Deuterium REDOR: Principles and Applications for Distance Measurements

Article abstract of DOI:10.1006/jmre.1999.1710

The application of short composite pulse schemes (90°_x-90°_y-90°_x and 90°_x-180°_y-90°_x) to the rotational echo double-resonance (REDOR) spectroscopy of X-²H (X: spin 1/2, observed)

Full text of DOI:10.1006/jmre.1999.1710

Journal of Magnetic Resonance 138, 54–65 (1999)
Article ID jmre.1999.1710, available online at http://www.idealibrary.com on
Deuterium REDOR: Principles and Applications
for Distance Measurements
,
1
I. Sack,* A. Goldbourt,† S. Vega,† and G. Buntkowsky*
*
Institut f u¨ r Organische Chemie der Freien Universit a¨ t Berlin, Takustrasse 3, D-14195 Berlin, Germany; and †Weizmann Institute, Rehovot, Israel
Received July 27, 1998; revised December 15, 1998
1
2
The application of short composite pulse schemes (90°
and 90°–180°–90°
) to the rotational echo double-resonance
REDOR) spectroscopy of X– H (X: spin , observed) systems with
–90°
–90°
x
y
x
can be used to determine X– H or X– H distances between an
1
x
y
x
I ϭ nucleus X and a neighboring proton or deuteron, namely,
2
2
1
2
(
the chemical shift and the nuclear magnetic dipolar interaction.
The value of the chemical shift interaction is very sensitive to
structural features such as the primary and secondary structure
of molecules. For example, it has been shown recently that
isotropic N chemical shielding can be used to determine
N– H or N– H distances (7, 8). However, there is no easy
direct way to interpret the CS data in terms of nuclear dis-
large deuterium quadrupolar interactions has been studied exper-
imentally and theoretically and compared with simple 180° pulse
schemes. The basic properties of the composite pulses on the
deuterium nuclei have been elucidated, using average Hamilto-
nian theory, and exact simulations of the experiments have been
achieved by stepwise integration of the equation of motion of the
15
15
1
15
2
15
2
density matrix. REDOR experiments were performed on N– H
13
2
2
in doubly labeled acetanilide and on C– H in singly H-labeled tances, because calibration measurements on systems with
acetanilide. The most efficient REDOR dephasing was observed well-known distances are needed to correlate these data to
when 90°
dephasing due to simple 180° deuterium pulses is about a factor of
less efficient than the dephasing due to the composite pulse
–180°
–90°
composite pulses were used. It is found that the distances. Nuclear dipolar interactions, on the other hand, give
x
y
x
direct geometrical information about intra- or intermolecular
2
1
distances. Thus the study of these interactions between H–X
2
sequences and thus the range of couplings observable by X– H
2
and H–X is generally much more informative than CS data.
REDOR is enlarged toward weaker couplings, i.e., larger dis-
2
For simple systems consisting of pairwise H–X-labeled
2
15
tances. From these experiments the H– N dipolar coupling be-
2
13
molecules, the values of dipolar interactions can be obtained by
simply recording the static NMR powder spectra and analyzing
their lineshapes (9, 10). It is advantageous to record the X-spin
tween the amino deuteron and the amino nitrogen and the H– C
dipolar couplings between the amino deuteron and the ␣ and ␤
carbons have been elucidated and the corresponding distances
have been determined. The distance data from REDOR are in
spectra, because it is nearly impossible to elucidate any dipolar
2
good agreement with data from X-ray and neutron diffraction, couplings from broad H spectra, because of the strong qua-
showing the power of the method. © 1999 Academic Press
drupolar interactions of the deuterons. In many systems the
Key Words: H REDOR; composite pulses; acetanilide; C– H dipolar interactions are much smaller than the chemical shift
2
13
2
1
5
2
distances; N– H distances.
anisotropy of the X spin, making it difficult to resolve the
dipolar interaction in static solid state NMR spectra. This
problem was already overcome in the sixties by the introduc-
tion of the spin echo double-resonance experiment (SEDOR)
INTRODUCTION
(
(
11) technique and by separated local field experiments
12, 13).
When more complicated systems with several inequivalent
In recent years solid state nuclear magnetic resonance
NMR) methods (1–4) have become an important tool for
(
structure determination in solid systems, in particular when
these systems exist in an amorphous or disordered phase,
which cannot easily be studied by X-ray diffraction techniques.
Moreover, solid state NMR has become, in conjunction with
inelastic neutron scattering, the primary technique for studying
proton or deuterium positions in solids, because of the well-
known problem in locating hydrogen atoms of X-ray scatter-
ing. Some recent examples are the localization of protons or
deuterons in H-bonded systems (5, 6).
deuterons and/or X nuclei are investigated, or when it is not
possible to label only the interesting H positions, special
2
high-resolution solid state NMR techniques can be employed.
By far the most important of these techniques is the rotational
echo double-resonance (REDOR) experiment (14, 15), which
monitors the dipolar interaction of spin pairs during magic
angle spinning (MAS), often combined with proton–X cross-
polarization (CP-MAS) (16) to enhance the X-spin signals. The
dipolar REDOR dephasing of the signals is accomplished by
the application of two 180° inversion pulses during each rotor
period.
There are two main interactions in NMR spectroscopy that
1
To whom correspondence should be addressed.
5
4
1
090-7807/99 $30.00
Copyright © 1999 by Academic Press
All rights of reproduction in any form reserved.

