Direct conversion of EPR dipolar time evolution data to distance distributions

Article abstract of DOI:10.1006/jmre.2001.2498

Shallow electron spin echo envelope modulations due to dipole-dipole couplings between electron spins provide information on the radial distribution function of the spins in disordered systems while angular correlations between spin pairs are negligible. Under these conditions and in the absence of orientational selection, the dipolar time evolution data can be quantitatively simulated for arbitrary radial distribution functions by shell factorization, i.e., by performing the orientational average separately for thin spherical shells and multiplying the signals of all the shells. For distances below 5 nm, a linear superposition of the signals of the shells is sufficient. The dipolar time evolution data can be separated into this linear contribution and a nonlinear background. The linear contribution can then be converted directly to a radial distribution function. For a series of shape-persistent and flexible biradicals with end-to-end distances between 2 and 5 nm, shell factorization and direct conversion of the data are in good agreement with each other and with force-field computations of the end-to-end distances. The neglect of orientation selection does not cause significant distortions of the determined distance distributions.

Full text of DOI:10.1006/jmre.2001.2498

Journal of Magnetic Resonance 155, 72–82 (2002)
doi:10.1006/jmre.2001.2498, available online at http://www.idealibrary.com on
Direct Conversion of EPR Dipolar Time Evolution Data
to Distance Distributions
Gunnar Jeschke, Achim Koch, Ulrich Jonas, and Adelheid Godt
Max-Planck-Institut fu¨r Polymerforschung, Postfach 3148, 55028 Mainz, Germany
Received August 20, 2001; revised November 26, 2001; published online February 8, 2002
Shallow electron spin echo envelope modulations due to dipole–
measured by modern pulse techniques (5) that separate the
dipole–dipole interaction of the electron spins from all the other
interactions (6–17). Using one of these techniques and suit-
ably functionalized nitroxide spin probes, we have recently suc-
ceeded in measuring spin–spin distance distributions between
1.8 and 8 nm (18). We could thus study how the size of and
the distances between ionic clusters in block copolymer-based
ionomers depend on polymer chain length (19). In these investi-
gations we could fit simple models for the distance distribution
to the experimental dipolar time evolution data. Such analysis of
data for disordered systems in terms of simple models was also
performed by other authors (20–22). For general applications it
would be much more favorable to directly convert such data to a
distance distribution without assuming any explicit model. En-
couraging early attempts to introduce such a method (8, 23, 24)
have not been followed up for a long time and did not find appli-
cations despite a recent surge in such measurements (1). In the
present work, we discuss which assumptions have to be made
to convert dipolar evolution data to a radial distribution func-
tion G(r). We then present the shell factorization model for the
simulation of dipolar time evolution data for arbitrary distance
distributions and derive the mathematics for a simple and fast
direct conversion procedure. Finally, this procedure is tested on
experimental data for both shape-persistent and flexible nitrox-
ide biradicals with end-to-end distances between 2 and 5 nm.
dipole couplings between electron spins provide information on the
radial distribution function of the spins in disordered systems while
angular correlations between spin pairs are negligible. Under these
conditions and in the absence of orientational selection, the dipolar
time evolution data can be quantitatively simulated for arbitrary
radial distribution functions by shell factorization, i.e., by perform-
ing the orientational average separately for thin spherical shells
and multiplying the signals of all the shells. For distances below
5 nm, a linear superposition of the signals of the shells is sufficient.
The dipolar time evolution data can be separated into this linear
contribution and a nonlinear background. The linear contribution
can then be converted directly to a radial distribution function. For
a series of shape-persistent and flexible biradicals with end-to-end
distances between 2 and 5 nm, shell factorization and direct con-
version of the data are in good agreement with each other and with
force-field computations of the end-to-end distances. The neglect
of orientation selection does not cause significant distortions of the
determined distance distributions. ꢀ 2002 Elsevier Science (USA)
C
Key Words: pulse EPR; solid state; distance measurements;
nitroxide; radial distribution function.
INTRODUCTION
Electron paramagnetic resonance (EPR) spectroscopy is one
of only a few methods that can precisely measure distances in
the range between 1 and 10 nm in disordered systems (1). For
complex systems, EPR spectroscopy may be the only practical
method, sincetechniquessuchassmall-anglex-rayscattering(2)
and small-angle neutron scattering (3) suffer from insufficient
contrast, and fluorescence resonance energy transfer (4) depends
on pairs of different fluorescence labels. Attaching two different
labels may be synthetically more demanding than attaching two
like paramagnetic labels. Furthermore, the larger fluorescence
labels may cause larger structure perturbations in the system. As
the characterization of structures in the distance range between 1
and 10 nm is crucial for both the development of nanomaterials
and the understanding of the function of biological systems,
further development of methodology for distance measurements
between electron spins is of great interest.
THEORY
Relation between Dipolar Time Evolution and the Radial
Distribution Function
Consider experiments such as three-pulse DEER (6, 9), four-
pulse DEER (10, 11), and double-quantum EPR (13–15) for
which the dipolar time evolution signal for a single pair (i, k)
of localized electron spins with distance rik is given by
V (t) = V0[1 − λik(1 − cos (ωikt))],
[1]
where the modulation depth λik quantifies the fraction of the
echo signal that is due to excited spin pairs (i, k) and
While continuous-wave EPR is well suited to the distance
range between 1 and 2 nm (1), larger distances have to be
(ik)
dd
ωik = ω (3 cos2 θik − 1)
[2]
72
1090-7807/02 $35.00
ꢀC
2002 Elsevier Science (USA)
All rights reserved.

