Tutorial in calculation of IR & NMR spectra (i.e. measuring nuclear vibrations and spins) using the GAUSSIAN03 computational chemistry package. Following an introduction to spectroscopy in general, each of the two measurement types is presented in sequence. For each one, we review the theory before presenting the calculation scheme. We then present the relative strengths and limitations (with respect to other measurements), and then compare the calculation method with experimentation. We close each of the two subjects with an advanced topic: Raman IR spectroscopy (and depolarization ratio), and indirect dipole coupling (a.k.a. spin-spin coupling). I've also made the last part available as a standalone presentation: http://www.slideshare.net/InonSharony/nmr-spinspin-splitting-using-gaussian03.

1. Calculation of IR & NMR SpectraCalculation of IR & NMR Spectra
(Measuring nuclear vibrations and spins)(Measuring nuclear vibrations and spins)
Inon SharonyInon Sharony
20122012

2. Lecture OutlineLecture Outline EMEM spectrumspectrum
 IRIR – vibrations of nuclei on the electronic PES– vibrations of nuclei on the electronic PES
 TheoryTheory
 Calculation schemeCalculation scheme
 Strengths & limitationsStrengths & limitations
 Calculations vs. experimentCalculations vs. experiment
 Advanced topic:Advanced topic: RamanRaman spectroscopyspectroscopy
 NMRNMR – effect of electronic environment on– effect of electronic environment on
nuclear spin transitionsnuclear spin transitions
 TheoryTheory
 Calculation ofCalculation of shielding tensorshielding tensor
 Calculation vs. experimentCalculation vs. experiment
 Advanced topic:Advanced topic: Calculation ofCalculation of spin-spin couplingspin-spin coupling

3. EM spectrumEM spectrum

4. Electromagnetic (EM) Spectrum
• Examples: X rays, UV, visible light, IR,
microwaves, and radio waves.
• Frequency and wavelength are inversely
proportional:
c = λν
(c is the speed of light)
Energy per photon = hν, where h is Planck’s
constant.

6. IR spectroscopyIR spectroscopy

7. IR OutlineIR Outline
 TheoryTheory
 Harmonic approximationHarmonic approximation
 Normal modesNormal modes
 State-spaceState-space
 Calculation schemeCalculation scheme
 PESPES
 HessianHessian
 DiagonalizationDiagonalization
 Strengths & limitationsStrengths & limitations
 Calculation vs. experimentsCalculation vs. experiments
 Raman spectroscopy and depolarization ratioRaman spectroscopy and depolarization ratio

8.  Infrared (IR) spectroscopy measures theInfrared (IR) spectroscopy measures the
bond vibration frequencies in a moleculebond vibration frequencies in a molecule
and is used to determine the functionaland is used to determine the functional
group and to confirm molecule-widegroup and to confirm molecule-wide
structure (“fingerprint”).structure (“fingerprint”).

9. IR: Some theoryIR: Some theory
H=HH=Hee+H+Hnn
ΨΨ== ΨΨee ΨΨnn
(T(Tkk+E+Eee(R))(R)) ΨΨnn=E=Enn ΨΨnn
Born-Oppenheimer calculation of the PES:

10. Separation of Vibrational and
Rotational motion
(with good accuracy)(with good accuracy)
HHnn=H=Hvv+H+Hrr
ΨΨnn==ΨΨvvΨΨrr
We are interested in the vibrational spectrum

11. Harmonic oscillator

12. In the vicinity of re , the potential looks like a HO!!!
Diatomic molecule: 1-D PES

13. Harmonic approximation nearHarmonic approximation near
the energy minimumthe energy minimum
0

14. Molecular vibrationsMolecular vibrations

15. How can we extractHow can we extract
the vibrational frequencies (the vibrational frequencies (ωω))
if the potential is known?if the potential is known?
That’s the potential from ourThat’s the potential from our
previous calculations!!!previous calculations!!!

