Theory

The Kohn-Sham eigenvalues of an approximate density functional are not electron removal and addition energies. They are the ingredients of a fictitious non-interacting system, and with the semilocal functionals in common use they are also contaminated by self-interaction: the energy of an orbital drifts as you change its occupancy, when it should not. This is why band gaps computed from Kohn-Sham eigenvalues are too small, and why those eigenvalues are unreliable as a spectrum.

Koopmans functionals repair this [3, 10]. They are corrective functionals, built on top of a base density functional,

\[E^\text{Koopmans} = E^\text{DFT} + \sum_i \alpha_i \Pi^u_i\]

where the correction \(\Pi^u_i\) is constructed so that the energy of orbital \(i\) is linear in its occupancy \(f_i\). Imposing that condition on every orbital in the system — the generalized piecewise linearity condition — makes each orbital energy equal to the total-energy difference for adding or removing an electron from that orbital. The orbital energies become quantities you can compare with an experimental spectrum.

The \(\alpha_i\) are screening parameters, and they account for the relaxation of everything else in the system when the occupancy of orbital \(i\) changes. They are computed from first principles, rather than being fitted to experiment or to some other level of theory. This is why a Koopmans calculation is a workflow rather than a single run: we need to determine the \(\alpha_i\), either from a series of constrained total-energy calculations [25], or from linear response [6, 7]. The tutorials show examples for both.

The result is spectral accuracy comparable to many-body perturbation theory at a much lower computational cost, while staying within a functional formulation with a well-defined total energy.

Going deeper

For the theory in full — the derivation of the functionals, the different flavors (KI, KIPZ), the role of the variational orbitals, the algorithms, and more, see [18].

The references page also lists many papers that you might be interested to read.