Concerted proton transfer in homogeneous and heterogeneous cyclic hydrogen-bonded clusters of H₂O, HF, and HCl

Abstract

This work examines various homogeneous and heterogeneous clusters of hydrogen-bonded (HF)_x(H₂O)_y(HCl)_z complexes, where x + y + z = 3 or 4. For each unique cyclic structure associated with the xyz permutations, the geometries of the reactant, product, and transition state (TS) for concerted proton transfer (CPT) were fully optimized using second-order Møller-Plesset perturbation theory (MP2) with a mixed basis set consisting of cc-pVTZ for H atoms and aug-cc-pVTZ for O, Cl, and F atoms (denoted haTZ). Harmonic vibrational frequencies were also computed at the same level of theory to verify the nature of the stationary points. Intrinsic reaction coordinate (IRC) calculations confirm that all transition states reported herein connect the two minima associated with the CPT process, and these reaction profiles provide a means to assess the associated barrier width. Single-point energies were computed on the optimized structures using an explicitly correlated coupled cluster method with an analogous quadruple-ζ basis set (CCSD(T)-F12/haQZ-F12) to determine the dissociation energy (D_e) of each trimer and tetramer minimum, as well as the barrier height (ΔE^‡) and energy difference between the reactants and products (ΔE) associated with the CPT process in each cluster. The energetics calculated utilizing this methodology were compared to CCSD(T) benchmark data for the homogeneous clusters (Y. Xue, T. M. Sexton, J. Yang and G. S. Tschumper, Phys. Chem. Chem. Phys., 2024, 26, 12483–12494, DOI: https://doi.org/10.1039/D4CP00422A), and deviations never exceeded 0.1 kcal mol⁻¹. For the heterogeneous trimer systems, (H₂O)₂(HCl)₁ had the smallest ΔE^‡ at only 11.9 kcal mol⁻¹, just below the corresponding D_e of 13.8 kcal mol⁻¹. Among the heterogeneous tetramers, six systems were identified with an even smaller ΔE^‡ (7.9 to 11.8 kcal mol⁻¹) and a larger D_e (19.8 to 30.8 kcal mol⁻¹). Among those, (HF)₃(H₂O)₁ has one of the narrowest barrier widths of any tetramer system, based upon MP2/haTZ IRC profiles.

---

1 Introduction

Proton transfer reactions can be found in a wide variety of contexts in chemistry, ranging from simple Brønsted-Lowry acid/base neutralizations^1–3 to proton-coupled electron transfer reactions.^4–6 The proton transfer reaction is vital in measuring and improving the efficiency of fuel cells,^7–10 and proton transfer via proton pumps in biological cells is a critical process for maintaining electrochemical gradients.^11–15 The transfer of one or more protons between different molecules commonly occurs along coordinates associated with intermolecular hydrogen bonds. In the tautomerization of base pairs in DNA, for example, the process can involve just a single H⁺ or multiple protons moving either directly along the interbase hydrogen bonds or indirectly through a network of hydrogen bonds formed with solvent water molecules.^16–19 When two or more protons are involved, the transfer mechanism can occur in a concerted manner or via a stepwise process.^20–24

The proton transfer process in various hydrogen-bonded systems has been extensively studied.^25–34 Small carboxylic acid dimers have emerged as prototype systems for concerted proton transfer (CPT) involving a pair of H⁺ ions, as they are amenable to both sophisticated experimental and theoretical analysis.^35–43 The analogous CPT process for 3, 4, and 5 protons was recently examined in a computational study of the homogeneous trimers, tetramers, and pentamers of HF, H₂O, and HCl.⁴⁴ In that study, the barrier heights of the proton transfer reaction, dissociation energy of the minima, and harmonic vibrational frequencies of the minima and transition states (TSs) in each system were calculated utilizing high levels of electron correlation with very large basis sets.

Several studies have not only extended the investigation of proton transfer in (H₂O)_n⁴⁵ and (HF)_n⁴⁶ to larger clusters but also expanded the work to related systems such as clusters of (H₂S)_m(H₂O)_n.⁴⁷ However, in contrast to the thorough investigation of homogeneous (HF)_n, (H₂O)_n, and (HCl)_n, relatively few studies⁴⁸ have examined the heterogeneous clusters of these molecules. This work presents an extension of the previous benchmark study, in which proton transfer of both homogeneous and heterogeneous clusters of (HF)_x(H₂O)_y(HCl)_z, where x + y + z = 3 or 4, is investigated. Larger clusters are not examined here because numerous experimental and computational studies have found that the dissociated ion products of heterogeneous H₂O and HCl clusters are more stable than the undissociated fragments for y + z ≥ 5, (H₂O)_y(HCl)_z → (H₂O)_y−m(HCl)_z−m(H₃O⁺)_m(Cl⁻)_m,^49–68 where m is an integer between 1 and the smaller of y and z. This behavior of small gas-phase heterogeneous clusters containing a strong acid like HCl or a weak acid like HF stands in stark contrast to the reactivity of these species in bulk solution, where complete or partial dissociation occurs. Similar interesting differences relative to an aqueous environment have also been observed for homo- and heterogeneous clusters containing H₂SO₄^69–71 or HCOOH.^72–74

2 Computational methods

Burnside's lemma⁷⁵ was utilized to systematically account for the unique permutations of each (HF)_x(H₂O)_y(HCl)_z cluster for which x + y + z = 3 or 4 in this study. The number of permutations is equivalent to the “number of necklaces with n beads of 3 colors, allowing turning over,” as defined by the Online Encyclopedia of Integer Sequences (OEIS). The number of permutations for an arbitrary system size is given by OEIS A027671.⁷⁶ For the trimers, there are 10 possible permutations, and for the tetramers there are 21. For heterogeneous systems with two or three H₂O fragments, there are additional variations to consider due to the different relative orientations that can be adopted by the free hydrogen atoms about the (pseudo-) plane formed by the heavy atoms (O, F, and Cl).