DISTANCE DISTRIBUTIONS FROM DIPOLAR TIME EVOLUTION DATA
73
distribution function p(rik) ∝ 4πr2. It has been demonstrated
earlier (25, 26) that in this case the system is fully characterized
by the concentration C of spins. The dipolar time evolution is
a monoexponential decay V (t) = V0 exp(−t/Thom) with time
constant
with
µ0 gi gkµ2B
(ik)
dd
ω
=
[3]
ri3k
-
4πh
is the dipolar evolution frequency. Without losing generality
we may treat spin i as an observer spin (A spin) and spin k
as a pumped spin (B spin). The parameters gi and gk are the
electron g values of the two spins, and θik is the angle between
the static field vector B0 and the vector connecting the loci of
the two spins. Henceforth we assume gi = gk = g for all i, k.
In a sample containing multispin systems, the signal for any
A spin is thus given by
√
-
9 3h
Thom
=
,
[5]
2πg2µ2Bµ0λC
where the modulation depth parameter λ is the fraction of
excited B spins.
For systems with intermediate order, the structure is charac-
terized by the distance distribution function p(r) = 4πr2G(r),
where G(r)istheradialdistributionfunction, andbycorrelations
ꢀ
between the θik. By expanding the product on the right-hand
Vi (t) = V0,i
[1 − λik(1 − cos (ωikt))]
[4]
ꢁ
k=i
side of Eq. [4] and computing i Vi (t) we find that the effect of
the distance distribution on the echo signal is linear in λ, while
the effect of correlations between m angles θik scales with λm.
For λ 1, which is usually fulfilled in DEER experiments on
nitroxide spin probes, the dipolar time evolution is thus domi-
nated by effects of the radial distribution function G(r), even if
significant correlations between the θik exist.
and the total signal is the sum of the signals for all A spins
ꢁ
i Vi (t). The macroscopic sample can be described as an en-
semble of a very large number of such multispin systems, which
are in turn configurations of n spins that are completely charac-
terized by n − 1 pairs of distances and angles (rik, θik). For
a given experiment, we only need to consider B spins with
rmin ≤ rik ≤ rmax(tmax), where rmin is determined by the band-
width of the pulses, as discussed below, and rmax is a distance for
which the dipolar time evolution does not lead to a perceptible
echo decay within the observation time of the experiment. In
the following we assume macroscopic disorder, i.e., the orien-
tation of the molecular frame is not correlated to the orientation
of B0.
Validity of the Linear Approximation and Shell
Factorization Model
ThedirectuseofEq. [4]fornumericalcomputationsofdipolar
time evolution data from a given model for the system is not
feasible. As the most simple approximation we can expand the
product on the right-hand side of this equation and consider
only the constant term and the terms linear in the λik. All effects
of angular correlations are thus neglected. If we perform the
ensemble average over the macroscopically disordered sample
and neglect orientation selection, we can express the result as
For a discussion of structure determination of such macro-
ꢁ
scopically disordered systems from
i Vi (t) it is useful to
examine first the two limiting cases of complete local order
and complete disorder. For complete order at the length scale
rmin ≤ r ≤ rmax, as is encountered in a powder of ideal crystal-
lites, all A spins see the same configuration of n − 1 B spins. A
Cartesian local frame is defined by the A spin and the first two
B spins. The relevant parameters are the distances ri1 and ri2 be-
tween the A spin the two B spins and the the angle between the
two A–B spin–spin vectors. Each additional B spin adds three
parameters, the distance rik and the polar angles θk and φk in the
local frame. Hence the total number of parameters characteriz-

ꢃ
π/2
rmax
ꢂ
ꢂ
2

Vlin(t) = 1 −
4πr λG(r) 1 −
cos (3 cos2 θ − 1)
rmin
0


ꢄ
g2µ2Bµ0 1
×
t
sin θ dθ dr.
[6]
ri3
-
4πh
ing such a structure with complete local order is 3(n − 2). Due Unfortunately, this approximation does not exhibit the proper
to the general symmetry properties of the spin Hamiltonian and asymptotic behavior for rmax → ∞ and does not reproduce the
due to resolution limitations it may be impossible to determine exponential decay with the time constant given in Eq. [5] for a
all these parameters from V (t). On the other hand it may be homogeneous distribution of spins. The reason for this failure
possible to correlate the local frame of the spin distribution to becomes obvious by considering the frequency dependence of
the molecular frame determined by the g or hyperfine tensor by the modulation depth. For large distances the dipolar modulation
making use of orientation selection (16, 17). In the following, frequency asymptotically approaches zero, while p(r) ∝ r2. As
we neglect orientation selection effects. Their influence on the a consequence, the concept of a finite modulation depth becomes
apparent distance distribution will be considered in the experi- obsolete at zero frequency. In other words, at sufficiently long
mental section.
distances the problem can no longer be treated in terms of iso-
Complete disorder corresponds to a homogeneous distribu- lated spin pairs. The linear approximation in Eq. [6] can thus
tion of spins with no correlation between the θik and a distance only be used up to a certain distance rlin.