DEUTERIUM REDOR
55
So far, most REDOR studies have been concerned with IS
1
pairs of spin- nuclei, the main reason being that the large
2
1
quadrupolar interaction of spins with S Ͼ 2 complicates the
interpretation of the REDOR data, because of the inefficient
inversion of the 180° pulses on these nuclei. For spin pairs with
S Ͼ 1 it is now common to apply the rotational echo adiabatic
pulse double-resonance (REAPDOR) method (17, 18).
2
1
2
For the study of the spin pair X– H (with S ϭ and I ϭ 1),
FIG. 1. Molecular structure of acetanilide.
two REDOR schemes have been suggested: the observation of
2
the H nucleus, while all 180° pulses are applied to the X
2
nucleus (19), or the observation of the X nucleus, while the H
addition, average Hamiltonian theory is used to understand the
main features of the X-spin REDOR decays.
nucleus is pulsed (19–23). The advantage of the first experi-
ment is that only moderate B fields that are larger than the
1
Acetanilide (Fig. 1) has been chosen for several reasons: (i)
it has only one exchangeable hydrogen, which makes specific
labeling easy; (ii) it is a molecule with a large deuterium
quadrupolar interaction (146 kHz) and with different types of
carbons, making it a good example for this type of experiment;
CSA dispersion are sufficient for spin inversion. In this
2
REDOR experiment the presence of H quadrupolar interaction
2
of order 100–200 kHz results in H MAS spectra with ca.
2
0–40 rotational sidebands for moderate spinning frequencies
of ϳ5 kHz. The widths of these sidebands make it difficult to
(
iii) it has a known crystal structure (28–30), allowing a
2
resolve the spectra of different H sites in the sample. Spectral
comparison of NMR and diffraction results; and (iv) it has
favorable NMR parameters, such as relatively short relaxation
2
resolution between different H sites therefore requires selec-
13
tive deuterium labeling. However, this is feasible only in some
fortunate cases, while for the majority of systems it is scarcely
times and well-separated C spectral lines (9, 10).
The rest of the article is organized as follows. After this
achieved. For example, most of the hydrogens involved in the Introduction, the theoretical background necessary for under-
2
formation of H bridges are easily exchanged and selective standing REDOR measurements on X– H spin systems is
2
labeling of X– H spin pairs is impossible or at least a severe given, followed by the numerical methods used. After an
synthetic problem.
overview of the experimental setup, experimental results are
From these considerations it follows that REDOR experi- presented, discussed, and summarized.
2
ments that monitor the X nuclei and irradiate the H nuclei hold
2
promise for X– H REDOR spectroscopy (19–23). In experi-
THEORY
ments of these types, it is possible to measure the dipolar
couplings even when several inequivalent deuterons are Pulse Sequences
present. However, the problem with this approach is that very
As mentioned before, the main problem of X-spin-detected
X– H REDOR experiments is the relatively low B1 field
2
strong B fields are necessary for faithfully inverting the H
2
1
nuclei, something that is not available in commercial triple-
tuned probes.
strength that can be obtained in our triple-tuned probes for the
2
excitation of the deuterons. In our case the H B field was
1
The application of phase cycling schemes (24), such as XY-8
or XY-16, is not sufficient to compensate for the influence of
the large quadrupolar broadening on the REDOR decays.
These schemes are used mainly to correct pulse imperfections
and the effects of small CS values. A solution to this problem
is to use composite inversion pulses (25–27). To improve the
X-spin REDOR dephasing, composite pulses can be used to
substitute the simple 180° pulses on the deuterium spins. For
nonrotating samples there are several elaborate composite
pulse schemes available for the inversion of quadrupolar spin-1
nuclei (26, 27). For rotating samples only the short composite
between 30 and 40 kHz, which is about four to five times
2
smaller than the H quadrupolar interaction in our samples.
2
This leads to poor excitation of the H NMR spectrum. This
situation is a little improved in MAS experiments, where the
quadrupolar frequencies are time-dependent during the pulses;
however, in rotor-synchronized irradiation schemes the spins
are always excited when they experience the same frequencies.
To improve this excitation profile we performed experiments
and calculations with composite pulse sequences. In the liter-
ature one can find a variety of composite pulse sequences that
2
are efficient for the inversion of H nuclei in nonrotating
sequences are suitable, due to the time restrictions implied by samples (26, 27). The application of these sequences in MAS
the sample rotation.
experiments is generally not feasible when the length of the
We have explored this experimentally by performing studies sequence becomes comparable to the rotation period. We
1
5
2
13
2
of N– H in doubly labeled and C– H in singly and doubly therefore chose to use the simple composite sequences 90 –
x
2
labeled acetanilide. To analyze the REDOR decay signals we 180 –90 and 90 –90 –90 for the inversion of the H nuclei.
y
x
x
y
x
have simulated the evolution of these data, using a numerical The actual pulse schemes used for the irradiation in our
integration of the equation of motion of the density matrix. In REDOR experiments are shown in Fig. 2.

5
6
SACK ET AL.
variety of dipolar interactions REDOR decay curves were
calculated and compared with the experimental results. Be-
cause of the quadrupolar interaction and the relatively low RF
fields, the universal REDOR ⌬S/S curve (14, 15) cannot be
used in this case.
The spin propagators for one rotation period of the different
steps of the experiment have been calculated by stepwise
numerical integration (⌬␶ ϭ 1 ␮s constant time steps) of the
spin density matrix equation. The REDOR decay signals were
evaluated as a function of the number of rotor cycles by
repeated application of these propagators. The B field strength
1
in the calculations was chosen in such a way that the products
2
of the experimental duration of the RF pulses and the B field
FIG. 2. Pulse sequences used for the X– H REDOR experiments.
1
are equal to ␲ or ␲/2 (i.e., ␥B n⌬␶ ϭ ␲ or ␲/2) for spins on
1
resonance. For the powder integration, summations of the
signals from single crystallites, defined by their Euler angles in
the rotor frame, were performed using the method of Cheng et
al. (31). The set with 200 triples of Euler angles was used to
simulate the data in the course of the data-fitting process. This
was sufficient to give a good approximation of the experimen-
tal data in a reasonable CPU time. The final simulations were
performed with the set of 1154 angle triples. The typical time
for a simulation of a decay curve is approximately 750 s on a
Pentium 200. The errors of the determined dipolar couplings
have been estimated by determining upper and lower limits of
the dipolar coupling strengths such that 70% of the experimen-
tal data are enclosed within the limiting curves.
1
A heteronuclear spin pair with a spin- nucleus I (for exam-
2
1
3
15
2
ple, C or N) coupled to a spin-1 nucleus S (for example, H)
in a solid has a high-field MAS NMR Hamiltonian of the form
1
3
2
z
H͑t͒ ϭ ␻ ͑t͒ ⅐ 2I S ϩ ␻ ͑t͒͑3S Ϫ 2͒ ϩ H ͑t͒, [1]
D
z
z
Q
RF
where the last term represents a set of RF pulses applied to both
spins and the time-dependent coefficients are due to the sample
spinning; ␻ (t) is the quadrupolar interaction and ␻ (t) the
dipolar coupling between the I and S spins. Terms correspond-
ing to off-resonance and chemical shift anisotropy parameters
are not essential for the forthcoming discussion and are there-
fore omitted.
Q
D
The Average Hamiltonian
The dipolar coupling coefficient ␻ (t) depends on the dis-
D
tance between the two spins and the time-dependent angle ␽(t)
Further insight into the effect of the application of composite
between the internuclear vector and the external magnetic field: pulses to the S-spin system can be gained by analyzing the
average Hamiltonians of the spin system. To evaluate the
I
S
response of the spin system to the Hamiltonian in Eq. [1] it is
most convenient to eliminate the RF term by transforming it to
the toggling frame (32). This can easily be accomplished by
transforming the linear and bilinear operators in Eq. [1] ac-
cording to the applied pulse sequence with
␮0
␥ ␥
3
2
␻D͑t͒ ϭ
ប
͑3 cos͑␽͑t͒͒ Ϫ 1͒
4
r
D
2
ϭ
͑3 cos͑␽͑t͒͒ Ϫ 1͒.
[2]
2
Detection of the D coefficient provides the internuclear dis-
tance. The sample spinning removes the effect of D on the
MAS FID signal by averaging the dipolar interaction to zero.
The effect can be recovered by applying a set of RF pulses that
interferes with the averaging of the dipolar interaction in the
course of the MAS experiment and results in an effective
dipolar signal decay.
Iz͑t͒ ϭ
͸
f ͑t͒I , Sz͑t͒ ϭ
͸
g ͑t͒S ,
[3]
[4]
p
p
p
p
pϭx,y,z
pϭx,y,z
resulting in the toggling frame Hamiltonian (33, 34)
T
H ϭ
͸
␻ ͑t͒ f ͑t͒ g ͑t͒
D p q
p,qϭx,y,z
Numerical Methods
1
3
ϫ 2I I ϩ
͸
␻ ͑t͒͑3g ͑t͒S S Ϫ 2͒,
Q pq p q
p
q
2
p,qϭx,y,z
For full quantitative evaluation of the X– H REDOR data,
numerical simulations are necessary. These are done by solv-
ing the equation of motion of the density matrix of the two-spin with g (t) ϭ g (t)g (t).
pq
p
q
system considering all experimental parameters, such as the
For XY-4 REDOR experiments, with ideal 180° pulses
sample rotation frequency, the RF irradiation pulse lengths and applied only to the S spin, the g (t) coefficient becomes a
z
intensities, and the quadrupolar interaction strength. For a simple step function equal to Ϯ1, changing sign at the posi-

