¹⁴N NQR spectroscopy reveals the proton position in N–H⋯N bonds: a case study with proton sponges

Abstract

The position of protons in hydrogen bonds is often uncertain to some degree, as the technique most often used for structure determination, X-ray diffraction, is sensitive to electron density, which is not particularly abundant around protons. In hydrogen bonds, protons introduce an additional problem: the potential for proton motion is inherently anharmonic and thus requires the consideration of nuclear quantum effects (NQEs). Here, we demonstrate that ¹⁴N NQR spectroscopy is able to rather accurately determine proton positions in N–H⋯N bonds, in certain cases with an accuracy comparable to that of X-ray and neutron diffraction at room temperature. We first derive, using ab initio calculations considering also the NQEs, a relation between the proton distance from the bond midpoint and the difference between the quadrupole coupling constants for the two nitrogen sites. The found relation is linear with a proportionality constant of 0.108 Å MHz⁻¹ for tertiary amine nitrogens. Then, we validate our theoretical calculations experimentally, using several 1,8-bis(dimethylamino)naphthalene (DMAN) complexes.

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Introduction

Hydrogen bonds are ubiquitous in organic molecules and their crystals and, because they are stronger than van der Waals interactions, they often substantially influence material properties. The position of heavy atoms in the bond (N, O, F) is usually well known, however, the position of protons is known to a lesser extent, with an accuracy changing from one system to another. This inaccuracy is a characteristic of the principal technique for structure determination in solids: X-ray diffraction. Namely, X-ray diffraction is sensitive to electron density, which is low around protons, but more importantly, this electron density deviates significantly from spherical symmetry.¹ Advanced refinement techniques, like difference Fourier maps² and Hirshfeld atom refinement,³ provide better results, but they also require higher quality experimental data or include extensive calculations. Protons in hydrogen bonds introduce a further problem for X-ray diffraction. The small proton mass makes hydrogen motion inherently quantum mechanical in nature, and effects such as zero-point motion and tunneling, commonly referred to as nuclear quantum effects (NQEs), become important.^4–8 When protons move in a harmonic potential, NQEs are rather easily taken into account by augmenting the atomic displacement parameters. However, the potential for proton motion in hydrogen bonds is very often anharmonic, in many cases even of a double well type, e.g. when the donor and acceptor atoms have similar proton affinities.⁵ In such situations, the anisotropy of the atomic displacement is large and non-symmetric, and in general, difficult to properly account for, especially because protons make only a small contribution to the X-ray intensity.

The first alternative to X-ray diffraction is neutron diffraction, which provides accurate hydrogen bond geometries, since neutrons are scattered by nuclei. However, the availability of appropriate infrastructure is limited and thus the method is used less often.

One of the more successful other techniques for structure determination, especially for organic molecules, is Nuclear Magnetic Resonance (NMR). This technique is not ideal to study hydrogen bonds in solids but it is nevertheless often used.^9–15 Here, the main issues are: crowded ¹H NMR spectra,¹⁶ substitution of the hydrogen bond proton with ²H modifies bond geometry, and ¹⁷O and ¹⁵N have a very low natural abundance,^17–19 while ¹⁴N has broad spectral lines, due to its large nuclear quadrupole moment. Quadrupole nuclei, like ²H, ¹⁴N and ¹⁷O, can be also observed indirectly, e.g. with various double resonance techniques,²⁰ providing the nuclear quadrupole resonance (NQR) frequencies rather than chemical shifts, but the sensitivity of such experiments is almost always low. Other nuclei not part of the hydrogen bond, mainly ¹³C and sometimes ³⁵Cl, are also used, but their NMR/NQR parameters may not easily be related to the hydrogen bond geometry. Nevertheless, the hydrogen bond geometry is an important parameter used in many ab initio studies so that research in this field is still intense.

In this paper, we show how ¹⁴N NQR spectroscopy is used to determine the proton position in intramolecular N–H⋯N hydrogen bonds. The motivation for this work is the following: The hydrogen bond proton is known to have a large impact on the electric field gradient (EFG) tensors at the position of the donor and acceptor atoms.²¹ In the N–H⋯N bond, the nitrogen EFG is roughly a superposition of two contributions due to two types of nitrogen valence electrons. The first contribution comes from the nitrogen lone pair, which is part of the hydrogen bond, and its orbital is strongly influenced by the hydrogen bond proton. A proton closer to the nitrogen atom decreases this contribution, whereas a more distant proton increases it. The second contribution to the EFG comes from the electrons that do not participate in the hydrogen bond. The orbitals of these electrons are mainly defined by the functional groups attached to the nitrogen atoms and unaffected by the position of the hydrogen bond proton. This contribution can then be considered as constant across some similar samples. Thus, by measuring the EFG at both nitrogen sites, which can easy be carried out with NQR spectroscopy, one should obtain some quantitative information on the proton position within the N–H⋯N bond.

