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4. Allene, C3H4, has point group symmetry Dzd, with character table provided below. It can be embedded in a rectangular prism as shown, with Cartesian axes as defined here. C₂A Dzd E 254 Cz 2C₂' 20d A1 1 1 1 1 1 1H- Az 1 1 1 -1 -1 H S4 C C= C B1 1 -1 1 1 -1 H Z B₂ 1 -1 1 -1 1 4 2H E 2 0 -200 (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. Here, |;) = |1s;) (the 1s orbital on atom Hi). (Note that the two C₂' axes from the character table have been labeled C₂ and C₂", and the σd' plane contains the z-axis, H1 and Hz, while the σd" plane contains the z-axis, H3 and H4.) R = Ê SA S₁³ Cz C₂' C₂" 8d8d" Ŕ$1 Φι Фэ ΦΑ $2 ΦΑ Φι Ŕ$2 $2 Фа $3 Φι $3 Фг Ŕ$3 $3 Фг Φι ΦΑ Фг ΦΑ Ŕ4 $4 Φι Фг Фз Φι Фэ (b) Find the reducible representation for the symmetry operations applied to the set of four 1s orbitals in allene. (c) Reduce the representation found for the 1s orbitals into its component irreducible representations.

4. Allene, C3H4, has point group symmetry Dzd, with character table provided below. It can be embedded in a rectangular prism as shown, with Cartesian axes as defined here. C₂A Dzd E 254 Cz 2C₂' 20d A1 1 1 1 1 1 1H- Az 1 1 1 -1 -1 H S4 C C= C B1 1 -1 1 1 -1 H Z B₂ 1 -1 1 -1 1 4 2H E 2 0 -200 (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. Here, |;) = |1s;) (the 1s orbital on atom Hi). (Note that the two C₂' axes from the character table have been labeled C₂ and C₂", and the σd' plane contains the z-axis, H1 and Hz, while the σd" plane contains the z-axis, H3 and H4.) R = Ê SA S₁³ Cz C₂' C₂" 8d8d" Ŕ$1 Φι Фэ ΦΑ $2 ΦΑ Φι Ŕ$2 $2 Фа $3 Φι $3 Фг Ŕ$3 $3 Фг Φι ΦΑ Фг ΦΑ Ŕ4 $4 Φι Фг Фз Φι Фэ (b) Find the reducible representation for the symmetry operations applied to the set of four 1s orbitals in allene. (c) Reduce the representation found for the 1s orbitals into its component irreducible representations.

Physical Chemistry
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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4. Allene, C3H4, has point group symmetry Dzd, with character table provided below. It can be embedded in a rectangular prism as shown, with Cartesian axes as defined here. C₂A Dzd E 254 Cz 2C₂' 20d A1 1 1 1 1 1 1H- Az 1 1 1 -1 -1 H S4 C C= C B1 1 -1 1 1 -1 H Z B₂ 1 -1 1 -1 1 4 2H E 2 0 -200 (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. Here, |;) = |1s;) (the 1s orbital on atom Hi). (Note that the two C₂' axes from the character table have been labeled C₂ and C₂", and the σd' plane contains the z-axis, H1 and Hz, while the σd" plane contains the z-axis, H3 and H4.) R = Ê SA S₁³ Cz C₂' C₂" 8d8d" Ŕ$1 Φι Фэ ΦΑ $2 ΦΑ Φι Ŕ$2 $2 Фа $3 Φι $3 Фг Ŕ$3 $3 Фг Φι ΦΑ Фг ΦΑ Ŕ4 $4 Φι Фг Фз Φι Фэ (b) Find the reducible representation for the symmetry operations applied to the set of four 1s orbitals in allene. (c) Reduce the representation found for the 1s orbitals into its component irreducible representations.
Transcribed Image Text:4. Allene, C3H4, has point group symmetry Dzd, with character table provided below. It can be embedded in a rectangular prism as shown, with Cartesian axes as defined here. C₂A Dzd E 254 Cz 2C₂' 20d A1 1 1 1 1 1 1H- Az 1 1 1 -1 -1 H S4 C C= C B1 1 -1 1 1 -1 H Z B₂ 1 -1 1 -1 1 4 2H E 2 0 -200 (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. Here, |;) = |1s;) (the 1s orbital on atom Hi). (Note that the two C₂' axes from the character table have been labeled C₂ and C₂", and the σd' plane contains the z-axis, H1 and Hz, while the σd" plane contains the z-axis, H3 and H4.) R = Ê SA S₁³ Cz C₂' C₂" 8d8d" Ŕ$1 Φι Фэ ΦΑ $2 ΦΑ Φι Ŕ$2 $2 Фа $3 Φι $3 Фг Ŕ$3 $3 Фг Φι ΦΑ Фг ΦΑ Ŕ4 $4 Φι Фг Фз Φι Фэ (b) Find the reducible representation for the symmetry operations applied to the set of four 1s orbitals in allene. (c) Reduce the representation found for the 1s orbitals into its component irreducible representations.
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