A Theoretical Investigation of 2D MAXY- J RES NMR Spectroscopy of CD

A Theoretical Investigation of 2D MAXY- J RES NMR Spectroscopy of CD

CD

Groups

˙Irfan S¸aka and Azmi Genc¸ten

Department of Physics, Faculty of Arts and Sciences, Ondokuz Mayıs University, 55139, Samsun, Turkey

Reprint requests to A. G.; Fax: +90 362 4576081; E-mail: gencten@omu.edu.tr Z. Naturforsch.63a,339 – 350 (2008); received September 4, 2007

There exists a variety of multiple-pulse NMR experiments for spectral editing of complex molecules in solution. Maximum quantum correlation NMR (MAXY NMR) spectroscopy is one of the techniques for distinguishing CHngroups by editing¹H NMR spectra. Spectral assignments of 2D homonuclearJ-resolved NMR spectroscopy become too difficult, due to complex overlapping spec- tra. In order to overcome this problem a new technique called 2D MAXY-JRES NMR spectroscopy, which is the combination of MAXY NMR and homonuclearJ-resolved NMR spectroscopy, is used.

In this study, product operator description of 2D MAXY-JRES NMR spectroscopy is performed for ISnIS_m(I=I=¹₂;S=S=1;n=1,2,3;m=1,2) multi-spin systems. Experimental suggestions are made and simulated spectra are presented for the 2D MAXY-JRES NMR experiment. Therefore, it is shown that 2D MAXY-JRES NMR spectroscopy can be used to distinguish CD, CD2and CD3

groups from each other in CDnCDmgroups.

Key words:NMR; MAXY-JRES; Product Operator Formalism; Spin-1.

1. Introduction

There exists a large number of homo- and het- eronuclear multiple-pulse 1D, 2D and 3D nuclear magnetic resonance (NMR) experiments. Maximum quantum correlation (MAXY) NMR spectroscopy is one of the techniques for distinguishing CH_n groups by editing¹H NMR spectra [1−7]. ¹³C NMR spec- troscopy is also widely used in editing methods such as DEPT [8] and SEMUT [9]. Editing and selec- tive detection methods for¹H NMR spectroscopy are reviewed in [10]. In homonuclear J-resolved NMR spectroscopy, the chemical shift and spin-spin cou- pling parameters are resolved along the two different axes for the same kind of coupled nuclei such as¹H.

Sometimes, spectral assignments of 2D homonuclear J-resolved NMR spectroscopy become too difficult, due to complex overlapping spectra. In order to over- come this problem a new technique called 2D MAXY- JRES NMR spectroscopy, which is the combination of MAXY and homonuclearJ-resolved NMR spec- troscopy, is used [3]. This technique is used to dis- tinguish CH_nCH_mgroups in complex samples such as biological fluids [3].²H NMR spectroscopy became a powerful technique for different applications such

0932–0784 / 08 / 0500–0339 $ 06.00 c2008 Verlag der Zeitschrift f¨ur Naturforschung, T ¨ubingen·http://znaturforsch.com

as deuterium labelling and structure elucidation [11].

Therefore, 2D MAXY-JRES NMR spectroscopy can be also used for the identification of CD_nCD_mgroups in complex samples.

As NMR is a quantum mechanical phenomenon, nu- clear spin systems can be treated by quantum mechani- cal methods. The product operator formalism, as a sim- ple quantum mechanical method, has been developed for the analytical description of multiple-pulse NMR experiments of weakly coupled spin systems in liquids having spin-¹/²and spin-1 nuclei [12 – 22]. In this for- malism, the spin operators themselves and their direct products, called product operators, are used. By us- ing the product operator theory, SEMUT NMR spec- troscopy is applied to CD_n groups [23]. Experimen- tal and theoretical investigations of¹³C DEPT NMR spectroscopy of CD_n systems are presented in [24].