74
JESCHKE ET AL.
In previous work, this problem was usually fixed by factor-
ization of the signal into one part corresponding to isolated spin
pairs with a well-defined distance (intrapair contribution) and
a part corresponding to interactions between spins in different
pairs (interpair contribution). To compute the interpair contri-
bution, a homogeneous distribution of the pairs in space was as-
sumed (6, 9). This contribution, which extends to r >rlin, could
thus be discribed by Eq. [5]. However, the distance distribution
in many systems of interest cannot be approximated by such
a simple model. To perform simulations for arbitrary distance
distributions, nonlinearityatlongdistancesduetomultispincon-
tributions must be accounted for in a more general way.
FIG. 1. Simulation of the dipolar time evolution signal V (t) for a homo-
geneous distribution of radicals with g = ge and λc = 1 mmol L−1 by the shell
factorization model (rmin = 1.5 nm, rmax = 40 nm, ꢀr = 0.02 nm). (a) Simulated
signal (solid line) and difference between the simulated signal and the exponen-
tially decaying signal with time constant Thom = 1.0027 µs (Eq. [5]) computed
from a closed analytical expression (dashed line). (b) Difference between nu-
merical and analytical simulation magnified by a factor of 100.
In numerical computations this problem can be solved by
substituting the integral over the pair correlation function by a
product of a finite number of spherical shells with radii rmin
rk ≤ rmax and uniform thickness ꢀr. By choosing ꢀr so that
≤
4πrk2ꢀrλG(rk) 1,
[7]
By fitting an exponential decay to the data from the numerical
simulation we obtain Thom = 1.0041 µs in good agreement with
the theoretical value of 1.0027 µs. Except for the discretization
of r, the shell factorization model is also in full agreement with
the analytical formula for an ensemble of isolated spin pairs
with a common distance r. The approximations in this model
thus hold for both limiting cases of a distance distribution, the
homogeneous distribution and a distribution consisting of a sin-
gle δ peak. Any distance distribution is an intermediate case
between these two limits; therefore, the approximations should
also apply.
The shell factorization model can thus be used to simulate the
corresponding dipolar time evolution data for arbitrary distance
distributions, as long as ꢀr can be made small enough so that
Eq. [7] applies over the whole interval (rmin, rmax). Such com-
putations are much less time consuming than Monte Carlo sim-
ulations which treat the multispin contribution explicitly (20),
which is particularly important for nonlinear fitting of exper-
imental data to model distributions with several parameters.
Furthermore, by comparing such simulations with computa-
tions according to the simple linear approximation (Eq. [6]),
we can estimate rlin. Depending on the modulation depth λ
and the required precision in the simulation (signal-to-noise ra-
for all k, the linear approximation is valid within each shell. Such
a choice is usually possible for disordered systems. We can thus
perform the integration over θ (powder average) for each shell
separately
π/2
ꢂ
Vk(t) =
cos [(3 cos2 θ − 1)ωdd(rk) t] sin θ dθ,
[8]
0
and compute the signal as the product of the signals of all the
shells
ꢀ
V (t) =
[1 − 4πr2λꢀrG(rk)(1 − Vk(t))].
[9]
k
This shell factorization model exhibits the proper asymptotic
behavior for r → ∞, as Vk approaches V (t) ≡ 1 for ω → 0.
∞
To compute the dipolar time evolution signal for tmax ≤ 8 µs we
use rmin = 1.5 nm, rmax = 40 nm, and ꢀr = 0.02 nm. For typical
concentrations c = C/NA < 5 mmol L−1, such numerical simu-
lations for a homogeneous distribution of radicals (G(r) ≡ C)
are in nice agreement with Eq. [5]. Figure 1 shows a numerical
simulation by the shell factorization model with λc = 1 mmol
L−1 and g = ge and its difference from a simulation based on
Eq. [5]. The difference at early times is due to the cutoff toward
short distances (r <rmin) and is also expected in experimental
data (for a discussion, see also Ref. 27). In contrast, the cutoff
toward longer distances (r >rmax) does not lead to any signifi-
cant difference, as was also checked by using larger rmax. Thus,
for a given maximum observation time tmax, spins at larger dis-
tio, required precision for the estimated G(r)), we find rlin
4 · · · 6 nm.
≈
As was pointed out by a referee, we may expand the product
in Eq. [9]; let ꢀr → 0, and replace the series by an exponential,
resulting in
V (t) = exp[Vlin(t) − 1],
[10]
tances than the appropriate rmax do not significantly contribute where Vlin(t) is given by Eq. [6]. Note that plots of ln V (t) have
to the dipolar time evolution signal, as was already discussed been used extensively in earlier work (9, 22). If the signal-to-
qualitatively in Ref. (6). For a homogeneous distribution, the noise ratio is sufficient, the direct transformation of the linear
experiments thus measure local concentrations on a nanoscopic contribution described in a later section should thus also be ap-
length scale.
plicable to ln V (t) + 1.

DISTANCE DISTRIBUTIONS FROM DIPOLAR TIME EVOLUTION DATA
75
a time t1 at which the low-frequency parts of pairs at r <rlin
have completely decayed and the linear contribution is a sum
of high-frequency oscillations (6). Spin pairs at long distances
do not contribute high frequencies, but their contributions at low
frequencies is observed at all times. We can thus fit the nonlinear
background for t > t1 by a low-order polynomial
Separation of Linear and Nonlinear Contributions
A “model-free” conversion of dipolar time evolution data to
a radial distribution function is expected to be much simpler
within the linear approximation than for data containing nonlin-
ear contributions. While the shell factorization model is gener-
ally applicable, we here consider direct conversion only for the
linear part of the distance distribution in cases where it is well
separated from the nonlinear part. Formally, we can write the
signal as the product of the nonlinear and linear contributions
q
ꢅ
B(t) =
ai ti ≈ Vnlin(1 − λ)
[12]
i=1



rlin
ꢂ
with order q and may reasonably expect that an extrapolation of
this fit to the range 0 ≤ t ≤ tlin provides a good approximation of
the nonlinear contributions at early times. Such a fit with a low-
order polynomial is superior to a fit by an exponential, as it can
compensate for peaks in the distance distribution at r >rlin, as
long as such peaks are well separated from the peaks at r < rlin.
The linear contribution is then given by
2



V (t) = Vnlin 1 − λ 1 − 4πr G(r)Vr (t) dr
,
[11]
rmin
where Vr (t) is the dipolar evolution function for a spin pair with
distance r. To find a good approximation of Vnlin, we may note
that spin pairs at short distances contribute components at large
frequencies over the whole time range, but components at low
frequencies only at short times (Fig. 2). Therefore, we can find
rlin
ꢂ
V (t) − B(t)
B(t)
λ
Vlin(t) =
≈
4πr2G(r)Vr (t) dr. [13]
1 − λ
rmin
This elimination of the nonlinear contribution is demonstrated in
Figs. 2a and 2b for a simulated signal corresponding to a distance
distribution with one Gaussian peak at 2.8 nm with a variance
of 0.1 nm and a homogeneous background with λc = 0.3 mmol
L
−1. With t1 = 0.5 µs a third-order polynomial (q = 3) provides
a good fit of the background (Fig. 2a), and the subtraction of the
polynomialandsubsequentdivisionbythepolynomialyieldsthe
separated linear contribution to the time-domain signal, which
corresponds to only the Gaussian peak (Fig. 2b). The inset in
Fig. 2b shows that a Fourier transform of this separated linear
contribution is in satisfying agreement with the expected Pake
pattern. The procedure still works reasonably well if the distance
distribution features a peak at longer distances. For a homoge-
noeus background with λc = 0.15 mmol L−1 and an additional
Gaussian peak at 7 nm with a variance of 0.25 nm and an in-
tegral intensity that is 3.9 times larger than that for the peak at
2.8 nm, the nonlinear contribution due to both the homogeneous
background and the peak at 7 nm can be fit by a fifth-order poly-
nomial, again with t1 = 0.5 µs (Fig. 2c). The extracted linear
contribution (Fig. 2d) is in reasonable agreement with the one
for a purely homogeneous background (Fig. 2b).
FIG. 2. Removal of nonlinear contributions by subtraction of a low-order
polynomial. (a) Simulation of V (t) by the shell factorization model with a dis-
tance distribution with a homogeneous background with λc = 0.3 mmol L−1
and a Gaussian peak at r = 2.8 nm with a width of 0.1 nm. The dashed line
is a fit with a third-order polynomial for t ≥ 0.5 µs. (b) The difference of
the simulated signal in (a) and the fitted cubic baseline corresponds approx-
imately to the contribution of only the Gaussian peak. The inset shows the
dipolar spectrum obtained by a Fourier transformation of the difference signal.
(c) Simulation of V (t) by the shell factorization model with a distance distribu-
tion with a homogeneous background with λc = 0.15 mmol L−1 and two Gaus-
sian peaks at r1 = 2.8 nm with a width of 0.1 nm and at r2 = 7 nm with a width of
0.25 nm and 3.9 times larger integral intensity. The dashed line is a fit with a fifth-
order polynomial for t ≥ 0.5 µs. (d) The difference of the simulated signal in
(c) and the fitted fifth-order polynomial corresponds approximately to the con-
tribution of only the Gaussian peak at 2.8 nm (linear contribution). The inset
shows the theoretical distance distribution corresponding to only the peak at
2.8 nm (dashed line) together with the result of a direct transformation of the
linear contribution to distance domain (solid line).
Assuming an exact separation of linear and nonlinear parts,
the discretized mathematical model for the linear contribution
is thus
K
ꢅ
Vlin(t) =
λꢂG(rk)Vk(t),
[14]
k=1
where k runs from 1 to klin with r1 =rmin and rK =rlin and
λꢂ = λ/(1 − λ). Deviations due to the limited precision of the

