G. wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 lue to the counter- rotating field the term"non-reso- than their mutual scalar coupling J; this situation nant effect was introduced [34]. Since this additional often referred to as weak spin-spin coupling whereas field may be rather strong and close to the resonance strong spin-spin coupling specifies the case where frequency, this effect can become quite large and is close to or even smaller than J. With these assump- needs to be compensated [35](Section 2.2.1) tions simple rules can be calculated which describe the evolution of spin operators under the action of 2. 1. 2. Operators, coherence, and product operator chemical shift, J coupling and RF pulses. The form formalism alism combines the exact quantum mechanical treat- The description of NMr experiments by the Bloch ment with an illustrative classical interpretation and is equations and by magnetization vectors in the rotating the basis for the development of many NMR experi- frame has significant limitations particularly for the ments. However, some parts of experimental schemes description of multipulse experiments. On the other for example ToCSY sequences(Section 4.2.1), can hand, a full quantum mechanical treatment which only be described with a full quantum mechanical describes the state of the system by calculation of treatment. Calculations with the formalism are not the time evolution of the density operator under the fficult, but many terms may have to be treated an action of the appropriate Hamiltonian can be cumber aplementations of the formalism within computer some. In a quantum mechanical description an R programs are very helpful in such situations [36,37 pulse applied to the equilibrium state creates a coher- Although most of the experiments applied in bio- ent superposition of eigenstates which differ in their molecular NMR correlate three or more spins, the magnetic quantum number by one, often simply majority of interactions can still be understood based referred to as a coherence. In more complex experi- on an analysis of two spins. For two spins I and S the ments the magnetic quantum numbers between states operator basis for the formalism contains 16 elements differ by a value g different from one, leadi Two sets of basis operator, cartesian and shift opera a q quantum coherence with at least q spins involved. tors, have proven very useful for the description of However, only in-phase single quantum coherences experimental schemes and are used in parallel. The (Table 2)are observable and correspond to the classi- two basis sets and the nomenclature used to character- al magnetization detected during the acquisition of ize individual states are summarized in Table 2 [28] an NMR experiment. Multiple quantum coherences The operators /, and S are identical in the two basis cannot be observed directly, but they influence the sets and a simple relationship exists between the two spin state and this information can be transferred to other cartesian and shift operators bservable magnetization In a step towards a full quantum mechanical treat- lx=(7++/)/2 I=l+il (4) ment, the product operator formalism for spin , nuclei as introduced [28]. In this approach it is assumed I-ilv at there is no relaxation and that the difference in the Three operators, which represent the action of the resonance frequency Ar of two nuclei is much larger Hamiltonians for chemical shift, scalar coupling and Table 2 Product operator basis for a two-spin system Cartesian operator basis Nomenclature Shift operator basis Longitudinal magnetization 、l,SS In-phase transverse magnetization 1,,S+,S 2/S:2/S Anti-phase I spin magnetization 2/+S,2S 25/252 Anti-phase S spin magnetization 1, 2S I. 2/Sx2Sx21/S,2/.S 2/*5+,2/S, 2/S,2/"S 21.S ongitudinal two-spin order
G. Wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)793-275 RF pulse, act on these basis operators and may trans- cartesian product operator may be associated with form them into other operators within the basis set. several coherence orders, for example 21_S, describes this way the spin states created during an NMr a linear combination of both DQC and zero quantum experiment can be described and the observable coherence(ZQC). The cartesian x and y components magnetization calculated. The operator formalism of a multiple quantum coherence are given by linear can be summarized by simple rules [28] which are combinations of the shift or cartesian operators. For listed for both basis systems in Appendix B for further example, the ZQCs and DQCs of a two-spin system reference. Operators transform individually under can be described as follows these rules even in products of operators except for (ZQC) =(2/.