Vibrational spectroscopy

Kenneth RuudUniversity of Tromsø

• U

NIV

ERSITETET

•

I TROMSØ

October 29 2010

Outline

Comparing electronic and vibrational spectroscopiesThe force fieldInfrared spectroscopyRaman spectoscopyVibrational circular dichroism (VCD)Raman optical activityCoherent anti-Stokes Raman Scattering (CARS)

Comparing electronic and vibrational spectroscopies

So far, response theory has been used largely in theelectronic domainExceptions were the pure and zero-point vibrationalcorrectionsAs for electronic spectroscopies, absorption and scatteringprocesses occur in the vibrational domain (correspondingto infrared energies rather than UV/Vis)In principle, electronic spectroscopies can be transferedinto the vibrational domain, creating new spectroscopiesInstrumental limitations in optics and detectors mayprevent some methods, but also enable others

Some notable differences

The vibrational eigenfunctions differ from those of theelectronic wave functionsThe properties of the vibrational wave functions impliesthat excitations can in general only occur to the lowestexcited vibrational state of a vibrational modeA frequency scan will record energies corresponding toabsorption in different vibrational modesAs for electronic spectroscopy, the transition momentsdetermines intensitiesNote: Also responses in the electron density may inducevibrational transitions (not only the multipole moments)Transition moments determined by the geometrydependence of the electronic properties (Taylorexpansions)

Force field and normal coordinatesWe recall that we would use as a starting point a quadraticexpansion of the electronic potential (at the equilibriumgeometry)

V = V0 +3N∑

i,j=1

(∂2

∂xi∂xj

)xjxj + . . . = V0 +

3N∑i,j=1

kijxjxj + . . .

We introduce mass-weighted coordinates as qi =√

mixi

The potential then becomes (V0 is constant and can beignored)

V =12

∑i,j

Kijqiqj Kij =∂2V∂qi∂qj

The classical energy for the total energy is then

EK =12

3N∑i

q2i +

12

3N∑ij=1

Kijqiqj

Normal coordinatesSolving the equation for the nuclear motion in a quadraticpotential much easier if the equation is separableIntroduce normal coordinates Q as the basis whichsimultaneously diagonalizes the kinetic energy and thepotential

E =12

3N−6∑i

Q2i +

12

3N−6∑i

λiQ2i

The equation is now separable, and we identify theequation for each normal mode as that of the harmonicoscillatorThe total vibrational wave function is a product of harmonicoscillators, one for each normal mode

ψ =∏

i

ψi (Qi) E =∑

i

(vi +

12

)~ωi

Note: The total ground-state vibrational wave function isalways totally symmetric

Infrared spectroscopy

When a molecule absorbs light in the UV/Vis region, theoscillator strength is

fn0 =4π3

∆En0 |〈0 |µ|n〉|2

The key quantity determining the oscillator strength is thedipole transition momentIn a similar manner, we may consider a vibrationaltransition dipole moment for a normal mode k , using asalways perturbation theory

〈0 |µ (Q)|n〉 = 〈0 |µ (Re)| kn〉+3N−6∑k=1

∂µ

∂Qk

∣∣∣∣Re

〈0 |Qk | kn〉+ . . .

The basis for infrared spectroscopy

Some important things to note on IR

Gross selection rule: Vibrational modes will only be IRactive if the dipole moment changes during the vibrationCorollary: Molecular vibrations preserving a center ofinversion will not be IR activeWe recall that assuming harmonic oscillator wave functions

〈km |Qk | kn〉 6= 0 m = n ± 1

Specific selection rule: ∆n = ±1In a molecule with 3N − 6(5) normal coordinates, how dowe identify the IR active modes?

