabilized by H-bonding (σ- or π-type) in which HF is the donor. The electron-donating methyl group makes DMS a better H-bond acceptor than methanethiol; hence the 1.4 kcal/mol higher stability of 5ad vs. 5aa. In addition, a stronger dispersion force is expected in the HF-DMS complex. Tetramethylammonium interacts with HF in either mono- (5ai), bi- (5aj), or tri-dentate (5ak) fashion with the latter being the most stable. The smaller size of methylammonium results in complex 5ah being 3.7 kcal/mol more stable than 5ak. The interaction energies of HF with methylguanidinium (5an) and with dimethyl phosphate (5as) are 1.1 and 8.5 kcal/mol less favorable than those for methylammonium (5ah) and acetate (5t), respectively, likely because the charge is more delocalized in the former ions.¹¹⁹

Lastly, we find the C─H⋯F interactions in HF complexes with toluene (6a, 6b), 3-methylindole (6d, 6e), and 4-methylphenol (6h, 6i) to be favorable but weak (E^CP between −0.3 and −0.9 kcal/mol). The minimum-energy conformers of HF complexes with toluene (6c) and 3-methylindoel (6g) are stabilized by π-type H-bonding (F─H-⋯π), which also contributes to the stability of some conformers of HF complexes with 4-methylphenol(ate) (6k, 6m, 6n) and 4-methylimidazole (6q). The higher stability of 6j vs 6l, and of 6p vs 6r, is again explained by the fact that HF is a better H-bond donor than acceptor.

As shown in Tables 6-7, our MM force field for HF results in interaction energies for lipid and amino acid side-chain analogs that are in very good agreement with ab initio results; the average unsigned error of E^MM over E^CP interaction energies is 1.2 kcal/mol. Nevertheless, the generic parameter set does underestimate the stability of the minimum-energy HF complexes with methanethiol (5s) and DMS (5v), most probably due to an inadequate description of the induced dipole moment in the highly polarizable S atom.¹¹⁹ This error can be minimized, however, by using pair-specific LJ parameters for the F and S atoms ϵ_FS = 1.7 kcal/mol and R_min,FS = 3.25 Å), optimized against ab initio potential-energy curves (Figure 8).

Figure 8. — Potential-energy curves (PEC) calculated with MP2(full)/6-311++G(3df,3pd) (black solid trace) and with the optimized (solid red trace) and default CHARMM FF (dotted red trace) for HF complexes with DMS. The selected intermolecular coordinate is shown in Figure 6.

3.6. Structure and Thermodynamic Properties of Liquid HF at 296 K

To further evaluate our MM model, the structure of liquid HF was investigated by analyzing g_FF(r), g_FH(^r), and g_HH(r) radial-distribution functions (RDFs) computed from 10-ns MD trajectories, for a system of 500 HF molecules at 296 K and 1.184 atm (Figure 9). Analogous data were calculated for the CJ, JV and K models. Figure 9 also shows calculated RDFs from previous ab initio MD simulations,⁴⁸ as well as RDFs deduced from neutron diffraction measurements at the same pressure and temperature conditions.⁴³ (The degree to which the latter are impacted by experimental noise or other source of error is unclear; the striking multiplicity of small peaks, even at large distances, and the zero-values of some minima, are highly atypical features. Hence, we focus this comparison on the first peaks/minima, which are likely to be the most reliable features.)

The ab initio MD simulations⁴⁸ grossly underestimate the position of the first peak in the experimental g_FF(r) function (2.19 vs 2.52 Å). In comparison, our model and the CJ model slightly overestimates the position of this peak (2.63 and 2.66 Å, respectively). Integrating up to 2.80 Å, the first minima in the experimental curve, yields 1.8, 1.7, 2.4, and 2.3 HF molecules for the current, CJ, JV, and K models, respectively. Integration up to 4.05 Å yields 7.7 HF molecules for all models.

The first intermolecular F⋯H peak in the g_FH(r) function is best reproduced by ab initio MD simulations (Figure 9). Both the JV and K models largely overestimate the intensity of the first intermolecular F⋯H peak and underestimate its equilibrium position (1.45 and 1.51 Å compared to 1.60 Å from experiment). In comparison, our model and the CJ model correctly reproduce the peak height but slightly overestimates its position (1.73 and 1.75 Å, respectively). Integrating the g_FH(r) function to its first minima (2.33, 2.37, 2.09, and 2.15 Å for the current, CJ, JV, and K models) results in 2 H atoms (one intramolecular and one intermolecular) around each F atom.

