Analytical descriptions of DEPT NMR spectroscopy for ISn(I = 1,S = 1; n = 1, 2, 3, 4) spin systems

Analytical descriptions of DEPT NMR spectroscopy for IS_n(I = 1,S = 1; n = 1, 2, 3, 4) spin systems

Irfan Ş aka

Department of Physics, Faculty of Arts and Sciences, Ondokuz Mayis University, 55139, Samsun, Turkey, E-mail: [e-mail]

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 IS_n(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 ¹⁴ND_n (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 abundance of 99.64%, the magnetic moment is lower than ¹³C ( γ_13Cγ_14N 1.8 ), ¹H ( γ_1H/ γ_14N 7.0 ) and ²H ( γ_2H/γ_14N 2.2 ) nuclei, and it is a spin-1 isotope. Because 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 polarization 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 nuclei [6-9]. The well-known cross polarization transfer methods are Distortionless Enhancement by Polarization Transfer (DEPT) and Insensitive Nuclei Enhanced by Polarization Transfer (INEPT). The DEPT pulse sequence, which offers to selective detection capability of CH, CH₂, CH₃ groups from each other, is a ¹³C-detected, edited-pulse and cross polarization 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 - ½, spin-1 and spin-3/2 nuclei [10-19]. However, it has been proposed that product operator theory can be used as a new approach for analytical description of solid-state NMR experiments under magic angle spinning (MAS) conditions [20,21]. A complete product operator theory for IS (I = ½ , S = 1) spin system and application to DEPT-HMQC (Heteronuclear Multiple Quantum Correlation) NMR experiment has been presented in our previous study [22]. Analytical descriptions of INADEQATE (Incredible Natural Abundance Double Quantum Transfer Experiment) and DQC (Double Quantum Correlation) NMR experiments 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 IS_n (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 ¹⁴ND_n groups if the experiment is performed for the suggested edited-pulse angles.

2. THEORY

The product operator formalism is the expansion of the density matrix operator in terms of matrix representation of angular momentum operators for individual spins. For IS (I=1, S=1) spin system, nine Cartesian spin angular momentum operators for I=1 are E_I, I_x, I_y, I_z, , [ I_x, I_z ]₊ , [ I_y, I_z ]₊ , [ I_x, I_y ]₊ and (- ) [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 system. Depending on the pulse experiment, ( I_x² - I_y² ) Cartesian spin angular momentum operator is separated into two spin angular momentum operators as I_x² and I_y² . In this case, there should be 10×10 = 100 product operators for this spin system. 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 IS_n spin systems, the total Hamiltonian consists of r.f. pulse, chemical shift and spin-spin coupling Hamiltonians can be written as

Time dependence of the density matrix is governed by Liouville-von Neumann equation:

When the Hamiltonian is time independent, the solution of the Equation (2) is

where σ(0) is the density matrix at t=0. After employing the Hausdorff formula [15]

evolutions of the product operators under the r.f. pulse, chemical 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 Hamiltonian ( H_J= 2πJI_zS_z ) are known and they are given in following equations [11, 12, 23, 24]:

In these equations I_yz = [I_y, I_z ]₊ , I_xz = [I_x, I_z ]₊ , S_xz = [S_x ,S_z ]₊ , S_yz = [S_y ,S_z ]₊ and S_xy = [S_x ,S_y ]₊ 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 I_y, is

Where σ(t) is the density matrix operator calculated from Equation (4) at any time. Since 〈I_y〉 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 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 experiment for IS_n (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 labeled with numbers and ¹⁴N is treated as spin I and ²H as spin S.

Starting from the density matrix operator at thermal equilibrium, 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 follow 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 implementation and evolutions of the product operators under the Hamiltonians [27].