An exercise in symmetry: Vibrations in CO2

Carbon dioxide has nine nuclear degrees of freedom, ofwhich three correspond to translations and two to rotationsStep 1: Identify the number of Cartesian atomicdisplacements that transform into themselves (±) underthe symmetry operations of the molecule

E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz)9 -3 -1 -1 -3 1 3 3

Step 2: Determine the number of functions in eachirreducible representation: al = 1

h∑

R χ∗l (R)χ (R)

Ag B1g B2g B3g Au B1u B2u B3u1 0 1 1 0 2 2 2

An exercise in symmetry: Vibrations in CO2

Carbon dioxide has nine nuclear degrees of freedom, ofwhich three correspond to translations and two to rotationsStep 1: Identify the number of Cartesian atomicdisplacements that transform into themselves (±) underthe symmetry operations of the molecule

E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz)9 -3 -1 -1 -3 1 3 3

Step 2: Determine the number of functions in eachirreducible representation: al = 1

h∑

R χ∗l (R)χ (R)

Ag B1g B2g B3g Au B1u B2u B3u1 0 1 1 0 2 2 2

An exercise in symmetry: Vibrations in CO2

Carbon dioxide has nine nuclear degrees of freedom, ofwhich three correspond to translations and two to rotationsStep 1: Identify the number of Cartesian atomicdisplacements that transform into themselves (±) underthe symmetry operations of the molecule

E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz)9 -3 -1 -1 -3 1 3 3

Step 2: Determine the number of functions in eachirreducible representation: al = 1

h∑

R χ∗l (R)χ (R)

Ag B1g B2g B3g Au B1u B2u B3u1 0 1 1 0 2 2 2

An exercise in symmetry: Vibrations in CO2

Carbon dioxide has nine nuclear degrees of freedom, ofwhich three correspond to translations and two to rotationsStep 1: Identify the number of Cartesian atomicdisplacements that transform into themselves (±) underthe symmetry operations of the molecule

E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz)9 -3 -1 -1 -3 1 3 3

Step 2: Determine the number of functions in eachirreducible representation: al = 1

h∑

R χ∗l (R)χ (R)

Ag B1g B2g B3g Au B1u B2u B3u1 0 1 1 0 2 2 2

Vibrations in water cont.

Step 3: Identify and remove translational motion:B1u + B2u + B3u ⇒ 1/0/1/1/0/1/1/1

Step 4: Identify and remove rotational motion:B2g + B3g ⇒ 1/0/0/0/0/1/1/1The four normal modes in water transform asΓA1 ⊕ ΓB1u ⊕ ΓB2u ⊕ ΓB3u

The components of the dipole moment spanΓB1u ⊕ ΓB2u ⊕ ΓB3u , thus three of the four vibrations are IRactive

Vibrations in water cont.

Step 3: Identify and remove translational motion:B1u + B2u + B3u ⇒ 1/0/1/1/0/1/1/1Step 4: Identify and remove rotational motion:B2g + B3g ⇒ 1/0/0/0/0/1/1/1

The four normal modes in water transform asΓA1 ⊕ ΓB1u ⊕ ΓB2u ⊕ ΓB3u

The components of the dipole moment spanΓB1u ⊕ ΓB2u ⊕ ΓB3u , thus three of the four vibrations are IRactive

Vibrations in water cont.

Step 3: Identify and remove translational motion:B1u + B2u + B3u ⇒ 1/0/1/1/0/1/1/1Step 4: Identify and remove rotational motion:B2g + B3g ⇒ 1/0/0/0/0/1/1/1The four normal modes in water transform asΓA1 ⊕ ΓB1u ⊕ ΓB2u ⊕ ΓB3u

The components of the dipole moment spanΓB1u ⊕ ΓB2u ⊕ ΓB3u , thus three of the four vibrations are IRactive

Raman spectroscopyIn the Raman process, laser light is directed at a molecule,which scatters the light (Rayleigh scattering)Some energy may be absorbed or emitted from themolecule due to “absorption” or “emission” from vibrationalstates

The scattering within the electronic states is described bythe polarizability, and we will be concerned with transitionsin the vibrational manifold

〈ψk |ααβ|ψl〉

Raman spectroscopy cont.

