2D NMR Nutation Analysis of Non-Thermal Polarization of Coupled Multi-Spin Systems

Article abstract of DOI:10.1524/zpch.217.12.1641.20479

A new, convenient method of analyzing the spin polarization of a non-equilibrium system of N coupled nuclei is described and applied to photo-reactions exhibiting chemically induced dynamic nuclear polarization (CIDNP). It is based on the Fourier analysis of the variation of NMR line intensities as a function of the radio frequency excitation pulse length. A relationship between the spectral components at various harmonic order and the alignment in the spin multiplet is established. In application to the Norrish type I photolysis of cyclodecanone we demonstrate that at low magnetic field the rate determining step in the reaction kinetics depends on the mutual orientation of at least four pairs of non-equivalent spins.

Full text of DOI:10.1524/zpch.217.12.1641.20479

Z. Phys. Chem. 217 (2003) 1641–1659
by Oldenbourg Wissenschaftsverlag, München
2D NMR Nutation Analysis of Non-Thermal
Polarization of Coupled Multi-Spin Systems
1
By K. L. Ivanov1, 2, K. Miesel3 , H.-M. Vieth3, , A. V. Yurkovskaya , and
∗
R. Z. Sagdeev1
1
International Tomography Center, Novosibirsk 630090, Russia
Novosibirsk State University, Novosibirsk 630090, Russia
Institut für Experimentalphysik, Freie Universität Berlin, Berlin, Germany
2
3
Dedicated to Prof. Dr. Hans-Heinrich Limbach on the occasion
of his 60th birthday
(Received May 31, 2003; accepted September 4, 2003)
CIDNP / Multi-Spin Order / Cyclic Ketones
A new, convenient method of analyzing the spin polarization of a non-equilibrium system
of N coupled nuclei is described and applied to photo-reactions exhibiting chemically
induced dynamic nuclear polarization (CIDNP). It is based on the Fourier analysis of
the variation of NMR line intensities as a function of the radio frequency excitation
pulse length. A relationship between the spectral components at various harmonic order
and the alignment in the spin multiplet is established. In application to the Norrish
type I photolysis of cyclodecanone we demonstrate that at low magnetic field the rate
determining step in the reaction kinetics depends on the mutual orientation of at least
four pairs of non-equivalent spins.
1. Introduction
NMR spectroscopy owes its appeal as a versatile analytic method to the high
content of information that is reflected in the position, the shape and intensity
pattern of the spectral lines as well as in the time evolution of the spin ob-
servables. Matching this diversity a large variety of experimental techniques
has been developed. Both aspects, developing new techniques and utilizing
the potential of NMR for solving problems in analytical chemistry, have been
a focus of Hans-Heinrich Limbach’s interest for several decades. In this con-
tribution we will deal with the intensity distribution in spectra of dynamically
polarized spin systems. While at thermal equilibrium the spectral intensities
*
Corresponding author. E-mail: hans-martin.vieth@physik.fu-berlin.de

1642
K. L. Ivanov et al.
correspond to the number of spins contributing to a particular line, for situ-
ations off equilibrium the response of the spin system to non-selective pulse
excitation is more complex and the different pattern of polarization has also to
be considered. As a rule the mechanism and the efficiency of forming the spin
polarization are closely related to the magnetic parameters and the dynamics
of the involved species. In many cases, the observed intensity distribution is
usable as a fingerprint of intermediate stages in the polarization process. The
analysis by evaluating the population numbers of the spin eigenstates by means
of Fourier Transform NMR (FT-NMR) faces difficulties. First, only for small
excitation pulse rotation angles ϕ = γB1τ (γ gyromagnetic ratio, B1 ampli-
tude of excitation pulse, and τ pulse duration) the spectral intensities of lines
are proportional to the population differences of the involved pair of levels as
obtained by slow passage NMR techniques (cw-NMR). This holds true even
for weakly coupled spins. Using a small excitation angle inevitably leads to
a small FID amplitude and thus to a low S/N ratio. T1 relaxation toward ther-
mal equilibrium prohibits consecutive acquisition and accumulation of a series
of signals excited at such a small angle, unless prior to each acquisition a full
polarization cycle is executed. Second, spin-spin coupling is often so small or
motional line broadening so large that resolution of neighboring lines is impos-
sible. In such cases the data are insufficient for the full analysis. An alternative
approach is to study the variation of the spectral intensity pattern as a function
of the rotation angle ϕ. In the present paper we show how this approach leads
to full characterization of the multi-nuclear spin order in molecules that contain
several coupled nuclei and illustrate the method by a prominent experimental
example.
Rotation angle dependent FT-NMR spectra, often termed nutation patterns,
have been studied experimentally and theoretically in a sizable number of
works [1–8]. In most cases the consideration was restricted to two-spin 1/2
systems because they show a relatively simple nutation that contains only sin ϕ
and sin 2ϕ terms. For larger spin groups the interpretation of experimental nu-
tation patterns and the evaluation of the relevant system parameters is rather
sophisticated. A spin system of N coupled spins 1/2 shows, in general, a su-
perposition of N terms with different angular variations. In the present article
we suggest a general method to analyze the nutation patterns of such multi-
nuclear spin systems. The approach follows the observation [3] that in a two
spin system the sin ϕ term carries the net magnetization of the nuclei, while
the sin 2ϕ term represents their non-equilibrium average mutual orientation.
A generalization of this idea to systems of N coupled spins is to record a se-
ries of spectra measured at stepwise increased rotation angle and decompose
this set by Fourier analysis into N harmonics. One can then represent the
result as a 2D spectrum where the second dimension corresponds to the har-
monic order of the periodic nutation pattern. In this case the k-th harmonic
component will reflect the mutual orientation of k nuclear spins (k-th spin
order).

