Answer to Question #143827 in Physical Chemistry for Booker

Question

Consider the carbonate ion, CO32-
i) Setup and solve the Huckel secular equations for the pi-electrons.
ii) Express the energies in terms of the Coulomb integrals]
iii) Determine the delocalization energy.

Answer 1

i) the molecular orbitals |ψ_i⟩ can be described as a linear combination of the 2p_z atomic orbital at carbon with corresponding c₁ coefficient and 2p_z atomic orbitals at oxygen with corresponding c₂, c₃ and c₄ coefficients

c₁(H₁₁-ES₁₁)+c₂(H₁₂-ES₁₂)+c₃(H₁₃-ES₁₃)+c₄(H₁₄-ES₁₄)=0

c₁(H₂₁-ES₂₁)+c₂(H₂₂-ES₂₂)+c₃(H₂₃-ES₂₃)+c₄(H₂₄-ES₂₄)=0

c₁(H₃₁-ES₃₁)+c₂(H₃₂-ES₃₂)+c₃(H₃₃-ES₃₃)+c₄(H₃₄-ES₃₄)=0

c₁(H₄₁-ES₄₁)+c₂(H₄₂-ES₄₂)+c₃(H₄₃-ES₄₃)+c₄(H₄₄-ES₄₄)=0

$\alpha$_i = H_ii

$\beta$ _ij = H_ij; $\beta$ _ij = $\beta$ _ji

S_ij = 0

S_ii = 1

only adjecent orbitals have interactions

c₁($\alpha$₁-E)+c₂($\beta$ ₁₂)+c₃($\beta$ ₁₃)+c₄($\beta$ ₁₄)=0

c₁($\beta$ ₂₁)+c₂($\alpha$₂-E)+0+0=0

c₁($\beta$ ₃₁)+0+c₃($\alpha$₃-E)+0=0

c₁($\beta$ ₄₁)+0+0+c₄($\alpha$₄-E)=0

$\beta$ ₁₂ = $\beta$ ₁₃=$\beta$ ₁₄=$\beta$

$\alpha$₂ = $\alpha$₃ = $\alpha$₄

$\begin{vmatrix} α1-E & β & β & β\\ β & α2-E & 0 & 0\\ β& 0 & α2-E & 0\\ β& 0 & 0 & α2-E\\ \end{vmatrix}$ = 0

$\alpha$₁$\alpha$₂³-3$\alpha$₂²$\beta$²-3$\alpha$₁$\alpha$₂²E-$\alpha$₂³E+6$\alpha$₂$\beta$ ²E+3$\alpha$₁$\alpha$₂E²+3$\alpha$₂²E²-3$\beta$²E²-$\alpha$₁E³-3$\alpha$₂E³+E⁴=0

Solving this system we obtain:

E₁=0.5($\alpha$₁+$\alpha$₂-($\alpha$₁²-2$\alpha$₁$\alpha$₂+$\alpha$₂²+12$\beta$²)^0.5)

E₂=E₃=$\alpha$₂

E₄=0.5($\alpha$₁+$\alpha$₂+($\alpha$₁²-2$\alpha$₁$\alpha$₂+$\alpha$₂²+12$\beta$²)^0.5)

ii) E₁=0.5($\alpha$₁+$\alpha$₂-($\alpha$₁²-2$\alpha$₁$\alpha$₂+$\alpha$₂²+12$\beta$²)^0.5)

E₂=E₃=$\alpha$₂

E₄=0.5($\alpha$₁+$\alpha$₂+($\alpha$₁²-2$\alpha$₁$\alpha$₂+$\alpha$₂²+12$\beta$²)^0.5)

iii) E(delocalized) = 0.5($\alpha$₁+$\alpha$₂-($\alpha$₁²-2$\alpha$₁$\alpha$₂+$\alpha$₂²+12$\beta$²)^0.5)

Let determine E(localized)

1-carbon, 2-oxygen with double bond

c₁(H₁₁-ES₁₁)+c₂(H₁₂-ES₁₂)=0

c₁(H₂₁-ES₂₁)+c₂(H₂₂-ES₂₂)=0

c₁($\alpha$₁-E)+c₂($\beta$ ₁₂)=0

c₁($\beta$ ₂₁)+c₂($\alpha$₂-E)=0

$\begin{vmatrix} α1-E & β \\ β & α2-E \\ \end{vmatrix}$ =0

$\alpha$₁$\alpha$₂-$\beta$²-$\alpha$₁E-$\alpha$₂E+E² = 0

E₁ = 0.5($\alpha$₁+$\alpha$₂-($\alpha$₁²-2$\alpha$₁$\alpha$₂+$\alpha$₂²+4$\beta$²)^0.5)

E₂ = 0.5($\alpha$₁+$\alpha$₂+($\alpha$₁²-2$\alpha$₁$\alpha$₂+$\alpha$₂²+4$\beta$²)^0.5)

E(delocalization) = 0.5($\alpha$₁+$\alpha$₂+($\alpha$₁²-2$\alpha$₁$\alpha$₂+$\alpha$₂²+4$\beta$²)^0.5)-0.5($\alpha$₁+$\alpha$₂-($\alpha$₁²-2$\alpha$₁$\alpha$₂+$\alpha$₂²+ 12$\beta$²)^0.5) = 0.5(-($\alpha$₁²-2$\alpha$₁$\alpha$₂+$\alpha$₂²+4$\beta$²)^0.5+($\alpha$₁²-2$\alpha$₁$\alpha$₂+$\alpha$₂²+ 12$\beta$²)^0.5)