Rung 3.5 density functionals: Another step on Jacob's ladder


  • Benjamin G. Janesko received a Ph. D. in 2005, working with David J. Yaron at Carnegie Mellon University. He performed postdoctoral research with Gustavo E. Scuseria at Rice University. Since 2009, he has been an assistant professor of chemistry at Texas Christian University. His group develops new approximate exchange-correlation functionals for density functional theory, and applies computational chemistry tools to heterogeneous catalysis, cross-coupling catalysis, conjugated polymers, and related problems in energy technology. [Color figure can be viewed in the online issue, which is available at]


Applications of density functional theory (DFT) to computational chemistry and solid-state physics rely on a “Jacob's Ladder” of progressively more complicated approximations to the many-body exchange-correlation (XC) density functional. Accurate, computationally tractable DFT calculations on large and periodic systems remain challenging for existing XC functionals. Simple XC functionals on the three lowest rungs of Jacob's Ladder are insufficiently accurate for many properties, while fourth-rung hybrid functionals incorporating nonlocal information can be prohibitively expensive. This perspective presents our work toward a compromise, a new class of “Rung 3.5” functionals that incorporate a linear dependence on the nonlocal one-particle density matrix. This work reviews these functionals' formal underpinning, numerical performance, and prospects for modeling solids and surfaces. © 2012 Wiley Periodicals, Inc.


Kohn-Sham density functional theory (DFT) provides a formally exact treatment of the ground-state electronic structure of many-electron systems. Practical DFT calculations must approximate the exchange-correlation (XC) density functional that captures all many-body effects.1 Development of accurate and computationally tractable approximate XC functionals has made DFT the preëminent electronic structure approximation in computational chemistry and solid-state physics. However, approximate XC functionals must always trade off accuracy versus computational cost. This perspective discusses our efforts toward a new compromise: functionals accurate enough for molecular thermochemistry and kinetics, and inexpensive enough to model solids and surfaces.

Perdew and Schmidt classify approximate XC functionals as “rungs” on a “Jacob's Ladder,” extending from Hartree theory to the “heaven” of chemical accuracy (Fig. 1).1 Higher rungs contain more complicated ingredients and can generally give higher accuracy. The first rung is the local spin density approximation (LSDA), in which the XC energy density at each point r is taken as that of a uniform electron gas (UEG) with electron density ρ(r) (spin dependence is suppressed for conciseness. All orbitals, densities, and related quantities are assumed to be equation image spin). Semilocal generalized gradient approximations2, 3 (GGAs) incorporate equation image. Meta-GGAs4 also include the density Laplacian and/or the kinetic energy density of the Kohn-Sham reference system equation image. This reference system places N noninteracting Fermions in orbitals equation image, with equation image equal to the real system's electron density. “Semilocal” is used here to distinguish GGAs and meta-GGAs from the LSDA. Other authors use “local” to denote first- through third-rung functionals. Semilocal functionals satisfy many exact constraints5, 6 and are computationally cheap enough to model solids and surfaces. But they systematically over-delocalize electrons,7, 8 giving overestimated heats of formation9 and molecule-surface adsorption energies, underestimated band gaps,10, 11 underestimated reaction barriers in molecules12, 13 and surfaces, overestimated magnetic exchange couplings,14 and related errors.

Figure 1.

(Left) Jacob's Ladder of approximate XC functionals. Our “Rung 3.5” functionals are intended to provide accuracy and computational expense intermediate between semilocal and hybrid approximations. (Right) Semilocal, Rung 3.5, hybrid, and exact exchange energy densities equation image in hydrogen atom. [Color figure can be viewed in the online issue, which is available at]

Fourth-rung “hybrid” XC functionals compensate these errors by incorporating a fraction of nonlocal exact exchange15–17

equation image(1)
equation image(2)

γ is the one-particle density matrix of the Kohn-Sham reference system, an implicit density functional obeying equation image. Admixture of a fraction of Eq. (1) tunes semilocal functionals' systematic overbinding.7, 8 This gives hybrid XC functionals a dramatically improved treatment of many properties.9, 12, 14

An instructive connection between semilocal and hybrid XC functionals comes from expressing a semilocal exchange functional in terms of a semilocal model for γ,

equation image(3)

The semilocal exchange hole equation image equation image, is localized by construction around the reference point r. This approximates localization of the XC hole in chemical bonds. Such localization ensures that, for example, the electrons in singlet H2 have a high probability being on different H atoms.7, 8 Admixing a fraction of delocalized exact exchange tunes this approximation.7 Semilocal exchange functionals are typically expressed in terms of models for the exchange energy density or exchange hole, rather than the one-particle density matrix itself.3 However, Eq. (3) is useful in what follows.

