Tutorial Session - Exercises. Problem 1: Dipolar Interaction in Water Moleclues

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1 Tutorial Session - Exercises Problem 1: Dipolar Interaction in Water Moleclues The goal of this exercise is to calculate the dipolar 1 H spectrum of the protons in an isolated water molecule which does not move. 1 H O 1 H Å Figure 1: Molecular geometry of the water molecule. γ H = rad s 1 T 1 γ T = rad s 1 T 1 µ 0 = 4π 10 7 Vs/Am = Js The homonuclear dipolar coupling Hamiltonian is given by Ĥ = µ 0γH ( cos Θ 1 } 4πr {{ Î1zÎz 1 } dipolar coupling constant ( I + k I n + I k I+ n. (1 1. Start with the single-spin operators in matrix representation and determine the matrix representation of the dipolar Hamiltonain. Calculate the Eigenvalues and allowed transitions. Also calculate the dipolar coupling constant.. Which orientation would a single isolated water molecule have to have with respect to an external field to show a maximum dipolar splitting? How big is the maximum splitting?. How does the spectrum of a powder-like sample containing water molecules in arbitrary orientations look like? 4. Which would be the longest 1 H- 1 H distance that could be detected by this experiment assuming a minimum spectral resolution of 100 Hz?

2 5. How large is the dipolar coupling if one of the protons is replaced by a tritium? What does the corresponing spectrum look like with respect to the homonuclear case? 6. Which interactions are present in the liquid state? What would the corresponding spectrum look like? Problem : Dipolar Coupling and the Spherical Tensor Notation 1. The laboratory-frame Hamiltonian can be expressed via the sum of scalar products between a spherical spatial- and a spin-tensor operator as Ĥ = l l ( 1 q A l,q T l, q q= l Starting from this expression, calculate the dipolar Hamiltonian (Hint: see additional expressions for the spherical tensors at the end of this exercise sheet. Show that the result is equivalent to the dipolar alphabet representation given as with Ĥ (k,n D = µ 0 γ k γ n (Â + ˆB + Ĉ + 4π r ˆD + Ê + ˆF kn cos Θ 1 Â = ÎkzÎnz ˆB = 1 cos (Î+ Θ 1 k Î n + Î k Î+ n sin Θ cos Θe iϕ Ĉ = (Î+ k Înz + ÎkzÎ+ n sin Θ cos Θe iϕ ˆD = (Î k Înz + ÎkzÎ n Ê = Î+ 1 sin Θe iϕ k Î+ n ˆF = 1 Î sin Θe iϕ k Î n. ( Assign to the components A, B, C, D, E, F the corresponding rank and order of the spin and spatial tensor.. Taking into account that only the q = 0 components are invariant under z-rotation, give a reason why all components except A and B are neglected in the secular approximation.. Calculate (k,n ˆT,+ = (Î+ 1 k Î+ n explicitly in matrix form. (

3 Problem : Single Crystals and MAS The aim of this problem is to calculate the resonance frequency of a 1 C spin in the carbonyl group of an Alanine single crystal. The components of the chemical-shift tensor in the principle axis system are ˆσ xx = 9 ppm, ˆσ yy = 184 ppm and ˆσ zz = 106 ppm. The orientation of the crystal is such that the vectors of the principle components lie along the x, y and z axis in the laboratory frame. The static magnetic field is along the laboratory z-axis ( B e z 1. Calculate the isotropic chemical shift σ iso.. Simplify general Hamiltonian describing the interaction of the chemical-shift tensor and the magnetic field, which is given as, and simplify it according to the high-field approximation. Ĥ = γ Î ˆσ B (4. The crystal is rotated from its original orientation by an angle γ around a rotation axis that has the polar coordiates φ = 0 and θ = cos 1 (1/ = The matrix describing this rotation is given as R(γ = 1 1 ( + cos γ sin γ (1 cos γ sin γ 1 sin γ cos γ (1 cos γ sin γ 1 (5 (1 + cos γ Calculate ˆσ zz(γ and express the result in the form ˆσ zz(γ = ˆσ iso + ˆσ 1 cos(γ + ˆσ cos(γ. What does ˆσ iso correspond to? (Keep in mind that ˆσ (γ = R(γˆσR( γ 4. Plot ˆσ zz(γ it as a function of γ. What happens if we rotate the sample continuously around the previously chosen axis (i.e. if γ(t = πω r t where ω r is the rotation frequency?

4 Problem 4 (Optional: The Tensor Product The tensor product of two spherical tensors A k and B k of rank k and k, respectively, can be expressed with irreducible tensors I K that follow the relation I KQ (k, k = k q= k k q = k kk qq KQ A kq B k q = ( 1 k k +Q K + 1 k k q= k q= k ( k k K q q A Q kq B k q (6 with kk qq KQ A kq B k q = ( 1k k +Q K + 1 as the so-called Clebsch-Gordan coefficients and k k q= k q= k ( k k K q q Q (7 (k + k K k k Q = q + q. (8 1. Name all components J KQ of the product A 1 B 1. How many components would there be for A B?. As an example, verify that J 00 = 1 (A 1, 1 B 1,1 + A 1,1 B 1, 1 A 1,0 B 1,0 (9 J 0 = 1 6 (A 1,0 B 1,0 + A 1,1 B 1, 1 + A 1, 1 B 1,1 (10 4

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6 6. Clebsch-Gordan coefficients 1 6. CLEBSCH-GORDAN COEFFICIENTS, SPHERICAL HARMONICS, AND d FUNCTIONS Note: A square-root sign is to be understood over every coefficient, e.g., for 8/15 read 8/15. Y 0 1 = 4π cos θ Y1 1 = sin θ eiφ 8π Y 0 = 5 4π ( cos θ 1 15 Y 1 = 8π Y = 1 4 sin θ cos θ eiφ 15 π sin θ e iφ Y m l = ( 1 m Y m l 4π d l m,0 = l + 1 Y l m e imφ d j m,m = ( 1m m d j m,m = d j m, m d 1 0,0 j 1 j m 1 m j 1 j JM = ( 1 J j 1 j j j 1 m m 1 j j 1 JM = cos θ d 1/ 1/,1/ = cos θ d 1/ 1/, 1/ = sin θ d 1 1,1 = 1 + cos θ d 1 1,0 = sin θ d 1 1, 1 = 1 cos θ d / /,/ = 1 + cos θ cos θ d / /,1/ = 1 + cos θ sin θ d / /, 1/ = 1 cos θ cos θ d / /, / = 1 cos θ sin θ d / 1/,1/ = cos θ 1 cos θ d / 1/, 1/ = cos θ + 1 sin θ ( 1 + cos θ d, = d,1 = 1 + cos θ sin θ 6 d,0 = 4 sin θ d, 1 = 1 cos θ sin θ ( 1 cos θ d, = d 1,1 = 1 + cos θ ( cos θ 1 d 1,0 = sin θ cos θ d 1, 1 = 1 cos θ ( cos θ + 1 d 0,0 = ( cos θ 1 Figure 6.1: The sign convention is that of Wigner (Group Theory, Academic Press, New York, 1959, also used by Condon and Shortley (The Theory of Atomic Spectra, Cambridge Univ. Press, New York, 195, Rose (Elementary Theory of Angular Momentum, Wiley, New York, 1957, and Cohen (Tables of the Clebsch-Gordan Coefficients, North American Rockwell Science Center, Thousand Oaks, Calif., 1974.