2D NMR Nutation Analysis of Non-Thermal Polarization . . .
1643
Typical examples of a process generating nuclear spin polarization far off
equilibrium are Para-Hydrogen Induced Polarization (PHIP) [9] and Chemi-
cally Induced Dynamic Nuclear Polarization (CIDNP) [10, 11]. PHIP (some-
times the acronyms ALTADENA [12] and PASADENA [13] are used) is ob-
served in hydrogenation reactions. It has its origin in the singlet multiplicity
of the precursor para-H2 molecule and thus gives rise to singlet correlation
of the corresponding pair of protons in the reaction products. CIDNP arises
at reactions with paramagnetic intermediates and results from reactivity dif-
ferences of radical pairs that depend upon the nuclear spin states [11, 14, 15]
and leads to the formation of high spin order. This polarization reveals itself in
anomalous intensities of the lines in the NMR-spectra of diamagnetic reaction
products. The intensity pattern is correlated with the magnetic properties of the
paramagnetic state, such as hyperfine coupling constants, electronic exchange
interaction, external magnetic field strength, and g-factors of the radical cen-
ters and thus provides information on the transient radicals. In many cases the
radical pairs recombine rapidly by means of hydrogen or electron transfer. As
a consequence, the CIDNP arising in these reactions contains evidence on the
elementary transfer event and allows to obtain detailed data on the reaction pro-
cess that are often unavailable by other techniques. Non-equilibrium nuclear
spins systems of various kind can be also prepared by applying suitable rf-pulse
sequences; [1] in this way such diverse phenomena as spin waves in linear
chains of nuclear spins [16] and algorithms in NMR quantum computation [17]
have been studied.
In the present article we consider in detail the theoretical treatment of the
two-dimensional harmonic analysis of CIDNP spectra of three coupled nuclei
and describe briefly the extension to N spins, before we apply our approach
to the analysis of the complicated CIDNP patterns arising in the course of the
photolytic reaction of cyclic ketones.
2. 2D harmonic analysis of spin order in polarized systems of
coupled nuclei
2.1 General case of the polarized system of N-spins
Here we shall present the analysis of the nutation patterns of thermally non-
equilibrium system of N weakly coupled spins 1/2. We shall restrict ourselves
to the so-called first order non-equilibrium states [1] where coherent superpo-
sition of states is absent and the nuclear density matrix, σ, is diagonal in the
ˆ
eigenbasis of the Hamiltonian. In principle, in the case of N coupled polarized
nuclei the total polarization, i.e. the distribution of population over the multi-
level system, is determined (when using the basis of products of single-spin
operators) not alone by all net polarizations
ꢀ
ꢁ
ꢂ
ꢃ
ˆ
ˆ
αz ˆ
Iαz = Tr I σ , (α = 1, . . . , N)
(2.1)

1644
K. L. Ivanov et al.
and the average mutual orientation of each pair of spins (multiplet polarization)
ꢀ
ꢁ
ꢂ
ꢃ
ˆ ˆ
ˆ ˆ
Iαz Iβz = Tr I I σ , (α, β = 1, . . . , N, α = β) ,
(2.2)
βz ˆ
αz
ˆ
but also by higher order entanglement. Here, Iαz is the operator of the pro-
jection of spin α on the Z axis. Correspondingly, it is necessary to determine
ꢀ
ꢁ
ꢀ
ꢁ
ˆ
ˆ ˆ
besides Iαz and Iαz Iβz also the following expectation values:
ꢀ
ꢁ
ꢂ
ꢃ
ˆ ˆ ˆ
ˆ ˆ ˆ
Iαz Iβz Iγz = Tr I I I σ ,
α, β, γ = 1, . . . , N,
α = β = γ
(2.3)
γz ˆ
αz βz
ꢀ
ꢁ
ꢂ
ꢃ
Nz ˆ
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ
Iαz Iβz Iγz Iδz = Tr I I I I σ , α, β, γ, δ = 1, . . . , N, α = β = γ = δ,
δz ˆ
αz βz γz
. . .
ꢀ
ꢁ
ꢂ
ꢃ
ˆ ˆ
ˆ
ˆ ˆ
ˆ
I1z I2z . . . INz = Tr I I . . . I σ .
1z 2z
For the k-th order multiplet effect (average mutual orientation of k spins) there
are CkN different combinations of the form (2.3) (here CkN are the binomial co-
efficients). To tackle the problem of determination of the quantities (2.1), (2.2),
(2.3) we suggest the method presented below.
As we deal exclusively with weakly-coupled systems of spin 1/2 nuclei
the eigenstates of the Hamiltonian can be characterized by the set of mag-
netic quantum numbers of the individual nuclei (namely, their projections on
the Z axis). Also, we will assume that the rf-excitation is homogeneous and
1
non-selective with respect to all H ꢁIzꢂ = 1 transitions. For systems of N
weakly-coupled non-selectively excited spins 1/2 the flip angle dependence of
the spectral line corresponding to the transition |iꢂ → | jꢂ between states |iꢂ and
| jꢂ (the nutation pattern) is given by the following equation [1, 2]:
ꢅ
ꢆ
ꢅ
ꢆ
ꢄ
2
2N−2
−2
ij,rs
ij,rs
ϕ
ϕ
(0)
r→s
L
i→ j(ϕ) = sin ϕ
sin
cos
L
.
(2.4)
2
2
(rs)
In this context it is important to note that the FT-MNR signal intensity does not
only depend on the population difference between states |iꢂ and | jꢂ, but on the
(0)
population differences between other states as well. Here Lr→s = (Ps − Pr )/2 is
the standard intensity of the fully selectively excited NMR transition |rꢂ → |sꢂ
determined by the difference between their populations,
is the so-called
ij,rs
“spin-flip number” [1] equal to the number of spin 1/2 flips, required to make
the state transitions |iꢂ → | jꢂ and |rꢂ → |sꢂ coincide. As a consequence, only
the signals corresponding to the so-called “parallel” transitions (i.e. transitions
that can be put into coincidence by flipping the spins) affect each other. The ex-
pression (2.4) of the N-spin non-equilibrium system can be written as a Fourier
series consisting of N sine terms:
N
ꢄ
(n)
i→ j
L
i→ j(ϕ) =
A
sin nϕ .
(2.5)
n=1

