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Vibrations of a Polyatomic Molecule
Benzene (D6h Symmetry)
564-17 Lec 29Mon-Wed27,29 Mar17
2
τd*integral, dipoleion x transitThe 21 ΨΨ∫∞
∞−
x
The word "totally symmetric" refers to a function that ALWAYS goes into itself upon ALL symmetry operations for the molecule. Typically called A1g or Ag, it is the irreducible representation type whoses characters are ALL = 1
Example: operator x = E1u , and ψ1 is Ag ground state, then need ψ2 = E1u i.e., only transition to E1u are dipole allowed. x and y are equivalent.
%chk=C:\564-17\benzene-vibs.chk#b3lyp/6-311++g(d,p) pop =reg opt freq
benzene b3lyp/6-311++g(d,p) pop =reg opt
0 1C 1.20809735 0.69749533 -0.00000000C 0.00000000 1.39499067 -0.00000000C -1.20809735 0.69749533 -0.00000000C -1.20809735 -0.69749533 -0.00000000C 0.00000000 -1.39499067 -0.00000000C 1.20809735 -0.69749533 -0.00000000H 2.16038781 1.24730049 -0.00000000H 0.00000000 2.49460097 -0.00000000H -2.16038781 1.24730049 -0.00000000H -2.16038781 -1.24730049 -0.00000000H 0.00000000 -2.49460097 -0.00000000H 2.16038781 -1.24730049 -0.00000000
Opt Frequency calculation for benzene with high correlation + large basis (required for accurate frequencies and normal modes)
1 2 3AU B3U AG(E2G)
410.7349 411.6070 622.2315
4 5 6B3G(E2G) B3U B2G
622.2429 685.7410 718.7617
7 8 9B1G B2G AU
860.6162 863.1904 986.4942
10 11 12B3U AG B2G
988.0685 1011.6087 1014.4673
13 14 15B1U B2U B1U
1023.3723 1059.1425 1059.4801
benzene#b3lyp/6-311++g(d,p) pop=reg opt(3 minutes on PC) Frequencies in cm-1 numbered by order of increasing frequency
16 17 18B2U B3G AG
1174.8619 1197.4179 1197.6082
19 20 21B2U B3G B2U
1337.3288 1381.2370 1510.5350
22 23 24B1U B3G AG
1510.8040 1634.4257 1634.4289
25 26 27B1U B3G AG
3157.3697 3166.8865 3166.9359
28 29 30B2U B1U AG
3182.5618 3182.6543 3192.7553
Figure from: "1B2u↔1A1g spectroscopy of jetcooled benzene: Single vibronic levelfluorescence studies"Thomas A. Stephenson, Patricia L. Radloff, and Stuart A. RiceCitation: J. Chem. Phys. 81, 1060 (1984); doi: 10.1063/1.447800
Figure from: "1B2u↔1A1g spectroscopy of jetcooled benzene: Single vibronic levelfluorescence studies"Thomas A. Stephenson, Patricia L. Radloff, and Stuart A. RiceCitation: J. Chem. Phys. 81, 1060 (1984); doi: 10.1063/1.447800
Numbered by mode convention
e2g x B2u = E1u
one-photon Herzberg-Teller active
two-photon Herzberg-Teller active
923 cm-1 ν1 a1g993 in ground Only a1g modesare FC-active(non-zero Franck-Condon
factors)
522 cm-1 ν6 e2g608 in ground
1570 cm-1 ν14 b2u1309 in ground
522 cm-1 ν6 e2g608 in ground
1599 cm-1 ν8 e2g1516 in ground
8
τd*integral, dipoleion x transitThe 21 ΨΨ∫∞
∞−
x
The word "totally symmetric" refers to a function that ALWAYS goes into itself upon ALL symmetry operations for the molecule. Typically called A1g or Ag, it is the irreducible representation type whoses characters are ALL = 1
Herzberg-Teller activity: operator x = E1u , and ψ1 is Ag ground state, then need ψ2 = e2g x B2u = E1u vibronic state which is formally dipole allowed. only 1 quantum of e2g vibrations make significant intensity
e2g x B2u = E1u only 1 quantum of e2g vibrations make significant intensity
37000 38000 39000 40000 41000 42000 430000.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0Ab
sorb
ance
cm-1
Benzene vapor, room temperature
610 (10
0)
60110
160
1102
60110
3
60110
4
601 (10
0)521 cm-1
608 cm-1
100 916.8 896.5
929.7
937
B2u 0
Ground A1g 0 606 cm-1 ν6 e2g
Electronic symmetry vibrationalsymmetry
Vibronicsymmetry
61 521 cm-1 ν6 e2g E1u
11 923 cm-1 ν1 a1g B2u
993 cm-1 ν1 a1g
0 a1g B2u
0-0 f
601
101f
601 10
1 601 10
2
61 11 e2g E1u
610
f= forbidden red = allowed
11060
1
61 12 e2g E1u
61 13 e2g E1u
Hot bands (have Boltzmannfactor.
