First Principles Calculations of NMR Chemical Shifts. Daniel Sebastiani

Transcription

1 First Principles Calculations of NMR Chemical Shifts Methods and Applications Daniel Sebastiani Approche théorique et expérimentale des phénomènes magnétiques et des spectroscopies associées Max Planck Institute for Polymer Research Mainz Germany 1

2 Outline Part I Introduction and principles of electronic structure calculations I. Introduction to NMR chemical shielding tensors Phenomenological approach II. Overview electronic structure methods HF, post-hf, DFT. Basis set types III. External fields: perturbation theory 2

3 Outline Part II Magnetic fields in electronic structure calculations I. Perturbation Theory for magnetic fields in particular: magnetic density functional perturbation theory II. Gauge invariance Dia- and paramagnetic currents Single gauge origin, GIAO, IGLO, CSGT III. Condensed phases: position operator problem 3

4 Outline Part III Applications I. Current densities II. Chemical shifts of hydrogen bonded systems: Water cluster Liquid water under standard and supercritical conditions Proton conducting materials: imidazole derivatives Chromophore: yellow dye 4

5 Nature of the chemical shielding External magnetic field B ext Electronic reaction: induced current j(r) = inhomogeneous magnetic field B ind (r) Nuclear spin µ Up/Down energy level splitting B ext j ind B ind Β=0 Β=Β 0 hω E = 2µ B = hω 5

6 Chemical shifts chemical bonding NMR shielding tensor σ: definition through induced field B tot (R) = B ext + B ind (R) σ(r) = Bind (R) B ext 1 Strong effect of chemical bonding Hydrogen atoms: H-bonds = NMR spectroscopy: unique characterization of local microscopic structure (liquid water) 6

7 Chemical shielding tensor σ(r) = B ind x (R) B ext x By ind B ext x Bz ind B ext x (R) (R) B ind x (R) B ext y By ind B ext y Bz ind B ext y (R) (R) B ind x (R) B ext z By ind B ext z Bz ind B ext z (R) (R) Tensor is not symmetric = symmetrization = diagonalization = Eigenvalues Isotropic shielding: Tr σ(r) Isotropic chemical shift: δ(r) = Trσ TMS Trσ(R) 7

8 First principles calculations: Electronic structure Methods Hartree-Fock Møller-Plesset Perturbation Theory Highly correlated methods CI, coupled cluster,... Density functional theory Basis sets Slater-type functions: Y m l exp r/a 0 Gaussian-type functions: Y m l exp (αr) 2 Plane waves: exp ig r 8

9 Kohn-Sham density functional theory (DFT) Central quantity: electronic density, total energy functional No empirical parameters E KS [{ϕ i }] = at i d 3 r ϕ i 2 ϕ i d 3 r d 3 r ρ(r)ρ(r ) r r q at d 3 ρ(r) r r R at + E xc[ρ] ρ(r) = i ϕ i (r) 2 9

10 DFT: Variational principle Variational principle: selfconsistent Kohn-Sham equations ϕ i ϕ j = δ ij δ δ ϕ i (r) (E KS[ϕ] Λ kj ϕ j ϕ k ) = 0 Iterative total energy minimization DFT: Invariant of orbital rotation ψ i = U ij ϕ j E[ϕ] = E[ψ] Ĥ [ρ] ϕ i = ε i ϕ i 10

11 Perturbation theory External perturbation changes the state of the system Expansions in powers of the perturbation (λ): Ĥ Ĥ(0) + λĥ(1) + λ 2 Ĥ (2) +... ϕ ϕ (0) + λϕ (1) +... E E (0) + λe (1) + λ 2 E (2)

12 Perturbation theory in DFT Perturbation expansion E [ϕ] = E (0) [ϕ] + λ E λ [ϕ] +... ϕ = ϕ (0) + λ ϕ λ +... [ ] ρ λ (r) = 2 R (r) ϕ λ i (r) ϕ (0) i Ĥ = Ĥ(0) + λ Ĥλ + ĤC [ρ λ ] +... E [ϕ] = E (0) [ϕ] + λ E λ [ϕ (0) ] λ2 E (2) [ϕ]... 12