2D NMR Nutation Analysis of Non-Thermal Polarization . . .
1645
(n)
The amplitude of the harmonics, A , are to be found as the coefficients of
i→ j
the Fourier series expansion of the periodic function Li→ j(ϕ). The expression
(n)
for the NMR line intensity at the n-th harmonic, A , is derived in the Ap-
i→ j
pendix. One may represent the flip angle dependent FT-NMR spectrum (2.5)
of a system off thermal equilibrium as a two dimensional one: the first coordi-
nate is the spin precession frequency while the second one is the number of the
harmonic, n. Hence, the original ϕ-dependent spectrum of the N spin system
can be decomposed in two dimensions into N spectra at the different harmon-
ics. It is important to note that the k1-th order multiplet polarization (2.3) does
not contribute to the k2-th harmonic if k2 > k1. Therefore, the N-th harmonic
ꢀ
ꢁ
ˆ ˆ
ˆ
contains purely N-th order multiplet polarization I1z I2z . . . INz , the (N −1)-
th multiplet polarizations can be extracted from the (N −1)-th harmonic and
so on. As a result, the expectation values of spin operators of different order
are resolved in the second dimension. Consequently, by analyzing the CIDNP
spectrum at all 1 ≤ n ≤ N one extracts all the quantities (2.1), (2.2), and (2.3)
that provide the desired information on the radical reaction pathways and the
magnetic properties (HFI constants, g-factors and electronic exchange interac-
tion) of the reactants.
The way suggested here to determine the net and multiplet CIDNP is by no
means unique. For systems of two spins 1/2 Vollenweider and Fischer [3] sug-
gested measuring the FT-NMR spectra at ϕ = π/4 and ϕ = 3π/4, because their
sum and their difference yield the net and multiplet polarization, respectively.
However, to extend this approach to a larger number of spins turns out to be-
come rather sophisticated; in general, one would have to measure at N exactly
set flip angles and perform a complicated superposition of the spectra. In prin-
ciple, the contributions of net and multiplet polarization can be also separated
by means of a two-dimensional COSY experiment [18], as it is widely used in
CIDNP investigations of protein structure [19]. In the presence of polarization
additional cross-peaks arise solely from the multiplet polarization [18]. While
such an analysis works well at N = 2, it meets a similar complication at larger
N: no cross peaks arise from multiplet polarization of an order higher than two.
To extract these quantities one would have to apply more pulses, thus increas-
ing the complexity of the COSY experiment. In contrast, our present approach
yields all data of interest in a straightforward way for systems with an arbitrary
number of polarized nuclei. At the same time, the experiment is rather simple
with just one variable rf-pulse necessary to record the data.
2.2 System of three coupled spins
As a demonstration of our approach we show an example of 2D harmonic
analysis of the nutation patterns for three weakly coupled spins 1/2 (AMX-
system). At thermal equilibrium the NMR spectrum shown in Fig. 1(a) consists
of three quartets centered at the frequencies σ1, σ2, σ3 (in units of the chemical
shift), where σi is the chemical shift of the corresponding nucleus. The system

1646
K. L. Ivanov et al.
Fig. 1. 2D harmonic analysis of a three-spin system (schematic). (a): (upper trace) spec-
trum in thermal equilibrium and (lower trace) CIDNP spectrum measured at ϕ = π/6, with
state populations calculated according to Adrian’s theory [20] (see text). (b): system of
eigenstates; states are enumerated according to Eq. (2.6); solid, long-dashed and short-
dashed lines show the “parallel” NMR transitions. (c): harmonic spectra at n = 1, 2, 3.
σi is the chemical shift of the i-th nuclei.
of eigenstates is presented in Fig. 1(b). Three groups of “parallel” transitions
that affect each other in the FT-NMR [1] are shown by the solid, long-dashed,
and short-dashed lines. Here we have chosen the following notation for the
eigenstates:
|1ꢂ = |α1α2α3ꢂ, |2ꢂ = |α1α2β3ꢂ, |3ꢂ = |α1β2α3ꢂ, |4ꢂ = |α1β2β3ꢂ, (2.6)
|5ꢂ = |β1α2α3ꢂ, |6ꢂ = |β1α2β3ꢂ, |7ꢂ = |β1β2α3ꢂ, |8ꢂ = |β1β2β3ꢂ .
Let us first consider the intensities of the lines corresponding to the
NMR transitions (flips) of the third spin. For instance, the expression for the

