Origin of Chemical Shifts
BCMB/CHEM 8190
Empirical Properties of Chemical Shift
υi (Hz) = γB0 (1-σi) /2π
σi, shielding constant dependent on electronic structure, is ~ 10-6.
Measurements are made relative to a reference peak (TMS).
Offsets given in terms of δ in parts per million, ppm, + downfield.
δi = (σref - σi ) x 106
or
δi = (( υi - υref )/υref) x 106
Ranges:
1H, 2H,
10 ppm;
13C, 15N, 31P,
300 ppm;
19F,
1000 ppm
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. Hence
electrons precess.
• Circulating current gives an opposing field.
B0
F
r
e- v
F = - (e/c) v x B0
B’ = -(e/c) (r x v) / r3 = -(e/cm) (r x p) / r3
• But, we actually need to treat electrons at QM level
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, 2p1, 2p0, 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:
Hz = - μ•B0, Ez = < α|- μ•B0 |α>
Begin with classical expression: substitute QM operators
μz = γ Iz (h/2π) (magnetic moment)
Ez = < α|-(γh/2π)IzB0 |α> = -½ γ(h/2π) B0<α*|α> = -½ γ(h/2π)B0
Quantum Expression for B’
• Have QM operator for linear momentum:
p0 = i(h/(2π))(∂/∂x + ∂/∂y +∂/∂z)
• But momentum in magnetic field has a “curl”
p = p0 + e(B x r) / (2c) = p0 + (e/c) A
A is the vector potential; A = (B x r)/2
B’ = -(e/cm) (r x p0) / r3 - e2 (r x A) / (r4c2m)
• Quantum mechanically:
B’ = <ψ0| -e (r x p0)/r3(cm) - e2(r x A)/(r3c2 m) |ψ0>
paramagnetic
diamagnetic
Diamagnetic Shifts
• Note: only the second term is proportional to B0 at first
order theory; this is the diamagnetic term; Lamb term
B’D = <ψ0| - e2(r x B0 x r)/(2r3c2 m) |ψ0>
Only interested in the z component:
k (x ⋅ (B0 x r)y – y ⋅ (B0 x r)x)
i
j
k
x
( Bxr ) x
y
( Bxr ) y
z
( Bxr ) z
B’D = - (e2/(2c2 m))B0 ∫ψ0*| (x2 + y2)/r3) |ψ0 dτ
Predictions: depends on electron density near to nucleus
opposes magnetic field (shields)
Examples: He
2 1s eσ = 59.93 x 10-6
Ne
10 eσ = 547 x 10-6
H
~2 1s eσ = ~60 x 10-6
HO- O withdraws ~10% ~6 ppm downfield
Paramagnetic Contribution to Shifts
• This comes from the first term – there was no
explicit B0 dependence – so carry to second order
electronic wave function is changed by field
• B0 can be in H’
ψ = ψ0 + Σn (<ψn| H’ |ψ0> / (En-E0)) ψn = ψ0 + ψ’
H0 = (1/(2m)) p02 + V … in absence of field
H = (1/(2m)) (p0 + (e/c)A) 2 + V … in presence
A = (B0 x r)/2 …. This introduces field dependence
H’ = (e/(2mc)) A⋅p0 … keeping most important term
Paramagnetic term continued
A⋅p0 = ((B0 x r)/2)⋅p0 ≡ B0⋅(r x p0)/2 = B0⋅(Lh/(2π))/2
= B0Lzh/(2π))/2
H’ = B0eh/(8πmc) Lz Hence ψ’ ∝ Σn (<ψn| LZ |ψ0> /ΔE)
B’P = <ψ0 + ψ’| (e/(cm))(r x p0)/r3 |ψ0 + ψ’>
= <ψ0 + ψ’| (e/(cm))(Lz)/r3 |ψ0 + ψ’>
Substituting ψ’ and saving only terms linear in B0
B’P ∝ Σn[(<ψ0|Lz|ψn><ψn|Lz/r3|ψ0>)/(En-E0) +
(<ψ0|Lz/r3|ψn><ψn|Lz|ψ0>)/(En-E0)]
Implications for Paramagnetic Term
• σP is negative (B’P ∝ - σP) … opposite to σD
• σP is zero unless LZ |ψ> is finite
hence, if only “s” orbitals populated, LZ |”s”> = 0
hence, small shift range for 1H
•
13C
has “p” orbitals (Lz |p1> = 1 p1) and finite σP
• Electron distribution must also be assymetric
otherwise, Σ Lz |p> = 0
hence, CH4 shift is small and resonance far upfield
13C
Example: Ethane vs Ethylene
• CH3-CH3 6 ppm,
CH2= CH2 123 ppm, Why?
