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Efficient theory of dipolar recoupling in solid-state nuclear magneticresonance of rotating solids using Floquet–Magnus expansion: Applicationon BABA and C7 radiofrequency pulse sequences

Eugene S. Mananga a,n,1, Alicia E. Reid b, Thibault Charpentier c

a Commissariat A L’ Energie Atomique, Neurospin/I2BM, Laboratoire de Resonance Magnetique Nucleaire, CEA-Saclay,

Bat 145, Point Courrier 156F-91191 Gif-sur-Yvette cedex, Franceb The City University Of New York, Medgar Evers College, 1638 Bedford Avenue, Brooklyn, NY 11225, USAc Commissariat A L’ Energie Atomique, Laboratoire de Structure et Dynamique par Resonance Magnetique, CEA/DSM/DRECAM/SCM–CNRS,

URA, 331, Saclay, 91191 Gif-Sur-Yvette cedex, France

a r t i c l e i n f o

Article history:

Received 19 July 2011

Received in revised form

20 September 2011Available online 1 December 2011

Keywords:

Average Hamiltonian theory

Floquet theory

Floquet–Magnus expansion

Magic angle spinning

BABA pulse sequence

C7 pulse sequence

a b s t r a c t

This article describes the use of an alternative expansion scheme called Floquet–Magnus expansion

(FME) to study the dynamics of spin system in solid-state NMR. The main tool used to describe the

effect of time-dependent interactions in NMR is the average Hamiltonian theory (AHT). However, some

NMR experiments, such as sample rotation and pulse crafting, seem to be more conveniently described

using the Floquet theory (FT). Here, we present the first report highlighting the basics of the Floquet–

Magnus expansion (FME) scheme and hint at its application on recoupling sequences that excite more

efficiently double-quantum coherences, namely BABA and C7 radiofrequency pulse sequences. The use

of Ln(t) functions available only in the FME scheme, allows the comparison of the efficiency of BABA

and C7 sequences.

& 2011 Elsevier Inc. All rights reserved.

1. Introduction

The power of solid-state NMR is based on its ability toelucidate molecular structure and dynamics in systems notamenable to characterization by other techniques. As such,solid-state NMR techniques have been applied to many studiessuch as determination of accurate intermolecular distances [1,2]and molecular torsion angles [3,4]. Several techniques to estimatethese parameters make use of double-quantum coherence (DQC).For a spin system subject to magic-angle spinning (MAS), the spinHamiltonian is the sum of the off-resonance, chemical shiftanisotropy (CSA), J coupling, and dipolar interaction. In thepresent article, we ignore effects of the off-resonance, chemicalshift anisotropy, and J couplings. We extend the description of thedouble-quantum (DQ) dynamics during dipolar recouplingbeyond the first-order limit, which generally assumes chemicalshift anisotropy effects negligible compared to large dipolarcouplings. We only consider and use the dipole–dipole couplingto generate DQC. Generally, experiments with DQC allow for

effective suppression of unwanted signals arising from naturallyabundant nuclei, while leaving signals arising from coupled spin-pairs [5]. Important features of any DQC excitation technique arethe excitation efficiency of DQC and the time scale on which theexcitation occurs. Therefore, much effort is spent on the design ofrecoupling sequences that are able to produce DQC with highefficiency and on a short time scale. DQ recoupling is very usefulin NMR spectroscopy, so success of this approach will require asolid understanding of the dynamics of recoupled multiple-spinsystems. In this work, we apply a promising tool for studying spindynamics, the Floquet–Magnus expansion (FME) developed byCasas et al. [6] and recently introduce to solid-state NMR byMananga et al. [7], to compare the performance of two importantrecoupling sequences (BABA and C7) [8,9]. These sequences allowhighly resolved DQ spectra to be obtained. To the best of ourknowledge, this paper represents the first report of such type.

The FME approach is the combination of the two majormethods used to describe the spin dynamics in solid-state NMR:the average Hamiltonian theory based on the Magnus expansionand the Floquet theory based on the Fourier expansion. The firstmethod, AHT, was developed by Waugh and Haeberlen in 1968[10] and is appropriate for stroboscopic sampling. However, thetechnique of AHT does not sufficiently describe the case of MAS[11,12] spectra because in this case, the signal is usually observed

Contents lists available at SciVerse ScienceDirect

journal homepage: www.elsevier.com/locate/ssnmr

Solid State Nuclear Magnetic Resonance

0926-2040/$ - see front matter & 2011 Elsevier Inc. All rights reserved.

doi:10.1016/j.ssnmr.2011.11.004

n Corresponding author. Fax: þ1 718 997 3349.

E-mail address: [email protected] (E.S. Mananga).1 Current address: Department of Physics, Queens College of the City

University of New York, 65-30 Kissena Blvd, Flushing, NY 11367, USA.

Solid State Nuclear Magnetic Resonance 41 (2012) 32–47


continuously with a time resolution much shorter than the rotorperiod [13–15]. The second method, FT, developed by Vega [16]and Maricq [17], provides a more universal approach for thedescription of the full time dependence of the response of aperiodically time-dependent system, but is most of the timeimpractical. However, the FT approach allows the computationof the full spinning sideband pattern that is of importance inmany MAS experimental circumstances to obtain information onanisotropic sample properties. The FME approach can be used tosolve a time-dependent linear differential equation, which is acentral problem in solid-state NMR and in quantum physics ingeneral. The FME scheme will be useful in the field of solid-stateNMR to (1) shed new lights on the established AHT and FT; (2)greatly simplify the calculation of higher order terms and (3)provide a more intuitive understanding of spin dynamics processes.

In the following sections of this paper, we describe theFloquet–Magnus expansion and explicitly give the first contribu-tions of the scheme with the addition of higher order effects(F3,L3(t)) as an improvement to the first two orders ðF1,L1ðtÞÞ and(F2,L2(t)). The FME scheme is then applied to BABA, BABA2 and C7pulse sequences with the inclusion of numerical analysis of thepulse sequences. A comparison of the efficiency of BABA, BABA2and C 7 sequences to produce DQ terms via the use of L1(t) andL2(t) functions is made. These operator functions facilitate theevaluation of the spin during or in between the RF pulses.

2. Theory

2.1. Floquet–Magnus expansion (FME) description

In the following, we use the FME approach to analyze the spindynamics evolving under the dipolar interaction subject totwo pulse sequences BABA, BABA2 and C7 shown, respectively,in Figs. 1–3. These recoupling pulse sequences are known to bevery efficient in exciting double-quantum coherences.

An illuminating approach for studying the dynamics of a spinsystem subject to a RF perturbation, given by the FME [6,7,19],can be to consider instead of the first and second order F1 and F2

that are identical to their counterparts in AHT and FT, but theL1.2(t) functions, available only in the FME scheme. This approachwill be shown to provide a new way for evaluating the spinevolution during ’’the time in between’’ through the Magnusexpansion of the operator connected to this part of the evolution.Using the Floquet theory [20–22], which is a branch of the theoryof ordinary differential equations relating to the class of solutionsof linear differential equations of the form

idU

dt¼HðtÞUðtÞ ð1Þ

with initial condition

Uð0Þ ¼ I, ð2Þ

we can write

UðtÞ ¼ PðtÞe�itF ð3Þ

where H(t), F and P(t) are n�n matrices. H(t) is a complex n�n

matrix-valued function and its matrix elements are integrableperiodic functions of time t with period denoted T. P(t) is aperiodic function of time with period T, i.e. P(t)¼P(tþT) and F isconstant. In NMR, this structure is exploited in many situationsincluding time-dependent periodic magnetic fields or samplespinning, which is the focus of this paper.

Using the exponential ansatz

PðtÞ ¼ e�iLðtÞ ðwith LðtþTÞ ¼LðtÞÞ, ð4Þ

we approximate solutions in the following form of the propagator

UðtÞ ¼ e�iLðtÞe�itF : ð5Þ

The function L(t) provides an alternative way for evaluatingthe spin behavior in between the stroboscopic observation points,and F is the time independent Hamiltonian that governs theevolution of the propagator U(t).

Considering the perturbation expansions for L(t) and F

LðtÞ ¼Xk ¼ 1

1

LkðtÞ, ð6Þ

F ¼Xk ¼ 1

1

Fk ð7Þ

with

Lkð0Þ ¼ 0, ð8Þ

Fig. 1. BABA pulse sequence.

Fig. 2. BABA2 pulse sequence.

Fig. 3. C7 pulse sequence.

E.S. Mananga et al. / Solid State Nuclear Magnetic Resonance 41 (2012) 32–47 33


for all k, and following the procedure described in Refs. [6,19], theabove propagator Eq. (5) gives

UðtÞ ¼ e�iP

k ¼ 1

1

LkðtÞ

e�itP

k ¼ 1

1

Fk

: ð9Þ

Casas et al. [19] derived explicitly, the following equations:

L1ðtÞ ¼

Z0

t

HðxÞdx�tF1, ð10Þ

F1 ¼1

T

Z0

T

HðxÞdx, ð11Þ

L2ðtÞ ¼1

2

Z0

t

HðxÞþF1,L1ðxÞ½ �dx�tF2, ð12Þ

F2 ¼1

2T

Z T

0HðxÞþF1,L1ðxÞ½ �dx: ð13Þ

The third order terms are also given in Appendix A0. Recently,we derived the general formula for the contribution of theFloquet–Magnus expansion [7] as follows:

LnðtÞ ¼

Z0

t

GnðtÞdt�tFnþLnð0Þ, ð14Þ

Fn ¼1

T

Z0

T

GnðtÞdt, ð15Þ

where Gn(t) functions are constructed using the FME recursivegeneration scheme [6,19]. The first two orders explicit formulaare

L1ðtÞ ¼L1ð0Þþ

Z0

t

G1ðtÞdt�tF1 ð16Þ

with

G1ðtÞ ¼HðtÞ, ð17Þ

L1ðTÞ ¼L1ð0Þ ð18Þ

yields

F1 ¼1

T

Z0

T

HðtÞdt: ð19Þ

Second order terms are

L2ðtÞ ¼

Z0

t

G2ðtÞdt�tF2þL2ð0Þ ð20Þ

with

G2ðtÞ ¼� i2 HðtÞþF1,L1ðtÞ½ �, ð21Þ

L2ðTÞ ¼L2ð0Þ ð22Þ

F2 ¼�i

2T

Z0

T

HðtÞþF1,L1ðtÞ½ �dt: ð23Þ

Ln(t)is the operator that describes the evolution within theperiod and Fn is the Hamiltonian governing the evolution atmultiple periods. The two operators Ln(t) and Fn are inter-dependent. While Floquet theory can also describe the time-evolution for arbitrary times, the FME approach however, canevaluate spin dynamics easier that FT and AHT. The additionalterms Ln(0)a0 (n¼1, 2, y) in our general expressions [7] differfrom the generally assumed Ln(0)¼0 [6]. However, in this article,

we use the expressions of Casas et al. [6] as first-order contribu-tions to the FME, where all Ln(0) are assumed to be zero. Notethat, the FME scheme is not restricted to dipolar or quadrupolarinteraction, and can be applied to any case.