DEUTERIUM REDOR
57
tions of the pulses (33, 34). All other coefficients are zero, matrix. This shows that the I-spin-detected REDOR signal
1
3
except f (t) ϭ 1 and g (t) ϭ 1, and therefore the Hamiltonian decays only to ,
z
zz
for this case becomes
REDOR
1
2
S
͑nT ͒ ϭ ϩ cos͑n ␻˜ T ͒,
[12]
R
3
3
D
R
T
1
3
2
z
H ͑t͒ ϭ ␻ ͑t͒ g ͑t͒ ⅐ 2I S ϩ ␻ ͑t͒͑3S Ϫ 2͒.
[5]
D
z
z
z
Q
2
5
leaving the Ix term of the density matrix untouched.
This Hamiltonian commutes with itself at all times and its
average Hamiltonian for one rotor cycle becomes (32)
In the case of pulses with finite lengths the form of the
toggling frame Hamiltonian becomes more complicated, and
the coefficients f (t) and g (t) have to be calculated explicitly.
In this publication we will discuss two examples of I-spin-
detected REDOR experiments on an I–S spin system with all
p
p
͑
0͒
H
ϭ ␻˜ I S with
D
z
z
TR/2
TR
2
pulses on the S ϭ 1 spin, implying that fx,y(t) ϵ 0 and f
z
(t) ϵ
␻
˜ ϭ
Ϫ
␻ ͑t͒dt ϩ
D
␻ ͑t͒dt ,
[6]
D
͵
͵
D
1
. In particular we will discuss the effect of the average
T2
ͩ
ͪ
0
TR/ 2
toggling frame Hamiltonian on the REDOR decay of the X-
spin signal.
where ␻˜ is the orientation-dependent REDOR frequency for a
D
single crystallite.
XY-4 REDOR
To proceed it is useful to rewrite the Hamiltonian in terms of
fictitious spin-half operators. Assuming a numbering of the
eigenstates as
First we consider the REDOR experiment with four XY-4
80° pulses applied to the S spin during two rotor cycles. In
1
Fig. 3 the time-dependent g (t) and g (t) coefficients (see
p
pq
also Appendix A for the explicit form of the g (t) and g (t))
p
pq
1
4
ϵ ͉␣, 1͘; 2 ϵ ͉␣, 0͘; 3 ϵ ͉␣, Ϫ1͘;
ϵ ͉␤, 1͘; 5 ϵ ͉␤, 0͘; 6 ϵ ͉␤, Ϫ1͘,
for the toggling frame Hamiltonian of Eq. [4] are shown.
A zero-order average Hamiltonian can then be obtained by
integration of its coefficients over the two rotor periods. The
resulting coefficients are functions of the exact forms of the
dipolar and quadrupolar coefficients. Simple inspection shows
that the terms involving g (t), g (t), g (t), g (t), and g (t)
[7]
the Hamiltonian in Eq. [1] can be expressed in the convenient
form (omitting the RF term)
x
y
xz
yz
xy
(
0)
␣
␤
do not contribute to H . The remaining terms result in an
H͑t͒ ϭ H ͑t͒ ϩ H ͑t͒
average Hamiltonian of the form
␣
13
z
2
3
12
z
23
z
H ͑t͒ ϭ 2␻ ͑t͒I ϩ ␻ ͑t͒͑I Ϫ I ͒
D
Q
͑
0͒
␣͑0͒
␤͑0͒
␤
46
z
2
3
45
z
56
z
H
ϭ H
ϩ H
H ͑t͒ ϭ Ϫ2␻ ͑t͒I ϩ ␻ ͑t͒͑I Ϫ I ͒
[8]
D
Q
␣
͑0͒
͑0͒
13
12
23
13
H
H
ϭ ␻˜ DI z ϩ ␻˜ ͑I z Ϫ I ͒ ϩ ␻˜ I
z
zz xx
x
and can be represented in matrix form as
␤
46
45
56
z
46
x
ϭ Ϫ ␻˜ DI z ϩ ␻˜ ͑I z Ϫ I ͒ ϩ ␻˜ I
[13]
[14]
zz
xx
␣
H ͑t͒
0
H͑t͒ ϭ
ͫ
␤
ͬ
,
[9] with
0
H ͑t͒
1
3
46
1
2
2
2
␣
␤
I x ϩ I x
ϭ
͑S Ϫ S ͒
x
y
where the H (t) and H (t) operators are 3 ϫ 3 matrices in the
1
2
1
2
12
23
z
45
z
56
z
2
z
S-spin basis states with m ϭ and m ϭ Ϫ , respectively. The
͑I z Ϫ I ͒ ϩ ͑I Ϫ I ͒ ϭ 3S Ϫ 2
I
I
linear I operator can be expanded as
x
and
1
4
25
36
Ix ϭ I x ϩ I x ϩ I x
[10]
2
TR
1
and is proportional to the initial density matrix ␳ after the
␻˜ ϭ
g ͑t͒␻ ͑t͒dt
0
D
z
D
͵
TR
CP-MAS excitation of the I spin. In this representation the
three terms of I connect states corresponding to m equal to 1,
0
x
S
2
TR
0
, and Ϫ1, respectively.
1
1
␻
˜
ϭ
ϭ
g ͑t͒ Ϫ ͑ g ͑t͒ ϩ g ͑t͒͒ ␻ ͑t͒dt
zz xx yy Q
zz
͵
͵
ͭ
ͮ
The zero-order Hamiltonian in Eq. [6] can be rewritten as
3TR
2
0
͑
0͒
13
z
46
z
14
z
36
z
H
ϭ ␻˜ ͑I Ϫ I ͒ ϭ ␻˜ ͑I Ϫ I ͒,
[11]
2TR
D
D
1
␻˜
͑ g ͑t͒ ϩ g ͑t͒͒␻ ͑t͒dt.
[15]
xx
xx
yy
Q
2
TR
1
4
36
influencing only the Ix and Ix parts of the initial density
0