16. Steps of calculationsSteps of calculations
1.1. Calculate the potential = VCalculate the potential = Vee (R)(R)
2.2. CalculateCalculate ωω22
3.3. Work done! We know the spectrumWork done! We know the spectrum

17. Polyatomic moleculesPolyatomic molecules
 Normal modes (classical)Normal modes (classical)
 Question: how many internal degrees of freedom forQuestion: how many internal degrees of freedom for
molecule with N atoms ?molecule with N atoms ?
Answer:Answer:
3N-5 for linear molecule3N-5 for linear molecule
3N-6 for nonlinear3N-6 for nonlinear

18. Normal modesNormal modes
For molecule with N atoms every vibration can beFor molecule with N atoms every vibration can be
expand as a sum ofexpand as a sum of 3N-63N-6 ((3N-53N-5) independent) independent
modes i.e. in the vicinity of the equilibriummodes i.e. in the vicinity of the equilibrium
geometry we havegeometry we have 3N-63N-6 independentindependent harmonicharmonic
oscillatorsoscillators
(with frequencies(with frequencies ωωii=1...(3N-6)=1...(3N-6)).).

19. Example: WaterExample: Water
Number of modes = 3x3-6 = 3

20. Transitions in state-spaceTransitions in state-space

22. Some technical details, or,Some technical details, or,
at the end of the day –at the end of the day –
how I can find your Normal Modes???how I can find your Normal Modes???

23. • The harmonic vibrational spectrum of the 3N-dim. PES:
• Solve the B-O electronic Hamiltonian at each nuclear configuration to
produce the PES, V(R):
• Create the force constant (k) matrix,
which is the matrix of second-order derivatives:
• The mass-weighted matrix is the Hessian:
• Diagonalize the Hessian to get eigenvalues, λk, and eigenvectors, ljk:
• Find the 3N roots of the secular equation
• Six of the roots should be zero
(rigid body degrees of freedom: Translation and rotation)
( ) ( ) ( ) ( ) ( ) ( ) ( )
21 1
2
e
n
n
e e e e e en n
R R
d V
V R V R V R R R V R R R R R
dR =
 
′ ′′= + − + − + + − + ÷
 
L L
( )ˆ
e NN e eH V Vψ ψ+ =
Normal Mode Calculation
2
2
.
1
ij
i ji j eq
k V
H
m R Rm m
ω
 ∂
= ⇒ = ÷ ÷∂ ∂ 
( )
3
, 1
0
N
ij ij k jk
i j
H lδ λ
=
− =∑
0ij ij kH δ λ− =
2
.i j eq
V
R R
 ∂
 ÷ ÷∂ ∂ 
2 2
kk
k
λω
ν
π π
= =
!

24. Normal Mode CalculationNormal Mode Calculation
Solve the electronicSolve the electronic
BO problem at eachBO problem at each
nuclear configurationnuclear configuration
to get PES.to get PES.
The Hessian is theThe Hessian is the
mass-weighted matrixmass-weighted matrix
of the second-orderof the second-order
derivatives.derivatives.
Diagonalize theDiagonalize the
Hessian by solvingHessian by solving
the secular equation,the secular equation,
finding 3N roots, sixfinding 3N roots, six
(or five) of which(or five) of which
aren’t vibrations.aren’t vibrations.
VibrationalVibrational
frequencies arefrequencies are
related to therelated to the
square root ofsquare root of
the eigenvalues.the eigenvalues.
• What would it mean if we gotWhat would it mean if we got too few non-zero rootstoo few non-zero roots??
• When would we get oneWhen would we get one negative vibrational frequencynegative vibrational frequency??

25. Stretching Frequencies

26. Molecular FingerprintMolecular Fingerprint
 Whole-molecule vibrations and bendingWhole-molecule vibrations and bending
vibrations are also quantized.vibrations are also quantized.
 No two molecules will give exactly the same IRNo two molecules will give exactly the same IR
spectrum (except enantiomers).spectrum (except enantiomers).
 Delocalized vibrations have lower energy (cf.Delocalized vibrations have lower energy (cf.
“particle in a box”):“particle in a box”):
 Simple stretching: 1600-3500 cmSimple stretching: 1600-3500 cm-1-1
..
 Complex vibrations: 600-1400 cmComplex vibrations: 600-1400 cm-1-1
,,
called the “fingerprint region.”called the “fingerprint region.”

27. An Alkane IR SpectrumAn Alkane IR Spectrum

28. Summary of IR AbsorptionsSummary of IR Absorptions

29. Strengths and LimitationsStrengths and Limitations
 IR alone cannotIR alone cannot determinedetermine a structure.a structure.
 Some signals may beSome signals may be ambiguousambiguous..
 Functional groupsFunctional groups are usually indicated.are usually indicated.
 TheThe absenceabsence of a signal is definite proof thatof a signal is definite proof that
the functional group isthe functional group is absentabsent..
 CorrespondenceCorrespondence with a known sample’s IRwith a known sample’s IR
spectrum confirms the identity of thespectrum confirms the identity of the
compound.compound.

