Calculation of IR & NMR Calculation of IR & NMR SpectraSpectra

(Measuring nuclear vibrations (Measuring nuclear vibrations and spins)and spins)

Inon SharonyInon Sharony20122012

Lecture OutlineLecture Outline EMEM spectrum spectrum IRIR – vibrations of nuclei on the – vibrations of nuclei on the

electronic PESelectronic PES TheoryTheory Calculation schemeCalculation scheme Strengths & limitationsStrengths & limitations Calculations vs. experimentCalculations vs. experiment Advanced topic:Advanced topic: RamanRaman spectroscopy spectroscopy

NMRNMR – effect of electronic environment – effect of electronic environment on nuclear spin transitionson nuclear spin transitions TheoryTheory Calculation of Calculation of shielding tensorshielding tensor Calculation vs. experimentCalculation vs. experiment Advanced topic:Advanced topic: Calculation of Calculation of spin-spin couplingspin-spin coupling

EM spectrumEM spectrum

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.

IR spectroscopy IR spectroscopy

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

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

IR: Some theoryIR: Some theoryH=HH=Hee+H+Hnn

ΨΨ= = ΨΨe e ΨΨnn

(T(Tkk+E+Eee(R)) (R)) ΨΨnn=E=En n ΨΨnn

Born-Oppenheimer calculation of the PES:

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

Harmonic oscillator

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

Diatomic molecule: 1-D PES

Harmonic approximation Harmonic approximation near the energy near the energy

minimumminimum

0

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 That’s the potential from our previous our previous

calculations!!! calculations!!!

Steps of calculationsSteps of calculations1.1. 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 Question: how many internal degrees of

freedom for molecule with N atoms ?freedom for molecule with N atoms ?

Answer: Answer: 3N-5 for linear 3N-5 for linear moleculemolecule3N-6 for nonlinear3N-6 for nonlinear

Normal modesNormal modes For molecule with N atoms every For molecule with N atoms every

vibration can be expand as a sum of vibration can be expand as a sum of 3N-63N-6 ( (3N-53N-5) independent modes i.e. in ) independent modes i.e. in the vicinity of the equilibrium the vicinity of the equilibrium geometry we have geometry we have 3N-6 3N-6 independentindependent harmonic oscillators harmonic oscillators

(with frequencies (with frequencies ωωii=1...(3N-6)=1...(3N-6)). ).

Example: WaterExample: Water

Number of modes = 3x3-6 = 3

Transitions in state-Transitions in state-spacespace

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 how I can find your Normal Modes???Modes???

• 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)The rest are vibrational modes.

21 12

e

nn

e e e e e en nR R

d VV R V R V R R R V R R R R RdR

ˆe NN e eH V V

Normal Mode Calculation

22

.

1ij

i ji j eq

k V Hm R Rmm

3

, 1

0N

ij ij k jki j

H l

0ij ij kH

2

.i j eq

VR R

2 2kk

k

!

Normal Mode CalculationNormal Mode Calculation

Solve the Solve the electronic BO electronic BO problem at each problem at each nuclear nuclear configuration to configuration to get PES.get PES.

The Hessian is The Hessian is the mass-the mass-weighted matrix weighted matrix of the second-of the second-order derivatives.order derivatives.

Diagonalize the Diagonalize the Hessian by Hessian by solving the solving the secular secular equation, equation, finding 3N finding 3N roots, six (or roots, six (or five) of which five) of which aren’t aren’t vibrations.vibrations.

Vibrational Vibrational frequencies frequencies are related are related to the to the square root square root of the of the eigenvalues.eigenvalues.

• What would it mean if we got What would it mean if we got too few non-zero rootstoo few non-zero roots??• When would we get one When would we get one negative vibrational frequencynegative vibrational frequency??

Stretching Frequencies

Molecular FingerprintMolecular Fingerprint

Whole-molecule vibrations and Whole-molecule vibrations and bending vibrations are also quantized.bending vibrations are also quantized.