We have analyzed this dependence theoretically, using ab initio calculations and vibrational analysis, as well as experimentally, with ¹⁴N NQR spectroscopy and published crystallographic data. We find that the accuracy of the method presented here is better than that of X-ray diffraction in certain cases, especially at room temperature. Compared to neutron diffraction, the advantage of ¹⁴N NQR spectroscopy is the possibility to use samples as synthesized, most importantly avoiding the preparation of single crystals.

This is not the first use of ¹⁴N NQR spectroscopy for N–H⋯N bond studies with an interest in proton transfer, however, the focus in the previous studies^20–22 was on molecular conformations associated with proton dynamics, rather than on proton position.

The compounds used in this study are a selection of 1,8-bis(dimethylamino)naphthalene (DMAN) salts, also known as proton sponges.^23,24 These compounds are suitable model compounds, as accurate crystal structures are available for many of them, whereas the proton is known to exhibit shifts in a wide range.^25–31

Theoretical part

DMAN salt geometry

DMAN salts, or complexes, consist of a DMAN-H⁺ cation and an anion usually of organic origin. The DMAN-H⁺ cation, shown in Fig. 1, contains an intramolecular N–H⋯N bond, which is incorporated in a rather rigid (naphthalene) skeleton of an almost C_2v symmetry. The skeleton geometry changes only slightly from one salt to another; all N–N distances are between 2.56 and 2.64 Å, which makes the N–H⋯N bond of a short, strong type,³² with a characteristic low barrier potential. In a hypothetically isolated cation, the proton would occupy a central position, given the C_2v symmetry, between the two nitrogen sites at some distance from the N–N axis since the bond is not linear. In the presence of anions, however, the symmetry of the bond is lifted by interaction with the anion and the proton shifts towards either nitrogen site by a variable amount, while its distance to the N–N axis and the N–H⋯N defined plane remain practically unaffected. The hydrogen bond in DMAN-H⁺ is relatively isolated, thus, the observed proton displacements are mainly due to the presence of the nearest anion, which is located above the N–H⋯N bond, with its most electronegative part, usually an oxygen ion, almost in the N–H⋯N plane.

Fig. 1 DMAN-H⁺ molecular scheme and the three N–H⋯N bond models.

The models

To better understand the relation between the proton displacement and the nitrogen EFG tensors in a N–H⋯N bond, we devised three models for isolated DMAN-H⁺ cations progressively increasing in complexity, as shown in Fig. 1.

Model I is a simple and popular model for linear N–H⋯N bonds^33–35 and it contains two ammonium molecules oriented head to head on a C_3v symmetry axis; the hydrogen bond proton is here located between the two nitrogen atoms. Within the model, the proton motion is constrained to the symmetry axis. This model has been studied from several points of view by different authors, and here, it should give us some additional insights related to NQR parameters. Nevertheless, this model is not very suitable to describe N–H⋯N bonds in molecules from the NQR perspective, as nitrogen in molecules is often bound to at least one carbon, which is known to appreciably modify ¹⁴N NQR parameters.

Model II is an evolution of Model I, where the hydrogen atoms have been replaced by methyl groups, thus obtaining two trimethylamine molecules oriented head to head. The symmetry of the model is preserved and the hydrogen bond proton is still confined to the symmetry axis. Model II should give us qualitatively similar results to Model I, but quantitatively relevant results for the DMAN-H⁺ cation.

Model III is a non-linear version of Model II, which is obtained by rotating the two trimethylamine molecules around the nitrogen atom into a configuration similar to the DMAN-H⁺ ion, which contains a pair of nearly parallel N–C bonds. The symmetry of Model III is C_s. The single mirror plane is defined by the two nitrogen atoms and the two, here parallel, N–C bonds. The symmetry of the model without the proton is C_2v, as for the naphthalene skeleton in proton sponges. The N–H⋯N bond in Model III is non-linear and the proton motion is still restricted, but now to the symmetry plane.

The geometry of all models is defined by the N–N distance r_NN with the proton coordinates x and z relative to the N–N midpoint, with x running along the N–N direction and z perpendicular to it. For linear N–H⋯N models (Model I and II), z = 0 at all times. The anion itself is not included explicitly in any model, but instead, its influence on the proton is modeled by the inclusion of an anion-specific perturbation potential. More complicated models for proton sponge cations have been treated by other authors, e.g. by Horbatenko et al.,³⁶ but none of these studies included NQR parameters. Here, we decided to keep our models as simple as possible, thus enabling us to extract some general properties.

Ab initio calculations

The first set of calculations involved using ab initio calculations for each model to obtain two quantities as a function of the model geometry (r_NN, x, z): (i) the cation potential energy surface (PES) E⁺_e and (ii) the EFG at both nitrogen sites V^L_μν and V^R_μν, for the left and right nitrogen site, respectively. The geometries of the ammonium and trimethylamine molecules were not altered during these calculations and were fixed to their optimized geometries.

