Article pubs.acs.org/crystal
Accurate Hydrogen Positions in Organic Crystals: Assessing a Quantum-Chemical Aide Volker L. Deringer, Veronika Hoepfner, and Richard Dronskowski* Institute of Inorganic Chemistry, RWTH Aachen University, Landoltweg 1, D-52056 Aachen, Germany S Supporting Information *
ABSTRACT: Organic molecules crystallize in manifold structures. The last few decades have seen the rise of high-resolution X-ray diffraction techniques that make the structures of even the most complex crystals easily accessible. Still, an intrinsic challenge lies in assigning hydrogen atoms’ positions from X-ray experiments alone. Quantum chemistry plays a fruitful, complementary role here, and so ab initio optimization techniques for organic crystals are on the rise as well. In this context, we review and evaluate a popular ab initio strategy based on plane-wave density-functional computations, namely, selectively relaxing H positions in an otherwise fixed cell. Our data show that such-optimized C−H, N−H, O−H, and B−H bond lengths coincide well with results from neutron diffractionthe experimental technique that sets the “gold standard” for H positions in molecular crystals but which is far less easily available. We have thus justified the use of a quantum-chemical aide with a broad variety of possible applications.
■
INTRODUCTION Understanding structure is a chemist’s first and most crucial step toward understanding reactivity.1 In more recent years, organic chemists ask for not only the structure of molecules they make but also for those of three-dimensionally extended organic crystals. In the latter case, the well-known “molecular” repertoire2NMR, IR, and rotational spectroscopy, together with mass-spectrometric measurementsneeds a helping crystallographic hand, despite its enormous power otherwise. Thus, one looks for tools from solid-state chemistry and crystallography such as X-ray or neutron diffraction of single crystals or, sometimes, even powder samples.3 Such methods led to the successful structural elucidation of DNA,4 biomolecules,5 and proteins,6 and are nowadays of paramount importance in characterizing (solid) organic compounds, some of which are yet to be made. Of the two techniques mentioned, X-ray diffraction has emerged as the most widespread and popular tool, and for good reasons.7 The effort of time and money is reasonable, and today a professional-grade CCD X-ray diffractometer together with powerful crystallographic computer programs can be found in most laboratories. Hydrogen positions, however, are difficult to resolve from such measurements for multiple reasons,7 and, as a consequence, covalent bonds involving hydrogen atoms are rendered “too short” (as is general consensus, but we will see exceptions from this rule of thumb in this paper). Besides a chemist’s universal quest for the most accurate data possible, there are cases where exact knowledge of the hydrogen positions is especially vital, for example, for a comparison with the uprising solid-state NMR measurements,8 or when dealing with hydrogen bonding which is strongly dependent on the covalent and noncovalent D−H···A bond lengths.9 Diffraction of X-rays alone cannot do the job here. © 2011 American Chemical Society
In due course, several ingenious but nonetheless heuristic workarounds have been developed: to name but two, X−H bond lengths are increased by a fixed amount10 or “restrained” (that is, computationally somewhat normalized directly within the refinement cycles) to a “common” value.11 The latter values have been extracted from highly accurate (see below) neutron measurements and are, in general, surely more reliable than the original X-ray-determined ones. However, such “normalized lengths” are available only as mean values for specific functional groups (methyl−H, aryl−H, etc.) but not for the particular system at hand. In contrast to such empirical corrections, an ab initio quantum-chemical way out of the hydrogen dilemma is available, namely, a plane-wave density-functional theory (DFT) based structural optimization which we will describe in a moment. Since the past decade, DFT is commonly applied to optimize hydrogen positions after a structure has been determined by X-ray diffractionnot only from single crystals12 but also from powder samples,13 which clearly indicates a variety of possible applications provided that the DFT optimization is successful. In some cases, one may even find hydrogen positions from scratch14 where they are downright unavailable from the experiment. However popular, this method has so far been used without quantitative assessment of its reliability, and where there has been discussion, it was thoughtful but qualitative (“to relax or not to relax?”) as by Yates et al. in 2005.13,15 We seek to provide this missing link in this paper and furthermore illustrate how an optimization based on plane-wave DFT is not only reliable but also straightforward in application. Received: November 15, 2011 Revised: December 15, 2011 Published: December 16, 2011 1014
dx.doi.org/10.1021/cg201505n | Cryst. Growth Des. 2012, 12, 1014−1021
