Symmetry-adapted MO Ex: C6H6 within Hückel approximation

Ψ(B) Ψ(E2b) Ψ(E2a) Ψ(E1a) Ψ(E1b) Ψ(A)

Heteronuclear diatomic molecule AB

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+

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

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

σ orbitals Benzene: C6H6 π orbitals

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;

, 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

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

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

Simultaneous equations =

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 …..) =