13 Perturbation theory in DFT unperturbed wavefunctions ϕ (0) known: min {ϕ} E[ϕ] min {ϕ (1) } E (2) [ ϕ (0), ϕ (1)] E (2) = ϕ (1) δ2 E (0) δϕ δϕ ϕ(1) + δe λ δϕ ϕ(1) orthogonality ϕ (0) j ϕ (1) k = 0 j, k 13

14 Perturbation theory in DFT (Ĥ(0) δ ij ε (0) ij Iterative calculation ) ϕ λ j + ĤC [ρ λ ] ϕ(0) i = Ĥλ ϕ (0) i Formal solution ϕ λ i = Ĝij Ĥλ ϕ (0) j 14

15 Magnetic field perturbation Magnetic field perturbation: vector potential A A = 1 2 (r R g) B Ĥ λ = e mˆp  = i he 2m B (ˆr R g) ˆ Cyclic variable: gauge origin R g Perturbation Hamiltonian purely imaginary = ρ λ = 0 15

16 Magnetic field perturbation Resulting electronic current density: ĵ r = e [ ] ˆπ r r + r r ˆπ 2m ĵ r = e 2m j(r ) = k [ (ˆp eâ) r r + r r (ˆp eâ) ϕ (0) k ĵ(2) r ϕ (0) k = j dia (r ) + j para (r ) ] + 2 ϕ(0) k ĵ(1) r ϕ (1) k Dia- and paramagnetic contributions: zero and first order wavefunctions 16

17 The Gauge origin problem Gauge origin R g theoretically not relevant In practice: very important: j dia (r ) R 2 g GIAO: Gauge Including Atomic Orbitals IGLO: Individual Gauges for Localized Orbitals CSGT: Continuous Set of Gauge Transformations: R g = r IGAIM: Individual Gauges for Atoms In Molecules 17

18 Magnetic field under periodic boundary conditions Basis set: plane waves (approach from condensed matter physics) Single unit cell (window) taken as a representative for the full crystal All quantities defined in reciprocal space (periodic operators) Position operator ˆr not periodic non-periodic perturbation Hamiltonian Ĥλ 18

19 PBC: Individual ˆr-operators for localized orbitals Localized Wannier orbitals ϕ i via unitary rotation: ϕ i = U ij ψ j orbital centers of charge d i Idea: Individual position operators ϕ b (x) ϕ ^r (x) b a (x) ^a r (x) 0 d d L 2L b a (x) 19

20 Magnetic fields in electronic structure Variational principle electronic response orbitals Perturbation Hamiltonian Ĥλ : Â = 1 2 (ˆr R g) B Response orbitals electronic ring currents Ring currents NMR chemical shielding Reference to standard NMR chemical shift 20

21 Electronic current density j k (r ) = ϕ (0) k ĵr ( ĵ r = e 2m ϕ (α) k [ ˆp r r + r r ˆp ϕ(β) k + ϕ( ) ] k ) modulus of current j B-field along Oz 21

22 Current and induced magnetic field in graphite Electronic current density j Induced magnetic field B z Identification of atom-centered and aromatic current densities Nucleus independent chemical shift maps 22

23 Isolated molecules Isolated organic molecules, 1 H and 13 C chemical shifts Comparison with Gaussian 98 calculation, (converged basis set DFT/BLYP) σ H [ppm] - calc C 6 H 6 C 2 H 4 H 2 O C 2 H 6 C 2 H 2 Gaussian (DFT) this work MPL method CH 4 σ C [ppm] - calc C 6 H 6 C 2 H 4 C 2 H 2 Gaussian (DFT) this work MPL method C 2 H 6 CH σ H [ppm] - exp σ C [ppm] - exp 23

