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Vibrational spectroscopy
Kenneth RuudUniversity of Tromsø
• U
NIV
ERSITETET
•
I TROMSØ
July 3 2015
Outline
Comparing electronic and vibrational spectroscopies
The force fieldInfrared spectroscopy
Raman spectoscopy
AnharmonicitiesVibrational circular dichroism (VCD)Raman optical activity
Coherent anti-Stokes Raman Scattering (CARS)
Comparing electronic and vibrational spectroscopies
So far, response theory has been used largely in the electronic domain
Exceptions were the pure and zero-point vibrational corrections
As for electronic spectroscopies, absorption and scattering processes occur in thevibrational domain (corresponding to infrared energies rather than UV/Vis)
In principle, electronic spectroscopies can be transfered into the vibrationaldomain, creating new spectroscopies
Instrumental limitations in optics and detectors may prevent some methods, butalso enable others
Some notable differences
The vibrational eigenfunctions differ from those of the electronic wave functions
The properties of the vibrational wave functions implies that excitations can ingeneral only occur to the lowest excited vibrational state of a vibrational mode
A frequency scan will record energies corresponding to absorption in differentvibrational modesAs for electronic spectroscopy, the transition moments determines intensities
Note: Also responses in the electron density may induce vibrational transitions(not only the multipole moments)
Transition moments can in principle be determined by the geometry dependenceof the electronic properties (Taylor expansions)
For multiphoton/multidimensional vibrational spectroscopies, analysis of thepoles and residues of the pure vibrational contributions may be morestraightforward
Force field and normal coordinates
We recall that we would use as a starting point a quadratic expansion of theelectronic potential (at the equilibrium geometry)
V = V0 +3N∑
i,j=1
(∂2
∂xi∂xj
)xj xj + . . . = V0 +
3N∑i,j=1
kij xj xj + . . .
We introduce mass-weighted coordinates as qi =√
mi xi
The potential then becomes (V0 is constant and can be ignored)
V =12
∑i,j
Kij qi qj Kij =∂2V∂qi∂qj
The classical energy for the total energy is then
EK =12
3N∑i
q2i +
12
3N∑ij=1
Kij qi qj
Normal coordinates
Solving the equation for the nuclear motion in a quadratic potential much easierif the equation is separable
Introduce normal coordinates Q as the basis which simultaneously diagonalizesthe kinetic energy and the potential
E =12
3N−6∑i
Q2i +
12
3N−6∑i
λi Q2i
The equation is now separable, and we identify the equation for each normalmode as that of the harmonic oscillatorThe total vibrational wave function is a product of harmonic oscillators, one foreach normal mode
ψ =∏
i
ψi (Qi ) E =∑
i
(vi +
12
)~ωi
Note: The total ground-state vibrational wave function is always totallysymmetric
Infrared spectroscopy
When a molecule absorbs light in the UV/Vis region, the absorption crosssection is
Ai→f =πωfi NA
3ε0~c|µfi |2
The key quantity determining the oscillator strength is the dipole transitionmomentIn a similar manner, we may consider a vibrational transition dipole moment fora normal mode k , using as always 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
An alternative approach: Residue analysis
We have throughout this summer school emphasized the strengths of theresponse theory approach
We recall that transition moments can be identified as residues of responsefunctions of different ordersMany (if not all) vibrational spectroscopies can be identified from a residueanalysis of the pure vibrational contributions to the response functions
Let us consider the pure vibrational contribution to the polarizability
[µ2]0,0
=∑
a
∂µα
∂Qa
∂µβ
∂Qa
12ωa
[1
(ωa − ω)+
1(ωa + ω)
]
The poles in this expression represents excitation energies to vibrationallyexcited states, the residues the transition moments
limω→ωk
(ωk − ω)[µ2]0,0
=1
2ωk
(∂µα
∂Qk
)(∂µβ
∂Qk
)
An alternative approach: Residue analysis
We have throughout this summer school emphasized the strengths of theresponse theory approach
We recall that transition moments can be identified as residues of responsefunctions of different ordersMany (if not all) vibrational spectroscopies can be identified from a residueanalysis of the pure vibrational contributions to the response functions
Let us consider the pure vibrational contribution to the polarizability
[µ2]0,0
=∑
a
∂µα
∂Qa
∂µβ
∂Qa
12ωa
[1
(ωa − ω)+
1(ωa + ω)
]
The poles in this expression represents excitation energies to vibrationallyexcited states, the residues the transition moments
limω→ωk
(ωk − ω)[µ2]0,0
=1
2ωk
(∂µα
∂Qk
)(∂µβ
∂Qk
)
Some important things to note on IR
Gross selection rule: Vibrational modes will only be IR active if the dipolemoment changes during the vibration
Corollary: Molecular vibrations preserving a center of inversion will not be IRactiveWe recall that assuming harmonic oscillator wave functions
〈km |Qk | kn〉 6= 0 m = n ± 1
Specific selection rule: ∆n = ±1
In a molecule with 3N − 6(5) normal coordinates, how do we identify the IRactive modes?