For CH_n and CD_n groups, the product operator the- ory of a 2D DEPTJ-resolved NMR experiment is in- vestigated [25, 26]. Product operator descriptions of MAXY and MAXY-HMQC NMR experiments of CD_n groups are also reported in our recent studies [27, 28].

The product operator theory for spin-3/2 and its ap- plication to 2D J-resolved NMR spectroscopy is re- ported [29]. For spin systems having spin-3/2 nuclei,

by using the product operator formalism, theoreti- cal investigations of 3DJ-resolved, DEPT, SEMUT, INEPT and RINEPT NMR experiments are also pre- sented [30 – 32]. The product operator theory of high resolution solution state double quantum spectroscopy of two spin-1 AX systems are investigated [21, 33, 34].

By using the product operator theory in NMR experi- ments, experimental results can be confirmed and also new experimental suggestions can be made [19, 23]. In the present study experimental suggestions are made.

In the present study, first, evolutions of some prod- uct operators under the homonuclear scalar spin-spin coupling Hamiltonian for aSS (S=1, S=1) spin system are obtained. Then, by using the product op- erator formalism, theoretical results are obtained for the 2D MAXY-JRES NMR experiment of IS_nIS_m (I=I=¹/²;S=S=1;n=1,2,3;m=1,2) multi- spin systems. By using the obtained theoretical re- sults, experimental suggestions are presented, and sim- ulated spectra of various groups are obtained for the 2D MAXY-JRES NMR experiment. To the best of our knowledge, this will be the first product operator de- scription of a 2D MAXY-JRES NMR experiment of CD_nCD_mgroups. As a result, it is shown that one can perform this experiment to distinguish CD, CD₂ and CD₃groups in CD_nCD_msystems.

2. Theory

The product operator theory is the expansion of the density matrix operator in terms of matrix represen- tation of angular momentum operators for individual spins. For the 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 mo- mentum operators forS=1,E_S,S_x,S_y,S_z,S²_z,[S_x,S_z]+, [S_y,S_z]+,[S_x,S_y]+,(S²_x−S²_y), can be easily found [35].

So, 4×9=36 product operators for theIS (I=¹/², S=1) spin system are obtained with direct products of these angular momentum operators. For theIS(I=1, S=1) spin system there will be 9×9=81 product operators.

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

dσ dt = i

¯

h[σ,H], (1)

where [σ,H] is the commutator of the density ma- trix,σ, and the Hamiltonian,H. When the Hamiltonian

is time-independent, the solution of the (1) is

σ(t) =exp(−iHt)σ(0)exp(iHt), (2) whereHis the total Hamiltonian which consists of ra- diofrequency (r. f.) pulse, chemical shift and spin-spin coupling Hamiltonians, andσ(0)is the density matrix att=0. After employing the Hausdorff formula [16]

exp(−iHt)Aexp(iHt) =A−(it)[H,A] +(it)²

2! [H,[H,A]]−(it)³

3! [H,[H,[H,A]]]+···, (3) evolutions of product operators under the r. f. pulse, chemical shift and spin-spin coupling Hamiltonians can be easily obtained [12, 16, 18]. A complete prod- uct operator theory for theIS(I=¹/²,S=1) spin sys- tem and its application to the DEPT-HMQC NMR ex- periment is presented in our recent study [36]. For the IS(I=1,S=1) spin system the evolutions of some product operators under the spin-spin coupling Hamil- tonian (HJ=2π^JIzS_z) are known; they are given as follows [21, 33, 34]:

I_xS_y HJt

−−→1

2I_xS_y(c_2J+1) +1

2I_yzS_xz(c_2J−1) +1

2(I_yS_yz−I_xzS_x)S_2J,

(4)

I_zS_x² HJt

−−→1

2I_zS_x²(c_4J+1)−1

2I_zS²_y(c_4J−1) +1

2I_z²S_xyS_4J,

(5)

I_zS_y² HJt

−−→1

2I_zS_y²(c_4J+1)−1

2I_zS²_x(c_4J−1)

−1

2I_z²S_xyS_4J.