All ab initio calculations were carried out at the MP2/6-311++G** level with the Firefly quantum chemical package,³⁷ which is partially based on the GAMESS program.³⁸

Vibrational analysis

In the second set of calculations, we considered the NQEs in the presence of anions. We first invoked the adiabatic approximation (by freezing r_NN) to find the proton ground state wavefunction ψ₀ in a cation–anion complex by solving the Schrödinger equation for an anion modified potential for proton motion, U = E⁺_e + δU, where δU is the anion contribution. Here, δU = E_xx had a simple form, which considered only an effective electric field E_x pointing in the direction of the coordinate x. The Schrödinger equation was solved for each value of r_NN and E_x separately. Finally, the found ψ₀ was used to calculate the expectation values of proton coordinates 〈x〉 and 〈z〉, and nitrogen EFG tensors 〈V^L_μν〉 and 〈V^R_μν〉. The presence of different anions was simulated by using different E_x, and in a similar way, the slightly different DMAN-H⁺ geometries were simulated with several r_NN. Note, 〈x〉, 〈z〉 (〈z〉 = 0 for Models I and II) and r_NN define here the experimentally observed hydrogen bond geometry while 〈V^L_μν〉 and 〈V^R_μν〉 define quantities measured by NQR spectroscopy. Within these calculations, the influence of E_x on either V^L_μν or V^R_μν is completely neglected, thus the anions do not directly contribute to the nitrogen EFG tensors.

The vibrational analysis was carried out with a matrix formalism in the program Mathematica. The proton wavefunctions were expanded on base functions of a harmonic oscillator. The numerically determined potentials were fitted with a polynomial function of coordinates x and z, with a 4th order for x and 2nd order for z. These functions were then used to determine the potential matrix elements. The proton wavefunction expansion coefficients were finally determined by solving the Schrödinger equation.

Experimental part

Samples

The polycrystalline DMAN complexes selected for the study are listed in Table 1. The complexes were prepared by the crystallization of a methanol solution of stoichiometric amounts of DMAN and the corresponding organic acid at room temperature. Only the 1 : 1 complex DMAN–chloranilic acid was left to crystallize at 4 °C. A complex was assumed to be successfully obtained when sharp ¹⁴N NQR lines at frequencies different from DMAN were observed and the proton spin–lattice relaxation time T₁ was a single component. The sharp NQR lines confirm the crystalline character of the substance, whereas the shifts of the NQR frequencies from the DMAN values confirm that the complex is formed. The single component proton T₁ is a sign that a single substance is present. Samples 1–7, with known crystallographic structures,^27–39 were also confirmed using powder X-ray diffraction. All reagents were purchased from Sigma-Aldrich.

Table 1 The 1,8-bis(dimethylamino)naphthalene (DMAN) complexes used in this study and the experimental geometric parameters of the hydrogen bonds together with ¹⁴N NQR data obtained in this work and the NQR predictions for samples 8–10

	DMAN complex	Method/temp.	r NN [Å]	x [Å]	z [Å]	N–H		N⋯H		N–H⋯N
						C Q [kHz]	η	C Q [kHz]	η	ΔCQ [kHz]
a Structure in ref. 28. b Structure in ref. 27. c Structure in ref. 39.
1	DMAN : benzoic acid (1 : 2)	X-ray^a/300 K	2.594	0.021	0.214	3550	0.132	3533	0.113	17
2	DMAN : 2-iodobenzoic acid (1 : 2)	X-ray^a/300 K	2.571	0.235	0.218	3950	0.138	3180	0.126	770
		Neutr.^a/300 K	2.575	0.079	0.267
3	DMAN : 4-iodobenzoic acid (1 : 2)	X-ray^a/200 K	2.602	0.272	0.266	4673	0.107	2473	0.202	2200
		Neutr.^b/200 K	2.602	0.236	0.279
4	DMAN : 2-fluorobenzoic acid (1 : 2)	X-ray^a/200 K	2.589	0.039	0.228	3707	0.119	3380	0.124	327
		Neutr.^b/200 K	2.595	0.016	0.273
5	DMAN : 4-hydroxybenzoic acid (1 : 2)	X-ray^c/100 K	2.566	0.051	0.209	3740	0.102	3420	0.099	320
6	DMAN : 2,3-dihydroxybenzoic acid (1 : 1)	X-ray^b/200 K	2.593	0.189	0.229	4213	0.095	2893	0.138	1320
		Neutr.^b/200 K	2.599	0.160	0.274
7	DMAN : chloranilic acid (1 : 1)	X-ray^b/100 K	2.610	0.299	0.245	4693	0.077	2420	0.124	2273
		Neutr.^b/200 K	2.621	0.228	0.271
8	DMAN : malonic acid (1 : 1)	NQR/300 K		0.171		4353	0.096	2820	0.177	1533
9	DMAN : 3,5-dinitrobenzoic acid (1 : 2)	NQR/300 K		0.086		3940	0.107	3180	0.113	760
10	DMAN : 5-chloro-2-hydroxybenzoic acid (1 : 2)	NQR/300 K		0.249		4593	0.070	2393	0.125	2200

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NQR spectroscopy

¹⁴N NQR frequencies were determined with the technique of cross-relaxation spectroscopy.⁴⁰ Here, ¹⁴N NQR levels are observed indirectly by their influence on ¹H relaxation of the whole sample. The found frequencies were then paired and used to determine the quadrupole coupling constant C_Q and the EFG asymmetry parameter η, where C_Q is linearly proportional to the largest EFG principal value.

