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Assessment of density functional methods with correct asymptotic behavior

Long-range corrected (LC) hybrid functionals and asymptotically corrected (AC) model potentials are two distinct density functional methods with correct asymptotic behavior. They are known to be accurate for properties that are sensitive to the asymptote of the exchange-correlation potential, such as the highest occupied molecular orbital energies and Rydberg excitation energies of molecules. To provide a comprehensive comparison, we investigate the performance of the two schemes and others on a very wide range of applications, including the asymptote problems, self-interaction-error problems, energy-gap problems, charge-transfer problems, and many others. The LC hybrid scheme is shown to consistently outperform the AC model potential scheme. In addition, to be consistent with the molecules collected in the IP131 database [Y.-S. Lin, C.-W. Tsai, G.-D. Li, and J.-D. Chai, J. Chem. Phys., 2012, 136, 154109], we expand the EA115 and FG115 databases to include, respectively, the vertical electron affinities and fundamental gaps of the additional 16 molecules, and develop a new database AE113 (113 atomization energies), consisting of accurate reference values for the atomization energies of the 113 molecules in IP131. These databases will be useful for assessing the accuracy of density functional methods.


I. INTRODUCTION
Over the past two decades, Kohn-Sham density functional theory (KS-DFT) [1,2] has been one of the most powerful approaches for the study of large ground-state systems due to its low computational cost and reasonable accuracy [3][4][5]. Its extension to the real-time domain, time-dependent density functional theory (TDDFT) [6], has also been actively developed for studying the time-dependent and excited-state properties of large systems [7][8][9][10].
In KS-DFT, the exact exchange-correlation (XC) energy functional E xc [ρ] remains unknown and needs to be approximated. Accurate approximations to E xc [ρ] have been successively developed to extend the applicability of KS-DFT to a wide range of systems. Among them, semilocal density functionals [11] perform satisfactorily for some applications but can produce erroneous results in situations where the accurate treatment of nonlocality of the XC hole is important [12][13][14].
One of the most important and challenging topics in KS-DFT is the asymptotic behavior of the XC potential v xc (r) = δE xc [ρ]/δρ(r). For finite systems, the exact v xc (r) should exhibit the Coulombic (−1/r) decay as r → ∞ [15][16][17][18]. However, owing to the severe self-interaction error (SIE) [19], the XC potential of semilocal functionals decays much faster than (−1/r) in the asymptotic region, yielding qualitatively incorrect predictions for properties sensitive to the asymptote of the XC potential, such as the highest occupied molecular orbital (HOMO) energies and high-lying (Rydberg) excitation energies of atoms and molecules [20,21].
To date, perhaps the most successful density functional methods in practice to include correct XC potential in the asymptotic region of molecules are provided by the long-range corrected (LC) hybrid scheme [22][23][24][25][26][27][28][29][30][31][32] and asymptotically corrected (AC) model potential scheme [33][34][35][36][37][38][39][40][41][42]. For the LC hybrid scheme, the nonlocal Hartree-Fock (HF) exchange for the long-range electron-electron interactions is added to a semilocal functional, which thereby resolves a significant part of the SIE problems and provides an AC XC potential. By contrast, for the AC model potential scheme, an AC XC potential is directly modeled and added to a semilocal functional. In principle, a model XC potential should be a functional derivative of some E xc [ρ]. However, several existing AC model potentials are found not to be functional derivatives. Consequently, when these AC model potentials are employed, several necessary conditions for a functional derivative, such as (i) minimization of the ground-state energy by the selfconsistent KS orbitals, (ii) path independence of the van Leeuwen-Baerends line integral [43], and (iii) net zero force and zero torque on the ground-state density, can be violated [44]. Besides, as these model potentials are not variationally stable, the associated XC energies are not well-defined, and the predicted properties need to be carefully interpreted. The proposals made by Staroverov and co-workers [45,46] are useful for transforming a model GGA exchange potential (which is not a functional derivative) into an exchange potential that has a parent GGA functional. However, for the LB94 potential (an AC model potential with no parent GGA functional) [33], the transformed potential no longer has the correct (−1/r) asymptote [46]. Further improvements on their scheme are needed to resolve this. Alternatively, one may adopt a density functional whose functional derivative has the correct (−1/r) asymptote [47], which should be more desirable than AC model potentials. However, in this work, we only focus on the general performance of the AC model potential scheme.
The rest of this paper is organized as follows. In Sec. II, we evaluate the performance of the LC hybrid scheme, AC model potential scheme, and other density functional methods, on a very wide range of applications, including the asymptote problems, SIE problems, energy-gap problems, charge-transfer problems, and many others. In Sec. III, we give our conclusions on the usefulness of these methods.