All structures reported herein have been fully optimized using second-order Møller–Plesset perturbation theory (MP2)⁷⁷ with a correlation consistent triple-ζ basis set augmented with diffuse functions on non-hydrogen atoms (cc-pVTZ for H, aug-cc-pVTZ for O, F, and Cl), hereafter denoted haTZ.⁷⁸ Correlation-consistent basis sets with additional tight d-functions (aug-cc-pV(X+d)Z) were found to have a negligible impact on the energetics of the (HCl)₃ and (HCl)₄ systems in the previous benchmark study⁴⁴ and thus have not been utilized here. Single-point energy computations were performed upon the MP2/haTZ optimized geometries using an explicitly correlated coupled-cluster method that includes all single and double substitutions as well as a perturbative estimate of the connected triple substitutions (CCSD(T)-F12, specifically with the F12b ansatz and unscaled triples contributions)^79–82 with Dunning's analogous correlation-consistent, explicitly correlated triple-ζ basis set, haTZ-F12⁸³ (using the default auxiliary basis sets in Molpro 2022). This CCSD(T)-F12/haQZ-F12//MP2/haTZ method was compared to benchmark data⁴⁴ to ensure its accuracy. Harmonic vibrational frequency computations at the MP2/haTZ level of theory were utilized to confirm the nature of each stationary point (1 imaginary frequency for transition states, 0 imaginary frequencies for minima). Additionally, intrinsic reaction coordinate (IRC) pathways were calculated at the same level of theory to determine whether the TS structures identified for each system connect directly to both the product and reactant minima in a concerted process or indirectly through an intermediate in a stepwise process.

The relative electronic energies of the reactant, product, and transition state structures for the CPT process have been computed for each system. In most systems, the reactant and product have equivalent structures, but when they differ, the reactant is defined as the lower-energy minimum. In the latter case, the relative energy of the product (ΔE) is obtained from the difference between their electronic energies, and in the former case, this quantity is simply zero. The electronic barrier height (ΔE^‡) for the transition state is determined in an analogous manner with respect to the reactant. The dissociation energies (D_e) for each system are calculated by subtracting the electronic energy of the hydrogen-bonded structure defined as the reactant from those of the corresponding fragments, which is equivalent to the relative energy of isolated monomers at their optimized geometries.

Gradients and Hessians were obtained analytically in all MP2/haTZ calculations. For geometry optimizations of the minima and transition state structures, the root mean square forces were converged to 1.0 × 10⁻⁵E_h⋅bohr⁻¹, and the thresholds adopted for the iterative SCF and CCSD procedures were sufficient to ensure the MP2 and CCSD(T) electronic energies were converged to at least 1.0 × 10⁻⁷E_h. All MP2 geometry optimization and harmonic vibrational frequency computations were performed in Gaussian16,⁸⁴ and all CCSD(T)-F12 single-point energy computations were performed in Molpro 2022.⁸⁵

3 Results and discussion

3.1 Barrier heights

3.1.1 Calibration. To verify that the CCSD(T)-F12/haQZ-F12//MP2/haTZ protocol utilized herein provides accurate barrier heights, the results obtained for not only the homogeneous trimers and tetramers, but also the pentamers, were compared to the benchmark data from ref. 44. The reported CCSD(T)-F12/haQZ-F12//CCSD(T)/aQZ values are nearly identical to estimates of the barrier heights at the complete basis set (CBS) limit, and that data can be used to directly assess the impact of the less demanding geometry optimization procedures adopted in the current study. For these nine homogeneous systems, the CCSD(T)/haQZ-F12 ΔE^‡ values computed with the MP2/haTZ and CCSD(T)/aQZ optimized structures differed by less than 0.1 kcal mol⁻¹.

3.1.2 Trimers. When considering the relative orientations of the free hydrogen atoms that do not participate in hydrogen bonding from the H₂O fragments when y = 2, there are 12 unique permutations for the (HF)_x(H₂O)_y(HCl)_z trimers, and a transition state for CPT has been identified for each. There is just one cluster (x = y = z = 1) in the set for which the CPT product is not equivalent to the corresponding reactant. The different structures of the reactant and product for this (HF)₁(H₂O)₁(HCl)₁ system can be seen in Fig. 1, along with the TS structure for CPT connecting the two minima. This results in a total of 13 unique minima and 12 transition states that have been characterized for the trimer systems. However, all results for the two structures with both free hydrogen atoms on the same side of the ring have been relegated to the SI because they correspond to local minima with slightly higher electronic energies and slightly larger CPT barrier heights. The manuscript only presents results for the lower-energy trimer isomers containing 2 H₂O fragments that have the dangling hydrogen atoms on opposite sides of the OOF or OOCl planes, leaving a total of 10 trimer systems for discussion in this section (e.g., data in Table 1). Within this subset of 10 trimer structures, those containing HF and H₂O fragments have been previously identified at the MP2/aug-cc-pVDZ level of theory,⁴⁸ and the MP2/haTZ optimized geometries reported here are very similar. The Cartesian coordinates and MP2/haTZ frequencies for all optimized trimer stationary points can be found in the SI.

Fig. 1 Reactant (left), transition state (Center), and product (right) for the (HF)₁(H₂O)₁(HCl)₁ CPT reaction, with the fragments (f1, f2, f3) labeled on the reactant structure.

Table 1 CCSD(T)-F12/haQZ-F12 relative energies of products, barrier heights of transition states for CPT, and dissociation energies of reactants (ΔE, ΔE^‡, and D_e, respectively, in kcal mol⁻¹) for each (HF)_x(H₂O)_y(HCl)_z trimer configuration, with the fragment labels (f1, f2, f3) denoting the direction of the hydrogen bond network in the reactant

f1	f2	f3	ΔE	ΔE^‡	D e
HF	HF	HF	0.0	20.6	15.3
HF	HF	HCl	0.0	22.6	11.5
HF	HCl	HCl	0.0	25.5	8.7
HF	HF	H2O	0.0	18.1	18.2
HF	H2O	H2O	0.0	21.7	18.2
HF	H2O	HCl	2.3	16.0	15.0
H2O	H2O	H2O	0.0	30.0	15.8
H2O	H2O	HCl	0.0	11.9	13.8
H2O	HCl	HCl	0.0	14.5	10.4
HCl	HCl	HCl	0.0	27.4	6.7

---

In the (HF)₁(H₂O)₁(HCl)₁ cluster, the minimum in which HF donates a hydrogen bond to H₂O and accepts a hydrogen bond from HCl (left side of Fig. 1) has a slightly lower CCSD(T)-F12/haQZ-F12 electronic energy than the minimum in which the donor and acceptor roles are reversed (right side of Fig. 1). This relative energy difference of 2.3 kcal mol⁻¹ is reported in Table 1, and the other entries in the ΔE column are zero because the reactant and product have equivalent structures. The fragment labels (f1, f2, f3) denote the direction of the hydrogen bond network in the reactant, with f1 donating a hydrogen bond to f2 and accepting a hydrogen bond from f3.

