Diagonal energy parameters and Koopmans' Theorem

The diagonal ($\varepsilon_{aa}$) and off-diagonal ($\varepsilon_{ab}$) ``energy'' parameters, are related to the Lagrange multipliers with $\varepsilon_{aa}=2\lambda_{aa}/w_a$ and $\varepsilon_{ab}= \lambda_{ab}/w_a$. Multiplying eq:genhf by $\P{a}{}$ and integrating, it is easy to show that

$$\varepsilon_{aa} = \frac{2}{w_a} \bar{\cal E}((n_al_a)^{w_a}) -\left(w_a-1\right)\sum_{k=0}^{2l_a} f_k(l_a)F^{k}(a,a).$$ (35)

In the special case where w_a=1, $\varepsilon_{aa}$ is twice the removal energy, or ionization energy. This is often referred to as Koopmans' Theorem but, as seen earlier, if a rotation of the radial basis leaves the wave function unchanged while transforming the matrix of energy parameters ($\varepsilon_{ab}$), the removal energies are extreme values obtained by setting the off-diagonal energy parameters to zero. For multiply occupied shells, $\varepsilon_{aa}$ is like an average removal energy, with a correction arising from the self-interaction.
 

2001-01-09