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). The order of magnitude of the shielding field is about 10 -6 times B 0 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//B 0 ). Electrons in p-orbitals behave of course differently (paramagnetic shift). “Very schematic” B0B0 B0B0 B0B0 B0B0 B0B0 BeBe 65

2. The chemical shift can be represented as a tensor in matrix form. Choosing a coordinate system with B 0 pointing in its z-direction: 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 (  ): And because, the only relevant component is in the direction of the main field. The magnitude of that component is : B0B0 Isotropic chemical shift “chemical shift anisotropy” B0B0 In liquids the anisotropy averages to zero 66

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

4. Carbon Chemical Shift Ranges * 68

5. http://ascaris.health.ufl.edu/classes/bch6746/2004_notes/lecture4onscreen.ppt Example of (de)shielding effects in the neighborhood of  -systems or double/triple bonds: “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,  -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. Random Coil Carbon and Proton Shifts of Amino Acids 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. 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. x y z 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: 4c. The vector model and the two level system 71

8. 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. x y z The Free Induction Decay : In the rotating frame: 4c. The vector model and the two level system 72 a spin-1/2 with three independent coefficients that behave like a vector and follow the Bloch equation

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

10. NMR on a spin-1/2 can be represented in a schematic way as: Spin evolution: RF pulses: 4d. The Spin-Spin interaction The interaction of two spins immediated by their overlapping wavefunctions is the Spin-Spin Interaction or j-coupling. To describe the interaction we will restrict ourselves here to the “secular” interaction only. This excludes the interaction between two neighboring equivalent spins. For example: A3A3 Ethanol proton spectrum X2X2 CH3CH2O-CH3CH2O- 74

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

12. The other coherences are: and the double and zero quantum coherences We can measure only the single 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 (0): : and the A-spin signal is 77

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

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

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