Physical Chemistry Third Edition

H The Hückel Method 1291

(3) we assume that forabHabhas one value, calledβ, whenaandbrepresent atoms

that are bonded to each other and thatHabvanishes when orbitalsaandbrepresent

atoms that are not bonded to each other. The secular equation for the allyl radical now

becomes

∣ ∣ ∣ ∣ ∣ ∣

α−Wβ 0

βα−Wβ

0 βα−W

∣ ∣ ∣ ∣ ∣ ∣

 0 (H-9)

We divide each element of the determinant byβand let

x

α−W

β

(H-10)

The secular equation becomes

∣ ∣ ∣ ∣ ∣ ∣

x 10

1 x 1

01 x

∣ ∣ ∣ ∣ ∣ ∣

 0 (H-11)

This determinant is expanded by minors using the top row:

x

∣

∣

∣

∣

x 1

1 x

∣

∣

∣

∣−^1

∣

∣

∣

∣

11

0 x

∣

∣

∣

∣+^0 ^0 (H-12a)

x(x^2 −1)−1(x−0)x^3 −x−x+ 0 x(x^2 −2) 0 (H-12b)

There are three roots, which we callx 1 ,x 2 , andx 3 :

x 1 −

√

2, x 2 0, x 3 

√

2 (H-13)

corresponding to

W 1 α+

√

2 β (H-14a)

W 2 α (H-14b)

W 3 α−

√

2 β (H-14c)

Sinceβturns out to be negative,W 1 is the lowest energy. For each of these roots, there

is a set of three equations for the threeccoefficients. Expressing these simultaneous

equations in terms ofx, we obtain

xc 1 +c 2  0 (H-15a)

c 1 +xc 2 +c 3  0 (H-15b)

c 2 +xc 3  0 (H-15c)

This represents one set of equations for each value ofx, so that we obtain one molecular

orbital for each value ofx. The resulting orbitals are

φ 1 c 1 (1)(ψ 1 +

√

2 ψ 2 +ψ 3 ) (H-16a)

φ 2 c 1 (2)(ψ 1 −ψ 3 ) (H-16b)

φ 3 c 1 (3)(ψ 1 −

√

2 ψ 2 +ψ 3 ) (H-16c)