Origin of Chemical Shifts BCMB/CHEM 8190

Transcription

1 Origin of Chemical Shifts BCMB/CHEM 8190

2 Empirical Properties of Chemical Shift υ i (Hz) = γb 0 (1-σ i ) /2π σ i, shielding constant dependent on electronic structure, is ~ Measurements are made relative to a reference peak (TMS). Offsets given in terms of δ in parts per million, ppm, + downfield. δ i = (σ - σ x 10 6 ref i ) or δ i = (( υ i - υ ref )/υ ref )x 10 6 Ranges: 1 H, 2 H, 10 ppm; 13 C, 15 N, 31 P, 300 ppm; 19 F, 1000 ppm

3 Ramsey s Equation for Chemical Shift Additional Reference: G.A. Webb, in Nuclear Magnetic Shielding and Molecular Structure, J.A. Tossel, ed. Nato Adv. Sci. Series (1993) 1-25 Physical origin: moving charges experience a force perpendicular to the trajectory. t Hence electrons precess. Circulating current gives an opposing field. B 0 r e - F =-(e/c) v x B F 0 B = -(e/c) (r x v) / r 3 = -(e/cm) (r x p) / r 3 v But, we actually need to treat electrons at QM level

4 Some Quantum Mechanics Fundamentals Expectation values correspond to observables: O = <ψ O ψ> = ψ* O ψ dτ O - an operator, ψ - a wave function (electronic or spin) Examples, wave functions: ψ = 1s, 2s, 2p 1, 2p 0, 2p -1 (electronic wave functions) ψ = α, β (one spin ½ ), αα, αβ, βα, ββ (two spins ½ ) All are solutions to Schrodinger s equation: H ψ = E ψ They are normalized: <ψ ψ> = ψ*ψ dτ = 1 Examples, Operators: Hamiltonian operator is special: <ψ H ψ> = E Zeeman Hamiltonian for nuclei in a magnetic field: H z = - μ B 0, E z = < α - μ B 0 α> Begin with classical expression: substitute QM operators μ z = γ I z (h/2π) (magnetic moment) E z = < α -(γh/2π)i z B 0 α> = -½ γ(h/2π) B 0 <α* α> = -½ γ(h/2π)b 0

5 Quantum Expression for B Have QM operator for linear momentum: p 0 =i(h/(2 i(h/(2π))( / x + / y + / z) But momentum in magnetic field has a curl p = p 0 + e(b x r) / (2c) = p 0 +( (e/c) )A A is the vector potential; A = (B x r)/2 B = -(e/cm) (r x p 0 ) / r 3 -e 2 (r x A) / (r 3 c 2 m) Quantum mechanically: B = <ψ 0 -e (r x p 0 )/r 3 (cm) - e 2 (r x A)/(r 3 c 2 m) ψ 0 > paramagnetic diamagnetic

6 Diamagnetic Shifts Note: only the second term is proportional to B 0 at first order theory; this is the diamagnetic term; Lamb term B D = <ψ 0 -e 2 (r x B 0 x r)/(2r 3 c 2 m) ψ 0 > Only interested in the z component: k (x (B 0 x r) y y (B 0 x r) x ) i x ( Bxr) x j y ( Bxr) y k z ( Bxr) z B =-(e 2 2 0* 2 +y 2 3 D /(2c m)) ψ (x )/r ) ψ 0 dτ Predictions: depends on electron density near to nucleus opposes magnetic field (shields) Examples: He 2 1s e- σ = x 10-6 Ne 10 e- σ = 547 x 10-6 H ~2 1s e- σ = ~60 x 10-6 HO- O withdraws ~10% ~6 ppm downfield

7 Paramagnetic Contribution to Shifts This comes from the first term there was no explicit B 0 dependence so carry to second order electronic wave function is changed by field B 0 can be in H ψ = ψ 0 + Σ n (<ψ n H ψ 0 > / (E n -E 0 )) ψ n = ψ 0 + ψ H 0 = (1/(2m)) ( p 02 + V in absence of field H = (1/(2m)) (p 0 + (e/c)a) 2 + V in presence A = (B 0 x r)/2. This introduces field dependence H = (e/(2mc)) A p 0 keeping most important term

8 Paramagnetic term continued A p 0 = ((B 0 x r)/2) p 0 B 0 (r x p 0 )/2 = B 0 (Lh/(2π))/2 = B 0 L z h/(2π))/2 H = B 0 eh/(8πmc) L z Hence ψ Σ n (<ψ n L Z ψ 0 > /ΔE) B P = <ψ 0 + ψ (e/(cm))(r x p 0 )/r 3 ψ 0 + ψ > = <ψ 0 + ψ (e/(cm))(l z )/r 3 ψ 0 + ψ > Substituting ψ and saving only terms linear in B 0 B P Σ n [(<ψ 0 L z ψ n ><ψ n L z /r 3 ψ 0 >)/(E n -E 0 ) + (<ψ 0 L z /r 3 ψ n ><ψ n L z ψ 0 >)/(E n -E 0 )]

