Singlet nuclear magnetic resonance of nearly-equivalent spins.

Abstract

Nuclear singlet states may display lifetimes that are an order of magnitude greater than conventional relaxation times. Existing methods for accessing these long-lived states require a resolved chemical shift difference between the nuclei involved. Here, we demonstrate a new method for accessing singlet states that works even when the nuclei are almost magnetically equivalent, such that the chemical shift difference is unresolved. The method involves trains of 180° pulses that are synchronized with the spin-spin coupling between the nuclei. We demonstrate experiments on the terminal glycine resonances of the tripeptide alanylglycylglycine (AGG) in aqueous solution, showing that the nuclear singlet order of this system is long-lived even when no resonant locking field is applied. Variation of the pulse sequence parameters allows the estimation of small chemical shift differences that are normally obscured by larger J-couplings.

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Singlet nuclear magnetic resonance of nearly-equivalent
spins


Journal: Physical Chemistry Chemical Physics
Manuscript ID: Draft
Article Type: Communication
Date Submitted by the
Author:
n/a
Complete List of Authors: Tayler, Michael; Southampton University, School of Chemistry
Levitt, Malcolm; Southampton University, School of Chemistry




Physical Chemistry Chemical Physics - For Review Only

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Singlet nuclear magnetic resonance of nearly-equivalent spins
Michael C. D. Tayleraand Malcolm H. Levitt∗a
Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX
First published on the web Xth XXXXXXXXXX 200X
DOI: 10.1039/b000000x
Nuclear singlet states may display lifetimes that are an order of magnitude greater than conventional relaxation times. Existing
methods for accessing these long-lived states require a resolved chemical shift difference between the nuclei involved. Here,
we demonstrate a new method for accessing singlet states that works even when the nuclei are almost magnetically equivalent,
such that the chemical shift difference is unresolved. The method involves trains of 180◦pulses that are synchronized with
the spin-spin coupling between the nuclei. We demonstrate experiments on the terminal glycine resonances of the tripeptide
alanylglycylglycine (AGG) in aqueous solution, showing that the nuclear singlet order of this system is long-lived even when
no resonant locking field is applied. Variation of the pulse sequence parameters allows the estimation of small chemical shift
differences that are normally obscured by larger J-couplings.
1Introduction
Nuclear singlet states are non-magnetic states of nuclear spin
pairs that may exhibit extraordinarily long lifetimes.1–3In the
case of15N-labeled nitrous oxide, for example, the singlet
state of the15N pair has a decay time constant of almost 25
minutes in solution.4The long lifetime property suggests ap-
plications to the study of slow motions, slow chemical ex-
change,5–8and the transport of hyperpolarized spin order.9,10
The rate of singlet decay has also been shown to carry infor-
mation on the locations of neighboring magnetic nuclei.11
Exploiting nuclear singlet order typically requires three dif-
ferent experimental stages, within each of which the res-
onance frequency difference induced by different chemical
shifts plays a key role: (i) nuclear magnetization is converted
into nuclear singlet order under the chemical shift asymmetry
of the nuclei; (ii) singlet order is then locked by suppressing
the chemical shift difference, so preserving it; (iii) the sin-
glet order is converted back to observable nuclear magnetiza-
tion, again by exploiting the chemical shift difference. The
long singlet lifetime is manifested during stage (ii). This re-
quires that singlet-to-triplet interconversion is suppressed, by
either transporting the sample into a region of low magnetic
field,1,12, keeping the sample in high magnetic field while ap-
plying resonant radiofrequency irradiation.2,13, or switching
the symmetry of the molecule using a chemical reaction10.
In this article we explore the situation in which the differ-
ence of nuclear resonance frequencies is very small, i.e. near
magnetic equivalence. If the chemical shifts of two nuclear
sites are so similar that the spin-spin coupling exceeds the dif-
ferenceinchemicalshiftfrequencies, thespinsystemissaidto
aSchool of Chemistry, University of Southampton, SO17 1BJ, UK. Fax:
+4423 8059 3781; Tel: +4423 8059 6753; E-mail: mhl@soton.ac.uk
be ‘extremely-strongly-coupled’, or ‘nearly-equivalent’. The
NMR spectrum of nearly-equivalent spin pairs displays a sin-
gle peak since the spin-spin coupling suppresses the effect of
the small chemical shift difference. This situation is found at
high magnetic fields when the chemical shifts of two protons
are extremely close, and is a rather widespread situation for
NMR in low magnetic fields.