39400 39450 39500 39550 39600 39650-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Benzene vapor, room temperature
Abso
rban
ce
cm-1
D
39400 39450 39500 39550 39600 39650-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Benzene vapor, room temperature
Absor
bance
cm-1
D
Figure from: "1B2u↔1A1g spectroscopy of jetcooled benzene: Single vibronic levelfluorescence studies"Thomas A. Stephenson, Patricia L. Radloff, and Stuart A. RiceCitation: J. Chem. Phys. 81, 1060 (1984); doi: 10.1063/1.447800
Benzene in coldjet (vib. temperatureis about 50 K.
60110
1 E2g
801 E2g
Figure from: "1B2u↔1A1gspectroscopy of jetcooled benzene: Single vibronic levelfluorescence studies"Thomas A. Stephenson, Patricia L. Radloff, and Stuart A. RiceCitation: J. Chem. Phys. 81, 1060 (1984); doi: 10.1063/1.447800
xy
x2-y2xy
s
fπ MOs of Benzene
HOMOs
LUMOs
2021
2223
1Lb , 1B2u
1La , 1B1u
1Ba,b ,1E1u
Ground
(xxy)(yx2–y2)
(yxy)(xx2–y2)
2-1/2 (xxy)-(yx2–y2)
2-1/2 (yxy)-(xx2–y2)
2-1/2 (yxy)+(xx2–y2)2-1/2 (xxy)+(yx2–y2)
After CIS
(Reduced) 1st order Transition Density is Product of Ground
and Excited statewavefunctions
(integrated over all but oneelectron)
A1g-B2uTransitionDensity
B2uVibrationMode 14
A1g-E1u
Transition DensityLooks like a dipole
E2g+ E2g
-E2g+ E2g
-
B2u- B1u
+
E1u- E1u
-E1u
+ E1u+
1,2,3 substitutiongives very weak one-photonabsorption
E2g vibrationsinduce weak one-photonabsorption
The previous slide has a wealth of information. It is answering the question: what perturbation symmetry will mix E1u(the x,y symmetry) into the one-photon forbidden B2u and B1u states to make them one-photon allowed.
It also answers the question: what perturbation symmetry will mix A1g and E2g (the x2 + y2, and x2 - y2 ,xy, symmetries) into the two-photon forbidden B2u and B1u states to make them two-photon allowed.(We will study two-photon absorption briefly in a later lecture to understand the reason for these symmetries)
One-photon absorption: <B2u|H’|E1u> ≠ 0 requires H’ = e2gB2u requires an e2g perturbation (H’) because B2u x e2g = E1u
experimentally, nu 6 is the primary Herzberg-Teller active vibration for benzene; it is weak because the transition density is on the atoms, not the bonds.
inductive perturbations due to substituents e.g., fluoro make strong allowedness in various patterns because the transition density is on the atoms: 1,4 difluoro will be 4 times more intense than mono-sustitution. 1,2,3 will provide no intensity although allowed by symmetry because of cancellations!
B1u also requires an e2g perturbation because B1u x e2g = E1unu 8 is very strongly HT-active, making B1u have molar extinction =10,000(!) because it is bond stretching.
inductive perturbations have little effect because the transition density is in the bonds, as shown on the next slide.
Two-photon absorption: <B2u|H’|A1g> or <B2u|H’|E2g> ≠ 0 requires H’ be B2u or E1u
the “Kekule” vibration perturbation overlaps the B2u transition density very strongly and mixes the B2u electronic state so strongly with the ground state that the frequencies are affected (pseudo Jahn-Teller effect) and creates extremely strong two-photon absorption.
The B1u state has weak vibrationally induced intensity, but inductive perturbations induce strong intensity because the transition density is on the atoms, as shown in a slide below.
Two-Photon Fluor. Excitation Spectrum
Benzene in hexane
Fluoro-Benzene in hexane
1401 (10
0)
100 Solvent induced
1500 cm-1
A1g-B2uTransitionDensity
A1g-B1uTransitionDensity
B2u
B1u