Reconstructing the adiabatic exchange-correlation kernel of time-dependent density-functional theory

The interacting and the Kohn-Sham static density-density response functions for different one-dimensional two-electron singlet systems are reconstructed numerically. From their inverse we obtain the exact static exchange-correlation kernel. This quantity represents the adiabatically exact approximation of the frequency-dependent exchange-correlation kernel that is crucial for time-dependent linear density-response theory. We investigate its performance for nonlocal perturbations and analyze its sum rule properties. We also compute the adiabatically exact transition energies that follow from the static kernel within linear-response theory.

Figure 1

(Color online) Interacting density-density response function ${\chi }_0\left(z,z^{\prime }\right)$ for helium.

Figure 2

(Color online) Noninteracting KS density-density response function ${\chi }_{\text{s},0}\left(z,z^{\prime }\right)$ for helium.

Figure 3

(Color online) Interacting density-density response function ${\chi }_0\left(z,z^{\prime }\right)$ for the A6-Hooke system.

Figure 4

(Color online) Noninteracting KS density-density response function ${\chi }_{\text{s},0}\left(z,z^{\prime }\right)$ for the A6-Hooke system.

Figure 5

Nonlocal potential perturbations $\delta v^{\text{exact}}\left(z\right)$ from Table adapted for the perturbation of $v_{\text{ext},\text{g}\text{s}}\left(z\right)$ of the helium atom (perturbation type 1–5: solid, dotted, dashed, dotted-dashed, triply-dotted-dashed).

Figure 6

Exact gs density response $\delta n_0^{\text{exact}}$ to the potential perturbations of Fig. 5 obtained from solving the interacting SE for the helium atom (perturbation type 1–5: solid, dotted, dashed, dotted-dashed, triply-dotted-dashed). The response calculation results $\delta n_0^{\text{resp}}$ according to Eq. (35) lie directly on top of the corresponding $\delta n_0^{\text{exact}}$.

Figure 7

Test of the sum rules (30, 31) for helium: ${\partial }_z{n}_0\left(z\right)$ from direct differentiation (solid) and as obtained from $\int {\chi }_0\left(z,z^{\prime }\right){\partial }_{z^{\prime }}{v}_{\text{ext},0}\left(z^{\prime }\right)d{z}^{\prime }$ (dotted) and $\int {\chi }_{\text{s},0}\left(z,z^{\prime }\right){\partial }_{z^{\prime }}{v}_{\text{s},0}\left(z^{\prime }\right)d{z}^{\prime }$ (dashed). As all curves lie on top of each other on this scale, the inset shows the (tiny) differences between first and second (dotted) and first and third (dashed) curves.

Figure 8

$\delta v_{\text{c},0}^{\text{exact}}$ and $\delta v_{\text{c},0}^{\text{resp}}$ corresponding to perturbations of the helium gs density caused by external potential perturbations of type 1 (solid), 2 (dotted), 3 (dashed), 4 (dotted-dashed), and 5 (triply-dotted-dashed) according to Table . All potentials are shifted by constants so that they are zero at the origin.

Figure 9

Test of the sum rule (34) for helium: ${\partial }_z{v}_{\text{xc},0}\left(z\right)$ from direct differentiation (solid) and as obtained from $\int f_{\text{xc},0}\left(z,z^{\prime }\right){\partial }_{z^{\prime }}{n}_0\left(z^{\prime }\right)d{z}^{\prime }$ (dotted). Deviations at the edges are due to the finite size of the inversion interval.

Figure 10

(Color online) Static correlation kernel for helium on a restricted grid of roughly $9\times 9\text{a}\text{.u}\text{.}$ For better visualization the xc kernel has been scaled according to $f_{\text{c},0}^{\text{scaled}}\left(z,z^{\prime }\right)=\text{arctan}\left[f_{\text{c},0}\left(z,z^{\prime }\right)\right]$.

Figure 11

Oscillator strengths of transition energies obtained from the bare KS values (dotted), EXX-Casida (dashed), AE-Casida (dotted-dashed), and from the exact eigenstates (solid) for helium. For each transition we plot only the dominant oscillator strength, i.e., either dipole or quadrupole. The quadrupole oscillator strengths have been rescaled for better visualization.

Figure 12

Helium dipole (solid) and quadrupole (dashed) power spectrum from AE real-time propagation of Ref. [12]. The vertical lines indicate the dipole (dotted) and quadrupole (dotted-dashed) transition energies obtained from the present AE Casida approach (cf. Fig. 11).

Figure 13

Oscillator strengths of transition energies obtained from the bare KS values (dotted), EXX-Casida (dashed), AE-Casida (dotted-dashed), and from the exact eigenstates (solid) for the A6-Hooke system. For each transition we plot only the dominant oscillator strength, i.e., either dipole or quadrupole. The quadrupole oscillator strengths have been rescaled for better visualization. The inset shows a magnification of the higher energy regime.

Figure 14

A6-Hooke dipole (solid) and quadrupole (dashed) power spectrum from AE real-time propagation of Ref. [12]. The vertical lines indicate the dipole (dotted) and quadrupole (dotted-dashed) transition energies obtained from the present AE Casida approach (cf. Fig. 13).