Near magnetic equivalence creates both difficulties and op-
portunities for singlet NMR. Stages (i) and (iii) of the ex-
perimental procedure become markedly more difficult in the
case of near-equivalence. Until recently, all NMR pulse se-
quences for converting magnetization into singlet order, and
back again, have required a chemical shift frequency differ-
ence that is larger than the spin-spin coupling.8Stage (ii),
on the other hand, is markedly easier in the case of near-
equivalence. It is much easier to suppress a small chemical
shift difference than a large one. Indeed, as shown below,
no intervention at all is required to sustain the long-lived sin-
glet state in the case of near-equivalence. This is an attractive
prospect for certain applications of singlet NMR.
We discuss a technique for stages (i) and (iii) that we have
previously used in the context of low-field NMR, where chem-
ical shift frequencies are vanishingly small.14In that case, ra-
diofrequency (RF) pulses applied entirely outside the NMR
magnet were used to transform, reversibly, the Zeeman mag-
netization of a spin pair into nuclear singlet order. The pro-
tocol uses ‘trains’ of spin echoes, with the pulse delay times
synchronized with the scalar spin-spin coupling, J. Only the
difference in the chemical shifts is important for triplet-singlet
conversion, since this is the mechanism that connects the sym-
metric and antisymmetric spin states. Here we show that this
method is applicable, generally, to pairs of coupled spins in
high field.
1–6 | 1
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2Theory
2.1 Singlet and triplet mixing
The singlet and triplet spin quantum states for a pair of spin-
1/2 nuclei are defined as
|S0? = (|αβ? − |βα?)/√2
|T0? = (|αβ? + |βα?)/√2
wherethesubscriptsindicatethevaluesoftheprojectionquan-
tum number, m, defined by (I1z+ I2z)|φm? = m¯ h|φm?.
The quantum mechanical evolution in this representation is
straightforward to follow because states with different m are
isolated from one another, or non-interconverting, since dur-
ing free evolution the two-spin Hamiltonian
|T+1? = |αα?
|T−1? = |ββ?
(1)
H = ω0
1I1z+ ω0
2I2z+ 2πJI1· I2
(2)
commutes with the total z angular momentum operator. Here
therotating-framechemicalshiftfrequenciesaredefinedω0
ω0(δj− δref) where δj(j = 1,2) are the chemical shifts of
the two sites and δrefis the chemical shift corresponding to the
RF irradiation frequency. ω0is the nuclear Larmor frequency
in the strong magnetic field.
Both the ‘outer’ triplet states with |m| = 1 are stationary
under free evolution, save for acquiring the time-dependent
phases exp[−i(πJ − mω0
energies. The states with m = 0 are the only ones that mix.
In the following we therefore regard |S0? and |T0? as a
closed two-level quantum system. A state vector, {?T0|ψ0?,
?S0|ψ0?}, is defined to represent the superposition- or ‘qubit’
state,15|ψ0?, for this system.
The evolution of a qubit state may be viewed geometrically
in the three-dimensional representation of its Hilbert space
known as the Bloch sphere.16The Bloch sphere is a map of all
qubits {cos(ξ/2), exp(−iφ)sin(ξ/2)} into points (ξ, φ) on
the surface of a unit sphere (figure 1a) so that unitary transfor-
mations between qubits are equivalent to rotations (to within
a phase factor). A rotation, Rn(ϕ), in the Bloch sphere that
rotates through an angle ϕ about the axis n = (nx,ny,nz)
can be shown to correspond with the matrix representation
j=
1− mω0
2)t], of their characteristic
Rn(ϕ) = exp[−iϕn · (X, Y, Z)/2]
where E,X,Y and Z are the Pauli matrices.16,17
?
?
The m = 0 qubit evolves according to the 2 × 2 matrix
representation of equation 2, which may be expressed as
?
(3)
E =
1
0
0
1
?
X =
?
0
1
1
0
?
?
Y =
0i
0
−i
?
Z =
?
1
0
0
−1
(4)
H ≡ −πJt
2
E +Ω
2
(X, Y, Z) · (sinθ, 0, cosθ)
?
