Origin of Chemical Shifts
BCMB/CHEM 8190
Empirical
p
Properties
p
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
f
force
perpendicular
di l tto th
the ttrajectory.
j t
H
Hence
electrons precess.
• Circulating
Ci l ti currentt gives
i
an opposing
i field.
fi ld
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
i ½ )), αα, αβ,
β βα,
β ββ (two
(t
spins
i 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
p
is special:
p
<ψ|
ψ| H |ψ
|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’
B
• Have QM operator for linear momentum:
p0 = i(h/(2
i(h/(2π))(∂/∂x
))(∂/∂x + ∂/∂y +∂/∂z)
• But momentum in magnetic field has a “curl”
p = p0 + e(B
(B x r)) / (2c)
(2 ) = 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) / (r3c2m)
• 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)) ∫ψ0*| (x2 + y2)/r3) |ψ0 dτ
B
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
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
) …. This introduces field dependence
p
A = ((B0 x r)/2
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/(2 ))/2
H’ = B0eh/(8πmc) Lz Hence ψ’ ∝ Σn (<ψn| LZ |ψ0> /ΔE)
B’P = <ψ0 + ψ’| (e/(cm))(r x p0)/r3 |ψ0 + ψ’>
= <ψ
< 0 + ψ’|’| (e/(cm))(Lz)/r3 |ψ
| 0 + 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-E
E0)]
Implications for Paramagnetic Term
• σP is negative (B’
(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
g 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-CH
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)(L
√
z| π*> )
• <ψ
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
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 10-66
-6 = 0 x 10-6
• σc-c
=
σ
+
σ
=
(200-200)
(200
200)
x
10
cc
D
P
What about Field Perpendicular to Plane?
B0
π* = (1/√2)(pzA-pzB)
π* = (1/√2)(p0A-p0B)
A
B
• Lz| p0> = 0;
0 th
therefore,
f
σP = 0
• σ⊥ = σD + σP = (200 +0) x 10-6
• Si
Similar
il ffor iin plane,
l
perpendicular
di l tto 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
p
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
pp
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
q
homology
gy JOURNAL OF BIOMOLECULAR
NMR 38: 289-302,2007
• 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
Sh Y
Y, L
Lange O
O, D
Delaglio
l li F
F, ett al.
l C
Consistent
i t t bli
blind
d
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
Other data can be combined with Chemical
Shifts – Protein Targets now up to 25 kDa
Comparison of traditional NMR structure and predicted
structure
t t
with
ith chemical
h i l shift
hift and
d RDC d
data.
t
Srivatsan, Lange, Rossi, et al. Science, 327:1014-1018, 2010