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

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

EM spectrumEM spectrumEM spectrumEM spectrum

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.

IR spectroscopy IR spectroscopy IR spectroscopy IR spectroscopy

�� 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”)..

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

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

Harmonic oscillator

Diatomic molecule: 1-D PES

In the vicinity of re , the potential looks like a HO!!!

Harmonic approximation near Harmonic approximation near

the energy minimumthe energy minimum

00

Molecular vibrationsMolecular vibrations

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!!!

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

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

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))). ).

Example: WaterExample: Water

Number of modes = 3x3-6 = 3

Transitions in stateTransitions in state--spacespace

• 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= =

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??

Stretching Frequencies

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.”

An Alkane IR SpectrumAn Alkane IR Spectrum

Summary of IR AbsorptionsSummary of IR Absorptions

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.

IR Calculation vs. ExperimentIR Calculation vs. Experiment

NMR spectroscopy NMR spectroscopy

NMR: Background

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

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

Nuclear Spin

�� 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

Example: protonExample: proton

Two Energy StatesTwo Energy States

Shielding by the Electronic Environment

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−

= −

− = −

− −⇒ = =

−

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.

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

Basic Calculation(single molecule in gas phase)

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::

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)

NMR: Summary