30. IR Calculation vs. ExperimentIR Calculation vs. Experiment

31. Raman SpectroscopyRaman Spectroscopy The elastic scattering (Rayleigh scattering) of a photon from a molecule is aThe elastic scattering (Rayleigh scattering) of a photon from a molecule is a
second-order process:second-order process:
 Photon with energy hPhoton with energy hνν is absorbed by a real energy level.is absorbed by a real energy level.
 Photon with energy hPhoton with energy hνν is emitted, returning the system to its original state.is emitted, returning the system to its original state.
 In contrast, Raman scattering is a third-order process by which aIn contrast, Raman scattering is a third-order process by which a photon isphoton is
inelastically scatteredinelastically scattered from, e.g. a molecule:from, e.g. a molecule:
 Photon with energy hPhoton with energy hνν is absorbed by virtual energy level.is absorbed by virtual energy level.
 Vibrational mode of energy ħVibrational mode of energy ħωω is populated (or de-populated), bringing the system to ais populated (or de-populated), bringing the system to a
non-virtual state.non-virtual state.
 Photon with energy hPhoton with energy hνν-ħ-ħωω (or hv+ħ(or hv+ħωω, respectively) is emitted., respectively) is emitted.
 The scattered photon may have less (or more) energy than the impinging photon.The scattered photon may have less (or more) energy than the impinging photon.
This is calledThis is called StokesStokes scattering (or anti-Stokes, respectively).scattering (or anti-Stokes, respectively).
 If the energy difference between the scattered and impinging photons is exactlyIf the energy difference between the scattered and impinging photons is exactly
equal to some linear combination of vibrational energies, the scattering isequal to some linear combination of vibrational energies, the scattering is
resonantresonant, and therefore has a much greater amplitude. This is known as chemical, and therefore has a much greater amplitude. This is known as chemical
amplification. Resonance Raman scattering should not to be confused withamplification. Resonance Raman scattering should not to be confused with
fluorescence.fluorescence.
 The intensity of the elastically scattered photons is stronger by someThe intensity of the elastically scattered photons is stronger by some seven ordersseven orders
of magnitudeof magnitude..

32. Raman SpectroscopyRaman Spectroscopy
 Regular scattering spectroscopy probes the coupling of theRegular scattering spectroscopy probes the coupling of the dipoledipole
moment to the electric field (for example, for an electric field withmoment to the electric field (for example, for an electric field with
frequency in the IR range).frequency in the IR range).
 Raman spectroscopy probes the coupling of a molecular vibration to theRaman spectroscopy probes the coupling of a molecular vibration to the
electric field, and is therefore a measure of the molecularelectric field, and is therefore a measure of the molecular polarizabilitypolarizability
associated with that vibrational mode. If the polarizability of a givenassociated with that vibrational mode. If the polarizability of a given
vibrational mode is non-zero, the mode is said to be Raman active.vibrational mode is non-zero, the mode is said to be Raman active.
 Raman spectroscopy is sensitive to the initial populations of theRaman spectroscopy is sensitive to the initial populations of the
molecular vibrational states, and can be used as a remote sensingmolecular vibrational states, and can be used as a remote sensing
thermometer:thermometer:

33. • The polarizability is a tensor of rank two (a matrix),
describing the response of the system polarization to
an external electric field:
• It can be broken down as a sum of:
– An isotropic part (scalar),
– A symmetric anisotropic part (symmetric matrix),
– An antisymmetric anisotropic part (antisym.mat.),
• In the axis system where the polarizability is
diagonalized, the depolarization ratio is defined
• The depolarization ratio is a measure of the symmetry
of a molecular vibrational mode:
Raman Spectroscopy
P Eα=
r r
.iso
α
.aniso
S α  
.aniso
A α  
( )
2
.
. .
3
45 4
aniso
depolarized
P aniso aniso
polarized
AI I
I I S A
α
ρ
α α
⊥
  
= = =
   +   P

34. • For a vibrational mode which is congruent with
the molecular symmetry group,
and then, .The molecule does not
polarize the Raman signal.
• For a vibrational mode which is not congruent
with any of the molecular symmetries,
and then, . The molecule de-polarizes
the Raman signal.
• Any other mode has . The Raman
signal is somewhat polarized by the molecule.
Raman Depolarization
.
0aniso
A α  = 
0Pρ =
.
0aniso
S α  = 
3
4Pρ =
30
4Pρ< <

