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Quantized field description of rotor frequency-driven dipolar recoupling
G. J. Boender, S. Vega, and H. J. M. de Groot
Citation: J. Chem. Phys. 112, 1096 (2000); doi: 10.1063/1.480664
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 112, NUMBER 3
15 JANUARY 2000
Quantized field description of rotor frequency-driven dipolar recoupling
G. J. Boendera)
Leiden Institute of Chemistry, Leiden University 2300 RA Leiden, The Netherlands
and Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel
S. Vega
Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel
H. J. M. de Groot
Leiden Institute of Chemistry, Leiden University 2300 RA Leiden, The Netherlands
共Received 25 August 1999; accepted 22 October 1999兲
A formalized many-particle nonrelativistic classical quantized field interpretation of magic angle
spinning 共MAS兲 nuclear magnetic resonance 共NMR兲 radio frequency-driven dipolar recoupling
共RFDR兲 is presented. A distinction is made between the MAS spin Hamiltonian and the associated
quantized field Hamiltonian. The interactions for a multispin system under MAS conditions are
described in the rotor angle frame using quantum rotor dynamics. In this quasiclassical theoretical
framework, the chemical shift, the dipolar interaction, and radio frequency terms of the Hamiltonian
are derived. The effect of a generalized RFDR train of ␲ pulses on a coupled spin system is
evaluated by creating a quantized field average dipolar-Hamiltonian formalism in the interaction
frame of the chemical shift and the sample spinning. This derivation shows the analogy between the
Hamiltonian in the quantized field and the normal rotating frame representation. The magnitude of
this Hamiltonian peaks around the rotational resonance conditions and has a width depending on the
number of rotor periods between the ␲ pulses. Its interaction strength can be very significant at the
n⫽0 condition, when the chemical shift anisotropies of the interacting spins are of the order of their
isotropic chemical shift differences. © 2000 American Institute of Physics.
关S0021-9606共00兲03203-7兴
Floquet description of MAS.17,18 This makes it possible to
disentangle the influence of the radiation field and the MAS
rotation, leading to the construction of a Floquet Hamiltonian
for the MAS NMR from first principles. Here it will be
shown that the quantized field description can be combined
with the average Hamiltonian approach in order to provide
us with a quasiclassical description of RFDR experiments
with a variable number of rotor periods between the
␲-pulses.19
The formal description of the RFDR experiment starts
with a brief summary of the quantum rotor dynamics in the
next section. The interaction of the quantized rotor field with
the nuclear spin is discussed in Sec. III. The chemical shielding of the magnetic field is calculated in this section and the
term in the quantized field Hamiltonian describing the dipolar interaction is put forward. In Sec. IV, the RF pulse is
included in the quantized field framework. Using the methodology discussed in Secs. II–IV, we analyze in Sec. V a
generalized RFDR sequence. In Sec. VI the polarization
transfer driven by the dipolar recoupling sequence is discussed and in Sec. VII the main conclusions are given. In the
appendix the transformation properties for the quantized field
operators constituting the spin-pair dipolar term in the multispin Hamiltonian are evaluated.
I. INTRODUCTION
Magic angle spinning 共MAS兲 nuclear magnetic resonance 共NMR兲 correlation spectroscopy in conjunction with
multispin isotope enrichment has been established recently
as a novel concept for refinement of electronic and spatial
structure in solids.1,2 This development was triggered by the
discovery of broadbanded pulse sequences for homonuclear
dipolar recoupling.3–11 In principle, MAS averages dipolar
interactions.12,13 Using dipolar recoupling techniques, homonuclear and heteronuclear dipolar correlation spectroscopy
can be performed and the NMR response of moderately sized
multispin clusters can be assigned.14,15 In addition, since the
dipolar interaction is strongly distance dependent, dipolar
correlation spectroscopy can be used for de novo structure
determination on an atomic scale of multispin labeled
systems.1,2 In this article we present a formal quantized field
interpretation of the seminal radio frequency driven dipolar
recoupling 共RFDR兲. The technique was originally proposed
for one refocusing ␲-pulse per single rotor period.5 RFDR is
straightforward to implement and robust, and can be applied
readily in high and ultra high fields. It has already been utilized for assignment and structure refinement of moderately
sized uniformly labeled biological aggregates.16
It has been shown that the effect of MAS on a nuclear
spin system in a solid can be described in terms of a quantized rotor field, leading to a physical interpretation of the
II. QUANTUM ROTOR DYNAMICS
MAS NMR spectroscopy is generally described with the
help of a spin Hamiltonian, in which the effect of the sample
a兲
Present address: ID-DLO, P.O. Box 65, 8200 AB Lelystad, The Netherlands.
0021-9606/2000/112(3)/1096/11/$17.00
1096
© 2000 American Institute of Physics
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J. Chem. Phys., Vol. 112, No. 3, 15 January 2000
Rotor frequency-driven dipolar recoupling
rotations on the nuclear spins is taken into account by a set
of periodically time dependent coefficients. The spin Hamiltonian of a single crystallite in the rotating frame can therefore be expanded in a Fourier series according to20
2
ĤS 共 t 兲 ⫽ĤS0 ⫹
兺
p⫽1
†
ĤSp exp共 ip 共 ␸ 0 ⫹ ␻ r t 兲兲 ⫹ĤSp
⫻exp共 ⫺ip 共 ␸ 0 ⫹ ␻ r t 兲兲 ,
â † 兩 l 典 ⫽ 共 l⫹1 兲 1/2兩 l⫹1 典 ,
共1兲
共2兲
where
L̂ z ⬘ ⫽â † â
共3兲
measures the orbital angular momentum. The energy of the
rotor is ␻ r l, which is very large on the energy scale of the
nuclear spin system.
It was shown that a complete description of the MAS for
a nuclear magnetic moment in a solid can be obtained within
a single set of reduced angular momentum states 兩 n 典 . By
taking l⫽n̄⫹n, with n the variation of the angular momentum around a large average value n̄, the orbital angular momentum states can be renumbered according to17
兩 n̄⫹n 典 → 兩 n 典 .
共4兲
To provide a description of the quasiclassical limit, a phase
operator ␸ˆ ⬘ is introduced that represents a continuous coordinate ␸⬘ with
exp共 i ␸ˆ ⬘ 兲 ⫽
â †
冑n̄
.
共5兲
and the z axis is the axis of rotation.
共7兲
According to Eq. 共5兲 the F p operators are multiquantum ladder operators with F̂ p 兩 n 典 ⫽ 兩 n⫹ p 典 and they can be interpreted as a repeated application of single quantum raising
and lowering operators, since F̂ p⫹q ⫽F̂ p F̂ q . 17
The commutator 关 L̂ z ⬘ ,F̂ p 兴 ⫽ pF̂ p implies that
exp共 i ␻ r tL̂ z ⬘ 兲 F̂ p exp共 ⫺i ␻ r tL̂ z ⬘ 兲
⫽1̂⫹i ␻ r t 关 L̂ z ⬘ ,F̂ p 兴 ⫹
共 i ␻ rt 兲2
关 L̂ z ⬘ , 关 L̂ z ⬘ ,F̂ p 兴兴 ⫹¯
2!
⫽exp共 ip ␻ r t 兲 F̂ p .
共8兲
In the quasiclassical limit, quantization of the rotor field is
therefore equivalent with the introduction of a phase operator
␸ˆ ⬘ in the rotor angle frame according to Eq. 共5兲. A quantized
field extension of the spin Hamiltonian in Eq. 共1兲 is then
2
ĤQ 共 t 兲 ⫽ĤS0 ⫹
兺
p⫽1
†
ĤSp exp共 ip 共 ␸ˆ ⬘ ⫹ ␻ r t 兲兲 ⫹ĤSp
⫻exp共 ⫺ip 共 ␸ˆ ⬘ ⫹ ␻ r t 兲兲 .