Among the homogeneous systems, (HF)₃ has the smallest barrier at 20.6 kcal mol⁻¹. Substituting any number of HCl fragments for HF fragments in this system (i.e. (HF)₂(HCl)₁ and (HF)₁(HCl)₂) results in an increased barrier height, but the barrier decreases when one H₂O is substituted. In general, the barrier heights in trimer systems with one or two H₂O fragments (i.e. y = 1 or 2) tend to be smaller than in systems without H₂O, with the exception of (HF)₁(H₂O)₂. The smallest barrier heights are in systems with both HCl and H₂O fragments. In the x = y = z = 1 trimer, the barrier height is 16.0 kcal mol⁻¹. This quantity decreases to 14.5 kcal mol⁻¹ for the (H₂O)₁(HCl)₂ system and to 11.9 kcal mol⁻¹ for the (H₂O)₂(HCl)₁ system. Based on these results, the (H₂O)₂(HCl)₁ system not only has the smallest CCSD(T)-F12/haQZ-F12 barrier for CPT, but it is also the only trimer for which the D_e exceeds the ΔE^‡ by an appreciable margin.

3.1.3 Tetramers. Transition states for proton transfer were identified for all 29 unique permutations (including relative orientations of the dangling hydrogen atoms that do not participate in hydrogen bonding) of the (HF)_x(H₂O)_y(HCl)_z systems for which x + y + z = 4. IRC computations indicate that 27 of them are transition states for concerted proton transfer, but 2 correspond to transition states for a stepwise proton transfer process (both for the (H₂O)₂(HCl)₂ systems with f1 = f3 = H₂O, differing only by the relative orientations of the free hydrogen atoms). Out of the other 27 systems for which a concerted pathway was established, 6 have a CPT product that is not equivalent to the corresponding reactant. One example is shown in Fig. 2 for the (HF)₁(H₂O)₂(HCl)₁ tetramer, where the different structures of the reactant and product can be seen along with the TS structure for CPT connecting the two minima. As a result, there are a total of 33 unique minima associated with the 27 transition states for CPT in the tetramer systems. As with the trimer systems, however, results for the higher energy isomers with slightly larger barrier heights that differ only by the relative orientations of the free hydrogen atoms have been relegated to the SI. This sequesters 6 minima and 5 TSs in systems with 2 H₂O fragments that have both free hydrogen atoms on the same side of the ring, as well as 4 minima and 2 TSs in systems with 3 H₂O fragments that have adjacent dangling hydrogen atoms on the same side of the ring. That curation process leaves 20 systems with a CPT TS connecting 23 unique minima for discussion (e.g., data in Table 2). Within this subset of 20 tetramer structures, those containing HF and H₂O fragments have already been characterized at the MP2/aug-cc-pVDZ level of theory,⁴⁸ and the MP2/haTZ optimized geometries reported here are very similar. The Cartesian coordinates and associated MP2/haTZ harmonic vibrational frequencies for all optimized tetramer stationary points can be found in the SI.

Fig. 2 Reactant (left), transition state (Center), and product (right) for the non-alternating (HF)₁(H₂O)₂(HCl)₁ CPT reaction, with the fragments (f1, f2, f3, f4) labeled on the reactant structure.

Table 2 CCSD(T)-F12/haQZ-F12 relative energies of products, barrier heights of transition states for CPT, and dissociation energies of reactants (ΔE, ΔE^‡, and D_e, respectively, in kcal mol⁻¹) for each (HF)_x(H₂O)_y(HCl)_z tetramer configuration, with the fragment labels (f1, f2, f3, f4) denoting the direction of the hydrogen bond network in the reactant

f1	f2	f3	f4	ΔE	ΔE^‡	D e
HF	HF	HF	HF	0.0	14.7	27.9
HF	HF	HF	HCl	0.0	17.6	22.0
HF	HF	HF	H2O	0.0	11.8	30.8
HF	HF	HCl	HCl	0.0	21.5	17.5
HF	HCl	HF	HCl	0.0	23.1	16.5
HF	HF	H2O	HCl	2.7	13.0	25.8
HF	HCl	HF	H2O	0.0	10.0	24.5
HF	HF	H2O	H2O	0.0	13.4	31.2
HF	H2O	HF	H2O	0.0	14.2	32.3
HF	HCl	HCl	HCl	0.0	26.6	13.3
HF	H2O	HCl	HCl	2.5	13.5	20.9
HF	HCl	H2O	HCl	0.0	13.3	18.5
HF	H2O	H2O	HCl	2.4	9.9	26.0
HF	H2O	HCl	H2O	0.0	8.1	25.9
HF	H2O	H2O	H2O	0.0	18.4	30.2
H2O	H2O	H2O	H2O	0.0	26.8	27.4
H2O	H2O	H2O	HCl	0.0	7.9	24.1
H2O	H2O	HCl	HCl	0.0	8.7	19.9
H2O	HCl	HCl	HCl	0.0	14.9	15.1
HCl	HCl	HCl	HCl	0.0	30.9	10.4

---

The three tetramers included in this discussion for which the relative energies of the reactant and product structures are not 0 include all non-alternating trinary clusters (i.e. for xyz = 211, 121 and 112, when the 2 fragments of the same type are adjacent). In all three systems, the structures in which an HF fragment donates a hydrogen bond to an H₂O fragment and accepts a hydrogen bond from an HCl fragment have a CCSD(T)-F12/haQZ-F12 electronic energy roughly 2.5 kcal mol⁻¹ lower than the minimum in which the donor and acceptor roles are reversed. These relative energy differences (ΔE) are reported in Table 2.

