Acta Cryst. (2014). A70, 309–316 doi:10.1107/S2053273314010626 309 research papers Acta Crystallographica Section A Foundations and Advances ISSN 2053-2733 Received 6 March 2014 Accepted 9 May 2014 On the temperature dependence of H-Uiso in the riding hydrogen model Jens Lu¨bben,a Christian Volkmann,a Simon Grabowsky,b Alison Edwards,c Wolfgang Morgenroth,d Francesca P. A. Fabbiani,e George M. Sheldricka and Birger Dittrichf* aInstitut fu¨r Anorganische Chemie, Georg-August-Universita¨t, Tammannstrasse 4, D-37077 Go¨ttingen, Germany, bSchool of Chemistry and Biochemistry, Stirling Highway 35, WA-6009 Crawley, Australia, cBragg Institute, Australian Nuclear Science and Technology Organisation, Locked Bag 2001, Kirrawee DC, NSW 2232, Australia, dInstitut fu¨r Geowissenschaften, Abteilung Kristallographie, Goethe-Universita¨t, Altenho¨ferallee 1, 60438 Frankfurt am Main, Germany, eGZG, Abteilung Kristallographie, Georg-August Universita¨t, Goldschmidtstrasse 1, 37077 Go¨ttingen, Germany, and fInstitut fu¨r Anorganische und Angewandte Chemie, Martin-Luther-King-Platz 6, 20146 Hamburg, Germany. Correspondence e-mail: birger.dittrich@chemie.uni-hamburg.de The temperature dependence of H-Uiso in N-acetyl-l-4-hydroxyproline mono- hydrate is investigated. Imposing a constant temperature-independent multi- plier of 1.2 or 1.5 for the riding hydrogen model is found to be inaccurate, and severely underestimates H-Uiso below 100 K. Neutron diffraction data at temperatures of 9, 150, 200 and 250 K provide benchmark results for this study. X-ray diffraction data to high resolution, collected at temperatures of 9, 30, 50, 75, 100, 150, 200 and 250 K (synchrotron and home source), reproduce neutron results only when evaluated by aspherical-atom refinement models, since these take into account bonding and lone-pair electron density; both invariom and Hirshfeld-atom refinement models enable a more precise determination of the magnitude of H-atom displacements than independent-atom model refinements. Experimental efforts are complemented by computing displacement parameters following the TLS+ONIOM approach. A satisfactory agreement between all approaches is found. 1. Introduction The riding hydrogen model is widely used in refining small- molecule X-ray diffraction data. Three positional and one isotropic displacement parameter can be constrained to a ‘parent atom’ that the H atom is ‘riding’ on, improving the data-to-parameter ratio and ensuring a chemically meaningful geometry. Alternatively, a single isotropic displacement parameter per riding H atom can be included in the least- squares refinement model while still constraining hydrogen positional parameters. Predicted H-atom positions usually lead to comparable figures of merit to a free refinement of H-atom positional parameters. This holds even for high-quality X-ray data, extending far into reciprocal space, since the scattering contribution of hydrogen is small and limited in resolution. Therefore predicted positions, e.g. by SHELXL (Sheldrick, 2008), have also been used for ‘invariom’ (Dittrich et al., 2004) aspherical-atom refinements (Schu¨rmann et al., 2012; Pro¨pper et al., 2013). Such H-atom treatment, in combination with elongating X—H vectors to bond distances computed by quantum chemical optimizations of model compounds, provides structures of high quality from conventional diffraction data. As stated above, the riding hydrogen model can include constraints for isotropic hydrogen displacement parameters. Ratios of 1.2 and 1.5 of H-Uiso with respect to the Ueq of the parent atom are being used in most refinement programs today. These ratios had been empirically derived for use with room-temperature data. However, most of today’s data sets are collected at temperatures of 100 K or lower, making full use of reduced thermal motion, e.g. to reduce the bias of anisotropic displacement parameters on bond distances (Busing & Levy, 1964). We will show that the ratio of H-Uiso/ X-Ueq is temperature dependent, which indirectly follows from Bu¨rgi & Capelli (2000). Therefore constant H-Uiso multipliers are inaccurate; the simple remedy of using temperature-dependent multipliers is proposed herein. Taking into account the temperature dependence of riding hydrogen treatments of H-Uiso is a detail of increasing importance in X-ray diffraction, as experimental data quality is improving with modern detectors and X-ray sources. Taking the effect into account