Chapter 10, Problem 10C.1P

Interpretation Introduction

Interpretation:

The irreducible representations of four $\text{H1s}$ orbitals of ${\text{CH}}_{\text{4}}$ span have to be stated.  Whether there are $s$ and $p$ orbitals of the central $\text{C}$ atom that might form molecular orbitals with $\text{H1s}$ orbitals or not has to be identified.  Whether $\text{3d}$ orbitals in ${\text{SiH}}_{\text{4}}$ could play a role in forming molecular orbitals by overlapping with the $\text{H1s}$ orbitals or not has to be stated.

Concept introduction:

The characters of the irreducible representations of the given point group can be multiplied by each other.  The only condition is the characters of the same symmetry operations are multiplied together.  The multiplication of the characters is commutative.

The great orthogonality theorem for the reducible representation can be expressed as shown below.

  $a_{\Gamma }=\frac{1}{h}{\displaystyle \sum _{\begin{array}{l}\text{all}\text{classes}\\ \text{of}\text{point}\text{group}\end{array}}N\cdot {\chi }_{\Gamma }\cdot {\chi }_{\text{linear}\text{combo}}}$