6.2 Using symmetry to simplify the calculations6.2.1 Butadiene: The point group isC2hoyEC2iC2h: The four P (p,) AOs can be divided into1two sets of basis, (Φ, Φ) and (@, Φ), and can be111Rzx2:y2;z2;xyAg1-11-1B.Rx;R,dealt with separately.XZ;yz11-1-1AuZ. Both sets transform as A, @ BgBu-1-111x;y·For the basis (Φ,Φ),2T00-2=A, @B,z transforms like Au:A,=(Φ,+Φ) is z-likeyz transforms like Bg:Sb,=(一i+Φ) is yz-like!? Similarly the basis (Φ, Φ) gives rise to two SOs,0A=(Φ,+Φ) & 0B,=(-Φ,+Φ3)Note: The normalization coefficients for these SOs have the same value, 1/V2
6.2 Using symmetry to simplify the calculations 6.2.1 Butadiene • The four p (pz ) AOs can be divided into 2 0 0 -2 = Au Bg 𝜽𝑨 𝒖 = (2+3 ) & 𝜽𝑩 𝒈 = (2+3 ) • The point group is C2h. • Both sets transform as Au Bg . • For the basis (1 , 4 ), two sets of basis, (1 , 4 ) and (2 , 3 ), and can be dealt with separately. z transforms like Au ; yz transforms like Bg ; 𝜽𝑨 𝒖 =(1+4 ) is z-like. 𝜽𝑩 𝒈 =(1+4 ) is yz-like! • Similarly the basis (2 , 3 ) gives rise to two SOs, Note: The normalization coefficients for these SOs have the same value, 1/ 2
6.2.1 ButadieneNowwehavefournormalizedSOs:p =(Φ2 +Φ3)A, Ba =(Φ1 +Φ4)· Only SOs of the same symmetryinteract. The symmetry analysishas reduced the problem to thetwo-way overlap of , and b, andB, c = (-Φ1 + Φ4)a = (-Φ2 + Φ3)the two-way overlap of e, and d· At this stage, the secular equations can be developed by thinking about formingMOs by the linear combination of any other kind of orbitals, such as symmetry orbitals
6.2.1 Butadiene • Now we have four normalized SOs: • Only SOs of the same symmetry interact. The symmetry analysis has reduced the problem to the two-way overlap of θa and θb , and the two-way overlap of θc and θd . • At this stage, the secular equations can be developed by thinking about forming MOs by the linear combination of any other kind of orbitals, such as symmetry orbitals. Au Bg 𝜃𝑎 = 1 2 (𝛷1 + 𝛷4) 𝜃𝑐 = 1 2 (𝛷1 + 𝛷4) 𝜃𝑏 = 1 2 (𝛷2 + 𝛷3) 𝜃𝑑 = 1 2 (−𝛷2 + 𝛷3)