allows removal of a resolution- dependent systematic error that would otherwise only affect low-resolution data, which is where the hydrogen scattering contributes. While the effect of underestimating H-Uiso might seem unimportant when only looking at the R factor (which is practically unchanged), the effect can be frequently detected when aspherical scattering factors,1 which take into account bonding and lone-pair electron-density distribution, are used for least-squares refinement of positional and atomic displa- cement parameters (ADPs).2 For charge-density studies, where the aim is to adjust the scattering factor via multipole parameters to the X-ray data, the anisotropic description of atomic displacements should be used. Hydrogen ADPs are usually estimated in such studies. Munshi et al. (2008) have compared competing approaches for such estimates, and the SHADE (simple hydrogen anisotropic displacement esti- mator) server (Madsen, 2006) is the approach most frequently used for that purpose today. Since the focus of this work is the most frequently used isotropic treatment of hydrogen displacements, we will not discuss the anisotropic description here. 2. Experimental Single crystals of the compound N-acetyl-l-4-hydroxyproline monohydrate (NACH2O) were grown by slow evaporation of saturated solutions prepared in hot acetone. Crystals grow to sizes suitable for neutron diffraction. A series of multi- temperature X-ray diffraction data collections3 at 9, 30, 50 and 75 K on the same specimen with dimensions of 0.34  0.28  0.28 mm (0.5 mm pinhole) were collected at the DORIS beamline D3 at the HASYLAB/DESY synchrotron in Hamburg. The experimental setup consisted of an Oxford Diffraction open-flow helium gas-stream cooling device, a Huber type 512 four-circle diffractometer and a 165 mmMAR CCD detector. A wavelength of 0.5166 A˚ and a detector distance of 40.3 mm were chosen, allowing a high resolution of d ¼ 0:50 A˚ or sin = of 1.0 A˚1 to be reached with a single detector setting. The XDS program (Kabsch, 2010) was used for data integration and scaling. Standard deviations of the unit-cell parameters (Fig. 1) were obtained by calculating the variance of intermediate cells during integration. A detector correction (Johnas et al., 2006) was applied to properly correct for the effect of oblique incidence (Wu et al., 2002) on the measured intensities. An empirical absorption correction was not performed at this short wavelength; Friedel opposites were merged. The structural model, cell settings but not the atom notation of the original structure determination by Hospital et al. (1979) as given in the CIF file of the Cambridge Structural Database refcode NAHYPL were used as input. Preliminary least-squares refinements were initi- alized with this model and performed with the program SHELXL (Sheldrick, 2008). Data sets at 100, 150 and 200 and 250 K were collected on an Xcalibur S diffractometer equipped with an Mo K sealed tube. Here an analytical absorption correction was performed following the method of Clark & Reid (1995) as implemented in the program CRYSALIS RED (Oxford Diffraction Ltd, 2006) employed for data reduction; Friedel mates were not merged. A second specimen was used for these four higher temperatures. High-resolution data (sin =  1) were again measured with the exception of the data set at 250 K. Neutron diffraction data were collected at the OPAL reactor on the Koala beamline at ANSTO, the Australian Nuclear Science and Technology Organization, in Lucas Heights, Australia. Laue neutron data were collected for a single specimen 1.8  1.4  0.5 mm using an unmonochro- mated thermal neutron beam on KOALA (Edwards, 2011) at 9, 150, 200 and 250 K. Each data set comprises 16, 12, 12 and ten images (each exposure = 42 min) accumulated on the image plate and from which intensities were extracted using LaueG (Piltz, 2011) employing unit-cell dimensions from the corresponding X-ray determination. The CRYSTALS program (Betteridge et al., 2003) was used for the refinement of positions and ADPs for all atoms. An isotropic extinction parameter was required at 9 K due to good crystal quality and comparably large specimen size for the neutron data. CCDC 977814–977817 contain the supplementary crystallographic information for the neutron data. These files can be obtained free of charge from the Cambridge Crystallographic Data research papers 310 Jens Lu¨bben et al.  