All measurements were carried out with a homebuilt instrument, where the sample is repeatedly moved from high to low field with a pneumatic unit. The ¹H Larmor frequency in a high field was 32 MHz.

Results

The DMAN salts used in this study, their known hydrogen bond parameters and the ¹⁴N EFG parameters determined here are summarized in Table 1. The geometry of the hydrogen bond (x and z relative to the N–N midpoint) was derived from the parameters r_NN, r_N–H and r_N⋯H reported in other studies. The EFG tensor is specified with two parameters: C_Q and η. For Models I and II, the symmetry implies η = 0 and C_Q = e²q〈V_xx〉/h. For Model III, the symmetry implies η ≠ 0 and the largest EFG tensor principal component is no longer along the x direction but rather approximately along the direction of the nitrogen lone pair. The results of our ab initio calculations and vibrational analysis are shown in Fig. 2.

Fig. 2 (a) The calculated and the experimental interdependence between C^L_Q and C^R_Q for Models I–III. For Model II, we show only data for 〈x〉 ≤ 0, while for Model III, 〈x〉 ≥ 0. Experimentally, we cannot distinguish between the left and right nitrogen sites, so C^L_Q and C^R_Q can be interchanged with a simultaneous reversal of the sign for 〈x〉. Thus, the sign of 〈x〉 is experimentally undetermined. In (b), we show the theoretical dependence of 〈x〉 on ΔC_Q = C^L_Q − C^R_Q for Models II and III for the following r_NN: 2.54 Å, 2.6 Å and 2.64 Å. The lines for various r_NN almost coincide and are not marked separately. Here, we also show the crystallographically determined x and experimentally determined ΔC_Q for samples 1–7. The data for sample 2, where a large discrepancy betwen the X-ray and neutron proton positions is observed, is additionally marked in red. The red dashed line represents the dependence found with 1st order degenerate perturbation theory.

In Fig. 2a we first present the inter-dependencies between the quadrupole coupling constants for the left and right nitrogen atom, C^L_Q and C^R_Q for a range of E_x and a selection of r_NN. The values for E_x (not shown) were chosen so that |〈x〉| ≤ 0.27 Å, where the limiting cases roughly correspond to the largest experimentally observed proton displacements. Similarly, r_NN was chosen from the range of experimentally found r_NN.

Qualitatively, the behavior is similar for all three models. When E_x = 0, which corresponds to an isolated cation, 〈x〉 = 0 and the two quadrupole coupling constants are equal, C^L_Q = C^R_Q ≡ C⁰_Q. For all three models, C⁰_Q is linearly dependent on r_NN, decreasing when r_NN increases. This behavior cannot be expected a priori as C⁰_Q depends on the balance between two opposing contributions. When r_NN increases, the proton, now further away from both nitrogen sites, becomes less efficient in attracting the two lone pairs, at the same time, however, the repulsion between the two lone pairs also diminishes and allows them to move further away from their nitrogen sites, i.e. closer to the proton. This r_NN dependence is also the reason for smaller C⁰_Q values for Model III compared to Model II, as the proton path between the two nitrogen sites in Model III is longer than r_NN. The C⁰_Q values in Model I are significantly smaller than those in Models II and III. This is expected and is due to differences in functional groups attached to the nitrogen atoms. In addition, for Model I, we expect a small C⁰_Q based on symmetry alone. The environment around the nitrogen atom is here rather symmetric as there are four hydrogen atoms directly bound to a nitrogen atom, which are arranged in a nearly tetrahedral structure. In a perfect tetrahedral environment, like in the ammonium cation, the symmetry would imply C_Q = 0.

The application of E_x induces a shift of the proton in the direction of E_x, thus 〈x〉 ≠ 0, with the magnitude of 〈x〉 increasing with field strength. At the same time, C^L_Q increases while C^R_Q decreases, or vice versa, all of which is as expected. A rather unexpected but interesting observation is, however, that C^L_Q + C^R_Q for a specific r_NN is practically independent of E_x and thus 〈x〉. This relation can be very useful, for example, to determine C⁰_Q ≈ ½(C^L_Q + C^R_Q) from a measurement at any E_x, i.e. from any cation–anion complex. Determining C⁰_Q by other means would be rather difficult, especially due to its r_NN dependence. Another feature observed in Fig. 2a is the almost equidistant contour lines for 〈x〉. This can be used to determine 〈x〉 empirically, e.g. by a comparison with a suitable reference.

To assess the usefulness of these ab initio calculations, we show our experimentally determined C^L_Q and C^R_Q values for all samples in Fig. 2a. The experimental points are within a reasonable range of theoretical values (Models II and III) for 〈x〉, but the predicted r_NN values are unrealistically short. This disagreement can be ascribed to a mismatch in functional groups attached to the nitrogen, which affect its C⁰_Q. A simple approach to compensate for an inexact C⁰_Q is to introduce ΔC_Q = C^L_Q − C^R_Q, which is roughly C⁰_Q independent, as functional groups shift both C^L_Q and C^R_Q by the same amount.

