DENSITY OPERATOR FORMALISM FOR SPIN SYSTEMS WITH PERMUTATION SYMMETRY IN MULTIPLE PULSE AND 2D NMR EXPERIMENTS (Ⅱ)——MULTIPLE QUANTUM PRODUCT OPERATOR FORMALISM
LIU AI-ZHUO, PEI FENG-KUI, 中国科学院长春应用化学研究所,长春,130022
Based on the Symmetry-Adapted Product Operator (SPO), the Moltiple Quantum Coherence Product Operator (MQCPO) formalism is introduced in this paper. The modified density operator theory is universal for the description of modern multiple pulse and two-dimensional NMR experiments performed on InS(I = 1/2, 5 = 1/2; n = an arbitray integer number) spin systems. In fact, MQCPO is a Hermitian combination of irredusible tensor operators for a compsite particle, and reveals symmetry properties of the spin system in a different aspect from SAPO. There exist simple linear relations between MQCPO and SAPO for In(I = 1/2, n = 2,3) spin systems, the SAPO forms of MQCPO are listed as a usual example. MQCPO shows a simple cyclic commutation relationship with the free procession Hamiltonian, but not with the Hamiltonian of a pulse effect. Therefore, MQCPO is convenient for description of evolution period. Using the linear relations between MQCPO and SAPO, as well as the permutation properties for the coordinate indices of Cartesian components in a SAPO, the description of a pulse action tarns out to be easy and direct under the "z"-representation. Analyses of the pulse sequences for heteronuclear spectra editing and spin topology filtration have shown the power of this formalism.
Based on the Symmetry-Adapted Product Operator (SPO), the Moltiple Quantum Coherence Product Operator (MQCPO) formalism is introduced in this paper. The modified density operator theory is universal for the description of modern multiple pulse and two-dimensional NMR experiments performed on InS(I = 1/2, 5 = 1/2; n = an arbitray integer number) spin systems. In fact, MQCPO is a Hermitian combination of irredusible tensor operators for a compsite particle, and reveals symmetry properties of the spin system in a different aspect from SAPO. There exist simple linear relations between MQCPO and SAPO for In(I = 1/2, n = 2,3) spin systems, the SAPO forms of MQCPO are listed as a usual example. MQCPO shows a simple cyclic commutation relationship with the free procession Hamiltonian, but not with the Hamiltonian of a pulse effect. Therefore, MQCPO is convenient for description of evolution period. Using the linear relations between MQCPO and SAPO, as well as the permutation properties for the coordinate indices of Cartesian components in a SAPO, the description of a pulse action tarns out to be easy and direct under the "z"-representation. Analyses of the pulse sequences for heteronuclear spectra editing and spin topology filtration have shown the power of this formalism.
A formalism is presented that describes the time behavior of the spin density matrix of a nuclear spin system with arbitrary spin in terms of fictitious spin −(1/2) operators. This formalism is an extension of that used earlier for nuclei with spin I=1. For a spin system with n eigenstates we define for every pair of eigenstates ‖i〉 and ‖j〉 three operators Ii−jp, with p=x, y, and z, according to the three 2×2 Pauli matrices σx, σy, and σz. These operators together constitute a complete set of n2−1 independent Hermitian operators, and we can write the n×n density matrix and the spin Hamiltonian of the system in terms of the Ii−jp operators. The commutation relations among the operators make it possible in many cases to solve the equation of motion of the density matrix analytically. Three examples of the use of the Ii−jp operators are presented. Firstly a system of noninteracting spins with I=1 is considered. The Ii−jp operators for this case are compared with the Iq,k operator defined earlier. The cw signal intensities of the single and double quantum transitions for the I=1 spin system are derived. Secondly a spin system with I=3/2 is described as a simple extension of the use of the Ii−jp operators. The excitation and the detection of the triple quantum coherence are discussed for noninteracting nuclei in solids. Finally the double quantum effects in a system of two interacting spins with I=1/2 are discussed.
Multiple quantum (MQ) nuclear magnetic resonance (NMR) spin dynamics are investigated analytically in infinite one-dimensional (1D) chains of spins 1/2. The representation of spin 1/2 operators with fermion field operators allows to calculate exactly the spin density operator, and hence NMR observables, under a variety of different conditions for 1D spin systems. The exact expressions are valid for all times and for a macroscopic number of coupled spins. The calculations for a 1D spin system initially at thermal equilibrium, and evolving under a 2-quantum/2-spin average dipolar Hamiltonian, in the presence of nearest-neighbor dipolar interactions yield MQ NMR spectra with 0- and 2-quantum coherences only. For a nonequilibrium initial condition with transverse magnetization, the analogous spin dynamics calculations produce MQ NMR spectra with all possible coherences of odd orders. Calculations at the level of perturbation theory, which include next-nearest-neighbor dipolar interactions, generate MQ spectra with higher even order coherences for equilibrium initial condition and evolution under a 2-quantum/2-spin propagator. Consideration of multiple spin correlations, 0-quantum coherences, and rf pulse imperfections are also presented. The relevance and implications of these theoretical results for comparison with the recent MQ NMR experiments of Yesinowski et al. on materials with quasi-one-dimensional distributions of spins, and for MQ NMR of solids in general are discussed.