2D NMR Nutation Analysis of Non-Thermal Polarization . . .
1647
Fig. 2. Photolysis of cyclodecanone (C-10). Reaction scheme (top), thermal spectrum (in-
sert) and CIDNP spectrum measured at ϕ = π/4 (bottom). Lines of product I at right and
of product II at left part of the spectral range. α, β, γ, δ, and denote corresponding pro-
tons of ketone I.
|α1α2α3ꢂ → |α1α2β3ꢂ (|1ꢂ → |2ꢂ) transition intensity is as follows [1]:
ꢅ
1
2
ϕ
ϕ
ϕ
L1→2
=
sin ϕ cos4 (P1 − P2)+cos2 sin2 (P3 − P4 + P5 − P6)
2
2
2
ꢆ
ϕ
+sin4 (P7 − P8) .
(2.7)
2

1648
K. L. Ivanov et al.
Fig. 3. CIDNP spectra of cylodecanone taken at different rotation angle ϕ.
One should note that in reference [1] there is a misprint (extra cos2 ϕ2 ) in the for-
mula corresponding to (2.7). In accordance with the general formula (2.4) the
intensity of the line that corresponds to the transition |1ꢂ → |2ꢂ does not only
depend on the population difference P1 − P2 but on the population differences
of “parallel” transitions (|3ꢂ → |4ꢂ, |5ꢂ → |6ꢂ, |7ꢂ → |8ꢂ) as well [1]. We can
write (2.7) as a sum of three Fourier harmonics:
(2)
1→2
(3)
1→2
L1→2 = A(11→) 2 sin ϕ+ A
sin 2ϕ+ A
sin 3ϕ .
(2.8)
The intensities of the harmonics, A(1n→) 2, are as follows:
5(P1 − P2 + P7 − P8)+3(P3 − P4 + P5 − P6)
(1)
1→2
A
A
A
=
=
=
,
(2.9)
32
P1 − P2 − P7 + P8
(2)
1→2
,
8
P1 − P2 − P3 + P4 − P5 + P6 + P7 − P8
(3)
1→2
,
32
The operators of interest involving the third spin are given for the AMX-system
as:
1
ˆ
I3z = Diag{1, −1, 1, −1, 1, −1, 1, −1} ,
2
1
ˆ ˆ
I1z I3z = Diag{1, −1, 1, −1, −1, 1, −1, 1} ,
(2.10)
4
1
ˆ ˆ
I2z I3z = Diag{1, −1, −1, 1, 1, −1, −1, 1} ,
4
1
ˆ ˆ ˆ
I1z I2z I3z = Diag{1, −1, −1, 1, −1, 1, 1, −1} .
8

2D NMR Nutation Analysis of Non-Thermal Polarization . . .
1649
Here the notation of states is taken as in (2.6). The corresponding expectation
values are:
ꢀ
ꢁ
ꢁ
ꢁ
ꢁ
P1 − P2 + P3 − P4 + P5 − P6 + P7 − P8
ˆ
I3z
=
=
=
=
,
,
,
.
(2.11)
2
ꢀ
ꢀ
P1 − P2 + P3 − P4 − P5 + P6 − P7 + P8
ˆ ˆ
I1z I3z
4
P1 − P2 − P3 + P4 + P5 − P6 − P7 + P8
ˆ ˆ
I2z I3z
4
ꢀ
P1 − P2 − P3 + P4 − P5 + P6 + P7 − P8
ˆ ˆ ˆ
I1z I2z I3z
8
By substituting (2.11) we can rewrite (2.9) as follows:
ꢀ
ꢁ
ꢀ
ꢁ
ꢀ
ꢁ
ꢀ
ꢁ
ꢀ
ꢁ
ˆ
ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ
ˆ ˆ ˆ
I3z + I1z I2z I3z
I1z I3z + I2z I3z
I1z I2z I3z
(1)
1→2
(2)
1→2
(3)
1→2
A
=
, A
=
, A
=
.
4
4
4
(2.12)
Hence, as expected, the intensity of the third harmonics is completely deter-
ꢀ
ꢁ
ˆ ˆ ˆ
mined by the triple multiplet polarization, I1z I2z I3z . As it was in the case
of two nuclei the intensity of the second harmonics depends on the multi-
plet polarization of two spins. Here we consider the NMR transitions of the
third spin, therefore the relevant two-spin multiplet polarization is the mu-
ꢀ
ꢁ
ˆ ˆ
tual entanglement of the third spin with the two other spins, i.e. I1z I3z and
I2z I3z . The first Fourier harmonics, obviously, reflects the net polarization of
ꢀ
ꢁ
ˆ ˆ
the third spin. The additional term containing the triple multiplet polarization
is not essential here and can be easily removed by adding up the intensities of
the transitions |1ꢂ → |2ꢂ, |3ꢂ → |4ꢂ, |5ꢂ → |6ꢂ, |7ꢂ → |8ꢂ (all flips of the third
spin), see text below. The respective sign of the contributions to the multiplet
polarization, I1z I3z and I2z I3z , to each line at the second harmonics depends
on the projections of the first and the second nuclei. A similar analysis can be
done for the other transitions of the third spin: |3ꢂ → |4ꢂ, |5ꢂ → |6ꢂ, |7ꢂ → |8ꢂ.
For all other transitions of the third spin the intensities of the harmonics are
given by:
ꢀ
ꢁ
ꢀ
ꢁ
ˆ ˆ
ˆ ˆ
ꢀ
ꢁ
ꢀ
ꢁ
ꢀ
ꢁ
ꢀ
ꢁ
ꢀ
ꢁ
ˆ
ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ
ˆ ˆ ˆ
I3z − I1z I2z I3z
I1z I3z − I2z I3z
I1z I2z I3z
(1)
3→4
(2)
3→4
(3)
3→4
A
=
, A
=
, A
= −
,
4
4
4
(2.13)
ꢀ
ꢁ
ꢀ
ꢁ
ꢀ
ꢀ
ꢁ
ꢀ
ꢁ
ꢀ
ꢁ
ˆ
ˆ ˆ ˆ
4
ˆ ˆ ˆ
4
ˆ ˆ
ˆ ˆ
ˆ ˆ ˆ
I1z I2z I3z
I3z − I1z I2z I3z
I1z I3z − I2z I3z
(1)
5→6
(2)
5→6
(3)
A
A
=
=
, A
, A
= −
= −
, A
, A
= −
,
5→6
4
4
ꢀ
ꢁ
ꢀ
ꢁ
ꢁ
ꢀ
ꢁ
ꢀ
ꢁ
ˆ
ˆ ˆ
ˆ ˆ
ˆ ˆ ˆ
I3z + I1z I2z I3z
I1z I3z + I2z I3z
I1z I2z I3z
(1)
7→8
(2)
7→8
(3)
=
.
7→8
4
4

