4. Allene, C3H4, has point group symmetry Dzd, with character table provided below. It can beembedded in a rectangular prism as shown, with Cartesian axes as defined here.C₂ADzdE254 Cz 2C₂'20dA1111111H-Az111-1-1HS4CC=CB11-111-1HZB₂1-11-1142HE20-200(a) Consider the four 1s orbitals on the hydrogen atoms of allene (numbered as shown). Thesymmetry transformations of these functions are shown on the table below with two ofthe symmetry operations omitted. Find the results for the two missing sets of results(shaded red) and complete the table.Here, |;) = |1s;) (the 1s orbital on atom Hi). (Note that the two C₂' axes from thecharacter table have been labeled C₂ and C₂", and the σd' plane contains the z-axis, H1and Hz, while the σd" plane contains the z-axis, H3 and H4.)R =ÊSAS₁³ CzC₂'C₂" 8d8d"Ŕ$1 Φι ФэΦΑ $2ΦΑΦιŔ$2 $2Фа$3Φι$3ФгŔ$3 $3 ФгΦιΦΑФгΦΑŔ4 $4 Φι ФгФзΦιФэ(b) Find the reducible representation for the symmetry operations applied to the set of four1s orbitals in allene.(c) Reduce the representation found for the 1s orbitals into its component irreduciblerepresentations.

Answered: 4. Allene, C3H4, has point group…

Physical Chemistry

2nd Edition

ISBN:9781133958437

Author:Ball, David W. (david Warren), BAER, Tomas

Publisher:Ball, David W. (david Warren), BAER, Tomas

Chapter13: Introduction To Symmetry In Quantum Mechanics

Section: Chapter Questions

Problem 13.21E

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Allene, C3H4, has point group symmetry D2d, with character table provided below. It can be embedded in a rectangular prism as shown, with Cartesian axes as defined here. 2d E 2S4 A₁ 1 1 A2 1 1 B₁ 1 -1 B₂ 1 −1 E 2 0 1 H الے 2H C₂₁ C: X C₂" 3 H Â0₁ $1 R$₂ $2 Â$3 03 Ⓡ$4 ΦΑ CH S4 D2d $3 ФА (a) Consider the four 1s orbitals on the hydrogen atoms of allene (numbered as shown). The symmetry transformations of these functions are shown on the table below with two of the symmetry operations omitted. Find the results for the two missing sets of results (shaded red) and complete the table. C₂ 2C₂' 1 1 1 -1 1 1 1 -1 -2 0 0 Here, ₁) = |1s;) (the 1s orbital on atom H;). (Note that the two C₂' axes from the character table have been labeled C₂ and C₂", and the od plane contains the z-axis, H₁ and H₂, while the od" plane contains the z-axis, H3 and H4.) ÂR = Ê SA S4²³ C₂ Ĉ₂' Ĉ₂" ôd' ôa" ΦΑ $2 $3 Φ1 2 Φι $3 Φι $2 ΦΑ 03 04 $4 $3 Φ1 $2 $2 Φ1 20d 1 -1 -1 (b) Find the reducible representation for the…
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