Excitation energies from time-dependent density-functional formalism for small systems

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Abstract
As a test of the time-dependent local-density approximation (TDLDA), we study the 1Σ+ g → 1 Σ+u excitation of H2 as a function of the nuclear distance d .W e f ind rather accurate results for intermediate d but not for small and large d .A t larged, TDLDA fails due to the strong non-locality and energy dependence of the exact functional. The spin-dependent formalism gives a qualitative improvement for large d. To analyze the results, we compare with the 2s → 2p excitation for He, Li and Be for small and intermediate d and with the Hubbard model for large d. The fairly accurate results for Li and Be are related to the accuracy of the ground-state formalism for a few electrons. The traditional density functional (DF) formalism (1) is limited to ground-state properties. The time-dependent density-functional formalism (2, 3) can, however (3-5), be used to obtain excitation energies. This method has in particular been applied to atoms and molecules (4,6-8). It has been found that already the time-dependent local density approximation (TDLDA) gives rather accurate results for these systems. The purpose of this paper is to obtain a better understanding of the TDLDA. We therefore apply the TDLDA to the H2 molecule and study the accuracy as a function of the nuclear separation d. Separations away from equilibrium are interesting for the study of dynamical properties of molecules. This system can also give some indications of the accuracy of TDLDA for atoms at surfaces in the so-called surface molecule limit (9). H2 is a good test case, since there are essentially exact results (10) available, with which we can compare. We find that the accuracy is rather good for intermediate d, while it is less good for small d. For large d the TDLDA fails qualitatively. The spin-dependent version (TDLSDA) gives a qualitative