1650
K. L. Ivanov et al.
Hence, the third harmonics at any line immediately yields direct information
ꢀ
ꢁ
ˆ ˆ ˆ
about the triple multiplet effect, I1z I2z I3z . From the second harmonics one may
ꢀ
ꢁ
ꢀ
ꢁ
ˆ ˆ
ˆ ˆ
easily extract I1z I3z and I2z I3z
:
ꢀ
ꢁ
ꢇ
(2)
(2)
(2)
ˆ ˆ
I1z I3z = 2 A1→2 + A3→4 = 2Aα α →α β
,
.
(2.14)
1
3
1
3
3
ꢀ
ꢁ
ꢇ
(2)
(2)
(2)
2
ˆ ˆ
I2z I3z = 2 A1→2 + A5→6 = 2Aα α →α β
Such a way of extracting the desired quantities is, in fact, the averaging over the
states of the “extra” nuclei as is indicated by the symbol on the right hand side
of Eq. (2.14). The total intensity of the spectrum at n = 1 allows one to deter-
mine the net polarization of the corresponding spin. For instance, summing all
3
2
ꢀ
ꢁ
ˆ
line intensities of the third spin transitions at n = 1 we obtain I3z
:
ꢀ
ꢁ
(2)
(2)
(2)
(2)
ˆ
I3z = A1→2 + A3→4 + A5→6 + A7→8
This result is independent of any multiplet CIDNP contribution.
.
(2.15)
The remaining 8 NMR lines corresponding to flips of the first and sec-
ond spins can be analyzed in the same way and the three residual quantities of
ˆ
ˆ
ˆ ˆ
interest, ꢁI1zꢂ, ꢁI2zꢂ, ꢁI1z I2zꢂ, can thus be extracted.
The CIDNP spectrum of three coupled spins 1/2 at ϕ = π/6 is shown in
Fig. 1(a), bottom. In this example we used the following population numbers
√
√
√
3,
of the nuclear states (2.6): P1 = Pααα
=
7, P2 = Pααβ
=
5, P3 = Pαβα
=
√
P4 = Pαββ = 1, P5 = Pβαα = 1, P6 = Pβαβ = 1, P7 = Pββα
=
3, and P8 =
√
Pβββ
=
5. They correspond to CIDNP resulting from geminate radical pair
recombination in liquid solution, and were calculated according to Adrian’s
model [20]. For this model calculation we assumed that (1) in the radical pair
all three nuclei belong to the first radical, (2) the relationship between their
HFI constants is a1 = 3a3, a2 = 2a3, (3) the external magnetic field Bpol is much
larger than a1, a2, a3 and (4) 2(g1 − g2)Bpol = a3 (where g1, g2 are the g-factors
of the first and second radical, respectively). The 2D NMR spectrum of the
three coupled spins 1/2 is shown in Fig. 1(c). The intensity integrated over all
transitions of each spin at n = 1 is equal to its net polarization (2.15). The spec-
trum at n = 2 yields the second order multiplet CIDNP (2.14), i.e., the three
ꢀ
ꢁ ꢀ
ꢁ
ꢀ
ꢁ
ˆ ˆ
ˆ ˆ
ˆ ˆ
quantities I1z I2z , I1z I3z , and I2z I3z . At n = 3 we obtain the pure spectrum of
the third-order multiplet CIDNP (2.12).
Systems with a larger number of weakly-coupled polarized spins can be
analyzed in the same way. The k-th order multiplet polarization does not
manifest itself in the harmonics of any order higher than k. By analyzing subse-
quently the spectra at N harmonics from n = N to n = 1 one obtains the values
of the multiplet effects of all orders and all net polarizations. All contributions
of higher order multiplet polarization to lower order harmonics can be eas-
ily removed by proper averaging over the corresponding nuclei as we showed
in (2.15).

