3 Multidimensional NMR Spectroscopy C Gerd Gemmecker 1999 Models used for the description of NMR experiments 1. energy level diagram: only for polarisations, not dependent phenom 2. classical treatment(BLOCH EQUATIONS: only for isolated spins(no J coupling!) 3. vektor diagram: pictorial representation of the classical approach(same limitations) 4. quantum mechanical treatment (density matrix): rather complicated; however, using appropriate simplifications and definitions-the product operators-a fairly easy and correct description of most experiments is possible 3.1. BLOCH equations The behaviour of isolated spins can be described by classical differential equations dM/dt=yM(t)x B(t)-R(M(t)-Mol [3-1 with Mo being the BOLTZMANN equilibrium magnetization and r the relaxation matrix R 01/T 2 0 0 The external magnetic field consists of the static field Bo and the oscillating r f. field Brf B(t)=Bo+ Br =Bcos(ot+φ)ex [3-3] The time-dependent behaviour of the magnetization vector corresponds to rotations in space(plus relaxation), with the Bx and B components derived from r f. pulses and B, from the static field dMz/dt =yBx My -yByMx-(Mz-MO)1 [3-4] dM/dt=yBy -Mx/2 [3-5] dMy/dt=yBZMx-y BxMz-MV/T2 [3-6] Product operators To include coupling a special quantum mechanical treatment has to be chosen for description. An operator, called the spin density matrix p(t), completely describes the state of a large ensemble of
26 3 Multidimensional NMR Spectroscopy © Gerd Gemmecker, 1999 Models used for the description of NMR experiments 1. energy level diagram: only for polarisations, not for time-dependent phenomena 2. classical treatment (BLOCH EQUATIONS): only for isolated spins (no J coupling!) 3. vektor diagram: pictorial representation of the classical approach (same limitations) 4. quantum mechanical treatment (density matrix): rather complicated; however, using appropriate simplifications and definitions – the product operators – a fairly easy and correct description of most experiments is possible 3.1. BLOCH Equations The behaviour of isolated spins can be described by classical differential equations: dM/dt = gM(t) x B(t) - R{M(t) -M0 } [3-1] with M0 being the BOLTZMANN equilibrium magnetization and R the relaxation matrix: x y z R = ë ê é û ú ù 1/T2 0 0 0 1/T2 0 0 0 1/T1 The external magnetic field consists of the static field B0 and the oscillating r.f. field Brf : B(t) = B0 + Brf [3-2] Brf = B1 cos(wt + f)ex [3-3] The time-dependent behaviour of the magnetization vector corresponds to rotations in space (plus relaxation), with the Bx and By components derived from r.f. pulses and Bz from the static field: dMz /dt = gBxMy - gByMx -(Mz -M0 )/T1 [3-4] dMx /dt = gByMz - gBzMy - Mx /T2 [3-5] dMy /dt = gBzMx - gBxMz - My /T2 [3-6] Product operators To include coupling a special quantum mechanical treatment has to be chosen for description. An operator, called the spin density matrix r(t), completely describes the state of a large ensemble of
spins. All observable(and non-observable) physical values can be extracted by multiplying the density matrix with their appropriate operator and then calculating the trace of the resulting matrix The time-dependent evolution of the system is calculated by unitary transformations(corresponding to"rotations")of the density matrix operator with the proper Hamiltonian H (including r f. pulses, chemical shift evolution, J coupling etc. p(t')=expiHt) p(t)exp(-iHt (for calculations these exponential operators have to be expanded into a Taylor series) The density operator can be written als linear combination of a set of basis operators. Two specific bases turn out to be useful for NMR experiments the real Cartesian product operators Ix, Iy and Iz(useful for description of observable magnetization and effects of r.f. pulses, J coupling and chemical shift)and the complex single-element basis set I, I, I and I(raising / lowering operators, useful for coherence order selection/ phase cycling/gradient selection) Cartesian Product operators Lit. O.W. Sorensen et al. (1983), Prog. NMR. Spectr: 16, 163-192 ngle spin operators correspond to magnetization of single spins, analogous to the classical macroscopic magnetization Mx, My, Mz (in-phase coherence, observable) ( polarisation, not observable Two-spin operators 2I1xl2:, 211y12=, 211z12x, 211212 (antiphase coherence, not observable) (longitudinal two-spin order, not observable) 211x12x, 2l1yl2x, 211xl2y, 2llyl2y (multiquantum coherence, not observable
27 spins. All observable (and non-observable) physical values can be extracted by multiplying the density matrix with their appropriate operator and then calculating the trace of the resulting matrix. The time-dependent evolution of the system is calculated by unitary transformations (corresponding to "rotations") of the density matrix operator with the proper Hamiltonian H (including r.f. pulses, chemical shift evolution, J coupling etc.): r(t') = exp{iHt} r(t) exp{-iHt} (for calculations these exponential operators have to be expanded into a Taylor series). The density operator can be written als linear combination of a set of basis operators. Two specific bases turn out to be useful for NMR experiments: - the real Cartesian product operators Ix, Iy and Iz (useful for description of observable magnetization and effects of r.f. pulses, J coupling and chemical shift) and - the complex single-element basis set I+ , I- , Ia and Ib (raising / lowering operators, useful for coherence order selection / phase cycling / gradient selection). Cartesian Product operators Lit. O.W. Sørensen et al. (1983), Prog. NMR. Spectr. 16, 163-192 Single spin operators correspond to magnetization of single spins, analogous to the classical macroscopic magnetization Mx , My , Mz . Ix , Iy (in-phase coherence, observable) Iz (z polarisation, not observable) Two-spin operators 2I1xI2z , 2I1yI2z , 2I1zI2x , 2I1zI2y (antiphase coherence, not observable) 2I1z I2z (longitudinal two-spin order, not observable) 2I1xI2x , 2I1yI2x , 2I1xI2y , 2I1yI2y (multiquantum coherence, not observable)
