# Prediction of NMR parameters in the solid-state

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1 1J PO Prediction of NMR parameters in the solid-state Jonathan Yates Materials Modelling Laboratory, Oxford Materials 2J PSi 1

2 Nuclear Magnetic Resonance Applied Field Zero Field B Population differences! NMR signal is proportional to population difference For 1H in a 9.4 T field at 298 K 1 For H in 9.4T at 298K ω= γ B magnetogyric ratio fundamental nuclear constant! "E = 2.65 # J nupper/nlower = exp ( "E / kt) nupper/nlower = Sensitivity is an issue! Need Large γ and/or B Populations differences in NMR are very small sensitivity is alway

3 Common Isotopes Isotope Spin ϒ 10 7 T -1 rad s -1 Freq 9.4T Abundance % 1 H 1/ C 1/ Si 1/ P 1/

4 Common Isotopes Isotope Spin ϒ Freq 10 7 T -1 rad s T Abundance % 1 H 1/ C 1/ Si 1/ P 1/ O 5/ Ti 7/

5 Superconducting magnets Solid-State NMR Spectrometers 600 MHz 14.1 T ~ 800,000 Image courtesy Sharon Ashbrook (St Andrews) 400 MHz 9.4 T ~ 300,000

6 Magnetic Shielding Flurbiprofen non-steriodal anti-inflamatory δ iso = (ω ω ref) 10 6 ω ref chemical shift (ppm) δ iso = σ ref σ iso Each distinct C atom experiences a different magnetic field and resonates at a unique frequency. Measure the change wrt a standard (for 13 C this is liquid tetramethylsilane) magnetic shielding (from calc)

7 Induced Current To compute the chemical shifts we just need to calculate the current induced by the external magnetic field Biot-Savart B in (r) = 1 c d 3 r j(r ) r r r r 3. Obtain current within perturbation theory (linear response) O = O (0) + O (1) + O(B 2 ) Bin = -σ B0 note: σ is a rank 2 tensor

8 Planewaves and Pseudopotentials To simulate periodic systems planewaves are a convenient choice: however describing the tightly bound core states, and oscillatory part of the valence states close to the nucleus are prohibitively expensive. We must approximate... Frozen Core Approximation Core electrons taken from free atom fixed during calculation Pseudopotential Approximation Valence electrons experience weak effective potential in the core region Note: Typically these two approximations are used together. But this does not have to be the case. Some codes can employ ʻself-consistentʼ pseudopotentials which allow the core states to ʻrelaxʼ to their specific environment.

9 Atomic States Si

10 Approximations Overcoming the previous approximations Frozen Core Approximation Contribution of core electrons to shielding is not chemically sensitive 1s states in Carbon contribute ~200ppm in diamond, benzene, proteins ie core states contribute to shielding - but not shift. Pseudopotential Approximation Use PAW method to fix-up valence wavefunction in the core region

11 pseudo = T, Projector Augmented Waves = T all-electron T = 1 + R,n[ φ R,n φ R,n ] p R,n T φ R,n φ R,n - + pseudo = all-electron GIPAW Gauge-Including Projector Augmented Waves Modification of PAW by Pickard and Mauri for systems in an external magnetic field - plays a role similar to GIAO in quantum chemistry techniques

12 GIPAW vs Gaussian test on small molecules n.b. v. big Gaussian basis sets JRY, C. Pickard, F. Mauri PRB 76, (2007) Convergence of shielding with #planewaves GIPAW A theory for solid-state NMR with GIPAW using uncorrected valence GIPAW - USP (ppm) GIPAW - USP (ppm) (a) All Electron Shielding (ppm) (b) All Electron Shielding (ppm) H C F Si P H C F Si P

13 First-principles NMR Crystal Structure X-ray, Neutron diffraction Cambridge Structural Database C C H Calculate forces refine structure 311.0ppm Calculate chemical shifts 316.9ppm