II. RESULTS AND DISCUSSION
For a comprehensive comparison of various density functional methods, we examine the performance of the local spin density approximation (LSDA) [48], generalized gradient approximations (GGAs) (PBE [49] and BLYP [50]), meta-generalized gradient approximations (MGGAs) (M06L [51] and VS98 [52]), global hybrid functionals (B3LYP [53] and M06-2X [54]), LC hybrid functionals (ωB97 [27], ωB97X [27], and ωB97X-D [28]), and AC model potentials (LB94 [33] and LBα [35]), on various test sets involving the 223 atomization energies (AEs) of the G3/99 set [55], the 40 ionization potentials (IPs), 25 electron affinities (EAs), and 8 proton affinities (PAs) of the G2-1 set [56], the 76 barrier heights (BHs) of the NHTBH38/04 and HTBH38/04 sets [57], the 22 noncovalent interactions of the S22 set [58], the additional 48 AEs of the G3/05 set [59], the 113 AEs of the AE113 database [60], the 131 vertical IPs of the IP131 database [31], the 131 vertical EAs of the EA131 database [31,60], the 131 fundamental gaps (FGs) of the FG131 database [31,60], two dissociation curves of symmetric radical cations, 19 valence excitation energies, 23 Rydberg excitation energies, and one long-range charge-transfer excitation curve of two well-separated molecules. As discussed in Ref. [31], each vertical EA can be evaluated by two different ways, and each FG can be evaluated by three different ways, so there are in total 1386 pieces of data in our test sets, which are very large and diverse. Unspecified detailed information of the test sets, basis sets, and numerical grids used is given in Ref. [31]. The LB94 (or LBα) potential can be expressed as a linear combination of the LSDA exchange potential, the LSDA correlation potential, and a gradient-dependent exchange potential (e.g., see Eq. (55) of Ref. [33]). Due to the inclusion of the gradient-dependent exchange potential, the LB94 (or LBα) potential is not a functional derivative [44,46]. In this work, the exchange energy from the LB94 (or LBα) exchange potential is calculated by the popular Levy-Perdew virial relation [61]: while the correlation energy from the LB94 (or LBα) correlation potential is directly calculated by the LSDA correlation energy functional (i.e., its parent functional).
All calculations are performed with a development version of Q-Chem 3.2 [62]. Spin-restricted theory is used for singlet states and spin-unrestricted theory for others, unless noted otherwise. For the binding energies of the weakly bound systems, the counterpoise correction [63] is employed to reduce basis set superposition error (BSSE). The error for each entry is defined as (error = theoretical value − reference value). The notation used for characterizing statistical errors is as follows: Mean signed errors (MSEs), mean absolute errors (MAEs), and root-mean-square (rms) errors.
As shown, ωB97, ωB97X, and ωB97X-D perform the best, as they all have very flexible functional forms that were parametrized using this training set. Although M06-2X was not particularly parametrized using this training set, it also performs very well. B3LYP performs similarly to M06L and VS98, and significantly outperforms LSDA, PBE, BLYP, LB94, and LBα. Due to the lack of E xc [ρ], LB94 and LBα perform the worst (even worse than LSDA). Therefore, one should avoid using the AC model potential scheme for the calculation of total energies and related properties. Here, we examine the performance of these functionals on the additional 48 AEs of the G3/05 set [59], using the G3LargeXP basis set [66] and the extra fine grid EML (99,590). As can be seen in Table II, ωB97, ωB97X, and ωB97X-D remain the best in performance, even if they are not particularly parametrized using this set. On the other hand, due to the lack of E xc [ρ], LB94 and LBα remain the worst in performance.  [60] for the AEs of the 113 molecules in IP131.
