# 7.3: The Hückel Parameterization

In the most simplified embodiment of the above orbital-level model, the following additional approximations are introduced.

## Approximation 1: Diagonal Component

The diagonal values $$\langle \chi_\mu | \frac{-\hbar^2}{2m_e}\nabla^2 + V|\chi_\mu \rangle$$, which are usually denoted $$\alpha_\mu$$, are taken to be equal to the energy of an electron in the atomic orbital $$\chi_\mu$$ and, as such, are evaluated in terms of atomic ionization energies (IP's) and electron affinities (EA's):

$\langle\chi_\mu|\dfrac{-\hbar^2}{2m_e}\nabla^2 + V|\chi_\mu\rangle = -IP_\mu,$

for atomic orbitals that are occupied in the atom, and

$\langle\chi_\mu|\dfrac{-\hbar^2}{2m_e}\nabla^2 + V|\chi_\mu\rangle = -EA_\mu,$

for atomic orbitals that are not occupied in the atom.

These approximations assume that contributions in V arising from coulombic attraction to nuclei other than the one on which $$\chi_\mu$$ is located, and repulsions from the core, lone-pair, and valence electron clouds surrounding these other nuclei cancel to an extent that $$\langle\chi_\mu|V|\chi_\mu\rangle$$ contains only potentials from the atom on which $$\chi_\mu$$ sits.

It should be noted that the IP's and EA's of valence-state orbitals are not identical to the experimentally measured IP's and EA's of the corresponding atom, but can be obtained from such information. For example, the 2p valence-state IP (VSIP) for a Carbon atom is the energy difference associated with the hypothetical process $C(1s^22s2p_x2p_y2p_z) \rightarrow C^+(1s^22s2p_x2p_y).$ If the energy differences for the "promotion" of C $C(1s^22s^22p_x2p_y) \rightarrow C(1s^22s2p_x2p_y2p_z); \Delta E_C$ and for the promotion of $$C^+$$ $C^+(1s^22s^22p_x) \rightarrow C^+(1s^22s2p_x2p_y); \Delta E_C^+$ are known, the desired VSIP is given by: $IP_{2p_z} = IP_C + \Delta E_C + - \Delta E_C.$ The EA of the 2p orbital is obtained from the $C(1s^22s^22p_x2p_y) \rightarrow C^-(1s^22s^22p_x2p_y2p_z)$ energy gap, which means that $$EA_{2p_z} = EA_C$$ . Some common IP's of valence 2p orbitals in eV are as follows: C (11.16), N (14.12), $$N^+$$ (28.71), O (17.70), $$O^+$$ (31.42), $$F^+$$ (37.28).

## Approximation 2: Nearest Neighbors Approximation

The off-diagonal elements $$\langle\chi_\nu | \frac{-\hbar^2}{2m_e}\nabla^2 + V|\chi_\mu \rangle$$ are taken as zero if $$\chi_\mu \text{ and } \chi_nu$$ belong to the same atom because the atomic orbitals are assumed to have been constructed to diagonalize the one-electron hamiltonian appropriate to an electron moving in that atom. They are set equal to a parameter denoted $$\beta_{\mu,\nu} \text{ if } \chi_\mu \text{ and } \chi_\nu$$ reside on neighboring atoms that are chemically bonded. If $$c_m$$ and $$c_n$$ reside on atoms that are not bonded neighbors, then the off-diagonal matrix element is set equal to zero.

## Approximation 3: Off-Diagonal Component

The geometry dependence of the $$\beta_{\mu,\nu}$$ parameters is often approximated by assuming that $$\beta_{\mu,\nu}$$ is proportional to the overlap $$S_{\mu,\nu}$$ between the corresponding atomic orbitals:

$\beta_{\mu,\nu} = \beta^o_{\mu,\nu}S_{\mu,\nu}.$

Here $$\beta^o_{\mu,\nu}$$ is a constant (having energy units) characteristic of the bonding interaction between $$\chi_\mu \text{ and } \chi_\nu$$; its value is usually determined by forcing the molecular orbital energies obtained from such a qualitative orbital treatment to yield experimentally correct ionization potentials, bond dissociation energies, or electronic transition energies.

It is sometimes assumed that the overlap matrix $$S$$ is the identity matrix. This means that overlap between the orbitals is neglected

The three approximations above form the basis of the so-called Hückel model. Its implementation requires knowledge of the atomic $$\alpha_\mu$$ and $$\beta^0_{\mu,\nu}$$ values, which are eventually expressed in terms of experimental data, as well as a means of calculating the geometry dependence of the $$\beta_{\mu,\nu}$$'s (e.g., some method for computing overlap matrices $$S_{\mu,\nu}$$ ).