共9兲
It includes two independent degrees of freedom, ␸ˆ ⬘ and t,
which contrasts with the single time degree of freedom that
is used to generate the time-dependence in the spin Hamiltonian ĤS (t) in Eq. 共1兲.17 The ␸ˆ ⬘ degree of freedom is essential, to account for the infinitesimal phase variations due
to the quantum rotor dynamics, in which the rotor is described as a two dimensional harmonic oscillator or a rigid
rotor.21,22
Using the relations 共7兲 and 共8兲, it is possible to rewrite
the Eq. 共9兲, yielding
冉
ĤQ 共 t 兲 ⫽exp共 i ␻ r tL̂ z ⬘ 兲 ĤS0 ⫹
2
兺
p⫽1
†
ĤSp F̂ p ⫹ĤSp
F †p
冊
⫻exp共 ⫺i ␻ r tL̂ z ⬘ 兲 .
共10兲
A time-independent Hamiltonian ĤQ can be obtained by a
transformation to the spatial rotating frame, defined by
exp(⫺i␻rtL̂z⬘):17
ĤQ ⫽Ĥn ⫹Ĥi ⫹Ĥr
共11兲
consisting of a term associated with the sample rotation
Ĥr ⫽ ␻ r L̂ z ⬘ ,
共12兲
a pure nuclear spin Hamiltonian term independent of F p
The relation between the phase operator and the angular momentum operator L̂ z ⬘ , is the canonical commutation relation
according to
关 ␸ˆ ⬘ ,L̂ z ⬘ 兴 ⫽i,
MAS is a rotation with a single normal frequency mode
␻ r . The interaction between the spin system and the sample
rotation can be interpreted in terms of p-quantum interactions with the single mode rotor field with a frequency ␻ r .
For a p-quantum transition, a Fourier operator is defined by
F̂ p ⫽exp共 ip ␸ˆ ⬘ 兲 .
in which ␻ r is the spinning speed and ĤSp the Fourier components of the Hamiltonian. These components differ from
crystallite to crystallite in the sample, while ␸ 0 is a constant
initial phase that may vary between experiments.
In the quantized field description the concept of a
nuclear spin Hamiltonian is extended with quantum rotor
dynamics.17 The interaction of the spins with the environment due to the MAS rotation is included in the Hamiltonian,
by using a set of quantum rotor dynamics operators L̂ z ⬘ , F̂ p ,
and F̂ †p , to replace the phenomenological time-dependent
phase of the rotation ␸ 0 ⫹ ␻ r t in Eq. 共1兲.
Recently, it was shown that rotor field quantization offers a self-consistent framework to describe the evolution of
the spin system with conservation of energy.17 The total orbital angular momentum l of a forced rotation of a rotor with
eigenstates 兩 lm 典 in the quasiclassical limit defined by l→⬁
and l⫽m, corresponds with the number of boson quasiparticles. The associated rotor field is a vector field that can be
constructed using boson annihilation and creation operators
â 兩 l 典 ⫽l 1/2兩 l⫺1 典 ,
1097
共6兲
Ĥn ⫽ĤS0 ,
共13兲
and an interaction term depending on F p and F †p
2
Ĥi ⫽
兺
p⫽1
†
ĤSp F̂ p ⫹ĤSp
F̂ †p ,
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共14兲
1098
J. Chem. Phys., Vol. 112, No. 3, 15 January 2000
Boender, Vega, and de Groot
which describes the effect of the rotor dynamics on the spin
interactions. Since ĤQ is time independent its unitary evolution operator is
Û 共 t 兲 ⫽exp共 ⫺iĤQ t 兲 .
共15兲
The effect of the phenomenological time dependence on
the spins is included by means of the time-independent orbital angular momentum ladder operators F̂ p ,F̂ †p . These
have a physical basis and have been identified as the Fourier
operators in the Floquet description of MAS NMR.17 The
ĤQ Hamiltonian operates in the product space spanned by
the spin states and the 共reduced兲 angular momentum states.
The commutation relation between the Fourier operators and
the orbital angular momentum operator 关 L̂ z ⬘ ,F̂ p 兴 ⫽ pF̂ p allows for energy exchange between the rotor and the spin
system. In addition, it provides a route to coherent superpositions of states involving the quantum rotor dynamics, since
in general 关 Ĥr ,Ĥi 兴 ⫽0.
III. THE INTERACTION HAMILTONIAN
The Fourier components of the spin Hamiltonian 共1兲,
containing chemical shift and dipolar interactions, can be
expressed in terms of contractions between Âk , the irreducible tensor operators of rank k in real space, and T̂k , the
irreducible tensor operators of rank k in spin space.23,24 In the
spin rotating frame only the part of the interactions that commute with the Zeeman Hamiltonian have to be taken into
account. An interaction term of the spin-Hamiltonian in the
spin rotating frame can therefore be written as
ĤS 共 t 兲 ⫽
兺k A k0共 t 兲 T̂ k0 .
共16兲
The time dependent spatial part can be expanded in a Fourier
series according to17
k
in which W 00
are some coefficients and Î 0 spin operators
commuting with Î z . The quantized field Hamiltonian becomes then
k
k
Î 0 ⫹
ĤQ 共 t 兲 ⫽Z 00
兺
p⫽⫺k
exp共 ip 共 ␻ r t⫹ ␸ 0 兲兲 V kp0 ,
共17兲
k
k
兺 兺
s⫽⫺k r⫽⫺k
冉
k
k
⵮ D rs
A kr
共 ␣ c , ␤ c , ␥ c 兲 D sp
共␣r ,␤r ,␥r兲
⫻D kp0 0,␤ m ,
冊
␲
.
4
共18兲
⵮ coefficients are the elements of the spatial tensor
The A kr
describing the coefficients of the interaction in the principal
axis system 共PAS兲. In Eq. 共18兲 three sets of Euler angles of
the Wigner coefficients D ki j ( ␣ , ␤ , ␥ ) describe the transformation from the PAS to a crystal frame ( ␣ c , ␤ c , ␥ c ), from this
crystal frame to the rotor angle frame ( ␣ r , ␤ r , ␥ r ), and finally from the rotor angle frame to the laboratory frame
(0,␤ m , ␲ /4). 23
The tensor operators in spin space can be written as
k
Î 0
T̂ k0 ⫽W 00
共19兲
共20兲
with
k
Z kp0 ⫽V kp0 W 00
共21兲
and the fact that the Hamiltonian is Hermitian requires that
Z ⫺p0 ⫽Z *
p0 . After the transformation to the spatial rotating
frame the interaction Hamiltonian in Eq. 共14兲 yields
k
Hi ⫽
兺
p⫽1
共 Z kp0 F̂ p ⫹Z kp0* F̂ †p 兲 Î 0 .
共22兲
Here we have used again the Fourier operators to include the
quantum rotor dynamics, according to Eq. 共7兲.
For a dipolar coupled spin system of like spins under
MAS, the Hamiltonian consists of a chemical shift term and
a dipolar term. Thus the nuclear spin Hamiltonian and the
interaction Hamiltonian can be written as
D
Ĥn ⫽ĤCS
n ⫹Ĥn ,
D
Ĥi ⫽ĤCS
i ⫹Ĥi .
共23兲
With the external magnetic field along the z axis, the spin
tensor operators in the truncated isotropic chemical shift
(k⫽0) and chemical shift anisotropy 共CSA兲 (k⫽2) Hamiltonian in Eq. 共16兲 are for a spin i23
i
T̂ 00
⫽⫺
i
⫽
T̂ 20
with
V kp0 ⫽
Z kp0 Î 0 exp共 ip 共 ␸ˆ ⬘ ⫹ ␻ r t 兲
⫹Z kp0* Î 0 exp共 ⫺ip 共 ␸ˆ ⬘ ⫹ ␻ r t 兲
k
A k0 共 t 兲 ⫽
兺
p⫽1
1
冑3
冑
␥ H 0 Î zi ,
共24兲
2
␥ H 0 Î zi .