2D NMR Nutation Analysis of Non-Thermal Polarization . . .
1651
3. 2D nutation analysis of the CIDNP spectra of multinuclear
systems
As a rule CIDNP (non-Boltzmann population of the nuclear spin states of
diamagnetic reaction products) is formed in reactions that involve radical inter-
mediates. In many cases electronic singlet–triplet inter-system crossing is the
kinetic bottleneck of radical pair recombination [11]. The rate of the singlet–
triplet conversion is governed by magnetic interactions (Zeeman interaction
and HFI) and depends on the nuclear configuration of the radicals. Therefore,
diamagnetic reaction products are either depleted or enriched by one or another
nuclear spin state, i.e. the reaction product acquires nuclear polarization that
can be much larger than Boltzmann polarization [11]. CIDNP investigations
allow one to determine the radical reaction pathways and to measure the mag-
netic interactions in transient short-lived radical species that are often beyond
the reach of EPR spectroscopy.
3.1 Experimental
Usually CIDNP experiments are carried out in the probe of the NMR spec-
trometer at a fixed strength of the external magnetic field B0. Quite often, such
a practice is not favorable for studying reaction dynamics because CIDNP ex-
hibits a well pronounced dependence on the external magnetic field strength
determined by the magnetic parameters of the radical species. In many cases
the strength of the NMR spectrometer field is not suitable for CIDNP meas-
urements. To avoid such limitation and measure, in particular, polarization
patterns near B = 0, we make in the present work use of the field cycling
high resolution NMR spectrometer recently built at the FU Berlin [5, 6]. A de-
tailed description of the experimental setup is given elsewhere [5, 6, 21]. The
mechanical field cycling set-up allows the detection of high resolution NMR-
spectra under permanent slow sample-rotation (0–150 Hz) at 7 T and polariza-
tion at any desired magnetic field strength between earth magnetic field and
7 T. At a field below 0.1 T, where a high homogeneity of the magnetic field
across the sample volume is required, the low field strength is set by con-
trol of the electric current through a pair of additional Helmholtz coils placed
under the cryomagnet. A CIDNP experiment that employs mechanical field
cycling contains three consecutive steps: (1) CIDNP generation by means of
photoreactions at the desired field Bpol; (2) transfer of polarized reaction prod-
ucts to the observation field B0 of the NMR spectrometer (switching of the
external magnetic field from Bpol to B0); (3) measurement of the FT-NMR
at B = B0. In the current experiments CIDNP was generated at an external
field near Bpol = 0 (“zero-field”). After the adiabatic transfer (≈ 500 ms) into
the cryomagnet (step 2), the length of the rf-excitation pulse in the NMR de-
tection (step 3) was varied in order to obtain the nutation patterns shown in
Fig. 3. Since the field change occurs adiabatically, the populations of the in-
dividual nuclear spin eigenstates are conserved. For the NMR-detection the

1652
K. L. Ivanov et al.
rf-amplitudes were adjusted, sufficiently high for exciting the whole spectrum
homogeneously, but low enough to allow small steps in ϕ for resolving har-
monics of high order.
During step 1 the sample is irradiated by a XeCl eximer laser at 308 nm
with a repetition rate of 50 Hz and energy of up to 100 mJ/pulse through a flex-
ible liquid light guide with a 90◦ prism on the top. This provides constant
irradiation conditions, and in previous investigations the technique allowed
the detailed determination of CIDNP field dependencies in photo-reactions of
amino acids, dipeptides [6–8, 21, 22] as well as in rigid and flexible biradi-
cals [21, 23]. The samples containing ≈ 0.06 M of cyclodecanone (Aldrich) in
CDCl3 (Aldrich) were prepared to yield an optical density of 0.7 at 308 nm
for the 4-mm optical pathway inside the NMR sample tube. All samples were
purged with pure nitrogen gas and sealed in a standard 5 mm Pyrex NMR tube.
In order to prevent vortex formation and sample shaking during the transfer,
a Teflon plug was inserted into the tube on top of the liquid. All chemicals were
used as received.
3.2 2D nutations analysis of the CIDNP in the photo-reactions of cyclic
ketones
In order to illustrate the applicability of the suggested approach to the analy-
sis of multi-nuclear spin systems, we choose a reaction where hydrogen atom
transfer plays a key role for the product formation and strong CIDNP is formed.
It is the Norrish type I reaction of the cyclic aliphatic ketone cyclodecanone
(C10H18O). The scheme of the photolysis is shown in Fig. 2. This reaction pro-
ceeds via α-cleavage of the C–C bond in the triplet excited state of the carbonyl
compound, resulting in the formation of a triplet acyl-alkyl biradical. The re-
action is convenient for the analysis of its nutation patterns, as it occurs with
restoration of the initial compound I. Another reaction product seen in the
CIDNP spectrum is dec-9-enal (product II in Fig. 2) formed via H-atom ab-
straction from β position at the alkyl end of the biradical by the acyl group [24].
Since the reactive state of the biradical is the singlet state, the singlet–triplet
conversion of the biradical is a necessary reaction step. This transition is spin
forbidden, but it becomes allowed due to weak magnetic interactions in the in-
termediate: the difference in Zeeman frequencies of the two unpaired electrons
and their hyperfine interaction with the magnetic nuclei.
The main peculiarity of reactions involving biradicals in comparison with
ordinary radical pairs in solution is that the radical centers do not have the pos-
sibility to diffuse apart but stay linked by the polymethylene chain. As a result
the electronic exchange interaction J, which determines the splitting between
singlet and triplet states, is non-zero during the lifetime of the biradical. The
presence of an external magnetic field leads to the splitting of the triplet sub-
levels according to the projections of the electronic and nuclear spins and
affects the rate of the singlet–triplet interconversion. Three different magnetic

