Polyatomic Molecule with high symmetry:

Benzene D6h Symmetry

564-15 Lec 28Mon 27Mar17

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 = E1uone-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

522 cm-1 ν6 e2g608 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 (100)

601101 601102

601103

601104

601 (100)521 cm-1

608 cm-1

100 916.8 896.5929.7

937

B2u 0

Ground A1g 0 606 cm-1 ν6 e2g

Electronic symmetry vibrationalsymmetryVibronicsymmetry

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

601 101 601 102

61 11 e2g E1u

610

f= forbidden red = allowed

110601

61 12 e2g E1u

61 13 e2g E1u

Hot bands (have Boltzmannfactor.

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

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