Temperature dependence of H-Uiso Acta Cryst. (2014). A70, 309–316 Figure 1 Temperature dependence of the lattice constants of the X-ray data of N- acetyl-l-hydroxyproline monohydrate. Unit-cell parameters and volume are normalized to the lowest data point at 9 K. E.s.d’s are also plotted (but may not be visible when small). Connecting lines are only guidelines for the eye. 1 This was only tested for scattering factors from the generalized invariom database (Dittrich et al., 2013), not for those from the UBDB2011 (Jarzembska & Dominiak, 2012), the ELMAM2 (Domagala et al., 2012) nor the SBFA (Hathwar et al., 2011) libraries. For modelling hydrogen scattering, theoretically derived databases have the advantage of higher precision, since experimental scattering factors for hydrogen can only be reliably determined to the dipolar level of the multipole expansion. 2 Since displacement parameters used in this work are either isotropic or anisotropic, we use the abbreviation ADPs, which was recommended to be used only for anisotropic displacement parameters (Trueblood et al., 1996), in a different manner here. 3 Post-analysis of the temperature and volume dependence of unit-cell parameters showed that the data point at 67 K (as indicated on the low-T device) was an outlier, probably due to inaccuracies caused by heating the cold stream of He gas to higher temperatures. We have corrected this temperature to 75 K, as derived from a plot of the increase of the unit-cell volume with temperature. Another reason for the deviating behaviour might be rotational disorder and this is discussed later on. Centre via http://www.ccdc.cam.ac.uk/data_request/cif. A depiction of the molecule with its atomic numbering scheme and anisotropic ADPs at 9 K from neutron diffraction is shown in Fig. 2. 3. Methods H-Uiso/X-Ueq ratios reported here were derived from four independent methods. Benchmark results for NACH2O were obtained from multi-temperature neutron diffraction. Values for the hydrogen ADPs from multi-temperature single-crystal X-ray diffraction evaluated with the independent-atom model (IAM) cannot reach the accuracy achievable by neutron diffraction. To improve the physical significance of ADPs and their accuracy from X-ray diffraction (Jelsch et al., 1998; Dittrich et al., 2008), we therefore performed aspherical-atom refinements [either Hirshfeld-atom (Jayatilaka & Dittrich, 2008) or invariom refinement (Dittrich et al., 2004), see below]. QM/MM and MO/MO quantum mechanical cluster calculations (for details of how to run such computations see Dittrich et al., 2012) yield normal modes within the ‘molecular Einstein approximation’. These were combined with a TLS fit in the TLS+ONIOM approach (Whitten & Spackman, 2006) and were subsequently converted to give anisotropic ADPs for H atoms. Such computations were performed to comple- ment the experimental results (see x3.3). 3.1. Aspherical-atom refinements Two types of aspherical-atom refinements were performed: in invariom refinement (Dittrich et al., 2005, 2006, 2013) the molecular electron-density distribution is reconstructed from Hansen/Coppens’ multipole-model parameter values (Hansen & Coppens, 1978) as tabulated in the generalized invariom database (Dittrich et al., 2013). In Hirshfeld-atom refinement (HAR) (Jayatilaka & Dittrich, 2008) the electron density of the asymmetric unit is obtained by a single-point energy calculation. Invariom refinements. For structure models based on invariom refinements a least-squares refinement of positions and displacement parameters was carried out using the program XDLSM of the XD2006 package (Volkov et al., 2006). The program INVARIOMTOOL (Hu¨bschle et al., 2007) was used to set up XD system files for that purpose. Refinement was against F2 with a SHELXL-type weighting scheme, and the R1 factor was calculated for all reflections with F> 4ðFÞ. Crystallographic details are given in Table 1. CCDC 990102–990109 contain the supplementary crystal- lographic data for the X-ray structures. CIF files including intensities are only provided for the invariom refinements, since the same intensities were also used for HAR. Scattering factors, their local atomic site symmetry and invariom names as well as the model compounds these were derived from are given in Table 2. H-atom positions were initially calculated with SHELXL. In invariom refinement the X—H bond distances were then elongated during initial scale- factor refinement to optimized bond distances of