76
JESCHKE ET AL.
separation are mainly expected for distances close to rlin. In the depart from f (t) = t. For a discretized kernel, the normalization
following we omit the prime in λꢂ.
constants c(ωm) are then given by
Discrete Integral Transformation dePakeing
ꢆ
∞
ꢂ
The conversion of the dipolar evolution data to the distance
distribution or radial distribution function thus boils down to an
inversion of Eq. [14] with known Vk(t). This turns out be an
ill-posed problem. However, since the dipolar frequency ωdd is
a strictly monotonous function of the distance r, we can rewrite
Eq. [14] as
cm = c(ωm) = 1
Vm2(t)t dt.
[19]
0
Furthermore we define the cross-talk matrix d for the approxi-
mative kernel by its elements
ꢅ
∞
ꢂ
Vlin(t) =
P(ωl)Vl(t),
[15]
1
dml
=
Vl(t)Vm(t)t dt.
[20]
l
cm
0
with
Due to normalization the diagonal elements of the cross-talk ma-
trix are all unity. However, the matrix is not symmteric, since in
general cm = cl for m = l. For an exact kernel, the orthogonality
condition, Eq. [17], would hold, and all off-diagonal elements
of the cross-talk matrix would be zero. Nonzero off-diagonal el-
ements correspond to an erroneous detection of frequency com-
ponent ωm due to the presence of another frequency component
ωl. The result P(0) of the discrete approximative Pake transfor-
mation
π/2
ꢂ
Vl(t) =
cos [(3 cos2 θ − 1)ωlt] sin θ dθ,
[16]
0
where the P(ωl) quantify a discrete distribution of dipolar fre-
quencies at appropriate sampling points ωl (see below). The dis-
tance distribution can be obtained from P(ωl) by a mapping of
the ωl to rl using Eq. [3] and by a proper scaling that is discussed
in the following section. The inversion of Eq. [15] is a so-called
dePakeing problem, for which several solutions of increasing
sophistication have been given in the literature (29–33). The
general dePakeing problem is still ill posed. However, we shall
show now that for our special case of a continuous distribution
P(ω) with a resolution limited by the conformational freedom
of the spin-carrying molecules, Eq. [15] can be converted to a
reasonably well-posed problem by choosing a suitable set of
ωl. This problem can be solved by an integral transformation
in analogy to the Fourier transformation that is widely used in
magnetic resonance spectroscopy. The approach is close in spirit
ꢅ
1
Pm(0)
=
Vlin(ti )Vm(ti )ti ,
[21]
cm
i
where ti runs from 0 to tmax, is then related to the true dipolar
frequency distribution P by
dP = P(0)
.
[22]
to the REDOR transformation used in solid-state NMR (34) but By solving this set of linear equations for P, the true distribution
implemented in a different way.
is obtained.
The inversion of Eq. [15] by an integral transformation re-
quires a kernel K(ωddt) so that the orthogonality condition
Such a cross-talk-corrected approximative integral transfor-
mation is in principle generally applicable for the solution of
inverse linear problems. However, the quality of the approxima-
tive kernel is still crucial, as it determines if the system of linear
equations, Eq. [22], corresponds to an ill-posed problem or not.
This can be quantified by computing the condition number of the
cross-talk matrix, which is the ratio of its largest to its smallest
singular value.
∞
ꢂ
Vl(ωlt)K(ωddt) dt = δ(ωl − ωdd)
[17]
0
is fulfilled. Instead of deriving K(ωddt) analytically, we may
start from an approximative kernel
The construction of a good approximative kernel involves the
optimization of f (t) and the choice of a suitable set of ωm,
i.e., proper discretization. For our case, it can be shown that
f (t) = t is a good choice, as it minimizes the cross talk between
K(ωddt) = c(ωdd)V (ωddt) f (t),
[18]
where f (t) does not depend on ωdd and c(ωdd) is a normalization dipolar frequency zero and any other frequency. This property
constant that does not depend on t. Examples for integral trans- is crucial for practical applications, as the separation of V (t)
forms of this kind are the Fourier transformation with f (t) ≡ 1 into a linear and a nonlinear contribution may leave a small
and the Bessel transformation with f (t) = t (35). As dipolar time erroneous constant contribution in Vlin. To secure this property
evolution data are closely related to Bessel functions, we also and approximate orthogonality for neighboring frequencies ωm

