Density functional with full exact exchange, balanced nonlocality of correlation, and constraint satisfaction

We construct a nonlocal density functional approximation with full exact exchange, while preserving the constraint-satisfaction approach and justified error cancellations of simpler semilocal functionals. This is achieved by interpolating between different approximations suitable for two extreme regions of the electron density. In a “normal” region, the exact exchange-correlation hole density around an electron is semilocal because its spatial range is reduced by correlation and because it integrates over a narrow range to $-1$. These regions are well described by popular semilocal approximations (many of which have been constructed nonempirically), because of proper accuracy for a slowly varying density or because of error cancellation between exchange and correlation. “Abnormal” regions, where nonlocality is unveiled, include those in which exchange can dominate correlation (one-electron, nonuniform high density, and rapidly varying limits), and those open subsystems of fluctuating electron number over which the exact exchange-correlation hole integrates to a value greater than $-1$. Regions between these extremes are described by a hybrid functional mixing exact and semilocal exchange energy densities locally, i.e., with a mixing fraction that is a function of position $\mathbf{r}$ and a functional of the density. Because our mixing fraction tends to 1 in the high-density limit, we employ full exact exchange according to the rigorous definition of the exchange component of any exchange-correlation energy functional. Use of full exact exchange permits the satisfaction of many exact constraints, but the nonlocality of exchange also requires balanced nonlocality of correlation. We find that this nonlocality can demand at least five empirical parameters, corresponding roughly to the four kinds of abnormal regions. Our local hybrid functional is perhaps the first accurate fourth-rung density functional or hyper-generalized gradient approximation, with full exact exchange, that is size-consistent in the way that simpler functionals are. It satisfies other known exact constraints, including exactness for all one-electron densities, and provides an excellent fit to the 223 molecular enthalpies of formation of the G3/99 set and the 42 reaction barrier heights of the BH42/03 set, improving both (but especially the latter) over most semilocal functionals and global hybrids. Exact constraints, physical insights, and paradigm examples hopefully suppress “overfitting.”

Figure 1

Mixing fraction $a_1\left(u\right)$ of Eq. (19). The values of parameters $A$ and $B$ were fitted as explained in Sec. 5.

Figure 2

Function $f\left(v\right)$ of Eq. (23). The value of $C$ was found as explained in Sec. 5. $F$ is related to $C$ by Eq. (24).

Figure 3

Function $g\left(x\right)$ defined by Eq. (27). The values of parameters $D$ and $E$ were fitted as explained in Sec. 5.

Figure 4

PSTS hyper-GGA mixing fractions $a$ constructed for the Mg atom and isoelectronic ${\mathrm{Si}}^{2+}$ and ${\mathrm{Ca}}^{8+}$ ions using self-consistent TPSS orbitals obtained in Partridge’s uncontracted basis sets [76, 77, 78, 79]: $(20s,12p)$ for Mg; $(20s,15p)$ for Si; $(23s,16p)$ for Ca. The parameters of $a$ are given in Sec. 5. Here $\zeta =0$, so $a_2=0$ and $a=a_1$. The figure illustrates that our mixing fraction $a$ tends smoothly to 1 in the low-density tail, and slowly to 1 in the core in the high-density $(Z\to \infty )$ limit.

Figure 5

PSTS hyper-GGA mixing fraction $a$ constructed with self-consistent $\mathrm{TPSS}∕u\text{−}6\text{−}311++\mathrm{G}(3df,3pd)$ orbitals along the internuclear axis of a stretched ${\mathrm{F}}_2$ molecule ($z=0$ is the center of the bond). In all regions shown, $a_2=0$ and $a=a_1$. The parameters of $a$ are given in Sec. 5. Note that $a$ has a cusp at $z=0$ which seems unphysical but is limited to a very small region of three-dimensional space.

Figure 6

PSTS hyper-GGA mixing fractions $a_1$ and $a$ constructed with self-consistent $\mathrm{TPSS}∕u\text{−}6\text{−}311++\mathrm{G}(3df,3pd)$ orbitals along the internuclear axis of a stretched ${\mathrm{Ne}}_2^{+}$ molecular ion ($z=0$ is the center of the bond). The parameters of $a$ are given in Sec. 5. In some regions, $a_2>0$ and $a>a_1$.