5
8
SACK ET AL.
FIG. 3. Time-dependent g
p
(t) and gpq(t) coefficients ( p, q ϭ x, y, z) of the toggling frame Hamiltonian, Eq. [4]. Left, XY-4 REDOR; right, CPL REDOR.
An explicit description of the time dependence of the coefficients is given in Appendix A.
The finite pulse lengths influence the average Hamiltonian in frequencies and will generate I-spin REDOR signals that ap-
Eq. [6] by reducing the dipolar frequency ␻˜ D and by adding a proach the zero-quadrupolar REDOR decay curves. As ex-
quadrupolar and a double-quantum term. While the ␻˜ zz term pected these pulses result in an increase of the overall rate of
has no effect on the I-spin signal, the ␻˜ xx double-quantum term the powder REDOR decay. In addition they cause the REDOR
1
25
x
can have a large effect. For the extreme case ␻Q ӷ ␻ it signals to decrease beyond , indicating a decay of the I terms
1
3
2
5
follows that ␻˜ Ӷ ␻˜ xx and the double-quantum elements of the density matrix. In order for the Ix part of the I spins to
D
1
3,46
(0)
␣(0)
␻
˜ I
become the dominant term in H . In that case H
decay, additional terms must appear in the average Hamilto-
and the initial spin density matrix is hardly influenced nian. A term of the form Iz in H could cause this additional
by H . The exact zero-order solution for the problem is decay; however, this operator does not appear in the zero-order
calculated by diagonalizing the Hamiltonian. A straightforward Hamiltonian, because the quadrupolar terms in the H and H
xx
z
␤
(0)
25 (0)
Х H
(
0)
␣
␤
calculation reveals that the REDOR signal of a single crystal- blocks must have the same form. We therefore expect that
2
5
lite develops as (again S(0) ϭ 1)
other types of interaction terms influence the Ix coherence. To
demonstrate this we choose to describe the composite pulse
REDOR sequence,
1
3
2
3
REDOR
2
eff
2
S
͑2nT ͒ ϭ
ϩ
͑cos ␾ cos( ␻˜ D ⅐ 2nT ͒ ϩ sin ␾͒
R
R
1
1
2
͕
90 90 90 ͖ Ϫ T Ϫ ͕90 90 90 ͖ Ϫ T Ϫ ͕90 90 90 ͖
y
x
y
2
R
x
y
x
R
y
x
y
␻
˜
xx
Ϫ1
eff
␻ D
2
D
2
xx
tan ␾ ϭ
;
ϭ
ͱ
␻˜ ϩ ␻˜ .
[16]
1
2
1
2
␻˜
Ϫ T Ϫ ͕90 90 90 ͖ Ϫ T ,
[17]
D
R
x
y
x
R
2
5
The Ix term of the density matrix remains unaffected and the
T
and show in Fig. 3 the g (t) and g (t) coefficients of its H (t)
p
pq
1
2
2
2
powder signal reaches a value of 3 ϩ ͗sin ␾͘, where ͗sin ␾͘
3
(
see also Appendix A for the explicit form of g (t) and g (t)).
The toggling frame Hamiltonian in this case has the same form
as in Eq. [4], and the zero-order average Hamiltonian becomes
p
pq
is the powder average of the sine squares for all single crys-
tallites. In a powder sample the different crystallites will decay
with different rates and will reach different values, depending
on their ␻ (t) values. The overall decay of the REDOR pow-
Q
͑
0͒
␣͑0͒
␤͑0͒
H
ϭ H
ϩ H
der signal for large quadrupolar interactions will therefore slow
1
down and will not reach the 3 value, as expected for small
␣͑0͒
(0) 13
͑0͒
12
23
͑0͒ 13
H
ϭ ␻ I ϩ ␻ zz ͑I z Ϫ I ͒ ϩ ␻ I x
D z z xx
quadrupolar interactions.
͑
xy
0͒ 13
͑0͒
12
23
x
͑0͒
yz
12
23
y
ϩ ␻ I y ϩ ␻ xz ͑I x Ϫ I ͒ ϩ ␻ ͑I y Ϫ I ͒
Composite Pulse REDOR
␤
͑0͒
(0) 46
͑0͒
45
56
͑0͒ 46
H
ϭ Ϫ␻ I ϩ ␻ zz ͑I z Ϫ I ͒ ϩ ␻ I x
D
z
z
xx
The use of composite pulses will increase the inversion
efficiency of the 180° pulses on S spins with large quadrupolar
͑
0͒ 46
͑0͒
45
56
x
͑0͒
yz
45
56
y
ϩ ␻ I y ϩ ␻ xz ͑I x Ϫ I ͒ ϩ ␻ ͑I y Ϫ I ͒ [18]
xy

DEUTERIUM REDOR
59
off-diagonal terms are much smaller in the CPL case than in
the case of XY-4. According to the expression for the REDOR
signal in Eq. [16], this decrease will increase the overall
REDOR decay of the powder by a reduction of the values of ␾.
2
xz
2
yz
The SQ elements of magnitude ͌ ␻˜ ϩ ␻˜ in the CPL
2
5
case cause oscillations of the Ix term of ␳ . In the limit when
0
the SQ terms are larger than the dipolar terms, they will again
␣
(0)
␤(0)
equalize the two Hamiltonians H
and H
and no REDOR
oscillation can be expected. Thus the SQ matrix elements will
2
5
influence the Ix term only when they are of the same order of
magnitude as the dipolar interaction element ␻˜ . Then a co-
D
operative mechanism, involving both the dipolar interaction
and the quadrupolar interaction through the SQ elements, will
1
3
result in a REDOR decay beyond . This effect can be shown
quantitatively for a single crystallite by a diagonalization of the
␣
(0)
␤(0)
matrix representations of H
and H
of Eq. [18], and a
transformation of I to their diagonal representation. The re-
x
sults of such a calculation are shown in Fig. 5. In this figure the
zero-order time-independent part of the REDOR signal is plot-
ted as a function of Euler angle ␤ of the quadrupolar tensor in
the rotor frame. As can be seen, the contribution of the SQ
elements reduces this part of the signal (solid line) to values
1
below 3 for certain values of ␤. The exact shape of this line
depends on the polar angle ␪ between the dipolar vector and the
principal direction of the quadrupolar tensor (␪ ϭ 30° in our
case). The dashed curve shows the same contribution when the
SQ terms are neglected in the calculation. We see that the two
curves overlap in the regions where the dipolar term is small
(see ␻˜ D in Fig. 4). We should emphasize here that for large
FIG. 4. Magnitudes of the various matrix elements of the average Ham-
iltonians in Eqs. [13] and [18] are plotted for different orientations of the
quadrupolar principal direction (␤ is given wrt the rotor main direction). The
effect of the CPL sequence on two different B
vector oriented 60° from the quadrupolar principal direction. ␾ ϭ 0°, ␥ ϭ 10°,
ϭ 0°.
1
fields is shown for the dipolar
␣
with
1
1
2
23
x
45
x
56
x
͑
I x Ϫ I ͒ ϩ ͑I Ϫ I ͒ ϭ
͑S S ϩ S S ͒, [19]
x z z x
ͱ
2
i, j
where the Iy operators contribute to the imaginary part of the
i, j
matrix elements of I . The coefficients of Eq. [18] are inte-
x
grals of the same type as in Eq. [15].
To investigate the improvement of the REDOR dephasing
due to the application of the CPL (composite pulse) sequences,
we must thus examine the effect of the single-quantum (SQ)
FIG. 5. The normalized zero-order time-independent part of the REDOR
signal S(ϱ) is plotted as a function of the Euler angle ␤ of the quadrupolar
tensor in the rotor frame. The solid line shows the calculated S(ϱ) resulting
terms with ␻ , ␻ in Eq. [18] on the spin system. These terms from the zero-order average Hamiltonian in Eq. [18]. The dashed line repre-
xz
0)
yz
(
sents the same, while omitting the SQ terms in the calculation. The values of
appear in H in addition to the terms of Eq. [13]. In Fig. 4 the
magnitudes of the various matrix elements of the average
Hamiltonians in Eqs. [13] and [18] are compared for some
arbitrary crystallite orientation. This figure shows that the
the parameters used in the calculation were QCC ϭ 146 kHz for the I ϭ 1
2
15
spin, ␯
ϭ 30° for the polar angle between principal direction of the quadrupolar
tensor and the dipolar vector, ␯ ϭ 4 kHz for the spinning speed, and B ϭ 38
d
ϭ 1.64 kHz for the dipolar interaction constant between H and N,
␪
r
1
magnitudes of the coefficients of the double-quantum (DQ) kHz for the RF field strength.