DISTANCE DISTRIBUTIONS FROM DIPOLAR TIME EVOLUTION DATA
77
and ωm+1, we have to choose
quencydistributionwiththreeGaussianpeaksandaresolutionof
40 kHz and white noise with an amplitude of 0.2% of the signal
maximum was added (Fig. 3a). The superposition of Pake pat-
terns obtained by Fourier transformation of these data is shown
in Fig. 3b. The noise level is already significantly enhanced in the
approximative Pake transform P(0) (Fig. 3c), which is entirely
due to the resolution enhancement achieved here by multiplying
the data with f (t) = t. The deviation of P(0) from the original
distribution (dashed line) is caused by cross talk. Accordingly,
the cross-talk-corrected Pake transform P agrees nicely with
the original distribution. The cross-talk correction induces only
a small additional increase in the noise level, as predicted by the
small condition number of the cross-talk matrix. The apparent
noise suppression of the more sophisticated dePakeing methods
is thus due to the smoothing inherent in them.
ꢇ
ꢈ
π
1
4
ωm
=
m +
,
[23]
tmax
where m runs from 1 to N/2 − 2 and N is the number of points
in the discrete dipolar time evolution data. The data are thus
analyzed between a minimum dipolar frequency
5π
ωmin
=
[24]
4tmax
and a maximum dipolar frequency
2π(2N − 7)
8(N − 1)ꢀt
ωmax
=
,
[25]
Distance-Domain Smoothing
where ꢀt is the dwell time. For a dwell time of 8 ns and
tmax = 2 µs this corresponds to a distance scale from 1.19 to
5.50 nm, which covers the whole range of interest rmin ≤r ≤rlin.
The condition number for the corresponding cross-talk matrix
with ꢀt = 8 ns and N = 256 is 2.99. For typical experimental
conditions 1 µs ≤ tmax ≤ 8 µs, the condition number ranges be-
tween 2.98 and 3.13. Performed in this manner, dePakeing thus
does not involve the solution of an ill-posed problem.
Mapping the discrete distribution P(ω) defined at equally
spaced values of ω (Eq. [23]) to G(r) provides a distance dis-
tribution with an ordinate spacing proportional to r−4. In fact,
such a spacing in the distance domain is more appropriate than
an equal spacing, since for a given precision of the frequency
measurement the relative error in a distance measurement scales
with r4. Furthermore, we have to consider that P(ω) is actually
mapped to 4πr2G(r), so that for the discrete data we find
Nevertheless, dePaked spectra are significantly noisier than
the Fourier transform of the time-domain data as can be seen in
Fig. 3. A time-domain data set was simulated for a dipolar fre-
Gm = G(r(ωm)) = r−6 P(ωm).
[26]
In the absence of noise, elimination of nonlinear contribu-
tions, approximate Pake transformation, cross-talk correction,
and mapping yield a distance distribution which agrees almost
perfectly with the theoretical distribution as can be seen, for
example, in Fig. 2d.
However, the r−6 scaling of the amplitude increases the noise
at short distances dramatically, as can be seen in Fig. 4 where
the direct conversion of simulated noisy dipolar time evolution
data to a distance distribution is shown. Despite the excellent
signal-to-noise ratio of the original data (Figs. 4a and 4b), the
distance distribution appears to be ill defined for r < 2.2 nm
(Fig. 4c). Fortunately, this extreme scaling of the noise is
caused by imposing on the data an unrealistic distance reso-
lution at the lower end of the distance range and can thus be
avoided.
Closer inspection reveals that the r−4 ordinate spacing of the
distance distribution becomes unnecessarily fine at short dis-
tances where the precision of the distance measurement is no
longer determined by the precision of the frequency measure-
FIG. 3. Cross-talk-corrected integral transformation dePakeing. The orig-
inal distribution of dipolar frequencies ωdd consists of three Gaussian peaks ment, but rather by the unavoidable uncertainty of the spin–spin
at 1.5 MHz (variance σ = 0.3 MHz, intensity I = 1.0), 3 MHz (σ = 0.8 MHz,
I = 0.4), and 9 MHz (σ = 2.5 MHz, I = 0.7). (a) Simulated time-domain sig-
nal with 0.2% noise added. (b) Fourier transform of the time-domain signal.
(c) Approximate Pake transform P(0) (solid line) and original distribution
(dashed line). (d) Cross-talk-corrected Pake transform P (solid line) and original
distribution (dashed line).
distance due to conformational freedom and the spatial distribu-
tion of the wave function of the unpaired electron. In practice,
linewidths narrower than about 0.02 nm cannot be expected in
a distance distribution. It is then sensible to smooth the distance
distribution based on the assumption of a realistic minimum

78
JESCHKE ET AL.
ethynylene-based biradicals with distances between 2.8 and
5.0 nm has been described elsewhere (38). Their end-to-end
distances were measured before by DEER and SIFTER experi-
ments and were found to be in good agreement with the results
of a force-field conformer search (12). The biphenyl-based bi-
radical 3 with an end-to-end distance of approximately 2 nm
was synthesized as follows:
Benzidine (155 mg, 0.84 mmol), DMAP (342 mg,
2.80 mmol), and 3-carboxy-2,2,5,5-tetramethyl-3-pyrrolin-1-
yloxy (461 mg, 2.50 mmol) were dissolved in THF (12 mL).
While cooling in an ice bath, (516 mg, 2.50 mmol) dicyclohexyl-
carbodiimide in THF (5 mL) tetrahydrofuran was added drop-
wise with a syringe. After 4 h the formed precipitate was sepa-
rated by filtration and the yellow filtrate was washed with 2 mol
L−1 HCl and brine, dried (MgSO4), and concentrated. Column
chromatography (SiO2; dichloromethane/ethyl acetate 2 : 1)
gave 3 (160 mg, 37%) as pale yellow crystals, mp. 250–253◦C
(dec.). FD-MS: m/z = 516.8 (100%, M+).
FIG. 4. Elimination of nonlinear contributions, conversion to distance do-
The flexible oligomethylene biradical 4 was kindly supplied
by Ciba-Geigy, Basel. For the DEER measurements, toluene
solutionswithatotalbiradicalconcentrationof1mmolL−1 were
filledintotheEPRsampletubes, shock-frozenbyimmersioninto
main, and distance-domain smoothing. The original distance distribution G(r)
consists of two Gaussian peaks at 2.0 nm (variance σ = 0.2 nm) and 3.2 nm
(σ = 0.4 nm) with equal integral intensity and a homogeneous background with
λc = 0.06 mmol L−1. (a) Time-domain data simulated with the shell factor-
ization model. Noise with an rms amplitude of 0.05% of the maximum signal liquid nitrogen, and transferred to the cooled probehead. For the
amplitude was added. The dashed line is a fit of the nonlinear contribution by a
second-order polynomial. (b) Linear contribution to the time-domain data due to
distances r < rlin obtained by subtraction of the polynomial. The inset shows the
biradical with an end-to-end distance of 5 nm, a concentration
of 0.2 mmol L−1 was used.
Fourier transform. (c) Distance distribution obtained by a cross-talk-corrected
Pake transformation and mapping of P(ω) to 4πr2G(r) (solid line) and original
distance distribution (gray dashed line). (d) Distance distribution obtained as in
(c) but after convolution with a Gaussian function with a variance of 0.05 nm
(solid line) and original distance distribution (dashed line).
DEER Measurements
Dipolar time evolution data were obtained at X-band fre-
quencies with a Bruker Elexsys E 580 spectrometer using
the four-pulse DEER experiment (10, 11). All biradical sam-
ples were measured at a temperature T = 80 K with a Bruker
Flexline split-ring resonator ER 4118X-MS3, except for the
sample of the longest biradical 2b which was measured at
T = 15 K with a Flexline pulse ENDOR resonator ER 4118X-
MD5-EN. Both resonators were overcoupled to Q ≈ 100. Pump
pulses at a secondary microwave frequency were generated by
feeding the output of an HP 83508 sweep oscillator to one mi-
crowave pulse forming unit of the spectrometer. The pump fre-
quency was set to the maximum of the nitroxide spectrum, and
the observer frequency was 60 MHz higher unless specified oth-
erwise. Both the π/2 and π pulses had a length of 32 ns and the
dwell time was 8 ns.
peak width. Because of the r−4 ordinate spacing such a distance-
domain smoothing has a dramatic effect on noise at short dis-
tances, as can be seen in Fig. 4d where the data of Fig. 4c are
shown again, but now after convolution with a Gaussian line with
a variance of 0.05 nm. The original distance distribution (dashed
line) is now reproduced with satisfying precision throughout the
distance range of interest. Such distance-domain smoothing is a
transparent procedure in the sense that it imposes only a known,
and in many cases insignificant, broadening on the true dis-
tance distribution. Furthermore, it is analogous to procedures
that are well established in Fourier transform NMR and EPR
spectroscopy (5, 28). The influence of the smoothing inherent in
other kinds of data analysis, such as maximum entropy methods
(36) or Tikhonov regularization (37), may be more difficult to
estimate.
Comparison of Shell Factorization with Direct Conversion
For a critical test of distance resolution close to the lower
limit of the distance range we selected a biphenyl-based birad-
ical (for the structure, see inset in Fig. 5c). The background in
the dipolar evolution data can be fit nicely by a second-order
polynomial (Fig. 5a) with t1 = 320 ns. The cross-talk-corrected
Pake transform (Fig. 5b) exhibits a narrow peak at 1.92 nm,
EXPERIMENTAL RESULTS AND DISCUSSION
Model Systems
As model systems for distance distributions with narrow which is close to the expected spin–spin distance of 2 nm. An
peaks we used shape-persistent biradicals with end-to-end dis- additional extremely narrow peak is observed at approximately
tances between 2 and 5 nm. The synthesis of these phenylene- 1.5 nm. This peak and its negative wings are an artifact due to

