NMR Spectroscopy: A Quantum Phenomena Pascale Legault Département de Biochimie Université de Montréal Outline 1) Energy Diagrams and Vector Diagrams 2) Simple 1D Spectra 3) Beyond Simple 1D Spectra 4) The Spin Echo 5) Selective Population Transfer 6) The INEPT Experiment 1
1) Energy Diagrams and Vector Diagrams Energy Diagram and NOE Importance of NOE (nuclear Overhauser Effect): - Through-space transfer of magnetization in NOESY-type experiments - Provide distance constraints for structure calculations Like T 1 and T 2, NOE is a relaxation phenomenon: Key: A fluctuating interaction is capable of causing transition, just like the RF field. Example of interactions: chemical shift anisotropy, dipoledipole, etc. NOEs result from dipole-dipole interactions. 2
Energy Diagrams EXTREMELY useful for understanding energy transfer in certain experiments (e.g.: NOESY, SPT, INEPT, HSQC) At equilibrium for single spins: In the presence of B o the spin states α and β have energies 0 and related frequencies: E = - γi z B o = -mћγb o = mћω o The population of spins in each state is given by the Boltzmann equation. At T 273 K: N m /N [1+(mћγB o /k B T)]/(2I+1) where N m is the number of nuclei in the m th state (e.g. α) N is the total number of spins, k B is the Boltzmann constant For 1 H at 500 MHz, 273 K: N α /N 1/2+2.2*10-5 ; N β /N 1/2-2.2*10-5 There is a small excess in the α state... Energy Diagram at Equilibrium In detail: More simply: 3
Steady-State NOE Lets take an AX 1H spin system with A and X close in space, but not connected through bonds (J = 0). Lets saturate X with an RF field to equilibrate the population. At equilibrium: ( AX spin system) After saturation: Steady-State NOE Now, allow relaxation W0, W1, W2 are transition probabilities or rate constants W0 and W2 are 2-spin transitions ALLOWED for relaxation Lets say W0 is most efficient (large molecules) Now, A transition decreases by Δ: 4
NOE and Molecular Size For macromolecules W0 >> W2 This leads to negative NOE in steady-state NOE and positive crosspeaks in 2D NOESY spectra For small molecules W2 > W0 This leads to positive NOE in steady-state NOE and negative crosspeaks in 2D NOESY spectra When W2 W0 small or no NOE observed. NOE and Molecular Size ssnoe: main use today: 13 C detection with 1 H BB decoupling 5
Vector Diagrams They are EXTREMELY useful, but it is important to know that they have certain limitations i.e. difficult to explain NOE, 2nd order spectra, population transfer, zero or multiple quantum coherence, etc. For ease of representation, usually in the rotating frame (x', y', and z) instead of the laboratory frame (x, y, and z). Very important to know what is the frequency (ν) of the rotating frame. Vector Diagrams: Effect of a Pulse on the Longitudinal Magnetization (Mz) At equilibrium (longitudinal magnetization Mz): Bulk magnetization along z caused by B0 Excess population in the α state After 90, 270 pulses: B 1 field brings Mz to the transverse x'-y' plane After 180 pulses: B 1 field inverts Mz 6
Vector Diagrams: Effect of 90 and 270 Pulses on Mz Energy Diagrams: Effect of 90 and 270 Pulses on Mz The populations of the two states are now equal: 7
Vector Diagrams: Effect of a 180 Pulse on Mz Energy Diagrams: Effect of a 180 Pulses on Mz The populations of the two states are now inverted: 8
