Chemistry 4b Lecture & Spring Quarter 004 Instructor: Richard Roberts Two Dimensional NMR The experiments and spectra so far described are one dimensional in the sense that the FID obtained is a function of a single time ariable. It is possible to introduce a second time ariable by including an incrementable delay in the pulse sequence. This incrementable delay is typically called t ; the time period during which the FID is acquired is then known as t. We collect a series of FIDs corresponding to different alues of t. This set of FIDs is a function of both time ariables: S(t,t ). Fourier Transformation of these FIDS wrt t and transposition of the data matrix yields a set of interferograms G(t,ω ). Finally, FT wrt t yields a D NMR spectrum F(ω,ω ). D spectra are typically displayed as contour plots. The aboe description can be generalied to any number of dimensions by introducing additional incrementable delays in the pulse sequence. Practically, dimensionality greater than 4 is not useful. 3D and 4D experiments hae become common since about 990. Homonuclear D-J-Resoled Spectroscopy Good introduction to nd NMR, but no longer commonly used.
Chemistry 4b Lecture & Pulse sequence 90 o x Homonuclear τ 80 y o s(t ) RD t / Preparation / eolution t / t Detection focusing pulse echo Figure - Uncoupled Spins Consider the effect of this pulse sequence on two spins and. Initially, assume that they are not J-coupled so we can see the effect of the pulse sequence on chemical shifts. β β β α α β α α Define magnetiation ector fast ν 0 slow for each transition population difference of states inoled. Figure -
Chemistry 4b Lecture & 3 The effect of the π along the y axis of the rotating frame. x pulse is to bring both isochromats π Immediately after the pulse, and are in phase. Howeer, with time, the magnetiation in the xy plane will eole. Consider time t. Because of different precession frequencies of s, there will be a phase difference between them. φ = π ( ν 0 ) t
Chemistry 4b Lecture & 4 If at this time a π pulse is applied along y (by changing the transmitter phase), then φ B y φ and φ ( t = t ) = φ + φ = φ + π ( ν 0 ) t = 0 So at the end of the eolution period, i.e., t = t, the two magnetiation ectors are π refocused: the signal at time t is exactly the same as after the first pulse (except for homogeneous effects... T relaxation during the time period t ). ny dispersion due to magnetic field inhomogeneity is refocused as well. Result: chemical shifts are refocused by the pulse sequence for any t... i.e. chemical shifts are independent of t and will not show up in the spectrum in ω. Coupled Spins Now we want to look at how the J-coupling term of the Hamiltonian behaes under the pulse sequence. ssume and are now coupled, with coupling constant J: J I I (, because δ >> J) J J β α β α ν 0
Chemistry 4b Lecture & 5 Consider the effect of the pulse sequence on spins. Label the magnetiation ectors α and β corresponding to whether the coupling partner ( spin) is α οr β. Then fter π x pulse α β fter eolution for time t α φ = π J t β ν J ν + J α φ = π J t fter ( π) y pulse β y ( π) y pulse rotates the magnetiation components α, β to mirror image position wrt the y axis. In this case, the ( π) y pulse has an additional effect: the state of the coupled is inerted, so the precession frequencies of the two magnetiation ectors are swapped.
Chemistry 4b Lecture & 6 π J t α β +π J t π J t Result: t t=t, the components of magnetiation are separated by πjt. Thus, J- coupling is not refocused by spin echo. Instead, it causes the components of the magnetiation to oscillate as a function of t at frequencies πj and +πj. The time-domain NMR signal will be of the following form: st (,t )= M 0 e t / T e iπ Jt e t / T e i ( ω 0 π J)t + e t /T e iπ Jt e t /T e i ω 0 +π J ( )t In t, we detect both chemical shift and J-coupling (ω 0 ±πj terms aboe), while only the J-coupling shows up in t terms. ( ) = st (,t ) F ω,ω e iω t e iω t dt dt
Chemistry 4b Lecture & 7 ω Predict resonances at ω or + π J π J f w 0 + π J w 0 π J f + J ν 0 + J J ν 0 J + J only f 0 45 normal ν 0 f δ + J ( ) ν 0 massage data to simplify appearance f 0 J tilted ν 0 f ν 0 spin-spin collapsed along f axis, but oriented along f axis Figure -3
Chemistry 4b Lecture & 8 Correlation Experiments The D-J-resoled experiment is an exception, in that it correlates one property of a particular spin with another property of the same spin. Most nd experiments are correlation experiments, correlating chemical shifts of pairs of spins which are related by e.g. J-coupling, dipole-dipole interaction, or chemical exchange. They may be homonuclear (all H s) or heteronuclear ( 3 C H or 5 N H). Homonuclear: Correlation ia J-coupling COSY TOCSY Correlation ia Cross-relaxation NOESY COSY 90 90 t COSY gies crosspeaks between spins separated by two or three bounds only (icinal, geminal protons). TOCSY 90 90 90 t spin lock TOCSY allows mixing of all spins or a spin system. For example, you will see a TOCSY crosspeak between and in an M spin system where J M, J M 0, J = 0. n example: TOCSY, but no COSY CH 3 CH CH OH COSY and TOCSY
Chemistry 4b Lecture & 9 Dipole-Dipole Interactions Two magnetic dipoles will interact through space (contrast with the J-coupling interaction, which occurs only through the interening electron density of bonds). Detection of this interaction is eidence for spatial proximity of the spins (measureable for distances 5 Å) µ µ R Let s call spin J and spin S.... conentional nomenclature for what follows. µ E dd = µ R 3 3 ( µ R )( R µ ) R 5 = γ h I I R 3 3 ( I R )( R I ) R 5 = γ I γ S h I S R 3 3 ( I R )( R S ) R 5 ( for γ = γ ) ( generally) This interaction proides the main means by which spins- exchange magnetiation (T processes called cross-relaxation) or dissipate magnetiation to the lattice in longitudinal relaxation. The motion of I relatie to S sets up local field fluctuations at S (or, equialently, modulates E leels of S.). There exists a component of this modulation at S transition frequency, so the fluctuations can induce transitions.
Chemistry 4b Lecture & 0 Solomon Equations The Solomon equations are a phenomenological description of the relaxation behaior of the Z-components of I, S magnetiation in a two-spin dipolar-coupled system. d dt d dt I Z = S Z = II ( I Z I 0 ) T IS ( I Z I 0 ) T T IS S Z S 0 ( ) SS ( S Z S 0 ) T ββ where T II = W 0 + W I + W T SS = W 0 + W S + W T IS = W W 0 βα W S W I W 0 W αα WI W S αβ Here W s are transition probabilities, related to molecular geometry and motion by timedependent perturbation theory. W 0 = 0 h γ I γ S R 6 τ c ( ) τ c ω I ω S W I = 3 0 h γ I γ S R 6 τ c + ω I τ c W S = 3 0 h γ I γ S R 6 τ c + ω S τ c W Z = 3 5 h γ I γ S R 6 τ c ( ) τ c + ω I + ω S where τ c = rotation correlation time for R ( ω I τ c for τ c 0 9 S for H at 500 mh) Main Result If in a two spin system, there exists nonequilibrium Z-magnetiation, there will be magnetiation exchange between and.