Demystifying the Black Box: The Mechanics Behind Nuclear Magnetic Resonance Spectroscopy

1. Demystifying the Black Box:The Mechanics Behind Nuclear Magnetic Resonance Spectroscopy Kelsie Betsch Chem 381 Spring 2004

2. What is NMR good for? • Spectroscopic method widely used by chemists • Provides information about: • The number of magnetically distinct atoms of the type being studied • The immediate environment surrounding each type of nuclei

3. Overview • NMR involves transitions of the orientations of nuclear spins in magnetic fields • Examine the quantum-mechanical states of nuclear spins interacting with magnetic fields • Specific focus on hydrogen

4. Spin • Electron has intrinsic spin angular momentum • z component of ±ħ/2 • Spin of ½ • Nuclei also have intrinsic spin angular momenta, I • Spins not restricted to 1/2

5. Spin eigenvalue equations • Nuclear spin eigenvalue equations for protons: Î2=½(½ +1) ħ2 Î2=½ (½ +1) ħ2 Îz=½ħ Îz =½ħ () and () are spin functions • is a spin variable • ↔ Iz= ħ/2 and ↔ Iz= -ħ/2 • and  are orthonormal

6. Ah, physics • Motion of an electric charge around closed loop produces a magnetic dipole: μ = iA i= current (amperes) A= area of loop (m2) • Substitution of i=qv/2πr and A=πr2 μ = qrv/2 • Noncircular orbit μ= q(r×v)/2

7. Physics • Express μin terms of angular momentum, L • L = r×p and p =mv μ = (q/2m)L • Replace classical angular momentum with spin angular momentum, I μ = gN(q/2mN)I = gNβNI = I gN= nuclear g factor, βN= nuclear magneton, mN= mass of nucleus, = gN βN=magnetogyric ratio

8. Physics • Magnetic dipole wants to align itself with magnetic field • Potential energy, V, for the process V = -μ•B where F = q(v×B) • Take magnetic field to be in the z direction: V = -μzBz = -γBzIz

9. Dipping into Quantum Mechanics… • Replace Iz by its operator equivalent, Îz • Can now write the spin Hamiltonian Ĥ = -γBzÎz • Corresponding Schrödinger equation Ĥ = -γBzÎz  = E  • Wave functions are the spin eigenfunctions Îz 1 = - ħγm1Bz E = - ħγm1Bz

10. Energy differences • Interested in transitions between alignment with the field (m1= ½) and against the magnetic field (m1= -½) • Energy difference E = E(m1= -½) –E(m1= ½) = ħγBz • Note that  E depends upon strength of magnetic field

11. Condition for resonance • Sample is irradiated with electromagnetic radiation • When E matches the energy of the radiation: • The proton will make a transition from the lower energy state to the higher energy state, • The sample will absorb and give the NMR spectrum • Condition for resonance/absorption E = ħγBz= hν

12. Shielding • Frequency of associated transition: ν = γBz/2π Bz = magnetic field experienced by nucleus • Seems all protons would absorb at the same frequency • Account for magnetic field induced by moving electrons • Total magnetic field = sum of applied field and shielding field B0 = (2)/((1-))

13. Resonance Frequency and Chemical Shift • Resonance frequency H = ((γB0)/(2))(1- H) • Chemical shift H = ((H - TMS)/spectrometer)  106 • Degree of shielding  with  electron density • Greater electron density = smaller chemical shift • Deshielded – left, downfield, weak field • Well-shielded – right, upfield, strong field

14. Why does splitting occur? • Any given hydrogen is also acted upon by the magnetic field due to the magnetic dipoles of neighboring hydrogen nuclei • Effect is to split the signal of the given hydrogen nuclei into multiplets

15. A quantitative approach: Step 1 • Hamiltonian that accounts for spin-spin interaction Ĥ = -γB0(1- 1)Îz1- γB0(1- 2)Îz2+ (hJ12/ħ) Î1Î2 J12 = spin-spin coupling constant

16. Step 2: Perturbation theory • Assume first-order perturbation theory is adequate • Unperturbed Hamiltonian Ĥ(0) = -γB0(1- 1)Îz1- γB0(1- 2)Îz2 • Perturbation term Ĥ(1) = (hJ12/ħ) Î1Î2

17. Step 3: Solve Schrödinger Eqn • Unperturbed wave function 1 = (1)(2) 2 = β (1)(2) 3 = (1)β(2) 4 = β(1)β(2) • Energy equation through first order Ej = Ej(0) + d1d2 j*Ĥ(1) j • Solve unperturbed and perturbed portions separately

18. Step 3: Solve Schrödinger Eqn • For unperturbed part, recall Ĥ(0) j = Ej(0) j • For first-order corrections Hii(1) i = (hJ12/ħ)d1d2i*Î1Î2 i • Turns out that only z components contribute to first-order energies

19. Energies and selection rules • Only one type of nucleus at a time can undergo a transition

20. First-order Spectra • Resonance frequencies • Occur as a pair of two closely spaced lines  doublet • Condition for use of first-order perturbation theory J12 << 01- 2  • Leads to two separated doublets, which is called a first-order spectrum

21. The Case of Equivalent Protons • Similar calculations • Two shielding constants are equal • Equivalent, indistinguishable nulcei  wave functions are combinations • Spin-spin coupling constant effect cancels in the transition frequencies due to selection rules • Single proton resonance observed

22. Visiting the Variational Method • Second-order spectra can be calculated exactly • Same Hamiltonian • Linear combination of possible wave functions as trial function  = c11+c2  2+c3  3+c4  4

23. Variational Method • Minimize E= (d1d2*Ĥ ) / (d1d2*) • Secular determinant

24. First- or second-order? • Observed spectra depend upon the relative values of 01- 2  and J • J = 0  two separate singlets; two distinct hydrogen nuclei with no coupling • 1 = 2  two chemically equivalent protons with one signal • Cases between these conditions, the spectrum can varies • This is a second-orderspectrum

25. First- and Second-order Examples • Spectrum depends upon field strength, B, because  depends upon B

26. Conclusion • Classical physics behind Nuclear Magnetic Resonance Spectroscopy • Chemical shifts • Quantum mechanical methods used to determine spectra • Splitting patterns

27. References • D.A. McQuarrie, J.D. Simon, Physical Chemistry: A Molecular Approach, University Science books, CA. 1997. • D.L. Pavia, G.M. Lampman, G.S. Kriz, Introduction to Spectroscopy, 3rd ed. Thomson Learning, Inc. 2001. • F.L. Pilar, Elementary Quantum Chemsitry, 2nd Ed. Dover Publications, Inc. NY. 1990.