As described in Ref. [31], the reference values in EA131, FG131, and AE113 are obtained via the very accurate CCSD(T) theory (coupled-cluster theory with iterative singles and doubles and perturbative treatment of triple substitutions) [67], where the CCSD(T) correlation energies in the basis-set limit are extrapolated from calculations using the aug-cc-pVTZ and aug-cc-pVQZ basis sets [68]: with X = 3 and Y = 4 for the aug-cc-pVTZ and aug-cc-pVQZ basis sets, respectively. Note that the reference atomization energies for the AE113 database are calculated without the zero-point energy correction. For completeness of this work, the reference values of IP131, EA131, FG131, and AE113 are all given in the supplementary material [60].
The performance of the functionals is examined on the AE113 database, using the 6-311++G(3df,3pd) basis set and the fine grid EML (75,302). As shown in Table VIII, ωB97, ωB97X, and ωB97X-D have the best performance, while LB94 and LBα have the worst performance, which is consistent with their performance on the atomization energies of the G3/99 and G3/05 sets.
and EA(N), which is IP(N + 1) by its definition, is identical to the minus HOMO energy of the (N+1)electron system, where ǫ i (N) is the i-th KS orbital energy of the N-electron system.
Here, we evaluate the performance of the functionals on frontier orbital energies for the IP131 and EA131 databases using the 6-311++G(3df,3pd) basis set with the fine grid EML (75,302), and summarize the results in Table IV. For the IP131 database, IP(N) is calculated by −ǫ N (N). As shown, the LSDA, GGAs, and MGGAs severely underestimate the vertical IPs, due to the incorrect asymptotic behavior of the XC potential.
By contrast, the LC hybrid functionals and AC model potentials yield very accurate vertical IPs, due to the correct asymptote of the XC potential.
For the EA131 database, EA(N) is calculated by both −ǫ N +1 (N + 1) and −ǫ N +1 (N). For the calculations using −ǫ N +1 (N + 1) (the minus HOMO energy of the (N+1)-electron system), the LSDA, GGAs, and MGGAs seriously underestimate the vertical EAs, due to the incorrect asymptote of the XC potential, while the LC hybrid functionals and AC model potentials perform very well, due to the correct asymptote of the XC potential.
On the other hand, for the calculations using −ǫ N +1 (N) (the minus LUMO energy of the N-electron system), the situation becomes very different. Although the AC model potentials yield accurate ǫ N +1 (N) due to the correct asymptote, the vertical EA remains severely overestimated due to the lack of the DD.
In spite of the lack of the DD, there is a fortuitous cancellation of errors in the vertical EA predicted by the LSDA, GGAs, and MGGAs, as the ǫ N +1 (N) is (incorrectly) upshifted due to the incorrect asymptote of the XC potential, effectively incorporating part of the DD. As the DD is close to zero for LC hybrid functionals [86], i.e., ǫ N +1 (N + 1) ≈ ǫ N +1 (N), the LC hybrid functionals significantly outperform the others using this estimate.

