Hamiltonians with focus on Anisotropic interac4ons (Chemical Shi8 Anisotropy and Dipolar Interac4ons) The (solid- state NMR) world is anisotropic

Transcription

1 Hamiltonians with focus on Anisotropic interac4ons (Chemical Shi8 Anisotropy and Dipolar Interac4ons) The (solid- state NMR) world is anisotropic! Munich School November 2014!! Beat H. Meier Diffusion tensors in human brain

2 Schrödinger equation i! Ψ t = ĤΨ Hamilton Operator (describes the Energy funclon) EssenLally the Physics Ĥψ = Eψ WavefuncLon, describes the state of a quantum system Ψ(x,t) =ψ (x)exp( ie /! t)

3 Bild: wikipedia Harmonic Oscillator

4

5 Stern-Gerlach Experiment! F = (! µ )! B = 0 0 µ z B dz

6 Stern-Gerlach Experiment

7 Stern-Gerlach Experiment

8 Im Bohr-Modell ist auch der Drehimpuls quantisiert.

9 Observables  = ψ  ψ

10 Expansion in Basis Functions (orthonormal) ψ (t) = N i=1 c i (t) φ i

11 a

12 Schrödinger equa4on for the density operator Liouville- von Neumann equa4on d dt ˆr = 1! ˆ*, ˆr(t) Solu4on for 4me- independent ˆ* : ˆρ(t) = e i! ˆ*t ˆρ(0)e i! ˆ*t

13 The full system Hamiltonian is complicated

14 ...but can be reduced to the spin Hamiltonian ˆ*(s k ) = 1! ψ Ĥ ψ part For diamagnelc compounds: ˆ*(s k ) = ˆ* BI + ˆ* II + ˆ* Q Integrate over all space coordinates of electrons and over all spin coordinates of paired electrons (+ Born Oppenheimer)

15 Simplest case: the spin Hamiltonian in isotropic phase (and in the rotalng frame) * = Ω 1 Î 1z + Ω 2 Î 2z + 2πJ 12 Î 1!"!" Î2 Chemical Shielding J- Coupling Chemical environment, CoordinaLon, DisLncLon of chemical sites ConnecLvity through chemical bonds (via electrons), Dihedral Angles magenta quanlles or obtained by integralon over the other quantum degrees of freedom

16 Chemical shi8 Nice cover, but slll not trivial at reasonable precision...! Therefore: - Databank approaches! Shi[S, Shi[X, Sparta...

17 In matrix representa4on 17

18 18

19 19

20 Hamiltonians * = * Zeeman +* internal High- Field ApproximaLon: * Zeeman >> * internal - > InteracLon Frame (rotalng Frame) 20

21 Rota4ng Frame * = γ B Î Zeeman i 0 iz i * Zeeman = γ B 0 ˆFz One spin species * = e i* Zeemannt * e i* Zeemannt Int Int = e iγ B 0 ˆF z t * e iγ B ˆF 0 z t Int 21

22 Rotating-Frame Transformation dσ The Liouville von Neumann Equation: = i [*, σ] dt Transformation into the rotating frame with R e i*ˆ 0t e iω rffˆ zt = = : d ( Rˆ 1 σ' ˆ Rˆ ) = i [*ˆ Rˆ 1 σ' ˆ Rˆ Rˆ 1 σ' ˆ Rˆ *ˆ ] dt iω rf Fˆ zrˆ 1 σ' ˆ Rˆ Rˆ 1 d + σ' ˆ Rˆ + Rˆ 1 σ' ˆ iω dt rf Fˆ zrˆ = i [*ˆ Rˆ 1 σ' ˆ Rˆ Rˆ 1 σ' ˆ Rˆ *ˆ ] Rˆ 1 d σ' ˆ Rˆ = i [*ˆ Rˆ 1 σ' ˆ Rˆ ( Rˆ 1 σ' ˆ Rˆ *ˆ ω dt rf ( Fˆ zrˆ 1 σ' ˆ Rˆ Rˆ 1 σ' ˆ Fˆ zrˆ ))] d σ' ˆ = irˆ [ *ˆ Rˆ 1 σ' ˆ ( σ' ˆ Rˆ *ˆ Rˆ 1 ω dt rf ( Fˆ zσ' ˆ σ' ˆ Fˆ z ))] d σ' ˆ dt = i [ *' ˆ ω rf Fˆ z, σ' ˆ ] In the rotating frame we have to correct the Hamiltonian by * 0. Otherwise, the resulting Liouville-von-Neumann equation looks the same as in the laboratory frame: d σ' ˆ = [ *'' ˆ, σ' ˆ ]. dt 22 page 2