An exercise in symmetry: Vibrations in CO2
Table : Group multiplication table for the D2h point group.
E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz)Ag 1 1 1 1 1 1 1 1 x2, y2, z2
B1g 1 1 -1 -1 1 1 -1 -1 Rz , xyB2g 1 -1 1 -1 1 -1 1 -1 Ry , xzB3g 1 -1 -1 1 1 -1 -1 1 Rx , yzAu 1 1 1 1 -1 -1 -1 -1B1u 1 1 -1 -1 -1 -1 1 1 zB2u 1 -1 1 -1 -1 1 -1 1 yB3u 1 -1 -1 1 -1 1 1 -1 x
An exercise in symmetry: Vibrations in CO2
Carbon dioxide has nine nuclear degrees of freedom, of which three correspondto translations and two to rotationsStep 1: Identify the number of Cartesian atomic displacements that transforminto themselves (±) under the 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 each irreducible 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 CO2 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 CO2 transform as ΓA1 ⊕ ΓB1u ⊕ ΓB2u ⊕ ΓB3u
The components of the dipole moment span ΓB1u ⊕ ΓB2u ⊕ ΓB3u , thus three of thefour vibrations are IR active
Raman spectroscopy
In the Raman process, laser light is directed at a molecule, which scatters thelight (Rayleigh scattering)
Some energy may be absorbed or emitted from the molecule due to “absorption”or “emission” from vibrational states
The scattering within the electronic states is described by the polarizability, andwe will be concerned with transitions in 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 the polarizability tensor
Raman spectroscopy is thus largely governed by matrix elements of the form(Placzek/harmonic approximation)(
∂ααβ
∂Qk
)∣∣∣∣∆Qi =0
⟨ψki|Qk |ψkj
⟩
Residue analysis
lim(ω2+ω3)→ωk
(ωk − [ω2 + ω3])[α
2]0,0
= lim(ω2+ω3)→ωk
(ωk − [ω2 + ω3])∑
a
1
8ωa
(∂ααβ
∂Qa
)(∂αγδ
∂Qa
)[ 1
ωa − (ω2 + ω3)+
1
ωa + (ω2 + ω3)
]
=1
8ωk
(∂ααβ
∂Qk
)(∂αγδ
∂Qk
)
Gross selection rule: The polarizability must change during the vibration
Specific selection rule: ∆vk = ±1
Raman spectroscopy cont.
By now, we know how to proceed:Represent the vibrational states as harmonic oscillatorsTaylor expand the geometry dependence of the polarizability tensor
Raman spectroscopy is thus largely governed by matrix elements of the form(Placzek/harmonic approximation)(
∂ααβ
∂Qk
)∣∣∣∣∆Qi =0
⟨ψki|Qk |ψkj
⟩
Residue analysis
lim(ω2+ω3)→ωk
(ωk − [ω2 + ω3])[α
2]0,0
= lim(ω2+ω3)→ωk
(ωk − [ω2 + ω3])∑
a
1
8ωa
(∂ααβ
∂Qa
)(∂αγδ
∂Qa
)[ 1
ωa − (ω2 + ω3)+
1
ωa + (ω2 + ω3)
]
=1
8ωk
(∂ααβ
∂Qk
)(∂αγδ
∂Qk
)
Gross selection rule: The polarizability must change during the vibration
Specific selection rule: ∆vk = ±1
Raman spectroscopy cont.