DISTANCE DISTRIBUTIONS FROM DIPOLAR TIME EVOLUTION DATA
79
FIG. 5. Direct conversion of experimental dipolar time evolution data to a distance distribution for a shape-persistent biradical with a spin–spin distance of
1.92 nm. (a) Fitting of the nonlinear contribution. A second-order polynomial (dashed line) is fitted to the experimental data at t ≥ 320 ns and subtracted. The inset
shows the extracted linear contribution. (b) Distance distribution obtained by a cross-talk-corrected Pake transformation (no smoothing). The peak at approximately
1.5 nm is due to proton modulations. (c) Distance distribution obtained as in (b) but after convolution with a Gaussian function in the distance domain with a
variance of 0.02 nm. (d) Experimental data (solid line) and fit by the shell factorization model (dashed line) assuming a distance distribution consisting of a single
Gaussian peak at 1.92 nm with a variance of 0.05 nm and a homogeneous background with λc = 0.45 mmol L−1. The inset shows a comparison of the distance
distribution obtained by direct extraction (solid line), in the shell factorization model (dashed line), and from a force-field conformer search (gray histogram).
1H nuclear modulation. This artifact can be partially, but not due to orientational selection effects is not apparent even for this
completely, suppressed by adjusting the fixed interpulse delays system with low conformational freedom.
τ1 and τ2 to a blind spot of the modulations (10, 11). If no such
We have also compared the experimental distance distribution
precautions are taken, as in our case, the distance distribution to the theoretical distance distribution in an ensemble of 48 con-
is significantly distorted for r < 1.8 nm. If DEER is performed formers which was computed with the Merck Molecular Force
at higher frequencies, the artifact is shifted toward shorter dis- Field using the Titan software package (Wavefunction Inc.). To
tances and ultimately out of the accessible distance range. avoid difficulties with the parametrization of nitroxide groups,
For example, at Q-band frequencies (ν ≈ 35 GHz) the proton these groups were substituted by keto groups. The electron spin
Zeeman frequency corresponds to a distance of 1.1 nm.
was assumed to be localized at the center of the N–O bond. The
In this measurement, which took 14 h, the signal-to-noise distance between the centers of the two N–O bonds may be by
ratio is sufficient to dispense with distance-domain smoothing. approximately 0.07 nm shorter than the distance between the
=
Nevertheless, convolution with a Gaussian function with a width centers of the C O groups in the diketone, as it was found for
of 0.02 nm improves the appearance of the distribution for low the conformer with the lowest energy by a BLYP density func-
distances without broadening the peak at 1.92 nm (Fig. 5c). The tional geometry optimization with the ADF (39) package (basis
dipolar evolution data can also be fit quite well using the shell set IV, version 2000.02) as well as by a B3LYP density functional
factorization model and a distance distribution consisting of one geometry optimization (basis set 6–31G∗) with the Titan pack-
Gaussian peak at 1.92 nm with a variance of 0.05 nm and a age. The theoretical distance distribution, corresponding to a
homogeneous background (Fig. 5d). The remaining deviations Boltzmann distribution over the conformers at the melting point
are due to orientation selection. As can be seen in the inset, there of toluene, is displayed as a histogram in the inset in Fig. 5d. This
is a satisfying agreement between the distributions obtained by distribution is narrower than the experimental distribution and
direct conversion (solid line) and modeling with a Gaussian peak it is shifted by approximately 0.15 nm toward longer distances.
(dashed line). A significant distortion of the distance distribution This difference, which slightly exceeds the estimated combined

80
JESCHKE ET AL.
FIG. 6. Direct conversion of dipolar time evolution data to distance distributions for model compounds (solid lines) and fits with simple model distributions
within the shell factorization model (dashed lines). (a) Flexible oligo (methylene) biradical with an average end-to-end distance of 2.1 nm. (b) Equimolar
mixture (c = 0.5 mmol L−1 each) of two shape-persistent biradicals 1 and 2a with distances of 2.83 and 3.63 nm (for structures, see Ref. 12). (c) Dipolar
evolution data of the shape-persistent biradical 2b (c = 0.2 mmol L−1) with an end-to-end distance of 4.93 nm. The dash-dot line is a linear background fit
for t ≥ 1.2 µs and the dashed line the best shell factorization fit obtained with a distribution consisting of a Gaussian peak (r = 4.93 nm, σ = 0.19 nm) and a
homogeneous background. (d) Radial distribution function obtained by direct conversion of the data in (c) with Gaussian distance-domain smoothing (σ = 0.2 nm).
(e) Radial distribution function obtained by zero-filling to 2048 points and direct conversion with Gaussian distance-domain smoothing (σ = 0.1 nm). (f) Structures
of the shape-persistent biradicals 1, 2a, and 2b.