6
0
SACK ET AL.
2
FIG. 6. Static H spectrum of acetanilide recorded (experiment and sim-
2
1
ulation with Q ϭ 146 kHz and B ( H) ϭ 100 kHz).
1
3
FIG. 7.
C CP-MAS spectrum and spectral assignment of acetanilide,
dipolar interactions higher order effects become significant and measured under the same conditions as the REDOR spectra (the asterisks mark
spinning sidebands).
the powder signals can decay toward zero. This can be shown
by exact numerical simulations.
We can thus conclude that the CPL sequences indeed im-
bond vector, was taken and the dephasing was evaluated for a
prove the powder REDOR decay rate and its overall decay, the
dipolar coupling of 1.64 kHz. For the three pulse sequences a
first due to reduction of the DQ terms and the latter due to the
REDOR dephasing was found with an initial slope that is
appearance of SQ terms in the average toggling frame Hamil-
slower than the calculated dephasing caused by ␦ pulses on the
deuterons by a factor of about 3–4. Another significant differ-
tonian.
ence is that the S/S values approach zero in all the experi-
ments and calculations with real pulses but reach 3 for the ␦
0
RESULTS
1
2
pulses. The fastest dephasing is obtained for the 90
composite pulses, followed by results of the 90
pulses. The dephasing of the simple 180
cantly less efficient than the dephasing of the other two se-
quences. While the calculated REDOR decay curves of the
composite pulse sequences reproduce the experimental data
x
–180
y
–90
–90
x
X– H REDOR experiments have been performed on doubly
2
15
15
2
x
–90
y
x
H– N-labeled acetanilide (I) (X ϭ N) and on singly H
II)-labeled and doubly H– CO (III)-labeled acetanilide
X ϭ C). The N spectrum (not shown) of (I) consists of a
single line, caused by the amide nitrogen. The H quadrupolar
echo spectrum (Fig. 6) exhibits a typical H powder pattern
with a quadrupolar splitting of 146 kHz. Figure 7 displays the
natural abundant C CP-MAS spectrum of (II), marking the
2
13
x
pulses is signifi-
(
(
13 15
2
2
fairly closely, the simulated dephasing of the 180 pulse gives
x
1
3
an underestimated value for S/S for large ␶ values. A possible
0
lines where a REDOR effect was observed. The REDOR
2
15
experiments on the directly bonded spin pair H– N in (I)
were performed mainly to compare the efficiency of the dif-
2
13
ferent pulse sequences. In the H– C REDOR experiments on
II) the dephasing of the carbons, assigned in Fig. 7, was
accomplished by the use of the 90 –180 –90 composite pulse
(
x
y
x
scheme.
2
15
H– N REDOR Results
15
Figure 8 compares the simulated and experimental and N-
detected S/S REDOR results of (I) for the three pulse se-
0
quences presented in Fig. 2 with the theoretical REDOR decay
curves under the influence of ideal 180° ␦ pulses. The symbols
mark the experimental data and the solid lines the simulations
of the decay curves. The dashed line corresponds to the ␦-pulse
result. All experiments and calculations have been performed
1
5
2
FIG. 8. REDOR results on N– H doubly labeled acetanilide. The sym-
bols mark the experimental data and the solid lines the simulations of the decay
2
15
curves using Q ϭ 146 kHz, B
1 1
( H) ϭ 37 kHz, B ( N) ϭ 31 kHz, and D ϭ
1
5
with N on resonance. In the simulations an axially symmetric
quadrupolar tensor, with its symmetry axis parallel to the ND to ␦ pulses (i.e., B
1
.64 kHz. The dashed line compares these decays to the decay corresponding
2
1
( H) ӷ Q).

DEUTERIUM REDOR
61
carbons and of 300 Ϯ 40 Hz for the C carbons, corresponding
4
to a nuclear distance of 2.58 Ϯ 0.06 and 2.49 Ϯ 0.12 Å,
respectively.
DISCUSSION
1
2
The REDOR dephasing of a spin- nucleus coupled to a
deuterium spin-1 nucleus is rather insufficient in the case of a
large deuterium quadrupolar interaction and relatively weak
1
80 pulses applied to each rotor cycle on the deuterons.
REDOR and REAPDOR methods, in which all except for one
1
1
80 pulse are applied in the spin- channel, could provide
2
alternative approaches that increase the dipolar dephasing. In
this publication we have reported the use of 90 –180 –90 and
x
y
x
9
0 –90 –90 composite pulses that replace standard 180
x
y
x
1
3
2
FIG. 9.
ilide ␣-carbons C
solid line shows the simulation of the data using D ϭ 540 Hz, B
C natural abundance H-labeled REDOR results of the acetan-
pulses on the deuterons. We have shown experimentally and
theoretically that the application of these composite pulse
sequences improves the REDOR dephasing significantly. For
small and intermediate spinning frequencies (Ͻ10 kHz) and
RF amplitudes down to 20 kHz the CPL sequences can be
made much shorter than the rotor periods and can therefore be
easily implemented into the rotor cycle. They can cause an
increase of the overall REDOR decay rates of up to about two
times the rates obtained with 180 pulses. Using the CPL
1
and C . Both carbons exhibit the same dipolar coupling. The
2
2
1
( H) ϭ 33
1
3
1
kHz, B ( C) ϭ 33 kHz, and Q ϭ 146 kHz.
reason for the deviations at large ␶ values is an additional
dephasing due to intermolecular couplings. We found in the
calculations that the actual decay curves depend rather criti-
cally on the values of the quadrupolar frequency as well as on
the B field applied to the H atoms.
2
13
2
1
sequences we have measured in acetanilide C– H dipolar
couplings in the range from 250 to 550 Hz, corresponding to
2
13
H– C REDOR Results
13
2
C and H atom pairs that are separated by two or three
2
13
covalent bonds. The quantitative analysis of the experimental
data was performed by numerically solving the kinetic equa-
tion of the density matrix under the influence of the various
interaction terms in the spin-Hamiltonian and finite RF pulse
lengths. After careful calibration of all NMR parameters and
measurement of the relevant NMR interaction tensors it was
The carbon-detected H– C REDOR results can be divided
into the ␣ carbons C and C , which are directly bonded to the
amide nitrogen, and the ␤ carbons C and C , which are two
bonds away from the nitrogen. Figure 9 presents the results of
the H– C REDOR experiments on the ␣ carbons C and C .
1
2
3
4
2
13
1
2
13
The experiments were performed on a C-labeled sample (III)
at the ␣–CO position and on a nonlabeled sample (II). The
2
possible to directly evaluate the X– H REDOR results, without
9
0 –180 –90 composite pulse sequence on the deuterons was
x
y
x
employed for these experiments. The solid line shows the
result of a simulation of the REDOR decay curve using a
dipolar coupling of 540 Hz. To cover most of the experimental
points a spread of about 80 Hz is necessary. The dipolar
interaction of 540 Ϯ 40 Hz corresponds to a distance of 2.05 Ϯ
0
.05 Å.
Figure 10 presents the results from the natural abundance
C REDOR results of the ␤ carbons C and C in (II). As a
1
3
3
4
result of the less perfect suppression of off-resonance effects
due to the simple XY phase cycling there are some small-
amplitude oscillations in the dephasing visible in the decay
curve of C (Fig. 10), which do not cause any problems in the
4
evaluation of the data, however. Despite the fluctuations in the
experimental results due to the small REDOR dephasings, the
C carbons show a slightly faster decay than the C carbons.
4
3
This suggests a difference between the CD distances of these
two carbon atoms. The solid lines mark the average results of
the simulations that best fit the experimental results. These
simulations correspond to a coupling of 270 Ϯ 20 Hz for the C3 0.30 kHz (C
1
3
2
FIG. 10.
C natural abundance H-labeled REDOR results of the acetan-
and C . The carbons exhibit different dipolar couplings. The
) and D ϭ
1 1
), B ( H) ϭ 33 kHz, B ( C) ϭ 33 kHz, and Q ϭ 146 kHz.
ilide ␤ carbons C
3
4
solid lines show the simulation of the data using D ϭ 0.27 kHz (C
3
2
13
4

