Analytical Descriptions of DEPT NMR Spectroscopy for ISn(I=1,S=1; n=1, 2, 3, 4) Spin Systems

Abstract

A DEPT pulse sequence is well-known 13C-detected, edited-pulse and cross polarization transfer NMR experiment which offers to selective detection capability of CH, CH2 and CH3 groups from each other. The product operator theory is widely used for analytical descriptions of the cross polarization transfer NMR experiments for weakly coupled spin systems. In this study, analytical descriptions of the DEPT NMR experiment have been presented for ISn(I = 1,S = 1; n = 1, 2, 3, 4) spin systems by using product operator theory. Then a theoretical discussion and experimental suggestions were made. It has been investigated that this experiment can be used to edit 14N sub-spectra of partly or full deuterated 14NDn (n=1, 2, 3, 4) groups.

Full text

Brazilian Journal of Physics, vol. 38, no. 3A, September, 2008 323
Analytical Descriptions of DEPT NMR Spectroscopy for ISn(I=1,S=1; n=1,2,3,4)Spin Systems
˙
Irfan S¸aka
Department of Physics, Faculty of Arts and Sciences,
Ondokuz Mayis University, 55139, Samsun, Turkey,
E-mail:isaka@omu.edu.tr
(Received on 18 March, 2008)
A DEPT pulse sequence is well-known 13C-detected, edited-pulse and cross polarization transfer NMR exper-
iment which offers to selective detection capability of CH, CH2and CH3groups from each other. The product
operator theory is widely used for analytical descriptions of the cross polarization transfer NMR experiments
for weakly coupled spin systems. In this study, analytical descriptions of the DEPT NMR experiment have been
presented for ISn(I=1,S=1; n=1,2,3,4)spin systems by using product operator theory. Then a theoretical
discussion and experimental suggestions were made. It has been investigated that this experiment can be used
to edit 14N sub–spectra of partly or full deuterated 14 NDn(n=1, 2, 3, 4) groups.
Keywords: NMR; DEPT; Product operator theory; Deuterated nitrogen groups
1. INTRODUCTION
Although nitrogen–14 (14N) isotope has a natural abun-
dance of 99.64%, the magnetic moment is lower than
13C¡γ13 C±γ14N∼
=1.8¢,1H¡γ1H±γ14N∼
=7.0¢and 2H
¡γ2H±γ14N∼
=2.2¢nuclei, and it is a spin-1 isotope. Be-
cause of rapid quadrapolar relaxation spectral line–widths of
14N NMR signals are very broad [1]. In order to overcome
those unwanted circumstances, cross polarization transfers are
made to increase signal–to–noise ratio in NMR. The cross po-
larization transfers from high magnetic moment of nuclei to
low magnetic moment of nuclei are routine ways to increase
sensitive enhancement for heteronuclear weakly coupled spin
systems in liquid–state NMR experiments [2–5]. Last decade,
the cross polarization transfers in solid compounds have also
become a useful technique to increase the sensitivity of nu-
clei [6–9]. The well–known cross polarization transfer meth-
ods are Distortionless Enhancement by Polarization Trans-
fer (DEPT) and Insensitive Nuclei Enhanced by Polarization
Transfer (INEPT). The DEPT pulse sequence, which offers to
selective detection capability of CH, CH2, CH3groups from
each other, is a 13C-detected, edited-pulse and cross polariza-
tion transfer NMR experiment [10-12].
The product operator theory as a quantum mechanical
method is widely used for analytical description of the cross
polarization transfers on weakly coupled spin systems in
liquid–state NMR having spin −1
/
2, spin-1 and spin–3/2 nu-
clei [10–19]. However, it has been proposed that product op-
erator theory can be used as a new approach for analytical
description of solid–state NMR experiments under magic an-
gle spinning (MAS) conditions [20,21]. A complete product
operator theory for IS (I=1
/
2,S=1) spin system and applica-
tion to DEPT–HMQC (Heteronuclear Multiple Quantum Cor-
relation) NMR experiment has been presented in our previ-
ous study [22]. Analytical descriptions of INADEQATE (In-
credible Natural Abundance Double Quantum Transfer Ex-
periment) and DQC (Double Quantum Correlation) NMR ex-
periments have been presented for two–spin–1 AX system by
Chandrakumar and co–workers [12, 23, 24].
In this study, product operator descriptions of DEPT NMR
experiment have been presented for weakly coupled ISn(I=1,
S=1, n=1, 2, 3, 4) spin systems. It has been found that the
DEPT NMR experiment can be used to edit 14N sub–spectra
for 14NDngroups if the experiment is performed for the sug-
gested edited-pulse angles.
2. THEORY
The product operator formalism is the expansion of the den-
sity matrix operator in terms of matrix representation of angu-
lar momentum operators for individual spins. For IS (I=1,
S=1) spin system, nine Cartesian spin angular momentum op-
erators for I=1 are EI,Ix,Iy,Iz,I2
z,[Ix,Iz]+,[Iy,Iz]+,[Ix,Iy]+
and ¡I2
x−I2
y¢[25]. Similarly, there are also nine Cartesian
spin angular momentum operators for S=1. So, 9 ×9=81
product operators are obtained with direct products of these
spin angular momentum operators for IS (I=1, S=1) spin sys-
tem. Depending on the pulse experiment, ¡I2
x−I2
y¢Cartesian
spin angular momentum operator is separated into two spin
angular momentum operators as I2
xand I2
y. In this case, there
should be 10 ×10 =100 product operators for this spin sys-
tem. In this study they are used in separated form.
In a liquid–state and solid–state (under MAS conditions)
pulse NMR experiments of weakly coupled ISnspin systems,
the total Hamiltonian consists of r.f. pulse, chemical shift and
spin–spin coupling Hamiltonians can be written as
H=ΩIIz+
n
∑
i=1
ΩSSiz +2π
n
∑
i=1
JiIzSiz.(1)
Time dependence of the density matrix is governed by
Liouville-von Neumann equation:
dσ
dt =i
~[σ,H](2)
When the Hamiltonian is time independent, the solution of
the Equation (2) is