Fourth-rung hybrids have (at least) two important limitations in applications to solids and surfaces. First, the optimal admixture of exact exchange differs for different properties: equation image for thermochemistry,6, 15, 18 equation image for kinetics,5 and 100% in one-electron regions where Eq. (1) is the exact XC functional. Second, slow decay of equation image in equation image makes evaluation of Eq. (1) computationally expensive19, 20 and formally problematic21 in metals. These limitations have motivated recent work on improved semilocal functionals,22–25 screened hybrids26–28 that cut off the long-range part of Eq. (1), algorithms for evaluating exact exchange,29 and hyper-GGAs30–33 incorporating different fractions of exact exchange in different regions.

Rung 3.5 Functionals

Since 2009, we have explored approximations to address both of the aforementioned limitations. These are intended to provide an entirely new class of approximate XC functionals, a new “rung” on Jacob's Ladder intermediate between semilocal and hybrid approximations. Inspired by our density-matrix-based local mixing function,34, 35 “Rung 3.5” functionals incorporate an exchange energy

equation image(4)
equation image(5)

Equations (4) and (5) are qualitatively “intermediate” between exact exchange [Eq. (1)] and semilocal exchange [Eq. (3)].36 This approximation addresses both of the aforementioned limitations of hybrid functionals. First, equation image and its exchange hole7 decay rapidly in equation image, cutting off the long-range piece of Eq. (5) even in metals where equation image is delocalized. Second, proper design of equation image could “tune” the fraction and range of nonlocal information in different regions as in hyper-GGAs.30

Equation (5) has useful formal properties. Expanding the Kohn-Sham orbitals and equation image in appropriate basis sets permits analytic integration over equation image.36 Equation (5) correctly37 scales as λ for uniform density scaling equation image if equation image scales as equation image [Eq. (5) does not appear to satisfy nonuniform scaling].38 While Eq. (4) is not positive-semidefinite like Eq. (1), Rung 3.5 exchange obeys the exact bound equation image via the Cauchy-Schwarz inequality.34, 36 [ equation image and equation image are here defined by integrating the integrands of Eqs. (1) and (3) over equation image].

Much of our work to date has focused on global-hybrid-like Rung 3.5 functionals combining a constant fraction of Eq. (4) with a corresponding fraction of semilocal exchange and a standard, coupling-constant-averaged semilocal correlation functional. Such functionals can be derived by modifying the adiabatic connection derivation of global hybrids.15 Briefly, one expresses the adiabatic connection integrand equation image at intermediate coupling strength equation image in terms of the diagonal two-particle density matrix equation image corresponding to equation image, performs a cumulant expansion of equation image, interpolates the one-matrix contributions equation image between equation image and equation image, and invokes semilocal approximations for equation image, and the irreducible two-particle density matrix.39

One interesting aspect of Rung 3.5 exchange is its treatment of band gaps. In exact Kohn-Sham theory, the energy of the highest occupied Kohn-Sham orbital (HOMO) in a system with equation image electrons ( equation image) equals the exact difference between the energies of the N and equation image electron systems. However, the difference between the highest-occupied and lowest-unoccupied Kohn-Sham orbital energies of an N-electron system (the HOMO-LUMO gap) generally does not equal the difference between the system's ionization potential and electron affinity. This is a result of a derivative discontinuity in the exact XC potential equation image as a function of N.40, 41 Generalized Kohn-Sham (GKS) calculations with a global hybrid exchange functional equation image equation image instead use a nonlocal exchange potential

equation image(6)

This incorporates an approximate derivative discontinuity into the HOMO-LUMO gap.10 GKS calculations using Eq. (4) for exchange include a nonlocal exchange potential

equation image(7)

[We symmetrize Eqs. (4) and (5) in equation image to ensure a symmetric exchange potential].36 The nonlocal terms in Eqs. (7) and (6) are equal if equation image and equation image, suggesting that reasonable equation image can incorporate an approximate derivative discontinuity.