An operator formalism is presented which conveniently treats the interaction of a spin-1 nucleus with a weak radio frequency field. The Hamiltonian in the rotating frame is ℋ=−Δω Iz−ω1Ix+(1/3) ωQ[3 I2z−I (I+1)], where Δω is the resonance offset (Δω=ω0−ω), ω1 is the intensity of the rf field, and ωQ is the quadrupolar splitting. Nine fictitious spin−1/2 operators, Ip,i where p=x,y,z and i=1,2,3, are defined where p refers to the transition between two of the levels and i the Cartesian component. The operators, which are the generators of the group SU (3), satisfy spin-1/2 commutation relations [Ip,j, Ip,k]=i Ip,l, where j,k,l=1,2,3 or cyclic permutation. Thus each p defines a three-dimensional space termed p space. For irradiation near one of the quadrupolar satellites, for example, Δω=ωQ+δω with δω, ω1≪ωQ, it is shown that the effective Hamiltonian can be written ℋ?−δωIx,3−√2 ω1Ix,1, i.e., a fictitious spin-1/2 Hamiltonian in x space with effective magnetogyric ratio γ along the 3 (resonance offset) axis and √2 γ along the 1 (rf field) axis. For irradiation near the center we can effect double quantum transitions between m=±1. The formalism allows us to write the effective operators for these transitions. For example, if we take Δω=δω again with δω, ω1≪ωQ we find the effective double quantum (DQ) Hamiltonian H?−2 δω Iz,1−(ω21/ωQ) Iz,3. Thus the z space is referred to as the double quantum frame with effective magnetogyric ratio 2γ along the 1 (resonance offset) axis and (ω1/ωQ) γ along the 3 (rf field) axis. The limiting expressions are compared with exact calculations for arbitrary ω1 done by high speed computer. The theory is applied to various cases of irradiation including our previously reported technique of Fourier transform double quantum NMR. Various pulse sequences for preparing, storing, and maintaining the evolution of double quantum coherence are analyzed for single crystal and polycrystalline samples. Finally, the effects of rf phase on the double quantum phase are presented briefly, and the possibility of double quantum spin locking is analyzed.
Product operator formalism has been developed to evaluate, in closed analytical form, the time evolution for strongly coupled spin (I=1/2) systems. This formalismis based on two facts: (ⅰ) the Hamiltonian for a strongly coupled spin system is a zero-quantum operator and when an arbitrary zero-quantum operator acts on a p-quantum operator, what is yielded is still a p-quantum operator and can be expressed in terms of a linear combination of a complete p-quantum operator base set of the spin system; (ⅱ) the zeroquantum and unitary transformation leads the Hamiltonian to be only a linear combination of zero-quantum longitudinal magnetization and spin order operators. Thus the time evolution for the spin systems can be evaluated in closed analytical form. The formalism retains completely the original character of the product operator formalism and enlarges its applicability. It can deal with both strongly and weakly coupled spin (I=1/2) systems in united and closed analytical forms.
The irreducible representation under for spin clusters of identical Ii ≥ 1 spin systems are explicitly derived over |IMα> Hilbert space for the range of F z subspaces by using the combinatorial aspects of each ‘weight’ set (M 1 – M 4). This defines the invariance properties for each specific Fz , and consequently for the complete system over ∑Fz . Our investigations show that there are limiting , i.e. independent of Ii , for Fz > Izi for integer spins, or Fz > 2Izi , for half integer spins. Similar considerations apply to the irreducible representations that characterize like ki - | kqv≫ Liouville space for q > (k max – ki ), where k and ki refer to the tensor ranks of the total system and the Ii spin, respectively. The invariance properties of clusters under for I = 1, 3/2 and 5/2 are derived for both | IMα> and | kqv≫ spaces. A general {[Xtilde] i } form over the total Liouville space is given for arbitrary Ii . In the context of the multiquantum N.M.R. of spin clusters, these properties allow one to construct a density operator strictly under the dual groups; such tensors provide a valid -partitioned quantum-Liouville description, of both evolution and relaxation processes, which is applicable to non-magnetically equivalent spin systems, such as [AX]4 under . For bicluster systems specific aspects of recoupling theory, implicit in the evaluation of reduced matrix elements (RME), are examined; explicit basis and RME forms under are considered. Combinatorics are shown to play a large role in characterizing the various spaces, |IMα: Sn >, | kqv: Sn ≫ and , by allowing direct evaluations of the pertinent invariances and components under .
The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored briefly. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalization techniques.