Determining Hydrogen Atom Positions for Hydrogen Bonded

ARTICLE pubs.acs.org/crystal

Determining Hydrogen Atom Positions for Hydrogen Bonded Interactions: A Distance-Dependent Neutron-Normalized Method Matteo Lusi and Leonard J. Barbour* Department of Chemistry and Polymer Science, University of Stellenbosch, Stellenbosch 7600, South Africa

bS Supporting Information ABSTRACT: Crystal structures determined by neutron and X-ray diffraction were retrieved from the Cambridge Structural Database (CSD), and hydrogen-bond geometrical descriptors (distances and angles) have been pairwisely compared, confirming that the two techniques produce significantly different results for the determination of hydrogen atom positions. Inclusion of neutron-normalized data shows that normalization fails to correct for bond polarization when applied to H-bond interactions. Statistical analysis has been carried out, and an empirical method is suggested for calculating more accurate positions for the hydrogen atoms involved in hydrogen bond formation. The simplicity of the approach is promising for future implementation in common crystal structure analysis software packages. The results are presented with a view to opening a discussion on how to approach one of the main limitations of X-ray diffraction in the area of structural chemistry.

1. INTRODUCTION Despite being a powerful technique for structural chemistry, X-ray diffraction has some limitations when applied to the detection of light elements such as hydrogen. This is because X-rays are scattered by electrons, which are better approximated as centered on the nuclei of heavier atoms. Lighter elements (with lower electron density) are therefore not as “visible” to X-rays, and the above approximation becomes less accurate with regard to locating their nuclei. This is particularly true for hydrogen atoms, which are extremely important to supramolecular chemists and crystal engineers and which must be added on the basis of chemical knowledge. Whenever the location of hydrogen atoms is critical, neutron radiation, which is scattered by the atomic nuclei, is considerably more accurate and would be the method of choice. However, neutron radiation requires access to a suitable source that is only available at a limited number of highly specialized centers, and there is a further requirement for relatively large specimens, which could be difficult to prepare in some cases. Hydrogen atom coordinates can be freely refined against X-ray intensity data, as is generally the case for heavier elements, to positions that minimize the residual R factor, or they can be placed in geometrically determined positions at a given distance from their “parent” atoms (X) by using a so-called riding model.1 Indeed, most refinement software packages allow the user to fix the X H bond distance in the riding model to a value that minimizes the R factor, or to a “known” value determined by other techniques. The result of these approaches is that X-H distances generally appear to be shorter than they really are. Figure 1 shows a comparison between oxygen hydrogen distances measured by X-ray and neutron data, and highlights the systematic errors associated with incorrect location of hydrogen atom positions by means of X-ray diffraction. A generic CSD search for O H groups yielded 41412 and 196 hits for X-ray and neutron data, respectively. r 2011 American Chemical Society

Figure 1. Histogram of OH distances retrieved from (a) XD and (b) ND.

This observation, predicted in the 1960s,2 was verified in the 1970s3 and 1980s4 6 by comparing the covalent X H bond Received: August 19, 2011 Revised: October 13, 2011 Published: October 17, 2011 5515

dx.doi.org/10.1021/cg201087s | Cryst. Growth Des. 2011, 11, 5515–5521

Crystal Growth & Design

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distances determined by X-ray and neutron diffraction data (henceforth referred to as XD and ND). These results have recently been revised by Allen7 using the larger (and evergrowing) data sets available. As suggested by Jeffrey et al.8,9 and others,4,5,10 hydrogen atoms can be adjusted to more accurate positions by enforcing neutron normalized (NN) geometries, even though the overall structure residual (R factor) may increase. For qualitative analyses, the CSD offers the possibility of exporting data for which the X H bond lengths are automatically adjusted to NN values.11 However, this option would not correct for the placement of hydrogen atoms in the wrong direction, which may occur when they are subject to strong interactions. For example, a hydrogen atom involved in a hydrogen bond (i.e., a donor-H 3 3 3 acceptor interaction, or D H 3 3 3 A) can be strongly polarized by receiving electron density from the acceptor. In particular, its nucleus would shift closer to the acceptor atom, while its electron would move further away from it. In general, polarization need not be expected to simply occur along the covalent D H bond direction but also along the supramolecular H 3 3 3 A hydrogen bond. Its quantification is not straightforward and fixing the covalent bond distance to an arbitrarily determined average value might be incorrect. Such approaches may be useful for aesthetic reasons, but it should be remembered that these coordinates must be verified by other techniques (ND or computationally) in order to be utilized for describing structural phenomena or when calculating lattice energies and Hirschfeld surfaces. We have noticed an increasing tendency toward reporting D H and H 3 3 3 A distances, as well as D H 3 3 3 A angles (together with the more believable D 3 3 3 A distances), usually extracted from crystal information files (CIFs), without due consideration of the