Density Matrix Models

The semilocal model density matrix equation image in Eq. (5) is critical to Rung 3.5 functionals. equation image must simultaneously play roles akin to the semilocal exchange functional of global hybrids, the range separation of screened hybrids, and the local mixing function of local hybrids; all while permitting analytic integrations over equation image in Eq. (5) and over equation image in Eq. (7). We have thus focused on developing approximate equation image. Our initial studies36, 42 use a single-Gaussian local density approximation (LDA) model

equation image(8)

Here equation image, equation image, and equation image is the Fermi vector of a spin polarized UEG with total density equation image. Equation (8) is based on the exact KS density matrix of the UEG,

equation image(9)

equation image correctly scales as equation image with uniform density scaling and equals equation image at equation image. Its Gaussian dependence enables analytic integrations over equation image in Eq. (5) and equation image in Eq. (7) when the KS orbitals are expanded in a Gaussian basis set.36 Nonempirical parameter equation image ensures that Eq. (5) returns the exact exchange energy density in the UEG where equation image. We use “Π-LDA” to denote a simple XC functional combining LSDA correlation with Eq. (4) using equation image for equation image,

equation image(10)

equation image suffices to demonstrate some useful properties of Rung 3.5 functionals.36 However, it is a poor approximation for the real one-particle density matrix in regions of large density gradient. The integrand of Eq. (10) is invariably small in such regions, producing a systematic underestimate of exchange in density tails and a systematic overbinding like the LSDA itself.42 We first corrected this overbinding with a simple one-parameter ansatz, using Eq. (10) to correct a GGA in regions of small reduced density gradient equation image. The resulting “PBE+ equation image” functional combines the Perdew-Burke-Ernzerhof (PBE) GGA for correlation2 with exchange energy

equation image(11)

equation image is the PBE exchange enhancement factor.2 equation image with empirical parameter equation image.

Our recent work goes beyond the ansatz of Eq. (11) to explicitly gradient-dependent model density matrices. Expanding Eq. (2) about equation image yields

equation image(12)

Here “·” denotes the scalar (dot) product. We constructed a nonspherically-symmetric model density matrix that recovers this expansion to second order,39

equation image(13)

equation image is a two-Gaussian version of Eq. (8) constructed to reproduce equation image over a chemically relevant range of y.39 equation image is parameterized to an existing GGA for exchange:

equation image(14)
equation image(15)

Figure 1 of Ref.39 shows that this gradient dependence dramatically improves the agreement between γ and equation image in H atom. We use “Π-PBE” to denote the combination of PBE correlation and Eq. (4), with equation image obtained by fitting Eq. (13) to PBE exchange. The global-hybrid-like “Π1-PBE” functional combines PBE exchange with a constant fraction equation image of Rung 3.5 exchange built from Eq. (13). Π1-PBE is motivated by the adiabatic connection discussed above. Ref. [39] also introduced a spherically symmetric, three-parameter density matrix equation image constructed from the reduced gradient s and the reduced kinetic energy equation image. We use “Π-mGGA3” to denote the combination of Rung 3.5 exchange, using equation image for equation image, and PBE correlation. Such empirical models should give insight into future nonempirical Rung 3.5 approximations.

Figure 1 illustrates the Rung 3.5 exchange energy densities obtained from these different model density matrices. The figure plots the energy-weighted ↑-spin exchange energy density equation image in H atom. Calculations use Hartree-Fock orbitals evaluated in the large UGBS basis set.43 The exact exchange energy density is taken as minus the Coulomb energy density

equation image(16)

The LSDA and Π-LDA underestimate exchange at large r, as discussed above. PBE+ equation image recovers the PBE GGA at large r by construction. The equation image density matrix makes Π-PBE more accurate in the density tail, though the overall exchange energy is too negative. The empirical Π-mGGA3 functional combines a poor treatment of density tails with a too-negative exchange energy at moderate r, presumably giving some error compensation.

Numerical Tests

We have implemented these Rung 3.5 functionals into the development version of the Gaussian electronic structure package44 and a Mathematica code for atomic calculations.36 Numerical tests using these implementations offer insights into the prospects and limitations of Rung 3.5 functionals. The results confirm that Rung 3.5 functionals largely live up to their name, providing accuracy intermediate between semilocal and hybrid DFT approximations. The results also indicate areas for improvement.

Molecular thermochemistry, kinetics, and geometries

Table 1 presents mean absolute errors in the total energies per electron of atoms H-Ar, the heats of formation equation image of the 223 molecules in the G3/99 data set,39 and the barriers of 38 hydrogen transfer12 and nonhydrogen-transfer13 reactions. Calculations are compared against the PBE2 GGA, the Tao-Perdew-Staroverov-Scuseria (TPSS)4 meta-GGA, and the PBE016, 17 global hybrid. Computational details are as in Ref. [39]. The self-consistent Π-LDA and equation image results differ slightly from the post-PBE0 results in Ref. [42].