6
2
SACK ET AL.
TABLE 1
(ii) Fast vibrational motions of the molecule cause a partial
Summary of Experimental Data
averaging of the dipolar interaction, which results in a slight
decrease of the distance measured by NMR spectroscopy (5).
a
b
c
d
d
e
e
r/Å
r/Å
r/Å
r/Å
D/Hz
D/Hz
r/Å
Besides this small difference the structure of acetanilide
found from REDOR is in good agreement with the structure
determined by scattering methods.
C
1
C
2
C
3
C
4
N
2.132 1.936 2.039 2.019
2.053 1.972 2.064 2.051
2.603 2.474 2.513 2.499
2.438 2.476 2.506 2.504
564
538
297
295
540 Ϯ 40
540 Ϯ 40
270 Ϯ 20
300 Ϯ 40
1640 Ϯ 40
2.05 Ϯ 0.05
2.05 Ϯ 0.05
2.58 Ϯ 0.06
2.49 Ϯ 0.12
1.04 Ϯ 0.09
1.04 Ϯ 0.09
It must be emphasized that the calculated REDOR decay
1.08
0.896 1.020 1.007 1810
curves depend strongly on the value of the deuterium B field
1
f
1
640 Ϯ 40
strengths in addition to its dependence on pulse settings, the
dipolar interaction strengths, and quadrupolar couplings. They
also depend, although to a lesser extent, on the relative orien-
tation of the dipolar vector with respect to the quadrupolar
tensor. For example, in the coupling regime of C and C (300
a
b
c
d
e
f
X-ray 295 K (29).
X-ray 113 K (28).
Neutron diffraction 15 K (30).
Neutron diffraction 295 K (30).
3
4
2
2
NMR H REDOR.
Hz), changes in the H B field or quadrupolar coupling of Ϯ1
1
15
N NMR lineshape analysis 295 K (9, 10).
kHz lead to a difference in the determined dipolar coupling of
Ϯ5 Hz, i.e., ca. 2%. Since the observed REDOR curves are
practically featureless there are various sets of parameters
which are in principle capable of fitting the same decay curve.
Therefore, a direct quantitative evaluation of the distances,
using the CPL REDOR approach, is possible only if the ex-
perimental NMR parameters of the system are well known.
If this is not the case a second strategy can be employed. It
is evident that the decay curves of different carbons coupled to
the same H nucleus or to H nuclei in the same functional
group are not independent of each other and should be ana-
lyzed with the same strength of quadrupolar interaction. Hence,
it follows that the relative sizes of the couplings of spins to the
same type of deuteron can be elucidated even if the absolute
sizes of the couplings are difficult to determine. In most situ-
ations a calibration of these relative couplings to absolute
values can be performed by using either a calibration experi-
ment on a chemically similar reference compound, with deu-
terons in the same functional group, or a measurement of a
well-known distance in the studied molecule, for example, a
direct CD bond or a CND distance. In our system it would have
been possible, instead of simulating all decay curves as de-
scribed above, to use as an internal standard the known dis-
tance between the carbonyl carbon C1 and the deuteron.
the necessity of introducing any fit parameters or employing a
calibration measurement.
Table 1 compiles the distances obtained from our REDOR
experiments and compares them with distances obtained by
diffraction techniques. There are some discrepancies between
the diffraction data. While the new X-ray (28) and the neutron
diffraction data (30) state that the C D distance (see Fig. 1) is
1
2
2
shorter than the C D distance and the C D and C D distance
2
3
4
are essentially the same, the older X-ray study (29) states the
opposite, i.e., the C D distance is longer than the C D distance
1
2
and the C distance is longer than the C distance. In addition,
3
4
there are strong deviations between the reported NH distances.
These differences can probably be attributed to the well-known
difficulty of localizing hydrons with X-ray spectroscopy.
For a quantitative comparison to our REDOR data the neu-
tron diffraction data at 295 K (30) and NMR lineshape results
at room temperature are particularly important, since the posi-
tion of the hydrogens depends on temperature, as is evident
from the difference between distances obtained from neutron
diffraction at 15 and 295 K. Comparing the high-temperature
neutron diffraction data with our data shows good agreement.
The observed difference between the C D (2.019 Å) and C D
1
2
(2.051 Å) distances could not be resolved in our experiments
(
2.05 Ϯ 0.05 Å). A slight discrepancy between our NMR
CONCLUSION
results for the ND distance to the NH distance obtained from
neutron diffraction is observed. Neutron scattering measures a
NH distance of 1.007 Å (30) at T ϭ 295 K, while our REDOR
data give 1.04 Ϯ 0.01 Å, corroborating previous results from
NMR lineshape analyses (9, 10) on N– H doubly labeled
acetanilide.
There are two possible explanations for these differences
between the NMR data and neutron diffraction data, which are
not necessarily mutually exclusive:
It has been shown that, when composite pulses are employed
on the deuterons, X– H REDOR is a powerful technique for
2
1
5
2
studying molecular conformations of molecules, even for weak
2
2
H B1 fields and strong H quadrupolar interactions. The
investigated 90 –180 –90 and 90 –90 –90 CPL have
x
y
x
x
y
x
proven far superior to the simple 180 pulses, introducing an
approximately two times faster dipolar dephasing, thus effec-
tively enlarging the range of weak dipolar couplings measur-
.
. .
2
(
i) The hydron is part of the ONH O|CO hydrogen able by X– H REDOR spectroscopy. Using a careful calibra-
bond. It is well known that the positions of the hydron and the tion of all NMR parameters it is feasible to analyze these
1
2
deuteron in these bonds are different due to H/ H isotope experiments directly by numerical simulations. For quantita-
2
effects (8).
tive applications of X– H REDOR on compounds, where not