possible pitfalls that affect the accuracy of these parameters. We believe that the misplaced trust (by both novices and experienced crystallographers) in flawed analyses is due to the

Figure 3. Histogram of the O H distances (in O H 3 3 3 O interactions) determined using (a) X-ray diffraction and (b) neutron diffraction.

Figure 2. The geometry of a typical hydrogen bond.

Table 1. The H-Bond Pairs Considered: Number of Structures Solved by XD at Room Temperature/ND between 0 and 303 K O H

N H

3 3 3O 3 3 3N

35846/133/237

29360/85/167

7043/5/23

4692/12/24

3 3 3 Cl

2726/18/18

4121/21/26

Figure 4. Plot and linear fitting of O 3 3 3 O distances (in O H 3 3 3 O interactions) determined from XD (ordinate) and ND (abscissa).

Table 2. Comparison of the D H 3 3 3 A and A 3 3 3 D H Angles Measured in Structures Solved from ND, XD, and Normalized XD X-ray DH

STD

mean value

DHA

neutron STD

160.06

ADH

STD

DH

STD

14.11

DHA

Gaussian fitting: a exp( 0.5((x 2

STD

164.40

X-ray normalized ADH

STD

10.66

DHA

STD

159.26

ADH

STD

14.11

x0)/b)2)

R a

0.92 56.14

3.53

0.94 64.04

3.89

0.95 73.53

4.12

0.95 83.62

5.05

0.99 90.76

2.71

0.99 101.97

2.82

0.94 61.04

3.55

0.95 73.53

4.12

b

0.11

0.01

10.40

1.13

7.52

0.84

0.02

0.002

8.17

0.46

5.95

0.38

11.73

1.32

7.52

0.84

x0

0.88

0.01

170.20

1.05

6.11

0.82

0.97

0.002

172.80

0.44

4.12

0.41

170.16

1.27

6.11

0.82

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dx.doi.org/10.1021/cg201087s |Cryst. Growth Des. 2011, 11, 5515–5521


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Figure 5. (a) The solid triangle is determined by neutron diffraction. The dashed triangle would be observed if D H was simply foreshortened along the D H vector (i.e., resulting in the same A 3 3 3 D H angle and a wider D H 3 3 3 A angle). (b) The dashed triangle is determined by XD (smaller D H 3 3 3 A and larger A 3 3 3 D H angles than are realistic); the dotted triangle results from NN of XD geometries (i.e., resulting in smaller D H 3 3 3 A as compared to non-normalized D H distances).

Figure 7. Histograms of A 3 3 3 D H angles measured by (a) XD, (b) ND.

important role,13 15 the H-bond descriptors (i.e., distances and angles) defined in Figure 2 provide a good indication of the strength of the interaction. For this reason the complete geometry of the D H 3 3 3 A triangle is often reported for each hydrogen bonding interaction. It is therefore desirable to formulate a more accurate approximation of the H-bond geometry using data obtained from X-ray diffraction, which is still by far the most commonly applied diffraction technique. The aim of the present study is to examine H-bond geometries elucidated by both XD and ND and to then suggest an accessible method for establishing a more accurate description of these interactions by taking into account the bond distances that are amenable to XD.