23 Box I: Rotating Frame Rotating-Frame Representation By transforming into the rotating frame, we have: Changed the Hamiltonian, i.e., for ω rf = ω 0 Removed the time-dependence from the rf Hamiltonian., we have removed the Zeeman term. Usually, the remaining time-dependent terms are neglected. This approximation is called the secular approximation and must be justified on a case-by-case basis. Transformation rules for spin operators: ˆI' x Î x cos( ω rf t) + Î y sin( ω rf t) ˆI' y Î y cos( ω rf t) Î x sin( ω rf t) ˆI' z Î z Transformation of rf Hamiltonian: *ˆ rf () t = γ i B 1 ( cos( [ ω rf t]î x + sin[ ω rf t] )Î y ) i *' ˆ rf = i γ i B 1 ˆI' ix page 3 23

24 Rota4ng Frame Hamiltonians rf- irradialon chemical shi[ J- coupling * rf = γ i B 0 Î ix/y i i * CS = γ i σ (i) zz B 0 Î iz * J = 2π J ˆ! ij I i iˆ! i< j I j dipole- coupling * D = µ! 0 4π i< j γ i γ j r ij 3 3cos 2 θ ij 1 2 2ÎizÎ 1 jz 2 (I + i I j + I + i I j 24

25 The anisotropic interac4ons 25

26 An (almost) isotropic light eminng object: our sun

27 A strongly anisotropic light eminng object: a plasma lamp

28 How do we describe an anisotropic body or an interac4on? A) Specify, for each direclon, (ϕ, θ), the length of the vector from the origin to the surface (digilzed polar coordinates) B) Write the surface as a linear superposilon of simple surfaces! =

29 We have single valued simple surfaces and method B is asrac4ve! = A possible choice for the basis funclons (which should be orthogonal and normalized, orthonormal) are the spherical harmonics

30 We are lucky as our interac4ons can be explained with only a few basis func4ons! = other objects are more complex, e.g mathemalcal knots

31 An example Simple example: magne4c suscep4bility isotropic anisotropic M i = x ij H j M i j = x ij H j (note summalon convenlon)

32 Higher terms are not important in NMR M = χ H + χ (3) H H + χ (4) H H H +... i ij j ijk j k ijkl j k l Linear effects are always described by second rank tensors or by 3x3 matrices

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34 but in other spectroscopies

35 The anisotropy of the basic interac4ons can be described by 9 basis func4ons max. Which, in general, have an imaginary part though... A 00 A 10 A 11 A 1,- 1 A 20 A 21 A 2-1 A 22 A 2-2 Re lm

36 For Spherical tensors we will call A lm Rank Component

37 The chemical- shi8- anisotropy tensor (CSA) The external, applied field B 0 = μ 0 H is modified by the electron cloud to yield an new field B k at the position of nucleus k ˆ* s = γ k (Îkx k σ, Îky, Îkz ) (k ) xx (k ) σ yx (k ) σ zx (k ) σ xy (k ) σ yy (k ) σ zy (k ) σ xz (k ) σ yz (k ) σ zz 0 0 B 0

38 The chemical shi8: mechanisms Lamb- Shi[ ParamagneLc Effect Ringcurrent- Effect Anisotropic Neighbor- Effect

39 The chemical shi8: Hamiltonian * #"# In short: ˆ* = γ!" I σ (k ) B s k 0 k Because the Zeeman interaclon is dominant, we go to the rotalng frame and neglect non- secular components: ˆ*' S = γ σ k zz k (k )! B 0 Î kz - > NMR measures the zz- element of the chemical- shielding tensor in the laboratory frame of reference (z along field direclon)

40 The chemical shi8: single crystal spectra The resonance frequency is proporlonal to the zz- element of the shielding tensor in the laboratory frame. Because the chemical- shielding tensor is defined with respect to a molecule fixed coordinate system we must first transform it into the lab frame to obtain the resonance frequency by: ˆσ (k ) = R ˆσ (k ) R 1 MF

41 The chemical shi8: single crystal spectra A parlcular molecular fixed coordinate system is the principal axis system (PAS), where is diagonal: The diagonal values of this matrix are called the principal values of the chemical- shielding tensor, the direclon of the axis system (3 Euler angles), the principal direc8ons. The ordering of the principal values is chosen such that is the least shielded component, is the most shielded component, lies in between.