In the previous benchmarking study,⁴⁴ the (HF)₄ system was identified as having the lowest CPT barrier of any homogeneous system. We have identified numerous heterogeneous systems for which this barrier is smaller. Of the 20 featured tetramer systems for which a CPT reaction pathway was found, 11 systems had a barrier height lower than the 14.7 kcal mol⁻¹ barrier for (HF)₄. Moreover, all 11 systems have a D_e of at least 18.4 kcal mol⁻¹, which is consistently greater than the corresponding ΔE^‡, usually by a factor of 2–3. All 11 of these systems contain at least one H₂O fragment, with the systems displaying the four lowest barrier heights containing at least two, and the system with the absolute lowest containing three. However, the (HF)₁(H₂O)₃ system has a barrier height 3.7 kcal mol⁻¹ greater than (HF)₄, and (H₂O)₄ has a barrier height 12.1 kcal mol⁻¹ greater than (HF)₄.

3.2 Barrier widths

The barrier width is another important feature of the reaction profile, especially when tunneling may be involved.^86–90 The reaction pathway for the CPT process in each system has been examined via IRC calculations at the MP2/haTZ level of theory. From these pathways, estimates of the barrier width have been generated. Although the inflection points along the IRC can be used to estimate barrier widths for a variety of chemical processes, including for simple hydrogenation,⁹¹ addition,⁹² and proton transfer reactions,⁴⁷ this study adopts a metric based on the collision energy, which has been successfully used elsewhere in studies of rotamerization and tautomerization of hydrocarbons via the quantum tunneling of H atoms.^86,87,90 In each (HF)_x(H₂O)_y(HCl)_z cluster, the energy with which the reactant collides against the barrier (the collision energy) is associated with the zero-point energy of the concerted stretching motion of the covalent bonds to the H atoms involved in the proton transfer process (the in-phase HX stretching mode). The barrier widths are determined by using a simple interpolation scheme to find the reaction coordinate corresponding to the appropriate collision energy on both the reactant and product sides of the profile. To obtain each point, a linear interpolation is performed between the two closest points above and below the collision energy along the MP2/haTZ reaction profile.

Using MP2/haTZ harmonic vibrational frequencies for the trimers, the collision energy (ε) ranges from 5.3 kcal mol⁻¹ for the totally symmetric HF stretching mode in (HF)₃ at 3714 cm⁻¹ down to 3.6 kcal mol⁻¹ for the analogous mode in (H₂O)₂(HCl)₁ at 2512 cm⁻¹. The ε values are given for each trimer system in Table 3 along with the barrier width at the corresponding collision energy w(ε). Two additional estimates of the barrier width are provided at the smallest and largest collision energies, w(3.6) and w(5.3), to facilitate qualitative comparison between the different trimer systems. In a follow-up study, IRCs will be evaluated near the CCSD(T) complete basis set limit to enable a more rigorous evaluation of the barrier widths and realistic estimation of the associated tunneling rates.

Table 3 MP2/haTZ barrier heights (ΔE^‡), collision energies (ε, in kcal mol⁻¹), barrier widths at 3.6 kcal mol⁻¹, 5.3 kcal mol⁻¹, and ε (w(3.6), w(5.3), and w(ε), respectively, in bohr⋅amu^1/2), and imaginary harmonic vibrational frequencies for the CPT TS (ω_i, in cm⁻¹), for each (HF)_x(H₂O)_y(HCl)_z trimer configuration, with the fragment labels (f1, f2, f3) denoting the direction of the hydrogen bond network in the reactant

f1	f2	f3	ΔE^‡	ε	w(3.6)	w(ε)	w(5.3)	ω i
HF	HF	HF	18.1	5.3	3.3	2.6	2.6	1695i
HF	HF	HCl	19.9	4.1	4.3	4.0	3.4	1478i
HF	HCl	HCl	22.3	4.1	5.6	5.2	4.5	1509i
HF	HF	H2O	16.2	4.8	3.3	2.8	2.6	1152i
HF	H2O	H2O	19.4	4.9	3.7	3.2	3.0	1326i
HF	H2O	HCl	14.2	4.1	4.7	4.3	3.6	877i
H2O	H2O	H2O	26.5	5.1	4.4	3.7	3.7	1814i
H2O	H2O	HCl	10.5	3.6	3.7	3.7	2.9	643i
H2O	HCl	HCl	12.5	3.7	4.6	4.5	3.4	690i
HCl	HCl	HCl	23.5	4.1	6.8	6.4	5.5	1504i

---

The IRC profiles for three trimer systems are presented in Fig. 3 along with the window of collision energies between 3.6 and 5.3 kcal mol⁻¹ (denoted by horizontal lines). As can be seen from the w(3.6), w(ε), and w(5.3) columns in Table 3, as well as the blue square data points in Fig. 3, (HF)₃ has the narrowest barrier width in this region, ranging from 2.6 to 3.3 bohr⋅amu^1/2. (HF)₃ also has the lowest MP2/haTZ barrier height of the homogeneous trimers, which is consistent with the corresponding CCSD(T)-F12/haQZ-F12 data from the previous barrier height analysis (ΔE^‡ column in Tables 1 and 3).

Fig. 3 MP2/haTZ IRC profiles of (HF)₃, (HF)₂(H₂O)₁, and (H₂O)₂(HCl)₁ with horizontal lines denoting the range of collision energies (ε) for the 10 trimer systems in Table 3.

Although four heterogeneous trimers have been identified with barrier heights lower than (HF)₃, none have narrower MP2/haTZ barrier widths in this range of collision energies. For example, (H₂O)₂(HCl)₁ has the lowest barrier height in both Tables 1 and 3, but the corresponding barrier widths (2.9 to 3.7 bohr⋅amu^1/2) are slightly larger than those for (HF)₃. This sizeable decrease in barrier height and small increase in barrier width can be seen by comparing the IRC profiles for (HF)₃ and (H₂O)₂(HCl)₁ in Fig. 3 (blue square vs. green triangle data points, respectively). In contrast, the IRC profile for the (HF)₂(H₂O)₁ system (represented by orange circles in Fig. 3) has a barrier width much closer to that of (HF)₃, ranging from 2.6 to 3.3 bohr⋅amu^1/2, albeit with a barrier height only slightly lower than (HF)₃.