3
The spatial isotropic chemical shift tensor is of rank 0 with a
⵮ ⫽Ā i . The CSA interaction is a real symsingle element A 00
metric tensor interaction ␴ of rank 2 and the three elements
that characterize this tensor in its principal axis system are
for a spin i
⵮⫽
A 20
冑␦
3
2
i
,
1
⵮ ⫽ ␩␦ i ,
A 2⫾2
2
共25兲
i
i
⫹ ␴ iy y ⫹ ␴ zz
) the isotropic and ␦ i
with Ā i ⫽ ¯␴ i ⫽ 冑1/3( ␴ xx
i
i
⫽1/3( ␴ zz ⫺ ¯␴ ) the anisotropy parameter and ␩ i ⫽( ␴ iy y
i
⫺ ␴ xx
)/ ␦ i the asymmetry parameter of the interaction.25 The
values of these parameters can be inserted in Eq. 共18兲 to
evaluate the V kpq parameters, yielding
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J. Chem. Phys., Vol. 112, No. 3, 15 January 2000
Rotor frequency-driven dipolar recoupling
2
2
V 2i
p0 ⫽d p0 共 ␤ m 兲 exp共 i ␥ r p 兲
兺
s⫽⫺2
2
共 d sp
共 ␤ r 兲 exp共 i 共 ␣ r ⫹ ␥ ic 兲 s 兲
2
2
⫻ 共 d 0s
共 ␤ ic 兲 冑 32 ␦ i ⫹ 兵 exp共 i2 ␣ ic 兲 d 2s
共 ␤ ic 兲
2
⫹exp共 ⫺i2 ␣ ic 兲 d ⫺2s
共 ␤ ic 兲 其 21 ␩ i ␦ i 兲兲
共26兲
2i
2
⫽0, since for MAS d 00
( ␤ m )⫽0.
with V 00
Using Eqs. 共19兲 and 共22兲, the chemical shift contribution
to the interaction Hamiltonian can be constructed and yields
N
ĤCS
i ⫽
兺兺
i⫽1 p⫽1
共 Z ip F̂ p ⫹Z ip* F †p 兲 Î zi ,
共27兲
with Z ip ⫽(Z 2p0 ) i and
Z ip ⫽
冑␥
2
H 0 V 2i
p0 .
3
共28兲
The isotropic part is not affected by the spinning and yields
a contribution to the pure nuclear spin term
N
ĤCS
n ⫽
兺 ⌬ ␻¯ i Î zi ,
共29兲
i⫽1
i
i
with ⌬ ␻
¯ i ⫽⫺ ␥ H 0 1/3( ␴ xx
⫹ ␴ iy y ⫹ ␴ zz
).
The spin space tensor operator for the dipolar interaction
between a pair of spins (i, j) is
ij
⫽⫺
T̂ 20
冑
1 2
i j
i j
Î ⫺ ⫹Î ⫹
Î ⫺ 兲兲 .
2 ␥ ប 共 2Î zi Î zj ⫺ 21 共 Î ⫹
6
共30兲
j
The components of the spatial tensor V 2i
p0 for the spin pair
interaction can be obtained from Eq. 共26兲 with ␦ i substituted
by
␮ 0 ␥ 2ប
␦ i j ⫽⫺
,
2 ␲ r 3i j
and ␩ i substituted by ␩ i j ⫽0. Thus the dipolar term in the
interaction Hamiltonian is a quantized field contribution that
is bilinear in the spin operators,
2
ĤD
i ⫽
兺
兺 共 Z ipj F̂ p ⫹Z ipj *F̂ †p 兲共 2Î zi Î zj ⫺
i⬍ j p⫽1
1
2
i j
i j
Î ⫺ ⫹Î ⫹
Î ⫺ 兲兲
共 Î ⫹
共31兲
with the summation over all spin pairs i⬍ j.
Here, with Z ipj ⫽(Z 2p0 ) i j ,
Z ipj ⫽⫺
ĤD
n ⫽0. According to Eq. 共11兲 the time independent quantized field MAS Hamiltonian ĤQ of the coupled spin system
is the sum of Eqs. 共27兲, 共29兲, 共31兲, and 共12兲.
IV. RADIO FREQUENCY PULSES
An on-resonance RF field contributes to the spatial rotating frame Hamiltonian a nuclear spin Hamiltonian term of
the form:
ĤRF⫽ ␻ 1 共 Î x cos ␾ ⫹Î y sin ␾ 兲 .
2
C ipj
r 3i j
共34兲
The ␻ 1 is the nutation frequency and ␾ is the RF phase in
the rotating frame.23 The ĤRF operates only on the spin part
of the spin system and is independent of the sample spinning.
If the intensity of a short RF pulse is sufficiently strong then
it is possible to neglect all the other terms in the Hamiltonian
containing spin operators. During a pulse the effect of the
rotor should be considered to account for refocusing errors
and to achieve stable effective dipolar recoupling.26 The remaining terms in the quantized field Hamiltonian are then
RF
ĤQ
⫽ĤRF⫹Ĥr .
共35兲
RF
is time-independent and its evolution operator at
The ĤQ
the end of a pulse of length ␶ p becomes
Û 共 ␶ p 兲 ⫽exp共 ⫺i 共 ĤRF⫹Ĥr 兲 ␶ p 兲 .
共36兲
However, in the limit of an extremely short pulse ␶ p →0,
while ␪ ⫽ ␻ 1 ␶ p is kept constant, also the rotor dynamics during the pulse can be neglected and a pulse evolution operator
can be defined as
冉
N
P̂ 共 ␪ 兲 ⫽exp ⫺i
␪ 共 Î ix cos ␾ ⫹Î iy sin ␾ 兲
兺
i⫽1
冊
,
M
兿
m⫽1
P̂ m Û 共 ␶ m 兲 ,
with
␶ c⬅ 兺 ␶ m .
m⫽1
2
␮0 2 2
2
␥ បd p0 共 ␤ m 兲 exp共 i ␥ r p 兲 兺 d sp
共␤r兲
4␲
s⫽⫺2
2
⫻exp共 i 共 ␣ r ⫹ ␥ icj 兲 s 兲 d 0s
共 ␤ icj 兲
共38兲
M
and
C ipj ⫽
共37兲
affecting only the spin-operational part of the spin
system.27,28
During an interval of length ␶ m between such pulses,
P̂ m⫺1 and P̂ m , in a multi-pulse experiment the spin system
is governed by the time independent quantized field Hamiltonian ĤQ and the evolution of the spin system in this period
is described by the evolution operator Û( ␶ m )⫽exp
(⫺iĤQ ␶ m ). The overall evolution operator for a pulse sequence of M equally spaced pulses during a time of ␶ c is
C 共 ␶ 兲⫽
U
c
共32兲
1099
共33兲
with ␤ icj and ␥ icj the polar angles of the dipolar vector in the
crystal frame.
Since the dipolar interaction is traceless, the contribution
to the pure nuclear term due to the dipolar interaction is
共39兲
Schematic representations of such pulse sequences are
shown in Fig. 1.
A product of unitary transformations is again a unitary
transformation and it should be possible to express the entire
sequence in terms of an average quantized field Hamiltonian
ˆ
H̄Q according to
C ␶ 兲,
C 共 ␶ 兲 ⫽exp共 ⫺iH
U
c
Q c
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共40兲
1100
J. Chem. Phys., Vol. 112, No. 3, 15 January 2000
ˆ ⫽
H̃
0
再
Boender, Vega, and de Groot
CS
ĤCS
n ⫹Ĥi ⫹Ĥr ,
0⬍t⬍ 21 ␶ w ,
CS
⫺ĤCS
n ⫺Ĥi ⫹Ĥr ,
1
2 w
CS
ĤCS
n ⫹Ĥi ⫹Ĥr ,
3
2 w
␶ ⬍t⬍ 23 ␶ w ,
␶ ⬍t⬍2 ␶ w ,
共42兲
ˆ
D
H̃
1 ⫽Ĥi
where we assumed infinitely short ␲-pulses.