In Fig. 2b, we show a plot of 〈x〉 as a function of ΔC_Q for Models II and III. The observed relation between 〈x〉 and ΔC_Q is linear and, rather unexpectedly, also practically independent of r_NN. In addition, there is an almost negligible difference between data for Model II and III, in both cases 〈x〉/ΔC_Q = 0.108 Å MHz⁻¹. Also shown in Fig. 2b are our NQR experimental data and the experimentally determined x, found with X-ray and/or neutron diffraction, for samples 1–7. The neutron diffraction data agree very well with the theoretical predictions for all samples. This is rather impressive, as no fitting parameters have been used. The situation for X-ray diffraction data is somewhat different however. For samples 1 and 3–7, the data agree well with the theoretical predictions. For sample 2, however, we observe a large discrepancy. The structure for this sample has been investigated by two separate X-ray studies, both finding a similar proton position, which is obviously different from the neutron diffraction one.^41,42 The authors attribute this discrepancy to a difficult refinement of the X-ray structure due to a large amount of noise caused by the large iodine atom.

The theoretical lines presented in Fig. 2b were also used to predict 〈x〉 for samples 8–10 with yet undetermined structures. These results are also shown in Table 1.

The ratio 〈x〉/ΔC_Q is practically equal for Models II and III, however, we find a different value for Model I, 〈x〉/ΔC_Q = 0.135 Å MHz⁻¹, which is ∼25% larger than for Models II and III. This points to the conclusion that functional groups not only determine C⁰_Q but also influence 〈x〉/ΔC_Q as well. This raises the question, how accurate is the predicted 〈x〉 in the DMAN-H⁺ case when using 〈x〉/ΔC_Q determined for Model III, as there is a mismatch in one functional group. We have not pursued an investigation in this direction, however, based on very similar NQR parameters for all tertiary amines, 4.5 < C_Q < 5.5 MHz and a generally small η, we are quite confident in claiming that the relation 〈x〉/ΔC_Q = 0.108 Å MHz⁻¹, as determined here, is a good approximation for DMAN complexes and also for all other tertiary amines forming similar N–H⋯N bonds. With similar arguments, we anticipate that 〈x〉/ΔC_Q for primary and secondary amines should be in the range of 0.108–0.135 Å MHz⁻¹.

In contrast to 〈x〉, 〈z〉 cannot be determined reliably with NQR measurements. First of all, the variations of 〈z〉 are very small between the salts chosen here (see Table 1), and second, the influence of 〈z〉 on C_Q is also small, in fact, smaller than the typical NQR experimental error.

Discussion

We expected a linear relation between 〈x〉 and ΔC_Q but across a much smaller range, e.g. for ΔC_Q/C⁰_Q < 0.1, which is a typical value for 1st order perturbation theory. Instead, we found it to be accurate across a substantially larger range, up to ΔC_Q/C⁰_Q ≲ 0.7. Intrigued by this result and in order to assess whether this is just a coincidence, we analysed our models in some more depth.

The instantaneous EFG tensor for linear models (Models I and II) can be expanded in a series with respect to the proton coordinate x. For V_xx, which is here the only independent component, we can write

V_xx(x) = V⁽⁰⁾_xx + V⁽¹⁾_xxx + V⁽²⁾_xxx² +…+ V⁽ⁿ⁾_xxxⁿ (1)

where in practice, truncating the series at n = 3 or 5 is sufficient. Then, 〈ΔV_xx〉, i.e. the difference between 〈V_xx〉 for the left and right nitrogen, which is proportional to ΔC_Q, can be written as

〈ΔV_xx〉 = 2V⁽¹⁾_xx〈x〉 + 2V⁽³⁾_xx〈x³〉 + 2V⁽⁵⁾_xx〈x⁵〉 +… (2)

because of symmetry. The right hand side of eqn (2) is still not linear in 〈x〉, because 〈xⁿ〉 are not trivially related to 〈x〉 for an arbitrary potential U.

Next, we analysed the 〈xⁿ〉 interdependence for some potentials relevant to this study, the asymmetric double well potentials of the form

with asymmetry parameter ΔU, barrier height U₀ and a fixed x₀ = 0.2 Å. The results when ΔU is increasing from 0 are shown in Fig. 3 for four representative cases that are labeled by a dimensionless parameter E₀/U₀, where E₀ is the ground state energy for a symmetric potential (ΔU = 0). In all cases, all 〈xⁿ〉 increase with ΔU, but these relations (not shown) are not linear. The comparison between the corresponding 〈xⁿ〉 is however interesting.

Fig. 3 〈xⁿ〉 as a function of 〈x〉 for various asymmetric double well potentials as the assymetry (ΔU) gradually increases. The solid blue curves represent the proton potential, whereas the red curves represent the proton density.