+21,S,)=(+s"+/s+)(5) and transformed accordingly; however, they can be (DQC)=(21Sx-21S)=(+S++I S) treated consecutively when different couplings the same nucleus exist The cartesian operators transform more easily ZQC)=(2lS-21S2)=i+S-S+) under pulses and their single operators have a direct classical interpretation as magnetization vectors. The DQC)=(2IS, +21 S)=-i(I*S+-I-S) shift basis provides a useful alternative for the On the basis of the operator formalism the selection of description of the evolution due to chemical shift particular states using phase cycling of Rf pulses in and/or the influence of magnetic field gradients an NMR experiment can be rationalized Coherences (Section 2.3)and is better suited for the description present in an experiment can be classified into their of coherence orders and coherence pathways(Fig. 1) different orders or coherence levels which can be Their single operators describe a transition from the a represented in a pictorial way(Fig. 1)to visualize to the B state 1, or from the B to the a state, I the coherence transfer pathways in an experimental Hence, shift product operators are uniquely associated scheme [38,39). The order of coherence p corresponds with one coherence order, for example /*S* describes to the change q in the magnetic quantum number only double quantum coherence(DQC), whereas the between the two connected states [28]. Hence for n coupled spins the maximal coherence level that can be reached is n Free precession conserves the coherence order whereas pulses may cause coherences to be transferred from one order to another(Fig. 1). The pulse is proportional to its order a q quantum coher ence will experience a phase shift Ao of an rf pulse Fig. 1. Coherence level diagram. The coherence levels p are for as q4d. If the pulse results in a change in the coher mally represented by products of the shift operators/and/ which ence order of Ap, the corresponding phase shift are conserved during periods of free evolution. The application of radiofrequency pulses may transfer coherences from one order xperienced by the affected coherence will be (level)to another. The positions of three pulses are indicated in △φAp. Proper phase cycling of consecutive RF the figure by vertical arrows labelled rf rf, and rf. Thick lines pulses allows for selection of a specific succession represent the coherence pathway starting at the equilibrium state of coherence levels that define a coherence pathway (p= 0) passing through single quantum coherences after the first pulse(pl= 1), reaching double quanturn coherences after the second (Fig. I). The concept of coherence transfer pathways pulse (pl= 2)and ending as observable magnetization (p clarifies the role of phase cycling in NMR experi after the third pulse Only single quantum coherences (p!= 1)can ments and describes their action with a simple set of be observed. The spectrometer detects only one of these two rules [21, 24. 38, 40 A particular RF pulse can be coherence levels which is usually assumed to be p=-[38] and designed to select a certain difference in coherence hence all other coherence orders after the third pulse cannot be order Ap t nN(n=0. 1, 2,..)with a phase cycle tected and therefore are not drawn thin lines indicate alternative pathways which have to be suppressed if only the pathway indicated comprising N phase steps Ap of the same size equal to by thick lines should contribute to the signal measured at the end of 360/N. The N signals obtained must be summ together with the proper receiver phase -kApAp to
G. Wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 compensate for the phase change experienced by the The experiment in Fig. 2 starts at time point a in coherence,k takes on values (0, 1, 2, 3, .. An thermal equilibrium(Fig 3)where the populations on example for the design of a phase cycle using the upper Pu and the lower energy level P, across system is given in Section 4.2.1 with the discussion transition fulfil the Boltzmann distribution(Eq. (6) of the double quantum filter. a detailed discussion of Because AE in Eq (1) is typically much smaller than phase cycling can be found in most textbooks on kT we can approximate this exponential distribution NMR,e.g.[6,21,24,261 by the first term in a Taylor expansion 2. 1.3. Descriptive representations of experimental △E △E Pu/P=exp( h kT An NMr experiment can be graphically described to a limited extent based on a classical physical model where k is Boltzmann's constant and T the absolute using populations and magnetization vectors in the temperature. With Eq (1)the energies E1,E,E3,and rotating frame or based on quantum mechanical E4 of the four different energy levels in the system can principles using the product operator formalism. be calculated Both descriptions find widespread applications for =h(-n-Bs+J/2)/2 the discussion and development of NMR experiments The different representations are discussed on the E2=h(-v1+us -J/2)/2 basis of the scheme shown in Fig. 2. The application of this experiment to a system of two scalar coupled E3=h(n-s-J2)/2 spins I and s is described with four different represen tations in Fig 3 for each of the five time points a to e. E4=h(1+s +J/2)/2 Fig.3 represents energy level diagrams(E), the where vi and vs stand for the resonance frequencies of observable spectra( S), magnetization vector diagrams the I and S nuclei, respectively. The resonance fre V)and the notation in the product operator formalism quencies of nuclei with positive gyromagnetic ratio (O)showing the cartesian and the shift operator y such as protons and carbons are negative(Eq(2), basis. In Fig. 3 spins I and s are assumed to be proton and, hence, E, becomes the highest and E4 the lowest and carbon nuclei, respectively, with the size of energy(Fig 