Crystal Growth & Design
Article
How does one quantum-chemically “optimize” atomic positions? Following the Born−Oppenheimer approximation,16 the atomic cores’ motion is typically not included in Schrödinger’s stationary equation (Ĥ Ψ = EΨ), which would indeed be an arduous task. Instead, the Hellmann−Feynman theorem17 allows one to precisely calculate the forces F acting on the atoms in a crystal, starting either from the previously determined electronic wave function Ψ(r) alone or from the corresponding electronic density ρ(r) instead. Once those forces are known, the atoms may be shifted accordingly, and an electronicstructure calculation is performed for the new structure, leading, hopefully, to a better Ψ(r) (i.e., a lower energy). This cycle is repeated until the energy changes (or the residual forces) fall below a reasonable threshold as is sketched in Figure 1. While
are scattered not by the electronic density (like in the X-ray case) but by the nuclei; furthermore, the scattering cross section for neutrons is independent of the nucleus’ atomic number Z so that hydrogen atoms (with Z = 1) can still be found without problems (if one neglects incoherent scattering for the moment). Today, though, neutron measurements are only sparsely available due to their high cost; in addition, they require well-grown single crystals of enormous size up to 1 cm3which are rarely available, let alone for deuterated samples (which rule out incoherent scattering, but in turn forbid direct comparison because of the H/D mass difference). Nonetheless, if a theoretical method can stand up to this best experimental technique possible, then such a cross-check will put the calculations on a much more solid basis and further justify their use. Comparison of a large number of computationally optimized hydrogen positions to neutron data will thus be the topic of this paper’s final section.
■
RESULTS AND DISCUSSION A. Paracetamol, A Famous Test Case. Let us go in medias res and look at the analgesic drug paracetamol (N-acetylp-aminophenol) as an illustrative example. It appears in different polymorphs with different mechanical (i.e., tabletting) properties,19 and for reasons of brevity, we will only deal with the monoclinic “I” phase which is the stable one at ambient conditions.20,21 The X-ray-derived structure is shown in Figure 2a,
Figure 1. Schematic representation of the particular Hellmann− Feynman cycle employed here for quantum-chemical structure optimization of the urea crystal for which the inaccurate H positions have been iconized by question marks. Note how electronic-structure and force calculations alternate.
solid-state theorists commonly “relax”, so to speak, all atomic positions as well as the cell parameters, one may here opt to optimize the hydrogen atoms’ positions onlyheavier cores are thus held fixed at their experimentally determined coordinates, and we will investigate herein whether such an approach is justified. Taking on an experimental point of view once more, the aforementioned neutron diffraction measurements generally provide reliable hydrogen positions.3,18 This is because neutrons
Figure 2. (a) Crystal structure of paracetamol I in space group P21/a, viewed along the c axis. (b) Molecular structure, indicating bond lengths in the crystal as determined from X-ray diffraction (with standard deviations in parentheses; taken from ref 20) and quantumchemical optimization (below, in bold face).
and in the underlying study by Haisa et al.,20 hydrogen positions have been taken from refinement but otherwise unaltered. In particular, the authors applied none of the aforementioned 1015
dx.doi.org/10.1021/cg201505n | Cryst. Growth Des. 2012, 12, 1014−1021
Crystal Growth & Design
Article
“re-normalization” techniques so that paracetamol is a reasonable and “typical” test subject. Starting from the X-ray crystal structure, we optimized the hydrogen positions using a Hellmann−Feynman cycle as described above. Figure 2b compares the original C−H, N−H, and O−H bond lengths extracted from X-ray diffraction data20 with those obtained by our calculations. At one glance, paracetamol confirms the notion of “too short” X-ray bond lengths for hydrogen atoms; all of those bonds turn out longer after optimization, and the differences are between 0.02 Å and an enormous 0.22 Å. Also, paracetamol’s terminal −CH3 group nicely exemplifies a trend; namely, the three experimental Cmethyl−H bond lengths in the methyl group dif fer significantly and range from 0.88(5) to 1.07(5) Å. Computational optimization, much on the contrary, leads to almost interchangeable Cmethyl−H bond lengths around 1.10 Å which may well be regarded “equal” within reasonable error tolerances. This is in agreement with a more recent neutron study22 performed at 20 K which found Cmethyl−H distances in the range of 1.073(6)−1.090(4) Å, and it is also consistent with the expectation that chemically equal H atoms should not differ noticeably in their covalent bond lengthas long as they do not experience largely different environments. If one H of, say, an amino group takes part in a hydrogen bond while the