8-3 The Independent π-Electron Assumption

Chapter 8 The Simple H ¨uckel Method and Applications where Vne (i) represents the attraction between electron i and all the nuclei. These hamiltonians do indeed depend on the separate groups of electrons, but they leave out the operators for repulsion between σ and π electrons:

k

n

1

Hˆ − Hˆπ − Hˆσ =

(8-7)

rij

i=1 j =k+1

In short, the σ and π electrons really do interact with each other, and the fact that the HMO method does not explicitly include such interactions must be kept in mind when we consider the applicability of the method to certain problems. Some account of σ−π interactions is included implicitly in the method, as we shall see shortly.

8-3 The Independent π-Electron Assumption The HMO method assumes further that the wavefunction ψπ is a product of one-electron functions and that the hamiltonian Hˆπ is a sum of one-electron operators. Thus, for nπ electrons, ψπ (1, 2, . . . , n) = φi(1)φj (2) . . . φl(n)

(8-8)

Hˆπ (1, 2, . . . , n) = ˆ Hπ (1) + ˆ Hπ (2) + · · · + ˆ Hπ (n)

(8-9)

and φ∗i(1) ˆHπ(1)φi(1)dτ(1)

≡ Ei

(8-10) φ∗i(1)φi(1)dτ (1) It follows that the total π energy Eπ is a sum of one-electron energies: Eπ = Ei + Ej + · · · + El

(8-11)

This means that the π electrons are being treated as though they are independent of each other, since Ei depends only on φi and is not inﬂuenced by the presence or absence of an electron in φj . However, this cannot be correct because π electrons in fact interact strongly with each other. Once again, such interactions will be roughly accounted for in an implicit way by the HMO method.

The implicit inclusion of interelectronic interactions is possible because we never actually write down a detailed expression for the π one-electron hamiltonian operator

ˆ

Hπ (i). (We cannot write it down because it results from a π–σ separability assumption and an independent π -electron assumption, and both assumptions are incorrect.) ˆ

Hπ (i) is considered to be an “effective” one-electron operator—an operator that somehow includes the important physical interactions of the problem so that it can lead to a reasonably correct energy value Ei. A key point is that the HMO method ultimately evaluates Ei via parameters that are evaluated by appeal to experiment. Hence, it is a semiempirical method. Since the experimental numbers must include effects resulting from