36. NMR spectroscopyNMR spectroscopy

37. NMR OutlineNMR Outline TheoryTheory
 Nuclear Magnetic ResonanceNuclear Magnetic Resonance
 ShieldingShielding
 Reference compoundsReference compounds
 Chemical shiftChemical shift
 Spin-spin couplingSpin-spin coupling
 ExamplesExamples
 Calculation of shielding tensorCalculation of shielding tensor
 Magnetic vector potential in the HamiltonianMagnetic vector potential in the Hamiltonian
 Gauge invarianceGauge invariance
 Gauge-Independent Atomic Orbitals –Gauge-Independent Atomic Orbitals – GIAOGIAO
 Diamagnetic and paramagnetic contributionsDiamagnetic and paramagnetic contributions
 Calculation vs. experimentCalculation vs. experiment
 Calculation ofCalculation of spin-spin couplingspin-spin coupling
 Indirect dipole-dipole couplingsIndirect dipole-dipole couplings
 Fermi Contact interactionFermi Contact interaction
 Isotope effectIsotope effect
 J-couplingJ-coupling

38. NMR: Background

39. Some theorySome theory
 If theIf the nuclear spinnuclear spin I=0I=0, then the, then the nuclear angularnuclear angular
momentummomentum,, p=0p=0 (nucleus doesn’t “spin”).(nucleus doesn’t “spin”).
 IfIf I>0I>0 then the nuclear angular momentumthen the nuclear angular momentum
 Since the nucleus is charged and spinning,Since the nucleus is charged and spinning,
there is athere is a nuclear magnetic dipole momentnuclear magnetic dipole moment
Gyromagnetic ratio

40. Magnetic moment of a protonMagnetic moment of a proton
Nucleus g factorNucleus g factor
Length of vector =Length of vector =
In the absence of magnetic field allIn the absence of magnetic field all 22II+1+1 directions of the spindirections of the spin
are equiprobableare equiprobable

41. Nuclear Spin

42.  In theIn the presence of magnetic fieldpresence of magnetic field,,
there is anthere is an interactioninteraction between the field and thebetween the field and the
magnetic moment:magnetic moment:
 If the field is in theIf the field is in the z directionz direction
There areThere are 2I+12I+1 values ofvalues of IIzz There areThere are 2I+12I+1 values of energyvalues of energy

43. Example: protonExample: proton

44. Two Energy StatesTwo Energy States

46. Shielding by the Electronic Environment

47. Shielding and Resonance Frequency

54. NMR SignalsNMR Signals
 TheThe numbernumber of signals shows how manyof signals shows how many
different kinds of protons are present.different kinds of protons are present.
 TheThe locationlocation of the signals shows how shieldedof the signals shows how shielded
or deshielded the proton is.or deshielded the proton is.
 TheThe intensityintensity of the signal shows the numberof the signal shows the number
of protons of that type.of protons of that type.
 SignalSignal splittingsplitting shows the number of protonsshows the number of protons
on adjacent atoms.on adjacent atoms.

55. Calculation of the Shielding Tensor
(Von Vleck, Ramsey)
• Scheme: We compute the electronic structure given the external (homogeneous)
magnetic field, H, and get the self-consistent magnetic field, B (includes the effects of
the motion of the electrons themselves).
• The magnetic field enters the elec. Hamiltonian in the form of the vector potential, A.
– i is the index of the electrons.
• The Maxwell equations allow some freedom, e.g. in the choice of vector potential
gauge: since for any choice of φ.
– a is the index of the nuclei.
• This “gauge invariance” allows different choices of gauge:
– Continuous Set of Gauge Transformations (CSGT) -- requires large basis set.
– IGAIM -- atomic centers as gauge origins
– Single gauge Origin
– Gauge-Independent Atomic Orbital (GIAO) -- default (Hameka, 1962):
∗ The results are gauge invariant, and therefore independent of q. Eventually, q is taken
equal to Ra, but not before the derivation is completed. Since the origin of the gauge is
identical to the origin of the AOs, this choice is named GIAO.