For E₀/U₀ = 0.25, a potential with a relatively high barrier, we find a direct proportionality between 〈x³〉, 〈x⁵〉 and 〈x〉, which is accurate for 〈x〉 up to ∼x₀. In fact, all 〈xⁿ〉 with odd n are linearly proportional to 〈x〉, although the breakdown for larger n occurs at smaller 〈x〉. For shallower potentials, i.e. with larger E₀/U₀, the linear relation is still a good approximation at small 〈x〉, the breakdown, however, occurs at progressively smaller 〈x〉. But, even for E₀/U₀ = 2, the shallowest potential considered, the linearity between 〈x³〉 and 〈x〉 is accurate almost up to 〈x〉 ∼ x₀.

The behavior described in Fig. 3 is not limited to the potentials used here (eqn (3)), but rather to all asymmetric potentials whose symmetric part has an energy level structure suitable for a two level approximation. Let us consider an arbitrary symmetric potential, with energy levels E₀, E₁, E₂…, where E₁ − E₀ ≪ E₂ − E₁. For this potential, and in general for symmetric potentials, the ground state wavefunction, ψ₊, is symmetric, whereas the wavefunction for the first excited state, ψ₋, is antisymmetric. Adding an antisymmetric term to this potential (e.g. of size ΔU) mixes the ground state wavefunctions. For ΔU < E₂ − E₀, but not necessarily ΔU < E₁ − E₀, we can use 1st order degenerate perturbation theory to find the new ground state wavefunction ψ₀′ = cos(½ϕ)ψ₊ + sin(½ϕ)ψ₋, where ϕ is a mixing angle dependent on ΔU. The expectation values for ψ₀′, which we are interested in, have one very striking property, which is straightforward to derive using the symmetry of ψ_±. The expectation values for all antisymmetric operators are exactly proportional to sin(ϕ) and thus, proportional to each other. For example, 〈x〉 = sin (ϕ)〈ψ₊|x̂|ψ₋〉 while 〈x³〉 = sin (ϕ)〈ψ₊|x̂³|ψ₋〉. This proportionality is effective as long as ΔU < E₂ − E₀, which applies to all cases in Fig. 3 at not too large 〈x〉.

Thus, using the found linearities between 〈xⁿ〉 and 〈x〉, one can rewrite eqn (2) as

〈ΔV_xx〉 ≈ 2Ṽ⁽¹⁾_xx〈x〉 (4)

whereis now an effective parameter, which takes into account the NQEs. Note, eqn (5) depends only on the symmetric part of the potential and the corresponding V_xx expansion coefficients, both defined by the skeleton, but not on δU and thus the anion. In Fig. 2b, we show 〈x〉 versus ΔC_Q by using Ṽ⁽¹⁾_xx calculated for Model II with r_NN = 2.6 Å. The line perfectly coincides with the lines calculated without the recourse to the perturbation theory.

Some useful numerical parameters that we find for Model II with r_NN = 2.56 Å are: U₀ = 2.21 kcal mol⁻¹ and E₀ = 1.77 kcal mol⁻¹ so that E₀/U₀ = 0.80, whereas at the other end, for r_NN = 2.64 Å, we find: U₀ = 4.29 kcal mol⁻¹ and E₀ = 2.53 kcal mol⁻¹ so that E₀/U₀ = 0.56. Thus, the cases presented in Fig. 3 that are relevant to this study are primarily (b) and (c). At the moment, it is difficult to find appropriate barrier values for the DMAN-H⁺ ion, as there are no reliable measurements, whereas calculations yield different numbers, according to the theory level used.³⁶ Nevertheless, the 〈x〉/ΔC_Q ratio found here seems to be independent of barrier height, so that knowing which case in Fig. 3 represents the DMAN-H⁺ ion is rather irrelevant.

The above discussion is not directly applicable to Model III, i.e. to nonlinear N–H⋯N bonds, and understanding the 〈V_μν〉 dependence on 〈x〉 and 〈z〉 for Model III is less straightforward than it is for Models I and II. In Fig. 4a, we show the PES and the associated energy level structure for Model III with r_NN = 2.64 Å. Here, the presence of two low lying levels with nearly equal energies followed by a larger gap can be clearly seen and it justifies the application of the two level approximation. More complications are related to the EFG tensor. The instantaneous nitrogen EFG tensor now depends on two coordinates, x and z. The average values 〈V_μν〉 thus in principle depend on 〈x〉, 〈x²〉, 〈x³〉,…, 〈z〉, 〈z²〉, 〈z³〉…, as well as on mixed terms like 〈xz〉. However, upon inspecting V_μν for Model III in the region of interest, that is, in places with a non-negligible proton ground state density, we find that the major V_μν components are only weakly dependent on z. In Fig. 4b, we show the instantaneous ΔV_xx = V^L_xx − V^R_xx, whose average mainly determines ΔC_Q. Additionally, in Fig. 4b, we also show the proton ground state density and 〈x〉 for four different asymmetric potentials that are derived from the symmetric potential in Fig. 4a by adding an E_xx term. In these ΔV_xx plots, the weak z dependence in the region of interest can be clearly seen. Thus, it seems permissible to expand ΔV_xx only with respect to x even for Model III and afterwards use the same reasoning that led to eqn (4) for linear N–H⋯N models.