For a spin system subject to MAS, the spin Hamiltonian is thesum of the off-resonance, chemical shift anisotropy, J coupling,and dipolar interaction. In the present article, we ignore off-resonance, chemical shift anisotropy, and J couplings. The analysisof the removal of the chemical shift interactions is the subject of aforthcoming paper [25]. In that work, we follow the explicitmethod described by Tycko for the removal of chemical shiftinteractions [24]. The current work however, is restricted to thestudy of dipolar interaction (HD(t)), and the following substitutionmust be done in the above Eqs. (10)–(13):

HðtÞ ¼HDðtÞ: ð24Þ

The dipolar spin interaction MAS is given by

HDðtÞ ¼1

2

Xia j

oijDðtÞT

ij20, ð25Þ

where

Tij20 ¼

1ffiffi6p 2Iij

ZZ�IijXX�Iij

YY

h ið26Þ

and

oijDðtÞ ¼ bij

Xn ¼ �2

2

Cijnða

ij,bij,gijÞe�inoRt ¼ bij

Xn ¼ �2

2

Cijnða

ij,bijÞe�inðoRtþgijÞ:

ð27Þ

In the following description, we are working only with the firsttwo orders contribution terms of the FME. It can be shown that,for BABA and C7 recoupling sequences, the second order term issmall compared to the first order term.

2.2. Applications

Here we applied the first contribution terms of the Floquet–Magnus expansion to the dipolar Hamiltonian when irradiatedwith the BABA [8] and sevenfold symmetric radiofrequency pulsesequences [9]. The degree to which these pulse sequencesrecouple magnetic dipolar between nuclear spins is useful forpreparing and detecting double quantum coherence.

2.2.1. Application of the Floquet–Magnus expansion (FME) to BABA

pulse sequence of one rotor period (tR)

The broadband BABA pulse sequences shown in Fig.1 are builtby suitable modification of the basic pulse sequence [8]

900X�

tR

2�900

X

� �900

Y�tR

2�900

Y

� �h i,

where the 901 pulses in the middle of the pulse sequence andbetween different cycles are placed Back-to-Back (BABA). Asshown below, the timing of the BABA sequence is important forfull synchronization of the sample rotation, that is, for generatinga pure DQ Hamiltonian. Using the FME, we are interested incalculating the degree to which the dipolar Hamiltonian leads tothe maximum strength of the DQ Hamiltonian as a result ofirradiating an ensemble of dipolar-coupled spin-pairs with thebasic and the broadband BABA pulse sequence acting respectivelyon tR. In this paper, BABA pulse sequence acting on one rotorperiod tR is constructed from four rf pulses, with flip anglesp=2,p=2,p=2,p=2� �

and rf phases X,�X,Y ,�Yf g. The calculationbegins with the definition of the dipolar spin interaction MASHamiltonian and using Eqs. (25)–(27) where a, b and g are Eulerangles describing the orientation of a given molecule or crystallitein the MAS rotor. bij ¼ 3ðgIÞ

2_=R3ij is the coupling constant, where

E.S. Mananga et al. / Solid State Nuclear Magnetic Resonance 41 (2012) 32–4734


gI is the gyromagnetic ratio for nuclear spins of type I and Rij is theinternuclear distance. The coefficients Cij

n can be expressed as

Cijnða

ij,bijÞ ¼ d2

0,nðyMÞX2

n’ ¼ �2

ð�1Þn’Yij

2n’e�inaij

d2n�n’ðb

ijÞ, ð28Þ

where d2nmðb

ijÞ are second-rank reduced Wigner rotation matrix

elements, and yM is the magic angle, and d200ðyMÞ ¼ 0, Cij

0 ¼ 0 [23].

Yij2n and Tij

2n are two independent types of rotations:Yij2n is a basis

for an irreducible representation of rotations of the spatial

coordinates of the sample, and fYij2ng are second-rank spherical

harmonics; Tij2n is a basis for an irreducible representation of

rotations of the spin angular momenta, and {T2n} are second-rankirreducible tensor operators [24]. See Appendix A3 for a moredetailed description.

We can compute the toggling frame during each half of therotor period. We have

For: 0rto tR2 ,

~HDðtÞ ¼1

2

Xia j

oijD tð ÞRþX

p2

� �Tij

20RXp2

� �

¼1

2

Xia j

oijDðtÞ

1ffiffiffi6p 2Iij

YY�IijXX�Iij

ZZ

h i

¼1

2

Xia j

oijDðtÞH

ijYY , ð29Þ

where

HijYY ¼

1ffiffi6p 2Iij

YY�IijXX�Iij

ZZ

h ið30Þ

and for: tR2 r0ot

~HDðtÞ ¼1

2

Xia j

oijDðtÞR

þY

p2

� �Tij

20RYp2

� �

¼1

2

Xia j

oijDðtÞ

1ffiffiffi6p 2Iij

XX�IijZZ�Iij

YY

h i

¼1

2

Xia j

oijDðtÞH

ijXX , ð31Þ

where

HijXX ¼

1ffiffi6p 2Iij

XX�IijZZ�Iij

YY

h i: ð32Þ

Consider the BABA pulse sequence in the following represen-tation

In this new picture, the toggling frame is written as

~HDðtÞ ¼HYY ðtÞyY ðtÞþHXXðtÞyXðtÞ, ð33Þ

where

HaaðtÞ ¼1

2

Xia j

oijDðtÞH

ijaa, a¼ X,Y ,Z: ð34Þ

The function y(t) is represented as follows:

Eq. (33) is now written as

~HDðtÞ ¼HYY ðtÞyðtÞþHXXðtÞ½1�yðtÞ�: ð35Þ

The time-dependent function y(t) can be expressed in the formof the Fourier expansion

yðtÞ ¼Xþ1

n ¼ �1

an expð�inoRtÞ: ð36Þ

With an representing the time-independent Fourier coeffi-cients corresponding to the Fourier index n. The coefficient ak

can be obtained

ak ¼1

tR

Z tR

0yðtÞeikoRtdt: ð37Þ

The coefficients in the first half of the sequence are as follows:

aXk ¼

1

tR

Z tR=2

0eikoRtdt, ð38Þ

which are given explicitly by

aXo ¼

1

2, ð39Þ

aXk ¼

1

2pikðeikp�1Þ: ð40Þ

In the second half of the sequence, they are

aYk ¼

1

tR

Z tR

tR=2eikoRtdt, ð41Þ

which can also be written explicitly as

aYo ¼

1

2, ð42Þ

aYk ¼

1

2pikð1�eikpÞ: ð43Þ

Globally, we have

aXo ¼ aY

o ¼12, ð44Þ

aXk ¼�aY

k ¼ ak: ð45Þ

The toggling frame Eq. (35) is rewritten as

~HDðtÞ ¼1

2

Xia j

oijDðtÞ

Xþ1n ¼ �1

ane�inðoRtþgijÞ

!Hij

YY

þ1

2

Xia j

oijDðtÞð1�

Xþ1n ¼ �1

ane�inðoRtþgijÞÞHijXX

¼1

2

Xia j

oijDðtÞ a0þ

Xka0

ake�ik oRtþgijð Þ

!Hij

YY

þ1

2

Xia j

oijDðtÞð1�a0�

Xna0

ake�ikðoRtþgijÞÞHijXX

¼1

2

Xia j

oijDðtÞa0ðH

ijXXþHij

YY Þ



þ1

2

Xia j

oijDðtÞ

Xka0

ake�ikðoRtþgijÞðHijYY�Hij

XXÞ: ð46Þ

Note that

HijXXþHij

YY ¼1ffiffi6p ðIij

XXþ IijYY�2Iij

ZZÞ �1ffiffi6p Hij

ZQ ð47Þ

and

HijYY�Hij

XX ¼3ffiffi6p ðIij

YY�IijXXÞ �

3ffiffi6p Hij

DQ : ð48Þ

Considering Eqs. (47) and (48), Eq. (46) is written as

~HDðtÞ ¼1

2

Xia j

oijDðtÞa0

1ffiffiffi6p ðIij

XXþ IijYY�2Iij

ZZÞ

þ1

2

Xia j

oijDðtÞ

Xka0

ake�ikðoRtþgijÞ 3ffiffiffi6p ðIij

YY�IijXXÞ

¼ ~HZQ ,DðtÞþ ~HDQ ,DðtÞ, ð49Þ

where ~HZQ ,D is the first term and ~HDQ ,D is the second term of Eq.(49), i.e.

~HDQ ¼3

2ffiffiffi6p

Xia j

oijDðtÞ

Xka0

ake�ikðoRtþgijÞðIijYY�Iij

XXÞ: ð50Þ

Substituting Eq. (27) into Eq. (50), we have

~HDQ ¼3

2ffiffiffi6p

Xia j

bij

Xn ¼ �2

þ2

Cijn

Xka0

ake�ikðoRtþgijÞe�inðoRtþgijÞðIijYY�Iij

XXÞ

¼3

2ffiffiffi6p

Xia j

bij

Xn ¼ �2

þ2

Cijn

Xka0

ake�iðnþkÞðoRtþgijÞðIijYY�Iij

XXÞ: ð51Þ

Setting nþk¼m, we have

~HDQ ¼3

2ffiffiffi6p

Xia j

bij

Xn ¼ �2

þ2

Cijn

Xm ¼ �1

þ1

am�ne�imðoRtþgijÞðIijYY�Iij

XXÞ

¼3

2ffiffiffi6p

Xia j

bij

Xm ¼ �1

þ1

e�imðoRtþgijÞX

n ¼ �2

þ2

am�nCijnðI

ijYY�Iij

XXÞ: ð52Þ

This toggling Hamiltonian of the double quantum term can besplit in two parts as following: the zero order term correspondingto m¼0, which is the average Hamiltonian theory, and the higherorder terms.

