Density Functional Resonance Theory of Unbound Electronic Systems

Density functional resonance theory (DFRT) is a complex-scaled version of ground-state density functional theory (DFT) that allows one to calculate the in-principle exact resonance energies and lifetimes of metastable anions. In this formalism, the energy and lifetime of the lowest-energy resonance of unbound systems is encoded into a complex “density” that can be obtained via complex-coordinate scaling. This complex density is used as the primary variable in a DFRT calculation, just as the ground-state density would be used as the primary variable in DFT. As in DFT, there exists a mapping of the $N$-electron interacting system to a Kohn-Sham system of $N$ noninteracting particles. This mapping facilitates self-consistent calculations with an initial guess for the complex density, as illustrated with an exactly solvable model system.

Figure 1

Exact two-electron complex densities associated with the LER of $v(x)$ when using different scaling angles ($a=4$, $b=0.5$, $c=4$, $d=2$, and $\lambda =1$).

Figure 2

The real and imaginary parts of $E_{\theta }$ in the model Hamiltonian of Eq. (8) at various values of $\lambda$ (joined by lines) calculated exactly with complex scaling, a first-order correction to the noninteracting energy, and our DFRT exchange-only self-consistent method ($a=4$, $b=0.5$, $c=4$, $d=2$, and $\lambda =1$).

Figure 3

The real and imaginary parts of the complex KS potential for the LER of 2 soft-Coulomb interacting electrons in the model potential, Eq. (8). The dashed lines are the real and imaginary parts of the complex-scaled parent potential $\tilde{v}(x)$ ($\theta =0.35$, $a=4$, $b=0.5$, $c=4$, $d=2$, and $\lambda =1$).

Figure 4

The individual contributions to the Kohn-Sham potential from Hartree exchange and correlation ($\theta =0.35$, $a=4$, $b=0.5$, $c=4$, $d=2$, and $\lambda =1$).