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):

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

for atomic orbitals that are occupied in the atom, and

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

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 \[ C(1s^22s2p_x2p_y2p_z) \rightarrow C^+(1s^22s2p_x2p_y). \nonumber \] If the energy differences for the "promotion" of C \[ C(1s^22s^22p_x2p_y) \rightarrow C(1s^22s2p_x2p_y2p_z); \Delta E_C \nonumber \] and for the promotion of \(C^+\) \[ C^+(1s^22s^22p_x) \rightarrow C^+(1s^22s2p_x2p_y); \Delta E_C^+ \nonumber \] are known, the desired VSIP is given by: \[ IP_{2p_z} = IP_C + \Delta E_C + - \Delta E_C. \nonumber \] The EA of the 2p orbital is obtained from the \[ C(1s^22s^22p_x2p_y) \rightarrow C^-(1s^22s^22p_x2p_y2p_z) \nonumber \] 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:

\[ \beta_{\mu,\nu} = \beta^o_{\mu,\nu}S_{\mu,\nu}. \nonumber \]

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 \(S_{\mu,\nu}\) ).