DISTANCE DISTRIBUTIONS FROM DIPOLAR TIME EVOLUTION DATA
81
uncertainties of the experiment and the force-field computation, the structure, see Fig. 6f). In this case the end-to-end distance
might indicate a small exchange coupling between the two moi- of approximately 5 nm is close to rlin, so that a precise elim-
eties of the molecule.
ination of the nonlinear contributions is more difficult and the
For a flexible oligomethylene biradical with 10 methylene resolution in G(r) is severely limited. It is therefore crucial to
units, the nonlinear background can again be fit nicely with work at low concentrations to have smaller nonlinear contri-
t1 = 320 ns and a second-order polynomial. After direct con- butions and to measure the data in a longer time window. We
version, we obtain a much broader distance distribution with a found that a sufficient signal-to-noise ratio in a four-pulse DEER
maximum at 2.1 nm. Due to this broadening, distance-domain measurement could still be obtained in a measurement time of
smoothing with a Gaussian function with a variance of 0.04 nm 16 h with a biradical concentration of 0.2 mmol L−1 and with
becomes necessary, although the measurement time was the τ1 = 400 ns and τ2 = 5 µs (Fig. 6c). For this low concentration
same as for the biphenyl-based biradical. Furthermore, the arti- and long distance, the background was best fitted by a linear
fact at 1.5 nm due to proton modulations becomes more promi- function (q = 1) with t1 = 1.2 µs (dash-dot line). The shell fac-
nent. Nevertheless, the distance distribution obtained by direct torization fit (dashed line) again exhibits slight differences from
conversion (Fig. 6a) can be considered reliable for r > 1.8 nm. the experimental data that can be traced back to orientation se-
Again, the agreement between the distributions obtained by di- lection. The radial distribution function obtained by direct con-
rect conversion (solid line) and modeling with a Gaussian peak version and distance-domain smoothing with σ = 0.2 nm is in
using the shell factorization model (dashed line) is satisfying. general agreement with the one obtained by a shell factorization
Distortionsduetoorientationalselectionarenotexpectedforthis fit. However, for this long distance the resolution is rather poor
flexible biradical, as the orientations of the molecular frames of with direct conversion. Zero-filling of the experimental data to
the two nitroxide groups are not strongly correlated. At least part 2048 points before the integral transformation significantly im-
of the deviation between the two curves is due to the fact that proves resolution and agreement with the shell factorization fit
the end-to-end distance of these biradicals is not expected to ex- without introducing significant distortions (Fig. 6e). Apodiza-
hibit a Gaussian distribution in a G(r) plot (40, 41). The access tion of the data by a Hamming window before zero-filling causes
to experimental distance distributions, rather than only mean significant broadening in the Pake transform (data not shown).
distances, should allow for more detailed checks of theoretical Even with zero-filling, the agreement of peak width and peak
models for chain conformation statistics.
maximum remains worse than for the shorter distances. The
To compare shell factorization and direct conversion at some- apparent line broadening with direct conversion may indicate
what largerdistanceswe usedanequimolarmixtureof the shape- a slightly incomplete separation of linear and nonlinear con-
persistent biradicals 1 and 2a (c1 = c2a = 0.5 mmol L−1) with tributions. As predicted by theory, a distance of 5 nm may thus
end-to-end distances of approximately 2.8 and 3.6 nm (for the correspond to the upper limit for the direct conversion approach,
structures, see Fig. 6f). Because of the slower decay of the linear unless even lower concentrations and longer time windows can
contributions for the larger distances, we used t1 = 480 ns for be used.
the background correction, which was again performed with a
second-order polynomial. The results obtained by direct conver-
sion and by a shell factorization fit with a distance distribution
consisting of two Gaussian peaks and a homogeneous back-
CONCLUSION
In the limit of small modulation depths, angular correlations
ground agree well (Fig. 6b). The mean distances in the shell between spin pairs can be neglected and dipolar time evolution
factorization fit (r1 = 2.83 nm, r2a = 3.64 nm) also agree within data can be analyzed in terms of a spin–spin distance distri-
0.01 nm with our earlier fits of DEER data for these biradicals bution. For disordered systems, the dipolar time evolution data
which neglected any effects of nonlinearity (12). However, the can be quantitatively simulated by integrating the signal over
widths of the two peaks are now found to be approximately thin spherical shells and multiplying the signals of all the shells
equal (σ1 = 0.13 nm, σ2a = 0.12 nm), in contrast to the earlier (shell factorization model). The contribution due to electron spin
fits. If we consider that P(r) = 4πr2G(r) is the proper distribu- pairs with distances smaller than about 5 nm is closely approx-
tion function for a comparison of integral intensities, we find an imated by integration over the distance distribution, as used in
experimental ratio of c1 : c2a = 1.18, which is in satisfying agree- earlier work. To a good approximation, the nonlinear contri-
ment with the expected ratio of 1.0 given the error in weighing bution due to spins at longer distances can be eliminated from
only a few milligramms of the substances. A distortion due to the data by fitting a low-order polynomial to the data after a
orientational selection might be expected as a ghost peak at certain threshold time t1, extrapolating this polynomial to time
2.25 nm, corresponding to twice the dipolar frequency for the zero, subtracting it, and dividing the residual by the same poly-
peak at 2.83 nm. No such peak is seen, which indicates again nomial. The extracted linear contribution to the data can be di-
that orientation selection at X-band frequencies is too weak to rectly converted to a distribution of dipolar coupling frequencies
be recognized with this kind of data analysis.
by a cross-talk-corrected integral transformation and mapped to
Finally we have tested the agreement between direct conver- a distance distribution or to the radial distribution function G(r).
sion and shell factorization for biradical 2b of Ref. (38) (for Excessive noise in G(r) at short distances can be suppressed by