• σD is about the same for both, ~200 x 10-6
• σP ∝ Σn[(<ψ0|Lz|ψn><ψn|Lz/r3|ψ0>)/(En-E0) + ..
• Examine ψn = ΣicinφI, φI = 1sc, 2sc, 2pc0, 2pc+/-1, 1sH
• Only ps count, ΔE small is most important
• Consider first excited state: π* = (1/√2)(piA-piB)
Consider Field Parallel to C-C Bond
B0
A
B
π* = (1/√2)(pxA-pxB)
π* = (1/√2)((p1A+p-1A)-(p1B+p-1B))
• Lz| π*> = (ih√2/π)((p1A-p-1A)-(p1B-p-1B))/(2i)
= (ih√2/π)(pyA-pyB)>
• <ψ0|Lz| π*> is finite if pyA, pyB are populated in ψ0
• ψ0 must also be assymetric – look at MOs
Molecular Orbitals for Ethylene
ψ3, 3 nodes, 1 bond
ψ2, 2 nodes, 4 bonds
E
ψ1, 1 node, 5 bonds
• Fill with electrons: 2x6 for C, 4 for H = 16
• 4 in 1sC, 2 in π0 (⊥ to plane), 2 in C-C σ, 4 in C-H σ
• Implies 4 maximum in above
Calculating Paramagnetic Contribution
• Only ψ1, ψ2, contribute
• ψ1, is symmetric, implies <ψ1|Lz| π*> is zero
• ψ2, is asymmetric and counts
• σP = -(eh/(2πmc))2<(1/r3)>2p c22 ≅ -200 x 10-6
• σc-c = σD + σP = (200-200) x 10-6 = 0 x 10-6
What about Field Perpendicular to Plane?
B0
π* = (1/√2)(pzA-pzB)
π* = (1/√2)(p0A-p0B)
A
B
• Lz| p0> = 0; therefore, σP = 0
• σ⊥ = σD + σP = (200 +0) x 10-6
• Similar for in plane, perpendicular to pπs
• σ (predict) = ⎡0
⎤ (observe) ⎡− 20
⎢
⎢
⎢⎣
0
⎥
⎥
200⎥⎦
⎢
⎢
⎢⎣
⎤
⎥
120
⎥
200⎥⎦
• Isotopic shift = 1/3 Tr σ = 70-100 ppm below Me
• Waugh, Griffin, Wolff, JCP, 67 2387 (1977) – solids NMR
13C
Chemical Shift Calculations on Peptides
α helix, -57, -47; β sheet, -139, 135; Oldfield and Dios, JACS, 116, 5307 (1994)
13C
•
•
•
•
shifts and Peptide Geometry
Shifts relative to random coil with same amino acid
Spera and Bax, JACS, 113, 5490 (1991)
See also: Case, http://www.scripps.edu/mb/case (Shifts)
See also: Wishart, http://redpoll.pharmacy.ualberta.ca/shiftz
Remote Group Effects
B0
C
O
H
deshielded
B’
B’
H
shielded
benzene does the same thing
• σ’remote = Δχ/r3 (1-3cos2θ)
• Benzene protons are 2 ppm further downfield
• Johnson and Bovey, JCP, 29, 1012 (1962)
Shielding from a Benzene Ring
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: 289302,2007
• 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:
9615-9620, 2007
• Shen Y, Lange O, Delaglio F, et al. Consistent blind protein
structure generation from NMR chemical shift data,
PROCEEDINGS OF THE NATIONAL ACADEMY OF
SCIENCES OF THE UNITED STATES OF AMERICA 105:
4685-4690, 2008
• Han, B., Liu, Y. F., Ginzinger, S. W. & Wishart, D. S. (2011).
SHIFTX2: significantly improved protein chemical shift
prediction. Journal of Biomolecular NMR 50, 43-57.
Other data can be combined with Chemical
Shifts – Protein Targets now up to 25 kDa
Comparison of traditional NMR structure and predicted
structure with chemical shift and RDC data.
Srivatsan, Lange, Rossi, et al. Science, 327:1014-1018, 2010