~HDQ ¼3

2ffiffiffi6p

Xia j

bij

Xþ2

n ¼ �2

a�nCijnðI

ijYY�Iij

XXÞ

þ3

2ffiffiffi6p

Xia j

bij

Xþ1m ¼ �1

e�imðoRtþgijÞXþ2

n ¼ �2

am�nCijnðI

ijYY�Iij

XXÞ ð53Þ

writing Eq. (52) more explicitly lead to

~HDQ ¼3

2ffiffiffi6p

Xia j

bij

Xn ¼ �2

þ2

a�nCijn

8<:

9=;|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

AHT ¼ oijD

ðIijYY�Iij

XXÞ

þX

m ¼ �1

þ1

e�imðoRtþgijÞ 3

2ffiffiffi6p

Xia j

bij

Xn ¼ �2

þ2

am�nCijn

8<:

9=;|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ooijD4m

ðIijYY�Iij

XXÞ:

ð54Þ

Using Eqs. (39)–(45), we can write the following

a�n ¼�1

2pinðe�inp�1Þ, ð55-aÞ

a�1 ¼1

pi, ð55-bÞ

a1 ¼�1

pið55-cÞ

and

a�2 ¼ a2 ¼ 0: ð55-dÞ

A straightforward calculation gives the first order averageHamiltonian, which is also the first order contributions to theFloquet–Magnus expansion (Eq. (11)) F1 ¼

1T

R T0~HDðtÞdt,

which is calculated by integrating over ½0,tR� to obtain thefollowing result

F1 ¼H0¼

3

2iffiffiffi6p

p

Xia j

bijðCij1�Cij

�1ÞðIijYY�Iij

XXÞ: ð56Þ

Or more explicitly,

F1 ¼H0¼

3

2iffiffiffi6p

p

Xia j

bij Cij1ða

ij,bijÞe�igij

�Cij�1ða

ij,bijÞeþ igij

h iðIij

YY�IijXXÞ:

ð57Þ

We note that

Tij22þTij

2�2 ¼ IijXX�Iij

YY ð58Þ

and

singij ¼eigij�e�igij

2i: ð59Þ

The constants Cij1ða

ij,bijÞ and Cij

�1ðaij,bijÞ are derived in Appendix

A3. Consideringonly the central term in Eq. (172), we have

Cij1ða

ij,bijÞ ¼

1ffiffiffi3p Y20e�iaij

d210ðb

ijÞ ¼ Y20e�iaij 1ffiffiffi

3p ð�1Þ

ffiffiffi3

2

rsinbijcosbij

¼�1

2ffiffiffi2p Y20e�iaij

sin2bij, ð60Þ

where the Wigner d-matrix elements with swapper lower indicesfound with the relation

djnm ¼ ð�1Þm�ndj

mn ð61Þ

has been used. In a similar vein, the constant Cij�1ða

ij,bijÞ can also

be derived. If Eqs. (58) and (59) are substituted into Eq. (57), weobtained the direct comparison between Eq. (57) and the zero-order average Hamiltonian of the dipolar Hamiltonian calculatedby Feike et al. [8]:

~H0

D ¼HDQ ¼3

pffiffiffi2p

Xi4 j

Dij sinð2bijÞsinðgijÞðTij

22þTij2�2Þ: ð62Þ

Therefore, the first order F1 to the Floquet–Magnus expansionappears to be equivalent to the zero-order average Hamiltonian.Next, Eq. (10) can be computed as follows:

L1ðtÞ ¼

Z t

0

~HDðt0Þdt0�tF1 ¼

Z t

0

~HZQ ,Dðt0Þdt0 þ

Z t

0

~HDQ ðt0Þdt0�tF1

¼

Z t

0

~HZQ ,Dðt0Þdt0 þ

Z t

0

~HAHT ð0Þ ðt0Þdt0 þ

Z t

0

~HAHT ðmÞ ðt0Þdt0�tF1

¼

Z t

0

~HZQ ,Dðt0Þdt0 þ

Z t

0

~HAHT ðmÞ ðt0Þdt0, ð63Þ

which leads to

L1ðtÞ ¼1

4ffiffiffi6p

Xia j

bij Cij�2ða

ij,bijÞe2igij 1

2ioR

� ðe2ioRt�1Þ

�ðIij

XXþ IijYY�2Iij

ZZÞ

þ1

4ffiffiffi6p

Xia j

bij �Cij2ða

ij,bijÞe�2igij 1

2ioR

� ðe�2ioRt�1Þ

�ðIij

XXþ IijYY�2Iij

ZZÞ

þ1

4ffiffiffi6p

Xia j

bij �Cij1ða

ij,bijÞe�igij 1

ioR

� ðe�ioRt�1ÞþCij

�1ðaij,bijÞeigij



1

ioR

� ðeioRt�1Þ

�ðIij

XXþ IijYY�2Iij

ZZÞ�X

m ¼ �1

þ1 1

imoRðe�imoRt�1Þ

3

2ffiffiffi6p

Xia j

bije�imgij

Xn ¼ �2

þ2

am�nCijn

24

35ðIij

YY�IijXXÞ: ð64Þ

Next, we can calculate the second order terms due to thecontributions of the Floquet–Magnus expansion by using Eq. (13).If we write the following

~HDðtÞþF1 ¼HZHijZQþðH

0þHm

ÞHijDQþH0Hij

DQ ð65Þ

and

L1ðtÞ ¼ AHijZQþBHij

DQ ð66Þ

where the constants are

HZ ¼1

2

Xia j

oijDðtÞa0

1ffiffiffi6p , ð67Þ

H0¼

3

2ffiffiffi6p

Xia j

bij

Xn ¼ �2

þ2

a�nCijn, ð68Þ

Hm¼

Xm ¼ �1

þ1


2ffiffiffi6p

Xia j

bij

Xn ¼ �2

þ2

am�nCijn

24

35, ð69Þ

A¼1

4ffiffiffi6p

Xia j

Cij�2ða

ij,bijÞe2igij 1

2ioR

� ðe2ioRt�1Þ

�

þ1

4ffiffiffi6p

Xia j

�Cij2ða

ij,bijÞe�2igij 1

2ioR


�

þ1

4ffiffiffi6p

Xia j

�Cij1ða

ij,bijÞe�igij 1

ioR

� ðe�ioRt�1Þ

þCij�1ða

ij,bijÞeigij 1

ioR

� ðeioRt�1Þ

�, ð70Þ

B¼�X

m ¼ �1

þ1 1


3

2ffiffiffi6p

Xia j

bije�imgij

Xn ¼ �2

þ2

am�nCijn

24

35:ð71Þ

Eq. (13) leads to

F2 ¼1

2T

Z T

0

~HDðt0ÞþF1,L1ðt

0Þ

h idt0

¼1

2T

Z T

0HZBdt0�

Z T

0ð2H0

þHmÞAdt0

�Hij

ZQ ,HijDQ

h i¼

1

2T

Z T

0HZBdt0�

Z T

0ð2H0

þHmÞAdt0

�ð2 Iij

XX ,IijYY

h i�2½Iij

XX ,IijZZ �þ2½Iij

YY ,IijZZ �Þ: ð72Þ

The computation of each term of the above integral (seeAppendix A1) givesZ T

0HZBdt¼ 0, ð73Þ

Z T

0HmAdt¼ 0, ð74Þ

2

Z T

0H0Adt¼ 2H0

Z tR

0Adt, ð75Þ

Z0

tR

Adt¼1

4ffiffiffi6p

Xia j

Cij�2ða

ij,bijÞe2igij 1

2ioR

� ð�tRÞ

�

þ1

4ffiffiffi6p

Xia j

�Cij2ða

ij,bijÞe�2igij 1

2ioR

� ð�tRÞ

�

þ1

4ffiffiffi6p

Xia j

�Cij1ða

ij,bijÞe�igij 1

ioR

� ð�tRÞ

þCij�1ða

ij,bijÞeigij 1

ioR

� ð�tRÞ

�: ð76Þ

Substituting Eqs. (73)–(76) into Eq. (72), we obtain

F2 ¼3

2ffiffiffi6p

Xia j

bij

Xn ¼ �2

þ2

a�nCijn

1

4ffiffiffi6p Cij

�2ðaij,bijÞe2igij 1

2ioR

�

�Cij2ða

ij,bijÞe�2igij 1

2ioR

� �n Hij

ZQ ,HijDQ

h i

þ3

2ffiffiffi6p

Xia j

bij

Xn ¼ �2

þ2

a�nCijn

1

4ffiffiffi6p �Cij

1ðaij,bijÞe�igij 1

ioR

�

þCij�1ða

ij,bijÞeigij 1

ioR

� �n Hij

ZQ ,HijDQ

h ið77Þ

or more explicitly

F2 ¼�tR

64p2

Xia j

bijðCij1�Cij

�1Þ Cij�2ða

ij,bijÞe2igij

�Cij2ða

ij,bijÞe�2igij

h iHij

ZQ ,HijDQ

h i�

tR

32p2

Xia j

bijðCij1�Cij

�1Þ �Cij1ða

ij,bijÞe2igij

þCij�1ða

ij,bijÞe�2igij

h iHij

ZQ ,HijDQ

h i:

ð78Þ

Similarly, using Eq. (12), we can compute the second orderterms L2(t) due to the contributions of the Floquet–Magnusexpansion. We have

L2ðtÞ ¼1

2

Z t

0HZBdt0�

Z t

0ð2H0

þHmÞAdt0

�Hij

ZQ ,HijDQ

h i�tF2þL2ð0Þ:

ð79Þ

After calculations (see Appendix A1) we obtain

L2ðtÞ ¼�1

32o2R

Xia j

bij

X2

n ¼ �2

Cijn

!�

Xþ1m ¼ �1

1

m

Xia j

bije�imgij

Xþ2

n ¼ �2

am�nCijn

24

35

0@

1A

24

35

n1

ðnþmÞ

�Hij

ZQ ,HijDQ

h iþ�1

32o2R

Xia j

bij

X2

n ¼ �2

Cijn

!24

�Xþ1

m ¼ �1

1

m

Xia j

bije�imgij

Xþ2

n ¼ �2

am�nCijn

24

35

0@

1A#

n�1

ðnþmÞe�ðnþmÞioRt

�Hij

ZQ ,HijDQ

h iþ�1

32o2R

Xia j

bij

X2

n ¼ �2

Cijn

!24

�Xþ1

m ¼ �1

1

m

Xia j

bije�imgij

Xþ2

n ¼ �2

am�nCijn

24

35

0@

1A#

n1

nðe�inoRt�1Þ

�Hij

ZQ ,HijDQ

h iþ

1

16pio2

Xia j

bijðCij1�Cij

�1Þ

Xia j

1

4Cij�2e2igij

ðe2ioRt�1ÞþXia j

1

4Cij

2e�2igij

ðe�2ioRt�1Þ

24

35

n HijZQ ,Hij

DQ

h iþ

1

16pio2

Xia j

bijðCij1�Cij

�1Þ

Xia j

Cij1e�igij

ðe�ioRt�1ÞþXia j

Cij�1eigij

ðeioRt�1Þ

24

35

n HijZQ ,Hij

DQ

h iþ

1

64o2R

Xþ1m ¼ �1

e�imgijXia j

bij

Xþ2

n ¼ �2

am�nCijn

Cij�2e2igij

�Cij2e�2igij

�Cij1e�igij

þCij�1eigij

h in

eð2�mÞioRt

ð2�mÞ

�Hij

ZQ ,HijDQ

h iþ

1

64o2R

Xþ1m ¼ �1

e�imgijXia j

bij

Xþ2

n ¼ �2

am�nCijn

Cij�2e2igij

�Cij2e�2igij

�Cij1e�igij

þCij�1eigij

h in�1

ð2�mÞ

�Hij

ZQ ,HijDQ

h i

þ1

64o2R

Xþ1m ¼ �1

e�imgijXia j

bij

Xþ2

n ¼ �2

am�nCijn



Cij�2e2igij

�Cij2e�2igij

�Cij1e�igij

þCij�1eigij

h inðe�imoRt�1Þ

m

�Hij

ZQ ,HijDQ

h iþL2ð0Þ ma0,ma2,ma�n:

ð80Þ

3. Numerical analysis of BABA

For m¼1, consider a system of 2 spins. Only DQ terms areconsidered for the functions L1(t) and L2(t). We consider thesimple case where the rotations are: aij ¼ bij

¼ gij ¼ 0.The coefficients Cij

n are C1 ¼�1=ffiffiffi3p

sinYcosYe�iF, C�1¼0,C2 ¼ 1=2

ffiffiffi6pðsin2Ye�2iFÞ and C�2¼0. For example, with Y¼ p=4

and F¼0, the coefficients are C1 ¼�1=2ffiffiffi3p

and C2 ¼ 1=4ffiffiffi6p

. Thefunctions L1(t) and L2(t) are:

L1ðtÞ ¼1

8ffiffiffi2p

oR

i�1

pffiffiffi2p

� ðe�ioRt�1ÞðIij

YY�IijXXÞþL1ð0Þ ð81Þ

and

L2ðtÞ ¼1

768

�1

2ðe�2ioRt�1Þþðe�ioRt�1Þ

�þ

�

þi

192po2R

1

8ffiffiffi2p ðe�2ioRt�1Þ�ðe�ioRt�1Þ

�)ðIij

YY�IijXXÞ

�1

1536o2R

ð1�1

2ffiffiffi2p ÞðeioRtþe�ioRt�2Þ

( )ðIij

YY�IijXXÞþL2ð0Þ

ð82Þ

or writing these functions in terms of the rotor period, we have

L1ðtÞ

bijtR¼ BABA1ðjÞ ¼

1

16ffiffiffi2p

pi�

1

pffiffiffi2p

� ðe�i2pj�1ÞðIij

YY�IijXXÞ, ð83Þ

L2ðtÞ

b2ijt2

R

¼ BABA2ðjÞ ¼1

768

�1

2ðe�4pij�1Þþðe�i2pj�1Þ

��

þi

768p3

1

8ffiffiffi2p ðe�i4pj�1Þ�ðe�i2pj�1Þ

� ðIij

YY�IijXXÞ

�1

6144p2ð1�

1

2ffiffiffi2p Þðei2pjþe�i2pj�2Þ

� ðIij

YY�IijXXÞ, ð84Þ

where the variable is chosen to be a dimensionless numberj¼ t=tR. BABA functions (real, imaginary, and absolute parts)are plot versus the dimensionless number.

3.1. Analysis of the figures

The plots of Figs. 4a and b, respectively, show the dimension-less functions L1 and L2 for BABA pulse sequence versus thedimensionless number j. Fig. 4c shows the plot of both functionsL1(t) and L2(t) versus j. Due to the complexity of these functions(L1(t) and L2(t)), real, imaginary, and absolute parts are plottedseparately as function of j. These functions depend on the DQterms. Therefore, the study of the amplitude of DQ terms can beconsidered as a viable approach for controlling the complex spindynamics of a spin system evolving under the dipolar interactionof BABA pulse sequence. The plot can be considered as a quan-titative representation of the amplitude of the DQ coherence as afunction of j. The size of BABA1ðjÞ ¼L1ðtÞ=bijtR determine theamplitude of the DQ coherence, which indicates the degree ofefficiency of the scheme. Figs. 4a–c show, respectively, the graphsof the functions L1ðtÞ=bijtR ¼ BABA1ðjÞ, L2ðtÞ=b2

ijt2R ¼ BABA2ðjÞ

and (BABA1(j),BABA2(j)) as a function of j¼ t=tR. A closer lookat Fig. 3c shows that the magnitude of BABA2(j) is small com-paratively to the magnitude of BABA1(j), i.e. 9L2ðtÞ=b2

ijt2R9o9L1ðtÞ=

bijtR9 as expected. As a result, L2(t) will be less useful in many cases.We can also observe that all graphs are strictly monotonous.

This tells us that, the strength of the DQ terms increase continuouslywith time and no decoupling conditions occur in the BABA pulsesequence.

3.1.1. Broadband BABA 2 acting on multiple of two rotor periods

(2tR)

In this article, the broadband BABA2 pulse sequence [8] actingon multiple of two rotor periods 2tR is constructed from eighth rfpulses, with flip angles {p/2, p/2, y, p/2} and rf phases{X,�X,Y,�Y,�X,X,�Y,Y}. Consider the BABA2 pulse sequence inthe following representation

As described previously for BABA, the toggling frame in thisnew scheme is written similarly as in Eq. (33), in the followingform

~HDðtÞ ¼HYY ðtÞyYY ðtÞþHXXðtÞyXXðtÞ ð85Þ

with HaaðtÞ ¼ 12

Pia j

oijDðtÞH

ijaa as before (34).

The function y (t) is represented as follows:

ð86Þ

ð87Þ

Substituting Eqs. (86) and (87) into Eq. (85), we have thefollowing

~HDðtÞ ¼HYY ðtÞy2Y ðtÞþHXXðtÞy

2XðtÞ ¼HYY ðtÞy

2þHXXðtÞð1�yÞ2: ð88Þ

The Fourier coefficients of the time-dependent function y(t)expressed in Eq. (36) are now

ak ¼1

2tR

Z 2tR

0yðtÞeikoRt dt: ð89Þ

The Fourier coefficients aXk in the ’’X-direction’’ of the above

picture are computed as follows:

aXk ¼

1

2tR

Z tR=2

0eikoRt dtþ

Z 3ðtR=2Þ

tR

eikoRt dt

" #ð90-aÞ

and

aX0 ¼

1

2tR

Z tR=2

0dtþ

Z 3ðtR=2Þ

tR

dt

" #, ð90-bÞ

which give

aXk ¼

1

ik2p eikp=2�1þe�ikp=2�eikph i

ð91-aÞ

and

aXo ¼

12: ð91-bÞ



Similarly, in the ’’Y-direction’’ of the above picture, the Fouriercoefficients aY

k are computed as follows:

aYk ¼

1

2tR

Z tR

tR=2eikoRt dtþ

Z 2tR

3tR=2eikoRt dt

" #ð92-aÞ

and

aY0 ¼

1

2tR

Z tR

tR=2dtþ

Z 2tR

3tR=2dt

" #, ð92-bÞ

which give

aYk ¼

1

ik2p eikp�e�ikp=2þ1�eikp=2h i

ð93-aÞ

and

aYo ¼

12: ð93-bÞ

Finally, the following result is obtained

aXo ¼ aY

0 ¼ a0 ¼12 ð94-aÞ

and

aXk ¼�aY

k ¼ ak: ð94-bÞ

The toggling frame Eq. (88) can be evaluated as follows:

~HDðtÞ ¼1

2

Xia j

oijDðtÞ

Xn

ane�inðoRtþgij

Þ

!2

HijYY

þ1

2

Xia j

oijDðtÞ 1�

Xn

ane�inðoRtþgijÞ

!2

HijXX ð95Þ

and substituting Eqs. (91), (93) and (94) into (95), we have

~HDðtÞ ¼1

2

Xia j

oijDðtÞ a0þ

Xka0

ake�ikðoRtþgijÞ

!2

HijYY

þ1

2

Xia j

oijDðtÞ 1�a0�

Xka0

ake�ikðoRtþgijÞ

!2

HijXX

Fig. 4



¼1

2

Xia j

oijDðtÞ a2

0þXka0

ake�ik oRtþgijð Þ

!224

35ðHij

XXþHijYY Þ

þ1

2

Xia j

oijDðtÞ2a0

Xka0

ake�ikðoRtþgijÞðHijYY�Hij

XXÞ

¼ ~HZQ ,DðtÞþ ~HDQ ,DðtÞ: ð96Þ

Similar results than Eqs. (55) and (55-a, b, c) are obtained forthe double-quantum toggling Hamiltonian, but with change ofcoefficients in the AHT term as follows:

~HDQ ¼3

2ffiffiffi6p

Xia j

bij

Xn ¼ �2

þ2

a�nCijn

8<:

9=;|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

AHT ¼ oijD

ðIijYY�Iij

XXÞ

þX

m ¼ �1

þ1

e�imðoRtþgijÞfg3

2ffiffiffi6p

Xia j

bij

Xn ¼ �2

þ2

am�nCijn

8<:

9=;|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

fooijD4mg

ðIijYY�Iij

XXÞ:

ð97Þ

From Eqs. (91) and (94), we have

ak ¼1

ikp cosðkp2Þ�

1

2ð1þeikpÞ

�ð98Þ

which can also be written as

a�n ¼�1

inp cos np2

� ��

1

2ð1þe�inpÞ

�ð99Þ

and evaluated numerically

a�2 ¼1

pi, ð100-aÞ

a2 ¼�1

pi, ð100-bÞ

and

a�1 ¼ a1 ¼ 0: ð100-cÞ

Therefore, the first order contribution to the Floquet–Magnusexpansion F1 gives

F1 ¼~H

0¼

3

2iffiffiffi6p

p

Xia j

bij Cij2ða

ij,bijÞe�2igij

�Cij�2ða

ij,bijÞeþ2igij

h iðIij

YY�IijXXÞ:

ð101Þ

The constants Cij2ða

ij,bijÞ and Cij

�2ðaij,bijÞ are given by Eq. (173).