2D NMR Nutation Analysis of Non-Thermal Polarization . . .
1653
field ranges with respect to the size of the exchange integral have to be consid-
ered. Near the magnetic field of electronic level crossing, Bpol = 2|J|/γelectron
the conditions for the singlet–triplet transition are the most favorable [11]. In
,
our case the electron–nuclear T –Sβ transition (S, T+,−,0 denote the electronic
−α
spin states, α and β the projections of the nuclear spin) becomes efficient, lead-
ing to a depletion of α and enrichment of β spins in the products. Accordingly,
a characteristic maximum of emissive nuclear spin polarization was detected
in the field dependence of unsaturated aldehydes, formed in the photoreaction
of the homologous series of cyclic aliphatic ketones [25]. At high magnetic
field as is typical for NMR spectrometer magnets (Bpol ꢃ 2|J|/γelectron) an-
other polarization pathway becomes dominant. Here, CIDNP is formed by
S–T0 transitions and the polarization pattern is in accordance with the spin
density distribution in the biradicals. The main part of the hyperfine interac-
tion with the protons is associated with the alkyl end of the biradical and can
be estimated using the corresponding values of the analogous monoradicals
(−2.2 mT and 2.8 mT in α and β position to the alkyl end, respectively) [26].
Thus, only protons in α- and β-CH2 positions are polarized with the sign of
polarization corresponding to the sign of the HFI constants [27–29]. From the
analysis of polarization patterns and their field dependence one obtains infor-
mation about exchange and hyperfine interaction as well as about molecular
and spin dynamics [30, 31]. In the range of weak magnetic field, where the Bpol
is lower than the value of individual hyperfine coupling constants, all electron
and nuclear spin states are strongly coupled by hyperfine interaction. The se-
lectivity of the singlet–triplet transitions is governed by hyperfine interaction
and determined by the magnitude of the total nuclear spin, but not by its pro-
jection as it is at high magnetic field. The singlet–triplet transitions occur in
a way that the total (electron plus nuclear) spin is conserved, therefore elec-
tronic transitions are accompanied by nuclear spin flips. Since the selection
rules for transitions at low field are associated with the total nuclear spin it is
reasonable to expect the formation of high order multiplet CIDNP in this field
region.
The nuclear spin polarization, which is formed in the singlet–triplet con-
version of short lived intermediate biradicals, is conserved in the diamagnetic
reaction products for a time of the order of T1 (nuclear longitudinal relaxation
time) and can therefore be detected as anomalous intensities in the NMR spec-
tra taken immediately after the reaction. The aldehyde proton of dec-9-enal
has a very long relaxation time (of about 50 seconds), therefore it acquires
rather strong polarization upon long light irradiation. In contrast, the protons
of the cycloketone have a much shorter relaxation time, their polarization stays
smaller and was not studied in detail before. Due to the short field cycling time
of our set-up which is much shorter than T1 of the ketone protons we were able
to detect CIDNP not only for the aldehyde proton of product II, but also for all
the protons of the initial ketone and for the three protons attached to the car-
bon atoms in the double bond of II. The NMR spectrum of cyclodecanone at

1654
K. L. Ivanov et al.
thermal equilibrium and the CIDNP spectrum detected at low magnetic field
of 2 mT are shown in Fig. 2 with the assignment of lines. Due to molecular
conformational motion of the cyclic molecule the lines of protons in α- and
β-CH2 positions are broadened and the spin-spin couplings of γ δ and pro-
tons are only partially resolved. In contrast to the high field CIDNP, in the low
field region all lines of the ketone show polarization. In the NMR spectrum
taken 1 second after laser irradiation at Bpol = 2 mT the lines corresponding to
α-CH2 protons exhibit very strong absorption, while for β-CH2 an absorptive
signal of weaker intensity is detected. In spite of the fact that for the δ- and
-CH2 positions of the biradical the HFI constants are negligibly small (less
than 0.1 mT) the strongest emissive polarization is detected for these protons
and a weak emissive signal for γ-CH2 protons. This observation leads us to
conclusion that not only α- and β-CH2, but all protons of the biradical are in-
volved in the singlet–triplet conversion at this magnetic field. The symmetry
of the polarization pattern of cyclodecanone looks very similar to the CIDNP
pattern detected for the amino acids tyrosine [8] and histidine [7] at very low
magnetic field. Here, all the protons are substantially polarized, but in a way
that the total polarization integrated over the whole spectrum is equal to zero.
This similarity allows us to suggest that in the photo-reaction of cyclodecanone
we detect mutual entanglement of nearly all protons in the diamagnetic prod-
ucts irrespective of the size of the hyperfine interaction coupling constant. The
aliphatic region of the CIDNP spectra taken at different rotation angle ϕ is
shown in Fig. 3.
For the analysis of the multi-spin order in the polarization of cyclode-
canone we choose the signal of the α-CH2 protons because it has the largest
intensity in the CIDNP spectrum. Up to a harmonic order of ten signals well
above the noise level are detected, with the trend that at even orders the
intensity is higher than at neighboring odd orders. In Fig. 4 the spectra corres-
ponding to the harmonic orders from 1 to 6 are shown. The occurrence of so
many harmonics confirms the suggestion that all protons in the molecule are
entangled with each other. The spectral shape at the first harmonic almost co-
incides with that of NMR at thermal equilibrium. In the spectra at the higher
harmonics the lines that are non-resolved in ordinary NMR split in emissive
and absorptive components. The same analysis is made for the β-CH2 protons
of cyclodecanone, and intensities above the noise level are found for the first
six harmonics (not shown). The absorptive spectrum at n = 1 (net CIDNP spec-
trum) is actually the same as the spectrum at thermal equilibrium. At n > 1
the spectra reflect the mutual ordering of spins and exhibit “antiphase” struc-
ture inherent to the multiplet polarization. Likewise, the harmonic spectra of
the spin group of the methylene protons of the unsaturated aldehyde (II) cor-
responding to the first 5 harmonics for the =CH2 protons as shown in Fig. 5
exhibit strong multiplet order up to n = 3 and of smaller size up to n = 5. The
net CIDNP spectrum (n = 1) of the these protons is emissive, while the spectra
of higher order harmonics have an antiphase nature. This result unambigu-

2D NMR Nutation Analysis of Non-Thermal Polarization . . .
1655
Fig. 4. CIDNP spectra of α-CH2 protons of cyclodecanone at different harmonics. Num-
bers indicate the order of the harmonics.
ously supports the conclusion that not only back cyclization of the biradical
with the restoration of the initial compound, but also the reaction of the hydro-
gen atom transfer from the β position of the alkyl end to the acyl moiety of
the biradical occurs with the formation of polarization of high multiplet order.
The quantitative analysis of the full CIDNP nutation behavior at low mag-
netic field is under way in our laboratories and will be published in a separate
paper.

