Optimization of Effective Atom Centered Potentials for London Dispersion Forces in Density Functional Theory

We add an effective atom-centered nonlocal term to the exchange-correlation potential in order to cure the lack of London dispersion forces in standard density functional theory. Calibration of this long-range correction is performed using density functional perturbation theory and an arbitrary reference. Without any prior assignment of types and structures of molecular fragments, our corrected generalized gradient approximation density functional theory calculations yield correct equilibrium geometries and dissociation energies of argon-argon, benzene-benzene, graphite-graphite, and argon-benzene complexes.

Figure 1

In the inset of the left graph the additional $d$-channel projector $p_{21}(r)$ [Eq. (1)] is depicted in arbitrary units as a function of $r$, the minimum is $\approx 3.3Å$. Potential energy curves of the total energy of interaction ($E^{\mathrm{i}\mathrm{n}\mathrm{t}}=E^{\mathrm{D}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{r}}-2E^{\mathrm{M}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{r}}$) are plotted as a function of the distance $z$. ○ corresponds to normal ECPs, ◇ to the optimized ECP (OECP), and □ to MP2 data from 29. The experimental value for the interlayer distance and energy of interaction of two graphite sheets is marked by a cross. The left hand panel shows the curve for calibration. In atomic units the calibrated values of the additional $d$ channel for carbon are ${\sigma }_1=-0.00352$ and ${\sigma }_2=3.280$. The other two graphs represent results for the T-shaped configuration of benzene (middle panel) and for the slab of graphite (right hand panel) using the same values ${\sigma }_1$, ${\sigma }_2$. In the inset of the right hand panel the averaged interlayer distance $z$ is presented during first-principles BOMD using the normal ECP (dotted line) and the OECPs (continuous line).

Figure 2

For the argon-argon and the benzene-argon dimer potential energy curves of the total energy of interaction ($E^{\mathrm{i}\mathrm{n}\mathrm{t}}=E^{\mathrm{D}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{r}}-2E^{\mathrm{M}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{s}}$) are plotted as a function of the distance $z$. ○ corresponds to the normal ECPs, ◇ to the optimized ECP (OECP), and the MP2 results [27] for the interlayer distances and energies of interaction are marked by a cross. In atomic units the calibrated values of the additional $f$ channel for argon are ${\sigma }_1=-0.00206$ and ${\sigma }_2=2.902$. In the inset of the right hand panel the $C_{6v}$ symmetry axis $z$ is plotted versus $\Delta =\int dxdy[n^{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}(\mathbf{r})-n^{\mathrm{o}\mathrm{p}\mathrm{t}}(\mathbf{r})]$ (the differences between integrated $xy$ planes of the electron density of the benzene-argon dimer at equilibrium distance computed with the normal ECPs and the OECPs, respectively). The dotted lines show the position of the moieties: benzene (the molecular plane is perpendicular to $z$) is at 7.85 Å on the $z$ axis, and argon is at 11.26 Å.