14 Examples Steven Brown (Warwick) Maltose sugar used in brewing 13 C axis 1 H axis MAS-J-HMQC

15 Examples Maltose sugar used in brewing 13 C x xx x x x 1 H axis xx x x x MAS-J-HMQC x x - first principles

16 Examples 13 C axis 1 H axis Molecule to solid variation due to intermolecular interactions (weak hydrogen bonds) x - first principles J. Am. Chem. Soc (2005)

17 NMR Interactions Chemical Shift orbital currents Quadrupolar coupling (EFG) nuclei I>1/2 interact with local electric field gradients Function of charge density spin-spin coupling (eg J-coupling) splitting of resonance due to nucleus-nucleus interaction hard to observe in solids...

18 EFG Eigenvalues V αβ (r) = d 3 r Electric Field Gradients Function of the charge density - ie ground-state property. Also computed by all-electron codes such as Wien2k, Crystal charge density n(r) r r 3 V xx, V yy, V zz V zz > V yy > V xx [ δ αβ 3 (r α r α )(r β r β ) r r 2 ] Quadrupolar Coupling Asymmetry C Q = eqv zz h η Q = V xx V yy V zz Note: The quadrupolar moment, Q, is a nuclear property. Most recent values given in Year-2008 Standard Values of Nuclear Quadrupole Moments : P. Pyykkö, Mol. Phys. 106, (2008) But note Q appears as a simple scaling factor 17 O MAS Glutamic Acid. HCl

19 J-coupling Electron-mediated interaction of nuclear spins Nucleus A causes a magnetic field at nucleus B (and vice versa) A B No J-coupling with J-coupling J In solids J is rarely revealed in splitting of peaks in 1D spectra (anisotropic interactions broaden peaks) parallel anti-parallel Increasing use of techniques, eg those using spin-echo modulation, to measure J in solids.

20 J-coupling Orbital Dipole induces orbital motion of electrons (due to electron charge) This creates a localised magnetic field similar mechanism to NMR shielding Orbital Current Spin Dipole induces spin density (relative displacement of and electrons) This creates induced field via: Fermi-Contact (ʻsʼ like density at nucleus) Spin-dipolar (non-spherical distribution of spin) similar mechanism to EPR-hyperfine & Knight-shift Induced Spin density: C2H2 Total J = Orbital + Spin H -ve C C H +ve -ve

21 J-coupling Guanosine self-assembles into ribbons molecular electronics (FET) a 2h JN7a,N1b a 2h JN7b,N1a = 6.2±0.4 Hz (expt) 6.5 Hz (calc) 2h JN7a,N1b = 7.4±0.4 Hz (expt) 7.7 Hz (calc) J. Am. Chem. Soc. 130, (2008) b b 2h JN7b,N1a

22 Practical Details *.param file task : magres magres_task : shielding efg nmr chemical shift/shielding electric field gradient both Must use on-the-fly pseudopotentials Highly sensitive to geometry (optimise H X-ray positions) CONVERGE (basis cut-off & k-points)

23 *.castep file =========================================================== Chemical Shielding Tensor Nucleus Shielding tensor Species Ion Iso(ppm) Aniso(ppm) Asym H H H O O O O =========================================================== Anisotropy σ aniso = σ zz 1/2(σ xx σ yy ) Asymmetry η = 3(σ yy σ xx )/2σ aniso

24 *.magres file ============ Atom: O 1 ============ O 1 Coordinates A TOTAL Shielding Tensor O 1 Eigenvalue sigma_xx (ppm) O 1 Eigenvector sigma_xx O 1 Eigenvalue sigma_yy (ppm) O 1 Eigenvector sigma_yy O 1 Eigenvalue sigma_zz (ppm) O 1 Eigenvector sigma_zz O 1 Isotropic: (ppm) O 1 Anisotropy: (ppm) O 1 Asymmetry: Note: shielding tensor has a symmetric and an antisymmetric component. Typical NMR experiments are only sensitive to the symmetric part. Therefore we only diagonalise the symmetric part of the shielding tensor

25 Practical Things Some practical things to keep in mind Structure NMR is very sensitive to structure! XRD atom positions are often not sufficient for accurate simulation of NMR parameters Dynamics Atoms move - spectra are recorded at finite T. How to account for that in (static) calculations? Accuracy How good are our present (approximate) density functionals?

