Excited and Ionized States of Polymers

Abstract

It is well known that in the HF theory of a closed-shell atom or of a finite molecule, the singlet excitation energy from a filled level i to an unfilled level a is given by the expression

$$^1\Delta {E_{i \to a}} = \varepsilon _a^{HF} - \varepsilon _i^{HF} - {J_{ia}} + 2{K_{ia}}$$

(8.1)

where ε ^HF _a and ε ^HF _i are, respectively, the corresponding HF one-electron energies while J _ia and K _ia denote the Coulomb and exchange integrals between orbitals ψ _i and ψ _a respectively. It can easily be shown that if we have a very long chain, J _ia and K _ia tend to zero (see, e.g., Ladik⁽¹⁾) as the number of unit cells, N, approaches infinity:

$$\underset{N\to \infty }{\mathop{\lim }}\, {{J}_{ia}}=\underset{N\to \infty }{\mathop{\lim }}\, {{K}_{ia}}=\underset{N\to \infty }{\mathop{\lim }}\, \frac{\ln N}{N}\to 0$$

(8.2)

In this way we are left with too large an excitation energy, because equation (8.1) for an infinite system reduces to the HF gap (which is too large anyway).