No two molecules will give exactly the No two molecules will give exactly the same IR spectrum (except same IR spectrum (except enantiomers).enantiomers).

Delocalized vibrations have lower Delocalized vibrations have lower energy (cf. “particle in a box”):energy (cf. “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.”

An Alkane IR SpectrumAn Alkane IR Spectrum

Summary of IR Summary of IR AbsorptionsAbsorptions

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 of a signal is definite proof

that the functional group is that the functional group is absentabsent.. Correspondence Correspondence with a known sample’s with a known sample’s

IR spectrum confirms the identity of the IR spectrum confirms the identity of the compound. compound.

IR Calculation vs. IR Calculation vs. ExperimentExperiment

Raman SpectroscopyRaman Spectroscopy The elastic scattering (Rayleigh scattering) of a photon from a The elastic scattering (Rayleigh scattering) of a photon from a

molecule is a second-order process:molecule is a 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 a In contrast, Raman scattering is a third-order process by which a photon is inelastically scattered photon is inelastically 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 is populated (or de-populated), bringing the

system to a non-virtual state.system to a 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 The scattered photon may have less (or more) energy than the impinging photon. This is called impinging photon. This is called StokesStokes scattering (or anti-Stokes, scattering (or anti-Stokes, respectively).respectively).

If the energy difference between the scattered and impinging photons If the energy difference between the scattered and impinging photons is exactly equal to some linear combination of vibrational energies, the is exactly equal to some linear combination of vibrational energies, the scattering is scattering is resonantresonant, and therefore has a much greater amplitude. , and therefore has a much greater amplitude. This is known as chemical amplification. Resonance Raman scattering This is known as chemical amplification. Resonance Raman scattering should not to be confused with fluorescence.should not to be confused with fluorescence.

The intensity of the elastically scattered photons is stronger by some The intensity of the elastically scattered photons is stronger by some seven orders of magnitudeseven orders of magnitude..

Raman SpectroscopyRaman Spectroscopy Regular scattering spectroscopy probes the coupling of Regular scattering spectroscopy probes the coupling of

the the dipoledipole moment to the electric field (for example, moment to the electric field (for example, for an electric field with frequency in the IR range).for an electric field with frequency in the IR range).

Raman spectroscopy probes the coupling of a Raman spectroscopy probes the coupling of a molecular vibration to the electric field, and is molecular vibration to the electric field, and is therefore a measure of the molecular therefore a measure of the molecular polarizabilitypolarizability associated with that vibrational mode. If the associated with that vibrational mode. If the polarizability of a given vibrational mode is non-zero, polarizability of a given vibrational mode is non-zero, the mode is said to be Raman active.the mode is said to be Raman active.

Raman spectroscopy is sensitive to the initial Raman spectroscopy is sensitive to the initial populations of the molecular vibrational states, and populations of the molecular vibrational states, and can be used as a remote sensing thermometer:can be used as a remote sensing thermometer:

• 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 E

.iso.anisoS .anisoA

2.

. .

3

45 4

anisodepolarized

P aniso anisopolarized

AI II I S A

• 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

. 0anisoA 0P

. 0anisoS 34P

30 4P

NMR spectroscopy NMR spectroscopy

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 of Calculation of spin-spin couplingspin-spin coupling

Indirect dipole-dipole couplingsIndirect dipole-dipole couplings Fermi Contact interactionFermi Contact interaction Isotope effectIsotope effect J-couplingJ-coupling

NMR: Background

Some theorySome theory If the If the nuclear spinnuclear spin I=0I=0, then the , then the nuclear nuclear

angular momentumangular momentum, , p=0p=0 (nucleus (nucleus doesn’t “spin”).doesn’t “spin”).