DEUTERIUM REDOR
63
all NMR parameters are known exactly, it is necessary to conjunction with homebuilt notch filters. The RF of the
perform a reference experiment either on a standard compound observed channel was fed through a crossed-diode duplexer
2
or alternatively on a X– H pair with known distance in the connected to the detection preamplifier and through the
measured compounds themselves. As an experimental example filters into the probe. The other two channels were fed
the molecular geometry of acetanilide has been studied by this directly through the filters into the probe. The typical 90°
method and compared to diffraction data. In conclusion it has pulse width was 6.5 ␮s for all three channels, corresponding
2
been shown that X– H REDOR spectroscopy is well suited to to 38-kHz B field in frequency units. For the proton de-
1
1
the localization of hydrons in organic compounds.
coupling the H B was changed to 52 kHz to avoid un-
1
wanted cross-polarization by mismatching the Hartmann–
Hahn condition and to improve the decoupling efficiency.
This was sufficient to remove H–X dipolar line broadening.
EXPERIMENTAL
1
Sample
The repetition time of the experiments was 30 s for the
acetanilide. All MAS spectra were recorded at a rotation
frequency of 5 kHz, employing standard CYCLOPS phase
cycling for data acquisition and XY phase cycling for the
REDOR pulses. The rotation frequency was controlled using
a Doty spin rate controller. Deviations in the rotation fre-
quency were below 5 Hz. The spectra were measured by first
cross-polarizing the observed nucleus from the protons and
then recording the signal of the observed X nuclei under proton
decoupling. The application of phase cycling schemes (24),
like XY-8 or XY-16, in the dephasing channel, is not sufficient
to compensate for the influence of the large quadrupolar broad-
ening on the REDOR decays. We therefore used a composite
pulse approach in the dephasing channel in conjunction with
numerical simulation of the REDOR decay curves. The CPL
were phase cycled in the XY scheme and put mirror symmetric
with respect to the central echo scheme into the rotor periods.
As an advantage of this simpler phase cycling scheme fewer
cycle propagators have to be calculated in the numerical eval-
uation of the data. From these spectra the decay curves of the
individual lines were determined and the REDOR decay curves
were calculated by normalizing the decay employing the usual
reference experiment without pulses in the dephasing channel.
The synthesis of isotopically labeled acetanilide was per-
formed by the standard procedures (35) of adding a solution of
acetyl chloride (either unlabeled or 99% C(CO)-labeled from
Chemotrade, Leipzig, Germany) in chloroform to aniline (ei-
ther unlabeled or 99% N-labeled, also from Chemotrade)
dissolved in chloroform and stirred in an ice bath. The addition
was completed at a higher temperature of about 50°C. The
solution was stirred for 1 h and the excess solvent removed to
yield crystals of acetanilide which were recrystallized in water.
These crystals were selectively deuterated in the exchanging
amide position by dissolving twice in excess CH OD, waiting
for 2 h at room temperature, and then drying under vacuum.
After this procedure the deuterium labeling was better than
13
1
5
3
9
9%.
Due to the size of the acetanilide molecule, the expected
intermolecular dipolar couplings are relatively weak. For ex-
1
5
2
ample, the largest intermolecular N– H coupling can be es-
timated as 37 Hz from the crystal structure, which has only a
nonnegligible influence on the REDOR dephasing of the X
nucleus for very long recoupling times. Therefore no isotope
dilution of the samples was performed.
Spectrometer
A detailed discussion of our homebuilt three-channel
NMR spectrometer has been given recently (10). Here only
some salient features are given. All experiments were per-
formed at a field of 6.98 T, corresponding to a proton
resonance frequency of 297.8 MHz on a standard Oxford
wide-bore magnet (89 mm) equipped with a room-
APPENDIX A
Explicit Representation of the Functions
g (t), g (t), and g (t)
x
y
z
2
An XY-4 REDOR pulse sequence that is applied on the H
temperature shim unit. For the proton channel a Creative spins during two rotor periods is shown schematically at the
1
5
Electronics 1-kW class C amplifier was used. For the N or top of Table A1. The pulses with an intensity ␻ have a
p
1
3
2
C channel a 1-kW class AB and for the H channel a 2-kW length t ϭ ␲/␻ and are applied at t ϭ 0, T / 2, T , 3/ 2T .
p
p
R
R
R
class AB amplifier, both from AMT, were employed. All In the table the periodic g functions are defined for all time
p
amplifiers are equipped with RF blanking to suppress noise intervals ①–⑧. The coefficients gpq ϭ g g are used in the
p
q
2
during data acquisition. The static H spectrum of acetani- calculations.
2
2
lide was recorded with a homebuilt single-resonant H NMR
A composite pulse sequence that is applied on the H
probe using the 90 –90 echo sequence with B ( H) ϭ 100 spins during one rotor period is shown schematically at the
2
x
y
1
kHz. All other experiments were performed using a Bruker top of Table A2. The pulses with an intensity ␻ have a
p
triple-resonance NMR probe operating at room temperature. length t ϭ ␲/(2␻ ) and are applied at t ϭ 0, t / 2, t ,
p
p
p
p
To improve the mutual RF isolation of the three channels 3/ 2t , and T / 2, T / 2 ϩ T / 2, T / 2 ϩ t , T / 2 ϩ 3/ 2t .
p
R
R
R
R
p
R
p
commercial bandpass filters (Texscan) were employed in In the table the periodic g functions are defined for all time
p

6
4
SACK ET AL.
TABLE A1
Explicit Representation of the Functions g
for the XY-4 REDOR
basis set defined in Eq. [7]. This 6 ϫ 6 matrix H can be divided
into two 3 ϫ 3 matrices,
x
(t), g
y
(t), g (t)
z
␻
˜ ϩ ␻˜
0
Ϫ2 ␻˜ zz
0
␻˜ xx
①
②
③
④
⑤
⑥
⑦
⑧
D
zz
1
2
␣
͑0͒
H
H
ϭ
ϭ
0
0
ͩ
ͩ
ͪ
ͪ
t
py
T
R
/2 Ϫ t
p
t
px
T
R
/2 Ϫ t
p
t
py
T
R
/2 Ϫ t
p
t
px
T
R
/2 Ϫ t
2T
p
␻˜
Ϫ ␻˜ ϩ ␻˜
xx
D
zz
zz
0
T
R
/2
T
R
3T
R
/2
R
Ϫ ␻˜ ϩ ␻˜
0
␻˜ xx
0
␻˜ ϩ ␻˜
D
D
zz
1
2
XY-4 REDOR sequence
␤͑0͒
0
Ϫ2 ␻˜ zz
[A2]
␻˜
0
xx
g
z
g
x
g
v
①
②
③
④
⑤
⑥
⑦
⑧
cos(␻
p
/2*t)
0
0
p
Ϫsin(␻ /2*t)
with
Ϫ1
0
0
0
Ϫcos(␻
cos(␻
p
p
/2*(t Ϫ T
1
/2*(t Ϫ T
Ϫ1
R
R
/2)) Ϫsin(␻
p
/2*(t Ϫ T
R
/2))
␣
͑0͒
0
0
0
H
0
͑
0͒
H
ϭ
ͩ
␣͑0͒
ͪ
[A3]
[A4]
))
sin(␻
p
/2*(t Ϫ T
R
))
0
H
0
0
0
Ϫcos(␻
p
/2*(t Ϫ 3T
R
/2)) sin(␻
p
R
/2*(t Ϫ 3T /2))
1
0
and can be diagonalized by a matrix ⌬,
D
0
D
⌬
ϭ
ͩ 0
؊1
ͪ
intervals ①–⑧. The coefficients gpq ϭ g g are used in the
calculations.
p
q
Ϫ1
with ⌳ ϭ ⌬H⌬ ,
APPENDIX B
␣
͑0͒
␣͑0͒ Ϫ1
⌳
ϭ DH
D
Derivation of Eq. [16]
eff
␻
˜
ϩ ␻˜ zz
D
0
0
0
In this appendix we present a short derivation of the expres-
sion for the signal in Eq. [16]. The signal is equal to
1
ϭ
ͩ
0
0
Ϫ2 ␻˜ zz
ͪ
2
eff
0
Ϫ ␻˜ D ϩ ␻˜ zz
Ϫ
S͑t͒ ϭ Tr͑S ͑t͒S ͒,
[A1]
␤͑0͒
Ϫ1 ␤͑0͒
x
⌳
ϭ D
H
D
eff
D
where the time dependence of the initial density matrix S is
Ϫ␻˜
ϩ ␻˜ zz
0
0
x
1
determined by the average Hamiltonian, defined in Eq. [13].
This Hamiltonian can be represented in matrix form in the
ϭ
0
0
Ϫ2 ␻˜ zz
0
[A5]
ͩ
ͪ
2
eff
0
␻˜ ϩ ␻˜ zz
D
TABLE A2
Explicit Representation of the Functions g
x
(t), g
y
(t), g (t) for the CPL REDOR
z
①
②
③
④
⑤
⑥
⑦
⑧
t
p
/2
0
y
t
p
/2
x
t
p
/2
y
T
R
/2 Ϫ 3t
p
/3
t
T
p
/2
/2
y
t
p
/2
x
t
p
/2
y
R p
T /2 Ϫ 3t /3
R
T
R
Composite pulses
g
z
g
x
g
v
①
②
③
④
⑤
⑥
⑦
⑧
cos(␻
p
/2*t)
0
p
Ϫsin(␻ /2*t)
0
sin(␻
cos(␻
p
/2*(t Ϫ t
/2*(t Ϫ t
p
/2))
))
Ϫcos(␻
p
p
/2*(t Ϫ t /2)
Ϫsin(␻
Ϫcos(␻
p
/2*(t Ϫ t
Ϫ1
/2*(t Ϫ T
p
))
p
p
0
0
0
0
p
R
/2))
sin(␻
p
/2*(t Ϫ T
R
/2))
0
Ϫsin(␻
p
/2*(t Ϫ T
/2*(t Ϫ T
R
/2 Ϫ t
/2 Ϫ t
p
/2))
))
cos(␻
p
/2*(t Ϫ T
R
/2 Ϫ t /2))
p
sin(␻
p
/2*(t Ϫ T
R
/2 Ϫ t
p
))
Ϫcos(␻
p
R
p
0
0
1
0