Ab initio MD simulations⁴⁸ significantly overestimate the experimental position of the first peak in the g_HH(r) function (2.53 vs 2.03 Å). (Surprisingly, the experimental function goes to zero at 2.55 Å, which seems inconsistent with the g_FH(r) or the g_FF(r) functions.) The K model yields the best agreement with experiment. While the JV model underestimates the position of the first peak by ~0.1 Å, our and the CJ models overestimate it by 0.37 and 0.30 Å, respectively. Integrating the function from the additive models up to its minima (2.93, 2.93, 2.65, and 2.73 Å) results in 2.4, 2.4, 2.1, and 2.2 H atoms.

Based on this data, it can be thus concluded that at 296 K and 1.184 atm, liquid HF is characterized by a structure in which a given molecule is surrounded by ~8 HF molecules, one of which acts as an H-bond donor while another is an acceptor.

Experimental studies of liquid and solid HF (and its deuterated variant, DF) have suggested HF molecules are primarily arranged in zig-zaging molecular chains,^25,26,43,139 in contrast to the cyclic oligomers favored in the gas-phase. Such chains would explain the large dielectric constant of liquid HF (ε = 83.6 at 273.15 K)²³, which would be inconsistent with nonpolar cyclic structures. In Figure 9G-9H we show two snapshots from our MD simulations using the current model, with H-bonds identified using cut-off values of 2.2 Å for the H⋯F_A distance⁵⁵ and of 150° for the F_D─H⋯F_A angle. Note that smaller H-bond angles results in weak H-bonds and were mainly observed in strained three-membered rings (see section 3.3). Indeed, open-chain structures, rather than cyclic-rings, are observed to be predominant. These chains are randomly arranged and vary in length between 2 and 9 HF molecules, with 4–7 being the most common. The majority of HF molecules act as single H-bond donors and acceptors, with few acting as bi-acceptor, or non-H-bonded at all.

The density (ϱ), heat of vaporization to the ideal gas state (ΔH_vap), self-diffusion coefficient (D), isothermal compressibility (β_T), heat capacity at constant pressure (C_p), and dielectric constant (ε) of liquid HF at 296 K and 1.184 atm calculated with the various models and available experimental data are reported in Table 8. For all calculated properties the current model results in good agreement with experiment. The CJ model underestimates ΔH_vap and overestimates D, the JV model overestimates D and β_T, and the K model underestimated ΔH_vap but overestimates D and β_T. All models underestimate ε , but our model produces the best agreement with experiment (83.6 at 273.15 K)²³, in part due to the smaller dipole moment of monomeric HF in the CJ, JV and K models (Table 2). These models also predict less polar chains because they underestimate the angle θ_H–F⋯H between interacting molecules (Figure 2 and Table 3).

Table 8.

Calculated properties of liquid HF at 296 K and 1.184 atm and available experimental data.

Property	CJ model	JV model	K model	This work	Expt.
ϱ (g/cm³)	0.972±0.001	0.934±0.001	0.933±0.001	0.987±0.001	0.952,^a 0.962,^b 0.964,^c 0.991^d
ΔH_vap (kcal/mol)	6.70±0.01	7.11±0.01	6.32±0.01	7.54±0.01	7.22,^e 7.26,^f 7.46^g
D × 10⁻⁹ (m²/s)	14.1±0.1	13.4±0.5	16.7±0.3	11.7±0.2	10.0±1.5^h
β_T (MPa⁻¹)	5.4±0.1	11.8±0.1	12.0±0.1	5.0±0.1	5.87ⁱ
C_p (cal/K mol)	14.8±0.5	15.9±0.8	16.2±0.7	14.4±0.5	12.1^j, 17.0^k
ε	25.4±0.5	17.9±0.3	16.5±0.2	34.2±0.5	83.6^l

Open in a new tab

Ref. 22.

Ref. 27.

Ref. 3.

Ref. 4, at 292.7 K.

Refs. 21 and 24.

Ref. 21.

Ref. 5.

Ref. 30, at 298 K.

ⁱ

Ref. 28, at 292.15 K.