1656
K. L. Ivanov et al.
Fig. 5. CIDNP spectra of =CH2 protons of product II at different harmonics. Numbers
indicate the order of the harmonics.
4. Conclusion
The main concern of this article is to introduce Fourier decomposition of coher-
ent spin motion of N coupled nuclei under excitation by a non-selective rf pulse
as a method for analyzing the spin order of multi-spin systems. In this way we
obtain a two dimensional representation of spectra, where the first dimension
is the usual NMR frequency while the second dimension is related to the num-
ber of spins that are entangled. As we demonstrated this technique allows the
extraction of all the quantities required for a full characterization of the spin po-

2D NMR Nutation Analysis of Non-Thermal Polarization . . .
1657
larization (namely, nuclear net polarizations and multiplet polarizations of all
orders) of non-equilibrium systems.
To illustrate the method it was applied to the study of the low-field CIDNP
formed in the photolysis of cyclodecanone having an acyl-alkyl biradical as
short-lived intermediate. For two products of this reaction, the starting cyclic
ketone and the unsaturated aldehyde formed via intramolecular hydrogen atom
abstraction in the biradical stage, the 2D analysis of the nutation pattern is car-
ried out. Because of “antiphase” structure in spectra at higher harmonic order
neighboring lines become better resolved. In a straightforward way the contri-
butions from net and multiplet polarization to the total CIDNP are separated.
We anticipate exhaustive quantitative analysis of net and multiplet CIDNP
generated by means of singlet–triplet interconversion not only for flexible bi-
radicals, but also for photoinduced hydrogen or electron transfer reactions of
many biologically relevant systems. Strong nuclear polarization has been de-
tected in recent low-field CIDNP experiments on amino acids, peptides and
proteins [5–8]. Already from a qualitative analysis of the nutation pattern it is
obvious that spin orders of N > 2 contribute to spin polarization at low mag-
netic field. Our results demonstrate that in spin-selective chemistry at low field
the entanglement of several nuclear spins has a significant influence. The quan-
titative evaluation of all net and multiplet contributions to the total polarization
as described in this article will provide all the information that is obtainable
from dynamic polarization data and often unavailable from other experiments.
Acknowledgement
This work was supported by the DFG (project Vi 103/9), RFBR (projects
02-03-32765, 02-03-32166), Russian Ministry of High Education (Grant
2298.2003.3) and INTAS (Project No. 02-2126). A.V.Y. acknowledges the
Science-Support-Foundation (Russia) for financial support, K.L.I. is indebted
to the INTAS Young NIS Scientist Fellowship Programme (project YSF
2001/2-103). We are grateful to Prof. Nikita N. Lukzen (International Tomog-
raphy Center, Novosibirsk) for his genuine interest to this work and stimulating
discussions.
Appendix A
(n)
i→ j
Here we shall derive the expression for A (intensity of the n-th harmonics at
transition |iꢂ → | jꢂ).
Formula (2.4) can be rewritten as follows:
ꢅ
ꢆ
ꢅ
ꢆ
ꢄ
ꢄ
2
+1
2N−2
−1
ij,rs
ij,rs
ϕ
ϕ
(0)
r→s
Li→ j = 2
sin
cos
L
=
Aij,rs Lr(0→) s
.
2
2
rs
rs
(A.1)

1658
K. L. Ivanov et al.
The new quantity, Aij,rs, can be represented as follows:
ꢅ
ꢆ
ꢅ
ꢆ
2
+1
2N−2
−1
ij,rs
ij,rs
ϕ
ϕ
Aij,rs = 2 sin
cos
(A.2)
2
2
+1
ꢈ
ꢉ
ꢈ
ꢉ
2
2N−2
−1
ij,rs
ij,rs
eiϕ/2 −e−iϕ/2
eiϕ/2 +e−iϕ/2
= 2
.
2i
2
This result can be rewritten as a sum of 2N +1 exponential terms:
N
ꢄ
1
(n)
ij,rs
Aij,rs
where
=
a
exp(inϕ) ,
(A.3)
(A.4)
2i
n=−N
2N−2
−1
ij,rs
ij,rs
ꢄ
(−1)
(n)
a
=
(−1)N−m−n C2N−m−+n1C2mN−2
.
ij,rs
−1
ij,rs
ij,rs
4N−1
m=0
(n)
ij,rs
(−n)
(0)
ij,rs
One may prove that a = −a
(as a consequence, a = 0). Therefore, the
ij,rs
expression (A.3) is a sum of N sine terms:
N
ꢄ
(n)
ij,rs
Aij,rs
=
a
sin nϕ
(A.5)
n=1
and the general result (A.1) is recast as follows:
N
N
ꢄ
ꢄ ꢄ
(n)
i→ j
(n)
Li→ j
=
A
sin nϕ =
a
sin nϕ L(0)
,
ij,rs
r→s
(A.6)
n=1
n=1 rs
(n)
where a is given by (A.4).
ij,rs
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