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
31 Scalar coupling IJtl1zI2z I1xcos(πJt)+2l1yl2zsin(πJ tls.z I,Ncos(1t)-2I1 L2,sin(TJ1t) I, I2z IJtliz' Ily cos(T Jt)-211xI2zsin(TJt) [3-12] I1xcOs(T120)+ 21y l2 sin(T1D) INyI> πJt1zZl2 (i.e, Ilz does not evolve J coupling!) Jtl1zI2 2l12 Lysin(TJt)+ 211xI2 [3-13] 2I1xI> I sin(Tjt)+ Ixlz cos(jt)
31 Scalar coupling pJtI1zI2z I1x ¾¾¾¾¾® I1xcos(pJt) + 2I1yI2zsin(pJt) [3-11] pJtI1zI2z I1y ¾¾¾¾¾® I1ycos(pJt) - 2I1xI2zsin(pJt) [3-12] pJtI1zI2z I1z ¾¾¾¾¾® I1z (i.e., I1z does not evolve J coupling!) pJtI1zI2z 2I1xI2z ¾¾¾¾¾® I1ysin(pJt) + 2I1xI2zcos(pJt) [3-13]
The antiphase term 211yl2z can be re-written using the single-element operators Ia und IB 2l12z2=I1y2-ly2B341 (21,=IQ-IB, Ia+IB=1; I@ und IP are called polarization operators The antiphase state 2I1yI2z consists of two separate populations for one half of the molecules in the ensemble spin 1 is in +y coherence(when spin 2 is in the a state), for the other half spin 1 is in -y coherence(with spin 2 in the B state); " spin 1 is in antiphase with respect to spin 2 Such an antiphase state can develop from Iix when spin 1 is J-coupled to spin 2. This leads to a dublet for spin 1, i.e., it splits into two lines with an up- and downfield shift by/2, depending on the spin state of the coupling partner, spin 2. If we wait long enough(/2J), then the frequency difference of J between the dublet lines ( I2a and Iix I2B)has brought them 180 out of phase C"antiphase"), as shown in the vector diagram I,I,sinπJt 1, (L, COsTJt (I1cosπJt IsinπJt) lsinπJt) I,cosπJt This is an oscillation between Iix in-phase coherence and 2lly I2z antiphase coherence. The antiphase component evolves with sin(/y) and then refocusses back to -Iix in-phase coherence(after t=3) Single-element operators In some cases(phase cycling, gradient coherence selection) it is necessary to use operators with a defined coherence order(Eigenstates of coherence order ) Coherence order describes the changes in quantum numbers mz caused by the coherence. A spin-12 system(no coupling) can assume two coherent states: a transition from a(m2+/2)to B(m2-72), i.e., a change(coherence order)of-1
32 The antiphase term 2I1yI2z can be re-written using the single-element operators Ia und Ib: 2I1yI2z = I1yI2 a - I1yI2 b [3-14] (2Iz = Ia - Ib, Ia + Ib = 1; Ia und Ib are called polarization operators ) The antiphase state 2I1yI2z consists of two separate populations: for one half of the molecules in the ensemble spin 1 is in +y coherence (when spin 2 is in the a state), for the other half spin 1 is in -y coherence (with spin 2 in the b state); "spin 1 is in antiphase with respect to spin 2". Such an antiphase state can develop from I1x when spin 1 is J-coupled to spin 2. This leads to a dublet for spin 1, i.e., it splits into two lines with an up- and downfield shift by J /2, depending on the spin state of the coipling partner, spin 2. If we wait long enough ( 1 /2J ), then the frequency difference of J between the dublet lines (I1x I2a and I1x I2b ) has brought them 180° out of phase ("antiphase"), as shown in the vector diagram. This is an oscillation between I1x in-phase coherence and 2I1y I2z antiphase coherence. The antiphase component evolves with sin(pJt) and then refocusses back to -I1x in-phase coherence (after t= 1 /J ). Single-element operators In some cases (phase cycling, gradient coherence selection) it is necessary to use operators with a defined coherence order (Eigenstates of coherence order). Coherence order describes the changes in quantum numbers mz caused by the coherence. A spin-1 /2 system (no coupling) can assume two coherent states: a transition from a (mz=+1 /2) to b (mz=- 1 /2), i.e., a change (coherence order) of -1
This can be described by the lowering operator I=B><Bl(coherence order +1) The real Cartesian operators Ix and ly correspond to mixtures of both coherence orders, +l, although they are more useful for directly corresponding to the observable x and y components of the magnetization. Their relationship with the complex I operators is simple I=Ix+ lly sing operator Ix=2(I++n) T=Ix"ly lowering operator ly=-h2(+-n) I=/21+lz polarisation operator(a) h(2-1P pP=1h21-L2 polarisation operator(B) The effect of r f pulses(here: an x pulse with flip angle (p)on single-element operators is as follows +1+/cos2((/2)+1-/+sin2(p/2)(+/-iL, sin()) SIPCOS2{q/2}+ Iasin2{(/2}+(1/2)sin{o}[I+-i][3-16 Iacos2( /2)+ISin(c/2)-(1/2)sin([I+-il-l [3-17] Generally it is easier to calculate the effects of r.f. pulses on Cartesian operators and then use the conversion rules to get the single-element version