The evolution operator can be written in the toggling
ˆ
ˆ
ˆ
ˆ
frame as Ũ
(t)⫽Ũ 0 (t)Ũ 1 (t), with Ũ 1 (t) governed by the
equation of motion
FIG. 1. Schematic representation of the generalized RFDR sequence with
共A兲 ␶ w ⫽ ␶ r and 共B兲 ␶ w ⫽2 ␶ r . The sequences start with ramped cross polarization. The magnetization is rotated to the z axis and the recoupling sequence is applied. Prior to detection the polarization is transferred to the xy
plane.
as for spin-Hamiltonians.29 The evaluation of this average
Hamiltonian must be accomplished by using average Hamiltonian theory in the quantized field representation. This is
discussed in the following section.
V. ROTOR FREQUENCY-DRIVEN DIPOLAR
RECOUPLING
The RFDR sequence was analyzed previously using average Hamiltonian theory and Floquet theory.5,30,31 It utilizes
a train of rotor-synchronized ␲-pulses to invoke the dipolar
recoupling. In the original version one pulse per rotor period
is used, with a time window between the pulses ␶ w ⫽ ␶ r , with
␶ r the length of one rotor period. Here a generalization of
RFDR is discussed that was first introduced in Ref. 19. A
fixed number of rotor periods per pulse n w is used and ␶ w
⫽n w ␶ r is the time window between the pulses.19,26 In this
paper we restrict ourselves to even n w values and one RFDR
cycle contains two pulses 关 21 ␶ w ⫺ P̂ 1 ( ␲ )⫺ ␶ w ⫺ P̂ 2 ( ␲ )
⫺ 12 ␶ w 兴 in a cycle time of ␶ c ⫽2 ␶ w . The sequence allows for
narrow band RFDR and its frequency selectivity has been
discussed recently in Ref. 26. When the pulses are cyclic in
p
P̂ 1 P̂ 2 ⫽1̂, the expression for the averaged
the sense ⌸ m⫽1
quantized field Hamiltonian can be found by transforming
ĤQ to the ‘‘toggling frame,’’ were the RF pulses are
eliminated.27,28 This implies that the evolution operator in the
spatial rotating frame and the toggling frame are the same at
the end of each RFDR cycle.
The basis RFDR pulse cycle can be divided into three
periods 0⭐t⭐ 21 ␶ w , 21 ␶ w ⭐t⭐ 23 ␶ w , and 32 ␶ w ⭐t⭐2 ␶ w . The
effect of such a cycle on an ensemble of N dipolar coupled
like spins can be evaluated using the quantized field formalism in the toggling interaction frame of the chemical shift
and the sample spinning. If the ĤQ is divided according to
ĤQ ⫽Ĥ0 ⫹Ĥ1 , with
ˆ
dŨ 1 共 t 兲 ˆ
ˆ t ⫽0,
⫹iH̃int共 t 兲 Ũ
1共 兲
dt
with
ˆ
U 0 共 t 兲 ⫽exp共 ⫺iH̃0 t 兲
共44兲
ˆ
ˆ
ˆ ˆ
H̃int共 t 兲 ⫽Ũ ⫺1
0 共 t 兲 H̃1 Ũ 0 共 t 兲 ,
共45兲
and
ˆ 25
the Hamiltonian in the interaction frame of H̃
0.
ˆ
The evolution operator Ũ 1 (t) in this interaction frame at
ˆ
t⫽2 ␶ w becomes equal to the toggling frame Ũ (2 ␶ w ) when
ˆ
Ũ 0 (2 ␶ w )⫽1̂. This equality can easily be shown by the following calculation and implies that the spin evolution in the
original spatial rotating frame is governed by Û(2 ␶ w )
ˆ
ˆ
⫽Ũ 1 (2 ␶ w ). The Hamiltonian H̃0 in Eq. 共42兲 contains three
terms. The isotropic chemical shift term, proportional to HCS
n
in Eq. 共29兲, the CSA interaction terms, equal to ⫾ĤCS
i defined in Eq. 共27兲, and the rotor term Hr . The first term commutes with the additional terms and its evolution operator is
one at 2 ␶ w . Thus the evolution operator in Eq. 共44兲 only
depends on the two remaining and noncommuting terms:
ˆ
CS
CS
Ũ
0 共 2 ␶ w 兲 ⫽exp共 ⫺i/2共 Ĥi ⫹Ĥr 兲 ␶ w 兲 exp共 ⫺i 共 ⫺Ĥi
⫹Ĥr 兲 ␶ w 兲 exp共 ⫺i/2共 ĤCS
i ⫹Ĥr 兲 ␶ w 兲 .
⫺1
ĤCS
Ĥr D̂,
i ⫹Ĥr ⫽D̂
⫺1
⫺ĤCS
i ⫹Ĥr ⫽D̂Ĥr D̂
共47兲
which are derived in Appendix A. Insertion of these equations into Eq. 共46兲 and using that exp(iĤr ␶ w )⫽1̂ results in a
ˆ
periodic Ũ 0 (t) operator with
ˆ 2 ␶ ⫽D̂ exp ⫺i/2Ĥ ␶ D̂ ⫺1 D̂ ⫺1
Ũ
共
0共
w兲
r w兲
⫻exp共 ⫺iĤr ␶ w 兲 D̂D̂ exp共 ⫺i/2Ĥr ␶ w 兲 D̂ ⫺1 ⫽1̂ .
共48兲
共41兲
the toggling frame Hamiltonian becomes
共46兲
To show that this operator equals one we use the following
diagonalization conditions:
CS
Ĥ0 ⫽ĤCS
n ⫹Ĥi ⫹Ĥr ,
Ĥ1 ⫽ĤD
i ,
共43兲
Thus the CSA interaction is refocused at the end of a RFDR
cycle, as is well known for two ␲-pulses that are separated
by an integer multiple of rotor periods ␶ w ⫽n w ␶ c .
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J. Chem. Phys., Vol. 112, No. 3, 15 January 2000
Rotor frequency-driven dipolar recoupling
It is now possible to use the interaction Hamiltonian in
Eq. 共45兲 for the evaluation of the average RFDR Hamiltonian. The quantized interaction Hamiltonian is time dependent and is not anymore defined in the spatial rotating frame.
ˆ
This is due to the fact that the transformation operator Ũ 0 (t)
contains the rotor Hamiltonian term Ĥr . However, because
we are only interested in the spin response at the end of the
ˆ
RFDR cycles and Ũ
0 (t) is periodic, the time independent
average Hamiltonian can be considered as being defined in
the spatial rotating frame. We will restrict ourselves to the
calculation of the zero order average Hamiltonian assuming
ˆ
that 2 ␶ w 储 H̃int储 Ⰶ1.
VI. THE ZERO ORDER AVERAGE RFDR
HAMILTONIAN
The zero order average RFDR Hamiltonian of a homonuclear spin pair (i, j), evolving with the isotropic and anisotropic chemical shift interactions of both spins defined in
Eqs. 共27兲 and 共29兲, equals
ˆ ⫽ 1
H̄
Q
2␶w
冕
2␶w
0
ˆ dt⫽ 1
H̃
int
2␶w
冕
2␶w
0
ˆ ⫺1 t H̃
ˆ ˆ
Ũ
0 共 兲 1 Ũ 0 共 t 兲 dt
共49兲
with the two-spin dipolar interaction Hamiltonian in Eq.