DEUTERIUM REDOR
65
6
7
8
9
. C. G. Hoelger, F. Aguilar-Parilla, J. Elguero, O. Weintraub, S. Vega,
and H. H. Limbach, J. Magn. Res. A 120, 46 (1996).
cos ␾/2 0 Ϫsin ␾/2
D ϭ
0
1
0
0
[A6]
[A7]
ͩ
ͪ
. H. Benedict, C. Hoelger, F. Aguilar-Parrilla, W. P. Fehlhammer, M.
Wehlan, R. Janoschek, and H. H. Limbach, J. Mol. Struct. 378, 11 (1996).
sin ␾/2
cos ␾/2
. H. Benedict, H. H. Limbach, M. Wehlan, W. P. Fehlhammer, N. S.
Golubev, and R. Janoschek, J. Am. Chem. Soc. 120, 2939 (1998).
and
. M. D. Lumsden, R. E. Wasylishen, K. Eichele, M. Schindler, G. H.
Penner, W. P. Power, and R. D. Curtis, J. Am. Chem. Soc. 116,
1403 (1994).
␻
˜
xx
D
eff
2
2 1/ 2
␻˜
ϭ ͑͑ ␻˜ ͒ ϩ ͑ ␻˜ ͒ ͒
;
tan ␾ ϭ
.
D
D
xx
␻˜
1
1
0. G. Buntkowsky, I. Sack, H.-H. Limbach, B. Kling, and J. Fuhrhop,
J. Phys. Chem. B 101, 11265 (1997).
The matrix representation of the operator S has the form
x
1. M. Emshwiller, E. L. Hahn, and D. E. Kaplan, Phys. Rev. Lett. 118,
4
14 (1960).
1
0
I
d
0
12. M. E. Stoll, A. J. Vega, and R. W. Vaughan, J. Chem. Phys. 65, 4093
Sx ϭ ͩ ͪ
[A8]
2
I
d
(1976).
1
3. J. S. Waugh, Proc. Natl. Acad. Sci. USA 73, 1394 (1976).
with I a 3 ϫ 3 unit matrix.
14. T. Gullion and J. Schaefer, Adv. Magn. Opt. Reson. 13, 57 (1989).
d
The signal in Eq. [A1] can be evaluated by insertion of Eqs. 15. T. Gullion and J. Schaefer, J. Magn. Reson. 81, 196 (1989).
[
A4–A7] and t ϭ 2nTR:
16. J. Schaefer, E. O. Stejskal, and R. Buchdahl, Macromolecules 8,
91 (1975).
17. T. Gullion, Chem. Phys. Lett 246, 325 (1995).
2
Ϫi⌳t Ϫ1
i⌳t Ϫ1
Ϫ
S͑t͒ ϭ Tr͑⌬e
⌬
Sx⌬e
⌬
S ͒
1
1
2
8. L. Chopin, T. Gullion, and S. Vega, J. Am. Chem. Soc. 120, 4406 (1998).
9. A. Schmidt, R. A. McKay, and J. Schaefer, J. Magn. Res. 96, 644 (1992).
Ϫi⌳ ␤͑0͒t
Ϫ2 i⌳ ␣͑0͒
t
2
ϭ Tr͑e
D
e
D ͒,
[A9]
0. A. Schmidt, T. Kowalewski, and J. Schaefer, Macromolecules 26,
which has the explicit form of Eq. [16]:
1729 (1993).
2
1. C. A. Klug, P. Lani Lee, I. S. H. Lee, M. M. Kreevoy, R. Yaris, and
J. Schaefer, J. Phys. Chem. B 101, 8086 (1997).
1
3
2
3
2
eff
2
S͑t͒ ϭ ϩ ͑cos ␾ cos ␻˜ D t ϩ sin ␾͒.
[A10]
2
2
2
2. P. L. Lee and J. Schaefer, Macromolecules 28, 2577 (1995).
3. P. L. Lee and J. Schaefer, Macromolecules 28, 1921 (1995).
4. T. Gullion, and J. Schaefer, J. Magn. Reson. 92, 439 (1991).
ACKNOWLEDGMENTS
Support by the German Israel Foundation under G.I.F. Research Grant 25. Counsell, M. H. Levitt, and R. R. Ernst, J. Magn. Reson. 63, 133
I-297.092.05/93 and by the Sonderforschungsbereich 448 “Mesoskopisch
Strukturierte Verbundsysteme” of the Deutsche Forschungsgemeinschaft, as
well as the Fonds der Chemischen Industrie, Frankfurt, is gratefully acknowl-
edged.
(1985).
2
2
2
6. M. H. Levitt, D. Suter, and R. R. Ernst, J. Chem. Phys. 80, 3064 (1984).
7. M. H. Levitt, Prog. NMR Spectrosc. 18, 61 (1986).
8. H. J. Wasserman, R. R. Ryan, and S. P. Layne, Acta Crystallogr. 41,
7
83 (1985).
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