4. The Nuclear Magnetic Resonance Interactions

1. 4. The Nuclear Magnetic Resonance Interactions 4a. The Chemical Shift interaction The most important interaction for the utilization of NMR in chemistry is the “chemical shift”. It makes it possible to distinguish between chemically inequivalent nuclei. In an external magnetic field the surrounding electronic densities of the nuclei generate a field at the nuclear positions that point in most cases in a direction opposing the external magnetic field. This shielding field shifts the Larmor frequency according to: Thus it turns out that the magnitude of the additional field is proportional to the external field. The source of the induced field can be understood by considering the example of an electron in an s-rbital. The external magnetic field generates an overall current proportional to this field, and this current generates in turn a field in opposite direction (diamagnetic shift). B0 B0 Be <e> <e> Electrons in p-orbitals behave of course differently (paramagnetic shift) . “Very schematic” B0 B0 B0 The order of magnitude of the shielding field is about 10-6 times B0 and seems to be anisotropic. The orientation of the molecule in the external field determines the shift. The induced field can thus point in all directions depending on the relative orientation of the molecules but we must consider only its z-component (z//B0). 65

2. The chemical shift can be represented as a tensor in matrix form. Choosing a coordinate system with B0 pointing in its z-direction: B0 And because , the only relevant component is Or: The chemical shift can be represented as a tensor in matrix form. Choosing the Principle Axis System coordinate system on the molecule , with the direction of the field defined by polar angles (q,j): B0 And because , the only relevant component is in the direction of the main field. The magnitude of that component is : Isotropic chemical shift “chemical shift anisotropy” In liquids the anisotropy averages to zero 66

3. The shift is measured in terms of TMS gives one line at high field and is inert! For samples in CDCl3 solution. The d scale is relative to TMS at d=0. http://orgchem.colorado.edu/hndbksupport/nmrtheory/protonchemshift.html 67

4. Carbon Chemical Shift Ranges* 68

5. Random Coil Carbon and Proton Shifts of Amino Acids http://ascaris.health.ufl.edu/classes/bch6746/2004_notes/lecture4onscreen.ppt “Understanding” the chemical shift values is a subject on its own, and requires a combination of empirical facts, shielding and deshielding characteristics of functional groups in terms of their relative position, electronegativity, bond strength, p-character, and molecular motion. Today possible quantum mechanical calculations based on orbital structure, or Hartree-Fock and lately DFT, are possible to predict chemical shift values. Example of (de)shielding effects in the neighborhood of p-systems or double/triple bonds: 69

6. 4b. Chemical shift in solids In solids the chemical shift anisotropy (CSA) does not vanish and the spectral lines broaden in powders: CSA powder lineshapes: See: Multidimensional Solid-State NMR and Polymers; K. Schmidt-Rorr and H.W. Spiess Academic Press (1994) 70

7. z y x 4c. The vector model and the two level system Finally, each individual inequivalent nucleus is described by its own spin ensemble, with its own magnetization vector in its own rotating frame, its own off resonance and its own two-level spin system. The Free Induction Decay : In the rotating frame: There exists a correlation between the QM description of a two level system and the rotation of a vector in a Cartesian axis system. The x and y components of the vector are proportional to two functions of the coefficients of the eigenstates (coherence) and the z component to the difference in eigenstate probabilities (population). In the spin-1/2 case the x- and y-components are observables. RF pulses will change the coefficients of the wavefunction: 71

8. z y x 4c. The vector model and the two level system Finally, each individual inequivalent nucleus is described by its own spin ensemble, with its own magnetization vector in its own rotating frame, its own off resonance and its own two-level spin system. The Free Induction Decay : In the rotating frame: a spin-1/2 with three independent coefficients that behave like a vector and follow the Bloch equation 72

9. z y x Measurable x-y components of a spin system AX One spin -1/2 Two spins -1/2: “AX” X A A X Coherences population differences total 73

10. NMR on a spin-1/2 can be represented in a schematic way as: Spin evolution: RF pulses: For example: 4d. The Spin-Spin interaction The interaction of two spins immediated by their overlapping wavefunctions is the Spin-Spin Interaction or j-coupling. Ethanol proton spectrum CH3CH2O- X2 A3 To describe the interaction we will restrict ourselves here to the “secular” interaction only. This excludes the interaction between two neighboring equivalent spins. 74

11. Suppose two spins A and X with off resonance values and . In their rotating frames the energy level diagram looks like: {13;24} {12;34} 0 There are 4 wave functions and thus six possible coherences: and there are 6 “fictitious spin-1/2” systems with 18 “vector components” . A “vector” with 18 components: and are dependent 75

12. We can measure only the single quantum coherences: The other coherences are: and the double and zero quantum coherences 76

13. The j-coupling shifts the energies as follows: Making the spectra look like: and the spin evolution looks like For example when spin A at t =0 is in state <Ix>(0):: and the A-spin signal is 77

14. The extension to more coupled spins is straightforward: A – X2 jAX and jAX 2jAX jAX The number of A-lines in A-Xnis (n+1): the (n+1) multiplet A and X can be spins of the same type or of different types: 1H-1H or 13C-1H etc. (The energy level diagrams are evaluated in the rotating frames of all interacting spins) jCH Proton spectrum 2jCH carbon spectrum 13CH2 X A 78

15. Vicinal Coupling (3J, H-C-C-H) Karplus equation: 79

16. Time evolution of AX spin system A – signals: detectable t non-detectable FFT The spectrum of A 80