,
(5)
Fig. 1 Bloch vector representation of the m = 0 qubit, showing
time evolution under equation 2 for the starting state |ψ0? = |T0?
on introducing the following terms:
Ω=
?
arctan(ω0∆δ/2πJ)
(2πJ)2+ (ω0∆δ)2
(6)
(7)
θ=
The resulting propagator, U, in the basis of |S0? and |T0? is
U(t)=exp[−iHt]
=exp[+iπJt/2]R(sinθ,0,cosθ)(Ωt)
(8)
(9)
for the free evolution. This identifies with a rotation through
angle Ωt about an axis in the ‘XZ’ plane, in the Bloch sphere,
at azimuth θ to the vertical (figure 1b). Using |ψ0?(0) = |T0?
this visualization allows one to verify easily that:
• If the nuclei are weakly-coupled (i.e. when |2πJ| ?
|ω0∆δ|) then |θ| ≈ π/2 and Ω ≈ ω0∆δ; singlet and
triplet states interchange at a rate approximately equal to
the difference of the nuclear Larmor frequencies;
• For very strongly coupled nuclei, (where ω0∆δ ? 2πJ),
the angle θ is small and singlet-triplet mixing is sup-
pressed. In this regime it is difficult to achieve transfer
between the two states;
• For perfectly magnetically equivalent nuclei, such as in
zero magnetic field, or where chemical shifts are coinci-
dental, the angle θ is zero. The states |S0? and |T0? are
exact eigenstates and do not interconvert.
2.2Resonant spin echo transfer
Maximum transfer between the singlet and triplet states oc-
curs on leaving a time delay π/Ω, but the maximum transfer
amplitude is equal to sinθ and goes to zero as θ → 0. This is
unfavourable in the extreme strong coupling situation, for re-
gardless of evolution time the singlet and triplet states do not
interconvert to a significant extent.
Singlet-to-triplet mixing may be arranged to accumulate,
however, over the course of a spin echo of duration 2τ (two
2 |
1–6
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Fig. 2 Singlet and triplet mixing under extreme strong coupling
using a synchronized spin echo train. The trajactory of the Bloch
vector (thick curve) shows the cumulative conversion of |T0? into
|S0? due to four echoes, for the optimal interval τ = π/2Ω
evolution periods of equal length τ, bisected by a 180◦RF
pulse). The qubit state at the end of an echo is determined
by the transformation U(τ)RZ(π)U(τ), where the 180◦RF
pulse leads to a 180◦rotation about the Z-axis in the Bloch
sphere. This may be seen noting that a 180◦pulse inverts the
sign of |T0?, while leaving |S0? unchanged, and provided the
pulse is ideal, there are no transitions between m = 0 and
|m| = 1. The non-commuting RZ(π) and U(τ) lead to cu-
mulative singlet-triplet conversion at the end of the echo. The
optimal transfer occurs when τ is set to π/2Ω. In this case the
propagator is
U(τ)RZ(π)U(τ) = e(iπJτ)R(cosθ,0,−sinθ)(4θ)
(10)
which in the strong-coupling case is approximately a rotation
of the Bloch vector through angle 4θ about the X-axis, ne-
glecting the phase factor.
As figure 2 shows, n successive echoes generate an over-
all rotation through the angle 4nθ in the Bloch sphere. For
extremely-strongly coupled spins, the rotation axis is close
to the X-axis, so |T0? ⇔ |S0? conversion is achievable us-
ing back-to-back echoes with intervals τ = π/2Ω ≈ 1/4J,
and the echo number n chosen so that 4nθ is close to π. In
practice, relaxation losses may limit the maximum achievable
transfer.
Regardless of the coupling limit, the mixing rate depends
solely on the value of ω0∆δ, not J. As expected, the singlet-
triplet conversion becomes increasingly slow as the nuclei ap-
proach magnetic equivalence.
Similar rotations in the m = 0 Bloch sphere are induced
by modulation of the static magnetic field at the J-coupling
frequency.18
Fig. 3 The ‘Magnetization-to-Singlet’ (M2S) RF pulse sequence
2.3Preparation of long-lived states
Conversion between equilibrium Zeeman magnetization and
nuclear singlet order was discussed in ref14. We shall briefly
re-outline the density operator transformations induced by
the M2S (Magnetization-to-Singlet) pulse sequence,14which
contains two synchronized echo trains (see figure 3). To start,
we have |S0? and |T0? populations equal, with the equilib-
rium magnetization Izcorresponding to population imbalance
across the m = ±1 triplet levels:
|αα??αα| − |ββ??ββ| ≡ |T+1??T+1| − |T−1??T−1| (11)
The initial 90◦‘y’ pulse converts starting magnetization
into single-quantum coherences between the inner (m = 0)
and outer (|m| = 1) triplet states, namely
Ix≡ |T+1??T0| + |T0??T+1| + |T0??T−1| + |T−1??T0| (12)
Thesesingle-quantumcoherencesaretransformedintoones
between the singlet and outer triplet states, through a π rota-
tion in the Bloch sphere:
i(|T+1??S0| + |S0??T+1| − |S0??T−1| − |T−1??S0|) (13)
whichintheCartesianproductoperatorrepresentationisequal
to (I1y−I2y). This conversion exploits a J-synchronized spin
echo train for approximate interchange of |S0? and |T0?, so
the number of echoes required, n1, is the integer such that
4n1θ ≈ π. As the figure depicts, the 180◦
implemented as 90◦
the high-accuracy of the inversions required during the pulse
train.19
A following 90◦‘x’ pulse rotates the transverse magneti-
zation to (I1z− I2z), equivalent to zero-quantum coherence
between the m = 0 states.
