Time-dependent density-functional-theory calculation of the van der Waals coefficient $C_6$ of alkali-metal atoms Li, Na, K; alkali-metal dimers ${\mathrm{Li}}_2$, ${\mathrm{Na}}_2$, ${\mathrm{K}}_2$; sodium clusters ${\mathrm{Na}}_n$; and fullerene ${\text{C}}_{60}$

We have employed time-dependent density-functional theory (TDDFT) to calculate the long-range dipole-dipole dispersion coefficient (van der Waals coefficient) $C_6$ of alkali-metal atoms Li, Na, K; alkali-metal atom dimers ${\mathrm{Li}}_2$, ${\mathrm{Na}}_2$, ${\mathrm{K}}_2$; sodium clusters containing even number of atoms ranging from 2 to 20 atoms; and fullerene ${\text{C}}_{60}$. The dispersion coefficient is obtained via Casimir-Polder expression which relates it to the frequency dependent linear polarizability at imaginary frequencies. The frequency-dependent polarizabilities are calculated by employing TDDFT-based complete sum-over-states expressions for the atoms, and direct TDDFT linear response theory for the closed shell dimers and clusters.

Figure 1

Comparison of linear polarizability $\alpha \left(\omega \right)$ (in Hartree atomic units) of the Li atom as a function of frequency obtained by different methods: TDDFT (SAOP and LDA, this work), Møller-Plesset perturbation theory (Ref. 37), Hylleraas wave function approach (Ref. 47), and Random-phase approximation and SOS expression (Ref. 49). The lines joining the points were added to guide the eye.

Figure 2

Same as Fig. 1 but for Na atom. TDDFT (SAOP and LDA, this work), Møller-Plesset perturbation theory (Ref. 37), CI approach (Ref. 48), and random-phase approximation and SOS expression (Ref. 49). The lines joining the points were added to guide the eye.

Figure 3

Same as Fig. 1 but for K atom. TDDFT (SAOP and LDA, this work), Møller-Plesset perturbation theory (Ref. 37), and random-phase approximation and SOS expression (Ref. 49). The lines joining the points were added to guide the eye.

Figure 4

Comparison of TDDFT and London formula based results for the van der Waals coefficient $C_6$ corresponding to the alkali-atom-cluster pairs (a) $\mathrm{Li}\text{−}{\mathrm{Na}}_n$, (b) $\mathrm{Na}\text{−}{\mathrm{Na}}_n$, and (c) $\mathrm{K}\text{−}{\mathrm{Na}}_n$. The numbers for $C_6$ are in atomic units. The London formula results (represented by solid circles) were taken from Ref. 19. The lines joining the points were added to guide the eye.