the respec- tive model compound for the invariom assigned to the H atom. This new H-atom position was then constrained to have the same shift as the parent X atom. Only Uiso values were freely refined. This procedure was followed because it is also feasible when conventional data of lower resolution than the data studied here are available. Moreover, idealized H-atom posi- tions provided better input for the MO/MO cluster compu- tations (see x3.3), since idealized positions facilitate reaching convergence. Ratios of hydrogen Uiso to Ueq of the parent atom were then averaged for H atoms sharing the same invariom name using the program APD-TOOLKIT.4 For direct comparison with HAR, free refinement of H atoms was also performed and the results obtained (not shown) are very similar. Hirshfeld-atom refinement. In Hirshfeld refinement the electron density from single-point energy calculations is used and partitioned into atomic contributions using Hirshfeld’s fuzzy boundary partitioning scheme (Hirshfeld, 1977). Fourier transform (Jayatilaka, 1994) then gives aspherical atomic scattering factors. Atomic positions and ADPs are adjusted to best fit the experimental data using these scattering factors. In an improved implementation of HAR in the quantum crys- tallography program TONTO (Jayatilaka & Grimwood, 2003), cycles of molecular electron-density calculations, aspherical- atom partitioning and least-squares refinement are now iter- ated to convergence in an automatic manner (Capelli et al., 2014). The Hartree–Fock method was used in combination with the basis set cc-pVTZ (Dunning, 1989). A supermolecule cluster approach was chosen to calculate a wavefunction for both molecules of the asymmetric unit for use in HAR (Woinska et al., 2014). The structural model used in HAR included individual positional parameters and isotropic ADPs for H atoms. Ratios of hydrogen Uiso and Ueq of the parent atom were again averaged for H atoms sharing the same invariom name. As expected and shown before for three urea Acta Cryst. (2014). A70, 309–316 Jens Lu¨bben et al.  Temperature dependence of H-Uiso 311 research papers Figure 2 ADPs of N-acetyl-l-hydroxyproline monohydrate from neutron diffrac- tion at T = 9 K. Ellipsoids at 50% probability (Burnett & Johnson, 1996). 4 This new segmented-body TLS refinement program and its functionality will be described in a forthcoming paper. derivatives (Checin´ska et al., 2013), both types of aspherical- atom models, the Hansen & Coppens multipole model and HAR, give similar figures of merit and anisotropic ADPs of the non-H atoms with experimental X-ray data. 3.2. Neutron diffraction As mentioned before, the H-Ueq/X-Ueq ratios from neutron diffraction provide benchmark values for this study. One of the advantages of neutron diffraction is that the scattering lengths of the elements that correspond to atomic scattering factors in X-ray diffraction are constant. Stewart (1976) demonstrated that Uiso and Ueq from single-crystal X-ray and neutron diffraction differ, and that Ueq will be in between the arithmetic and geometric mean of the diagonal elements of the mean-square displacement matrix. Since we are interested in the ratio of hydrogen Uiso and Ueq of the parent atom, conventional least-squares adjustment can nevertheless provide relative reliable experimental estimates of atomic motion at a particular temperature. Equivalent isotropic displacements H-Ueq 5 (orthorhombic system) were obtained both by geometric and by arithmetric averaging the diagonal elements of the matrix of the anisotropic displacements of H atoms (Fischer & Tillmanns, 1988), and both give the same ratios within the estimated uncertainty. In contrast to the deposited structural model, refinements were evaluated without using split-atom sites to model rotational disorder in the methyl group above 150 K. Structural models are given in the supporting information.6 research papers 312 Jens Lu¨bben et al.  