Alan
Gregorovič
*^{a},
Tomaž
Apih
^{a},
Veselko
Žagar
^{a} and
Janez
Seliger
^{ab}
^{a}Institute “Jožef Stefan”, Jamova 39, 1000 Ljubljana, Slovenia. E-mail: alan.gregorovic@ijs.si
^{b}Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia

Received
14th August 2018
, Accepted 25th November 2018

First published on 27th November 2018

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 ^{14}N 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.

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 ^{1}H NMR spectra,^{16} substitution of the hydrogen bond proton with ^{2}H modifies bond geometry, and ^{17}O and ^{15}N have a very low natural abundance,^{17–19} while ^{14}N has broad spectral lines, due to its large nuclear quadrupole moment. Quadrupole nuclei, like ^{2}H, ^{14}N and ^{17}O, 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 ^{13}C and sometimes ^{35}Cl, 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 ^{14}N 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 ^{14}N 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 ^{14}N 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 ^{14}N 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}

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 ^{14}N 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.

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}

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.

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 | ΔC_{Q} [kHz] |
||||||

a Structure in ref. 28. b Structure in ref. 27. c Structure in ref. 39. | ||||||||||

1 | DMAN:benzoic acid (1:2) | X-ray/300 K^{a} |
2.594 | 0.021 | 0.214 | 3550 | 0.132 | 3533 | 0.113 | 17 |

2 | DMAN:2-iodobenzoic acid (1:2) | X-ray/300 K^{a} |
2.571 | 0.235 | 0.218 | 3950 | 0.138 | 3180 | 0.126 | 770 |

Neutr./300 K^{a} |
2.575 | 0.079 | 0.267 | |||||||

3 | DMAN:4-iodobenzoic acid (1:2) | X-ray/200 K^{a} |
2.602 | 0.272 | 0.266 | 4673 | 0.107 | 2473 | 0.202 | 2200 |

Neutr./200 K^{b} |
2.602 | 0.236 | 0.279 | |||||||

4 | DMAN:2-fluorobenzoic acid (1:2) | X-ray/200 K^{a} |
2.589 | 0.039 | 0.228 | 3707 | 0.119 | 3380 | 0.124 | 327 |

Neutr./200 K^{b} |
2.595 | 0.016 | 0.273 | |||||||

5 | DMAN:4-hydroxybenzoic acid (1:2) | X-ray/100 K^{c} |
2.566 | 0.051 | 0.209 | 3740 | 0.102 | 3420 | 0.099 | 320 |

6 | DMAN:2,3-dihydroxybenzoic acid (1:1) | X-ray/200 K^{b} |
2.593 | 0.189 | 0.229 | 4213 | 0.095 | 2893 | 0.138 | 1320 |

Neutr./200 K^{b} |
2.599 | 0.160 | 0.274 | |||||||

7 | DMAN:chloranilic acid (1:1) | X-ray/100 K^{b} |
2.610 | 0.299 | 0.245 | 4693 | 0.077 | 2420 | 0.124 | 2273 |

Neutr./200 K^{b} |
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 |

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 ^{1}H Larmor frequency in a high field was 32 MHz.

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} ≡ C^{0}_{Q}. For all three models, C^{0}_{Q} is linearly dependent on r_{NN}, decreasing when r_{NN} increases. This behavior cannot be expected a priori as C^{0}_{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 C^{0}_{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 C^{0}_{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 C^{0}_{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 C^{0}_{Q} ≈ ½(C^{L}_{Q} + C^{R}_{Q}) from a measurement at any E_{x}, i.e. from any cation–anion complex. Determining C^{0}_{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 C^{0}_{Q}. A simple approach to compensate for an inexact C^{0}_{Q} is to introduce ΔC_{Q} = C^{L}_{Q} − C^{R}_{Q}, which is roughly C^{0}_{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 C^{0}_{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.

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)}_{xx}x + V^{(2)}_{xx}x^{2} +…+ V^{(n)}_{xx}x^{n} | (1) |

〈ΔV_{xx}〉 = 2V^{(1)}_{xx}〈x〉 + 2V^{(3)}_{xx}〈x^{3}〉 + 2V^{(5)}_{xx}〈x^{5}〉 +… | (2) |

Next, we analysed the 〈x^{n}〉 interdependence for some potentials relevant to this study, the asymmetric double well potentials of the form

(3) |

For E_{0}/U_{0} = 0.25, a potential with a relatively high barrier, we find a direct proportionality between 〈x^{3}〉, 〈x^{5}〉 and 〈x〉, which is accurate for 〈x〉 up to ∼x_{0}. In fact, all 〈x^{n}〉 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 E_{0}/U_{0}, the linear relation is still a good approximation at small 〈x〉, the breakdown, however, occurs at progressively smaller 〈x〉. But, even for E_{0}/U_{0} = 2, the shallowest potential considered, the linearity between 〈x^{3}〉 and 〈x〉 is accurate almost up to 〈x〉 ∼ x_{0}.

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 E_{0}, E_{1}, E_{2}…, where E_{1} − E_{0} ≪ E_{2} − E_{1}. 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 < E_{2} − E_{0}, but not necessarily ΔU < E_{1} − E_{0}, we can use 1st order degenerate perturbation theory to find the new ground state wavefunction ψ_{0}′ = cos(½ϕ)ψ_{+} + sin(½ϕ)ψ_{−}, where ϕ is a mixing angle dependent on ΔU. The expectation values for ψ_{0}′, 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_{+}||ψ_{−}〉 while 〈x^{3}〉 = sin(ϕ)〈ψ_{+}|^{3}|ψ_{−}〉. This proportionality is effective as long as ΔU < E_{2} − E_{0}, which applies to all cases in Fig. 3 at not too large 〈x〉.

Thus, using the found linearities between 〈x^{n}〉 and 〈x〉, one can rewrite eqn (2) as

〈ΔV_{xx}〉 ≈ 2Ṽ^{(1)}_{xx}〈x〉 | (4) |

(5) |

Some useful numerical parameters that we find for Model II with r_{NN} = 2.56 Å are: U_{0} = 2.21 kcal mol^{−1} and E_{0} = 1.77 kcal mol^{−1} so that E_{0}/U_{0} = 0.80, whereas at the other end, for r_{NN} = 2.64 Å, we find: U_{0} = 4.29 kcal mol^{−1} and E_{0} = 2.53 kcal mol^{−1} so that E_{0}/U_{0} = 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〉, 〈x^{2}〉, 〈x^{3}〉,…, 〈z〉, 〈z^{2}〉, 〈z^{3}〉…, 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_{x}x 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.

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〉.

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