Atomic and Molecular Motion in Crystals

1. Atomic and Molecular Motion in Crystals Thammarat Aree, Silvia Capelli, Marcel Förtsch, Jürg Hauser Hans-Beat Bürgi Department of Chemistry and Biochemistry, University of Berne, Organic Chemistry Institute, University of Zürich hans-beat.buergi@krist.unibe.ch

2. What are Atomic Displacement Parameters I • ADPs represent atomic motion and disorder in terms of • mean square displacements from mean atomic positions. • A typical value of an ADP for an organic molecule at room temperature is • ~0.04 Å2, corresponding to an • rms amplitude of 0.2 Å • - to be compared with the length of a C-C bond, 1.50(1) Å • An isotropic displacement parameter U (a scalar quantity) implies equal displacements in all directions of space.

3. What are ADPs II Displacement parameters of atoms in general positions are usually anisotropic implying that mean square displacements are direction dependent. Anisotropic ADPs take the form of a 2nd order tensor U. U11 U12 U13 U = U12 U22 U23 U13 U23 U33 The eigenvectors of U indicate the principal displacementdirections: largest, intermediate and smallest. These quantities depend on all elements Uij of U.

4. n How are ADPs represented graphically? Equiprobability surface (ellipsoid) shows rms amplitudes in principal direction. The probability to find the atom in the enclosed volume depends on ‘const’. (vTU-1v = const) Mean-square amplitude (difference-)surface <u2(n)> = nTU n ,<Δu2(n)> = nT ΔU n Rms amplitude (difference-)surface (PEANUT) <u2(n)>1/2 = (nTU n)1/2 (|n| = 1)

5. ADPs account for ~2/3 of the numerical results of crystal structure analyses Proper correction of interatomic distances Discrimination between motion and disorder Low-frequency, large-amplitudes modes of vibration (including eigenvectors) Specific heat curves, enthalpies and entropies Isotope effects Why bother about ADPs?

6. What is the physical origin of ADPs? Time average Atomic displacements arising from dynamic processes faster than seconds, e.g. molecular vibrations, conformational equilibria, etc. Space average Different atomic positions in different unit cells, i.e. positional disorder smaller than the resolution limit (ca. 0.5 Å)

7. Models of motion Atomic Einstein or mean-field model (harmonic and isotropic or anisotropic) Generalized Einstein or molecular mean-field model (harmonic or quasi-harmonic) Lattice-dynamical model (harmonic)

8. P(Δx) Δx 1-D harmonic oscillator (atomic Einstein model) Total probability density is Gaussian P(v) = (2π)-3/2 (detU)-1/2 exp(-vTU-1v/2)

9. 2 <Δx > . s T d 0 q /2 e E T(K) T(K) T(K) Harmonic oscillator only Harm. Osc. with T-indep. Contrib. Anharm. Osc. and T-indep. Contrib. Temperature dependence of ADPs <Δx2> = /(2ω) coth (ω/2kBT) + ε Δx2 = kBT/(2)+ ε high T Δx2 = ħ/(2)+ ε low T e

10. Lower Limit Upper Limit Distance correctionDiatomic O-H Fragment from Chondrodite d(O-H) = 0.94Å Bond Length Correction: Δd = <Δz2> /(2dobs) Δz = ΔzH – ΔzO W.R. Busing, H.A. Levy, Acta Cryst 17 (1964) 142 <Δz2O> <Δz2H> F <Δz2> = <Δz2O> + <Δz2H> – 2<ΔzOΔzH> Upper Limit <Δz2O> + <Δz2H> + 2{<Δz2O><Δz2H>}1/2 Independent Motion <Δz2O> + <Δz2H> H riding on O – <Δz2O> + <Δz2H> Lower Limit <Δz2O> + <Δz2H> – 2{<Δz2O><Δz2H>}1/2

11. Normal mode analyses based on • Frequencies • GFL = Lλλi = ωi2 • usually underdetermined, but isotopic substitution • Mean-square amplitudes • ΣG-1L = Lδδi= (ħ/2ωi) coth(ħωi/2kT) • usually underdetermined, but temperature dependence of Σ

12. < > < > é ù D D D z 2 ( T ) z z ( T ) = + - d w - e O O H m 1 / 2 V ( 1 / , T) V' m 1 / 2 x ê ú < > < > D D D z z ( T ) z 2 ( T ) ë û O H H Temperature dependence of ADPs ADPs Generalized Einstein Model ADPs, determined at several T’s libration and translation (ω,V) + disorder (ε), (~temperature independent) Correlation ADPs <ΔxO ΔxH (T) > from model δi = (ħ/2ωi) coth(ħωi/2kT); H.B. Bürgi, S.C. Capelli, Acta Cryst., A56 (2000) 403

