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Calculation of IR & NMR SpectraCalculation of IR & NMR Spectra
((Measuring nuclear vibrations and spins)Measuring nuclear vibrations and spins)((Measuring nuclear vibrations and spins)Measuring nuclear vibrations and spins)
20132013
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Lecture OutlineLecture Outline�� EMEM spectrumspectrum
�� IRIR –– vibrations of nuclei on the electronic PESvibrations of nuclei on the electronic PES�� TheoryTheory
�� Calculation schemeCalculation scheme
�� Strengths &Strengths & limitationslimitations
�� Calculations vs.Calculations vs. experimentexperiment
�� NMRNMR –– effect of electronic environment on effect of electronic environment on
nuclear spin transitionsnuclear spin transitions�� TheoryTheory
�� Calculation of Calculation of shielding tensorshielding tensor
�� Calculation vs.Calculation vs. experimentexperiment
�� Advanced topic:Advanced topic: Calculation of Calculation of spinspin--spin couplingspin coupling
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EM spectrumEM spectrumEM spectrumEM spectrum
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Electromagnetic (EM) Spectrum
• Examples: X rays, UV, visible light, IR,
microwaves, and radio waves.
• Frequency and wavelength are inversely
proportional:proportional:
c = λν(c is the speed of light)
Energy per photon = hν, where h is Planck’s
constant.
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IR spectroscopy IR spectroscopy IR spectroscopy IR spectroscopy
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�� InfraredInfrared (IR)(IR) spectroscopyspectroscopy measuresmeasures thethebondbond vibrationvibration frequenciesfrequencies inin aa moleculemoleculeandand isis usedused toto determinedetermine thethe functionalfunctionalgroupgroup andand toto confirmconfirm moleculemolecule--widewidegroupgroup andand toto confirmconfirm moleculemolecule--widewidestructurestructure (“fingerprint”)(“fingerprint”)..
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IR: Some theoryIR: Some theory
H=HH=Hee+H+Hnn
ΨΨ= = ΨΨe e ΨΨnn
(T(T +E+E (R)) (R)) ΨΨ =E=E ΨΨ
Born-Oppenheimer calculation of the PES:
(T(Tkk+E+Eee(R)) (R)) ΨΨnn=E=En n ΨΨnn
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Separation of Vibrational and Rotational motion
((with good accuracy)with good accuracy)
HHnn=H=Hvv+H+Hrr
ΨΨnn==ΨΨvvΨΨrrΨΨnn==ΨΨvvΨΨrr
We are interested in the vibrational spectrum
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Harmonic oscillator
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Diatomic molecule: 1-D PES
In the vicinity of re , the potential looks like a HO!!!
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Harmonic approximation near Harmonic approximation near
the energy minimumthe energy minimum
00
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Molecular vibrationsMolecular vibrations
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How can we extract How 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 our That’s the potential from our
previous calculations!!! previous calculations!!!
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Steps of calculationsSteps of calculations
1.1. Calculate the potential = VCalculate the potential = Ve e (R)(R)
2.2. Calculate Calculate ωω22
3.3. Work done! We know the spectrumWork done! We know the spectrum
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Polyatomic moleculesPolyatomic molecules
�� Normal modes (classical)Normal modes (classical)
�� Question: how many internal degrees of freedom for Question: how many internal degrees of freedom for
molecule with N atoms ?molecule with N atoms ?
Answer: Answer: Answer: Answer:
33NN--5 5 for linear moleculefor linear molecule
33NN--6 6 for nonlinearfor nonlinear
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Normal modesNormal modes
For molecule with N atoms every vibration can be For molecule with N atoms every vibration can be
expand as a sum of expand as a sum of 33NN--66 ((33NN--55) independent ) independent
modes i.e. in the vicinity of the equilibrium modes i.e. in the vicinity of the equilibrium
geometry we have geometry we have 33NN--6 6 independentindependent harmonic harmonic geometry we have geometry we have 33NN--6 6 independentindependent harmonic harmonic
oscillators oscillators
(with frequencies (with frequencies ωωii==11...(...(33NN--66))). ).