Ref. 8, at 289.1 K.

Ref. 141.

Ref. 23, at 273.15 K.

Compared to liquid HF, liquid H₂O (at 298 K and 1 atm) is denser (ϱ = 0.997 g cm⁻³), and features a larger heat of vaporization (ΔH_vap = 10.52 kcal mol⁻¹) and heat capacity at constant pressure (C_p = 18.0 cal mol⁻¹ K⁻¹), but smaller isothermal compressibility (β_T = 4.6 MPa⁻¹) and self-diffusion coefficient (D = 2.27 × 10 m s⁻¹ m² s⁻¹).¹⁴⁰ That liquid H₂O is denser seems counterintuitive; given the smaller atomic volume of F relative to O and the larger mass of HF relative to H₂O, one might expect the density of liquid HF to be the greater. Likewise, the similar interaction energy of the (HF)₂ and (H₂O)₂ dimers (section 3.4) would indicate similar heats of vaporization for the two liquids. The observed differences between the two liquids can however be explained in terms of their structure. While HF molecules form unbranched H-bonded chains, H₂O molecules are arranged such that each molecule forms four H-bonds with neighboring molecules in a three-dimensional network.¹⁴⁰ This network makes water a more associated liquid; hence its larger ϱ, ΔH_vap, C_p, but smaller β_T and D as compared to liquid HF.

3.7. Transferability Across Thermodynamic Conditions

To test the transferability of our MM model for HF to other thermodynamic conditions, we performed MD simulations of liquid HF along the liquid-vapor coexistence curve in the 190–380 K temperature range and at the corresponding vapor pressures (0.006–12.182 atm).² Analogous calculations were carried out for the CJ, JV and K models. The calculated density, heat of vaporization to the ideal gas, self-diffusion coefficient, and isothermal compressibility are reported in Figure 10 and contrasted with available experimental data. The raw data are also provided in Supplementary Information (Tables S6-S9).

Figure 10. — (A) Density, (B) enthalpy of vaporization to the ideal gas, (C) self-diffusion coefficient, and (D) isothermal compressibility of liquid HF as a function of temperature, for the four MM models discussed in the text (colors). Available experimental data (Expt. 1 – Expt. 7) are shown in black or black and white symbols and are taken from references 22, 3, 27, 5, 21, 30, and 28, respectively.

Although our new model for HF does not outperform all other models for every single observable, it shows the best global agreement with the set of experimental data considered. The CJ model is satisfactory below 340 K, but above this temperature it yields very low densities and enthalpy of vaporizations, and too large diffusivity and isothermal compressibility, suggesting that the liquid-vapor phase boundary has been crossed. Similar results are obtained for the JV and K models above 300 K. Indeed, simulations above 320 K with these two models result in gas rather than liquid HF. In addition, these two models significantly overestimate the density of the liquid at low temperatures (Figure 10).

Franck et al.⁴⁰ have measured the density of supercritical HF in the temperature and pressure ranges of 573–973 K and 40–200 MPa. To further assess our model, we thus simulated 500 HF molecules at temperatures of 573, 673, 773, 873, and 973 K under the pressures of 40, 80, 120, 160, and 200 MPa. The calculated density at each of these thermodynamic conditions is given in Supplementary Information (Table S10). Figure 11 shows the density as a function of pressure at three temperatures (573, 773, and 973). The proposed model again yields the best agreement with the experiment, especially at high temperature and low pressure.

Figure 11. — Density of supercritical HF as a function of pressure as calculated from the four MM models discussed in the text (colors). The experimental curves (black) are taken from reference 40.

A decrease in density at higher temperatures reflects an increase in system size and the corresponding reduction in the number of H-bonding interactions between molecules. This is clear in the snapshots from MD simulations of 500 HF molecules at 773 K/200 MPa (density = 0.682 g/cm³) and at 973 K/40 MPa (density = 0.112 g/cm³), shown in Figure S1. Although H-bonding remains the stabilizing force of the liquid at the higher density, the length of the H-bonded chains is limited to a maximum of three molecules. H-bonding at the lowest density is a rare event and is limited to interaction between two molecules at the most.