For the IS spin system, the density matrix operator at thermal equilibrium is σ₀ = S_z . Then, the evolutions of density matrices under the Hamiltonians for each labeled point are obtained:

At above and following equations c_2J = cos(2πJτ) and s_2J = 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

is found. At above and following equations s_nθ = sin(nθ), c_nθ = cos(nθ) and s_I = sin(Ω_It). 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:

At any time during the experiment, the ensemble averaged expectation value of the spin angular momentum, 〈I_y〉 , is proportional to the magnitude of the y-magnetization and

is written. It represents the free induction decay (FID) signal of a multiple-pulse NMR on y-axis. Tr[I_yO] values of observable product operators, indicated by O, have been calculated by a home made computer program in Mathematica and results are given in Table 2 for IS_n (I=1, S=1; n=1, 2, 3, 4) spin systems. Using Table 2,

is obtained for IS spin system.

For the IS₂ spin system, the density matrix at the thermal equilibrium is σ₀ = S_1z + S_2z . At the end of the experiment forty-one observable terms are obtained as shown in following equation:

Using the Trace values in Table 2;

is found for the IS₂ spin system.

Applying the same procedure for the IS₃ and IS₄ spin systems, as one can guess huge amount of observable terms are obtained at the end of the DEPT experiment by using the computer program. Then, replacing the Trace values of observable terms in Table 2 we obtain

4. DISCUSSION

Tr[I_yσ₇] values obtained in Section 3 for IS, IS₂, IS₃ and IS₄ spin systems represent FID signals of DEPT NMR experiment for ¹⁴ND, ¹⁴ND₂, ¹⁴ND₃ and ¹⁴ND₄ groups, respectively. These Tr[I_yσ₇] values for ¹⁴ND_n (n=1, 2, 3, 4) spin systems can be generalized as following

The Tr[I_yσ₇] values can be normalized by multiplication with 3/(Tr(E)). Here E is the unity product operator for the corresponding spin system. Then, the normalized FID values become as follows:

These normalized Tr[I_yσ₇] values can be also written in a generalized form as

The plots of the normalized FID functions are presented in Fig. 2. As seen in this Figure, the relative signal intensities of ¹⁴ND, ¹⁴ND₂, ¹⁴ND₃ and ¹⁴ND₄ groups vary as functions of θ. The real relative signal intensities can be found from Eq. (18) for ¹⁴ND, ¹⁴ND₂, ¹⁴ND₃ and ¹⁴ND₄ groups 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 60º, all groups will give positive signal. For the pulse angle of 90º, only ¹⁴ND groups will be observed giving positive signal. For the pulse angle of 120º, ¹⁴ND and ¹⁴ND₃ groups will give positive signals and ¹⁴ND₂ and ¹⁴ND₄ groups will give negative signals. ¹⁴ND₃ group can be easily separated by comparison of spectra for θ=90º and θ=120º. To selectively enhance of ¹⁴ND₂ and ¹⁴ND₄ groups, one might collect FIDs at θ=60º and θ=120º and take the linear combinations:

According to these results, deuterated nitrogen groups can be separated from each other if the experiment is performed for the angles of 60º, 90º and 120º.

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 determine whether the ammonium ions are transferred as a whole leading to or partially deuterated ¹⁴ND₃H⁺ , and groups. In addition, a sample can contain some deuterated nitrogen groups such as ¹⁴ND₄- , -, ¹⁴ND₂H⁺- , ¹⁴ND₂- or ¹⁴NDH- containing molecules. As a result, a DEPT NMR experiment can be employed for determination of deuteration degree of nitrogen 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, analytical descriptions of DEPT NMR experiment have been presented for weakly coupled IS_n (I=1; S=1; n=1, 2, 3, 4) spin systems by using product operator theory. Thus a theoretical discussion and experimental suggestions for DEPT NMR spectroscopy have been performed in order to edit ¹⁴N signals of ¹⁴ND, ¹⁴ND₂, ¹⁴ND₃ and ¹⁴ND₄ groups.

Acknowledgments:

I would like to thank Dr. Sedat Gümüş for the helpful suggestions about the Mathematica program.

(Received on 18 March, 2008)

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Publication Dates

• Publication in this collection
  22 Sept 2008
• Date of issue
  Sept 2008

History

• Received
  18 Mar 2008