33 This can be described by the lowering operator I - = |b><b| (coherence order +1). The real Cartesian operators Ix and Iy correspond to mixtures of both coherence orders, ±1, although they are more useful for directly corresponding to the observable x and y components of the magnetization. Their relationship with the complex I ± operators is simple: I + = Ix + iIy raising operator Ix = 1 /2 (I+ + I- ) I - = Ix - iIy lowering operator Iy = - i /2 (I+ - I- ) I a = 1 /2 1 + Iz polarisation operator (a) Iz = 1 /2 (Ia - Ib ) I b = 1 /2 1 - Iz polarisation operator (b) 1 = Ia + Ib The effect of r.f. pulses (here: an x pulse with flip angle j) on single-element operators is as follows: jx I +/- ¾¾¾®I +/-cos2{j/2} + I-/+sin2{j/2} (+/- iIz sin{j}) [3-15] jx I b ¾¾¾®I bcos2{j/2} + Iasin2{j/2} + (1/2)sin{j}[I+ - iI- ] [3-16] jx Ia ¾¾¾®Iacos2{j/2} + Ibsin2{j/2} - (1/2)sin{j}[I+ - iI- ] [3-17] Generally it is easier to calculate the effects of r.f. pulses on Cartesian operators and then use the conversion rules to get the single-element version
Signal phase, In-phase and antiphase signals For a single spin I, one gets the following signal during acquisition with receiver reference phase x l、→>Icos(g2t)+ Iy sin(g2t) Iy>ly cos(Q2t)-I sin(S2t These two signals are 90 out of phase(also after FT), which is indicated by one Ix component having a sine. the other one a cosine modulation For a spin I, coupled to another spin I one gets the following signal during acquisition(neglecting chemical shift evolution) Ix> Ix cos(2t)+ ly sin(g2t) I cos(Ωt)cos(πJt)+2 Ily 12z cos(2tsin(πJt ly sin(Q2t)cos(t)-2 I1x I2z sin(Q2t) Sin(πJt) the detected x component corresponds to an in-phase dublet with splitting J, i.e., lines with intensity /2 at positions(Q2+12)and(Q2-12) t 2兀/2t) cos a cos B=//2 [ cos(a+B)+ cos(a-B) 2 I1x 12z>2 Iix lz cos(@2t)+ 2 Ily 12z sin(S2t)> 2 IIx lz cos(Ωt)cos(πJt)+ Ly cos( Q2t)sin(πJt) 2 lly I2z sin(Q2t)sin(TJt)-I sin( @2t) sin(TJt) this tiem the x component corresponds to an anti-phase dublet with splitting J, i.e., lines with intensities of /2 and -/2 at positions(Q2+12)and(Q2-12)
34 Signal phase, In-phase and antiphase signals For a single spin I1 one gets the following signal during acquisition with receiver reference phase x: Ix ® Ix cos (Wt ) + Iy sin (Wt) Iy ® Iy cos (Wt ) - Ix sin (Wt) These two signals are 90° out of phase (also after FT), which is indicated by one Ix component having a sine, the other one a cosine modulation. For a spin I1 coupled to another spin I2 one gets the following signal during acquisition (neglecting chemical shift evolution): Ix ® Ix cos (Wt) + Iy sin (Wt) ® Ix cos (Wt) cos (pJt) + 2 I1y I2z cos (Wt) sin(pJt) + Iy sin (Wt) cos (pJt) - 2 I1x I2z sin (Wt) sin(pJt) the detected x component corresponds to an in-phase dublet with splitting J, i.e., lines with intensity 1 /2 at positions (W + J /2) and (W - J /2) ( pJt = 2p J /2 t ). cos a cos b = 1 /2 [cos (a+b) + cos (a-b)] 2 I1x I2z ® 2 I1x I2z cos (Wt) + 2 I1y I2z sin (Wt) ® 2 I1x I2z cos (Wt) cos(pJt) + Iy cos (Wt) sin (pJt) + 2 I1y I2z sin (Wt) sin(pJt) - Ix sin (Wt) sin (pJt) this tiem the x component corresponds to an anti-phase dublet with splitting J, i.e., lines with intensities of 1 /2 and -1 /2 at positions (W + J /2) and (W - J /2) :
sin a sinβ=h2[cos(a+β)-cos(a-B In the same way, Iy leads to an in-phase dublet 90 out of phase(dispersive)and 2 Ily I2z to a ispersive anti-phase dublet di Some applications 1. For solvent signal suppression in ID spectra, the Jump-Return sequence can be used 90°(x)-τ-90°(x)- acquisition Calculate the excitation profile with product operator formalism 2. What happens to chemical shift evolution during this sequence, and what about J coupling? Calculate 90°(x)-τ-180°(x)-τ 90°(x)-τ-180°(u)-τ
35 sin a sin b = 1 /2 [cos (a+b) - cos (a-b)] In the same way, Iy leads to an in-phase dublet 90° out of phase (=dispersive) and 2 I1y I2z to a dispersive anti-phase dublet. Some applications 1. For solvent signal suppression in 1D spectra, the Jump-Return sequence can be used: 90°(x) - t - 90°(x) - acquisition Calculate the excitation profile with product operator formalism! 2. What happens to chemical shift evolution during this sequence, and what about J coupling? Calculate! 90°(x) - t - 180°(x) - t - 90°(x) - t - 180°(y) - t -