3.8. Solvation of HF in Liquid Water and of H₂O in Liquid Hydrogen Fluoride

As noted above, combination of default parameters for the TIP3P model for H₂O and our generic MM model for HF leads to an overestimation of the strength of HF-H₂O interactions. Accordingly, at 298.15 K and 1 atm the default FF yields a hydration free energy for HF, ΔG_hyd, of −7.4 ± 0.1 kcal/mol (ΔG_elec = −9.5, ΔG_disp = −2.3 and ΔG_rep= 4.4 kcal/mol), which is 1.5 kcal/mol larger than the experimental value for undissociated HF (−5.9 kcal/mol).⁸² However, this error can be corrected through specific LJ parameters optimized for the F-O interaction (ϵ_FO = 0.1513 kcal/mol and R_min,FO = 3.510 Å); these pair-specific parameters improve the agreement with ab initio-calculated interaction energies for HF-H₂O clusters (Table 5), and also correct the calculated hydration free energy (ΔG_elec = −8.2, ΔG_disp = −2.4 and ΔG_rep= 4.7 kcal/mol, for a total ΔG_hyd of −5.9 ± 0.1 kcal/mol).

HF is the only weak acid in the hydrogen halide series. Indeed, we observed no ionization for any of the binary H₂O-HF clusters studied here through ab initio methods, in keeping with previous analyses.^88,89 Consistent with this result, Zhang et al.⁸⁵ recently isolated the H₂O·HF pair in a fullerene cage, and using NMR, showed that no proton transfer occurs even at 140 °C. Dissociation of HF in water does occur, however, generating ions such as H₃O⁺, F⁻, and HF₂⁻.^79,80,82,83 Nevertheless, HF behaves as a non-ideal weak acid, in that its degree of dissociation in water increases as its concentration increases. The weak acidic nature of HF in diluted solutions has been attributed to strong H-bonding between HF and H₂O.⁸⁸ Enhanced dissociation in concentrated solutions is likely explained by the nature of the interactions formed by the resulting fluoride ions, which would be stabilized by stronger F─H⋯F⁻ as compared to HO─H⋯F⁻ hydrogen bonding (section 3.1). In fact, experiments have revealed the presence of H₂F⁻ and H₃F⁻ in aqueous HF solutions.^79,80,82-86 Investigating the degree of HF dissociation would require quantum-mechanical MD simulations, which are beyond the scope of this work. Here, we use classical MD simulations with our MM model for HF and the optimized LJ parameters for F-O atom pairs to investigate instead the solvation structure and thermodynamic properties of molecular HF in liquid water and of H₂O in liquid HF at 298.15 K and 1.0 atm; in both cases, a single solute molecule is immersed in 500 solvent molecules. The results are analyzed in terms of interatomic RDFs (Figure 12), and compared with earlier studies based on ab initio quantum mechanical charge-field (QMCF) MD simulations⁹¹ and multistate empirical-valence-bond (MS-EVB) MD simulations.⁹²

For HF in water, the g_FO(r) function exhibits a sharp peak centered at 2.58 Å followed by a less intense and broad peak centered at 3.00 Å (Figure 12A). These distances are similar to those found in structures wfd (2.63 Å) and wfc (2.98 Å) of the H₂O·HF dimer and thus reflect an HF molecule forming strong and weak H-bonds with surrounding water molecules as donor or acceptor, respectively. Integration of the first peak (up to the first minimum at 2.77 Å) yields ~1.2 H₂O molecules. The second peak is much broader and does not decay at a clear minimum, and thus a precise estimate of the number of water molecules in the first solvation shell of HF is not possible. Integration up to 3.8 Å, however, results in a first-shell coordination number of 7 H₂O molecules. In agreement with our results, the g_FO(r) calculated for undissociated HF with MS-EVB MD simulations⁹² reveals peaks at 2.50 and 3.00 Å (albeit with larger and lower intensity than those obtained from our model). Consistent with a poorly defined first hydration shell, QMCF MD simulations revealed 3–9 water molecules around HF, with 4–7 as the most probable coordination numbers.⁹¹

The g_{FH_w}(r) function (Figure 12A) shows two peaks centered at 2.03 and 3.23 Å; the corresponding minima are at 2.4 and 3.8 Å. The position and height of these peaks are also in good agreement with QMCF MD results.⁹¹ These peaks correspond to water molecules acting as H-bond donor and acceptor, respectively. The low intensity of the first peak reflects the weakness of the O─H⋯F H-bond. Integration up to 2.4 Å results in 1.4 H atoms in direct contact with F, and integration to 3.8 Å results in 14 H atoms, consistent with 7 H₂O molecules around HF.