By now, we know how to proceed:Represent the vibrational states as harmonic oscillatorsTaylor expand the geometry dependence of thepolarizability tensor

Raman spectroscopy is thus largely governed by matrixelements of the form (Placzek/harmonic approximation)(

∂ααβ∂Qk

)∣∣∣∣∆Qi =0

⟨ψki |Qk |ψkj

⟩Gross selection rule: The polarizability must change duringthe vibrationSpecific selection rule: ∆vk = ±1

The symmetry analysis of CO2

We recall that the normal coordinates of CO2 transform asΓA1 ⊕ ΓB1u ⊕ ΓB2u ⊕ ΓB3u

The components of the polarizability tensor transform asΓA1 ⊕ ΓB1g ⊕ ΓB2g ⊕ ΓB3g

Thus, only one of the modes (the symmetric stretch) isRaman activeGeneral rule: For molecules with an inversion center, novibrational mode can be both IR and Raman active

Comparison to experimentDetailed analysis of the Raman scattering process showsthat the observable quantities depend on experimentalsetup in most cases (forward/backward/right-anglescattering)L. D. Barron, Molecular Light Scattering and Optical Activity (2204)

The detailed expressions are not important from acomputational point of viewThe scattering intensities are determined by isotropic andanisotropic invariants defined as

α2 =19ααααββ

β (α)2 =12

(3ααβααβ − ααααββ)

The Raman scattering cross section is then proportional to

45α2 + 4β (α)2

Anharmonicity

We discussed in connection with pure and zero-pointvibration correction the possibility of electric andmechanical anharmonicitiesMay also occur for vibrational transitions, and allow newvibrational bands to appearFrom electric anharmonicity, we will for instance havecombination bands, in which two vibrational modes areexcited simultaneously

〈1k1l |µα|0k0l〉 =

(∂2µα∂Qk∂Ql

)0〈1k |Qk |0k 〉 〈1l |Ql |0l〉

Will in general be weak if it is at all symmetry allowed

Fermi resonancesMechanical anaharmonicity can also lead to combinationbands⟨

2k0l

∣∣∣V (3)kkl

∣∣∣0k1l

⟩=

12

(∂3V

∂Q2k∂Ql

)0

⟨2k

∣∣∣Q2k

∣∣∣0k

⟩〈0l |Ql |1l〉

Of particular interest here is when 2ωk ≈ ωl , as happens inCO2

Vibrational analogue of configuration interactionWeak overtones can steal intensity from an allowedfundamental⇒ 1 strong fundamental and 1 weak overtonereplaced by two medium-intensity bands shifted in energy

Vibrational Circular DichroismWe recall that circular dichroism was given by the rotatorystrength tensor

Rn0αβ ∝ 〈0 |µα|n〉 〈n |mβ|0〉

Taking the expression into the vibrational domain, usingthe standard tricks, we obtain

Rk0αβ ∝

⟨0∣∣∣∣ ∂µα∂Qk

∣∣∣∣ k⟩⟨k∣∣∣∣∂mβ

∂Qk

∣∣∣∣0⟩Note here that the magnetic (and electric) dipole momentalso includes a corresponding nuclear contributionProblem: The gradient of the magnetic dipole moment willbe zero, and thus there is no VCD intensityHowever, this is a consequence of our adoption of theBorn–Oppenheimer approximationWe recall that we get an induced magnetic moment fromthe molecular rotation due to the decoupling of theelectrons and the nuclei

Vibrationally induced magnetic momentLet us consider non-Born–Oppenheimer effects from thedecoupling of electrons and nuclei during molecularvibrationsWe recall that we in the Born–Oppenheimer approximationignored the contributions

H(1)N =

∑A,α

1MA

⟨χk ′ | PAαχk

⟩δk ,k ′PAα + 〈χk ′ | TNχk 〉

Let us now consider this as a perturbation to our vibronicwave function with a harmonic-oscillator description of thevibration

ΨpertKk = Ψ

(0)KK + Ψ

(1)Kk = Ψ

(0)KK +

∑(L,l) 6=(0,0)

⟨ΨKk

∣∣Hel + TN∣∣ΨLl

⟩EKk − ELl

ΨLl

The magnetic dipole moment can then be evaluated usingthis perturbed wave function

Vibrationally induced magnetic momentBy evaluating the expression⟨

ΨpertKk |mβ|Ψpert

Kk

⟩The leading-order, non-vanishing contribution to thevibrationally induced magnetic moment (the atomic axialtensor) is

