Density functional theory beyond the linear regime: Validating an adiabatic local density approximation

We present a local density approximation (LDA) for one-dimensional (1D) systems interacting via the soft-Coulomb interaction based on quantum Monte Carlo calculations. Results for the ground-state energies and ionization potentials of finite 1D systems show excellent agreement with exact calculations obtained by exploiting the mapping of an $N$-electron system in $d$ dimensions onto a single electron in $N\times d$ dimensions, properly symmetrized by the Young diagrams. We conclude that 1D LDA is of the same quality as its three-dimensional (3D) counterpart, and we infer conclusions about 3D LDA. The linear and nonlinear time-dependent responses of 1D model systems using LDA, exact exchange, and the exact solution are investigated and show very good agreement in both cases, except for the well-known problem of missing double excitations. Consequently, the 3D LDA is expected to be of good quality beyond the linear response. In addition, the 1D LDA should prove useful in modeling the interaction of atoms with strong laser fields, where this specific 1D model is often used.

Figure 1

Binding energy per atom of the one-dimensional hydrogen molecule as a function of the distance between the two ions; exact and LDA calculations for the singlet ground state and the first triplet state.

Figure 2

Linear (top) and nonlinear (bottom) spectra of Be${}^{2+}$ comparing the exact and the 1D LDA calculation. The inset in the bottom figure shows a zoom into the region from $2.7$ to $3.0$ Ha.