9 Implications for Paramagnetic Term σ P is negative (B P - σ P ) opposite to σ D σ P is zero unless L Z ψ> is finite hence, if only s orbitals populated, L Z s > = 0 hence, small shift range for 1 H 13 C has p orbitals (L z p 1 > = 1 p 1 ) and finite σ P Electron distribution must also be assymetric otherwise, Σ L z p> = 0 hence, CH 4 shift is small and resonance far upfield

10 13 C Example: Ethane vs Ethylene CH 3 -CH 3 6 ppm, CH 2 = CH ppm, Why? σ D is about the same for both, ~200 x 10-6 σ P Σ n [(<ψ 0 L z ψ n ><ψ n L z /r 3 ψ 0 >)/(E n -E 0 ) +.. Examine ψ n = Σ i c in φ I, φ I = 1s c, 2s c, 2p c0, 2p c+/-1, 1s H Only ps count, ΔE small is most important Consider first excited state: π * = (1/ 2)(p ia -p ib )

11 Consider Field Parallel to C-C Bond B 0 π* = (1/ 2)(p xa -p xb ) A B π* = (1/ 2)((p 1A +p -1A )-(p 1B +p -1B )) L z π*> = (ih 2/π)((p 1A -p -1A )-(p 1B -p -1B ))/(2i) = (ih 2/π)(L z π*> ) <ψ 0 L z π*> is finite if p ya, p yb are populated in ψ 0 0 z ya yb 0 ψ 0 must also be assymetric look at MOs

12 Molecular Orbitals for Ethylene ψ 3, 3 nodes, 1 bond ψ 2, 2 nodes, 4 bonds E ψ 1, 1 node, 5 bonds Fill with electrons: 2x6 for C, 4 for H = 16 4 in 1s C, 2 in π 0 ( to plane), 2 in C-C σ, 4 in C-H σ Implies 4 maximum in above

13 Calculating Paramagnetic Contribution Only ψ 1 ψ 2 contribute Only ψ 1, ψ 2, contribute ψ 1, is symmetric, implies <ψ 1 L z π*> is zero ψ 2, is asymmetric and counts σ 2 P = -(eh/(2πmc)) 2 <(1/r 3 )> 2p c x 10-6 σ c = + = ( ) -6 = 0x10-6 c-c σ D σ P 200) x 10

14 What about Field Perpendicular to Plane? A B B 0 π* = (1/ 2)(p za -p zb ) π* = (1/ 2)(p 0A -p 0B ) L z p 0 > = 0; therefore, σ P =0 σ = σ D + σ P = (200 +0) x 10-6 Similar il for in plane, perpendicular to p π s σ (predict) = 0 (observe) Isotopic shift = 1/3 Tr σ = ppm below Me Waugh, Griffin, Wolff, JCP, (1977) solids NMR

15 13 C Chemical Shift Calculations on Peptides α helix, -57, -47; β sheet, -139, 135; Oldfield and Dios, JACS, 116, 5307 (1994)

16 13 C shifts and Peptide Geometry Shifts relative to random coil with same amino acid Spera and Bax, JACS, 113,, 5490 (1991) See also: Case, (Shifts) See also: Wishart,

17 Remote Group Effects B 0 C O H deshielded B B H shielded benzene does the same thing σ remote = Δχ/r 3 (1-3cos 2 θ) Benzene protons are 2 ppm further downfield Johnson and Bovey, JCP, 29, 1012 (1962)

18 Shielding from a Benzene Ring

19 Recent Applications of Chemical Shift to Protein Structure Determination Shen Y, Bax A, Protein backbone chemical shifts predicted from searching a database for torsion angle and sequence homology JOURNAL OF BIOMOLECULAR NMR 38: ,2007 Cavalli A, Salvatella X, Dobson CM, et al. Protein structure determination from NMR chemical shifts PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA 104: , 2007 Shen Y, Lange O, Delaglio li F, et al. Consistent t blind protein structure generation from NMR chemical shift data, PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA 105: , 2008

20 Other data can be combined with Chemical Shifts Protein Targets now up to 25 kda Comparison of traditional NMR structure and predicted structure t with chemical shift and RDC data. Srivatsan, Lange, Rossi, et al. Science, 327: , 2010