Vector Diagrams: Effect of a Pulse on the Transverse Magnetization (Mx', My') The transverse magnetization (Mx', My') is not at equilibrium : Bulk magnetization in the x'-y' plane Equal populations in the α and β states Vector Diagrams: Effect of 90 and 180 Pulses on Transverse Magnetization (My' only) 9
Vector Diagrams: Effect of 90 and 180 Pulses on Transverse Magnetization (Mx' only) 2) Simple 1D Spectra 10
Vector Diagrams: Transverse Magnetization: Where Does it Come From? Lets consider a simple 1 H 1D experiment Vector Diagrams: Simple 1 H 1D Experiment 11
Vector Diagrams: Effect of 90 x Pulses on Transverse Magnetization with Mx' and My' Vector Diagrams: Effect of 180 Pulses on Transverse Magnetization with Mx' and My' 12
3) Beyond Simple 1D Spectra Beyond Simple 1D Spectra Simple 1D spectra are not always sufficient for assigning spectra and determining structure even for small organic compounds. The main problems are: 1) Resonance assignment 2) Low S/N in insensitive nuclei with low natural abundance (e.g. 13C and 15N) 3) No correlation information Example: Neuraminic acid derivative 1 13
Beyond Simple 1D Spectra We would also like to use the following information: 1) 13 C- 1 H correlations 2) The number of protons attach to one carbon 3) 1 H- 1 H correlations (through-bond and through-space) 4) 13 C- 13 C correlations etc. Solution: Complex pulse sequences, which use multiple pulses, delays and decoupling schemes to transfer magnetization Various pulses: hard pulses: 90 x, 90 y, 180 x, 180 y, etc. selective pulses: 90 x, 90 y, 180 x, 180 y, etc. pulse field gradients Various delays: fixed or variable delays Decoupling: for selective or broadband decoupling Magnetization Transfer Via J-coupling (Through-bond) Via NOE (Through-space) Via chemical exchange (dynamics) 14
Analyzing the Effect of Complex Pulse Sequences Various Tools to Represent Magnetization Transfer Block Equations Energy Diagrams Vector Diagrams Density Matrix Product Operator 4) The Spin Echo 15
The Spin-Echo Spins echoes are widely used as part of larger pulse sequences to refocus the effects of: 1) unwanted chemical shift precession 2) magnet inhomogeneity 3) heteronuclear J coupling The spin-echo does not refocus homonuclear J coupling. The spin-echo pulse sequence can be used to measure the relaxation parameter T 2 ; it does not refocus the effect of T 2 relaxation. The Spin-Echo in Vector Diagram: the Non-Coupled Single Spin Case Example: 1 H in CHCl 3 (not 13 C-labeled) with ν H = ν rf + 100 Hz 1 H: 90 x- τα - 180 y - τα (echo) Detected Signal after FT: 16
The Spin-Echo in Vector Diagram: the Non-Coupled Single Spin Case Example: 1 H in CHCl 3 (not 13 C-labeled) with ν H = ν rf + 100 Hz 1 H: 90 x- τα - 180 x - τα (echo) Detected Signal after FT: Note that the intensity is plotted relatively to the positive signal on the previous page. In practice, this signal would be drawn as a positive signal by adjusting the zero order phase correction by 180. The Spin-Echo in Vector Diagram: the Non-Coupled Single Spin Case Conclusions: 1) Chemical shift evolution (precession) is refocused by the spin-echo 2) Similarly the spin-echo refocuses magnet inhomogeneity (ΔB o ): The magnetic field B o is not perfectly homogeneous throughout the volume of the sample, therefore not all nuclei experience the same magnetic field. The small differences in magnetic field (ΔB o ) across the sample volume causes nuclei that are chemically equivalent to precess at different rate. 17
