Group Theory  Contents -  Valence-Bond Theory

Symmetry

The idea is to let symmetry help in the determination of molecular orbitals. We have already seen that in H₂O, ( ${\rm 1s}_A+{\rm 1s}_B)$ mixes with 2p_zO and $({\rm 1s}_A-{\rm 1s}_B)$ mixes with 2p_yO. Combinations such as ${\rm 1s}_A\pm{\rm 1s}_B$ are called symmetry orbitals. Note that symmetry orbitals can only be combinations of the same type (s,p..) on different atoms. Note also that

$$\langle {\rm 1s}_A+{\rm 1s}_B\vert{\rm 1s}_A-{\rm 1s}_B\rangle=1-1+S_{AB}-S_{BA}=0$$ (46)

 demonstrates that symmetry orbitals of different symmetry do not mix.
Because of this the secular matrix ${\bf F-\epsilon S}$ for H₂O has the form
 

	1sO	2sO	2pzO	1sA+1sB	2pyO	1sA-1sB	2pxO
1sO	..	..	..	..	0	0	0
2sO	..	..	..	..	0	0	0
2pzO	..	..	..	..	0	0	0
1sA+1sB	..	..	..	..	0	0	0
2pyO	0	0	0	0	..	..	0
1sA-1sB	0	0	0	0	..	..	0
2pxO	0	0	0	0	0	0	..

 
In otherwords, the secular problem has been reduced from a 7$\times$7 to three problems, 4$\times$4,2$\times$2 and 1$\times$1.    
 Group Theory  Contents -  Valence-Bond Theory

1998-09-23