MK ,αβ = IK ,αβ + JK ,αβ

IK ,αβ =

⟨∂Ψel

0∂RK ,α

∣∣∣∣∣ ∂Ψel0

∂Bβ

⟩∣∣∣∣∣R=R0,B=0

JK ,αβ =i4

∑γ

εαβγZK RKγ

Denoting the dipole gradient (atomic polar tensor) asPK ,αβ, the VCD rotational strength for normal mode k is

R (0→ 1)k = Im |PK ·MK |

Comments on VCD

The first observations of VCD was made in the early 70s,and the theoretical foundation developed by Stephens in1985 (J. Phys. Chem. 89, 748 (1985))

VCD gained momentum around the mid 90s due to thedevelopment of rigorous ab initio methods, and inparticular with DFT implementations, and commercialinstrumentationK.L.Bak et al., J. Chem. Phys. 98, 8873 (1993)

F. J. Devlin et al., J. Am. Chem. Soc. 118, 6327 (1996)

The research group of Stephens has actively calibrated thecomputational requirement, leading to the recommendationof a TZ2P basis with the B3LYP of B3PW91 functionalsThe method has been approved by the FDA as anexperimental method for verifying enantiomeric excess.

An example of the powers of VCD: Troger’s base

Absolute configuration of the (+)-Troger’s base determinedto be (R,R) by circular dichroism (and empirical rules foranalysis (octant rule))X-ray crystallography would however suggest theconfiguration to be (S,S)?Can VCD contribute?

(+)-(R,R)-Troger’s base: Experimental and theoreticalVCD spectra

A. Aamouche, F. J. Devlin and P. J. Stephens, J. Am. Chem. Soc. 122, 2346 (2000)

Raman Optical ActivityTheoretical foundation developed by Barron and Buckingham inthe early 70s (L. D. Barron and A. D. Buckingham, Mol. Phys. 20 (1971) 1111)

The Raman analogue of VCDCircular Scattering Intensity Differences determined by threetensorsThe electric dipole polarizability

ααβ = 2Xn 6=0

ωn0Re [〈0 |µα| n〉 〈n |µβ | 0〉]

ω2n0 − ω2

= −〈〈µα;µβ〉〉ω

The electric dipole–electric quadrupole polarizability

Aαβγ = 2Xn 6=0

ωn0Re [〈0 |µα| n〉 〈n |Θβγ | 0〉]

ω2n0 − ω2

= −〈〈µα; Θβγ〉〉ω

The electric dipole–magnetic dipole polarizability

G′αβ = −2Xn 6=0

ωIm [〈0 |µα| n〉 〈n |mβ | 0〉]

ω2n0 − ω2

= −i 〈〈µα; mβ〉〉ω

Vibrational matrix elements

In ROA we are interested in the changes in the previouslydefined linear response functions during a vibrational transitionIf we assume the Placzek approximation (double-harmonicapproximation), the relevant quantities can be approximated as

〈ν0 |ααβ | ν1p〉 〈ν1p |ααβ | ν0〉 =1

2ωp

(∂ααβ

∂Qp

)∣∣∣∣re

(∂ααβ

∂Qp

)∣∣∣∣re

〈ν0 |ααβ | ν1p〉⟨ν1p∣∣G′αβ

∣∣ ν0⟩

=1

2ωp

(∂ααβ

∂Qp

)∣∣∣∣re

(∂G′αβ

∂Qp

)∣∣∣∣re

〈ν0 |ααβ | ν1p〉 〈ν1p |εαγδAγδβ | ν0〉 =1

2ωp

(∂ααβ

∂Qp

)∣∣∣∣re

εαγδ

(∂Aγδβ

∂Qp

)∣∣∣∣re

Scattering Intensity Differences

We will compare our results with the scattering intensitydifference between left and right circularly polarized lightFor scattered light of α polarization

∆α = IRα − IL

α

For the specific cases of right-angle and backscattering, thisequation is reduced to

∆z (90◦) =1c

(6β (G′)2 − 2β (A)2

)∆z (180◦) =

1c

(24β (G′)2

+ 8β (A)2)

Scattering Intensity Differences

The intensity differences are determined using the anisotropicinvariants

β (α)2 =12

(3ααβααβ − ααααββ)