324 ˙
Irfan S¸ aka
TABLE 1. The evolution of some product operators under the weak spin-spin coupling Hamiltonian for IS (I=1, S=1) spin system where
Ixz = [Ix,Iz]+,Iyz = [Iy,Iz]+,Ixy = [Ix,Iy]+,Sxz = [Sx,Sz]+,Syz = [Sy,Sz]+,Sxy = [Sx,Sy]+and S2
x−y=¡S2
x−S2
y¢short notations are used.
Product
Operator
The evolution under the HJ
IxS2
x1
2IxS2
x(c2J+1)−1
2IxS2
y(c2J−1) + 1
2IxS2
z(c2J−1) + 1
2¡IxzSxy +IySz¢s2J
IxS2
y1
2IxS2
y(c2J+1)−1
2IxS2
x(c2J−1) + 1
2IxS2
z(c2J−1)−1
2¡IxzSxy −IySz¢s2J
IyS2
x1
2IyS2
x(c2J+1)−1
2IyS2
y(c2J−1) + 1
2IyS2
z(c2J−1)−1
2¡IyzSxy −IxSz¢s2J
IyS2
y1
2IyS2
y(c2J+1)−1
2IyS2
x(c2J−1) + 1
2IyS2
z(c2J−1) + 1
2¡IyzSxy +IxSz¢s2J
IxSxz 1
2IxSxz (c2J+1)−1
2IyzSy(c2J−1) + 1
2¡IySx+IxzSyz¢s2J
IxSyz 1
2IxSyz (c2J+1) + 1
2IyzSx(c2J−1) + 1
2¡IySy−IxzSxz¢s2J
IySxz 1
2IySxz (c2J+1) + 1
2IxzSy(c2J−1)−1
2¡IxSx−IyzSyz¢s2J
IySyz 1
2IySyz (c2J+1)−1
2IxzSx(c2J−1)−1
2¡IxSy+IyzSxz¢s2J
IxzSxz 1
2IxzSxz (c2J+1)−1
2IySy(c2J−1) + 1
2¡IxSyz +IyzSx¢s2J
IxzSyz 1
2IxzSyz (c2J+1)−1
2IxSy(c2J−1) + 1
2¡IyzSy−IxSxz¢s2J
IxzSxy IxzSxy c2J−IxS2
x−ys2J
IyzSyz 1
2IyzSyz (c2J+1)−1
2IxSx(c2J−1)−1
2¡IySxz +IxzSy¢s2J
IyzSxy IyzSxy c2J−IyS2
x−ys2J
σ(t) = exp(−iHt)σ(0)exp(iHt).(3)
where σ(0) is the density matrix at t=0. After employing the
Hausdorff formula [15]
exp(−iHt)Aexp (iHt) = A−(it)[H,A] + (it )2
2! [H,[H,A]]
−(it)3
3! [H,[H,[H,A]]] + ··· ,
(4)
evolutions of the product operators under the r.f. pulse, chem-
ical shift and spin-spin coupling Hamiltonians can be easily
obtained. For IS(I=1,S=1)spin system the evolutions of
some product operators under the spin-spin coupling Hamilto-
nian (HJ=2πJIzSz)are known and they are given in follow-
ing equations [11, 12, 23, 24]:
Sx
HJt
−→ Sx+I2
zSx(c2J−1) + IzSys2J(5a)
Sy
HJt
−→ Sy+I2
zSy(c2J−1)−IzSxs2J(5b)
IxSy
HJt
−→ 1
2IxSy(c2J+1) + 1
2IyzSxz (c2J−1) +
1
2(IySyz −IxzSx)s2J(5c)
IxSz
HJt
−→ IxSzc2J+IyS2
zs2J(5d)
IxS2
z
HJt
−→ IxS2
zc2J+IySzs2J(5e)
IySz
HJt
−→ IySzc2J−IxS2
zs2J(5f)
IyS2
z
HJt
−→ IyS2
zc2J−IxSzs2J(5g)
IzS2
x
HJt
−→ 1
2IzS2
x(c4J+1)−1
2IzS2
y(c4J−1) + 1
2I2
zSxys4J(5h)
IzS2
y
HJt
−→ 1
2IzS2
y(c4J+1)−1
2IzS2
x(c4J−1)−1
2I2
zSxys4J(5i)
In these equations Iyz = [Iy,Iz]+,Ixz = [Ix,Iz]+,
Sxz = [Sx,Sz]+,Syz = [Sy,Sz]+and Sxy = [Sx,Sy]+short
notations are used. Evolutions of the nine Cartesian spin
angular momentum operators under the r.f. pulse and the
chemical shift Hamiltonians have been presented in our
previous works for spin–1 [22, 26]. For IS (I=1, S=1) spin
system, evolutions of some product operators under the
spin-spin coupling Hamiltonian are given in Table 1.
At any time during the NMR experiments, the ensemble
averaged expectation value of the spin angular momentum,
e.g. for Iy,is
My(t)∝hIyi≡Tr [Iyσ(t)].(6)
Where σ(t)is the density matrix operator calculated from
Equation (4) at any time. Since hIyiis proportional to the mag-
nitude of the y–magnetization, it represents the signal detected

