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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.