Figure 6. Histograms of D H 3 3 3 A angles measured by (a) XD, (b) ND, and (c) normalization of XD.

widespread use of software that automatically extracts such data from CIF files to produce seemingly credible tables that can be readily imported into structural reports. Hydrogen bonds are closely linked to crystal engineering.12 Even though it is clear that other factors must also play an

2. METHODOLOGY Typical H-bond donor acceptor pairs were considered in which D 3 3 3 A < (sum of VdW radii). For each type of hydrogen bond, crystal structures determined from XD and ND with R values < 0.10 were retrieved from the CSD (version 5.31 February 2010). Structures with disorder or errors were excluded, as well as those solved from powder samples or collected under nonambient pressures. When comparing XD and ND structures, the space group and unit cell dimensions were checked to ensure that the same polymorph was considered in each case. Multiple instances of the same structure were not excluded from the analysis. When more than one H-bond was present in a crystal structure, D 3 3 3 A distances were manually ordered to ensure that the same interaction was compared between XD and ND. Further subdivision of the data was made to distinguish between structures measured at different temperatures (as indicated in the discussion and summarized in Table 1). Student and Wilcoxon matched-pair signed-rank tests were used 5517



Figure 8. (a) D H distances vs D 3 3 3 A from ND (red), XD (orange), and NN XD (green); (b) polynomial fitting of the neutron data: secondorder (violet), third-order (orange), fourth-order (green).

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Figure 10. Plot of D H and H 3 3 3 A vs D 3 3 3 A distances (red and blue, respectively) from ND for D 3 3 3 A < 2.80 Å at (a) room temperature and (b) temperatures between 0 and 303 K. The polynomial fitting for the data at room temperature (orange) coincides with that for temperatures between 0 and 303 K (black).

to quantify the significance of the statistical findings. Both tests indicated a >0.9999 probability of systematic errors between XD and ND for each of the cases examined. CSD Vista, Microsoft Office 2007 Suite, and Sigma Plot 11.0 were used to analyze the results and to render the graphs.

3. RESULTS AND DISCUSSION

Figure 9. (a) A 3 3 3 H distances from ND (blue), XD (orange), and NN XD (green) vs D 3 3 3 A; (b) polynomial fitting of the neutron data: second-order (violet), third-order (orange), fourth-order (green).

3.1. Comparing O H 3 3 3 O Bonds. We extracted a total of 133 structures determined at room temperature using ND that have at least one O H 3 3 3 O interaction and R < 0.10. These include water molecules, charged H-bonds, and ionic species. The same query produced >35000 structures for XD. Of these, 128 are present in both lists and a total of 339 O H 3 3 3 O interactions were compared. The histogram of the O H distance shows a trend similar to that seen in Figure 1, and the Gaussian fits (Table 2) for XD and ND are centered on 0.87 and 0.97 Å, respectively (Figure 3a,b); O H neutron normalized values are set to 0.983 Å. As expected, there is excellent agreement between the D 3 3 3 A distances for the two data sets (Figure 4).6 By combining the observations described above, and assuming that the electron density of the hydrogen atom is centered along the D H vector, it follows that a related disagreement should also be observed for the H-bond angles. In particular, if we assume the O H direction to be accurate, it is reasonable to expect that the D H 3 3 3 A angle (commonly used as the H-bond angle descriptor) would increase with foreshortening of the O H distance, whereas the A 3 3 3 D H angle should be unaffected (Figure 5). Interestingly, Figure 6 shows that this is not 5518



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Figure 11. Plot of D H and H 3 3 3 A vs D 3 3 3 A distances (red and blue, respectively) from ND between 0 and 303 K for (a) O H 3 3 3 O interactions with D 3 3 3 A < 2.80 Å, (b) O H 3 3 3 N, (c) O H 3 3 3 Cl, (d) N H 3 3 3 O interactions with D 3 3 3 A < 2.80 Å, (e) N H 3 3 3 N, and (f) N H 3 3 3 Cl. The violet series in (d) represents the data-points that were excluded from the analysis.