共31兲. Simplifying the expressions by defining the operators,
Û r 共 t 兲 ⫽exp共 ⫺iĤr t 兲 ,
冉
2
Û CS
n 共 t 兲 ⫽exp ⫺i
兺 ⌬ ␻¯ i Î zi t
i⫽1
冊
共50兲
and using the diagonalization operators D̂⫽D̂ i D̂ j of the two
spins, the integral of the average Hamiltonian can be written
as
1
ˆ
H̄Q ⫽
2␶w
冕
1/2␶ w
⫺1/2␶ w
⫺1
dt Û CS
共t兲
n
⫻兵 D̂Û r 共 t 兲 D̂ ⫺1 其 Ĥ1 兵 D̂Û r⫺1 共 t 兲 D̂ ⫺1 其 Û CS
n 共t兲
⫺1
⫹Û CS
Û r 共 t 兲 D̂ 其 Ĥ1
n 共 t 兲 兵 D̂
⫻ 兵 D ⫺1 Û r⫺1 共 t 兲 D̂ 其 U ⫺1
0 共 t 兲.
共51兲
Here we used the fact that during the three time intervals of
the RDFR cycle the Hamiltonians are time independent. In
the appendix the integral in Eq. 共51兲 is calculated by expanding the diagonalization operator. The result yields an expresˆ
sion for H̄Q of the form:
ˆ
H̄Q ⫽⫺
⫽⫺
冉
兺冉
兺
i⬍ j
i⬍ j
⬁
d 0i j ⫹
兺
k⫽1
d ki j F̂ k ⫹d ki j* F̂ †k
⬁
d 0i j ⫹
兺
k⫽1
1
r 3i j
⬁
兺
n⫽⫺⬁
1
2
i j
i j
Î ⫹ 兲
Î ⫺ ⫹Î ⫺
共 Î ⫹
d ki j F̂ k ⫹d ki j* F̂ †k 共 Î ix Î xj ⫹Î iy Î yj 兲 , 共52兲
with
d ki j ⫽
冊
冊
ij
M kn
sinc
冉
2n w 共 ⌬ ␻
¯ i ⫺⌬ ␻
¯ j ⫺n ␻ r 兲
␻r
冊
共53兲
1101
and
2
ij
M kn
⫽⫺
兺
with
⬁
ij
⫽
Km
ij
i j*
i j*
ij
C ipj 共 K n⫹k
K n⫹p
⫹K n⫺k
K n⫺p
兲
p⫽⫺2
兺
k⫽⫺⬁
冉
Jk ⫺
兩 Z i2 ⫺Z 2j 兩
␻r
冊 冉
J m⫺2k ⫺2
兩 Z i1 ⫺Z 1j 兩
⫻exp 共 i 共 m⫺2k 兲 ␾ i1j ⫹ik ␾ i2j 兲 ,
␻r
共54兲
冊
共55兲
where
is given by Eq. 共33兲 and
in Eq. 共32兲. In Eq.
共52兲 a summation over all spin pairs in a coupled homoC can induce polarizanuclear spin system is added. The H
Q
5
tion transfer between the nuclei. The rate of transfer
strongly depends on the internuclear distance d ki j ⬀1/r 3i j , according to Eq. 共53兲. This provides the selectivity necessary
for correlation spectroscopy in isotope enriched solids.
Introducing the deviation from the rotational resonance
¯ ni j ⫽(⌬ ␻
¯i
condition by the off-rotational resonance value ␦ ␻
j
⫺⌬ ␻
¯ ⫺n ␻ r ), we see that the sinc-function in Eq. 共53兲 has a
¯ ni j ⫽0,
maximum at every rotational resonance condition ␦ ␻
¯ ni j
including the n⫽0 rotational resonance condition. For ␦ ␻
⫽0 the sinc-function decreases and can become zero for a
certain value. If we define the bandwidth of the RFDR as the
difference between the two zero points around the maximum
value, the bandwidth is19
C ipj
B w⫽
Z ipj
␻r
.
nw
共56兲
This implies that the bandwidth can be narrowed by increasing the number of rotor periods per pulse. In the limit of
large windows between the pulses (n w →⬁) this bandwidth
goes to zero. In this case only at the rotational resonance
condition the zero order average Hamiltonian recovers the
dipolar interaction. With adding ␲-pulses according to the
extended RFDR scheme, the rotational resonance condition
becomes broader and spectral selectivity can be considered.
In this way, the rotor frequency drives the transfer in RFDR
and not the radio frequency.
In Eq. 共53兲 also the n⫽0 rotational resonance is included. This implies that also for small isotropic shift differences the dipolar interaction can be recovered. In the early
treatment of the RFDR, this possibility for efficient transfer
with small isotropic shift differences has been overlooked,
since the shift anisotropy was generally excluded from the
earlier analytical treatments. In contrast, in our quantized
field analysis the chemical shift anisotropy is included and is
ij
in Eq. 共55兲. This theoretical outcome is
embedded in the K m
experimentally confirmed, because efficient transfer around
n⫽0 has been observed in chlorophylls.14,16 It can be concluded that RFDR is considerably more broadband than
originally thought.
VI. POLARIZATION TRANSFER
In Eq. 共52兲 the quantized field average Hamiltonian is
given for the RFDR cycle. This Hamiltonian can be used to
describe the evolution of the spin system after n m cycles via
the evolution operator
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1102
J. Chem. Phys., Vol. 112, No. 3, 15 January 2000
C 共 ␶ 兲 ⫽ 关 exp共 ⫺i2H
C ␶ 兲兴 n m ⫽exp共 ⫺iH
C ␶ 兲,
U
m
Q w
Q m
Boender, Vega, and de Groot
共57兲
with ␶ m ⫽2n m ␶ w . This propagator promotes polarization
transfer from nucleus i to nucleus j. Within the multi-spin
formalism presented here, multiple pathways are implicitly
covered, since the polarization transfer can be direct
C
H
Q
i→ j,
or can be relayed, e.g.,
C
C
H
Q HQ
i→ k→ j .
The discrimination between direct and relayed transfer has
attracted considerable attention in recent years.32 In a general
treatment, the evolution of the spin system and the transfer of
polarization can be described in terms of the evolution of the
density operator
C 共 ␶ 兲 ␳ˆ 共 0 兲 U
C ⫺1 共 ␶ 兲 .
␳ˆ 共 ␶ m 兲 ⫽U
m
m
共58兲
The density operator in the quantized field representation
describes the combination of rotor and spin system. This
approach is very demanding, because the total process of
creation and annihilation of particles via the Fourier operators in combination with the multiple spin processes has to
be taken into account. Therefore, to simplify matters we
project the quantized Hamiltonian onto an average spin
C via a ‘‘dequantization’’ of the phase: F̂
Hamiltonian H
S
k
⫽exp(ik␸ˆ ⬘)→exp(ik␸0). This is justified in the simple case
we intend to consider, provided we are in the limit of a large
number of rotor quanta. The average Hamiltonian in Eq. 共52兲
becomes then
C ⫽⫺
H
S
with
d̄ i j ⫽
冉
d̄ i j 共 Î ix Î xj ⫹Î iy Î yj 兲 ,
兺
i⬍ j
⬁
d 0i j ⫹
兺
k⫽1
共59兲
冊
d ki j exp共 ik ␸ 0 兲 ⫹d ki j* exp共 ⫺ik ␸ 0 兲 . 共60兲
If we restrict ourselves to a two spin system then we avoid
relayed polarization transfer. A convenient way of describing
a two-spin system is using the single-transition operators33,34
I z23⫽ 21 共 Î z1 ⫺Î z2 兲 ,
1 2
1 2
I 23
x ⫽ 共 Î x Î x ⫹Î y Î y 兲 ,
共62兲
1 2
1 2
I 23
y ⫽ 共 Î y Î x ⫺Î x Î y 兲 .