The Spin-Echo in Vector Diagram: Simple Case of Heteronuclear Coupling Example: a two-spin AX system with A = 1 H and X = 13 C in CHCl 3 ( 13 C-labeled) with ν rf = ν H. 1 J AX = 209 Hz 1 H: 90 x- τα - 180 x - τα (echo)- Acquisition time Detected Signal after FT: More on AX Spin Systems by Energy Diagrams Example: a two-spin AX system with A = 1 H and X = 13 C CHCl 3 (carbon is 13 C-labeled) with ν rf = ν H. 1 J AX = 209 Hz Essentially equal population differences for the α and β 13 C transitions Population diferences: αα to αβ transition: (N + ΔH + ΔC) - (N + ΔH) = ΔC βα to ββ transition: (N + ΔC) - (N ) = ΔC αα to βα transition: (N + ΔH + ΔC) - (N + ΔC) = ΔH αβ to ββ transition: (N + ΔH) - (N ) = ΔH 18
The Spin-Echo in Energy Diagram: Simple Case of Heteronuclear Coupling Two different Larmor frequencies as a result of C-H coupling ν ( 13 CHαCl3) = νc - 1/2*JCH ν ( 13 CHβCl3) = νc + 1/2*JCH with JCH = 209 Hz and δ = 77.7 ppm (center of the doublet) 77.7 δ (ppm) In the first delay τ of the spin-echo experiment, a phase angle Θ is created between these two vectors Θ = 2πJCH*τ Examples: If τ = 0 than Θ = 0, if τ = 1/(4J) than Θ = π/2 = 90, etc. The Spin-Echo: Simple Case of Heteronuclear Coupling Conclusions: Heteronuclear coupling is refocused by the spin-echo (180 x -> with inversion of magnetization) 19
5) Selective Population Transfer Sensitivity Problem in NMR Sensitivity problem in NMR: ε = electromagnetic induction force in detection coil ε Nγ 3 h 2 B o2 I(I+1)/(3k B T) Small S/N in spectra of insensitive nuclei with low natural abundance (e.g. 13 C, 15 N) is a main problem in NMR spectroscopy of organic molecules. Example: [ ε ( 13 C)/ ε ( 1 H)] = (1.1% * 1) / (100% * 4 3 ) = 1/5818 One would need to record ~33 million (5818 2 ) more scans in a 1D 13 C spectrum to get equal S/N than in a 1D 1 H spectrum! Solutions to this problem are: 1) Get more sample 2) Isotope labeling (may be expensive and not practical) 3) Record spectrum at higher field (B o ) 4) Record spectrum at lower temperature (not significant) 5) Special NMR experiments 20
Selective Population Transfer (SPI Experiment) Advantage of SPI: Very useful to explain the principle of Population Transfer that provides a means to "recover" one of the γ factor. Disadvantage of SPI: Not very practical because selective pulses are used. Selective Population Transfer (SPI Experiment) Lets consider the two-spin AX system ( 13 CHCl3) with A= 1 H = sensitive nuclei and X= 13 C = insensitive nuclei A) At equilibrium: N4 = N N3 = N + ΔC N2 = N + ΔH N1 = N + ΔC + ΔH N2 - N4 N1 - N3 = ΔH N3 - N4 N1 - N2 = ΔC ΔH = 4 * ΔC For 13 C spectrum: X1 transition: N3 N4 = ΔC X2 transition: N1 N2 = ΔC 21
Selective Population Transfer (SPI Experiment) B) After a selective 180 pulse exciting the A2 transition: The populations of N1 and N3 are inverted: N4 = N N3 = N + ΔC + ΔH N2 = N + ΔH N1 = N + ΔC X1 transition: N3 N4 = ΔC + ΔH = 5ΔC X2 transition: N1 N2 = ΔC - ΔH = -3ΔC Selective Population Transfer (SPI Experiment) C) After a selective 180 pulse exciting the A1 transition: The populations of N2 and N4 are inverted: N4 = N + ΔH N3 = N + ΔC N2 = N N1 = N + ΔC + ΔH X1 transition: N3 N4 = ΔC - ΔH = -3ΔC X2 transition: N1 N2 = ΔC + ΔH = 5ΔC After selective inversion of the A1 or A2 transition, the signal amplification factors for the spectra of X are given by: 1 + γa / γx and 1 - γa / γx 22
6) The INEPT Experiment The INEPT Experiment INEPT: Insensitive Nuclei Enhanced by Polarization Transfer Polarization transfer achieved using non-selective pulses A) Pulse sequence in the 1 H and 13 C channels (Note: without carbon pulses, this is a spin-echo experiment on 1 H!) 23