Brazilian Journal of Physics, vol. 38, no. 3A, September, 2008 325
on y−axis. So, in order to estimate the free induction decay
(FID) signal of a multi-pulse NMR experiment, density matrix
operator should be obtained at the end of the experiment.
3. RESULTS
For the analytical descriptions of the DEPT NMR experi-
ment for ISn(I=1, S=1; n=1, 2, 3, 4) spin systems, the pulse
sequence given in Fig. 1 is used. As shown in this figure, the
density matrix operator at each stage of the experiment is la-
beled with numbers and 14N is treated as spin Iand 2H as
spin S.
Starting from the density matrix operator at thermal equi-
librium, one should apply the required Hamiltonians during
the pulse sequence and obtain the density matrix operator at
the end of the experiment. For multi–spin–1 systems, to fol-
low these processes by hand becomes too difficult. In order to
overcome this problem a home made computer program has
been written in Mathematica which is very flexible for imple-
mentation and evolutions of the product operators under the
Hamiltonians [27].
For the IS spin system, the density matrix operator at ther-
mal equilibrium is σ0=Sz. Then, the evolutions of density
matrices under the Hamiltonians for each labeled point are
obtained:
σ0
90◦
x(S)
−→ σ1=−Sy,(7)
σ1
2πJIzSzτ
−→ σ2=−Sy+I2
zSy+I2
zSyc2J+IzSxs2J,(8)
σ2
180◦
x(S);90◦
x(I)
−→ σ3=Sy−I2
ySy+I2
ySyc2J−IySxs2J.(9)
At above and following equations c2J=cos(2πJτ)and
s2J=sin(2πJτ). In density matrix operator, only the terms
with observable product operators are kept as they are the only
ones that contribute to the signal on y−axis detection. In the
last step
σ7=1
2IyS2
z(1+c2J)s2
2JsθsI+1
4IyS2
x(1+c2J)s2
2Js2θsI
+1
4IyS2
y(1−c2J)s2
2Js2θsI−1
4IyS2
z(1+c2J)s2
2Js2θsI
(10)
is found. At above and following equations snθ=sin(nθ),
cnθ=cos(nθ)and sI=sin(ΩIt). If the evolution period is set
to τ=1±(2J), there is not any observable term in Eq.(10). For
the choice of the evolution period as τ=1±(4J), we obtain
following expression for spin-I:
σ7=1
2IyS2
zsθsI+1
4IyS2
xs2θsI+1
4IyS2
ys2θsI−1
4IyS2
zs2θsI(11)
At any time during the experiment, the ensemble averaged
expectation value of the spin angular momentum, hIyi, is pro-
portional to the magnitude of the y–magnetization and
My(t)∝hIyi=Tr [Iyσ(t)] (12)
is written. It represents the free induction decay (FID) signal
of a multiple-pulse NMR on y-axis. Tr[IyO]values of observ-
able product operators, indicated by O, have been calculated
by a home made computer program in Mathematica and re-
sults are given in Table 2 for ISn(I=1, S=1; n=1, 2, 3, 4) spin
systems. Using Table 2,
My(t)∝hIyi(IS) = Tr[Iyσ7]=(2sθ+s2θ)sI(13)
is obtained for IS spin system.
For the IS2spin system, the density matrix at the thermal
equilibrium is σ0=S1z+S2z. At the end of the experiment
forty–one observable terms are obtained as shown in follow-
ing equation:
σ7=1
2ÃIyS2
1z+IyS2
2z−1
4IyS2
1zS2
2x−1
4IyS2
1zS2
2y
−1
4IyS2
1xS2
2z−1
4IyS2
1yS2
2z−3
2IyS2
1zS2
2z!sθsI
+1
4