26 Structure Anomeric Carbon Crystallises as: 80% α-maltose 20% β-maltose Compute spectra for both structures separately and superimpose

27 Structure Anomeric Carbon After relaxing neighbouring atom O1ʼ which had large force / thermal ellipsoid

28 Dynamics MLF peptide 13 C rms error Static relaxed structure ab-initio MD classical MD 4.2ppm 3.3ppm 3.0ppm 2 pm 5 μm J. Am. Chem. Soc. 132, 3993 (2010) Ab-initio MD Captures large (20ppm) fluctuations on fs timescale Classical MD Samples rotomeric states half of sites have errors <1ppm - limitations of PBE

29 Magnetic Resonance File Format Aim Provide a complete archival format for a first principles calculation of NMR parameters Visualisation Ab-initio Abinit, CASTEP QE, Wien2k etc file-format Spin-simulation Database

30 Jmol viewer Open-source Crystal-viewer Multi-platform (Java) Can embed in website New developments display selected values on structure convert shielding to chemical shift switch isotopes visualization of tensors (D, J, σ, EFG) relative orientations (Euler angles) select spin system for export

31 Database Ab-initio calculations provide NMR parameters for all atoms in the system Papers often report only selected results Store full information as ab-initio nmr file-format in a database Allows later retrieval and processing of data Establish structural trends

32 Consultation Draft specification and examples Feedback and Comments to by 19th October 2012

33 Getting more information NMR Books Good Introduction Nuclear Magnetic Resonance (Oxford Chemistry Primers) P. J. Hore More advanced Spin Dynamics: Basics of Nuclear Magnetic Resonance Malcolm H. Levitt Solid state NMR Solid-State NMR: Basic Principles and Practice David Apperley, Robin Harris, Paul Hodgkinson Useful survey of applications Multinuclear Solid-State Nuclear Magnetic Resonance of Inorganic Materials Kenneth J.D. MacKenzie, M.E. Smith Recent Review Articles Recent advances in solid-state NMR spectroscopy of spin I = 1/2 nuclei Anne Lesage, Phys. Chem. Chem. Phys., 2009, 11, 6876 Recent advances in solid-state NMR spectroscopy of quadrupolar nuclei Sharon E. Ashbrook, Phys. Chem. Chem. Phys., 2009, 11, 6892

34 Getting more information GIPAW Theory A more in depth introduction to the theory ( JRY for a copy) Computations of Magnetic Resonance Parameters for Crystalline Systems: Principles Jonathan R. Yates, Chris J. Pickard Encyclopedia of Magnetic Resonance (2008) doi: / emrstm1009 Exhaustive (and exhausting!) review of applications and theory ( JRY for a copy) First-Principle Calculation of NMR Parameters Using the GIPAW (Gauge Including Projector Augmented Wave) Method: a Chemist's Point of View Bonhomme, Gervais, Babonneau, Coelho, Pourpoint, Azais, Ashbrook, Griffin, Yates, Mauri, Pickard, Chemical Reviews (in press 2012) Original Theory Papers: All-electron magnetic response with pseudopotentials: NMR chemical shifts, Chris J. Pickard, and Francesco Mauri. Phys. Rev. B, 63, (2001) Calculation of NMR Chemical Shifts for extended systems using Ultrasoft Pseudopotentials Jonathan R. Yates, Chris J. Pickard, and Francesco Mauri. Physical Review B 76, (2007) A First Principles Theory of Nuclear Magnetic Resonance J-Coupling in solid-state systems Sian A. Joyce, Jonathan R. Yates, Chris J. Pickard, Francesco Mauri J. Chem. Phys. 127, (2007)

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