Table 1. Mean absolute errors in the total energies per electron of atoms H-Ar (mH/electron), the G3/99 data set of 223 equation image (kcal/mol), and the HTBH38/04 and NHTBH38/04 sets of reaction barriers (kcal/mol).
equation image24.

The aforementioned limitations of equation image in large-gradient regions makes Π-LDA give insufficiently negative total atomic energies and significant overbinding. All other Rung 3.5 methods improve on Π-LDA. equation image and Π1-PBE predict thermochemistry and kinetics with accuracy intermediate between semilocal and hybrid functionals, befitting their “Rung 3.5” designation. Ref. [39] also showed that Π-PBE and Π-mGGA3 provide reasonable accuracy for geometries of small molecules and odd-electron bonds. These results confirm that Rung 3.5 functionals incorporate chemically meaningful nonlocal information. They also point to the importance of careful design of equation image, particularly in regions of large density gradient.

Magnetic exchange couplings

Simulations of magnetic materials often invoke a phenomenological Heisenberg Hamiltonian with isolated spins equation image interacting via magnetic exchange couplings equation image. Approximate equation image can be obtained from DFT calculations of the difference between high-spin triplet and symmetry-broken singlet energies of a system with two magnetic centers. Semilocal DFT's systematic over-delocalization yields overestimates exchange couplings, an error ameliorated by global or range-separated hybrids.45–48 We recently collaborated with the Peralta group at Central Michigan University to test Rung 3.5 functionals for magnetic exchange couplings.14 Calculations on representative transition metal dimers and found that equation image predicted magnetic couplings with accuracy intermediate between semilocal and hybrid DFT. Even Π-LDA outperformed all tested semilocal functionals, remarkable given its poor performance for molecular thermochemistry (Table 1). Both Π-LDA and equation image correctly made the spins more localized than semilocal DFT, providing a physical rationale for the improved exchange couplings. Π-LDA and equation image also systematically overestimated the magnetic coupling of ferromagnetic complexes. Qualitatively similar trends were seen in GGAs with reduced gradient dependence, suggesting that this error arose from Rung 3.5 functionals' aforementioned deficiencies in large-gradient regions.

Semiconductor band gaps

As discussed above, the nonlocal Rung 3.5 exchange potential of Eq. (7) is expected to provide an approximate derivative discontinuity similar to Eq. (6). HOMO-LUMO gaps from Rung 3.5 GKS calculations should partially correct the systematic bandgap underestimate of semilocal DFT. We explored this effect by calculating the bandgaps of conjugated polymers. Ref. [49] benchmarked Rung 3.5 functionals on the bandgaps of isolated, infinite chains of twelve semiconducting polymers. All Rung 3.5 functionals gave larger (and more accurate) bandgaps than any tested semilocal functional, including the TPSS4 and M06-L25 meta-GGAs. equation image and equation image gave additional improvements.39 While the Rung 3.5 bandgaps were smaller than those predicted by global or screened hybrids, this result further demonstrates that Eq. (6) includes useful nonlocal information.


The “Rung 3.5” approach offers a new route to including nonlocal information into approximate XC functionals. The formal and computational results discussed here show that Rung 3.5 functionals have the potential to be more accurate than semilocal DFT, while remaining applicable to systems beyond the reach of global hybrids.

These results also suggest several opportunities for future work. Our existing implementation is adequate for proof-of-concept but has an unnecessary computational overhead. We are exploring faster implementations via auxiliary basis expansions of equation image, and implementation into plane-wave codes. We are developing more sophisticated equation image, including nonempirical extensions of equation image and a equation image based on the Becke-Roussel model.50 These might mitigate Rung 3.5 functionals' remaining errors in large-gradient regions. Connections between semilocal models for γ and the noninteracting kinetic energy may let us leverage insights from orbital-free DFT.51–53 We are also considering functionals analogous to hyper-GGAs that use Eq. (5) to satisfy known exact constraints5, 30 Overall, “Rung 3.5” functionals seem to be one interesting direction in the continued search for faster, more accurate approximations in DFT.22, 26, 30

Biographical Information

original image

Benjamin G. Janesko received a Ph.D. in 2005, working with David J. Yaron at Carnegie Mellon University. He performed postdoctoral research with Gustavo E. Scuseria at Rice University. Since 2009, he has been an assistant professor of chemistry at Texas Christian University. His group develops new approximate exchange-correlation functionals for density functional theory, and applies computational chemistry tools to heterogeneous catalysis, cross-coupling catalysis, conjugated polymers, and related problems in energy technology. [Color figure can be viewed in the online issue, which is available at]