IyS2
1x+IyS2
2x+IyS2
1y+IyS2
2y−IyS2
1z−IyS2
2z
−1
2IyS2
1xS2
2x−1
2IyS2
1yS2
2x−1
2IyS2
1xS2
2y−1
2IyS2
1yS2
2y
−1
2IyS2
1zS2
2x−1
2IyS2
1zS2
2y−1
2IyS2
1xS2
2z−1
2IyS2
1yS2
2z
+7
2IyS2
1zS2
2z






s2θsI
+1
8³IyS2
1zS2
2x+IyS2
1zS2
2y+IyS2
1xS2
2z+IyS2
1yS2
2z−2IyS2
1zS2
2z´c2θsθsI
+1
4³IyS2
1zS2
2x+IyS2
1zS2
2y+IyS2
1xS2
2z+IyS2
1yS2
2z−2IyS2
1zS2
2z´cθs2θsI
+1
8ÃIyS2
1xS2
2x+IyS2
1yS2
2x+IyS2
1xS2
2y+IyS2
1yS2
2y+IyS2
1zS2
2z
−IyS2
1zS2
2x−IyS2
1zS2
2y−IyS2
1xS2
2z−IyS2
1yS2
2z!c2θs2θsI
(14)
Using the Trace values in Table 2;

326 ˙
Irfan S¸ aka
FIG. 1: DEPT NMR pulse sequence for the cross polarization transfer from 2H (S=1) nuclei to 14N (I=1) nuclei. τ: Evolution period for
optimum polarization transfer, t: acquisition period, BB: Broad Band for decouple.
TABLE 2. Results of the Tr £IyO¤calculations for some of the observable product operators in
ISn(I=1, S=1; n=1, 2, 3, 4) spin systems (i=x,y,z;j=x,y,z;k=x,y,zandl=x,y,z).
Spin system Operator (O)Tr£IyO¤
IS IyS2
i4
IS2
IyS2
1i;IyS2
2j12
IyS2
1iS2
2j8
IS3
IyS2
1i;IyS2
2j;IyS2
3k36
IyS2
1iS2
2j;IyS2
1iS2
3k;IyS2
2jS2
3k24
IyS2
1iS2
2jS2
3k16
IS4
IyS2
1i;IyS2
2j;IyS2
3k;IyS2
4l108
IyS2
1iS2
2j;IyS2
1iS2
3k;IyS2
1iS2
4l;IyS2
2jS2
3k;IyS2
2jS2
4l;IyS2
3kS2
4l72
IyS2
1iS2
2jS2
3k;IyS2
1iS2
2jS2
4l;IyS2
1iS2
3kS2
4l;IyS2
2jS2
3kS2
4l48
IyS2
1iS2
2jS2
3kS2
4l32
My(t)∝hIyi(IS2) = Tr[Iyσ7] = (1+4cθ+c2θ)(2sθ+s2θ)sI(15)
is found for the IS2spin system.
Applying the same procedure for the IS3and IS4spin sys-
tems, as one can guess huge amount of observable terms are
obtained at the end of the DEPT experiment by using the com-
puter program. Then, replacing the Trace values of observable
terms in Table 2 we obtain
My(t)∝hIyi(IS3) = 3
4(1+4cθ+c2θ)2(2sθ+s2θ)sI,(16)
My(t)∝hIyi(IS4) = 1
2(1+4cθ+c2θ)3(2sθ+s2θ)sI.(17)
4. DISCUSSION
Tr[Iyσ7]values obtained in Section 3 for IS,IS2,IS3and
IS4spin systems represent FID signals of DEPT NMR exper-
iment for 14ND, 14 ND2,14ND3and 14ND4groups, respec-
tively. These Tr[Iyσ7]values for 14NDn(n=1, 2, 3, 4) spin
systems can be generalized as following
hIyi¡14NDn¢=n
2n−1(1+4cθ+c2θ)n−1(2sθ+s2θ)sI.(18)
The Tr[Iyσ7]values can be normalized by multiplication
with 3±(Tr(E)). Here Eis the unity product operator for the
corresponding spin system. Then, the normalized FID values
become as follows:

