Question

When using an independent particle model, an MO ψ is expressed in LCAO form, i.e. as a linear combination of atomic orbitals {ϕ_1, ϕ_2, . . . , ϕ_K}, (1)
and the expansion coefficients are determined from secular equation: the Hamiltonian is the one electron operator h and the “orbital energy” ε. Find the MOs and orbital energies for a linear
polyene (Cn Hn+2), taking ϕ_µ to be 2p_z AOs on C and including only nearest-neighbor matrix
element, with
(ϕ_µ|h|ϕ_µ) = α, (ϕ_µ|h|ϕ_µ±1) = β (2)
and with the neglect of the overlap.
Give explicitly the MOs and orbital energies for N = 3 and N = 4.

Answer 1

Answer on Question #82730 - Chemistry - Physical Chemistry

Question:

When using an independent particle model, an MO $\psi$ is expressed in LCAO form, i.e. as a linear combination of atomic orbitals $\{\varphi_{-}1,\varphi_{-}2,\dots ,\varphi_{-}K\}$, (1)

and the expansion coefficients are determined from secular equation: the Hamiltonian is the one electron operator $h$ and the "orbital energy" $\varepsilon$. Find the MOs and orbital energies for a linear

polyene (Cn Hn+2), taking $\varphi_{-}\mu$ to be 2p_z AOs on C and including only nearest-neighbor matrix element, with

$$(\varphi_ {-} \mu | h | \varphi_ {-} \mu) = \alpha , (\varphi_ {-} \mu | h | \varphi_ {-} \mu \pm 1) = \beta (2)$$

and with the neglect of the overlap.

Give explicitly the MOs and orbital energies for $N = 3$ and $N = 4$.

Solution:

$$\left\langle \varphi_ {i} \right| \mathbf {H} \left| \varphi_ {i} \right\rangle = \alpha_ {i}$$ $$\left\langle \varphi_ {i} \right| \mathbf {H} \left| \varphi_ {i} \right\rangle = \beta_ {i j}$$ $$\left\langle \varphi_ {i} \right| \varphi_ {j} \rangle = S _ {i j}$$

where $\alpha_{i}$ is termed the Coulomb integral, $\beta_{ij}$ the resonance integral and $S_{ij}$ the overlap integral. We are using normalized AOs, so $S_{ii}$ . Furthermore, the two atoms are identical, so

$$\alpha 1 = \alpha 2;$$ $$\beta 1 2 = \beta 2 1;$$ $$(c _ {1} ^ {2} + c _ {2} ^ {2}) \alpha + 2 c _ {1} c _ {2} \beta - E (c _ {1} ^ {2} + c _ {2} ^ {2} + 2 c _ {1} c _ {2} S) = 0$$ $$\frac {\partial E}{\partial c _ {1}} = \frac {\partial E}{\partial c _ {2}} = 0$$ $$(\alpha - E) c _ {1} + (\beta - E S) c _ {2} = 0$$ $$(\beta - E S) c _ {1} + (\alpha - E) c _ {2} = 0$$ $$\left| \begin{array}{c c} \alpha - E & \beta - E S \\ \beta - E S & \alpha - E \end{array} \right| = (\alpha - E) ^ {2} - (\beta - E S) ^ {2} = 0$$ $$E _ {1} = \frac {\alpha + \beta}{1 + S} \quad \text{and} \quad E _ {2} = \frac {\alpha - \beta}{1 - S}$$ $$\left\langle \Psi_ {i} \right| \Psi_ {i} \rangle = c _ {i 1} ^ {2} + c _ {i 2} ^ {2} + 2 c _ {i 1} c _ {i 2} S = 1$$ $$\Psi_ {1} = \frac {1}{\sqrt {2 (1 + S)}} \left(\varphi_ {1} + \varphi_ {2}\right) \quad \text{and} \quad \Psi_ {2} = \frac {1}{\sqrt {2 (1 + S)}} \left(\varphi_ {1} - \varphi_ {2}\right)$$

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