Fig. 4 (a) The proton PES for Model III and r_NN = 2.64 Å. (b) The spatial dependence of ΔV_xx = V^L_xx − V^R_xx (black solid lines) and the proton ground state density (red) for four values of 〈x〉 as a result of four E_x. The solid red line is a boundary of an area with 95% proton probability. The contours of ΔV_xx are labeled with the corresponding values of in MHz.

There are a few more points that need some attention. In our analysis, we assumed that the anion influences our models only by modifying the PES, through δU, here E_x, but not by affecting V_μν directly. This is an approximation that substantially reduces the computational time, as otherwise, we would have to make ab initio calculations for each instance of δU separately. To assess the plausibility of this approximation, we made an additional set of calculations, only for a few cases, by adding E_x to our ab initio calculations. The values change, but the change is small, on the order of a few percent. In another set of calculations, we added a point charge to our models instead of E_x, in an effort to more closely represent the anion. By moving the point charge around, we obtained different δU. Again, the values change, this time by a slightly larger amount, but not large enough to justify the extra computational time. This last set of calculations also offered the possibility to inspect the influence of the point charge (the anion) on the PES. We found that the uniform field approximation with E_x is acceptable in the region of interest, i.e. between the two minima. Nevertheless, the inclusion of any antisymmetric terms (with respect to x) in δU would not affect the dependencies presented in Fig. 3.

Our analysis completely neglects N–N vibrations. This simplification was necessary, as for our models, the PES attains a minimum at very large r_NN; for example, r_NN = 2.68 Å for Model I and r_NN = 2.88 Å for Model II. Nevertheless, we made an attempt to estimate the influence of N–N vibrations on 〈V_xx〉 and 〈x〉 by using a Gaussian wavefunction for these vibrations, with an average of r_NN = 2.6 Å and a root mean square of 0.1 Å. We find a practically negligible difference between these averaged values and the non-averaged values for r_NN = 2.6 Å. We should emphasize here that although it is tempting to directly average the lines presented in Fig. 2a and b, this is not the correct procedure. One must average quantities at the same δU, rather than the same 〈x〉.

Conclusions

We have theoretically and experimentally investigated the intramolecular N–H⋯N hydrogen bond in DMAN complexes from the ¹⁴N NQR perspective. The aim was to obtain quantitative information on proton displacement from the bond midpoint, which happens in the presence of various anions. Using ab initio calculations and vibrational analysis, we found that there is a very simple linear relation between the proton 〈x〉 and the difference between the quadrupole coupling constants of the two ¹⁴N nuclei (ΔC_Q). The ratio is 〈x〉/ΔC_Q = 0.108 Å MHz⁻¹ for tertiary amines and is rather insensitive to the exact geometry of the N–H⋯N bond, i.e. the N–N distance and the NHN angle. Unfortunately, ¹⁴N NQR spectroscopy is also insensitive to displacements in the direction perpendicular to N–N, so that the proton 〈z〉 cannot be determined. The theoretical predictions were also verified experimentally for several DMAN complexes, by comparing the X-ray and/or neutron diffraction proton positions with the predictions from the ¹⁴N NQR measurements performed here. The agreement is excellent, especially in view of the fact that no fitting parameters have been used. The relation found here is not limited to DMAN complexes, but can also be used in other systems with intramolecular N–H⋯N bonds if the potential for proton motion satisfies certain conditions.

Conflicts of interest

There are no conflicts to declare.