The average Hamiltonian in Eq. 共59兲 can be expressed in
those operators, according to
C ⫽⫺
H
S
d̄ i j Î 23
兺
x .
i⬍ j
共63兲
If the spin density operator at the beginning of the RFDR
period is
␳ˆ 共 0 兲 ⫽ 共 Î z1 ⫺Î z2 兲 ⫽2I z23 ,
共64兲
the average Hamiltonian will lead to rotation in the zy plane
around the x axis of this space of single-transition operators,
according to
␳ˆ 共 ␶ m 兲 ⫽ 共 2Î z23 cos 共 d̄ 12␶ m 兲 ⫹2Î 23
y sin 共 d̄ 12␶ m 兲
共65兲
which is consistent with earlier findings.5,31 However, in d̄ i j
also the CSA and the narrow band RFDR sequence are taken
into account. This implies that by measuring d̄ i j as a function
of the n w in principle not only information can be deduced
about the internuclear distance, but also about the relative
orientation of the CSA tensors of the two sites.35
This is illustrated in Fig. 2 with simulations of a RFDR
experiment on two 13C spins resonating with a Larmor frequency of 125.7 MHz and separated by ⬃1.4 Å with a small
isotropic shift difference of 14 ppm, corresponding with 1.76
kHz. The orientation of the PAS of the CSA for the second
spin is defined with respect to the PAS of the CSA of the first
spin and changed by varying the Euler angle 0⭐␤⭐180. For
␤⫽0 the two tensors are aligned and the dipolar vector is
along the axis corresponding with the largest principle component of the spin 1. The points are evaluated at three different spinning speeds ␯ r ⫽ ␻ r /2␲ assuming an initial condition for the density matrix 具 2Î z1 典 and calculating the 具 2Î z2 典
after a constant mixing time ␶ m ⫽2.66 ms.
At a low speed ␯ r ⫽6 kHz we calculate efficient transfer
for n w ⫽1 with 具 2Î z2 典 approaching the maximum of ⬃0.5
over a broad range and it depends on the value of ␤. The
transfer efficiency drops for n w ⫽2 and the angular dependence is enhanced. The narrow band character of the RFDR
is nicely illustrated for n w ⫽4. Here the bandwidth is
B w /2␲ ⫽1.5 kHz, less than the isotropic shift difference
(⌬ ␻
¯ 1 ⫺⌬ ␻
¯ 2 )/2␲ ⫽1.76 kHz and the 具 2Î z2 典 ⬃0.1, much less
than for the n w ⫽1 and n w ⫽2 at the same spinning speed.
This reflects the decrease of the bandwidth for increasing
nw .
With a spinning speed of 12 kHz, we calculate again a
considerable dependence of the transfer efficiency on the
angle ␤, which is most pronounced around ␤⫽30° and
␤⫽150°. The 具 2Î z2 典 goes through a maximum around
␤⫽90°. The relative variation of the transfer with ␤ is strongest for n w ⫽4. The 具 2Î z2 典 peaks around ␤⫽90° and it almost
vanishes for ␤⫽0° or ␤⫽180°. These calculations suggest
new and interesting possibilities for collecting angular information by measuring the transfer rate for different n w , using
the generalized RFDR treatment discussed here for the
analysis. At 24 kHz, the transfer characteristics change
again. At the high speed of 24 kHz also for the n w ⫽4 the
B w /2␲ ⫽6 kHz is sufficient to cover the isotropic shift difference of 1.76 kHz. In addition, the spinning speed of 24
kHz is sufficient to attenuate the CSA’s, since ␦ 1 /2␲
ij
⫽⫺11.3 kHz and ␦ 2 /2␲ ⫽⫺11.7 kHz. The M kn
in Eq. 共54兲
depend on the spinning speed. They vanish in the limit of
␻ r →⬁. Therefore the transfer is less efficient at high speed,
with 具 2Î z2 典 below 0.2 for v r ⫽24 kHz. This illustrates that
close to the n⫽0 rotational resonance condition the presence
of a dipolar interaction alone is insuffient for polarization
transfer and that there is a prominent effect of the CSA on
ij
.
the transfer efficiency via the M kn
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J. Chem. Phys., Vol. 112, No. 3, 15 January 2000
Rotor frequency-driven dipolar recoupling
1103
field approach, from which the Floquet approach can be derived, can be combined with average Hamiltonian theory. In
the earlier Floquet treatment of the RFDR, an effective
Hamiltonian was formulated, with the pulses included in the
sequence.30 In the quantized field formalism the effect of the
MAS rotation and the rotation itself are disentangled and the
resulting time-independent Hamiltonian can be treated in an
average Hamiltonian formalism by adding the pulses afterwards. In this way average Hamiltonian theory and Floquet
theory are fully compatible, since average Hamiltonian
theory can be applied to the time dependent quantized field
Hamiltonian in the toggling frame.
C in Eq. 共52兲 induces polarization transfer beThe H
Q
tween nuclei in a multiple spin system.5 The probability of
transfer strongly depends on the internuclear distance ␦ ki j
⬀1/r 3 and this makes correlation spectroscopy in isotope enriched solids possible. The probability of polarization transfer has a maximum at every rotational resonance condition
ij
¯␴ i ⫺ ¯␴ j ⫽n ␻ r . Since the CSA is embedded in the K m
also
the n⫽0 rotational resonance condition is included.
According to our treatment, the driving force in both
RFDR and rotational resonance is the rotor frequency and
not the radio frequency. This suggests that a physically more
appropriate expansion of the RFDR abbreviation would be
‘‘rotor frequency-driven dipolar recoupling.’’ The RFDR
bandwidth can be changed by changing the window between
the ␲-pulses. This additional variable can be used to obtain
information about relative orientation of the CSA tensors.
ACKNOWLEDGMENTS
This research was supported by the foundation of Life
sciences 共ALW兲, financed by the Netherlands Organization
for Scientific Research 共NWO兲 and part of this research was
supported by The Israel Science Foundation. G.J.B. and
H.J.M.dG are recipients of NWO TALENT and PIONIER
awards, respectively.
FIG. 2. Examples of magnetization transfer within a dipolar coupled spin
pair. The calculations are performed for two spins separated by 14 ppm and
resonating at 125.7 MHz. A set of parameters was used that correspond to
the CSA’s that can be observed, e.g., aromatic or ethylenic carbons: for the
first spin an anisotropy ␦ 1 ⫽⫺90 ppm was used with an asymmetry parameter ␩⫽0.41, while ␦ 2 ⫽⫺93 ppm with ␩⫽0.73. In the evaluation of the
signals ␸ 0 ⫽0 and 300 orientations are taken for each point. The relative
orientation of the tensors is defined by the Euler angles for the second spin
( ␣ c2 , ␤ c2 , ␥ c2 ), following Eq. 共26兲. In this calculation ␣ c2 ⫽ ␥ c2 ⫽0 and ␤ c2
⫽ ␤ 0 is varied. The Euler angles for the transformation of the PAS of the
first spin and for the transformation of the dipolar frame to the crystal frame
according to Eq. 共33兲 are taken zero. This corresponds with an orientation of
the dipolar vector parallel to the most shielded direction of the first spin I z1 .
The powder integration is performed over ( ␣ r , ␤ r , ␥ r ). The points are
evaluated after ␶ mix⫽2.66 msec by assuming an initial condition for the
density matrix 2I z1 and following the magnitude of 具 2I z2 典 . The maximum
value we can expect is 具 2I z2 典 ⬃0.5 in a powder. XY 8 phase cycling was used
for the full RFDR 共n w ⫽1, squares兲, XY 4 for n w ⫽2 共diamonds兲, and XX̄ for
n w ⫽4 共circles兲.