(6)

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]+, andS_xy= [S_x,S_y]+. For this spin system the evolutions of some other product oper- ators under the spin-spin coupling Hamiltonian are ob- tained and presented in the following section. At any time during the experiment, the ensemble averaged ex- pectation value of the spin angular momentum, e. g. for I_y, is

I_y=Tr[I_yσ(t)], (7) whereσ(t) is the density matrix operator calculated from (2) at any time. AsI_yis proportional to the mag- nitude of they-magnetization, it represents the signal

Fig. 1. Pulse sequence of the 2D MAXY- JRES NMR experiment [3].

detected on they-axis. So, in order to estimate the free induction decay (FID) signal of a multiple-pulse NMR experiment, the density matrix operator should be ob- tained at the end of the experiment.

3. Evolutions of Some Product Operators under the Spin-Spin Coupling Hamiltonian

As the homonuclear spin-spin coupling occurs be- tween two spin-1 nuclei (S and S spin) duringt₁ in the 2D MAXY-JRES NMR experiment, we need to obtain the evolutions of some product operators un- der the homonuclear scalar spin-spin coupling Hamil- tonian,H(S−S) =2π^JSSS_zS_z. By using the Hausdorff formula, evolution of theS²_xproduct operator under the spin-spin coupling Hamiltonian can be obtained as

S²_x HSSt

−−−→exp(−2iπ^JSzS_zt)S²_xexp(2iπ^JSzS_zt)

=S²_x−(i2π^Jt)A(1) +(i2π^Jt)² 2! A(2)

−(i2π^Jt)³

3! A(3) +...,

(8)

A(1) = [S_zS_z,S²_x], (9) A(2) = [S_zS_z,A(1)], (10) A(3) = [S_zS_z,A(2)]. (11) Using commutation relations andS_iⁿ≡S_iⁿ⁻²(n≥3) for spin-1,

S_x² HSSt

−−−→S²_x−1 2

(4π^Jt)²

2! (S²_x−S_y²)S²_z +...

+1 2

(4π^Jt)[Sx,Sy]+S_z

−(4π^Jt)³

3! [Sx,Sy]+S_z+...

(12)

is obtained. In order to get the generalized form for the evolution of the S²_x product operator under the spin- spin coupling Hamiltonian, some additions and sub- tractions can be made; then one obtains the following

Table 1. Evolutions of some product operators under the spin-spin coupling Hamiltonian (HSS =2πJSSSzS_z).Sxz= [Sx,Sz]₊,Syz= [Sy,Sz]₊, andSxy= [Sx,Sy]₊.

Product Evolution of product operator under the spin-spin operator coupling Hamiltonian

S²_x S²_x+¹₂(S²_x−S²_y)S_z²(c4J−1) +¹₂SxyS_zs4J

S²_y S²_y−¹₂(S²x−S²_y)Sz²(c4J−1)−¹₂SxyS_zs4J

Sxz Sxz+SxzS_z²(c2J−1) +SyzS_zs2J

Syz Syz+SyzS_z²(c2J−1)−SxzS_zs2J

Sxy Sxy+SxyS²_z(c4J+1)−(S²_x−S²_y)S_zs4J

shorthand notation form:

S²_x ^2πJSzS

zt

−−−−−→S²_x+1

2(S²_x−S_y²)S²_z(c_4J−1)+1

2S_xySzs_4J. (13) Here, and in the following equations,S_xz = [S_x,S_z]+, S_yz = [S_y,S_z]+,S_xy= [S_x,S_y]+, c_nJ =cos(nπ^Jt1) and snJ=sin(nπ^Jt1). For the remaining product operators, the similar procedure is applied and the obtained re- sults are given in Table 1. Similar evolutions can be easily obtained for the product operatorsS²_x,S²_y,S_xz , S_yz , andS_xyunder the scalar spin-spin coupling Hamil- tonian.