Temperature dependence of H-Uiso Acta Cryst. (2014). A70, 309–316 Table 1 Crystal data of N-acetyl-l-4-hydroxyproline monohydrate from invariom refinements. GoF = goodness of fit; GoFW = goodness of fit (weighted). Crystal data Chemical formula C7H10NO4H2O Formula weight 191.18 Cell setting, space group Orthorhombic, P212121 Temperature (K) 9 30 50 75 100 150 200 250 a (A˚) 9.854 (3) 9.853 (4) 9.866 (7) 9.884 (6) 9.9026 (2) 9.9408 (2) 9.9748 (2) 10.0123 (2) b (A˚) 9.249 (3) 9.251 (5) 9.250 (7) 9.253 (6) 9.2485 (2) 9.2479 (2) 9.2492 (2) 9.2556 (2) c (A˚) 10.144 (2) 10.145 (2) 10.149 (6) 10.155 (3) 10.1662 (2) 10.1875 (2) 10.2103 (2) 10.2441 (2) V (A˚3) 924.5 (4) 924.7 (7) 926.2 (11) 928.7 (9) 931.06 (3) 936.55 (3) 941.99 (3) 949.32 (3) Z, F(000) 4, 408 Dx (Mg m 3) 1.374 1.373 1.371 1.367 1.364 1.356 1.348 1.338 Radiation type Synchrotron Synchrotron Synchrotron Synchrotron Mo K Mo K Mo K Mo K  (mm1) 0.070 0.061 0.061 0.061 0.116 0.116 0.115 0.114 Crystal form, colour Rectangular, colourless Rectangular, colourless Crystal size (mm) 0.34  0.28 0.28 0.54  0.27  0.14 Data collection Diffractometer Huber Type 512 Oxford Diffraction Xcalibur S Data-collection method ’ scans ! and ’ scans Absorption correction None Analytical Tmin, Tmax n/a n/a n/a n/a 0.959/0.986 0.954/0.987 0.960/0.989 0.956/0.989 No. of measured reflections 45747 25178 44258 45127 39746 30837 30309 17876 No. of independent reflections 8304 7775 7803 7809 10866 7826 7829 4420 No. of observed reflections 7885 7262 7372 7255 7744 5673 4860 3245 Criterion for observed reflections Fo > 4ðFoÞ Rint (%) 0.040 0.038 0.055 0.051 0.037 0.039 0.039 0.020 max ( ), sinð=Þmax 31.90, 1.022 31.88, 1.000 31.90, 1.000 31.87, 1.000 53.28, 1.132 53.32, 1.000 53.31, 1.069 36.25, 0.833 Invariom refinement Refinement on F2 R1 ½I> 2ðIÞ 0.026 0.028 0.026 0.028 0.031 0.029 0.029 0.025 No. of reflections 7885 7262 7372 7255 7744 5673 4860 3245 No. of parameters 131 H-atom treatment Invarioms: calculated H position, bond-length elongated, Uiso refined; HAR: all parameters adjusted Weighting scheme 1/2ðF2oÞ + [] (P = 13F 2 o + 2 3F 2 c ) [0.06P2+ 0.04P] [0.04P2+ 0.05P] [0.04P2+ 0.04P] [0.05P2+ 0.02P] [0.04P2+ 0.07P] [0.05P2+ 0.05P] [0.06P2+ 0.04P] [0.04P2+ 0.08P] GoF 1.76 1.44 1.48 2.02 2.81 2.96 2.12 3.84 GoFW 0.96 0.95 0.94 1.00 0.81 0.84 0.82 0.81 max, min (e A˚ 3) 0.36/0.25 0.32/0.22 0.27/0.21 0.30/0.25 0.36/0.21 0.25/0.16 0.20/0.17 0.16/0.12 5 We have also refined isotropic ADPs for H atoms using the neutron data instead of the anisotropic description. Results show the same trends within the standard deviations of our experiments, but since the figures of merit are worse we chose to use H-Ueq for neutron data and H-Uisofor the X-ray data. 6 Supporting information is available from the IUCr electronic archives (Reference: KX5033). 3.3. Theoretical computations A quantum mechanical cluster computation was performed to complement the experimental results. The computation was initiated using the experimental geometry from invariom refinement at the lowest temperature of 9 K with idealized hydrogen positions and elongated X—H distances. The method/basis set for optimizing these model compounds was B3LYP/D95++(3df,3pd). The utility program BAERLAUCH (Dittrich et al., 2012) was used to generate a cluster of 17 asymmetric units packed around a central unit. The water solvent molecule was optimized together with the main molecule. Preliminary QM/MM calculations [HF/6-31G(d,p): UFF] ensured that this cluster size leads to convergence and is suitable to reproduce experimental ADPs at low temperature. Calculations to obtain final results employed the MO/MO basis-set combination B3LYP/cc-pVTZ:B3LYP/3-21G. Only the central molecule was optimized, whereas the surrounding 16 asymmetric units were kept at fixed positions. Normal modes were calculated and transformed to Cartesian atomic displacements after optimization. On the basis of the discussion by Dunitz et al. (1988), the temperature dependence of atomic motion can be described in analogy to a Boltzmann-type distribution of the harmonic oscillator. Atomic motion at higher temperatures can there- fore be estimated by the formula given by Blessing (1995): hu2!i ¼ h- 2!m coth h- ! 2kBT   : ð1Þ Although the molecular Einstein approach underlying the MO/MO calculations is not able to take into account lattice vibrations with acceptable accuracy, such a cluster calculation can provide a H/parent-atom Uiso=Ueq ratio, which is however dominated by internal atomic motion. Estimates so derived predict a higher ratio than the experimentally observed ratios from neutron and X-ray diffraction and require a Ueq scale factor. To reach agreement between theory and experiment, and to take the temperature dependence into account, it was therefore necessary to go back to the TLS+ONIOM approach (Whitten & Spackman, 2006) and to include the experimental TLS contribution, treating the whole asymmetric unit as a rigid body. In this process the internal atomic relative displa- cement predicted by the MO/MO computation was subtracted from the experimental ADP data at a given temperature prior to the TLS fit (Schomaker & Trueblood, 1968). Both TLS fit and subtraction were performed by the program APD- TOOLKIT. A more sophisticated (and computationally more demanding) theoretical method based on periodic computa- tions of different-sized unit-cell assemblies was studied by Madsen et al. (2013); for reproducing temperature depen- dence the TLS+ONIOM approach was sufficient. 