VII. CONCLUDING REMARKS
We have shown that the quantized field description can
be used to analyze pulse sequences including MAS on a
multiple spin system. It is demonstrated that the quantized
APPENDIX
In this appendix the expression for the zero order average Hamiltonian in Eq. 共52兲 is derived. To do so the interaction Hamiltonian of a two-spin system must be calculated
by applying the transformation of Eq. 共45兲:
ˆ t ⫽Ũ
ˆ ⫺1 t H̃
ˆ ˆ
H̃
int共 兲
0 共 兲 1 Ũ 0 共 t 兲 .
共A1兲
The transformation operator is defined by the sum of the
isotropic chemical shift Hamiltonians
ˆ
H̃CS
n 共t兲
⫽
再
¯ i ⫺⌬ ␻
¯ j 兲共 I zi ⫺I zj 兲 ,
1/2共 ⌬ ␻
0⬍t⬍1/2␶ w ,
¯ i ⫺⌬ j 兲共 I zi ⫺I zj 兲 ,
⫺1/2共 ⌬ ␻
1/2␶ w ⬍t⬍3/2␶ w ,
¯ i ⫺⌬ ␻
¯ j 兲共 I zi ⫺I zj 兲 ,
1/2共 ⌬ ␻
3/2␶ w ⬍t⬍2 ␶ w ,
共A2兲
and the CSA interaction term plus the rotor Hamiltonian
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1104
J. Chem. Phys., Vol. 112, No. 3, 15 January 2000
Boender, Vega, and de Groot
The dipolar interaction Hamiltonian from Eq. 共31兲 has the
form
ˆ
H̃CS
i 共 t 兲 ⫹Ĥr
⫽
冦
2
兺 兺
k⫽i, j p⫽1
共 Z kp F̂ p ⫹Z kp* F̂ †p 兲 I zk ⫹Ĥr ,
0⬍t⬍1/2␶ w ,
ˆ (t) and H̃
ˆCS(t)⫹Ĥ commute and so
The Hamiltonians H̃CS
r
n
i
ˆCS
CS
CS
do Û n (t) and Û i (t). In addition, the H̃i (t)⫹Ĥr is block
diagonal and can be diagonalized according to Refs. 33 and
36 with (k⫽i, j),
2
⫺
兺 兺
k⫽i, j p⫽1
共 Z kp F̂ p ⫹Z kp* F̂ †p 兲 I zi ⫹Ĥr ,
1/2␶ w ⬍t⬍3/2␶ w ,
2
兺 兺
k⫽i, j p⫽1
共 Z kp F̂ p ⫹Z kp* F̂ †p 兲 I zi ⫹Ĥr ,
ˆ ⫽ Z i j F̂ ⫹Z i j * F̂ † 2Î i Î j ⫺1/2 Î ⫹ Î ⫺ ⫹Î ⫺ Î ⫹ . 共A6兲
H̃
共 i j
i 共 p p
p
p 兲共
z z
i j 兲兲
3/2␶ w ⬍t⬍2 ␶ w ,
共A3兲
2
兺
p⫽1
as
共 Z kp F̂ p ⫹Z kp* F̂ †p 兲 Î zk ⫹Ĥr ⫽D̂ k Ĥr D̂ ⫺1
k ,
共A7兲
2
ˆ
CS
Ũ 0 共 t 兲 ⫽Û CS
n 共 t 兲 Û i 共 t 兲 ,
with
冉 冕
冉 冕
Û CS
n 共 t 兲 ⫽exp ⫺i
Û CS
i 共 t 兲 ⫽exp
ˆ t ⫽
H̃
int共 兲
再
⫺i
t
0
t
0
共A4兲
冊
with
再 兺冉
2
ˆCS ␶ d ␶ ,
H̃
n 共 兲
D̂ k ⫽exp ⫺
冊
共A5兲
ˆ
共 H̃CS
i 共 ␶ 兲 ⫹Ĥr 兲 d ␶ .
冕
1/2␶ w
⫺1/2␶ w
⫻D̂Ĥ1 D̂ ⫺1 Û r⫺1 共 t 兲 D̂Û CS
n 共 t 兲 其 dt.
共A10兲
To evaluate this integral we must consider the following
computational steps: 共a兲 calculation of Â⫽D̂ ⫺1 Ĥ1 D̂ and
 ⬘ ⫽D̂Ĥ1 D̂ ⫺1 by expanding the diagonalization operators;
共b兲 application of the rotor operator Â(t)⫽Û r (t)ÂÛ r⫺1 and
 ⬘ (t)⫽Û r (t) ⬘ Û r⫺1 (t); 共c兲 transformation of Â(t) and
 ⬘ (t) to B̂(t)⫽D̂Â(t)D̂ ⫺1 and B̂ ⬘ (t)⫽D̂ ⫺1  ⬘ (t)D̂;
共d兲 and application of the isotropic chemical shift operator
CS⫺1
Ĉ(t)⫽Û CS
B̂ ⬘ (t)Û CS
and Ĉ ⬘ (t)⫽Û CS⫺1
n (t)B̂(t)Û n
n
n (t).
To perform the first step we calculate the D-transformations
of the terms of Ĥ1 :
Z kp*
p␻r
冊冎
F̂ †p Î zk .
F̂ p Î zi Î zj D̂⫽F̂ p Î zi Î zj ,
共A9兲
i j
i j
D̂ ⫺1 F̂ p Î ⫹
Î D̂⫽⌬ˆ i j⫺1 F̂ p Î ⫹
Î ⫺
with
再 兺冉
⌬ ⫽exp ⫺
ij
Z ip ⫺Z pj
p␻r
p⫽1
⫺
Z ip* ⫺Z pj*
p␻r
冊冎
共A12兲
and ⌬ˆ i j ⫽⌬ˆ ji⫺1 and ⌬ˆ i j⫹ ⫽⌬ˆ i j⫺1 . This exponent can be expanded in a Fourier series using Z ip ⫺Z pj ⫽ 兩 Z ip
⫺Z pj 兩 exp(i␾ijp ):
⬁
2
⌬ˆ i j ⫽
冉
⫻ ⫺
p␻r
⬁
F̂ ⫺p
1
冊
冉
Z ip ⫺Z pj
p␻r
⬁
冊 冉
⬁
兿 兺 兺
p⫽1 m⫽0 l⫽⫺⬁
m⫹l
冊
n
1
m!
m
1
兿 兺 兺
p⫽1 m⫽0 l⫽⫺m m! 共 m⫹l 兲 !
⫻ ⫺
⫽
冉
Z ip ⫺Z pj
1
⫺
F̂ p
n!