The g_{H_FO}(r) function (Figure 12B) features a sharp, intense peak at 1.67 Å with minimum at 2.25 Å, which reflects a strong F─H⋯O H-bond, followed by a broad peak centered at 3.7 Å with minimum at 5.0 Å. Integration up to 2.3 Å shows a single H₂O molecule bound to H. QMCF MD simulations predict two peaks centered at slightly shorter distances (1.62 and 3.49 Å) but similarly yield a single F─H⋯O hydrogen bond per HF.⁹¹ Lastly, the g_{H_FH_W}(r) function (Figure 12B) peaks at 2.35 Å, in good agreement with MS-EVB MD simulations (2.26 Å).

In summary, our model for HF in liquid water yields a first hydration shell containing ~7 H₂O, one of which acts as the acceptor for a strong H-bond. The other six molecules act as transient H-bond donors, one or two at a time. A simulation snapshot illustrating this configuration is shown in Figure 12C.

The RDFs for H₂O in liquid hydrogen fluoride (Figure 12D-E) resemble the corresponding data for HF in water. The g_OF(r) exhibits a peak at 2.58 Å with a minimum at 2.77 Å followed by a broader peak at 3.0 Å and minimum at 3.7 Å. Integration of the RDF up to the two minima yields 1.2 and ~6 HF molecules around H₂O. The g_{OH_F}(r) function has a sharp peak with a maximum and minimum at 1.67 and 2.3 Å followed by a broad peak centered at 3.7 Å. Integration to 2.3 Å yields 1.1 H_F. The g_{H_WF}(r) displays peaks at 2.03 and 3.3 Å with minima at 2.5 and 3.7 Å. Integration yields 0.8 and 5.5 HF molecules. As noted, the low intensity of the g_{H_WF}(r) function relative to the g_{OH_F}(r) function is evidence that the F─H⋯O H-bond is stronger than O─H⋯F. lastly, the g_{H_WH_F}(r) function displays a peak at 2.37 Å and a minimum at 2.8 Å. This data show that an H₂O molecule is surrounded by 6 HF molecules, with one acting as a strong H-bond donor. The other five molecules in the first shell act as transient H-bond acceptors, two at a time (Figure 12F).

3.9. Permeation of HF through a Phospholipid Membrane

HF is distinct from other, stronger acids in that it is much more able to penetrate and diffuse within biological tissues. Once internalized, intracellular HF will dissociate readily (at pH ~ 7), releasing highly reactive fluoride anions. The cytotoxicity of HF thus stems from its ability to passively diffuse across phospholipid membranes. To qualitatively illustrate how the proposed MM model might be used in future investigations of this process of trans-bilayer diffusion, we calculated five 60-ns molecular dynamics trajectories for a POPC lipid bilayer flanked by equimolar aqueous HF solutions (Figure 13A). The simulations corroborate the notion that HF can easily partition into and across the membrane; indeed, the free-energy barrier one might expect at the bilayer center is only about 3.5 kcal/mol, or 1.2 kcal/mol lower than for H₂O, i.e. more crossable by almost a factor of 10. In large part, this increased permeability stems from the ability of HF to form strong H-bonds (as donor) with the carbonyl groups in the phospholipid ester linkages; compared with H₂O, this H-bonding ability permits HF to penetrate about 10 Å deeper into the membrane at no free-energy cost (Figure 13B). Consistent with these results, we observe tens of full trans-bilayer crossings, in both directions (Figure 13C); specifically, an average of about 25 full crossings per trajectory (minimum of 21 and maximum of 32), i.e. a permeation rate of 0.4 ns⁻¹ under the simulated conditions. Interestingly, we observe that the most prevalent mechanism is one in which one HF molecule and one H₂O molecule become paired at the solvent-membrane interface (with HF acting as the H-bond donor), and both traverse the hydrophobic core of the bilayer concurrently, and then dissociate on the other side (Figure 13D). In this mechanism, the HF-H₂O pair diffuses within the bilayer core for hundreds of picoseconds. An alternative mechanism observed in our trajectories is one in which isolated HF quickly traverses the bilayer core after residing in the ester region; the crossing time in this case is in the order of tens of picoseconds (Figure 13E). We observe few instances of other processes, e.g. involving two HF molecules or an HF molecule and two H₂O molecules. In physiological conditions, with HF much more diluted than in our proof-of-concept simulations, it is highly probable that water-mediated permeation mechanisms will be dominant.