Table 3. Suggested Formulae for Determining D H and H 3 3 3 A Distances As a Function of D 3 3 3 A (given as x) for a Selection of Commonly Observed Hydrogen Bonds16 O H 6.0990x3 + 49.8309x2

N H 20.5863x3 + 168.0642x2

3 3 3O

D H=

3 3 3N

58.6376x + 159.2668x 143.5384 H 3 3 3 A = 7.2512x D H = 3.5974x3 + 31.6627x2 92.9293x + 91.9034

H 3 3 3 A = 4.2718x2 + 24.4210x 33.1052 D H = 0.7770x2 4.7164x + 8.1749

H 3 3 3 A = 2.1699x3 19.7600x2 + 60.9661x D H = 0.1900x + 1.5595

H 3 3 3 A = 0.9805x2 + 6.9476x D H = 0.1532x + 1.5148

3 3 3 Cl

3

H3 3 3A =

135.8206x + 124.4715

D H=

2

2.1935x2 + 15.0415x

61.5666

23.3696

the case for the data under consideration: aside from the poor linear correlation (Figure S2, Supporting Information), D H 3 3 3 A angles measured in structures determined by XD are smaller (by about 4.5°, on average) than those measured in structures determined by ND (Table 2). Similarly, the apparent A 3 3 3 D H

H 3 3 3 A = 1.7342x

457.4145x + 416.0801

10.0157

3.3214

angles are slightly wider for XD (by about 3.5°), indicating that the electron density, as revealed by X-rays, is not simply closer to the donor atom but also farther away from the acceptor (Figure 7). This is indeed confirmed by NN data, for which the D H distance is fixed at 0.983 Å along the original D H 5519


Crystal Growth & Design direction: D H 3 3 3 A angles are 5.1° smaller (on average) than those observed by ND (Figures 6 and 7). Student and Wilcoxon matched-pair signed-rank tests for the conjecture that angles determined by XD (both with and without NN) are equal to those determined by ND indicate >0.9999 probability of a systematic error. Qualitative considerations suggest that this “pulling” effect, which can be measured from ND, is stronger for shorter D 3 3 3 A interactions and wider D H 3 3 3 A angles. On the contrary, X-rays “measure” shorter D H distances for shorter D 3 3 3 A interactions (Figure S4, Supporting Information). 3.2. Estimation of H-Bond Descriptors in O H 3 3 3 O Interactions. In order to correct for the systematic errors described above, it would be convenient to use an experimental procedure that takes into account how D H changes in relation to D 3 3 3 A. Figure 8 shows how D H and H 3 3 3 A are related to D 3 3 3 A. Neutron normalization is currently equivalent to a linear fitting of the data points with the simple equation D H = 0.983 and, as seen in the figure, the correlation (R2) is very low; higher order polynomials provide far better fits. The situation is similar when considering the relationship between D 3 3 3 A and H 3 3 3 A (Figure 9). The 339 interactions considered in this study can be fitted to second-order polynomials with R2 values of 0.68 for D H vs D 3 3 3 A and 0.80 for A 3 3 3 H vs D 3 3 3 A. Better fits are observed with third- and fourth-order polynomials and for shorter interactions (i.e., D 3 3 3 A < 2.80 Å); the inclusion of data from crystal structures collected at low temperature seems to have a small influence on the shape of the curve (Figure 10). With these relationships in hand, it is possible to estimate D-H and H 3 3 3 A from D 3 3 3 A distances, which can be measured accurately using XD. Given the above relationships, the knowledge of one distance allows one to calculate the values for D H and H 3 3 3 A, and it is even tempting to use the three interatomic distances to calculate the relevant angles by applying the cosine law: B2 = A2 + C2 2AC cos(β). However, using this formula, relatively small errors in the distances would yield relatively large errors in the calculated angles. D H 3 3 3 A angles calculated in this manner do not correlate better with the ND than those for XD and the correlation is even worse for A 3 3 3 D H angles (Figure S5, Supporting Information). 3.3. Other Common Interactions. Using the approach described above, H-bond descriptors can also be calculated for other donor acceptor pairs. The most prominent H-bond interactions are presented in Figure 11, and the respective equations are given in Table 3. It is interesting to note that N H 3 3 3 O interactions appear to follow two different trends: for shorter contacts (D 3 3 3 A < 2.9 Å) the AH distances increase continuously with D 3 3 3 A, as expected. The same trend persists for some of the longer interactions as well, although a second distribution of points is observed with H 3 3 3 A values >2.1 Å. This behavior might suggest that, despite the D 3 3 3 A separation being less than the sum of the van der Waals radii, these interactions are not H-bonds and they were therefore excluded from the analysis.