Among the tetramers, the MP2/haTZ collision energy (ε) ranges from 4.9 kcal mol⁻¹ for the totally symmetric OH stretching mode in (H₂O)₄ at 3401 cm⁻¹ down to 3.1 kcal mol⁻¹ for the analogous mode in (H₂O)₃(HCl)₁ at 2166 cm⁻¹. The ε values, along with the barrier width at that collision energy, w(ε), are provided in Table 4. As with the trimers, two additional estimates of the barrier width at the smallest and largest tetramer collision energies, w(3.1) and w(4.9), are provided to facilitate qualitative comparison between the different tetramer systems.

Table 4 MP2/haTZ barrier heights (ΔE^‡), collision energies (ε, in kcal mol⁻¹), barrier widths at 3.1 kcal mol⁻¹, 4.9 kcal mol⁻¹, and ε (w(3.1), w(4.9), and w(ε), respectively, in bohr⋅amu^1/2), and imaginary harmonic vibrational frequencies for the CPT TS (ω_i, in cm⁻¹), for each (HF)_x(H₂O)_y(HCl)_z tetramer configuration, with the fragment labels (f1, f2, f3, f4) denoting the direction of the hydrogen bond network in the reactant

f1	f2	f3	f4	ΔE^‡	ε	w(3.1)	w(ε)	w(4.9)	ω i
HF	HF	HF	HF	11.8	4.8	2.9	2.1	2.0	1441i
HF	HF	HF	HCl	14.5	3.9	4.2	3.7	3.1	1243i
HF	HF	HF	H2O	9.6	4.2	2.8	2.2	1.9	850i
HF	HF	HCl	HCl	18.0	4.0	5.8	5.1	4.5	1281i
HF	HCl	HF	HCl	19.4	4.0	6.1	5.4	4.8	1363i
HF	HF	H2O	HCl	10.5	3.9	5.0	4.1	3.3	1055i
HF	HCl	HF	H2O	8.4	3.8	3.6	3.2	2.5	107i
HF	HF	H2O	H2O	11.1	4.1	3.2	2.7	2.3	926i
HF	H2O	HF	H2O	11.8	4.4	3.2	2.5	2.3	976i
HF	HCl	HCl	HCl	22.5	4.0	7.7	6.9	6.2	1396i
HF	H2O	HCl	HCl	11.2	3.9	6.4	5.4	4.4	378i
HF	HCl	H2O	HCl	10.7	3.4	4.9	4.6	3.5	918i
HF	H2O	H2O	HCl	8.0	3.8	4.7	3.9	2.9	338i
HF	H2O	HCl	H2O	6.4	3.2	3.1	3.0	1.7	552i
HF	H2O	H2O	H2O	15.7	4.5	3.9	3.2	3.0	1100i
H2O	H2O	H2O	H2O	22.9	4.9	4.9	4.0	4.0	1653i
H2O	H2O	H2O	HCl	6.5	3.1	3.6	3.6	2.0	344i
H2O	H2O	HCl	HCl	6.7	3.2	4.2	4.1	2.2	612i
H2O	HCl	HCl	HCl	12.2	3.5	6.1	5.7	4.6	536i
HCl	HCl	HCl	HCl	26.2	4.0	9.3	8.3	7.6	1467i

---

The IRC profiles for three tetramer systems are presented in Fig. 4 along with the window of collision energies between 3.1 and 4.9 kcal mol⁻¹ (denoted by horizontal lines). From the w(3.1), w(ε), and w(4.9) columns in Table 4, (HF)₄ has some of the narrowest barrier widths in this region, ranging from 2.0 to 2.9 bohr⋅amu^1/2, with the lowest w(ε) value. (HF)₄ also has the lowest MP2/haTZ barrier height of the homogeneous tetramers, consistent with the corresponding CCSD(T)-F12/haQZ-F12 data from Table 2.

Fig. 4 MP2/haTZ IRC profiles of (HF)₄, (HF)₃(H₂O)₁, and (H₂O)₃(HCl)₁ with horizontal lines denoting the range of collision energies (ε) for the 20 trimer systems in Table 4.

This study has identified eleven heterogeneous tetramers of interest with barrier heights lower than (HF)₄ near the CCSD(T)/CBS limit (ΔE^‡ column in Table 2), but only one of these systems, (HF)₃(H₂O)₁, also has consistently narrower MP2/haTZ barrier widths in the relevant range of collision energies, ranging from 1.9 to 2.8 bohr⋅amu^1/2 (w(3.1), w(ε), and w(4.9) columns in Table 4). This decrease in both barrier height and barrier width can be seen by comparing the IRC profiles for (HF)₄ and (HF)₃(H₂O)₁ in Fig. 4 (blue square vs. orange circle data points, respectively). In contrast, the system with the lowest barrier height ((H₂O)₃(HCl)₁, represented by green triangle data points in Fig. 4) has a barrier width comparable to (HF)₄ and (HF)₃(H₂O)₁ at the higher end of the range of relevant collision energies (w(4.9) in Table 4), whereas it is noticeably wider in the lower range of collision energies (w(3.1) in Table 4). These two features (widths comparable to (HF)₄ at higher collision energies but broader at lower collision energies) are common to many of the tetramer systems with low barrier heights (ΔE^‡ ≤ 10 kcal mol⁻¹).

Although the magnitude of the imaginary harmonic vibrational frequency associated with the CPT TS can provide some information about the curvature in that region of the reaction pathway, it does not provide a consistent measure of the barrier width. For example, the (HF)₄ and (HF)₃(H₂O)₁ IRC profiles have very similar barrier widths, as can be seen in Fig. 4, but the MP2/haTZ imaginary frequencies of their transition states for CPT (last column in Table 4) differ by nearly 600i cm⁻¹ (1441i and 850i cm⁻¹, respectively). In contrast, Fig. 5 shows the IRC profiles for another pair of tetramer systems with significantly different barrier widths but similar imaginary frequencies for their CPT transition states (within 16i cm⁻¹). The TS for CPT in the isomer of (HF)₁(H₂O)₂(HCl)₁, where f2 = f4, has an MP2/haTZ harmonic imaginary frequency of 552i cm⁻¹, which is very similar to the value of 536i cm⁻¹ for the (H₂O)₁(HCl)₃ system. However, the barrier width for the former is approximately half that for the latter in the relevant range of collision energies.