other does not, their bond length will be different in experiment and (necessarily) in theory. Let us stay with the example of paracetamol for a moment to investigate some practical aspects. Hydrogens OnlyOr Not? Two important questions of methodology have already been raised in the introduction, the first one concerning the “relaxation” itself. As said before, such an optimization of atomic positions usually affects the entire cell, and it may be argued that this should be mandatory because the potential energy surface (PES) depends on all atomic coordinates, and they must all be considered to find the PES minimum, at least in theory. In contrast to thisand in accord with chemical reasoningrecent studies prefer to only relax hydrogen positions.12b We will stay with paracetamol to compare both approaches directly for the first time. Table 1
hold the three-dimensional structure together.20 For both hydrogen bonds, their H···acceptor distances are overestimated in the X-ray study as a direct consequence of the too short covalent donor−H lengths, of course. Uncritically adopting those bond lengths would introduce a significant error if one aimed to study the hydrogen bonding network in detail.9c Turning now to the computational techniques, the resulting X−H bond lengths as well as the hydrogen bond geometries arrive at practically the same value. This justifies the use of “selective” optimization which bears two advantages: First, we stick closer to the experimental datakeep in mind that a “full” relaxation would also change the heavier atoms’ positions, and it is not the goal of such an optimization to replace but to complement the measurement because the latter relates more closely to observable quantities. Second, the computational cost is most significantly reduced: In the exemplary case of paracetamol, a full structural optimization required 44 h of computational time on a given machine, whereas an optimization of only the hydrogen atoms took less than 11 ha gain in efficiency of more than 300% without accuracy losses.23 B. What is PossibleAnd What Not? Hydrogen Positions “From Scratch”. Often, hydrogen positions are not even known approximately because, for example, they cannot be determined in the vicinity of heavy metal atoms.14 An important question is thus how well computational optimization can find hydrogen positions “from scratch”. To resolve this question, we performed a computer experiment: starting from the X-ray-determined structure of paracetamol, we displaced the hydrogen atoms randomly, changing their covalent bond length individually by an amount of Δd, and the angle by Δα. Then we optimized the so-obtained artificial crystal structures and checked if the correct position of that H atom had been recovered in the process. In this context, we consider a hydrogen position “recovered” if it is no more than 0.01 Å away from the original optimized position. Figure 3 illustrates the definitions
Table 1. Bond Lengths in Å and Selected Angles in the Paracetamol Structurea d(Cmethyl−H)
d(Carom−H)
d(N−H) d(NH···O) ∠(NH···O) d(O−H) d(OH···O) ∠(OH···O)
X-ray diffraction
full relaxation
selective relaxation
0.88(5) 0.95(5) 1.07(5) 0.93(4) 0.95(3) 0.96(4) 0.98(3) 0.90(3) 2.05(3) 165(4)° 0.89(3) 1.80(4) 165(4)°
1.097 1.097 1.099 1.086 1.089 1.092 1.091 1.030 1.914 166.7° 1.007 1.645 165.6°
1.097 1.100 1.098 1.086 1.089 1.092 1.091 1.029 1.926 165.5° 1.004 1.680 165.2°
Figure 3. “Random” positioning of hydrogen atoms: Results for various random displacements of hydrogen atoms in paracetamol. Blue squares indicate that the atom’s position has been recovered by subsequent optimization (i.e., it deviates from the “ideal” optimized structure by less than 0.01 Å). Red squares indicate that the position could not be recovered.
of Δd and Δα together with the results of our computer experiment. From these, we learn that hydrogen positions are correctly recovered (indicated by blue squares) even if the bond is artificially shortened by up to 0.3 Å, lengthened by up to 0.9 Å, and bent away from its original direction by up to 45°. In other words, the “error tolerance” of the optimization technique is far beyond the limits of chemical reasoning, and any educated guess by a chemist leads to a correct result in the subsequent
a
Values taken from X-ray diffraction measurements (ref 20) with standard deviation in parentheses and obtained quantum-chemically by relaxation of all (“full”) and hydrogen atoms only (“selective”). For hydrogen bonds, the distance “···” between hydrogen and acceptor atom is given.
lists results obtained from both optimization techniques, investigating the bond lengths already discussed and, in addition, the two types of hydrogen bonds (N−H···O and O−H···O) which 1016
dx.doi.org/10.1021/cg201505n | Cryst. Growth Des. 2012, 12, 1014−1021
Crystal Growth & Design
Article
Table 2. Bond Lengths d (Å) and Selected Angles in the Paracetamol Structure, Starting from Experimental Reports That Utilize Various Methods and Temperatures (CSD Reference Codes Are Given)a X-ray diffraction 150 K HXACAN0421 d(C
−H)
methyl
d(Carom−H)
d(N−H) d(NH···O)