13. Distance Correctionsd(O-H)corr=d(O-H)obs+[<Δz2O>+<Δz2H> – 2<ΔzOΔzH>]/[2d(O-H)obs] Upper limit Indep. Motion T-DEPENDANCE Riding motion Lower Limit Observed Average O-H distance 0.976 Å Vibration frequencies ┴ to O–H bond 888, 338 cm-1 Vibration frequencies ║ to O–H bond 3514, 263 cm-1 M. Kunz, G. A. Lager, H.-B. Bürgi, M. T. Fernandez-Diaz, Phys. Chem. Minerals33(2006) 17–27

14. Rigid-body and segmented rigid-body motions ADPs pertain to single atoms and say nothing about correlated displacements of different atoms D.W.J. Cruickshank, Acta Cryst. 9 (1956) 754 TL model of rigid body motion (>600 citations); Assumption: libration axes intersect in a point V. Schomaker, K.N. Trueblood, Acta Cryst. B24 (1968) 63 TLS model of rigid body motion with screw coupling (~2000 citations); Limitation: Trace(S) indeterminate; Assumption: Trace(S) = 0 V. Schomaker, K.N. Trueblood, Acta Cryst. B54 (1998) 507 TLSφ model, rigid body motion and internal rotation (φ); Limitations: Trace(S) and correlation between internal rotation and parallel libration are indeterminate; Assumption: Trace(S) = 0, <φl||> = 0

15. G.A. Jeffrey, J.R. Ruble, R.K. Mullan, J.A. Pople, Proc. R. Soc. London, A414 (1987) 47 O. Ermer, Angew. Chem., Int. Edit., 26 (1987) 782 Centrosymmetric super-position of two cyclo-hexatriene molecules? (1.35 and 1.45 Å) Concerning the structure of benzene 123 K Uiso(C) ~ 0.023 Å2 (shown: * 2.5) 15 K Uiso(C) ~ 0.008 Å2 (shown: * 2.5) Rms displacements U of C6D6 fromneutron diffraction

16. Temperature dependence of ADPs Vibrations of a molecule in its crystal field Σx(T) = A * g * V * δ(1/ω,T) * V’ * g’ * A’ + εx ADPs (blue) determine parameters of model (red) ADPs, determined experimentally at several temperatures Low frequency, soft vibrations (ω), e.g. librations, translations and deformations (V) Intramolecular, hard vibrationsand disorder (ε) (~temperature independent), H.B. Bürgi, S.C. Capelli, Acta Cryst., A56 (2000) 403

17. 15 K Uiso(C) ~ 0.008 Å2 (shown: *2.5) (ADPobs – ADPcalc)1/2 |av diff| ~ 0.0002 Å2 (shown: *2.5*5) εC ~ 0.0007 Å2zero-point motion or disorder? (shown: *2.5) Results for Benzene, C6D6 Zero point motion from neutron diffraction and a benchmark force field (*104Å2) C(bond) C(ip) C(oop) D(bond) D(ip) D(oop) Diffraction 14(1) 7(1) 15(1) 52(1) 83(1) 110(2) Force Field 13 8 16 44 89 133

18. Uobs – Ucalc(harmonic) Uobs – Ucalc(anharmonic) Anharmonic motion one quasi-harmonic translational and one quasi-harmonic vibrational frequency ωeff (T) = ω0 [1 - G V(T) / V0] Grüneisen constant G = 2.3, elastic n-diffraction G = 2.2 – 2.5, inelastic n-scattering Duckworth et al., Acta Cryst. A26 (1970) 263, Kampermann et al., Acta Cryst. A51 (1995) 489 Dolling et al., Proc. R.. Soc. Lond. A319 (1970) 209

19. CV , CP of hexamethylenetetramine Translation νD = 1.5-1.732 νE Libration Internal vibrations B3LYP/6-311+G(2d,p) Thermodynamics A0=0.0163 K mol cal-1 Tm: melting point Cp(T) - CV(T) = T χ2(T) V(T) / κ(T) Nernst-Lindemann approximation Cp(T) - CV(T) = 3 R A0 T CV(T)/Tm

20. CV , CP of hexamethylenetetramine Good agreement between calorimetric and diffraction results Possibility to measure compressibility κ(T) by diffraction

21. Some conclusions • Molecular Einstein model seems physically sound and practically useful for gauging the quality of ADPs • dynamical information obtained with moderate experimental investment compared to measuring phonon dispersion curves (but also less information) • probing atomic displacment patterns (eigenvectors) unlike spectroscopies which probe frequencies (eigenvalues) • Problem: optimal combination of experimental and calculated data

22. Summary • What are ADPs? • Measure ADPs as a function of temperature • Distance corrections without riding assumptions • Vibration frequencies and displacement patterns (generalized Einstein model) • Distinguish motion from disorder • Anharmonicity (frequencies decrease with increasing T) • Estimate specific heats (cV)