L1(t) is computed by using Eq. (10) and the result is similar to the

solution of BABA (Eq. (64)) by substituting coefficients Cij1ða

ij,bijÞ

and Cij�1ða

ij,bijÞ, respectively, by Cij

2ðaij,bijÞ and Cij

�2ðaij,bijÞ. We

obtain

L1ðtÞ ¼1

4ffiffiffi6p

Xia j

Cij�1ða

ij,bijÞeigij 1

ioR

� ðeioRt�1Þ

�ðIij

XXþ IijYY�2Iij

ZZÞ

þ1

4ffiffiffi6p

Xia j

�Cij1ða

ij,bijÞe�igij 1

ioR

� ðe�ioRt�1Þ

�ðIij

XXþ IijYY�2Iij

ZZÞ

þ1

4ffiffiffi6p

Xia j

�Cij2ða

ij,bijÞe�2igij 1

2ioR


þCij�2ða

ij,bijÞe2igij 1

2ioR

� ðe2ioRt�1Þ

�ðIij

XXþ IijYY�2Iij

ZZÞ

�X

m ¼ �1

þ1 1


3

2ffiffiffi6p

Xia j

bije�imgij

Xn ¼ �2

þ2

am�nCijn

24

35

ðIijYY�Iij

XXÞ: ð102Þ

The second order term F2 due to the contribution of the Floquet–Magnus expansion can also be easily derived from Eq. (78) bymaking a similar substitution as above and we obtain

F2 ¼�tR

64p2

Xia j

bijðCij2�Cij

�2Þ Cij�1ða

ij,bijÞeigij

�Cij1ða

ij,bijÞe�igij

h iHij

ZQ ,HijDQ

h i

�tR

32p2

Xia j

bijðCij2�Cij

�2Þ �Cij2ða

ij,bijÞe�2igij

þCij�2ða

ij,bijÞe2igij

h iHij

ZQ ,HijDQ

h i:

ð103Þ

The second order term L2(t) due to the contributions of theFloquet–Magnus expansion can also be calculated using Eq. (12),but it is less useful. We obtain the following

L2ðtÞ ¼�1

32o2R

Xia j

bij

Xn ¼ �2

2

Cijn

!24

� �X

m ¼ �1

þ1 1

m

Xia j

bije�imgij

Xn ¼ �2

þ2

am�nCijn

24

35

0@

1A35

�1

ðnþmÞ

�Hij

ZQ ,HijDQ

h iþ�1

32o2R

Xia j

bij

Xn ¼ �2

2

Cijn

!24

� �X

m ¼ �1

þ1 1

m

Xia j

bije�imgij

Xn ¼ �2

þ2

am�nCijn

24

35

0@

1A35

��1

ðnþmÞe�ðnþmÞioRt

�Hij

ZQ ,HijDQ

h i

þ�1

32o2R

Xia j

bij

Xn ¼ �2

2

Cijn

!24

� �X

m ¼ �1

þ1 1

m

Xia j

bije�imgij

Xn ¼ �2

þ2

am�nCijn

24

35

0@

1A35

�1

nðe�inoRt�1Þ

�Hij

ZQ ,HijDQ

h i

þ1

16pio2

Xia j

bijðCij2�Cij

�2ÞXia j

1

4Cij�1eigij

ðeioRt�1Þ

24

þXia j

1

4Cij

1e�igij

ðe�ioRt�1Þ

35 Hij

ZQ ,HijDQ

h i

þ1

16pio2

Xia j

bijðCij2�Cij

�2ÞXia j

Cij2e�2igij

ðe�2ioRt�1Þ

24

þXia j

Cij�2e2igij

ðe2ioRt�1Þ

35 Hij

ZQ ,HijDQ

h i

þ1

64o2R

Xm ¼ �1

þ1

e�imgijXia j

bij

Xn ¼ �2

þ2

am�nCijn

� Cij�1eigij

�Cij1e�igij

�Cij2e�2igij

þCij�2e2igij

h i eð2�mÞioRt

ð2�mÞ

�Hij

ZQ ,HijDQ

h i

þ1

64o2R

Xm ¼ �1

þ1

e�imgijXia j

bij

Xn ¼ �2

þ2

am�nCijn

� Cij�1eigij

�Cij1e�igij

�Cij2e�2igij

þCij�2e2igij

h i�1

ð2�mÞ

�Hij

ZQ ,HijDQ

h i

þ1

64o2R

Xm ¼ �1

þ1

e�imgijXia j

bij

Xn ¼ �2

þ2

am�nCijn

� Cij�1eigij

�Cij1e�igij

�Cij2e�2igij

þCij�2e2igij

h i�ðe�imoRt�1Þ

m

�Hij

ZQ ,HijDQ

h ima0, ma2, ma�n: ð104Þ



3.1.2. Comparison of both BABA and BABA 2 pulse sequences

The amplitude of the BABA and BABA2 DQ Hamiltoniandepend on the angle g, a fact which is supposed [5,8] to lead toa comparatively low efficiency for multiple quantum excitation inpowder samples, since most molecular orientations experiencepoor dipolar recoupling. Nevertheless, not only is the functionaldependence on g important, but also the strength of the DQHamiltonian that is tailored from the secular dipolar Hamiltonianby these BABA sequences. It is remarkable that, based on thestrength of the dipolar DQ Hamiltonian, comparing Eqs. (56) and(101), the BABA2 pulse sequence should give more efficientrecoupling than the BABA pulse sequence.

3.1.3. Application of the Floquet–Magnus expansion to C7

The structure of the C7 cycles is shown in Fig. 3. In thisrecoupling sequence, seven RF cycles are timed to occupy tworotational periods, 2tR [9]. The overall phase of each consecutivecycle, j, is incremented in steps of 2ðp=7Þ. The RF amplitudethroughout the sequence satisfies the condition os

rf ¼ 7oR. Thetwo rotational periods, each of duration 2p=oR, are subdividedinto seven equal segments of duration tC ¼ 2tR=7. In this article,each element Cj is comprised of two RF pulses, both with flipangle oIStc=2¼ 2p, but with RF phases differing by p, i.e.Cj¼2pj2pjþp in conventional pulse sequence notation. Thetheory behind the C7 cycles is summarized as: a sequence of nradiofrequency cycles Cjp is considered in the general situation,with p¼0, y, n�1. p is timed to occupy N rotational periods.Each cycle C has duration tC ¼NtR=n and an overall RF phasejp ¼ 2pp=n. With the initial RF sequence starting at a time t0,then the pth RF cycle Cjp runs between time points t0

p and t0pþ1,

where t0p ¼ t0þptc . In this scheme of C7 sequences, the general

form of the Hamiltonian of the spin-pair system expressed in theinteraction frame of the RF field is given by

~HDðtÞ ¼X

Q

Xlmlm

~oQlmlmðt�t0

pÞeiðNm�mÞjp TQ

lm, ð105Þ

with

jp ¼ 2p p

n: ð106Þ

Q represents the indexes of different types of interaction (Q¼ i,j or ij), l is the rank of the interaction with respect to rotations ofthe sample, m is the spatial rotational component (m¼� l, y, l), lis the rank of the interaction with respect to rotations of the spinpolarizations and m is the spin rotational component (m¼�l, ..., l).The amplitudes of the interaction frame Hamiltonian terms aregiven in general by

~oQlmlmðtÞ ¼ imdl

m0ð�brf ðtÞÞoQl0lmeimoRðtþ t0Þ, ð107Þ

where dlm0 is a reduced Wigner function and brf is the overall spin

rotation angle induced by the RF field given for the specific sequenceCj ¼ 2pj2pjþp, by

brf ðtÞ ¼ 4p ttc

if 0rto tc

2ð108Þ

and

brf ðtÞ ¼ 4p 1�ttc

� iftc

2rtotc: ð109Þ

In this article, we are concerned with the spin-pair interacting(Q¼ ij), the DQ dipolar recoupling (l¼2, m¼2), and the directdipole–dipole coupling space of rank 2 (l¼2). We are workingwith the first type of g-encoded DQ recoupling sequence calledm¼1. In summary, the sequence is C71

2 with, the following

Q ¼ ij, l¼ 2, m¼ 72, l¼ 2, m¼ 71: ð110Þ

Substituting these new values into Eq. (107), we obtain

~oij2722mðtÞ ¼ i2d2

20ð�brf ðtÞÞoij202meimoRðtþ t0Þ ð111Þ

where

oij202m ¼ ðo

ij202�mÞ ¼

ffiffiffi6p

bijd20mðbPRÞe

�imgPR d2m0ðbRLÞ ð112Þ

and

bij ¼�m0gigj_

4pr�3ij

: ð113Þ

bij is the dipole–dipole coupling constant. bRL is the angle betweenthe spinning axis and the main field (bRL ¼ arctan

ffiffiffi2p

, magic-anglespinning), with

m¼ 0, oij2020 ¼o

i1020 ¼ 0: ð114Þ

bPR and gPR are Euler angles specifying the relative orientation ofthe principal axis frame of the dipolar interaction P and a rotor-fixed frame R. Both angles (bPR and gPR) are random variables in apowder sample. The Hamiltonian of the spin-pair system,expressed in the toggling frame of the RF field is written as

