Huckel Theory
(Very briefly – its in almost any text book, Atkins for example – and its very simple.)
Planar Conjugated Hydrocarbons.
Symmetry separates basis set, (molecule in x, y plane):
H 1s, C 2s, 2px, 2py – mirror plane symmetric
C 2pz – mirror plane antisymmetric
Basis set for π m.o.s, φn = C 2pz on C atom n
π molecular orbital: ψπ = Σncnφn
Huckel approximations:
Hamiltonian Integrals:
Hmn = α if m = n
= β if m is bonded to n
= 0 otherwise
Zero overlap approximation:
Smn = 1 if m = n
= 0 if m ≠ n
Ethene: n = 1, 2
Secular Equations:
n = 1 : (H11 + ε S11) c1 + (H21 + ε S21) c2 = 0
n = 2 : (H12 + ε S12) c1 + (H22 + ε S22) c2 = 0
Huckel approx. and zero overlap approx:
n = 1 : (α + ε) c1 + (β) c2 = 0
n = 2 : (β) c1 + (α + ε) c2 = 0
divide by β throughout and set x = (α + ε)/β
n = 1 : x c1 + c2 = 0
n = 2 : c1 + x c2 = 0
multiply out secular determinant and show:
x = +1 or -1
for x = -1, ε = (α + β): c1 = c2
normalization (and zero overlap) c12 + c22 = 1
c1 = c2 = 1/21/2
for x = 1, ε = (α – β): c1= - c2
normalization (and zero overlap)
c1 = - c2 = 1/21/2
Butadiene:
n = 1, 2, 3, 4
Secular Equations:
divide by β throughout and set x = (α + ε)/β
secular determinant:
multiply out to show:
4 values of x
4 mo energies
Normalization:
etc. for the other three mo.s, e.g.