The INEPT Experiment B) Vector diagrams showing the 1 H magnetization vectors (CHCl3) The INEPT Experiment B) Vector diagrams showing the 1 H magnetization vectors (CHCl3) a: MH Cα and MH Cβ are of approximately equal populations b: ν ( 13 CαHCl3) = νh JCH/2 and ν ( 13 CβHCl3) = νh + JCH/2 c- d: until then just like beginning of a spin-echo experiment on 1 H e: Effect of 13 C 180 : - phase of 180 doesn t matter (x or y), MC from z to z - inverts population between N1 and N2 and between N3 and N4 - MH Cα becomes MH Cβ and MH Cβ becomes MH Cα f: JCH continue to evolve instead of being refocused during the next τ delay g: 1 H 90 pulse rotates MH Cα to +z and MH Cβ to z 24
The INEPT Experiment Limitation of vector diagrams: We can t pursue our analysis at this poing (g) without trying to understand what happens in terms of the energy diagram... The INEPT Experiment: Vector and Energy Diagrams C) Energy diagrams showing the population transfer (CHCl3) g: 1 H 90 pulse rotates MH Cα to +z and MH Cβ to z The populations of N2 and N4 are inverted: N4 = N + ΔH N3 = N + ΔC N2 = N N1 = N + ΔC + ΔH X1 transition: N3 N4 = ΔC - ΔH = -3ΔC X2 transition: N1 N2 = ΔC + ΔH = 5ΔC 25
The INEPT Experiment: Vector and Energy Diagrams D) Vector diagrams showing the 13 C magnetization vectors g : Note that MC Hα is in its original position, but that MC Hβ is inverted h: The 90 x pulse on 13 C create transverse magnetization components which are observable The INEPT Experiment: Vector and Energy Diagrams E) The natural I spin magnetization in the INEPT experiment: In many applications, the contribution from the natural 13 C magnetization (ΔC) is unwanted. There are multiple ways to remove it: 1) Presaturate 13 C at the start of the pulse sequence 2) Apply a 90 13 C pulse followed by a gradient pulse at the start of the pulse sequence In cases 1) and 2) the populations at point a are: N4 = N + ΔC/2; N3 = N + ΔC/2 N2 = N + ΔC/2 + ΔH; N1 = N + ΔC/2 + ΔH The populations at point g are (N2 and N4 inverted): N4 = N + ΔC/2 + ΔH; N3 = N + ΔC/2 N2 = N + ΔC/2; N1 = N + ΔC/2 + ΔH X1 transition: N3 N4 = -ΔH = -4ΔC X2 transition: N1 N2 = ΔH = 4ΔC 3) By phase cycling 26
The INEPT Experiment: Phase Cycling The INEPT Experiment: Vector and Energy Diagrams F) Signal enhancement in the INEPT spectra Nucleus Maximum NOE Polarization Transfer 31 P 2.24 2.47 13 C 2.99 3.98 (~4) 15 N 3.94 9.87 (~10) 27
Exercises (due in a week from now) 1) Steady State NOE and energy diagrams. For small molecules W2 is the most efficient relaxation mechanism. Use an energy diagram to explain relaxation in a small molecule after excitation of spin A of an AX system, where A and X are close in space but not J coupled. Also draw the expected 1D spectra before and after saturation of the A resonance. 2) Vector diagrams and spin echo. Using vector diagrams, show that the spin echo sequence 90x-τ-180x- τ refocuses Bo field inhomogeneity. To represent inhomogeneity, use three vectors to represent 3 spins (one that rotates with the rotating frame, one a little faster, and one a little slower than the rotating frame). Label you Cartesian axes, indicate the direction of rotation of the spins, and show all the steps explicitly. 1) Use vector and energy diagrams to show the fate of magnetization in the modified INEPT sequence: (label your axes and show each step) 28