24 Example system: Water cluster Water cluster: water molecule surrounded by 6 neighbors Strong hydrogen bonding, nonsymmetric geometry 24

25 Example system: Water cluster Hydrogen bonding effects strongly affect the proton chemical shieldings Large range of individual shieldings 25

26 Extended system: liquid water Most important solvent on earth Complex, dynamic hydrogen bonding Configuration: single snapshot from molecular dynamics Complex hydrogen bonding, strong electrostatic effects NMR experiment: average over entire phase space 32 water molecules at ρ=1g/cm 3, under periodic boundary conditions 26

27 Supercritical water: hydrogen bond network CPCHFT 110 (8) (2002) ISSN Vol. 3 No. 8 August 16, 2002 D55711 ab-initio MD: 3 9ps, 32 molecules P.L. Silvestrelli et al., Chem.Phys.Lett. 277, 478 (1997) M. Boero et al., Phys.Rev.Lett. 85, 3245 (2000) NMR sampling: 3 30 configurations proton shifts 2001 Physics NOBEL LECTURE in this issue 8/2002 Concept: Conductance Calculations for Real Nanosystems (F. Grossmann) Highlight: Terahertz Biosensing Technology (X.-C. Zhang) Conference Report: Femtochemistry V (M. Chergui) Experimental data: N. Matubayashi et al., Phys.Rev.Lett. 78, 2573 (1997) 27

28 Supercritical water: chemical shift distributions δ H 6 5 [ppm] δ H 6 5 [ppm] δ H 6 5 [ppm] ρ=1 g/cm 3, T =303K ρ=0.73 g/cm 3, T =653K ρ=0.32 g/cm 3, T =647K Standard conditions: broad Gaussian distribution, continuous presence of hydrogen bonding Supercritical states: narrow distribution, hydrogen bonding tails 28

29 Supercritical water: gas liquid shift Qualitatively reduced hydrogen bond network in supercritical water 6 5 calculated δ liq calculated δ liq (this work) (MPL) Excellent agreement with experiment Slight overestimation of H-bond strength at T BLYP overbinding? Insufficient relaxation? = confirmation of simulation δ H [ppm] experimental δ liq ρ [g / cm 3 ] 29

30 Ice Ih: gas solid shift Ice Ih: hexagonal lattice with structural disorder 16 molecules unit cell, full relaxation Experimental/computed HNMR shifts [ppm]: Exp Exp MPL this work

31 Crystalline imidazole Molecular hydrogen-bonded crystal experimental (a) (b) calculated (crystal) Very good reproduction of experimental spectrum (c) [ppm] calculated (molecule) HNMR: π-electron proton interactions, mobile imidazole 31

32 Crystalline Imidazole-PEO Imidazole [Ethyleneoxide] 2 Imidazole Strongly hydrogen bonded dimers, complex packing structure Anisotropic proton conductivity (fuel cell membranes) 32

33 Crystalline Imidazole-PEO: NMR spectra Particular hydrogen bonding: two types of high-field resonances, intra-pair / inter-pair Partly amorphous regions (10ppm): mobile Imidazole-PEO molecules Packing effect at 0ppm Quantitative reproduction of experimental spectrum top: experimental middle: calculated (crystal) bottom: calculated (molecule) 33

34 Chromophore crystal: yellow dye Material for photographic films Unusual CH O bond unusual packing effects 244 atoms / unit cell 34

35 Chromophore NMR spectrum Full resolution of experimental spectrum, unique assignment of resonances Strong packing effects from aromatic ring currents: CH 3 Ar, ArH Ar H-bonding too weak (9ppm): insufficient geometry optimization, temperature effects Starting point for polycrystalline phase top: experimental bottom: calculated 35

36 Conclusion NMR chemical shifts from ab-initio calculations Gas-phase, liquid, amorphous and crystalline systems Assignment of experimental shift peaks to specific atoms Verification of conformational possibilities by their NMR pattern Strong dependency on geometric parameters (bonds, angles,... ) Quantification of hydrogen bonding 36