Figure 13. — Simulated permeation of HF through a phospholipid membrane. (A) Simulation system, comprising a POPC bilayer (gray) flanked by 1:1 HF (magenta): H₂O (red) solutions. The Z-axis is perpendicular to the mid-plane of the bilayer. (B) Apparent free-energy of membrane partioning, for HF and H₂O, based on all simulation data (five 60-ns trajectories). (C) Trans-bilayer crossings observed in one of the trajectories, both for HF and H₂O. (D) Snapshots of a crossing whereby HF is paired with H₂O (the corresponding time-trace is indicated in panel C). The elapsed time from one snapshot to the next is indicated. (E) Same as (D), for an alternative mechanism whereby HF crosses the bilayer core alone.

4. Conclusions

Hydrogen fluoride is extensively produced and widely used as a fluorinating agent. Pure and aqueous HF solutions have many industrial applications.^9-11 These solutions are however toxic and highly corrosive,^9,10 making computer-simulations thereof a useful research tool. While several molecular-mechanics models have been reported previously,^{44,47,51,52,58,69,74,75,97} important properties of HF in both the gas and liquid state are not faithfully captured by existing models; when tested in a broad set of thermodynamic conditions, some of these models are markedly inaccurate. As HF is a weak acid, especially in dilute solutions, it can exist in the undissociated form in some biological settings.^95,96 The nature of HF interactions with proteins and membranes, however, had been unexplored to date.

We have characterized HF, (HF)₂, and HF·H₂O with multiple state-of-the-art ab initio methods, including CCSD(T)/aug-cc-pV5Z, which is computationally very demanding. The calculations explain a variety of non-trivial fundamental properties, including why HF is a very good H-bond donor but a poor acceptor and why HF favors non-linear interaction geometries despite being diatomic. Our analysis of HF and H₂O interactions also reveals a hierarchy of H-bonding strengths following the sequence F─H⋯O >> O─H⋯O ≥ F─H⋯F >> O─H⋯F.

Quantum-mechanical analysis of (HF)_3–7, (H₂O)₂·HF, and (HF)₂·H₂O clusters, and of HF complexes with analogs of the amino acid side chains and lipid head groups also revealed interesting findings. For example, that there exists an H-bonding cooperativity when HF/H₂O acts as a single donor and/or acceptor, while anti-cooperativity occurs when HF/H₂O accepts or donates two H-bonds. Consistent with existing experimental data, our results show that nonpolar monocyclic structures of (HF)_3–7 clusters are energetically favored in the gas phase, rather than polar zig-zag structures. Also in the context of side chain interactions HF is found to be a strong H-bond donor but weak acceptor, leading to non-trivial differences; for example, F─H⋯NH₂CH₃ H-bonds (neutral lysine) are significantly stronger than (CH₃NH₃)⁺⋯ F─H interactions (protonated lysine, also arginine).

A three-site nonpolarizable molecular-mechanics model for HF was developed and calibrated primarily against the ab initio data. The resulting model yields liquid-state properties in better agreement with experiments than previously reported models,^44,50,74 including structure, density, enthalpy of vaporization, self-diffusion coefficient, isothermal compressibility, and dielectric constant. The model is also transferable to liquid HF at high temperatures, where other models cross the liquid-vapor coexistence curve and predict gaseous HF. It also is suitable for simulating HF in the supercritical state. Compared to a tetrahedral H-bonding structure in liquid water, liquid hydrogen fluoride is predominantly composed of unbranched chains of H-bonds, making the latter a less associated fluid. This in turns results in a smaller density, heat of vaporization, and heat capacity but larger diffusivity and compressibility of liquid HF compared to liquid H₂O.

The proposed model was calibrated to reproduce the hydration free energy for HF; the resulting hydration structure of HF was found to be in very good agreement with high-end ab initio MD simulations. The model was also used to investigate the solvation structure of H₂O in liquid HF, to our knowledge for the first time. Consistent with gas-phase results, F─H⋯O H-bonds in aqueous solutions are much stronger than O─H⋯F bonds. In its first solvation shell, the HF molecule is surrounded by ~8 H₂O molecules while the H₂O molecule is surrounded by ~6 HF molecules.

