Physical Chemistry Third Edition

890 21 The Electronic Structure of Polyatomic Molecules

EXAMPLE21.11

Without calculations, predict the energies and the shapes of the delocalized pi molecular

orbitals fortrans-1,3-butadiene.

Solution

With the planar molecule located in thexyplane, there are four unhybridized 2pzcarbon

orbitals to act as basis functions for the delocalized orbitals. We assume that the LCAO-

MOs have different numbers of vertical nodal surfaces in addition to the nodal plane in the

xyplane: no nodes, one node, two nodes, and three nodes. There is only one symmetrical

way to have each number of nodes, with the lowest-energy orbital not having any verti-

cal nodal surfaces, the next-lowest-energy orbital having one vertical nodal surface, and so

on. Since there is only one obvious way to have each number of nodes, the energy levels

are nondegenerate. Figure 21.10 shows the results of a Hückel calculation, bearing out our

predictions.

(a) (b)

4

3

2

Orbital energy 

1

 2 1.62

 2 0.62

 1 0.62

 1 1.62

Figure 21.10 Results of the Hückel Molecular Orbital Calculation for 1,3-Butadiene.

(a) The orbital regions for the 1,3-butadiene Hückel molecular orbitals. (b) The orbital ener-

gies of these orbitals. In part a, positive regions are in black and negative regions are in

gray.