** Journal of Modern Physics ** Vol. 2 No. 7 (2011) , Article ID: 5791 , 5 pages DOI:10.4236/jmp.2011.27084

2D J–INEPT NMR Spectroscopy for CD_{n} Groups: A Theorsssical Study

Ondokuz Mayıs University, Samsun, Turkey

E-mail: Gencten@omu.edu.tr, Isaka@omu.edu.tr

Received March 15, 2011; revised May 13, 2011; accepted June 1, 2011

**Keywords:** NMR, 2D J-INEPT, Product Operator Formalism, CD_{n} Groups

ABSTRACT

2D J-INEPT NMR experiment is a combination of heteronusclear 2D J-Resolved and INEPT experiments. In this study, 2D J-INEPT experiment was analytically investigated by using product operator theory for weakly coupled IS_{n} (I = 1/2, S = 1; n = 1, 2, 3) spin systems. The obtained theoretical results represent the FID values of CD, CD_{2} and CD_{3} groups. In order to make Fourier transform of the obtained FID values, a Maple program is used and then simulated spectra for of 2D J-INEPT experiment are obtained for CD, CD_{2} and CD_{3} groups. It is found that 2D J-INEPT is a useful experiment for both polarisation transfer and 2D J-resolved spectral assignment for CD_{n} groups in complex molecules.

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 ^{13}C 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 IS_{2} (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 IS_{n} (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 IS_{n} (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_{,} CD_{2} and CD_{3} groups. Simulated spectra of 2D J-INEPT NMR spectroscopy are explained in detail for CD_{n} groups. It is shown that this experiment can be used for the polarization transfer and J-resolved spectral assignment of CD_{n} 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; E_{I}, I_{x}, I_{y}, I_{z} and nine Cartesian spin angular momentum operators for S=1; E_{S}S_{x}, S_{y}, S_{z}, , ,

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 I_{y}, 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. ^{13}C is treated as spin I and D (^{2}H) 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 IS_{n} (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. IS_{2} Spin System

For IS_{2} 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. IS_{3} Spin System

For IS_{3} 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, IS_{2} and IS_{3} spin systems are given in Eqs. (10), (13) and (14), respectively. They represent the FID signals of 2D J-INEPT NMR spectroscopy for CD_{n}_{ }groups. By using values, simulated spectra of this experiment are obtained and they are given in Figures 2-4 for CD, CD_{2} and CD_{3} 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 CD_{2} group.

Figure 4. Simulated spectrum of 2D J-INEPT NMR experiment for CD_{3} group.

spectra, ^{13}C chemical shift values of CD, CD_{2} and CD_{3} groups were assumed to be 75 ppm, 50 ppm and 25 ppm, respectively. Spin-spin coupling constants between ^{13}C and ^{2}D nuclei for all CD, CD_{2} and CD_{3} 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, CD_{2} and CD_{3} groups from each other and also to determine spin-spin coupling constant between ^{13}C and ^{2}D nuclei in CD_{n} groups.

4. Conclusions

In this study, by using product operator theory, analytical description of 2D J-INEPT NMR experiment is presented for CD_{n} groups. The obtained theoretical results represent the FID values of CD_{n} groups. Then, in order to obtain the simulated spectra for CD, CD_{2} and CD_{3} 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 CD_{n} groups. It is shown that, by using 2D J-INEPT NMR experiment, CD_{n} groups can be identified from each other and ^{1}J(C,D) coupling constants can be determined in deuterated complex molecules.

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