Fig. 5 (H₂O)₁(HCl)₃ and (HF)₁(H₂O)₂(HCl)₁, where f2 = f4, have very different MP2/haTZ IRC profiles but very similar imaginary frequencies associated with their transition states for CPT (536i and 552i, respectively).

Our previous study of the corresponding homogeneous trimers, tetramers and pentamers attempted to establish relationships between the imaginary vibrational frequencies, barrier heights and other properties of the clusters. Although some interesting correlations were identified, the sample size was too limited to draw any definitive conclusions. Unfortunately, we have not identified any good connections between the associated data for the heterogeneous clusters in the present investigation that might provide new insight into the underlying chemical physics of these CPT processes. As such, the information related to this correlation analysis has been relegated to the SI.

4 Conclusions

A total of 39 transition states for concerted proton transfer (CPT) in cyclic hydrogen bonded (HF)_x(H₂O)_y(HCl)_z systems, where x + y + z = 3 or 4, have been characterized in this study, along with 39 reactant minima and 7 distinct product minima. However, the results for 9 of these pathways are only presented in the SI because they involve higher-energy conformational isomers that merely permute the relative orientations of the free hydrogen atoms in heterogeneous systems containing 2 or 3 H₂O fragments. Features of the CPT process have been carefully compared for the remaining 30 systems, such as barrier heights, barrier widths, dissociation energies, and relative energies of the reactants and products. MP2/haTZ IRC calculations confirm that each TS directly connects the reactant and product along a low-energy CPT pathway and provide the means to assess barrier widths for each system. The protocol employed here to calculate the barrier heights and other energetic quantities (specifically, CCSD(T)-F12/haQZ-F12 energies on MP2/haTZ geometries) should provide reasonable estimates of these quantities near the CCSD(T) CBS limit, based on a comparison to benchmark data from ref. 44 for homogeneous HF, H₂O, and HCl clusters.

Among the trimers, there are four systems that display a barrier height lower than (HF)₃ (ΔE^‡ = 20.6 kcal mol⁻¹), these being (H₂O)₂(HCl)₁, (H₂O)₁(HCl)₂, (HF)₁(H₂O)₁(HCl)₁, and (HF)₂(H₂O)₁, ranging from 11.9 to 18.1 kcal mol⁻¹. Of these, the system containing only HF and H₂O fragments has a narrower (albeit higher) barrier than those containing 1 or more HCl fragments. For the tetramers, five heterogeneous systems were identified that had even smaller ΔE^‡ values (≤10 kcal mol⁻¹) and larger D_e values (around 20 to 25 kcal mol⁻¹) than the trimer systems. These tetramers are (H₂O)₃(HCl)₁, the isomer of (HF)₁(H₂O)₂(HCl)₁ where f2 = f3, the isomer of (HF)₁(H₂O)₂(HCl)₁ where f2 = f4, the isomer of (H₂O)₂(HCl)₂ where f1 = f2, and the isomer of (HF)₂(H₂O)₁(HCl)₁ where f1 = f3, with barrier heights of 7.9, 9.9, 8.1, 8.7, and 10.0 kcal mol⁻¹, respectively, and corresponding dissociation energies of 24.1, 26.0, 25.9, 19.9, and 24.5 kcal mol⁻¹ near the CCSD(T) CBS limit. These ΔE^‡ values are smaller than those of (HF)₄, which ref. 44 identified as having the lowest barrier height and largest dissociation energy (14.7 and 27.9 kcal mol⁻¹, respectively) of all homogeneous systems studied therein. Although the barriers for CPT in this group of heterogeneous tetramers are appreciably lower than in (HF)₄, the MP2/haTZ IRC profiles indicate they are wider. However, the barrier for (HF)₃(H₂O)₁ is both lower and narrower than (HF)₄. The ΔE^‡ for (HF)₃(H₂O)₁ is 11.8 kcal mol⁻¹ near the CCSD(T) CBS limit, and the corresponding D_e is more than 2.5 times larger (30.8 kcal mol⁻¹).

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting this article have been included as part of the supplementary information (SI). Supplementary information: energetics of isomers at the CCSD(T)-F12/haQZ-F12 level of theory generated by permuting relative orientations of free hydrogen atoms in heterogeneous systems when y = 2 or 3 are provided, as well as Cartesian coordinates and harmonic vibrational frequencies for all stationary points optimized at the MP2/haTZ level of theory, and the MP2/haTZ energy at each point along the reaction pathway as determined by the IRC calculation for each system. See DOI: https://doi.org/10.1039/d5cp03769g.

Acknowledgements

This work was supported by the National Science Foundation (grant number CHE-2452726). Some computations for this work were performed on the high-performance computing infrastructure provided by Research Support Solutions at Missouri University of Science and Technology, https://doi.org/10.71674/PH64-N397 (last accessed September 15th, 2025).