References

1. P. Müller, R. Herbst-Irmer, A. Spek, T. Schneider and M. Sawaya, Crystal Structure Refinement: A Crystallographer's Guide to Shelxl, Oxford University Press, New York, 2006, vol. 8, ch. 3.
2. A. Parkin, S. M. Harte, A. E. Goetab and C. C. Wilson, New J. Chem., 2004, 28, 718.
3. M. Woinska, S. Grabowsky, P. M. Dominiak, K. Wozniak and D. Jayatilaka, Sci. Adv., 2016, 2, e1600192.
4. X.-Z. Li, B. Walker and A. Michaelides, PNAS, 2011, 108, 6369.
5. D. V. Moreno, S. A. González and A. Reyes, J. Chem. Phys., 2011, 134, 024115.
6. R. H. McKenzie, C. Bekker, B. Athokpam and S. G. Ramesh, J. Chem. Phys., 2014, 140, 174508.
7. D. M. Wilkins, D. E. Manolopoulos, S. Pipolo, D. Laage and J. T. Hynes, J. Phys. Chem. Lett., 2017, 8, 2602.
8. T. E. Markland and M. Ceriotti, Nat. Rev. Chem., 2018, 2, 0109.
9. E. Brunner and U. Stemberg, Prog. Nucl. Magn. Reson. Spectrosc., 1998, 32, 21–57.
10. R. K. Harris, P. Y. Ghi, R. B. Hammond, C.-Y. Ma and K. J. Roberts, Chem. Commun., 2003, 2834.
11. A. E. Aliev and K. D. M. Harris, Supramolecular Assembly via Hydrogen Bonds I, Structure and Bonding, Springer, Berlin, Heidelberg, 2004, vol. 108, pp. 1–53.
12. J. R. Yates, S. E. Dobbins, C. J. Pickard, F. Mauri, P. Y. Ghi and R. K. Harris, Phys. Chem. Chem. Phys., 2005, 7, 1402.
13. M. Baias, C. M. Widdifield, J.-N. Dumez, H. P. G. Thompson, T. G. Cooper, E. Salager, S. Bassil, R. S. Stein, A. Lesage, G. M. Day and L. Emsley, Phys. Chem. Chem. Phys., 2013, 15, 8069.
14. C. M. Widdifield, H. Robson and P. Hodgkinson, Chem. Commun., 2016, 52, 6685.
15. M. G. Siskos, M. I. Choudhary and I. P. Gerothanassis, Molecules, 2017, 22, 415.
16. T. Emmler, S. Gieschler, H. H. Limbach and G. Buntkowsky, J. Mol. Struct., 2004, 700, 29–38.
17. H. Benedict, H.-H. Limbach, M. Wehlan, W.-P. Fehlhammer, N. S. Golubev and R. Janoschek, J. Am. Chem. Soc., 1998, 120, 2939–2950.
18. A. Brinkmann and A. P. M. Kentgens, J. Am. Chem. Soc., 2006, 128, 14758–14759.
19. Z. J. Li, Y. Abramov, J. Bordner, J. Leonard, A. Medek and A. V. Trask, J. Am. Chem. Soc., 2006, 128, 8199–8210.
20. J. Seliger and V. Žagar, J. Phys. Chem. A, 2011, 115, 11652.
21. J. Seliger and V. Žagar, Phys. Chem. Chem. Phys., 2014, 16, 18141–18147.
22. J. Seliger and V. Žagar, J. Phys. Chem. C, 2013, 117, 20193–20200.
23. H. A. Staab and T. Saupe, Angew. Chem., Int. Ed. Engl., 1988, 27, 865.
24. A. Llamas-Saiz, C. Foces-Foces and J. Elguero, J. Mol. Struct., 1994, 328, 297.
25. P. R. Mallinson, G. T. Smith, C. C. Wilson, E. Grech and K. Wozniak, J. Am. Chem. Soc., 2003, 125, 4259.
26. A. Parkin, K. Wozniak and C. C. Wilson, Cryst. Growth Des., 2007, 7, 1393.
27. A. O. F. Jones, A. A. Kallay, H. Lloyd, G. J. McIntyre, C. C. Wilson and L. H. Thomas, Cryst. Growth Des., 2016, 16, 2123.
28. L. H. Thomas, A. O. F. Jones, A. A. Kallay, G. J. McIntyre and C. C. Wilson, Cryst. Growth Des., 2016, 16, 2112.
29. C. Lopez, R. M. Claramunt, A. L. Llamas-Saiz, C. Foces-Foces, J. Elguero, I. Sobrados, F. A. Parilla and H.-H. Limbach, New J. Chem., 1996, 20, 523.
30. C. Lopez, P. Lorente, R. M. Claramunt, J. Marin, C. Foces-Foces, A. L. Llamas-Saiz, J. Elguero and H.-H. Limbach, Ber. Bunsen-Ges. Phys. Chem., 1998, 102, 414.
31. R. Anulewicz, I. Wawer, T. M. Krygowski, F. Männle and H.-H. Limbach, J. Am. Chem. Soc., 1997, 119, 12223.
32. C. L. Perrin and J. B. Nielson, Annu. Rev. Phys. Chem., 1997, 48, 511.
33. S. Ikuta, J. Chem. Phys., 1987, 87, 1900.
34. J. A. Platts and K. E. Laiding, J. Phys. Chem., 1995, 99, 6497.
35. T. Asada, H. Haraguchi and K. Kitaura, J. Phys. Chem. A, 2001, 105, 7423.
36. Y. Horbatenko and S. F. Vyboishchikov, ChemPhysChem, 2011, 12, 1118.
37. A. A. Granovsky, Firefly Version 8.1, available at http://classic.chem.msu.su/gran/firefly/index.html.
38. M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. Su, T. L. Windus, M. Dupuis and J. A. Montgomery, J. Comput. Chem., 1993, 14, 1347.
39. L. K. Saunders, H. Nowell, H. C. E. Spencer, L. E. Hatcher, H. J. Shepherd, L. H. Thomas, C. L. Jones, S. J. Teat, P. R. Raithby and C. C. Wilson, CrystEngComm, 2018, 20, 3074.
40. D. Stephenson and J. A. S. Smith, Proc. R. Soc. London, Ser. A, 1988, 416, 149.
41. A. A. Kallay, PhD thesis, University of Glasgow, 2011.
42. L. K. Saunders, PhD thesis, University of Bath, 2016.

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