82
JESCHKE ET AL.
14. S. Saxena and J. H. Freed, J. Chem. Phys. 107, 1317 (1997).
15. P. Borbat and J. H. Freed, Chem. Phys. Lett. 313, 145 (1999).
16. R. G. Larsen and D. J. Singel, J. Chem. Phys. 98, 5134 (1993).
convolution with a Gaussian function of constant width in the
distance domain. Experiments on shape-persistent and flexible
biradicals with end-to-end distances between 2 and 5 nm con-
firm that radial distribution functions obtained by such a direct
conversion are in satisfying agreement with radial distribution
functions obtained by fits with the shell factorization model and
with theoretical expectations. As the latter model considers non-
linear effects directly and quantitatively, this demonstrates that
nonlinear effects can be neglected for r ꢀ 5 nm and that the
contributions due to pairs with larger spin–spin distances are
effectively eliminated by the procedure described above.
17. A. G. Maryasov, Yu. D. Tsvetkov, and J. Raap, Appl. Magn. Reson. 14, 101
(1998).
18. M. Pannier, V. Scha¨dler, M. Scho¨ps, U. Wiesner, G. Jeschke, and H. W.
Spiess, Macromolecules 33, 7812 (2000).
19. M. Pannier, M. Scho¨ps, V. Scha¨dler, U. Wiesner, G. Jeschke, and H. W.
Spiess, Macromolecules 34, 5555 (2001).
20. A. M. Raitsimring and V. V. Tregub, Chem. Phys. 77, 123 (1983).
21. V. V. Kurshev, A. M. Raitsimring, and T. Ichikawa, J. Phys. Chem. 95, 3565
(1991).
22. A. D. Milov, A. G. Maryasov, Y. D. Tsvetkov, and J. Raap, Chem. Phys.
Lett. 303, 135 (1999).
ACKNOWLEDGMENTS
23. A. Y. Pusep and N. V. Shokirev, Opt. Spektrosk. 57, 792 (1984); Opt.
Spectrosc. 57, 482 (1984).
The authors thank Christian Bauer for technical support and Dr. C. Kro¨hnke
(Ciba-Geigy AG) for supplying the oligomethylene biradical. Detailed com-
ments by an anonymous referee are gratefully acknowledged.
24. A. M. Raitsimring, L. A. Rapatskii, and N. V. Shokhirev, Chem. Phys. 105,
117 (1986).
25. J. R. Klauder and P. W. Anderson, Phys. Rev. 125, 912 (1962).
REFERENCES
26. A. M. Raitsimring, K. M. Salikhov, B. A. Umanskii, and Yu. D. Tsvetkov,
Sov. Phys. Solid State 16, 492 (1974).
1. L. J. Berliner, S. S. Eaton, and G. R. Eaton (Eds.), “Biological Magnetic
Resonance,” Vol. 19, Kluwer, New York (2001).
27. A. D. Milov and Yu. D. Tsvetkov, Appl. Magn. Reson. 18, 217 (2000).
28. R. R. Ernst, G. Bodenhausen, and A. Wokaun, “Principles of Nuclear Mag-
netic Resonance in One and Two Dimensions,” Clarendon, Oxford (1987).
2. O. Kratky and O. Glatter (Eds.), “Smaller Angle X-ray Scattering,”
Academic Press, London (1982).
29. M. Bloom, J. H. Davis, and A. L. MacKay, Chem. Phys. Lett. 80, 198 (1981).
30. E. Sternin, M. Bloom, and A. L. MacKay, J. Magn. Reson. 55, 274 (1983).
3. J. S. Higgins and H. C. Benoit, “Polymers and Neutron Scattering,”
Clarendon Press, Oxford (1994).
31. K. Whitall, E. Sternin, M. Bloom, and A. L. MacKay, J. Magn. Reson. 84,
4. P. Wu and L. Brand, Anal. Biochem. 218, 1–13 (1994).
64 (1989).
5. A. Schweiger and G. Jeschke, “Principles of Electron Paramagnetic Reso-
nance,” Oxford University Press, Oxford (2001).
32. M. A. McCabe and S. R. Wassall, J. Magn. Reson. B 106, 80 (1995).
33. H. Scha¨fer, B. Ma¨dler, and F. Volke, J. Magn. Reson. A 116, 145 (1995).
6. A. D. Milov, K. M. Salikhov, and M. D. Shirov, Fiz. Tverd. Tela (Leningrad)
23, 957 (1981).
34. K. T. Mueller, T. P. Jarvie, D. J. Aurentz, and B. W. Roberts, Chem. Phys.
Lett. 242, 535 (1995).
7. A. D. Milov, A. B. Ponomarev, and Yu. Tsvetkov, Chem. Phys. Lett. 110,
35. D. Lemoine, J. Chem. Phys. 101, 3936 (1994).
67 (1984).
36. J. L. Hoch and A. S. Stern, “NMR Data Processing,” Wiley–Liss, New York
8. A. M. Raitsimring and K. M. Salikhov, Bull. Magn. Reson. 7, 184
(1996).
(1985).
37. A. N. Tikhonov and V. Y. Arsenin, “Solutions of III-Posed Problems,” Wiley,
9. A. D. Milov, A. G. Maryasov, and Yu. D. Tsvetkov, Appl. Magn. Reson. 15,
New York (1977).
107 (1998).
38. A. Godt, C. Franzen, S. Veit, V. Enkelmann, M. Pannier, and G. Jeschke,
J. Org. Chem. 65, 7575 (2000).
10. M. Pannier, S. Veit, A. Godt, G. Jeschke, and H. W. Spiess, J. Magn. Reson.
142, 331–340 (2000).
39. C. Fonseca Guerra, J. G. Snijders, G. te Velde, and E. J. Baerends, Theor.
Chem. Acc. 99, 391 (1998).
11. G. Jeschke, M. Pannier, and H. W. Spiess, in “Biological Magnetic Reso-
nance” (L. J. Berliner, S. S. Eaton, and G. R. Eaton, Eds.), Vol. 19, Chap. II,
Kluwer, New York (2001).
40. V. Pfannebecker, H. Klos, M. Hubrich, T. Volkmer, A. Heuer, U. Wiesner,
and H. W. Spiess, J. Phys. Chem. 100, 13428 (1996).
12. G. Jeschke, M. Pannier, A. Godt, and H. W. Spiess, Chem. Phys. Lett. 331,
243 (2000).
41. P. J. Flory, “Statistical Mechanics of Chain Molecules,” Wiley–Interscience,
New York (1969).
13. S. Saxena and J. H. Freed, Chem. Phys. Lett. 251, 102 (1996).