References

1. R. H. Morris, Chem. Rev., 2016, 116, 8588–8654.
2. J. N. Brönsted, Recl. Trav. Chim. Pays-Bas, 1923, 42, 718–728.
3. T. M. Lowry, J. Soc. Chem. Ind., 1923, 42, 43–47.
4. C. Costentin, Chem. Rev., 2008, 108, 2145–2179.
5. C. Costentin, M. Robert and J.-M. Savéant, Acc. Chem. Res., 2010, 43, 1019–1029.
6. R. E. Warburton, A. V. Soudackov and S. Hammes-Schiffer, Chem. Rev., 2022, 122, 10599–10650.
7. D. B. Spry and M. D. Fayer, J. Phys. Chem. B, 2009, 113, 10210–10221.
8. Y. Zhou, J. Yang, H. Su, J. Zeng, S. P. Jiang and W. A. Goddard, J. Am. Chem. Soc., 2014, 136, 4954–4964.
9. P. Yao, J. Zhang, Q. Qiu, Y. Zhao, F. Yu and Y. Li, Adv. Energy Mater., 2025, 15, 2403335.
10. R. Ram, L. Xia, H. Benzidi, A. Guha, V. Golovanova, A. G. Manjón, D. L. Rauret, P. S. Berman, M. Dimitropoulos, B. Mundet, E. Pastor, V. Celorrio, C. A. Mesa, A. M. Das, A. Pinilla-Sánchez, S. Giménez, J. Arbiol, N. López and F. P. G. de Arquer, Science, 2024, 384, 1373–1380.
11. L. Wang, Q. Yu and Y. Zhang, Chem. Eng. J., 2025, 504, 158914.
12. A.-N. Bondar and J. C. Smith, Biophys. J., 2025, 124, 3813–3826.
13. J. E. Besaw, W.-L. Ou, T. Morizumi, B. T. Eger, J. D. Sanchez Vasquez, J. H. Chu, A. Harris, L. S. Brown, R. D. Miller and O. P. Ernst, J. Biol. Chem., 2020, 295, 14793–14804.
14. S. A. Siletsky, Int. J. Mol. Sci., 2023, 24, 9070.
15. R. Hussein, A. Graça, J. Forsman, A. O. Aydin, M. Hall, J. Gaetcke, P. Chernev, P. Wendler, H. Dobbek, J. Messinger, A. Zouni and W. P. Schröder, Science, 2024, 384, 1349–1355.
16. T. Matsui, Y. Shigeta and K. Hirao, J. Phys. Chem. B, 2007, 111, 1176–1181.
17. D. Jacquemin, J. Zúñiga, A. Requena and J. P. Céron-Carrasco, Acc. Chem. Res., 2014, 47, 2467–2474.
18. B. King, M. Winokan, P. Stevenson, J. Al-Khalili, L. Slocombe and M. Sacchi, J. Phys. Chem. B, 2023, 127, 4220–4228.
19. R. Srivastava, Front. Chem., 2019, 7, 536.
20. Y.-H. Cheng, Y.-C. Zhu, X.-Z. Li and W. Fang, Chin. Phys. B, 2023, 32, 018201.
21. L. Briccolani-Bandini, E. Brémond, M. Pagliai, G. Cardini, I. Ciofini and C. Adamo, J. Comput. Chem., 2023, 44, 2308–2318.
22. Q. Cui and M. Karplus, J. Phys. Chem. B, 2003, 107, 1071–1078.
23. J. M. Mayer and I. J. Rhile, Biochim. Biophys. Acta, 2004, 1655, 51–58.
24. S. Takeuchi and T. Tahara, Proc. Natl. Acad. Sci. U. S. A., 2007, 104, 5285–5290.
25. L. T. F. d M. Camargo, R. Signini, A. C. C. Rodrigues, Y. F. Lopes and A. J. Camargo, J. Phys. Chem. B, 2020, 124, 6986–6997.
26. C. Li and J. M. J. Swanson, J. Phys. Chem. B, 2020, 124, 5696–5708.
27. Y. Zeng, A. Li and T. Yan, J. Phys. Chem. B, 2020, 124, 1817–1823.
28. S. A. Fischer and D. Gunlycke, J. Phys. Chem. B, 2019, 123, 5536–5544.
29. A. T. Nasrabadi and L. D. Gelb, J. Phys. Chem. B, 2018, 122, 5961–5971.
30. M. Hasani, S. A. Amin, J. L. Yarger, S. K. Davidowski and C. A. Angell, J. Phys. Chem. B, 2019, 123, 1815–1821.
31. M. A. Bowring, L. R. Bradshaw, G. A. Parada, T. P. Pollock, R. J. Fernández-Terán, S. S. Kolmar, B. Q. Mercado, C. W. Schlenker, D. R. Gamelin and J. M. Mayer, J. Am. Chem. Soc., 2018, 140, 7449–7452.
32. J. N. Schrauben, M. Cattaneo, T. C. Day, A. L. Tenderholt and J. M. Mayer, J. Am. Chem. Soc., 2012, 134, 16635–16645.
33. Z. Smedarchina, W. Siebrand, A. Fernández-Ramos and Q. Cui, J. Am. Chem. Soc., 2003, 125, 243–251.
34. N. Sülzner, G. Jung and P. Nuernberger, Chem. Sci., 2025, 16, 1560–1596.
35. K. Töpfer, S. Käser and M. Meuwly, Phys. Chem. Chem. Phys., 2022, 24, 13869–13882.
36. H. Tachikawa, J. Phys. Chem. A, 2020, 124, 3048–3054.
37. S. F. Sutton, C. H. Rotteger, C. K. Jarman, P. Tarakeshwar and S. G. Sayres, J. Phys. Chem. Lett., 2023, 14, 8306–8311.
38. S. Meddeb-Limem and A. Ben Fredj, RSC Adv., 2024, 14, 23184–23203.
39. Monu, B. K. Oram, A. Kothari and B. Bandyopadhyay, Comput. Theor. Chem., 2025, 1248, 115164.
40. U. Lourderaj, K. Giri and N. Sathyamurthy, J. Phys. Chem. A, 2006, 110, 2709–2717.
41. G. Feng, L. B. Favero, A. Maris, A. Vigorito, W. Caminati and R. Meyer, J. Am. Chem. Soc., 2012, 134, 19281–19286.
42. C. S. Tautermann, A. F. Voegele and K. R. Liedl, J. Chem. Phys., 2004, 120, 631–637.
43. E. Plackett, C. Robertson, A. De Matos Loja, H. McGhee, G. Karras, I. V. Sazanovich, R. A. Ingle, M. J. Paterson and R. S. Minns, J. Chem. Phys., 2024, 160, 124311.
44. Y. Xue, T. M. Sexton, J. Yang and G. S. Tschumper, Phys. Chem. Chem. Phys., 2024, 26, 12483–12494.
45. N. Elahi and C. D. Zeinalipour-Yazdi, Comput. Theor. Chem., 2025, 1244, 115061.