Brazilian Journal of Physics, vol. 38, no. 3A, September, 2008 327
FIG. 2: The relative signal intensity plots of DEPT NMR spectroscopy for 14NDngroups as functions of θ.
hIyi¡14ND¢=1
3(2sθ+s2θ)sI,(19)
hIyi¡14ND2¢=1
9(1+4cθ+c2θ)(2sθ+s2θ)sI,(20)
hIyi¡14ND3¢=1
36 (1+4cθ+c2θ)2(2sθ+s2θ)sI,(21)
hIyi¡14ND4¢=1
162 (1+4cθ+c2θ)3(2sθ+s2θ)sI.(22)
These normalized Tr[Iyσ7]values can be also written in a
generalized form as
hIyi¡14NDn¢=n
2n−13n(1+4cθ+c2θ)n−1(2sθ+s2θ)sI.
(23)
TABLE 3. The real relative signal intensities of DEPT NMR
experiment of 14NDngroups for several pulse angles of θ.
Spin System θ=60◦θ=90◦θ=120◦
IS(14ND) 3√3
22√3
2
IS2(14ND2)15√3
40−3√3
4
IS3(14ND3)225√3
32 027√3
32
IS4(14ND4)375√3
32 0−27√3
32
The plots of the normalized FID functions are presented in
Fig. 2. As seen in this Figure, the relative signal intensities of
14ND, 14 ND2,14ND3and 14ND4groups vary as functions of
θ. The real relative signal intensities can be found from Eq.
(18) for 14ND, 14 ND2,14ND3and 14ND4groups for several
pulse angles which are given in Table 3. As seen in Fig. 2 and
Table 3, when the experiment is performed for the angle of
60 ˚ , all groups will give positive signal. For the pulse angle
of 90 ˚ , only 14ND groups will be observed giving positive
signal. For the pulse angle of 120 ˚ , 14ND and 14ND3groups
will give positive signals and 14ND2and 14ND4groups will
give negative signals. 14 ND3group can be easily separated by
comparison of spectra for θ=90 ˚ and θ=120 ˚ . To selectively
enhance of 14ND2and 14 ND4groups, one might collect FIDs
at θ=60 ˚ and θ=120 ˚ and take the linear combinations:
IND2=FID(60◦) + 5FID(120◦) = 0,
IND4=FID(60◦) + 5FID(120◦) = 15√3
2.
According to these results, deuterated nitrogen groups can
be separated from each other if the experiment is performed
for the angles of 60 ˚ , 90 ˚ and 120 ˚ .
Studies on full or partially deuterated nitrogen molecules
by using liquid- and solid-state NMR spectroscopies offer
new features in biological and material science [28–32]. In
determination of deuteration degree, it is of interest to de-
termine whether the ammonium ions (14NH+
4)are trans-
ferred as a whole leading to 14ND+
4or partially deuterated
14ND3H+,14 ND2H+
2and 14NDH+
3groups. In addition, a
sample can contain some deuterated nitrogen groups such as

328 ˙
Irfan S¸ aka
14ND4−,14 ND+
3−,14ND2H+−,14 ND2−or 14NDH−con-
taining molecules. As a result, a DEPT NMR experiment can
be employed for determination of deuteration degree of ni-
trogen groups in molecules if one uses the suggested angels
found in this study.
5. CONCLUSION
The cross polarization transfers between nuclei became a
useful technique to increase NMR signal intensity in both
liquid– and solid–state NMR experiments. In this study, ana-
lytical descriptions of DEPT NMR experiment have been pre-
sented for weakly coupled ISn(I=1; S=1; n=1, 2, 3, 4) spin
systems by using product operator theory. Thus a theoreti-
cal discussion and experimental suggestions for DEPT NMR
spectroscopy have been performed in order to edit 14N signals
of 14ND, 14 ND2,14ND3and 14ND4groups.
Acknowledgments:
I would like to thank Dr. Sedat G¨
um¨
us¸ for the helpful sug-
gestions about the Mathematica program.
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