Question

4. Consider four tetrahedrally directed hybrid orbitals represented by vectors A,B,C and D drawn inside a cube as shown in the figures below. Derive a set of five 4x4 transformation matrices (fill in blank spaces below) that transform the positions A, B, C and D under the symmetry operations of the Td point group (E, ?3, C2, S4, Sd) as shown. The 4x1 matrices below represents the position A, B, C and D before and after each operation. Use the characters of each transformation matrix to formulate the reducible representation It by which the vectors transform under each operation (Fill in the blanks under It space bottom right below). B C C D D C Othrough Al C B D A A Ta E 8C3 3C2 6S4 6\sigma_d B B = D \Gamma_t D C Use the character table for Td point group to reduce \Gamma_t to two irreducible symmetry species (do it by inspection if you can). Assign \"s\", \"p\", and \"d\" orbitals to their corresponding symmetry species (reducible representations). Which sets of orbitals is involved in the hybridization scheme to form SiH$_4$, whose structure is the same as that in the Figures above. How does the character for each operation in \Gamma_t relate to the number of vectors that are not shifted (moved) by the operation? Explain.

          4. Consider four tetrahedrally directed hybrid orbitals represented by vectors A,B,C and D drawn inside a
cube as shown in the figures below. Derive a set of five 4x4 transformation matrices (fill in blank spaces
below) that transform the positions A, B, C and D under the symmetry operations of the Td point group (E, ?3,
C2, S4, Sd) as shown. The 4x1 matrices below represents the position A, B, C and D before and after each
operation. Use the characters of each transformation matrix to formulate the reducible representation It by
which the vectors transform under each operation (Fill in the blanks under It space bottom right below).
B
C
C
D
D
C
Othrough Al
C
B
D
A
A
Ta
E
8C3
3C2
6S4
6\sigma_d
B
B
= D
\Gamma_t
D
C
Use the character table for Td point group to reduce \Gamma_t to two irreducible symmetry species (do it by
inspection if you can). Assign \"s\", \"p\", and \"d\" orbitals to their corresponding symmetry species (reducible
representations). Which sets of orbitals is involved in the hybridization scheme to form SiH$_4$, whose structure
is the same as that in the Figures above. How does the character for each operation in \Gamma_t relate to the
number of vectors that are not shifted (moved) by the operation? Explain.

4. Consider four tetrahedrally directed hybrid orbitals represented by vectors A,B,C and D drawn inside a
cube as shown in the figures below. Derive a set of five 4x4 transformation matrices (fill in blank spaces
below) that transform the positions A, B, C and D under the symmetry operations of the Td point group (E, ?3,
C2, S4, Sd) as shown. The 4x1 matrices below represents the position A, B, C and D before and after each
operation. Use the characters of each transformation matrix to formulate the reducible representation It by
which the vectors transform under each operation (Fill in the blanks under It space bottom right below).
B
C
C
D
D
C
Othrough Al
C
B
D
A
A
Ta
E
8C3
3C2
6S4
6B
B
= D
D
C
Use the character table for Td point group to reduce to two irreducible symmetry species (do it by
inspection if you can). Assign s̈,̈ p̈,̈ and d̈örbitals to their corresponding symmetry species (reducible
representations). Which sets of orbitals is involved in the hybridization scheme to form SiH4, whose structure
is the same as that in the Figures above. How does the character for each operation in relate to the
number of vectors that are not shifted (moved) by the operation? Explain.

Added by Victor Manuel O.

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