~HpðtÞ ¼X

ij

Xmm

~o ij2m2mðtÞe

i2pðNm�mÞpnTij2m: ð115Þ

The spin dynamics under the pulse sequence may also beinvestigated by using an alternative method such as the FT orFME, which have advantages in some circumstances. The calcula-tion of F1 (Eq. (11)) over the pth cycle (t0

p rtot0pþ1) for the first

order contribution to the FME leads to

F1p ¼1

tC

Z tC

0

~HpðtÞdt¼Xijm

ei2pðNm�mÞpnTij2mð

1

tCÞ

Z tC

0

~o ij2m2mðtÞdt,

ð116Þ

where

~oij2m2m ¼

1

tC

Z tC

0

~o ij2m2mðtÞdt: ð117Þ

Accounting for F1 over the whole sequence, we have

F1 ¼1

n

Xn�1

p ¼ 0

F1p ¼1

n

Xn�1

p ¼ 0

Xijmm

~oij2m2mTij

2mei2pðNm�mÞðp=nÞ

¼1

n

Xijmm

~o ij2m2mTij

2m

Xn�1

p ¼ 0

ei2pðNm�mÞðp=nÞ, ð118Þ

which produces

F1 ¼1

n

Xij21

~oij2221Tij

22

Xn�1

p ¼ 0

ei2pðN�2Þðp=nÞ

þ1

n

Xij�2�1

~oij2�22�1Tij

2�2

Xn�1

p ¼ 0

ei2pð�Nþ2Þðp=nÞ

¼1

7

Xij21

~oij2221Tij

22þ1

7

Xij�2�1

~oij2�22�1Tij

2�2: ð119Þ

We can evaluate ~oij2221 by using Eq. (117)

~oij2221 ¼

1

2tR

Z 2tR=7

0

~oij2221ðtÞdtþ

Z 4tR=7

2tR7

~oij2221ðtÞdt

"

þ

Z 6tR=7

4tR7

~oij2221ðtÞdtþ

Z 8tR=7

6tR7

~o ij2221ðtÞdt

#

þ1

2tR

Z 10tR=7

8tR=7

~o ij2221ðtÞdtþ

Z 12tR=7

10tR=7

~o ij2221ðtÞdt

"

þ

Z 14tR=7

12tR=7

~oij2221ðtÞdt

#: ð120Þ



With ~o ij2221ðtÞ obtained from Eq. (111)

~o ij2221ðtÞ ¼ i2d2

20ð�brf ðtÞÞoij2021eioRðtþ t0Þ

¼ 916sinð2bRLÞbij sin2

ðbrf ÞeioRteiðoRt0�gPRÞ sinð2bPRÞ, ð121Þ

where the relevant Wigner matrix elements (see Appendix A2)have been used. ~oij

2221ðtÞ Can be written in the following form forintegration purposes

~o ij2221ðtÞ ¼ Asin2

ðbrf ÞeioRt ¼�A

4ðe2ibrf þe�2ibrf�2Þeiort ð122Þ

with

A¼ 916sinð2bRLÞbije

iðoRt0�gPRÞ sinð2bPRÞ: ð123Þ

The result of each individual integral term of the Eq. (120) isZ 2tR=7

0

~o ij2221ðtÞdt¼

A7tC784

8peip=14þ i

780

� : ð124Þ

Using the fact that

sinð2arctanðxÞÞ ¼2x

1þx2, ð125� aÞ

i.e.

sinð2bRLÞ ¼ sinð2arctanffiffiffi2pÞ¼ 2

3

ffiffiffi2p

, ð125� bÞ

we have

A¼ 38

ffiffiffi2p

bijeiðoRt0�gPRÞ sinð2bPRÞ ð126Þ

and finally, we obtain

~o ij2221 ¼

343

520pffiffiffi2p ðeiðp=14Þ þ iÞbije

iðoRt0�gPRÞ sinð2bPRÞ ¼ ð ~oij2�22�1Þn:

ð127Þ

Lee et al. [9] derived a similar expression. Using the firstcontributions to the FME Eqs. (10)–(13) we obtain the followingresults

F1 ¼1

7

Xij

343ðiþeiðp=14ÞÞ

520pffiffiffi2p bije

iðoRt0�gPRÞ sinð2bPRÞTij22,

þ1

7

Xij

343ð�iþe�iðp=14ÞÞ


�iðoRt0�gPRÞ sinð2bPRÞTij2�2, ð128Þ

L1ðtÞ ¼1

7

Xij

21ffiffiffi2p

itC

64pbije

iðoRt0�gPRÞsinð2bPRÞ

�1

30e

i60p7

ttC�

1

26e

i52p7

ttC�e

i4p7

ttC

� Tij

22

þ1

7

Xij

343itC


iðoRt0�gPRÞsinð2bPRÞTij22

þ1

7

Xij

�21ffiffiffi2p

itC

64pbije�iðoRt0�gPRÞsinð2bPRÞ

�1

30e�i60p

7ttC�

1

26e�i52p

7ttC�e

�i4p7

ttC

� Tij

2�2

þ1

7

Xij

�343itC


�iðoRt0�gPRÞsinð2bPRÞTij2�2�tF1, ð129Þ

F2 ¼X

ij

�449A2Aei10ðp=7Þ

7200poR�

2pA2Aei4ðp=7Þffiffiffi2p

3549poR

þ152212pA2A

798525ffiffiffi2p

poR

�49iA2A

1365oR�pA2B2

7oR

�Tij

22,Tij2�2

h i

þX

ij

49A2

454272oRð1�ei8ðp=7ÞÞþ

343A2

p76050oRð1�ei4ðp=7ÞÞ

"

�7B2

4poRð4i

p7Þei4ðp=7Þ

�Tij

22,Tij2�2

h i

þX

ij

ip61A2

11760oRþ

7B2

1800poRð60i

p7Þei4ðp=7Þ

"

�7B2

1352poRð52i

p7Þei4ðp=7Þ�C:C

�Tij

22,Tij2�2

h iþ

1

2A2þ

49A

390pðiþeiðp=14ÞÞ

�Tij

22,L1ð0Þh i

�1

2An

2þ49A

390p ð�iþe�iðp=14ÞÞ

�Tij

2�2,L1ð0Þh i

ð130Þ

and

L2ðtÞ ¼X

ij

anijA2

2400ffiffiffi2p

o2R

ðe�i15oRt�1Þ

"

�3an

ijA2

5408ffiffiffi2p

o2R

ðe�i13oRt�1Þ�3an

ijA2

16ffiffiffi2p

o2R

ðe�ioRt�1Þ

#nTij

22Tij2�2

�X

ij

7ianijA2t

130ffiffiffi2p

oR

þA2B2t2

2þ

3ffiffiffi2p

aijA

163072o2R

ðei2oRt�1Þ

"

�3ffiffiffi2p

aijA

43904o2R

ðei14oRt�1Þ

#nTij

22Tij2�2

þX

ij

�ffiffiffi2p

aijA

878080o2R

ðe�i28oRt�1Þþ3ffiffiffi2p

aijA

2119936o2R

ðe�i26oRt�1Þ

"

þ3ffiffiffi2p

aijA

43904o2R

ðe�i14oRt�1Þ

#nTij

22Tij2�2

þX

ij

ffiffiffi2p

aijA

219520o2R

ðe�i14oRt�1Þ�

ffiffiffi2p

aijA

163072o2R

ðe�i12oRt�1Þ

"

þ49aijA

7800ffiffiffi2p

o2R

ðe�i15oRt�1Þ

#nTij

22Tij2�2

�X

ij

49aijA

6760ffiffiffi2p

o2R

ðe�i13oRt�1Þþ49aijA

260ffiffiffi2p

o2R

ðeioRt�1Þ

"

þB2A

14o2R

ðioRtÞeioRt

#nTij

22Tij2�2

þX

ij

B2A

6300o2R

ði15oRtÞei15oRt�B2A

4732o2R

ði13oRtÞe�i13oRt

" #

� Tij22,Tij

2�2

h i�C:C Tij

22,Tij2�2

h i�tF2

þ7A2t

2�

A

i120oRðei15oRt�1Þþ

A

i104oRðe�i13oRt�1Þ

�

� Tij22,L1ð0Þ

h iþ

A

4ioRðeioRt�1Þ

�Tij

22,L1ð0Þh i

þC:C Tij2�2,L1ð0Þ

h i,

ð131Þ


aij ¼ bijeiðoRt0�gij

PRÞ sinð2bPRÞ, ð132Þ

A¼ 38

ffiffiffi2p

aij, ð133Þ

A2 ¼1

7

Xij


520pffiffiffi2p aij, B2 ¼ An

2 ð134Þ

with

Tij22 ¼

1

2Iiþ Ijþ , ð135Þ

Tij2�2 ¼

1

2Ii�Ij� ð136Þ

and C.C. is the complex conjugated.



4. Numerical analysis of C7

The functions L1(t) and L2(t) written in terms of the rotorperiod (real, imaginary and absolute part) are plotted versus thedimensionless number j¼ t=tR. Assuming a system of two spins,we have:

F1

aij¼

1

7


520pffiffiffi2p Tij

22þ1

7

343ð�iþe�iðp=14ÞÞ

520pffiffiffi2p Tij

2�2, ð137Þ

L1ðtÞ

aijtR¼ C71ðjÞ ¼

3i

112ffiffiffi2p

p1

30ei30pj�

1

26ei26pj�ei2pj

�

þ49

520ffiffiffi2p

p2i

7�jðiþeðip=4ÞÞ

� �Tij

22

þ�3i

112ffiffiffi2p

p1

30e�i30pj�

1

26e�i26pj�e�i2pj

�

þ49

520ffiffiffi2p

p�2i

7�jð�iþeð�ip=4ÞÞ

� �Tij

2�2, ð138Þ

F2 ¼F2

a2ijtR¼�449A2Aei10ðp=7Þ

14400p2�

A2Aei4ðp=7Þffiffiffi2p

3549þ

76106A2A

798525ffiffiffi2p

�49iA2A

2730p �A2B2

14

�½Tij

22,Tij2�2�|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

DQ

þ49A2

908544p ð1�ei8ðp=7ÞÞþ343A2

p2152100ð1�ei4ðp=7ÞÞ

"

