Computational Chemis..

66 4 Hückel Molecular Orbital Theory4.13 The Eigenvector Calculation of the Secular MatrixThe expansion of any molecular orbital over a basis set φ k , ψ = ∑a k φ k leads toka set of arbitrary expansion coefficients a k , which we optimize by imposing the conditionsof optimization, ∂E = ∂E = ∂E = ...= ∂E = ...= ∂E = 0, to find the∂a 1 ∂a 2 ∂a 3 ∂a k ∂a nenergy minimum in an n-dimensional vector space by calculating the eigenvector.The eigenvector calculation using MATLAB is quite simple. The entries are givenas follows:>> A=[0 1 0;1 0 1;0 1 0];>> [V,D] = eig(A)V =0.5000 -0.7071 0.5000-0.7071 0.0000 0.70710.5000 0.7071 0.5000D =-1.4142 0 00 -0.0000 00 0 1.4142The elements of the diagonal in the d matrix correspond to the eigenvalues. Theeigenvector of the matrix with −1.414 as the eigenvalue is:⎡⎣ 0.5000 ⎤−0.7071⎦0.5000The eigenvector of the matrix with 0 as the eigenvalue is:⎡⎣ −0.7071⎤0.0000 ⎦ .0.70714.14 The Chemical Applications of Hückel’s MOTThe Hückel results show some interesting features for conjugated hydrocarbonskeeping alternate double and single bonds [2, 3]:1. The orbital energies are in pairs of equal magnitude and opposite signs. Thismeans that if there is an odd number of orbitals, there must be an orbital energyof zero (a non-bonding orbital) that pairs with itself. (An example is the benzylradical in Table 4.1).