Analytic density functionals with initial-state dependence and memory

We analytically construct the wave function that, for a given initial state, produces a prescribed density for a quantum ring with two noninteracting particles in a singlet state. In this case the initial state is completely determined by the initial density, the initial time derivative of the density and a single integer that characterizes the (angular) momentum of the system. We then give an exact analytic expression for the exchange-correlation potential that relates two noninteracting systems with different initial states. This is used to demonstrate how the Kohn-Sham procedure predicts the density of a reference system without the need of solving the reference system's Schrödinger equation. We further numerically construct the exchange-correlation potential for an analytically solvable system of two electrons on a quantum ring with a squared cosine two-body interaction. For the same case we derive an explicit analytic expression for the exchange-correlation kernel and analyze its frequency dependence (memory) in detail. We compare the result to simple adiabatic approximations and investigate the single-pole approximation. These approximations fail to describe the doubly excited states, but perform well in describing the singly excited states.

Figure 1

The density, x potential, and c potentials (all in a.u.) for $C_0^2=0.5$ and $m^{\prime }=0$ as well as $m^{\prime }=1$ ($\lambda =100$, $L=1$ a.u.). Note the change of scale between $m^{\prime }=0$ and $m^{\prime }=1$. Further note, that we used the gauge freedom of the potentials in order to set them to zero at $x=0$.

Figure 2

The real part of $f_{\mathrm{Hxc}}^k$, $f_{\mathrm{Hx}}^k$, and $f_{\mathrm{HLDA}}^k$ for $k=1,2$ ($\lambda =10$ and $L=1$ a.u.).

Figure 3

The real parts of ${(1-{\chi }_0{f}_{\mathrm{Hxc}})}^k$, ${(1-{\chi }_0{f}_{\mathrm{Hx}})}^k$, and ${(1-{\chi }_0{f}_{\mathrm{HLDA}})}^k$ for $k=1,2$ ($\lambda =10$ and $L=1$ a.u.). The bare KS resonances ${\omega }_1$ and ${\omega }_2$ are indicated with arrows pointing to their values on the frequency axis. We have also indicated the single-pole approximated resonances ${\Omega }_1^{+}$ and ${\Omega }_2^{+}$ (in this frequency range) by arrows pointing to their respective values.

Figure 4

The relative error ${\Delta }_k^{\pm }$ of the SPA and the relative error ${\delta }_k^{\pm }$ of the bare KS resonances for $\lambda =10$ and $L=1$ a.u.

Figure 5

The relative error ${\Delta }_k^{\pm }$ of the SPA and the relative error ${\delta }_k^{\pm }$ of the bare KS resonances for $\lambda =1000$ and $L=1$ a.u.