42 Experimental: spectra are orienta4on dependent Single- crystal 13 C NMR spectra of benzoic acid, labelled at the carboxylic carbon posilon. Recorded at 100 K. Vosegaard et. al. Journal of MagneLc Resonance 142, 379 (2000)

43 The chemical shi8: powder spectra For a powder sample, the FID (and the spectrum) is the weighted superposilon of the possible crystallite orientalons: [0.1] Because of the axial symmetry around the direclon of the applied field, the last rotalon which is around the direclon of does not influence the NMR signal and can be evaluated immediately in the above integral, leading to: [0.2]

44 The chemical shi8: powder spectra

45 Powder averages are usually calculated numerically The integral is replaced by a sum. The surface element for integralon must reflect the probability on the surface of a sphere. The best division on the surface gives the best spectrum for the smallest number of surface elements. Simple scheme: Lebedev scheme (LEB 301): M. Eden, M. H. Levir, J. Magn. Reson. 1998, 132, 220.

46 There are more second- rank interac4ons MagneLc dipole- dipole interaclon: Informa4on contents! Internuclear distances, Bond angles * Single crystal spectrum of A 2 spinsystem δ D (k,n) = 2 µ 0 γ k γ n! 4πr kn 3

47 This interac4on is anisotropic and leads to broad powder spectra

48 Anisotropic interac4ons and isotropic liquids These interac4ons vanish in isotropic liquid phase because of fast molecular tumbling H = Ω 1 Î1z + Ω 2 Î2z + 2π J 12!"!"! Î Î2 1 They are, however, the source of Relaxa4on (stochas4c 4me dependence)

49 Chemical Shielding InformaLon: - Chemical environment - 3 direclons in space

50 Powder- Lineshape: Anisotropy and (if present) asymmetry of the tensor. No orientalonal info

51 Magne4c Dipole- Dipole Interac4on Single Crystal Spectrum (Spin ½ coupled to another spin ½): InformaLon: - Through space distance - 1 direclon in space Size of the InteracLon as a funclon of orientalon with respect to external field: The dipolar interaclon is averaged to zero in isotropic liquids but can be indirectly observed through relaxalon processes

52 J- Coupling Single Crystal Spectrum (Spin ½ coupled to another spin ½): InformaLon: - through- bond conneclvity - dihedral angles

53 Tensors The chemical- shi[ tensor and the dipolar tensors introduced are cartesian tensors. The describe the angular dependence of the space part of under sample rotalons. In The spectrum of a single crystal is determined by the zz component of the tensor in the lab coordinate system (z- axis along the magnelc field)

54 2D Tensor- correla4on Spectroscopy Dipolar CSA Dipolar 1 Euler angle 2 Euler angles CSA 2 Euler angles 3 Euler angles

55 Further angle info by correlated spectroscopy (2) Procedure: EvoluLon: Measure resonance frequencies of involved spins Mixing: Transfer magnelzalon from inital spins through space or through bond to a neighbor DetecLon: Measure resonance frequencies in new configuralon single crystal powder Parern reveals angle β

56 Applica4on to structure determina4on Principle: MagneLsaLon (or polarisalon) prepared on one spin is transferred to neighbours by so called flip- flop processes which are mediated by the dipolar interaclon. The correlalon is established in the mixing Lme In 2D- real space: spin diffusion flip- flop

57 spin diffusion samples the local environment Crystalline Solids (or domains)! Quasi- equilibrium spectra Amorphous Solids (or domains)! Buildup rates from non- equilibrium spectra

58 and is behind many methods for structure determina4on in proteins...

59 Double rota4on removes also 4 th rank tensors

60 Wu, Sun, Pines, Samoson, Lippmaa: Na resonance