D. Fundamental gaps
The FG of a N-electron system is defined as E g ≡ IP(N) − EA(N). As detailedly described in Ref. [31], since IP(N) and EA(N) can both be calculated by their definitions, or by their relations to the frontier orbital energies, E g can be evaluated by ǫ N +1 (N) − ǫ N (N), ǫ N +1 (N + 1) − ǫ N (N), and Here, we examine the performance of the functionals on FGs for the FG131 database, using the 6-311++G(3df,3pd) basis set with the EML(75,302) grid, and summarize the results in Table V.
For the calculations using ǫ N +1 (N) − ǫ N (N) (the HOMO-LUMO gaps), the LSDA, GGAs, MGGAs, and AC model potentials perform very poorly, due to the lack of the DD. As mentioned previously, the hybrid functionals incorporate part of the DD in the GKS orbital energies [84][85][86][87][88], and therefore provide significant improvements, especially for the LC hybrid functionals.
For the calculations using ǫ N +1 (N + 1) − ǫ N (N) (the energy difference between the HOMOs of the N-and (N+1)-electron systems), the LC hybrid functionals and AC model potentials perform very

E. Dissociation of symmetric radical cations
Due to the severe SIEs of semilocal functionals, spurious fractional charge dissociation can occur [89], especially for symmetric charged radicals X + 2 , such as H + 2 [90] and He + 2 [91]. To examine the performance of the density functional methods upon the SIE problems, we perform spin-unrestricted calculations using the aug-cc-pVQZ basis set and a high-quality EML(250,590) grid. The DFT results are compared with results from HF theory, and the highly accurate CCSD(T) theory.
The dissociation curves of H + 2 and He + 2 are shown in Figs. 1 and 2, respectively. As can be seen, the LC hybrid scheme can remove the unphysical barriers of the dissociation curves, while the AC model potential scheme cannot. We emphasize that a SIE-corrected functional must be AC, while an AC functional (or model potential) may not be SIE-corrected.

Valence and Rydberg excitations
To assess the performance of the density functionals on valence and Rydberg excitation energies, we perform TDDFT calculations on five small molecules [92], including nitrogen gas (N 2 ), carbon monoxide taken from Ref. [92]. For the AC model potential scheme, the adiabatic LSDA kernel is adopted in our TDDFT calculations [33]. to the correct asymptote of the XC potential.

Long-range charge-transfer excitations
Semilocal functionals qualitatively fail to describe long-range charge-transfer (CT) excitations between a donor and an acceptor [93][94][95][96][97][98]. The correct CT excitation energy ω CT (R) should have the following asymptote [93]: where IP D is the IP of the donor, EA A is the EA of the acceptor, and R is their separation. As the (−1/R) dependency, which is the consequence of Coulomb interaction, can be accurately described by the nonlocal HF exchange [93], LC hybrid functionals, containing the full long-range HF exchange, can yield accurate CT excitation energies [27,28,31,32].
On the other hand, within the framework of TDDFT, Hellgren and Gross showed that the discontinuity of the XC kernel is essential for the accurate description of CT excitations [98]. They showed that the divergency of the discontinuity as r → ∞ can produce a divergent kernel in the dissociation limit, which is essential for the asymptote of CT excitation energy (see Eq. (8)) [95,98].

III. CONCLUSIONS
In conclusion, we have examined the performance of the LC hybrid scheme and AC model potential scheme on a very wide range of applications. Due to the correct XC potential in the asymptotic region, both the LC hybrid scheme and AC model potential scheme are reliably accurate for properties related ω CT (R)(eV) FIG. 4. Relative excitation energy for the lowest CT excitation of C 2 H 4 · · · C 2 F 4 dimer along the intermolecular distance R (inÅ). The excitation energy at 5Å is set to zero for each method.
to the asymptote, such as the HOMO energies and Rydberg excitation energies of molecules.
Relative to the AC model potential scheme, the LC hybrid scheme has several advantages. First, the LC hybrid scheme shows significant improvements in the SIE problems, such as dissociation of symmetric radical cations, while the AC model potential scheme fails to resolve the SIE problems. Secondly, as the LC hybrid scheme belongs to the GKS method, the fundamental gap can be accurately predicted by the HOMO-LUMO gap. For the AC model potential scheme, the HOMO-LUMO gap is a poor approximation of the fundamental gap, due to the lack of the DD. Thirdly, the long-range HF exchange kernel in the LC hybrid scheme is essential for the accurate description of the long-range CT excitations between two well-separated molecules, which cannot be described by the AC model potential scheme due to the lack of the discontinuity of the XC kernel. Finally, the LC hybrid scheme yields accurate predictions for atomization energies, barrier heights, and noncovalent interactions. However, due to the lack of E xc [ρ], the AC model potential scheme performs very poorly for total energies and related properties.
Despite their similarity in correct asymptotic behavior, the AC model potential scheme may be accurate only for the asymptote problems, while the LC hybrid scheme, which has remedied several qualitative failures of semilocal functionals, could be reliably accurate for a very wide range of applications. Although only three LC hybrid functionals (ωB97, ωB97X, and ωB97X-D) are examined in this work, we expect that other LC hybrid functionals should also perform well for the properties sensitive to the long-range HF exchange, such as the asymptote problems, SIE problems, energy-gap problems, and charge-transfer problems. For the properties insensitive to the long-range HF exchange (e.g., atomization energies, noncovalent interactions), further comparisons should be made for other LC hybrid functionals (e.g., see Refs. [31,32]).