Combined with the CHARMM36 FF,¹²⁴ our generic HF model yields interaction energies for HF-protein and HF-lipid complexes with an average unsigned error of 1.2 kcal/mol relative to ab-initio methods. To improve the accuracy of the model for problematic interactions, pair-specific LJ parameters are introduced to better represent interactions between the strongly polar HF and the highly polarizable S atoms (cysteine, methionine). Finally, to illustrate how the proposed model may be readily used to examine biological problems of interest, we simulated the spontaneous diffusion of HF across a phospholipid membrane, i.e. the process at the onset of the cytotoxic action of HF. This proof-of-concept analysis nevertheless revealed some non-trivial insights, e.g. that HF partioning is greatly facilitated by H-bonding interactions with the ester layer, and that trans-bilayer HF crossings are likely to be in complex with H₂O. In future studies it will be of interest to quantify and further evaluate this process and its dependence on a range of factors e.g. the precise lipid composition of the membrane. It is hoped that the model presented here will also catalyze new research on the interactions between HF and membrane channels and enzymes.

Supplementary Material

Supplementary Information

NIHMS1736508-supplement-Supplementary_Information.pdf^{(1.2MB, pdf)}

ACKNOWLEDGMENTS

This work was funded by the Division of Intramural Research of the National Heart, Lung and Blood Institute (NHLBI). Computational resources were provided in part by Biowulf, the HPC facility of the National Institutes of Health (NIH).

Footnotes

The authors declare no competing financial interest.

Supporting Information

Interaction energies calculated with MP2(full)/6-311++G(d,p) and CCSD(T)/6-311++G(3df,3pd) of the HF complexes presented in Figures 3-6, calculated density, enthalpy of vaporization to the ideal gas, self-diffusion coefficient, and isothermal compressibility of liquid HF, atomic coordinates of the structures shown in Figures 2-6, and the FF parameters for HF and NBFIX parameters for its interactions with H₂O/CH₃SH/(CH₃)₂S.

This information is available free of charge via the Internet at http://pubs.acs.org.

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PERMALINK

A New Molecular-Mechanics Model for Simulations of Hydrogen Fluoride in Chemistry and Biology

Esam A Orabi

José D Faraldo-Gómez

Abstract

Graphical Abstract

1. Introduction

2. Computational Methodology

2.1. Ab initio Quantum Chemical Calculations

2.2. Development of a Molecular-Mechanics model for HF

Table 1.

2.3. Molecular Dynamics Simulations and Free-Energy Calculations

2.4. Thermodynamic and Transport Properties of HF

3. Results and Discussion

3.1. Molecular Properties of the Gaseous HF Monomer

Figure 1.

Table 2.

3.2. Properties of the Gaseous (HF)2 Dimer

Figure 2.

Table 3.

3.3. Structure and Energetics of (HF)n (n = 3–7) Gaseous Clusters

Figure 3.

Figure 4.

Table 4.

3.4. Structure and Energetics of (H2O)2, H2O·HF, (H2O)2·HF and (HF)2·H2O Clusters

Figure 5.

Table 5.

3.5. Structures and Energies of HF Complexes with Amino Acid and Lipid Analogs

Figure 6.

Table 6.

Figure 7.

Table 7.

Figure 8.

3.6. Structure and Thermodynamic Properties of Liquid HF at 296 K

Figure 9.

Table 8.

3.7. Transferability Across Thermodynamic Conditions

Figure 10.

Figure 11.

3.8. Solvation of HF in Liquid Water and of H2O in Liquid Hydrogen Fluoride

Figure 12.

3.9. Permeation of HF through a Phospholipid Membrane

Figure 13.

4. Conclusions

Supplementary Material

ACKNOWLEDGMENTS

Footnotes

References

Associated Data

Supplementary Materials

ACTIONS

PERMALINK

RESOURCES

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3.2. Properties of the Gaseous (HF)₂ Dimer

3.4. Structure and Energetics of (H₂O)₂, H₂O·HF, (H₂O)₂·HF and (HF)₂·H₂O Clusters

3.8. Solvation of HF in Liquid Water and of H₂O in Liquid Hydrogen Fluoride