By now, we know how to proceed:Represent the vibrational states as harmonic oscillatorsTaylor expand the geometry dependence of the polarizability tensor
Raman spectroscopy is thus largely governed by matrix elements of the form(Placzek/harmonic approximation)(
∂ααβ
∂Qk
)∣∣∣∣∆Qi =0
⟨ψki|Qk |ψkj
⟩
Residue analysis
lim(ω2+ω3)→ωk
(ωk − [ω2 + ω3])[α
2]0,0
= lim(ω2+ω3)→ωk
(ωk − [ω2 + ω3])∑
a
1
8ωa
(∂ααβ
∂Qa
)(∂αγδ
∂Qa
)[ 1
ωa − (ω2 + ω3)+
1
ωa + (ω2 + ω3)
]
=1
8ωk
(∂ααβ
∂Qk
)(∂αγδ
∂Qk
)
Gross selection rule: The polarizability must change during the vibration
Specific 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) is Raman active
General rule: For molecules with an inversion center, no vibrational mode can beboth IR and Raman active
Comparison to experiment
Detailed analysis of the Raman scattering process shows that the observablequantities depend on experimental setup in most cases(forward/backward/right-angle scattering)L. D. Barron, Molecular Light Scattering and Optical Activity (2204)
The detailed expressions are not important from a computational point of view
The scattering intensities are determined by isotropic and anisotropic invariantsdefined 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-point vibration correction thepossibility of electric and mechanical anharmonicities
May also occur for vibrational transitions, and allow new vibrational bands toappear
From electric anharmonicity, we will for instance have combination bands, inwhich two vibrational modes are excited simultaneously
〈1k 1l |µα| 0k 0l 〉 =
(∂2µα
∂Qk∂Ql
)0〈1k |Qk | 0k 〉 〈1l |Ql | 0l 〉
Will in general be weak if it is at all symmetry allowed
Fermi resonances
Mechanical anaharmonicity can also lead to combination bands
⟨2k 0l
∣∣∣V (3)kkl
∣∣∣ 0k 1l
⟩=
12
(∂3V
∂Q2k∂Ql
)0
⟨2k
∣∣∣Q2k
∣∣∣ 0k
⟩〈0l |Ql | 1l 〉
Of particular interest here is when 2ωk ≈ ωl , as happens in CO2
Vibrational analogue of configuration interaction
Weak overtones can steal intensity from an allowed fundamental⇒ 1 strongfundamental and 1 weak overtone replaced by two medium-intensity bandsshifted in energy
Multidimensional vibrational spectroscopiesHigh information content in multidimensional spectroscopies such asDOVE-FWM-2D-IR
〈〈µα;µω1β , µ
ω2γ , µ
ω3δ 〉〉IR =
[µ4]
+[µ2α
](E)+[µ2α
](I)+[α2]
+ [µβ]
W. Zhao and J. C. Wright, Phys. Rev. Lett. 84, 1411 (2000).
Note in particular that leading-order contributions to the odd-order terms[µ2α
]are anharmonicModeling multidimensional and nonlinear vibrational spectroscopies requiresthe evaluation of higher-order energy and property derivatives
Fundamental vibrational frequencies of CH4
M. Ringholm, D. Jonsson, R. Bast, B. Gao, A. J. Thorvaldsen, U. Ekstrom, T. Helgaker and K. Ruud,J. Chem. Phys.140, 034103 (2014).
No symmetry used, degenerate modes identical to within 10−5
B3LYP geometry and harmonic force field both for HF and B3LYP
6-31G works suprisingly well, no basis set effect beyond cc-pVTZ
Minor differences in anharmonicity corrections between HF and B3LYP (7-9% )
Why do analytic derivatives?
The easiest way to calculate energy derivatives would be to use numericaldifferentiation
6N calculations for first derivatives36N2 calculations for second derivatives208N3 calculations for third derivatives
Geometrical derivatives of the dipole moment of hydrogen fluoride at the CCSD/cc-pCVDZ level of theoryin atomic units, bond lengths optimized at the same level of theory (Req = 1.735686661 a.u.).
Computed from ∆R/A(
dµzdR
)R=Req
(d2µzdR2
)R=Req
(d3µzdR3
)R=Req
(d4µzdR4
)R=Req
µz 10−2 -0.31736764 0.17871386 0.763567 0.0750µz 10−3 -0.31741264 0.178712 0.764 . . .µz 10−4 -0.31741309 0.1787 . . . . . .µz 10−5 -0.3174131 0.18 . . . . . .
dµz /dR 10−2 . . . 0.17871606 0.76369391 0.0748442dµz /dR 10−3 . . . 0.17871165 0.763745 0.0714dµz /dR 10−4 . . . 0.17871157 0.7656 . . .dµz /dR 10−5 . . . 0.1787139 0.77 . . .
Fully analytic value -0.317413034 0.17811155 . . . . . .T.-C. Jagau, J. Gauss and K. Ruud, J. Chem. Phys. 139, 154106 (2013).
Numerical derivatives have limited and (non-systematic) numerical precision
Scaling for higher-order derivatives unfavorable
Why do analytic derivatives?