4. Results and Discussion

In this section, the analytical descriptions of 2D MAXY-JRES NMR spectroscopy ofIS_nIS_m (I=I=

1

2;S=S=1;n=1,2,3;m=1,2) multi-spin systems are presented by using the product operator theory.

This section is divided into two subsections. In the first one, theoretical results for the analytical description of the 2D MAXY-JRES experiment are obtained; in the second one, experimental suggestions and simulated spectra for several CD_nCD_mgroups are presented.

4.1. Product Operator Theory

For the product operator description of the 2D MAXY-JRES NMR experiment, the pulse sequence,

Table 2. Evolutions of nine angular momentum operators un- der the r. f. pulse Hamiltonian (θS_φ) at x- andy-axes for S=1 [36].cnθ =cos(nθ)andsnθ =sin(nθ).

Angular momen- φ Evolution under the tum operator r. f. pulse Hamiltonian

Sx x Sx

Sy x Syc_θ+Szs_θ Sz x Szc_θ−Sys_θ

[Sx,Sy]+ x [Sx,Sy]+c_θ+ [Sx,Sz]+s_θ [Sx,Sz]+ x [Sx,Sz]+c_θ−[Sx,Sz]+s_θ [Sy,Sz]+ x [Sy,Sz]+c2θ−(S_y²−S²_z)s2θ

S_x² x S²_x

S_y² x ¹₂S²_y(c2θ+1)−¹2S²_z(c2θ−1) +¹2[Sy,Sz]+s2θ S_z² x ¹₂S²_z(c2θ+1)−¹2S²_y(c2θ−1)−¹2[Sy,Sz]+s2θ Sx y Sxc_θ−Szs_θ

Sy y Sy

Sz y Szc_θ+Sxs_θ

[Sx,Sy]+ y [Sx,Sy]+c_θ+ [Sy,Sz]+s_θ [Sx,Sz]+ y [Sx,Sz]+c2θ+ (S²_x−S²_z)s2θ [Sy,Sz]+ y [Sy,Sz]+c_θ+ [Sx,Sy]+s_θ

S_x² y ^S₂^2x(c2θ+1)−^S2^2z(c2θ−1)−¹2[Sx,Sz]+s2θ

S_y² y S²_y

S_z² y ^S₂^2z(c2θ+1)−^S2^2x(c2θ−1) +¹2[Sx,Sz]+s2θ

Table 3. Evolutions of nine angular momentum operators un- der the chemical shift Hamiltonian (ΩSSz) for S=1 [36].

c_nS=cos(nΩSt)ands_nS=sin(nΩSt). Angular momentum Evolution under the operator chemical shift Hamiltonian Sx SxcS+SysS

Sy SycS−SxsS

Sz Sz

[Sx,Sy]+ [Sx,Sy]+c2S+ (S²_y−S²_x)s2S

[Sx,Sz]+ [Sx,Sz]+cS+ [Sy,Sz]+sS

[Sy,Sz]+ [Sy,Sz]+cS−[Sx,Sz]+sS

S_x^{2 1}₂S²_x(c2S+1)−¹2S²_y(c2S−1) +¹2[Sx,Sy]+s2S

S_y^{2 1}₂S²_y(c2S+1)−¹2S²_x(c2S−1)−¹2[Sx,Sy]+s2S

S_z² S²_z

illustrated in Fig. 1, is used [3], where the density ma- trix operator at each stage of the experiment is labelled with numbers.¹³C is treated as spins I and I, and

2H(D) as spinsS andS in Figure 1. In the pulse se- quence, the optimum value of ∆ is 1/(2J_IS),∆ is a short compensation delay,t₁ is the incremented evo- lution delay during homonuclear spin-spin coupling, which takes place betweenSandSspins andt₂is the acquisition time forS andS spins along y-axes [3].