If If I>0I>0 then the nuclear angular then the nuclear angular momentummomentum

Since the nucleus is charged and spinning,Since the nucleus is charged and spinning,there is a there is a nuclear magnetic dipole nuclear magnetic dipole momentmoment

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+1+1 directions of the spin are equiprobabledirections of the spin are equiprobable

Nuclear Spin

In the In the presence of magnetic fieldpresence of magnetic field,,there is an there is an interactioninteraction between the field between the field and the magnetic moment:and the magnetic moment:

If the field is in the If the field is in the z directionz direction

There are There are 2I+12I+1 values of values of IIzz There are There are 2I+12I+1 values of energy values of energy

Example: protonExample: proton

Two Energy StatesTwo Energy States

Shielding by the Electronic Environment

Shielding and Resonance Frequency

NMR SignalsNMR Signals

The The number number of signals shows how of signals shows how many different kinds of protons are many different kinds of protons are present.present.

The The location location of the signals shows how of the signals shows how shielded or deshielded the proton is.shielded or deshielded the proton is.

The The intensity intensity of the signal shows the of the signal shows the number of protons of that type.number of protons of that type.

Signal Signal splitting splitting shows the number of shows the number of protons on adjacent atoms. protons on adjacent atoms.

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 .

– 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.

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.

Calculation of the Shielding Tensor

Calculate Calculate zero-field zero-field SCFSCF

Choose gauge Choose gauge by which to by which to enter the enter the magnetic vector magnetic vector potentialpotential

Calculate new Calculate new SCF for non-SCF for non-zero field.zero field.

Use the zero-Use the zero-field SCF field SCF results as the results as the initial guessinitial guess

Calculate Calculate shielding shielding tensor, tensor, susceptibilitysusceptibility, etc. using , etc. using the the non-zero non-zero field electron field electron structurestructure

Basic Calculation(single molecule in gas phase)

Calculation vs. Experiment(single molecule in gas phase)

• Since the calculation is done on a static Since the calculation is done on a static molecule, molecule, no bond rotationsno bond rotations are possible are possible((numbernumber of sp of sp33 proton kinds may be proton kinds may be different, different, e.g. e.g. HH33CNHF).CNHF).

• The The location location of the signals is given relative of the signals is given relative to a reference material calculated to a reference material calculated separately, 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, TT11=∞).=∞).

• Signal Signal splitting splitting can be calculated can be calculated separately, separately, e.g. using G03:e.g. using G03:

Calc. of indirect dipole-dipole Calc. of indirect dipole-dipole couplingcoupling

Direct dipole-dipole coupling becomes Direct dipole-dipole coupling becomes negligible for negligible for closed-shell closed-shell systems at high systems at high temperature (existence of temperature (existence of intermolecular intermolecular collisionscollisions).).

GAUSSIAN keyword option GAUSSIAN keyword option NMR=SpinSpinNMR=SpinSpin

Calculates Calculates four four contributions to contributions to isotropic isotropic spin-spin-spin coupling:spin 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)

Fermi Contact (FC) Fermi Contact (FC) InteractionInteraction Of the four, FC is the largest contributionOf the four, FC is the largest contribution

The only AOs which are The only AOs which are not zero at the not zero at the nuclei nuclei have have s-symmetry s-symmetry (zero orbital angular (zero orbital angular momentum).momentum).

The The core core AOs are important, so we need to AOs are important, so we need to use a use a basis set basis set which includes more of which includes more of these.these.

We do this We do this only for FC only for FC – it isn’t necessary – it isn’t necessary for the other contributions.for the other contributions.

NMR=SpinSpin,NMR=SpinSpin,mixedmixed

Isotope Effect and J-Isotope Effect and J-couplingcoupling

GAUSSIAN gives two values for each GAUSSIAN gives two values for each coupling strength (in Hz):coupling strength (in Hz): K isotope K isotope independentindependent J isotope J isotope dependent dependent -- uses the isotopes -- uses the isotopes in the in the

input fileinput file Hence the name, J-couplingHence the name, J-coupling ³J couplings -- between nuclei connected by 3 ³J couplings -- between nuclei connected by 3

bonds 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 J couplings -- between nuclei connected by 4 bonds.bonds.

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

NMR: Summary