The easiest way to calculate energy derivatives would be to use numericaldifferentiation
6N calculations for first derivatives36N2 calculations for second derivatives208N3 calculations for third derivatives
Geometrical derivatives of the dipole moment of hydrogen fluoride at the CCSD/cc-pCVDZ level of theoryin atomic units, bond lengths optimized at the same level of theory (Req = 1.735686661 a.u.).
Computed from ∆R/A(
dµzdR
)R=Req
(d2µzdR2
)R=Req
(d3µzdR3
)R=Req
(d4µzdR4
)R=Req
µz 10−2 -0.31736764 0.17871386 0.763567 0.0750µz 10−3 -0.31741264 0.178712 0.764 . . .µz 10−4 -0.31741309 0.1787 . . . . . .µz 10−5 -0.3174131 0.18 . . . . . .
dµz /dR 10−2 . . . 0.17871606 0.76369391 0.0748442dµz /dR 10−3 . . . 0.17871165 0.763745 0.0714dµz /dR 10−4 . . . 0.17871157 0.7656 . . .dµz /dR 10−5 . . . 0.1787139 0.77 . . .
Fully analytic value -0.317413034 0.17811155 . . . . . .T.-C. Jagau, J. Gauss and K. Ruud, J. Chem. Phys. 139, 154106 (2013).
Numerical derivatives have limited and (non-systematic) numerical precision
Scaling for higher-order derivatives unfavorable
Numerical accuracy also an issue for spectra
1000 1200 1400 1600 1800
harm.
exp. freq.
analytic anharm.
2600 2800 3000 3200 3400
Wavenumber /cm−1
numeric (δx=10−2Å) anharm.
numeric (δx=5.10−3Å) anharm.
numeric (δx=10−3Å) anharm.
Methanimine, B3LYP/cc-pVTZ calculations, GVPT2/DVPT2 anharmonic corrections
Comparison to experiment, CH4
Mode ωB3LYP νHF νB3LYP νBLYP ωBLYP ωexp νexpcc-pVDZ
1 3146 2977 2988 2906 30682 3025 2887 2892 2817 29543 1530 1484 1488 1451 14944 1309 1264 1268 1233 1275
cc-pVQZ1 3127 2967 2979 2903 3055 3156.8 3022.52 3025 2896 2902 2833 2960 3025.5 2920.93 1558 1510 1514 1481 1524 1582.7 1532.44 1340 1293 1298 1267 1310 1367.4 1308.4
M. Ringholm, D. Jonsson, R. Bast, B. Gao, A. J. Thorvaldsen, U. Ekstrom, T. Helgaker and K. Ruud,J. Chem. Phys.140, 034103 (2014).
Basis set effect cc-pVDZ to cc-pVQZ for ωB3LYP is 0-31 cm−1: Scaling a poor basisset is not the right solution
Scaling factor B3LYP varies between 0.952-0.972, scaling acceptable foranharmonicities if the basis set is good
ωB3LYP in better agreement with experiment than νBLYP
Unless you have true anharmonic effects (Fermi resonances), first make sureyour basis set and functional is adequate
Comparison to experiment, C2H6
M. Ringholm, D. Jonsson, R. Bast, B. Gao, A. J. Thorvaldsen, U. Ekstrom, T. Helgaker and K. Ruud,J. Chem. Phys.140, 034103 (2014).
Experimental data: J. L. Duncan, R. A. Kelly, G. D. Nivellini and F. Tullini, J. Mol. Spectro sc. 98, 87 (1983).
