Response to Comment on “Density functional theory is straying from the path toward the exact functional”

Abstract

Kepp argues in his Comment, among other concerns, that the atomic densities we have considered are not relevant to molecular bonding. However, this does not change the main conclusion of our study, that unconstrained fitting of flexible functional forms can make a density functional more interpolative but less widely predictive.K epp argues in his Comment (1) that our Report (2) does not properly assess the accuracy of approximate functionals for molecules. Many studies have been published to assess the accuracy of such functionals for various applications. We have, instead, investigated the ability of these functionals to make predictions for electron densities-the property employed in computations of all other properties but almost never used to fit or assess functionals. We deliberately chose atomic systems because they are somewhat different from molecules and because they should be well described by the tested functional forms. We found that nonempirical or few-parameter empirical functionals were much more predictive than flexible (highly unconstrained) multiparameter empirical functionals. We will respond in detail to the Comment after clarifying the methodological issue in density functional theory (DFT) and more generally.There is a systematic, nonempirical way to improve approximations to the exact density functional for the exchange-correlation energy (3, 4): (i) Prove the existence of the exact functional and derive exact formal expressions for it. (ii) Discover mathematical properties of the exact functional (exact constraints), which include limits, scaling relations, equalities, and bounds. (iii) Develop approximate but computationally tractable forms for the approximations at various levels of flexibility, including the local spin density approximation (LSDA), generalized gradient approximations (GGAs), meta-GGAs, and hybrids tested in our paper. (iv) Impose the "exact constraints" from step ii on each form, as appropriate. (v) If a form still retains some flexibility, fit it to energies/densities of appropriate norms (4), systems for which the form can be expected to be highly accurate. (Bonded systems are not appropriate norms, because in them the standard approximations display an understood but uncontrollable error cancellation between exchange and correlation. Fitting to bonded systems could make a functional more interpolative but less widely predictive, in the sense of the next paragraph.) The period 1965 to 1979 focused on step i, the 1980s on step ii, 1965 to 1998 on step iii, and the 1970s to the present on step iv, whereas the first appropriate norm for step v was the uniform electron gas (1965). A recent example of steps i to v is the SCAN (strongly constrained and appropriately normed) meta-GGA (4), which respects all 17 exact constraints that a meta-GGA can and works well for diversely bonded molecules and materials [e.g., reference 5 in (2)]. Because SCAN was fitted only to energies of nonbonded systems, every bonding description from it is a genuine prediction. On the meta-GGA lev...

Doubly hybrid density functionals that correctly describe both density and energy for atoms

“…Medvedev et al (26,27) concluded that (i) functionals constructed with little or no empiricism tend to produce more accurate electron densities than highly empirical ones; (ii) at the level of little or no empiricism, the accuracy of the density tends to increase along with the complexity of the DFA as one ascends the Jacob's ladder from LDA to GGA to meta-GGA to hybrids; and (iii) recent DFAs on average are shifting away from the exact functional.…”

“…As energy and density are two sides of one coin, the exact functional shall yield the exact energy of the system from its exact density. Such a mismatch between energy and density suggests a departure from the exact functional (26,27). This argument adds to the long-standing debate on how DFAs can be improved.…”

“…However, Medvedev et al (26,27) showed in their recent reports that improvements of the first four rung DFAs in energies are not always on par with their improvements in densities. For a set of isolated atoms and atomic cations with 2, 4, or 10 electrons-i.e., Be 0 , B 3+ , B + , C 4+ , C 2+ , N 5+ , N 3+ , O 6+ , O 4+ , F 7+ , F 5+ , Ne 8+ , Ne 6+ , and Ne 0 -their results based on 128 DFAs demonstrated that some of the modern functionals on the fourth rung, in particular those based on advanced fitting schemes, produce electron densities that are even less accurate than those from LDAs on the first rung.…”

A review on non‐relativistic, fully numerical electronic structure calculations on atoms and diatomic molecules

“…Since atoms are the basic building block of molecules, a number of popular density functionals have been parametrized using ab initio data on noble gas atoms; these functionals have been used in turn as the starting point for the development of other functionals. A comparison of the results of density‐functional calculations to accurate ab initio data on atoms has, however, recently sparked controversy on the accuracy of a number of recently published, commonly used density functionals . This underlines that there is still room for improvement in DFT even for the relatively simple case of atomic studies.…”

ACCDB: A collection of chemistry databases for broad computational purposes

“…[], and for a recent debate, see Refs. []), we believe that databases will undoubtedly remain a fundamental tool for the assessment of the applicability, accuracy, and reliability of new and existing xc functionals in the years to come, regardless of the underlying wars between optimization philosophies (from first principle vs. parametrized). It is objectively true that more parameters in the xc functionals require more data in the training set (a statistical necessity to avoid over‐training), but the availability of larger training sets does not necessarily translate into more parametrized functionals, nor into more parameters in a functional …”