4. Results and discussion 4.1. Temperature dependence of Uiso/Ueq of riding hydrogen and parent atom from X-ray data Prior to further analysis, a way to distinguish H atoms and their chemical environment is required. One choice would be the well established SHELXL AFIX groups. However, this would not distinguish H atoms exhibiting a distinct vibrational behaviour in theoretical calculations, e.g. an OH group in ethyl alcohol and one in phenol. The invariom formalism (Dittrich et al., 2013) allows a finer distinction. Here H atoms that share the same invariom name are in the same covalent bonding environment and have the same number of next-nearest non- H neighbours, so it was used for classification throughout.7 Vibrational modes of individual invarioms (as derived from their model compounds) in other molecules will be investi- gated in a forthcoming study. We can now consider the ratio of hydrogen Uiso and parent- atom Ueq from X-ray diffraction at different temperatures. Initial observations with invariom refinements on d,l-serine (Dittrich et al., 2005) indicated a temperature dependence at very low temperatures. Subsequent tests using the IAM showed that the IAM does not provide the model precision required to obtain significant results (Thorn, 2012). Our first question was therefore whether the Uiso=Ueq ratios from aspherical-atom refinements on NACH2O are able to repro- duce the temperature dependence seen for d,l-serine. Fig. 3 shows such ratios for several hydrogen invarioms. Values were either obtained from invariom refinements with constrained riding hydrogen positions, but adjusted hydrogen Uiso (Fig. 3a), or by free least-squares refinement of positional and isotropic displacement parameters with HAR (Jayatilaka & Dittrich, 2008) (Fig. 3b). As one can see, the programs/ methods used, XD (Fig. 3a) (Volkov et al., 2006) and TONTO Acta Cryst. (2014). A70, 309–316 Jens Lu¨bben et al.  Temperature dependence of H-Uiso 313 research papers Table 2 Scattering factor assigned during invariom refinement with atom names, invariom names, local atomic site symmetry and model compounds they were derived from. Atom name Invariom name Local atomic site symmetry Model compound O1 O2c mm2 Formaldehyde O2 O1c1h mz Methanol O3 O1.5c[1.5n1c] mm2 Acetamide O4 O1c1h mz Methanol O5 O1h1h mm2 Water N1 N1.5c[1.5o1c]1c1c mz N,N-Dimethylacetamide C1 C2o1o1c mz Acetic acid C2 C1n1c1c1h mz 2-Aminopropane C3 C1c1c1h1h mm2 Propane C4 C1o1c1c1h mz 2-Propanol C5 C1n1c1h1h mz Ethylamine C6 C1.5o1.5n[1c1c]1c mz N,N-Dimethylacetamide C7 C1c1h1h1h 3m Ethane H1,2 H1o[1c] 6 Methanol H3 H1c[1n1c1c] 6 2-Aminopropane H4,5 H1c[1c1c1h] 6 Propane H6 H1c[1o1c1c] 6 2-Propanol H7,8 H1c[1n1c1h] 6 Ethylamine H9,10,11 H1c[1c1h1h] 6 Ethane H12,13 H1o[1h] 6 Water 7 It should be noted that differences due to hydrogen bonding are not taken into account in the invariom approach. However, such influences are an order of magnitude smaller than electron-density redistributions due to covalent bonding, which are not taken into account in the IAM model. (Jayatilaka & Grimwood, 2003) (Fig. 3b), give comparable results. Both refinements do indeed show the expected temperature dependence and even distinguish different hydrogen invarioms from X-ray data, although the standard deviation associated with each value is non-negligible (not shown for clarity).8 At very low temperatures the relative motion of H atoms relative to their parent atoms is clearly appreciably extended compared to higher temperatures. This temperature dependence can be understood by looking at the low- and high-temperature limits of equation (1) as well as the transition temperature between both limits. Such dependence can be understood by considering the evolution of ADPs of atoms of different mass with temperature (Bu¨rgi & Capelli, 2000). Division of the mass- and temperature- dependent functions f1ð!;T;m1Þ and f2ð!;T;m2Þ with masses m1