β (G′)2=

12(3ααβG′αβ − αααG′ββ

)β (A)2 =

12ωααβεαγδAγδβ

Implicit summation over repeated indices has been usedFirst ab initio calculations presented in 1990P. L. Polavarapu J. Phys. Chem. 94 8106 (1990)

Calculating ROA spectra

ROA requires the geometrical derivatives of several linearresponse tensor (thus formally a quadratic response function)Can be obtained using finite differences of the (mixed)polarizabilities with respect to nuclear distortionsFor a molecule with N atoms, this gives 6N + 1 propertycalculationsAnalytic calculations have recently become availableV. Liegeois, K. Ruud and B. Champagne. J. Chem. Phys. 127, 204105 (2007)Theoretical approaches to the calculation of Raman Optical Activity Spectra.K. Ruud and A. J. Thorvaldsen, Chirality, in press

The most critical component (as is the case for VCD) is anappropriately chosen force field (CIDs less sensitive to thecomputational level)Multilevel computational approaches with specially tailored basissets/force fields is an attractive approach

The powers of ROA: (R)-[2H1, 2H2,2H3]-neopentane

J. Haesler et al., Nature 446, 526 (2007)

ROA: Computational requirements

To include some electron correlation effects, we would like to doDFTWhich basis set and what functional?Test set constructed from:

Five molecules (methyloxirane, glycidol, epichlorhydrine,spiro[2,2]pentane-1,4-diene and σ-[4]-helicene)Three functionals: SVWN, BLYP, and B3LYP6 basis sets: cc-pVDZ, aug-cc-pVDZ, cc-pVTZ,aug-cc-pVTZ, sadlej and a basis by Zuber and Hug(3-21++G plus diffuse p on hydrogens) (G. Zuber and W. Hug,

J. Phys. Chem. A 108 (2004) 2108).90 ROA calculations with up to 21 atoms and 506 basis functionsM. Reiher, V. Liegeois, and K. Ruud. J. Phys. Chem. A, J. Phys. Chem. A 109, 7567 (2005)

The molecules

Analysis of the results

Comparison with experiment difficult due to shape of ROA andRaman bandsDFT/B3LYP using the aug-cc-pVTZ (aug-cc-pVDZ for thehelicene) used as a benchmarkDeviation for a basis set given as

δ (I) =

∑p

∣∣I trialp − Iref

p

∣∣∑p

∣∣Irefp∣∣

We also note the number of modes with incorrect signExperimental data kindly provided by prof. Werner Hug(University of Fribourg, Switzerland)

S-methyloxirane

M. Reiher, V. Liegeois, and K. Ruud. J. Phys. Chem. A 109, 7567 (2005)

(M)-spiro-[2,2]pentane-1,4-diene

M. Reiher, V. Liegeois, and K. Ruud, J. Phys. Chem. A 109, 7567 (2005)

ROA requirements

There appears to be no alternative to aug-cc-pVDZSadlej may work, but unreliableThe basis set of Zuber and Hug may work if used with a properforce fieldBLYP can in many cases be an alternative to B3LYP→ allowsfor efficient density-fitting (RI) techniquesBased on the assumption B3LYP is the best functional

M. Reiher, V. Liegeois, and K. Ruud. J. Phys. Chem. A, A 109, 7567 (2005)

Vibrational mode selection

Idea: Calculate relevant normal modes onlyMode-tracking—Selective calculation of vibrational frequenciesand normal modes from eigenpairs of the Hessian matrixthrough subspace iteration.Instead of

[H− λµ]Qµ = 0

solve

[H− λ(i)µ ]Q(i)

µ = r(i)µ

iteratively for a few eigenpairs by a Davidson-type calculation.

M. Reiher, J. Neugebauer, J. Chem. Phys. 118 2003, 1634–1641

(R,R)-Dialanine: ROA intensities

Intensities for amide I and amide II modes

Full (calculated) spectrum:

(R,R)-Dialanine: ROA intensities

Intensities for amide I and amide II modes

Full (calculated) spectrum:

Frequency region 1371–1502 cm−1

AKIRA converges within 4 iterations→ 24 additional single-pointscalculations→ compared to full analysis: 24+6 = 30 vs. 138 points