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Example: WaterExample: Water
Number of modes = 3x3-6 = 3
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Transitions in stateTransitions in state--spacespace
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• 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:
( ) ( ) ( )( ) ( )( ) ( )21 12
e
nn
e e e e e en n
R R
d VV R V R V R R R V R R R R R
dR=
′ ′′= + − + − + + − +
⋯ ⋯
( )ˆe NN e eH V Vψ ψ+ =
Normal Mode Calculation
2
.i j eq
V
R R
∂ ∂ ∂
!
• 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)The rest are vibrational modes.
22
.
1ij
i ji j eq
k VH
m R Rm mω
∂= ⇒ = ∂ ∂
( )3
, 1
0N
ij ij k jki j
H lδ λ=
− =∑0ij ij kH δ λ− =
.eq
2 2kk
k
λων
π π= =
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Normal Mode CalculationNormal Mode Calculation
Solve the electronic Solve the electronic The Hessian is the The Hessian is the Diagonalize the Diagonalize the Vibrational Vibrational Solve the electronic Solve the electronic
BO problem at each BO problem at each
nuclear configuration nuclear configuration
to get PES.to get PES.
The Hessian is the The Hessian is the
massmass--weighted matrix weighted matrix
of the secondof the second--order order
derivatives.derivatives.
Diagonalize the Diagonalize the
Hessian by solving Hessian by solving
the secular equation, the secular equation,
finding finding 33N roots, six N roots, six
(or five) of which (or five) of which
aren’t vibrations.aren’t vibrations.
Vibrational Vibrational
frequencies are frequencies are
related to the related to the
square root of square root of
the eigenvalues.the eigenvalues.
•• What would it mean if we got What would it mean if we got too few nontoo few non--zero rootszero roots??
•• When would we get one When would we get one negative vibrational frequencynegative vibrational frequency??
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Stretching Frequencies
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Molecular FingerprintMolecular Fingerprint
�� WholeWhole--molecule vibrations and bending molecule vibrations and bending
vibrations are also quantized.vibrations are also quantized.
�� No two molecules will give exactly the same IR No two molecules will give exactly the same IR
spectrum (except enantiomers).spectrum (except enantiomers).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: Simple stretching: 16001600--3500 3500 cmcm--11..
�� Complex vibrations: Complex vibrations: 600600--1400 1400 cmcm--11,,
called the “fingerprint region.”called the “fingerprint region.”
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An Alkane IR SpectrumAn Alkane IR Spectrum
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Summary of IR AbsorptionsSummary of IR Absorptions
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Strengths and LimitationsStrengths and Limitations
�� IR alone cannot IR alone cannot determine determine a structure.a structure.
�� Some signals may be Some signals may be ambiguousambiguous..
�� Functional groups Functional groups are usually indicated.are usually indicated.
The The absence absence of a signal is definite proof that of a signal is definite proof that �� The The absence absence of a signal is definite proof that of a signal is definite proof that
the functional group is the functional group is absentabsent..
�� Correspondence Correspondence with a known sample’s IR with a known sample’s IR
spectrum confirms the identity of the spectrum confirms the identity of the
compound. compound.
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IR Calculation vs. ExperimentIR Calculation vs. Experiment
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NMR spectroscopy NMR spectroscopy
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NMR: Background
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Some theorySome theory
�� If the If the nuclear spinnuclear spin I=I=00, then the , then the nuclear angular nuclear angular
momentummomentum, , p=p=00 (nucleus doesn’t “spin”).(nucleus doesn’t “spin”).
