Final answer:
The total wave function of a molecule, composed from molecular orbitals, must be antisymmetric for systems with multiple electrons. Group theory is employed to decompose the reducible representation of the wave function into irreducible representations, using characters from character tables and the reduction formula. Through this decomposition, basis functions that generate irreducible representations from the original product of MOs can be determined.
Step-by-step explanation:
In molecular orbital (MO) theory, the behavior of electrons within molecules is described by wave functions. These wave functions can be constructed from the combination of atomic wave functions, resulting in molecular orbitals that extend over the entire molecule. The total wave function for systems like the 1,3-butadiene molecule is generated by taking the product of such MOs. When dealing with the electronic states and their symmetries, especially in molecules with multiple electrons, the overall wave function must adhere to the requirements of the Pauli exclusion principle and must be antisymmetric under the exchange of two electrons.
During the application of group theory to molecular systems, the reducible representation created by the total wave function's symmetry characteristics can often be decomposed into a direct sum of irreducible representations. Each irreducible representation corresponds to a set of basis functions that transform in a specific way under symmetry operations of the molecule's point group. To distinguish between different irreducible representations from the overall wave function, character tables and the reduction formula are typically employed.
The reducible representation is broken down step by step using the characters from the character table corresponding to the symmetry operations of the molecule's point group. The characters provide information on how to express basis functions for each irreducible representation from the original product of MOs, ensuring that the resulting set of basis functions for each irreducible representation retains its symmetry properties. Through this mathematical procedure, one can extract individual molecular orbitals that adhere to specific symmetry rules required for the proper description of the molecule's electronic states.