1. Symmetry-adapted MO
Ex: C6H6 within Hückel approximation

2. Ψ(B)
Ψ(E2b)
Ψ(E2a)
Ψ(E1a)
Ψ(E1b)
Ψ(A)

3. Heteronuclear diatomic molecule AB

4. Heteronuclear diatomic molecule: αA ≠ αB
If αA = αB (homonuclear diatomic molecule), E± reduces to
If one assumes that S = 0 (zero overlap approximation),
E -
When lαA ≠ αBl >> 2 lβl, one can use the
approximation (1+x)1/2 ~1+x/2 αB αA E+

6. Molecular orbitals for polyatomic systems
Conjugated π-system: -C=C-C=C-
Consider the π-bond of ethylene (c=c)
Hückel approximation
Sij = 0 for all i ≠ j.
βij = 0 between non-neighboring atoms.
βij = β between neighboring atoms.
1. All diagonal elements: α – E
2. Off-diagonal elements between
non-neighboring atoms: β
3. All other off-diagonal elements: 0

7. Butadiene: c = c- c =c
A B C D
ψ = χA + χB+ χC+ χD
χA χB χC χD

8. σ orbitals
Benzene: C6H6
π orbitals

9. 1. HAA c1,A + HAB c1,B = (SAA c1,A + SAB c1,B)E1 = X11E1
For a two-atom system , e.g. CH2= CH2 1. 3. 4. 2.
Rewriting these 4 equations,
1. HAA c1,A + HAB c1,B = (SAA c1,A + SAB c1,B)E1 = X11E1
3. HAA c2,A + HAB c2,B = (SAA c2,A + SAB c2,B)E2 = X12E2
2. HBA c1,A + HBB c1,B = (SBA c1,A + SBB c1,B)E1 = X21E1
4. HBA c2,A + HBB c2,B = (SBA c2,A + SAB c2,B)E2 = X22E2
These equations can be concisely written using the following matrices;

10. , where X = Sc. Therefore, Hc =ScE
Hückel approximation S = S-1 = I
Hc = cE → c-1Hc = E (diagonal matrix) ; similarity transformation of H
E can be obtain by matrix diagonalization of H
Extended Hückel approximation: S ≠ I
c-1 (S-1H)c = c-1S-1ScE = E
E can be obtain by diagonalization of S-1H matrix

11. Review of matrices M Method to obtain M-1
If BA = I, then, B =A-1 , where B is called the inverse of A
Multipling B-1 on both sides of the original eq.,
B-1BA = B-1I and A = B-1 M
Method to obtain M-1
Form transpose MT
3. Construct
cofactor

12. Method to obtain M-1 Gauss-Jordan elimination method
I l A → same matrix operations until to diagonalize A → A-1l I

13. Simultaneous equations=

14. Eigenvalue equation For a n x n square matrix a
is an eigenvalue equation, where
λ (constant) is an eigenvalue and
x is an eigenvactor. There are n λ’s
and for each λ there is a corresponding x.
a =
for a nontrivial solution to exist
↔ Hc = cE (note the similarity)
X = (x1,x2,x3 …..) =