�� If If I>I>00 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 a there is a nuclear magnetic dipole momentnuclear magnetic dipole moment
Gyromagnetic ratio
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Magnetic moment of a proton Magnetic moment of a proton
Nucleus g factorNucleus g factor
Length of vector =Length of vector =
In the absence of magnetic field all In the absence of magnetic field all 22II++11directions of the spin directions of the spin
are equiprobableare equiprobable
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Nuclear Spin
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�� In the In the presence of magnetic fieldpresence of magnetic field,,
there is an there is an interactioninteraction between the field and the between the field and the
magnetic moment:magnetic moment:
�� If the field is in the If the field is in the z directionz direction
There are There are 22I+I+11 values of values of IIzz There are There are 22I+I+11 values of energyvalues of energy
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Example: protonExample: proton
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Two Energy StatesTwo Energy States
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Shielding by the Electronic Environment
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Shielding and Resonance Frequency
Shielding effects can be taken into account by the expression:B0 is the applied magnetic field strength and theσi is the shielding factor
0 0iB B Bσ= −
0 (1 ) [nucleus ]2i i
Bi
γν σ
π⇒ = −(1 ) [nucleus ]
2i i iν σπ
⇒ = −
0
6
0
chemical shift
(1 )2
( )2
11
0 in
ref ref
i ref ref i
i refi
ref i
ref ref
B
p
B
pm
γν σ
πγ
ν ν σ σπ
ν νσ
δσ σ
ν−
= −
− = −
− −⇒ = =
−
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NMR SignalsNMR Signals
�� The The number number of signals shows how many of signals shows how many
different kinds of protons are present.different kinds of protons are present.
�� The The location location of the signals shows how shielded of the signals shows how shielded
or deshielded the proton is.or deshielded the proton is.or deshielded the proton is.or deshielded the proton is.
�� The The intensity intensity of the signal shows the number of of the signal shows the number of
protons of that type.protons of that type.
�� Signal Signal splitting splitting shows the number of protons shows the number of protons
on adjacent atoms. on adjacent atoms.
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Calculation of the Shielding Tensor
{ |k>0,εk0} φ { |k>,εk} χ,σ
Calculate Calculate Choose gauge by Choose gauge by Calculate new SCF Calculate new SCF Calculate Calculate Calculate Calculate
zerozero--field SCFfield SCF
Choose gauge by Choose gauge by
which to enter the which to enter the
magnetic vector magnetic vector
potentialpotential
Calculate new SCF Calculate new SCF
for nonfor non--zero field.zero field.
Use the zeroUse the zero--field field
SCF results as the SCF results as the
initial guessinitial guess
Calculate Calculate
shielding tensor, shielding tensor,
susceptibility, etc. susceptibility, etc.
using the using the
nonnon--zero field zero field
electron structureelectron structure
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Basic Calculation(single molecule in gas phase)
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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 of spsp33 proton kinds may be proton kinds may be differentdifferent, ,
e.g. e.g. HH33CNHF).CNHF).
•• The The location location of the signals is given relative to a reference of the signals is given relative to a reference •• The The location location of the signals is given relative to a reference of 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 zero are zero
(no solvent or temperature effects, (no solvent or temperature effects, TT==00).).
•• Signal Signal splitting splitting can be calculated separately, can be calculated separately,
e.g. using Ge.g. using G0303::
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Calc. of indirect dipoleCalc. of indirect dipole--dipole couplingdipole coupling
�� Direct dipoleDirect dipole--dipole coupling becomes negligible for dipole coupling becomes negligible for closedclosed--shell shell systems at high temperature (existence of systems at high temperature (existence of intermolecular collisionsintermolecular collisions).).
�� GAUSSIAN keyword option GAUSSIAN keyword option NMR=SpinSpinNMR=SpinSpin
Calculates Calculates four four contributions to contributions to isotropic isotropic spinspin--spin spin coupling:coupling:
1.1. Paramagnetic spinParamagnetic spin--orbit coupling (PSO)orbit coupling (PSO)
2.2. Diamagnetic spinDiamagnetic spin--orbit coupling (DSO)orbit coupling (DSO)
3.3. SpinSpin--dipolar coupling (SD)dipolar coupling (SD)
4.4. Fermi contact interaction (FC)Fermi contact interaction (FC)
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NMR: Summary