�7B2

8pi4

7

� ei4ðp=7Þ

�½Tij


DQ

þi61A2

23520þ

7B2

1800p i30

7

� ei4ðp=7Þ�

7B2

1352p i26

7

� ei4ðp=7Þ

" #

½Tij22,Tij

2�2�|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}DQ

�C:C:½Tij22,Tij


ð139Þ

and

L2ðtÞ

a2ijt

2R

¼ C72ðjÞ ¼1

4p2

A2

2400ffiffiffi2p ðei30pj�1Þ�

3A2

5408ffiffiffi2p ðe�i26pj�1Þ

�3A2


�nn½Tij


DQ

�1

4p2

7iA22pj130

ffiffiffi2p þ2A2B2p2j2þ

3ffiffiffi2p

A

163072ei4pj�1� �"

�3ffiffiffi2p

A

43904ei28pj�e�i28pj� �#

nn½Tij22,Tij


þ1

4p2

�ffiffiffi2p

A

878080ðe�i56pj�1Þþ

3ffiffiffi2p

A

2119936ðe�i52pj�1Þ

"

þ

ffiffiffi2p

A

219520ðe�i28pj�1Þ

#nn½Tij


DQ

þ1

4p2�

ffiffiffi2p

A

163072ðe�i24pj�1Þþ

49A


"

þ49A


�nn½Tij


DQ

�1

4p2

49A

260ffiffiffi2p ðei2pj�1Þþ

B2A

14ði2pjÞei2pj

þB2A

6300o2R

ði30pjÞei30pj

#½Tij


DQ

þ1

4p2�

B2A

4732ði26pjÞe�i26pj

�½Tij


DQ

�C:C:½Tij22,Tij


�jF2 , ð140Þ


aij ¼ bijeiðoRt0�gij

PRÞ sinð2bPRÞ, ð141Þ

A¼ 38

ffiffiffi2p

, ð142Þ

A2 ¼1

7


520pffiffiffi2p , ð143-aÞ

B2 ¼ An

2 ð143-bÞ

and C.C. means complex conjugate.

4.1. Analysis of the figures

A similar analysis previously done for the BABA sequence canalso be extended to the C7 sequence. Figs. 5a–c show respectively,the graphs of the functions L1ðtÞ=aijtR ¼ C71ðjÞ, L2ðtÞ=a2

ijt2R ¼

C72ðjÞ and (C71(j),C72(j)) as a function of j¼ t=tR. A closerlook at Fig. 5c shows that the magnitude of C72(j) is small incomparison to the magnitude of C71(j), i.e. L2ðtÞ=a2

ijt2RoL1ðtÞ=

aijtR as expected. As a result, L2(t) will be less useful in manycases. We can also observe that all graphs are strictly mono-tonous. This tells us that, the ’’weight’’ of DQ terms increasescontinuously with time and no decoupling condition occurs inthis scheme.

5. Comparison between BABA and C7 pulse sequences

For a direct comparison between BABA and C7, if we consideronly the central term corresponding to n0 ¼0, and neglect all otherterms, we have

Cij1ða

ij,bijÞ ¼ �1

2ffiffi2p Y20e�iaij

sin2bijð144Þ

and from an earlier report [24], Y20 ¼ 1=ffiffiffi6pð3cos2Y�1Þ with Y

being one of the two angles that specify the direction of theinternuclear vector in polar coordinates in the laboratory axissystem. For example, for a direction corresponding to Y¼0,JH

0

DQJBABAC0:056singij units and JH0

DQJC7C0:040 units (seeAppendix A4). Eq. (175) shows that the amplitude of the BABADQ Hamiltonian depends on the angle gij, a fact that is supposedto lead to a comparatively low efficiency for MQ excitation inpowder samples [5,8,9]. Instead, for C7 pulse sequence, Eqs. (128)and (174) show that, the g-angle encoding takes place in thephase of the DQ Hamiltonian, resulting in a low sensitivity to theorientation of the internuclear vectors with respect to the rotorleading to a high overall efficiency for [9,26–30]. It is well knownin the NMR community that gamma-angle dependent sequencesreach about 50% conversion efficiency, while gamma-angle inde-pendent sequences reach around 70% [5,31,32]. Even though thetheoretical efficiency of the gamma-independent sequences ishigher, the choice of one recoupling sequence in favor of anotheris not obvious. Therefore, it is essential to consider several otherfactors that lead to the excitation efficiency such as the magni-tude of the homonuclear dipole–dipole coupling constant or theCSA interaction [5,31] which is ignored in this article. In addition,this work presents only the analysis of the original C7 pulsesequence (C72

1). The basic construction principle of the original C7allows considerable freedom over the choice of the element Cj.For instance, the new element referred to POST-C7 [5,29] shows



significant improvement to the robustness of the scheme withrespect to chemical shift offsets and rf inhomogeneity.

For example, considering the strength of the DQ Hamiltonian (forexample, the central term, n0 ¼0), the norms of the DQ Hamiltoniansare related by JH

0

DQJBABAC1:4singijJH0

DQJC7. This shows that bothsequences have about the same efficiency, but with the strength ofthe DQ Hamiltonian of the BABA sequence proportional to sin gij.However, unlike the BABA pulse sequence, the C7 pulse sequence isextremely insensitive to chemical shifts and rf field errors [5,8,29].Furthermore, if all terms in Eq. (175) are taken into account, thedynamics are more complex in BABA for given values of the anglesthat specify the direction of the internuclear vector. This maydescribe not only the buildup but also the destruction of DQcoherence during a rotor period. This yields the strength of the DQHamiltonian of the BABA sequence relatively smaller than that ofthe C7 sequence. As stated above, several other factors need to beconsidered as well for the comparison of both pulse sequences.Despite the fact that the CSA is ignored in this article, it is importantto mention that the BABA pulse sequence is not really useful whenthe chemical shift anisotropies are big [5,8].

The efficiency of the recoupling pulse sequences is indicatedby double quantum terms. Here, the magnitudes L1ðtÞ=bijtR forBABA and L1ðtÞ=aijtR for C7 determine the amplitude of the DQcoherence. This tells us about the potential degree to which thesequence is efficient. Therefore, the functions L1(t) and L2(t) areuseful to study the spin dynamics and can be used as a viableapproach to compare recoupling sequences. We have extendedthe description of DQ dynamics during dipolar recoupling beyondthe first-order limit, which generally assumes chemical shiftanisotropy effects are negligible comparatively to large dipolarcouplings. For the first order terms, we have showed that C7 is a gindependent recoupling sequence, since the norm of its DQHamiltonian is g-independent (JH

0

DQJC7¼ 0:040bij sinð2bPRÞ).

6. Conclusion

In this work, we use the Floquet–Magnus expansion approach todescribe the spin dynamics in solid-state NMR. Whereas the AHTand FT Hamiltonians are in general connected with stroboscopic

Fig. 5



detection schemes, the FME instead provides, at least in principle,the option to evaluate the spin evolution between the time points ofdetection. To validate such a description, we used the examples ofthe dipolar interaction in a MAS with the BABA and the original C7pulse sequences.

In contrast to the usual approaches (AHT and FT) used tocontrol the dynamics of spin systems in solid-state NMR, the FMEscheme overcomes the limitation of the stroboscopic detectionschemes and simplifies calculations of ’’the points in between’’,which is complicated using the FT [18]. In the FME approach, evenwhen the first and second order F1 and F2 are identical to theircounterparts in AHT and FT, the Ln(t) (n¼1, 2, 3, y) functionsprovide a simple approach for evaluating the spin evolutionduring ’’the time in between’’ through the Magnus expansion ofthe operator connected to this part of the evolution. Ln(t)functions are connected to the appearance of features like spin-ning sidebands in MAS. The evaluation of Ln(t) operator functionsare useful especially for the analysis of the non-stroboscopicevolution. For example, in the case of C7, for non-stroboscopicdetection scheme, they can be used to estimate the intensity ofthe spinning sidebands manifold. Higher order effects such as (F2,Ln(t)), (F3, L3(t)) can also be evaluated using FME approach easierthan in the case of AHT or FT.

We made an attempt to sketch out the FME technique andmost of the conclusions derived from these pulse sequences canbe obtained from AHT. This is in agreement with the AHTproviding that the stroboscopic detection points is a particularcase of FME. The FME approach in solid-state NMR spectroscopyprovides new aspects not present in AHT and FT, namely theresults of Eqs. (10), (12)–(148). The fact that the Fn(n¼1, 2, y)expressions can also be obtained by AHT and FT makes the FMEapproach unique through its expressions for Ln(t). We derived theexpressions for L1(t) and L2(t) that are unique additions for AHTand FT. These functions are used to evaluate the spin behaviorduring or in between the RF pulses. Unlike in AHT where anevaluation of nested commutators and their painful double, tripleand multiple integrals are required to obtain respectively thesecond, third and higher correction terms of a Hamiltonian, inFloquet–Magnus expansion only an evaluation of nested commu-tators with a single integral is required.

The paper describes the time evolution of the spin system at alltimes and offers a way to handle other similar approaches. We hopeto approach other interesting problems that are amenable to thetreatment presented here such as multi-mode Hamiltonian, rota-tional-resonance recoupling, continuous wave (CW) irradiation on asingle species, DARR and MIRROR recoupling, simultaneous CWirradiation on two different spin species, phase-alternating (XiX)irradiation on a single spin species, CW irradiation on one and (XiX)irradiation on a second spin species, phase-modulated Lee-Goldburgdecoupling, C-type and R-type sequences, TPPM decoupling, etc.

We recognize that the choice of the pulse sequences (BABA,C7) used in this paper may not be ideal due to lengthy calcula-tions using FME comparatively to the choice of simple examplesoutlined in a recent report [7]. The possibility of enhanced FMEperformance certainly deserves further attention and additionalquantitative work would demonstrate the utility of the Floquet–Magnus expansion in solid-state NMR.

Acknowledgments

This work was supported by the French National ResearchAgency (ANR) under the )DESIRE* project. Eugene S. Manangaacknowledges also support from the Commissariat �a l’EnergieA-tomique, NEUROSPIN, France.

Appendix

A0.