46. D. Patkar, M. B. Ahirwar, S. R. Gadre and M. M. Deshmukh, J. Phys. Chem. A, 2021, 125, 8836–8845.
47. Monu, B. K. Oram and B. Bandyopadhyay, Int. J. Quantum Chem., 2024, 124, e27442.
48. D. Patkar, M. B. Ahirwar, S. P. Shrivastava and M. M. Deshmukh, New J. Chem., 2022, 46, 2368–2379.
49. M. Svanberg, J. B. C. Petterson and K. Bolton, J. Phys. Chem. A, 2000, 104, 5787–5798.
50. J. S. Mancini and B. M. Bowman, J. Phys. Chem. Lett., 2014, 5, 2247–2253.
51. A. L. Sobolewski and W. Domcke, J. Phys. Chem. A, 2003, 107, 1557–1562.
52. D. E. Bacelo, R. C. Binning and Y. Ishikawa, J. Phys. Chem. A, 1999, 103, 4631–4640.
53. S. Odde, B. J. Mhin, S. Lee, H. M. Lee and K. S. Kim, J. Chem. Phys., 2004, 120, 9524–9535.
54. M. Masia, H. Forbert and D. Marx, J. Phys. Chem. A, 2007, 111, 12181–12191.
55. R. C. Tudela and D. Marx, Phys. Rev. Lett., 2017, 119, 223001.
56. M. Planas, C. Lee and J. J. Novoa, J. Phys. Chem., 1996, 100, 16495–16501.
57. C. Lee, C. Sosa, M. Planas and J. J. Novoa, J. Chem. Phys., 1996, 104, 7081–7085.
58. M. Tachikawa, K. Mori and Y. Osamura, Mol. Phys., 1999, 96, 1207–1215.
59. E. M. Cabaleiro-Lago, J. M. Hermida-Ramón and J. Rodríguez-Otero, J. Chem. Phys., 2002, 117, 3160–3168.
60. M. Boda and G. N. Patwari, Phys. Chem. Chem. Phys., 2017, 19, 7461–7464.
61. A. K. Samanta, Y. Wang, J. S. Mancini, J. M. Bowman and H. Reisler, Chem. Rev., 2016, 116, 4913–4937.
62. S. Re, Y. Osamura, Y. Suzuki and H. F. Schaefer, J. Chem. Phys., 1998, 109, 973–977.
63. H. Tachikawa, J. Phys. Chem. A, 2017, 121, 5237–5244.
64. S. Sugawara, T. Yoshikawa, T. Takayanagi and M. Tachikawa, Chem. Phys. Lett., 2011, 501, 238–244.
65. T. Takayanagi, K. Takahashi, A. Kakizaki, M. Shiga and M. Tachikawa, Chem. Phys., 2009, 358, 196–202.
66. A. Milet, C. Struniewicz, R. Moszynski and P. E. S. Wormer, J. Chem. Phys., 2001, 115, 349–356.
67. A. Vargas-Caamal, J. L. Cabellos, F. Ortiz-Chi, H. S. Rzepa, A. Restrepo and G. Merino, Chem. – Eur. J., 2016, 22, 2812–2818.
68. F. Xie, D. S. Tikhonov and M. Schnell, Science, 2024, 384, 1435–1440.
69. H. Sippola and P. Taskinen, J. Chem. Eng. Data, 2014, 59, 2389–2407.
70. H. Arstila, K. Laasonen and A. Laaksonen, J. Chem. Phys., 1998, 108, 1031–1039.
71. L. A. Kurfman, T. T. Odbadrakh and G. C. Shields, J. Phys. Chem. A, 2021, 125, 3169–3176.
72. J. M. Hermida Ramón and M. A. Ríos, Chem. Phys., 1999, 250, 155–169.
73. G. Schüürmann, Chem. Phys. Lett., 1999, 302, 471–479.
74. O. M. Longsworth, C. J. Bready and G. C. Shields, Environ. Sci.: Atmos., 2023, 3, 1335–1351.
75. W. Burnside, Proc. R. Soc. London, Ser. A, 1900, s1–33, 162–184.
76. OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, 2025, Published electronically at https://oeis.org/A027671.
77. C. Møller and M. S. Plesset, Phys. Rev., 1934, 46, 618–622.
78. T. H. Dunning, Jr., J. Chem. Phys., 1989, 90, 1007–1023.
79. R. Bartlett, Ann. Rev. Phys. Chem., 1981, 32, 359.
80. G. Purvis and R. Bartlett, J. Chem. Phys., 1982, 76, 1910.
81. T. B. Adler, G. Knizia and H. J. Werner, J. Chem. Phys., 2007, 127, 221106.
82. H. J. Werner, G. Knizia and F. R. Manby, Mol. Phys., 2011, 109, 407.
83. K. E. Yousaf and K. A. Peterson, J. Chem. Phys., 2008, 129, 184108.
84. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Izmaylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, T. A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman and D. J. Fox, Gaussian 16 Revision C.01, Gaussian Inc., Wallingford CT, 2016.
85. H.-J. Werner, P. J. Knowles, F. R. Manby, J. A. Black, K. Doll, A. Heßelmann, D. Kats, A. Köhn, T. Korona, D. A. Kreplin, Q. Ma, I. Miller, F. Thomas, A. Mitrushchenkov, K. A. Peterson, I. Polyak, G. Rauhut and M. Sibaev, J. Chem. Phys., 2020, 152, 144107.
86. S. Amiri, H. P. Reisenauer and P. R. Schreiner, J. Am. Chem. Soc., 2010, 132, 15902–15904.
87. G. Qiu and P. R. Schreiner, ACS Cent. Sci., 2023, 9, 2129–2137.
88. P. R. Schreiner, H. P. Reisenauer, D. Ley, D. Gerbig, C.-H. Wu and W. D. Allen, Science, 2011, 332, 1300–1303.
89. M. M. Linden, J. P. Wagner, B. Bernhardt, M. A. Bartlett, W. D. Allen and P. R. Schreiner, J. Phys. Chem. Lett., 2018, 9, 1663–1667.
90. P. R. Schreiner, J. P. Wagner, H. P. Reisenauer, D. Gerbig, D. Ley, J. Sarka, A. G. Császár, A. Vaughn and W. D. Allen, J. Am. Chem. Soc., 2015, 137, 7828–7834.
91. Z. Konkoli, E. Kraka and D. Cremer, J. Phys. Chem. A, 1997, 101, 1742–1757.
92. D. Cremer, A. Wu and E. Kraka, Phys. Chem. Chem. Phys., 2001, 3, 674–687.

---