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

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demystifying the black box the mechanics behind nuclear magnetic resonance spectroscopy

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

Kelsie Betsch

Chem 381

Spring 2004

what is nmr good for
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
  • 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
  • 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
spin eigenvalue equations
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
ah physics
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

  • 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

  • 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

dipping into quantum mechanics
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

energy differences
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
condition for resonance
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

  • 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-))

resonance frequency and chemical shift
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
why does splitting occur
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
a quantitative approach step 1
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

step 2 perturbation theory
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

step 3 solve schr dinger eqn
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
step 3 solve schr dinger eqn1
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
energies and selection rules
Energies and selection rules
  • Only one type of nucleus at a time can undergo a transition
first order spectra
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
the case of equivalent protons
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
visiting the variational method
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

variational method
Variational Method
  • Minimize

E= (d1d2*Ĥ ) / (d1d2*)

  • Secular determinant
first or second order
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
first and second order examples
First- and Second-order Examples
  • Spectrum depends upon field strength, B, because  depends upon B
  • Classical physics behind Nuclear Magnetic Resonance Spectroscopy
    • Chemical shifts
  • Quantum mechanical methods used to determine spectra
    • Splitting patterns
  • 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.