4. CONCLUSIONS As previously observed, hydrogen atom coordinates determined by X-ray diffraction are systematically closer to the donor atoms in H-bond synthons. Neutron normalization of the D H distance does not take the covalent bond elongation into account owing to the proximity of the H-bond acceptor, and thus fails to

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yield sensible positions of hydrogen atoms involved in H-bonds. Comparison of structures determined by both X-ray and neutron radiation has revealed that the former “locates” the hydrogen atom closer to the H-bond donor than it should be and also further away from the H-bond acceptor than it should be. Therefore, X-ray diffraction analysis appears to “push” the hydrogen atom away from the acceptor. X-ray analysis tends to underestimate the D H distance and D H 3 3 3 A angle and overestimates the H 3 3 3 A distance and A 3 3 3 D H angle. These observations suggest that extreme caution should be observed when reporting information on H-bonded systems or retrieving such data from the literature. Interpolation of the H-bond descriptors, extracted from structures determined by neutron diffraction data, provides the geometrical relationships required to correct the positions of hydrogen atoms involved in the most common H-bonds. We have presented a relatively simple empirical assessment that can be used to calculate D H and H 3 3 3 A distances as function of the D 3 3 3 A distance for the six most prominent H-bond interactions. The relatively low-order polynomials considered here provide a compromise between computational overheads (for large numbers of data) and goodness of fit with the experimental data. This simple distance-dependent neutron normalization method can easily be implemented in software packages for structure refinement, database management, and packing analysis. The aim of this study is not to provide a final answer, but to stimulate the discussion. We hope that this will eventually lead to consensus regarding the subtle problem of determining (or generating) the positions of hydrogen bonding hydrogen atoms with X-ray radiation.

’ ASSOCIATED CONTENT

bS

Supporting Information. A series of graphs that plot (S1) the values of D H and A 3 3 3 H distances measured by XD and NN XD vs ND; (S2) the values of D H 3 3 3 A angles as measured by XD and NN XD vs those measured by ND; (S3) the values of A 3 3 3 D H angles as measured by XD vs those measured by ND; (S4) the values of D H distances vs D H 3 3 3 A and A 3 3 3 D H angles measured for XD and ND; (S5) the values of D H 3 3 3 A and A 3 3 3 D H angles calculated using the cosine law vs those measured by ND. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*Tel: +27-21-808-3335. E-mail: [email protected].

’ ACKNOWLEDGMENT The authors thank the National Research Foundation (South Africa) and the Claude Leon Foundation for financial support of this work. ’ REFERENCES (1) Crystal Structure Refinement A Crystallographer’s Guide to SHELXL; M€uller, P., Ed.; Oxford University Press: Oxford, 2006. (2) Stewart, R. F.; Davidson, E. R.; Simpson, W. T. J. Chem. Phys. 1965, 42, 3175. (3) Hanson, J. C.; Sieker, L. C.; Jensen, L. H. Acta Crystallogr. Sect. B 1973, 29, 797. 5520



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(4) Taylor, R.; Kennard, O. Acc. Chem. Res. 1984, 17, 320. (5) Taylor, R.; Kennard, O.; Versichel, W. J. Am. Chem. Soc. 1983, 105, 5761. (6) Allen, F. Acta Crystallogr. Sect. B 1986, 42, 515. (7) Allen, F. H.; Bruno, I. J. Acta Crystallogr. Sect. B 2010, 66, 380. (8) Jeffrey, G. A.; Lewis, L. Carbohydr. Res. 1978, 60, 179. (9) Jeffrey, G. A.; Takagi, S. Acc. Chem. Res. 1978, 11, 264. (10) Steiner, T. Crystallogr. Rev. 2003, 9, 177. (11) See Conquest 1.12 user guide and tutorials 2010. (12) Desiraju, G. R. Angew. Chem., Int. Ed. 2007, 46, 8342. (13) Steiner, T. Chem. Commun. 1999, 313. (14) Braga, D.; D’Oria, E.; Grepioni, F.; Mota, F.; Novoa, J. J.; Rovira, C. Chem.—Eur. J. 2002, 8, 1173. (15) D’Oria, E.; Novoa, J. J. CrystEngComm 2008, 10, 423. (16) An Excel spreadsheet can be downloaded at http://academic. sun.ac.za/barbour/Software3.html.

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Determining Hydrogen Atom Positions for Hydrogen Bonded

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