3). For nuclei with a positive y value the representative vectors proportional to the correspond- a state(spin 2) has lower energy than the B state(spin ing populations, but qualitatively the figure applies ). The polarizations Mf and Mi are proportional to all nuclei with spin and positive gyromagnetic to the energy differences(E+- E2) and(E3-ED ratIo respectively, and they determine the intensity of the orresponding transitions 2+ 4(24)and 1+3(13). The consistent use of signs and transformation proper ties as presented in Fig. 3 may seem not to be of great importance and, indeed, has very often no direct experimental consequences. But there are situations where inconsistencies occur and the interpretation of Fig. 2. Sketch of the experimental scheme used for the discussion of data becomes confusing or wrong [31, 411 different representations shown in Fig. 3. The black narrow bars from the consistent illustration of different indicate 90 RF pulses. five time-points on the time axis I are representations for the description of an NMR experi denoted by the letters a, b. c, d and e. The RF pulses applied on- ment, Fig. 3 demonstrates that the scheme shown in sonance to the two species of nuclei I and S are indicated on the ig. 2 transfers polarization from proton to carbon nes marked with the corresponding letters, a particular pulse acts uclear species. The scalar coupling between the two pins. At time point d the proton polarization Mi is separated by the time inverted. As a consequence the populations across the rod(2). The phases of the RF pulses are indicated by x or y at arbon transitions 12 and 34 acquire a larger differ he top of the pulses, where x or y stand for the application of the B nce than at thermal equilibrium at time point a and held in the rotating frame along the positive r or y axis, respectively. hence, the experimental scheme allows measurement
G Wider/Progress in Nuclear MaGnetic Resonance Spectroscopy 32(1998)193-275 巴 5巴a E4点 E3 gmv变 三8 3J a
ider/Progress in Nuclear Resonance Spectroscopy 32(1998)193-275 of carbon spectra with higher sensitivity, Such polar- Fig. 4 shows a plot of J(a)as a function of the fre ization transfer experiments are extremely important quency w for the three correlation times T of 5, 10 and in heteronuclear NMR experiments and are discussed 20 ns. These correlation times represent the motion of further in Section 4.2.3 small, medium and large globular proteins in In addition, Fig ustrates that low fre 2.1. 4. Relaxation motions are especially effective in NMR rel processes for proteins. Using the concept of spectral Relaxation processes re-establish an equilibrium density functions the different behaviour of longitudi distribution of spin properties after a perturbation After a disturbance, the non-equilibrium state decays nal relaxation and relaxation of transverse magnetize the simplest case exponentially characterized by the tion can be rationalized. When considering onl spin-lattice relaxation time T. Re-establishing relaxation due to fluctuating dipolar interactions thermal equilibrium requires changes in the popu- caused by stochastic motion, the relaxation rate TI lation distribution of the spin states and lowers the is proportional to J(wo)since only stochastic magnetic energy of the spin system. Thus, it involves energy fields in the transverse plane at the resonance fre transfer from the spin system to its surroundings quency w, are able to interact with the transverse which are usually referred to as the lattice. Micro- magnetization components, bringing them back to scopically, relaxation is caused by fluctuating values J( o decrease for the three increasing values the z axis. For frequencies larger than 25 MHz the magnetic fields. Dynamical processes such as atomic of Te represented in Fig 4. The longitudinal relaxation actions and facilitate spin-lattice relaxation. The times, therefore, increase for increasing molecular extent of the overlap between the frequency spectrum very small Te the values J (wo)values get smaller aga of the motional process and the relevant resonance leading to an increase of T, compared to the value for lap is described by th density function J(w). Since J()is the Fourier trans- obtained when woTe=l, e.g. at 600 MHz for a Te of form of the time correlation function describing the 0.26 ns. Transverse relaxation shows a different notion, its functional form depends on the mechanism dependence on the molecular weight of the molecule of motion. An exponential correlation function with T2 relaxation not only depends on Jwo)but also on correlation time Te results in the spectral density func J(O) since the z components of stochastic magnetic tion[16,24,29 fields(zero frequency)reduce the phase coherence of transverse magnetization components which con 1(0)=31+ ( 8) sequently sum up to a smaller macroscopic magneti zation. Since J(o) increases monotonously with J() 0°s]|=20ms 4tc=lOns T. 5n 0.001 001 0102051250[109rads] 63280160320800v[MHz] 4. Plot of the spectral density function /(o)(Eq (8))versus the frequency w on a logarithmic scale, Three correlation times 5, 10 and 20 ns indicated which represent small, medium and large proteins. The frequency scale is given in units of rad s and in MHz