Starting from the density matrix operator at thermal equilibrium, one should apply the required Hamilto- nians during the pulse sequence and obtain the den- sity matrix operator at the end of the experiment. For

Table 4. Evolutions of 32 product operators under the spin- spin coupling Hamiltonian (2πJIzSz) of theIS(I=¹/²,S=1) spin system [36].cnJ=cos(nπJt)andsnJ=sin(nπJt).

Product Evolution under the spin-spin operator coupling Hamiltonian Ix IySzs2J+Ix(1+S²_z(c2J−1)) Iy −IxSzs2J+Iy(1+Sz²(c2J−1)) Sx SxcJ+2IzSysJ

Sy SycJ−2IzSxSJ

IxSz IxSzc2J+IyS²_zs2J

IySz IySzc2J−IxS²zs2J

IzSx IzSxcJ+¹2SysJ

IzSy IzSycJ−¹₂SxsJ

IxSx IxSxcJ+Iy[Sx,Sz]+sJ

IxSy IxSycJ+Iy[Sy,Sz]+sJ

IySx IySxcJ−Ix[Sx,Sz]+sJ

IySy IySycJ−Ix[Sy,Sz]+sJ

IxS²_z IxS²_zc2J+IySzs2J

IyS²_z IyS²_zc2J−IxSzs2J

IxS²_y IxS²_y+¹2IxS²_z(c2J−1) +¹2IySzs2J

IyS²_y IyS²_y+¹2IyS²_z(c2J−1) +¹2IxSzs2J

S²_x ¹₂S²_x(c2J+1)−¹2S²_y(c2J−1) +Iz[Sy,Sx]+s2J

S²_y ¹₂S²_y(c2J+1)−¹2S²_x(c2J−1)−Iz[Sy,Sx]+s2J

IzS²_x ¹₂IzS²_x(c2J+1)−¹₂IzS²_y(c2J−1) +¹₄[Sy,Sx]+s2J

IzS²_y ¹₂IzS²_y(c2J+1)−¹2IzS²_x(c2J−1)−¹4[Sy,Sx]+s2J

Ix[Sx,Sz]+ Ix[Sx,Sz]+cJ+IySxsJ

Ix[Sy,Sz]+ Ix[Sy,Sz]+cJ+IySysJ

Iy[Sx,Sz]+ Iy[Sx,Sz]+cJ−IxSxsJ

Iy[Sy,Sz]+ Iy[Sy,Sz]+cJ−IxSysJ

[Sx,Sz]+ [Sx,Sz]+cJ+2Iz[Sy,Sz]+sJ

[Sy,Sz]+ [Sy,Sz]+cJ+2Iz[Sx,Sz]+sJ

[Sx,Sy]+ [Sx,Sy]+c2J+2Iz(Sy²−S_x²)s2J

Iz[Sx,Sz]+ Iz[Sx,Sz]+cJ+¹2Sy,Sz+sJ

Iz[Sy,Sz]+ Iz[Sy,Sz]+cJ+¹2[Sx,Sz]+sJ

Iz[Sx,Sy]+ Iz[Sx,Sy]+c2J+¹2(S²_y−S_x²)s2J

IxS²_x IxS²_x+¹₂IxS²_z(c2J−1) +¹₂IySzs2J

IyS²_x IyS²_x+¹2IyS²_z(c2J−1)−¹2IxSzs2J

this experiment, r. f. pulse, chemical shift and spin- spin coupling Hamiltonians are applied. ForI=I=

1

2 spins, evolutions of angular momentum operators under the r. f. pulse Hamiltonian are known [12, 22].

In the case ofS=S=1 spins, evolutions of 9 angu- lar momentum operators under the r. f. pulse Hamilto- nian were obtained in our previous study [36]. They are presented in Table 2. Duringt₂, the chemical shift Hamiltonian for angular momentum operators ofS= S=1 spins is applied. Their evolutions are presented in Table 3 [36]. During∆and∆, the spin-spin coupling HamiltoniansHJ(I−S) =2π^JISI_zS_zandHJ(I−S) = 2π^JISI_zS_z are applied to the IS (I = ¹₂, S =1) and