xRF pulses may be
x180◦
y90◦
xcomposite rotations, to ensure
i(|T0??S0| + |S0??T0|) ≡ I1z− I2z
(14)
1–6 | 3
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Such coherences have been shown, in suitable cases, to be
slowly-relaxing compared to T1.20Long-lived coherences
(LLCs), as these are known, are conventionally prepared by
frequency-selective inversion yet such methods fail when the
pair chemical shift difference is unresolved. The above proves
a route to LLCs even for very strongly-coupled spin pairs.
The coherences of equation 14 are finally converted to pop-
ulation difference between the m = 0 states. Free evolution
for time τ = 1/4J first serves to phase shift these coherences,
so that they have opposite sign:
|T0??S0| − |S0??T0|.
(15)
A π/2 rotation in the Bloch space, i.e., the one that transforms
|S0? → (|S0? + |T0?)/√2, then creates a singlet-triplet popu-
lation difference:
|S0??S0| − |T0??T0|
(16)
In this transformation the appropriate number of echoes, n2,
should be approximately half n1, ensuring 4n2θ ≈ π/2.
The evolution of the singlet-triplet population difference is
described in refs.12–14. After equilibration of the triplet popu-
lations, the difference between the singlet population and the
average of the three triplet populations decays with the time
constant TS.
Conversion of singlet order into Zeeman magnetization,
the reverse process, should evidently be performed by ap-
plying the above transformations in reverse chronological or-
der. The reverse pulse sequence is denoted S2M (Singlet-to-
Magnetization).14
3Results and discussion
Figure 4 shows the structure of the tripeptide L-alanyl-glycyl-
glycine (AGG), which contains two isolated CH2units. The
pairs of protons within each of these units are chemically
nonequivalent, by virtue of the alanine chiral center that ren-
ders the molecule unsymmetrical.
stereocenter, however, generates only a very small chemical
The remoteness of the
Fig. 4 Proton NMR spectrum of AGG (20mM in D2O), 9.4 T.
Fig. 5 Singlet relaxation of the extremely-strongly coupled BB’
proton pair in AGG: (a) when no RF locking is applied; (b) during
forced magnetic equivalence under an RF field. An additional delay
of 3 seconds is provided in (b) in order to allow equilibration of the
triplet populations. Blocks ‘M2S’ and ‘S2M’ in the pulse diagrams
abbreviate the forward and reverse ‘Magnetization to Singlet’
transformations, respectively.
asymmetry, as deduced from the one-dimensional spectrum.
Consider the ‘central’ glycine proton pair, which is only three
atom centers away from the chiral center. The two CH2pro-
tons (labeled AA’) show a characteristic strong-coupling pat-
tern with |2JHH| = 18 Hz, ω0∆δ/2π = 14 Hz, correspond-
ing to a chemical shift difference ∆δ = (δ1− δ2) of 0.035
ppm. The terminal CH2resonance appears as just a single
peak with the chemical shifts of the nuclei unresolved (coin-
cidentally, the resonance lies on top of one of the peaks for
the other diastereotopic pair). These protons (labeled BB’)
are ‘nearly equivalent’ because the chiral center is remote, six
sigma bonds away along the peptide chain. The AA’ and BB’
spin pairs both have a T1of ≈ 0.75 s, as measured by satura-
tion recovery at 9.4 T on a degassed solution.
We have detected singlet order on the pair BB’ by using the
pulse sequences shown in figures 5a and 5b. At a field of 9.4 T,
Zeeman order was transported into singlet- triplet population
difference using the resonant echo method described through
4 |
1–6
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