Tð2Þ3 ¼X1

m ¼ 1

½Lm,Tð1Þ3�m� ¼ ½L1,T ð1Þ2 � ¼ ½L1,½L1,F1��, ð145Þ

L3

_

¼�F3�1

2ð L2,A½ �þ L2,F1½ �|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

L2 ,AþF1½ �

þ½L1,F2�Þ

þ1

12ð½L1,½L1,A��½L1,½L1,F1��|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}L1, L1,A½ �½ �þ L1, L1,�F1½ �½ �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

L1 , L1 ,A�F1½ �

Þ: ð146Þ

After integration, we obtain the following result

L3ðtÞ ¼ �tF3�1

2

Z t

0ð L2,AþF1½ �þ L1,F2½ �Þdt0

þ1

12

Z t

0L1, L1,A�F1½ �½ �dt’þL3ð0Þ ð147Þ

or

L3ðtÞ ¼1

2

Z t

0AþF1,L2½ �dt0 þ

1

2

Z t

0F2,L1½ �dt0

þ1

12

Z t

0L1, L1,A�F1½ �½ �dt0�tF3þL3ð0Þ: ð148Þ

We can write the following, as previously described [6]

F3 ¼O3ðTÞ

T, ð149Þ

where

O3ðTÞ ¼X2

j ¼ 0

Bj

j!

Z T

0SðjÞ3 ðtÞdt

¼B0

0!

Z T

0Sð0Þ3 ðtÞdtþ

B1

1!

Z T

0Sð1Þ3 ðtÞdtþ

B2

2!

Z T

0Sð2Þ3 ðtÞdt

¼�1

2

Z T

0Sð1Þ3 ðtÞdtþ

1

12

Z T

0Sð2Þ3 ðtÞdt: ð150Þ

Sð1Þ3 ¼W ð1Þ3 ðtÞþð�1Þð2ÞTð1Þ3 ðtÞ ¼W ð1Þ

3 ðtÞþTð1Þ3 ðtÞ ¼ ½L2,A�þ½L1,F2�þ½L2,F1�,

ð151Þ

Sð1Þ3 ¼ ½L2,AþF1�þ½L1,F2�, ð152Þ

Sð2Þ3 ¼W ð2Þ3 ðtÞþð�1Þð3ÞTð2Þ3 ðtÞ ¼W ð2Þ

3 ðtÞ�T ð2Þ3 ðtÞ

¼ ½L1,½L1,A��½L1,½L1,F1��|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}½L1 ,½L1 ,A�F1 ��

, ð153Þ

O3ðTÞ ¼1

2

Z T

0AþF1,L2½ �dtþ

1

2

Z T

0F2,L1½ �dtþ

1

12

Z T

0L1, L1,A�F1½ �½ �dt:

ð154Þ

Finally we obtain,

F3 ¼1

2T

Z T

0AþF1,L2½ �dtþ

1

2T

Z T

0F2,L1½ �dtþ

1

12T

Z T

0L1, L1,A�F1½ �½ �dt:

ð155-aÞ

We can also write the above Eq. (147) and (155) in thefollowing forms [7]

L3ðtÞ ¼

Z0

t

G3ðtÞdt�tF3þL3ð0Þ, ð155-bÞ

G3ðtÞ ¼� i2 HðtÞþF1,L2ðtÞ½ �� i

2 F2,L1ðtÞ½ �� 112 L1ðtÞ, L1ðtÞ,HðtÞ�F1½ �½ �,

ð155-cÞ



L3ðTÞ ¼L3ð0Þ ð155-dÞ

and

F3 ¼�i

2T

Z0

T

HðtÞþF1,L2ðtÞ½ �dt� i

2T

Z0

T

F2,L1ðtÞ½ �dt

�1

12T

Z0

T

L1ðtÞ, L1,HðtÞ�F1½ �½ �dt: ð155-eÞ

A1.

Z0

T

HZBdt¼

Z0

tR 1

2

Xia j

oijDðtÞa0

1ffiffiffi6p

0@

1A �

Xm ¼ �1

þ1 1


n3

2ffiffiffi6p

Xia j

bije�imgij

Xn ¼ �2

þ2

am�nCijn

24

351Adt

¼1

4ffiffiffi6p

Z0

tR Xia j

oijDðtÞ �

Xm ¼ �1

þ1 1


n3

2ffiffiffi6p

Xia j

bije�imgij

Xn ¼ �2

þ2

am�nCijn

24

351Adt

¼1

4ffiffiffi6p

Z0

tR Xia j

ðbij

Xn ¼ �2

2

Cijnða

ij,bij,gijÞe�inoRtÞ

�X

m ¼ �1

þ1 1


n3

2ffiffiffi6p

Xia j

bije�imgij

Xn ¼ �2

þ2

am�nCijn

24

351Adt

¼1

16

Xia j

ðbij

Xn ¼ �2

2

Cijnða

ij,bij,gijÞÞ �X

m ¼ �1

þ1 1

imoR

nXia j

bije�imgij

Xn ¼ �2

þ2

am�nCijn

24

35!Z

0

tR

e�inoRtðe�imoRt�1Þdt

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}� 0

¼ 0:

ð156Þ

Since

oR ¼2pN

tR, N¼ 0,1,2,. . . ð157Þ

Similarly,

Z T

0HmAdt¼

Z tR

0

Xþ1m ¼ �1


2ffiffiffi6p

Xia j

bij

Xþ2

n ¼ �2

am�nCijn

24

35

0@

1A

n1

4ffiffiffi6p Cij

�2ðaij,bijÞe2igij 1

2ioR

� ðe2ioRt�1Þ

�� dt

þ

Z tR

0

Xþ1m ¼ �1


2ffiffiffi6p

Xia j

bij

Xþ2

n ¼ �2

am�nCijn

24

35

0@

1A

n1

4ffiffiffi6p �Cij

2ðaij,bijÞe�2igij 1

2ioR

� ðe2ioRt�1Þ

�� dt

þ

Z tR

0

Xþ1m ¼ �1


2ffiffiffi6p

Xia j

bij

Xþ2

n ¼ �2

am�nCijn

24

35

0@

1A

n1

4ffiffiffi6p �Cij

1ðaij,bijÞe�igij 1

2ioR

� ðe2ioRt�1Þ

�� dt

þ

Z tR

0

Xþ1m ¼ �1


2ffiffiffi6p

Xia j

bij

Xþ2

n ¼ �2

am�nCijn

24

35

0@

1A

n1

4ffiffiffi6p Cij

�1ðaij,bijÞeigij 1

2ioR

� ðe2ioRt�1Þ

�� dt:¼ 0 ð158Þ

A2. Table of Wigner d-matrix elements for j¼1 and j¼2

djm0m ¼ ð�1Þm�m0dj

mm0 , ð159Þ

d20,0ðbÞ ¼

3cos2ðbÞ�1

2, ð160Þ

d20, 72ðbÞ ¼

ffiffi38

qsin2ðbÞ, ð161Þ

d20, 71ðbÞ ¼ 7

ffiffi38

qsinð2bÞ, ð162Þ

d220ð�brf Þ ¼

ffiffi38

qsin2ðbrf Þ, ð163Þ

d21,0ðbRLÞ ¼�d2

01ðbRLÞ ¼ �

ffiffi38

qsinð2bRLÞ, ð164Þ

d21, 71ðbÞ ¼

17cosðbÞ2

ð2cosðbÞ81Þ, ð165Þ

d22, 71ðbÞ ¼�

17cosðbÞ2

sinðbÞ, ð166Þ

d22, 72ðbÞ ¼

17cosðbÞ2

� 2

: ð167Þ

A3.

a. Second-rank spherical harmonics

Y272 ¼1

2sin2Ye72iF, ð168Þ

Y271 ¼�ð7sinYcosYe7 iFÞ, ð169Þ

Y20 ¼1ffiffi6p ð3cos2Y�1Þ: ð170Þ

b. Explicit expressions of the coefficients C ıjn

Cijnða

ij,bijÞ ¼ d2

0,nðyMÞX2

n’ ¼ �2

ð�1Þn0

Yij

2n’e�inaij

d2n�n’ðb

ijÞ

¼ d20,nðyMÞ½Y

ij2�2e�inaij

d2n,2ðb

ijÞ�Yij

2�1e�inaij

d2n,1ðb

ijÞ

þYij20e�inaij

d2n,0ðb

ijÞ�

þd20,nðyMÞ½�Yij

21e�inaij

d2n,�1ðb

ijÞþYij

22e�inaij

d2n,�2ðb

ijÞ�,

ð171Þ

Cij1ða

ij,bijÞ ¼

1ffiffiffi3p e�iaij

Y ij2�2d2

1,2ðbijÞ�Yij

2�1d21,1ðb

ijÞþYij

20d21,0ðb

ijÞ

h�Y21ijd2

ð1,�1ÞðbijÞþY22ijd2

ð1,�2ÞðbijÞ�, ð172Þ

Cij2ða

ij,bijÞ ¼

1ffiffiffi6p e�iaij

Y ij2�2d2

2,2ðbijÞ�Yij

2�1d22,1ðb

ijÞþYij

20d22,0ðb

ijÞ

h�Yij

21d22,�1ðb

ijÞþYij

22d22,�2ðb

ijÞ�: ð173Þ

Calculation of the magnitudes

a. The magnitude of the C7 average Hamiltonian

JH0

DQJC7 ¼343

520pffiffiffi2p j eiðp=14Þ þ i j bij sinð2bPRÞ



¼1

7

343

520p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þsin

p14

� �rbij sinð2bPRÞ

¼ 0:040bij sinð2bPRÞ: ð174Þ

b. The magnitude of the BABA average Hamiltonian using onlythe central term corresponding to n0 ¼0

J ~H0

DQJBABA ¼3

2ffiffiffi6p

p1

2ffiffiffi2p Y20bij sinðgijÞsinð2bPRÞ: ð175Þ

For a particular direction of the internuclear vector in polarcoordinates in the laboratory axis system corresponding forexample to Y¼0, and Y20 ¼ 2=

ffiffiffi6p

, we have

J ~H0

DQJBABA ¼3

2ffiffiffi6p

p1

2ffiffiffi2p

2ffiffiffi6p Y20bij sinðgijÞsinð2bPRÞ

C0:056bij sinðgijÞsinð2bPRÞ: ð176Þ

c. The magnitude of the BABA2 average Hamiltonian using onlythe central term corresponding to n0 ¼0

J ~H0

DQJBABA2 ¼3

2ffiffiffi6p

p1ffiffiffi6p Yij

20d220ðb

ijÞbij

¼3

2ffiffiffi6p

p1ffiffiffi6p Yij

20

ffiffiffi6p

4bijsinðgijÞsin2

ðbijÞ

¼3

8ffiffiffi6p

pYij

20bijsinðgijÞsin2ðbijÞ: ð177Þ

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