Vibrational Circular Dichroism
We recall that circular dichroism was given by the rotatory strength tensor
Rn0αβ ∝ 〈0 |µα| n〉
⟨n∣∣mβ∣∣ 0⟩
Taking the expression into the vibrational domain, using the standard tricks, weobtain
Rk0αβ ∝
⟨0∣∣∣∣∂µα∂Qk
∣∣∣∣ k⟩⟨k∣∣∣∣∂mβ∂Qk
∣∣∣∣ 0⟩Note here that the magnetic (and electric) dipole moment also includes acorresponding nuclear contribution
Problem: The gradient of the magnetic dipole moment will be zero, and thusthere is no VCD intensity
However, this is a consequence of our adoption of the Born–Oppenheimerapproximation
We recall that we get an induced magnetic moment from the molecular rotationdue to the decoupling of the electrons and the nuclei
Vibrationally induced magnetic moment
Let us consider non-Born–Oppenheimer effects from the decoupling of electronsand nuclei during molecular vibrations
We recall that we in the Born–Oppenheimer approximation ignored thecontributions
H(1)N =
∑A,α
1MA
⟨χk′ | PAαχk
⟩δk,k′ PAα + 〈χk′ | TNχk 〉
Let us now consider this as a perturbation to our vibronic wave function with aharmonic-oscillator description of the vibration
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 using this perturbed wavefunction
Vibrationally induced magnetic moment
By evaluating the expression ⟨Ψ
pertKk
∣∣mβ∣∣ΨpertKk
⟩The leading-order, non-vanishing contribution to the vibrationally inducedmagnetic moment (the atomic axial tensor) 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) as PK ,αβ , the VCD rotationalstrength 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 theoreticalfoundation developed by Stephens in 1985 (J. Phys. Chem. 89, 748 (1985))
VCD gained momentum around the mid 90s due to the development of rigorousab initio methods, and in particular 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 the computationalrequirement, leading to the recommendation of a TZ2P basis with the B3LYP ofB3PW91 functionalsThe method has been approved by the FDA as an experimental method forverifying enantiomeric excess.
An example of the powers of VCD: Troger’s base
Absolute configuration of the (+)-Troger’s base determined to be (R,R) bycircular dichroism (and empirical rules for analysis (octant rule))
X-ray crystallography would however suggest the configuration 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 = 2∑n 6=0
ωn0Re [〈0 |µα| n〉 〈n |µβ | 0〉]
ω2n0 − ω2
= −〈〈µα;µβ〉〉ω
The electric dipole–electric quadrupole polarizability
Aαβγ = 2∑n 6=0
ωn0Re [〈0 |µα| n〉 〈n |Θβγ | 0〉]
ω2n0 − ω2
= −〈〈µα; Θβγ〉〉ω
The electric dipole–magnetic dipole polarizability
G′αβ = −2∑n 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
⟩=
12ωp
(∂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)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 tailoredbasis sets/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→ allows forefficient 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
Coherent anti-Stokes Raman Scattering
Enhances the Raman signal by tuning two lasers to be in resonance with avibrational excitation
Determined by the second hyperpolarizability tensor
γα;β,γ,δ (− (2ω1 − ω2) ;ω1, ω1,−ω2)
If ω1 − ω2 corresponds to a resonance in the IR region, the tensor is dominatedproducts of vibrational matrix elements of the kind
〈0 |α (ωσ)| k〉 〈k |α (ωτ )| 0〉
We have used the resolution of the identity when summing over vibrationalstates for an electronic excited states
Coherent anti-Stokes Raman scattering
Assuming the double-harmonic approximation we can writethese matrix elements as
〈ν0 |ααβ (ωσ)| ν1p〉 〈ν1p |ααβ (ωτ )| ν0〉 =1
2ωp
(∂ααβ
∂Qp
)∣∣∣∣re
(∂ααβ
∂Qp
)∣∣∣∣re
The CARS signals are thus determined by the same tensors asthe Raman intensityHowever, the intensity a mixture of electronic and vibrationalhyperpolarizability contributions, since
I ∝ |γ|2 =[(γR
v + γIv
)+ γe
]2=(γR
v
)2+ 2γR
v γe + (γe)2 +(γI
v
)2.
A. J. Thorvaldsen, L. Ferrighi, K. Ruud, H. Agren, S. Coriani, P. Jørgensen, Phys. Chem. Chem. Phys. 11, 2293 (2009)
Computational details
Geometries and force fields calculated using B3LYP/cc-pVDZ
Polarizability derivatives calculated using Hartree–Fock and the cc-pVDZ basisset
ANTHRACENE TETRACENE
CHRYSENE
PYRENE1,2 BENZANTHRACENE
PERYLENEBENZO(E)PYRENE
BENZO(A)PYRENE
BENZO(GHI)PERYLENECORONENE
Pyrene
After lineshape analysis
Demonstrating the different selectivity: Benzonitrile
A. Mohammed et al., Chem. Phys. Lett. 485, 320 (2010)
Concluding remarks
Many absorption processes occuring in the electronic domain will have ananalogous process in the vibrational domain
Scattering processes may however also allow for new interaction mechanisms
The vibrational domain provides a wealth of information, with up to 3N − 6responses for a molecule with N atoms
Multiphoton processes in the vibrational domain opens for the possibility fordetailed information about a molecule to be obtainedVCD and ROA are both powerful methods for determining the absoluteconfiguration of chiral molecules when used in conjunction with theoreticalcalculationsMost critical computational success factor: The quality of the force field
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