In the Hückel treatment of the cyclopropenyl radical, the basis functions are the three 2pz atomic orbitals, which we denote by f₁, f₂, and f₃.

ψ = c₁ f₁ + c₂ f₂ + c₃ f₃.

The possible values of the orbital energyWare sought as a function of the c coefficients by solving the three simultaneous equations

xc₁ + c₂ + c₃ = 0,
c₁ + xc₂ + c₃ = 0,
c₁ + c₂ + xc₃ = 0,

where x = (α ˆ’ W)/β) and where α and β are certain integrals whose values are to be determined from empirical data.
(a) The determinant of the c coefficients must be set equal to zero in order for a nontrivial solution to exist. This is the secular equation. Solve the secular equation and obtain the orbital energies.
(b) Solve the three simultaneous equations, once for each value of x. Since there are only two independent equations, express c₂ and c₃ in terms of c₁.
(c) Impose the normalization condition c²₁ + c²₂ + c²₃ = 1 to find the values of the c coefficients for each value of W.
(d) Check your work by using Mathematica to find the eigenvalues and eigenvectors of the matrix

Transcribed Image Text:

х 1 1 1х 1 11х