Sums and differences of product operators 211x2x+ 2lly2y=11 12+1112 zero-quantum cohere 2I1 2y (not observable) 211xI2x-211v2v=1112+I1I2 louble-quantum conerence 2I1xl2y+2I12x=I112-I1 (not observable) The single-element operators I* and I- correspond to a transition from the mz=-/2 to the mz =+12 state and back, resp., hence"raising" and"lowering operator". Products of three and more operators ble Only the operators Ix and ly represent observable magnetization. However, other terms like antiphase magnetization 2 I1x I2z can evolve into observable terms during the acquisition period Pictorial representations of product operators (cf. the paper in Progr:. NMR Spectrosc. by Sorensen et al.) coherences L,1 polarisation 2I
28 Sums and differences of product operators 2 I1xI2x + 2 I1yI2y = I1 + I2 - + I1 - I2 + zero-quantum coherence 2 I1yI2x - 2 I1xI2y = I1 + I2 - - I1 - I2 + (not observable) 2 I1xI2x - 2 I1yI2y = I1 + I2 + + I1 - I2 - double-quantum coherence 2 I1xI2y + 2 I1yI2x = I1 + I2 + - I1 - I2 - (not observable) The single-element operators I+ and I- correspond to a transition from the mz = - 1 /2 to the mz = + 1 /2 state and back, resp., hence "raising" and "lowering operator". Products of three and more operators are also possible. Only the operators Ix and Iy represent observable magnetization. However, other terms like antiphase magnetization 2 I1x I2z can evolve into observable terms during the acquisition period. Pictorial representations of product operators (cf. the paper in Progr. NMR Spectrosc. by Sørensen et al.) aa aa aa aa aa aa aa aa ab ab ab ab ab ab ab ab ba ba ba ba ba ba ba ba bb bb bb bb bb bb bb bb Ix I1 z I2 z 2I I I +I 1 z 2 z 1 z 2 z Iy I I 1 x 2 z I I 1 y 2 z x x x x y y y y z z z z coherences polarisations
In the energy level diagrams for coherences, the single quantum coherences Ix and Iy are symbolically depicted as black and gray arrows. Both arrows in each two-spin scheme (for the coupling partner being a or B) belong to the same operator; in the vector diagrams these two species either align(for in-phase coherence)or a 180 out of phase(antiphase coherence). In the NMR spectrum, these two arrows transitions correspond to the two lines of the dublet caused by the coupling between the two spins. The term 211xI2z is called antiphase coherence of spin 1 with respect to spin 2 For the populations, filled circles represent a population surplus, empty circles a population deficit (with respect to an even distribution). Ilz and I2z are polarisations of one sort of spins only, Ilz+l2z is the normal BOLTZMANN equilibrium state, and 2 Iiz 12z is called longitudinal two-spin order(with the two spins in each molecule preferentially in the same spin state) Evolution of product operators Chemical shift Q21tIz I1xcos(Q2t)+lysin(Q2t) [3-7 2l2 I cos(S2(t)+ lysin(S2t) Q2tI1z IIvcos(Q2t)-IIxsin(Q2 S2 tI ( Q2 t)-Ixsin(Q2t)
29 In the energy level diagrams for coherences, the single quantum coherences Ix and Iy are symbolically depicted as black and gray arrows. Both arrows in each two-spin scheme (for the coupling partner being a or b) belong to the same operator; in the vector diagrams these two species either align (for in-phase coherence) or a 180° out of phase (antiphase coherence). In the NMR spectrum, these two arrows / transitions correspond to the two lines of the dublet caused by the J coupling between the two spins. The term 2I1xI2z is called antiphase coherence of spin 1 with respect to spin 2. For the populations, filled circles represent a population surplus, empty circles a population deficit (with respect to an even distribution). I1z and I2z are polarisations of one sort of spins only, I1z+I2z is the normal BOLTZMANN equilibrium state, and 2 I1z I2z is called longitudinal two-spin order (with the two spins in each molecule preferentially in the same spin state). Evolution of product operators Chemical shift W1 tIz I1x ¾¾¾® I1xcos(W1 t) + I1ysin(W1 t) [3-7] W1 tI1z I1y ¾¾¾® I1ycos(W1 t) - I1xsin(W1 t) [3-8]
Effect ofr. pulses 1z cosp+IIxsinp [3-9] BI l1xcos阝-I1zsnβ [3-10] ly The effects of x and pulses can be determined by cyclic permutation of x, y, and z. All rotations obey the"right-hand rule",i.e, with the thumb of the right(!)hand pointing in the direction of the r.f. pulse, the curvature of the four other fingers indicate the direction of the rotation
30 Effect of r.f. pulses bIy I1z ¾¾¾® I1zcosb + I1xsinb [3-9] bIy I1x ¾¾¾® I1xcosb - I1zsinb [3-10] bIy I1y ¾¾¾® I1y The effects of x and z pulses can be determined by cyclic permutation of x, y, and z. All rotations obey the "right-hand rule", i.e., with the thumb of the right (!) hand pointing in the direction of the r.f. pulse, the curvature of the four other fingers indicate the direction of the rotation