56. Calculation of the Shielding Tensor
(Von Vleck, Ramsey)
• Computed quantities (derived using the theory of variations):
– Magnetic susceptibility -- needs optional NMR=Susceptibility keyword option
∗ The first term on the RHS is the diamagnetic term and the second is the paramagnetic
term.
∗ |0> is the ground state and |k> is the k-th excited state, both with respect to the
perturbed Hamiltonian (i.e. for non-zero external magnetic field).
∗ M is the electron angular momentum operator.
– Shielding tensor
∗ La is the electron angular momentum operator, relative to the atom with index a , for
which the shielding is calculated.
∗ σ is, in general, a tensor of rank 2 (a matrix). In more complicated (anisotropic) NMR
experiments, directionality matters, and the shielding cannot be assumed a scalar.

57. Calculation of the Shielding Tensor
CalculateCalculate
zero-field SCFzero-field SCF
Choose gauge byChoose gauge by
which to enter thewhich to enter the
magnetic vectormagnetic vector
potentialpotential
Calculate new SCFCalculate new SCF
for non-zero field.for non-zero field.
Use the zero-fieldUse the zero-field
SCF results as theSCF results as the
initial guessinitial guess
CalculateCalculate
shielding tensor,shielding tensor,
susceptibility, etc.susceptibility, etc.
using theusing the
non-zero fieldnon-zero field
electron structureelectron structure

58. Basic Calculation
(single molecule in gas phase)

59. Calculation vs. Experiment
(single molecule in gas phase)
• Since the calculation is done on a static molecule,Since the calculation is done on a static molecule,
no bond rotationsno bond rotations are possibleare possible
((numbernumber of spof sp33
proton kinds may be different,proton kinds may be different,
e.g.e.g. HH33CNHF).CNHF).
• TheThe locationlocation of the signals is given relative to a referenceof the signals is given relative to a reference
material calculated separately,material calculated separately, at the same calculation levelat the same calculation level..
• LinewidthsLinewidths are zeroare zero
(no solvent or temperature effects,(no solvent or temperature effects, TT11=∞).=∞).
• SignalSignal splittingsplitting can be calculated separately,can be calculated separately,
e.g. using G03:e.g. using G03:

60. Calc. of indirect dipole-dipole couplingCalc. of indirect dipole-dipole coupling
 Direct dipole-dipole coupling becomes negligible forDirect dipole-dipole coupling becomes negligible for
closed-shellclosed-shell systems at high temperature (existence ofsystems at high temperature (existence of
intermolecular collisionsintermolecular collisions).).
 GAUSSIAN keyword optionGAUSSIAN keyword option NMR=SpinSpinNMR=SpinSpin
CalculatesCalculates fourfour contributions tocontributions to isotropicisotropic spin-spinspin-spin
coupling:coupling:
1.1. Paramagnetic spin-orbit coupling (PSO)Paramagnetic spin-orbit coupling (PSO)
2.2. Diamagnetic spin-orbit coupling (DSO)Diamagnetic spin-orbit coupling (DSO)
3.3. Spin-dipolar coupling (SD)Spin-dipolar coupling (SD)
4.4. Fermi contact interaction (FC)Fermi contact interaction (FC)

61. Fermi Contact (FC) InteractionFermi Contact (FC) Interaction
 Of the four, FC is the largest contributionOf the four, FC is the largest contribution
 The only AOs which areThe only AOs which are not zero at the nucleinot zero at the nuclei
havehave s-symmetrys-symmetry (zero orbital angular(zero orbital angular
momentum).momentum).
 TheThe corecore AOs are important, so we need to useAOs are important, so we need to use
aa basis setbasis set which includes more of these.which includes more of these.
 We do thisWe do this only for FConly for FC – it isn’t necessary for– it isn’t necessary for
the other contributions.the other contributions.
 NMR=SpinSpin,NMR=SpinSpin,mixedmixed

62. Isotope Effect and J-couplingIsotope Effect and J-coupling
 GAUSSIAN gives two values for each couplingGAUSSIAN gives two values for each coupling
strength (in Hz):strength (in Hz):
 K isotopeK isotope independentindependent
 J isotopeJ isotope dependentdependent -- uses the isotopes-- uses the isotopes in the input filein the input file
 Hence the name, J-couplingHence the name, J-coupling
 ³J couplings -- between nuclei connected by 3 bonds³J couplings -- between nuclei connected by 3 bonds
(e.g. proton-proton coupling in R-C(e.g. proton-proton coupling in R-CHH=C=CHH-R).-R).
 ⁴⁴J couplings -- between nuclei connected by 4 bonds.J couplings -- between nuclei connected by 4 bonds.

63. Example: Ethanol 6-31+G(d,p)Example: Ethanol 6-31+G(d,p)

65. NMR: Summary