1. Introduction
Polarization transfer from high natural abundance nucleus to low natural abundance nucleus is widely used for heteronuclear weakly coupled spin systems in liquid–state NMR experiments [1-3]. The most common examples for the polarization transfer experiments are Distortionless Enhancement by Polarization Transfer (DEPT) and Insensitive Nuclei Enhanced by Polarization Transfer (INEPT). They both are used to increase sensitive enhancement of 13C spectra from spin-1/2 or spin-1 nucleus [4,5]. In order to resolve the chemical shift and spin-spin coupling parameters along the two different axes heteronuclear 2D J-Resolved NMR spectroscopy is used. Sometimes, spectral assignments of 2D J-Resolved NMR spectra become too difficult due to complex overlapping spectra. In order to overcome this problem, 2D J-INEPT experiment [6], which is the combination of 2D J-Resolved and INEPT NMR experiments, can be used.
As NMR is a quantum mechanical phenomenon, the product operator theory as a quantum mechanical method is widely used for the analytical description of multipulse NMR experiments on weakly coupled spin systems in liquids having spin-
and spin-1 nuclei [7-18]. Analytical description of polarization transfer in INEPT experiment using product operator formalism has been presented for IS and IS2 (I = 1/2 and S = 1) spin systems [12]. So far, the product operator description of 2D J-INEPT NMR experiment is not investigated for heteronuclear weakly coupled ISn (I = 1/2; S = 1; n = 1, 2, 3) spin systems.
In this study, by using product operator formalism, analytical description of 2D J-INEPT NMR experiment is presented for heteronuclear weakly coupled ISn (I = 1/2; S = 1; n = 1, 2, 3) spin systems. Then, using the obtained theoretical results and a Maple program, the simulated spectra of the experiment are obtained for CD, CD2 and CD3 groups. Simulated spectra of 2D J-INEPT NMR spectroscopy are explained in detail for CDn groups. It is shown that this experiment can be used for the polarization transfer and J-resolved spectral assignment of CDn groups in complex molecules.
2. Theory
The product operator theory is the expansion of the density matrix operator in terms of matrix representation of angular momentum operators for individual spins. For IS (I = 1/2, S = 1) spin system, four Cartesian spin angular momentum operators for I = 1/2; EI, Ix, Iy, Iz and nine Cartesian spin angular momentum operators for S=1; ESSx, Sy, Sz, 
, 
,
can be easily found [19]. So, 
product operators are obtained with direct products of these angular momentum operators for IS (I = 1/2, S = 1) spin system.
Time dependency of the density matrix is given by [11]
(1)
where H is the total Hamiltonian which consists of radio frequency (r.f.) pulse, chemical shift and spin-spin coupling Hamiltonians and s(0) is the density matrix at t = 0. After employing the Hausdorff formula [11]
(2)
evolutions of product operators under the r.f. pulse, chemical shift and spin-spin coupling Hamiltonians can easily be obtained [7,11,13,16]. A complete product operator theory for IS (I = 1/2, S = 1) spin system and its application to some NMR experiments are presented elsewhere [16-18].
At any time during the experiment, the ensemble averaged expectation value of the spin angular momentum, e.g. for Iy, is
(3)
where 
is the density matrix operator calculated from Eq. (6) at any time. As 
is proportional to the magnitude of the y-magnetization, it represents the signal detected on y-axis. So, in order to estimate the free induction decay (FID) signal of a multiple-pulse NMR experiment, density matrix operator should be obtained at the end of the experiment.
3. Results and Discussion
In this study, the product operator formalism is used for the analytical description of 2D J-INEPT NMR experiment. Pulse sequence of 2D J-INEPT, shown in Figure 1, is used [6]. The density matrix operator at each stage of the experiment is labelled with numbers. 13C is treated as spin I and D (2H) as spin S. For the analytical descriptions of the experiment, we have written a computer program in Mathematica which is very flexible for the implementation and the evolutions of the product operators under the Hamiltonians [20].
3.1. IS Spin System
is the density matrix operator at thermal equilibrium for IS spin system. Evolutions of density matrix operator for each labelled point are obtained:
(4)
(5)
(6)
(7)
(8)
At the end of the experiment we obtain
(9)
In Equation (9), 
, 
, 
and
. In density matrix operator theory, only the last term of Equation (9) contributes to the signal as acquisition is taken along y-axes. It is necessary to obtain the 
values of observable product operators indicated by O. For ISn (I = 1/2, S = 1; n = 1, 2, 3) spin systems, 
values of all the observable product operators can be found elsewhere [16].
Using 
values and making trigonometric
Figure 1. Pulse sequence of 2D J-INEPT NMR experiment. Optimum t value is
. BB: Broad band decoupling [6].
expansions,
(10)
is obtained. This equation shows that the spin–spin coupling information appears on F1 axis and represents two coherences for I nucleus with phase of
. Therefore, they give doublets signals with opposite direction and no signal for central peak. The signals coordinate are
, 
and 
with intensity distribution of –1:0:1, respectively.
3.2. IS2 Spin System
For IS2 spin system, 
is the density matrix operator at thermal equilibrium:
(11)
The density matrix operator at the end of the experiment is
(12)
Using 
values;
(13)
is obtained. This equation represents five signals at the coordinates of
, 
, 
, 
and 
with the relative intensities of –2:–2:0:2:2, respectively.
3.3. IS3 Spin System
For IS3 spin system, applying the same procedure
(14)
is obtained. As seen in this equation, there exist seven signals at
,
,
, 
, 
, 
and 
coordinates with the relative intensities of –3:–6:–6:0:6:6:3, respectively.
3.4. Simulated Spectra
A computer program was written by Kanters et. al. for product operator description of NMR experiments and for the simulations of FID signals [21,22]. This is called Product Operator Formalism (POF.M) using Maple. In this study, in order to obtain the simulated spectra of the FID results, POF.M program is implemented for this experiment. 
values obtained for IS, IS2 and IS3 spin systems are given in Eqs. (10), (13) and (14), respectively. They represent the FID signals of 2D J-INEPT NMR spectroscopy for CDn groups. By using 
values, simulated spectra of this experiment are obtained and they are given in Figures 2-4 for CD, CD2 and CD3 groups, respectively. In simulated
Figure 2. Simulated spectrum of 2D J-INEPT NMR experiment for CD group.
Figure 3. Simulated spectrum of 2D J–INEPT NMR experiment for CD2 group.
Figure 4. Simulated spectrum of 2D J-INEPT NMR experiment for CD3 group.
spectra, 13C chemical shift values of CD, CD2 and CD3 groups were assumed to be 75 ppm, 50 ppm and 25 ppm, respectively. Spin-spin coupling constants between 13C and 2D nuclei for all CD, CD2 and CD3 groups were taken as 25 Hz. It can be seen from the theoretical results and the simulated spectra that 2D J-INEPT NMR experiment can be used to identify CD, CD2 and CD3 groups from each other and also to determine spin-spin coupling constant between 13C and 2D nuclei in CDn groups.
4. Conclusions
In this study, by using product operator theory, analytical description of 2D J-INEPT NMR experiment is presented for CDn groups. The obtained theoretical results represent the FID values of CDn groups. Then, in order to obtain the simulated spectra for CD, CD2 and CD3 groups, the Fourier transforms of the FID values are performed in Maple. Simulated spectra of 2D J-INEPT NMR spectroscopy are explained in detail for CDn groups. It is shown that, by using 2D J-INEPT NMR experiment, CDn groups can be identified from each other and 1J(C,D) coupling constants can be determined in deuterated complex molecules.