Enforcing the linear behavior of the total energy with hybrid functionals: Implications for charge transfer, interaction energies, and the random-phase approximation

We obtain the exchange parameter of hybrid functionals by imposing the fundamental condition of a piecewise linear total energy with respect to electron number. For the Perdew-Burke-Ernzerhof (PBE) hybrid family of exchange-correlation functionals (i.e., for an approximate generalized Kohn-Sham theory) this implies that (i) the highest occupied molecular orbital corresponds to the ionization potential $\left(I\right)$, (ii) the energy of the lowest unoccupied molecular orbital corresponds to the electron affinity $\left(A\right)$, and (iii) the energies of the frontier orbitals are constant as a function of their occupation. In agreement with a previous study [N. Sai et al., Phys. Rev. Lett. 106, 226403 (2011)], we find that these conditions are met for high values of the exact exchange admixture $\alpha$ and illustrate their importance for the tetrathiafulvalene-tetracyanoquinodimethane complex for which standard density functional theory functionals predict artificial electron transfer. We further assess the performance for atomization energies and weak interaction energies. We find that atomization energies are significantly underestimated compared to PBE or PBE0, whereas the description of weak interaction energies improves significantly if a $1/R^6$ van der Waals correction scheme is employed.

Figure 1

Summary of properties of the exact functional (left row), approximate functionals that suffer from the DSLE (middle row), and approximate functionals that are DSLE free (right row). The top panel shows the total energy, the middle panel the derivative of the total energy with respect to particle number, and the bottom panel the corresponding eigenvalues. Note that for mathematical clarity, there are two functions in each graph in the last two panels. They correspond to the two closed domains of definition $[N_0-1,N_0]$ and $[N,N_0+1]$, whereas the boundary values correspond to the one-sided limits. For example, $\partial E/\partial N\left(N_0\right)=\partial E/\partial N^{-}\left(N_0\right)$ if $E$ is considered over $[N_0-1,N_0]$ and $\partial E/\partial N^{+}\left(N_0\right)$ if $E$ is considered over $[N_0,N_0+1]$. $C^1$ refers to approximate functionals that are explicit and differentiable functionals of the first-order density matrix.

Figure 2

Illustration of the straight line behavior of the total energy $\left(E\right)$ (a) and its derivative (b).

Figure 3

Total energy for the TCNQ (a) and TTF (b) molecules as a function of partial occupation. (c), (d) Show the deviation from the straight line behavior; (e), (f) show GKS eigenvalues at partial occupation.

Figure 4

Transfer energy (a) and its derivative (b) for the TTF-TCNQ complex.

Figure 5

Binding energy (c) and Hirshfeld charge transfer (d) for the TTF-TCNQ complex as a function of distance. A sketch of the geometry is shown in (a) which was obtained by cutting out a dimer of the TTF-TCNQ interface. The reference distance $d=0$ corresponds to the equilibrium geometry the dimer would have in the periodic interface. (b) Shows a zoom of the transfer energy.

Figure 6

Charge density difference $\delta \rho =\rho \left(\text{dimer}\right)-\rho \left(\text{TTF}\right)-\rho \left(\text{TCNQ}\right)$ for PBE, $\mathrm{PBEh}({\alpha }^{*})$, and fully self-consistent $GW$ $\left(\mathrm{sc}GW\right)$. Also shown are the TTF-HOMO and TCNQ-LUMO orbitals. Blue corresponds to electron density depletion and red corresponds to electron density accumulation. The yellow and pink colors for the orbitals label the sign of the orbital.

Figure 7

Mean absolute percentage error (%) of the (a) TS-vdW scheme [30] and (b) rPT2 [32] for the S66 test set. The errors are relative to CCSD(T) reference data [26].

Figure 8

Binding curves for the pentane dimer (illustrated in the inset) obtained by the TS method (a), and its contributions from the (generalized) Kohn-Sham (G)KS functionals (b), the TS-vdW tail (c), and the TS-vdW tail where the $s_R$ parameter is set to one (d). The binding distance is given by the scaling parameter $d$ with respect to the equilibrium distance. The CCSD(T) reference curve was taken from Ref. [26].

Figure 9

Binding curve for the benzene dimer obtained by (a) rPT2, (b) EX+cRPA. (c) Shows the SOSEX and (d) the rSE correction. Also shown is the CCSD(T) reference curve, which was taken from Ref. [26]. The binding distance is given by the scaling parameter $d/d_0$ with respect to the equilibrium distance.

Figure 10

EX and RPA contributions for the benzene-benzene binding energy (a). Shown are the differences with respect to the PBE starting point [Eqs. (41) and (42)]. (b) Shows the rSE contributions to EX at the equilibrium distance $d/d_0=1$.

Figure 11

MAPE (%) for the G2 test set of atomization energies [27] for GKS, rPT2. Reference data are from Ref. [111]. The results were obtained from the Gaussian cc-pV6Z basis set [95].

Figure 12

MAPE (%) of the HTBH38 and NHTBH38 test sets for barrier heights. The errors are relative to reference data of Refs. [28, 29] for HTBH38 and NHTBH38, respectively. The results were obtained with the Gaussian cc-pV6Z basis set [95].

Figure 13

MAPE (%) for the G2 test set of atomization energies [27] for (a) functionals that use the same $\alpha$ value for molecules and atoms: PBE $\left(\alpha =0\right)$, $\text{PBE}0\left(\alpha =\frac{1}{4}\right)$, and $\mathrm{PBEh}({\alpha }_{\text{mol}}^{*})$ using the optimum $\alpha$ value of the molecule $\left({\alpha }_{\text{mol}}^{*}\right)$. (b) Shows the MAPE where $\alpha$ was individually optimized for molecules and atoms $\left({\alpha }_{\text{atom}}^{*}\right)$. PBE0@$\text{PBEh}\left({\alpha }^{*}\right)$ is the PBE0 energy expression evaluated for $\mathrm{PBEh}({\alpha }^{*})$ densities. Reference data are from Ref. [111].