p␻r
Z ip* ⫺Z pj*
⬁
2
⫽
⬁
兿 兺 兺
p⫽1 m⫽0 n⫽0
2
共A11兲
共A8兲
i j
i j
D̂F̂ p Î ⫹
Î ⫺ D̂ ⫺1 ⫽⌬ˆ i j F̂ p Î ⫹
Î ⫺ ,
D̂F̂ p Î zi Î zj D̂ ⫺1 ⫽F̂ p Î zi Î zj ,
D̂
p␻r
F̂ p ⫺
Insertion of these results into Eq. 共A1兲, defining D̂⫽D̂ i D̂ j
CS
with 关 D̂ i ,D̂ j 兴 ⫽0 and using 关 Û CS
n (t),Û i (t) 兴 ⫽0 we get
2
⫺1
Ĥ1 D̂Û r⫺1 共 t 兲
兵 Û CS
n 共 t 兲 D̂Û r 共 t 兲 D̂
⫻D̂ ⫺1 Û CS⫺1
共 t 兲 ⫹Û CS⫺1
共 t 兲 D̂ ⫺1 Û r 共 t 兲
n
n
⫺1
p⫽1
Z kp
⫺1
Û CS
Ĥ1 D̂Û r⫺1 共 t 兲 D̂ ⫺1 Û CS⫺1
共 t 兲,
n 共 t 兲 D̂Û r 共 t 兲 D̂
n
CS⫺1
⫺1
⫺1 ⫺1
Û n
共 t⫺ ␶ w 兲 D̂ Û r 共 t⫺ ␶ w 兲 D̂Ĥ1 D̂ Û r 共 t⫺ ␶ w 兲 D̂Û CS
n 共 t⫺ ␶ w 兲 ,
CS
⫺1
⫺1
⫺1 CS
Û n 共 t⫺2 ␶ w 兲 D̂Û r 共 t⫺2 ␶ w 兲 D̂ Ĥ1 D̂Û r 共 t⫺2 ␶ w 兲 D̂ Û n 共 t⫺2 ␶ w 兲 ,
respectively, for the three time intervals, where we assumed
that n w is even in ␶ w ⫽n w ␶ r , and Û r (1/2␶ w )⫽1̂. When we
change the variables of integration and take into account that
the Hamiltonians are constant during each time interval, the
average Hamiltonian can be rewritten as
1
ˆ
H̄Q ⫽
2␶w
兺
p⫽1
⫺ 共 Z kp F̂ p ⫹Z kp* F̂ †p 兲 Î zk ⫹Ĥr ⫽D̂ ⫺1
k Ĥr D̂ k
Z ip* ⫺Z pj*
p␻r
冊
m
F̂ lp
1
共 ⫺1 兲 m
m! ⌫ 共 m⫹l⫹1 兲
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J. Chem. Phys., Vol. 112, No. 3, 15 January 2000
冉
兩 Z ip ⫺Z pj 兩
⫻ ⫺
p␻r
⬁
⫽
⬁
兺 兺
k⫽⫺⬁ l⫽⫺⬁
冉
冉
冊
⫻ ⫺2
exp共 il ␾ ipj 兲 F̂ lp
␻r
冊
兩 Z i2 ⫺Z 2j 兩
2␻r
冊
Jl
1
ˆ
H̄Q ⫽⫺
4␶w
exp共 il ␾ i1j ⫹ik ␾ i2j 兲 F̂ l⫹2k
⫻
⬁
⫽
兺
共A13兲
ij
Km
F̂ m
m⫽⫺⬁
⬁
兺 Jk
k⫽⫺⬁
冉
⫺
兩 Z i2 ⫺Z 2j 兩
␻r
冊 冉
J m⫺2k ⫺2
兩 Z i1 ⫺Z 1j 兩
␻r
⫻exp共 i 共 m⫺2k 兲 ␾ i1j ⫹ik ␾ i2j 兲
Â⫽
再
兺
p⫽1
冊
共A14兲
兺
m⫽⫺⬁
ij
ij
⫺
F m⫹p ⫹Z ipj * K m
F m⫺p 兲 I ⫹
共共 Z ipj K m
i Ij
冎
i j*
i j*
⫹
⫹ 共 Z ipj K ⫺m
F m⫹p ⫹Z ipj * K ⫺m
F m⫺p 兲 I ⫺
i Ij 兲 ,
2
 ⬘ ⫽
兺
p⫽1
再
2 共 Z ipj F p ⫹Z ipj * F †p 兲 I zi I zj ⫺1/2
兺
m⫽⫺⬁
i j*
i j*
⫺
F m⫹p ⫹Z ipj * K ⫺m
F m⫺p 兲 I ⫹
共共 Z ipj K ⫺m
i Ij
冎
ij
ij
⫹
⫹ 共 Z ipj K m
F m⫹p ⫹Z ipj * K m
F m⫺p 兲 I ⫺
i Ij 兲 .
In the next step 共b兲 we use the fact that
U r 共 t 兲 F p U r⫺1 共 t 兲 ⫽F p exp共 ⫺ip ␻ r t 兲 ,
U r⫺1 共 t 兲 F p U r 共 t 兲 ⫽F p exp共 ip ␻ r t 兲
冉
i j*
i j
K ikj K n⫹p
F̂ k⫺n Î ⫹
Î ⫺
exp共 i 共 ⌬ ␻
¯ i ⫺⌬ ␻
¯ j ⫺n ␻ r 兲 t 兲 dt
冕 ␶␶
1
2 w
1
⫺2 w
exp共 i 共 ⌬ ␻
¯ j ⫺⌬ ␻
¯i
冕 ␶␶
1
2 w
1
⫺2 w
i j*
j i
⫺⌬ ␻
¯ i ⫹n ␻ r 兲 t 兲 dt⫹K ikj K n⫹p
F̂ k⫺n Î ⫹
Î ⫺
⫻exp共 i 共 ⌬ ␻
¯ i ⫺⌬ ␻
¯ j ⫺n ␻ r 兲 t 兲 dt
冊
exp共 i 共 ⌬ ␻
¯j
冕 ␶␶
1
2 w
1
⫺2 w
共A18兲
B. J. van Rossum et al., Spectrochim. Acta A 54, 1167 共1998兲.
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共A15兲
⬁
⫻
兺
k,n⫽⫺⬁
and the actual integration gives Eqs. 共52兲 and 共53兲. This
i j
i j
I ⫺ ⫹I ⫺
I ⫹)
Hamiltonian has only a (I ix I xj ⫹I iy I yj )⫽1/2(I ⫹
i j
term, since the integral of the I z I z coefficient, containing
only terms of exponents exp(⫾ip␻rt) is zero when ␶ w is a
multiple of the rotor period ␶ r .
2 共 Z ipj F p ⫹Z ipj * F †p 兲 I zi I zj ⫺1/2
⬁
⫻
1
2 w
1
⫺2 w
Z ipj
i j* i j
i j
⫹n ␻ r 兲 t 兲 dt⫹K ⫺k
K n⫺p F̂ k⫹n Î ⫹
Î ⫺
and J k (x) the kth order Bessel function of x. Thus in step 共a兲
the transformation results in
2
兺
兺
i⬍ j p⫽⫺2
冕 ␶␶
⬁
2
i j* i j
j i
K n⫺p F̂ k⫹n Î ⫹
Î ⫺
⫹K ⫺k
with
ij
⫽
Km
1105
in order to obtain expressions for C(t) and C ⬘ (t) from B(t)
and B ⬘ (t). Insertion of the results of these calculations in Eq.
共A10兲 yields for the zero order average Hamiltonian the following:
2m⫹l
J k ⫺2
兩 Z il ⫺Z lj 兩
Rotor frequency-driven dipolar recoupling
共A16兲
and replace all F p in Eq. 共A15兲 by F p exp(⫺ip␻r) and
F p exp(ip␻r) to obtain A(t) and A ⬘ (t), respectively. In step
共c兲 an additional D-transformation must be performed. The
results for B(t) and B ⬘ (t), obtained again by using Eqs.
共11兲–共13兲, are cumbersome and are therefore not shown
i j
I ⫾ contain multiples of
here. Their coefficients of I zi I zj and I ⫾
the K-parameters. In the last step the isotropic chemical shift
Hamiltonian is applied to the bilinear spin operators
⫽I zi I zj ,
U CS⫾1
共 t 兲 I zi I zj U CS⫿1
n
n
i j
i j
U CS⫾1
I ⫺ U CS⫿1
⫽I ⫹
I ⫺ exp共 ⫿ 共 ⌬ ␻ i ⫺⌬ ␻ j 兲 t 兲 ,
共 t 兲I⫹
n
n
共A17兲
i j
i j
U CS⫾1
I ⫹ U CS⫿1
⫽I ⫺
I ⫹ exp共 ⫾ 共 ⌬ ␻ i ⫺⌬ ␻ j 兲 t 兲
共 t 兲I⫺
n
n
2
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1106
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