Keywords

1 Introduction

1.1 Chemistry on Surfaces

Chemical reactions at surfaces and interfaces are central to many problems in chemistry. Topical examples include heterogeneous catalysis [1], surface-enhanced spectroscopies [2], battery technologies and hydrogen storage [3], and nanoscale devices [4]. Experimental studies of surface chemistry can be challenging. Measurements of molecules’ adsorption, desorption, and reaction on single crystal surfaces under ultrahigh vacuum have yielded important insights [1] recognized by the 2007 Nobel Prize in Chemistry [5]. However, connections to practical surface chemistries often requires bridging the “pressure gap” between ultrahigh vacuum and industrially relevant high pressures, and the “materials gap” between single crystal surfaces and industrially relevant nanoparticles or catalysts. These gaps can play critical roles in surface chemistry. To illustrate, the Fischer–Tropsch process [6] for converting synthesis gas (CO + H2) to long-chain hydrocarbons is catalyzed industrially by promoted and nanostructured cobalt [7] and iron [8] surfaces, but does not occur on single-crystalline surfaces under ultrahigh vacuum [9]. While there has been substantial progress in experiments bridging these gaps [10–12], experimental surface science remains challenging. Figure 3 of Maitlis [13] illustrates the contrast between well-defined experiments on homogeneous catalysts and the more “impressionistic” data available for heterogeneous catalysts.

1.2 Simulating Surface Chemistry

Simulations of molecules on surfaces have become essential tools for interpreting this impressionistic experimental data [14]. Such simulations typically require electronic structure calculations. Simulations of surface-enhanced spectroscopies combine classical electrodynamics models [15] with electronic structure calculations modeling how molecule–surface interactions shift a molecule’s vibrational spectrum [16]. Simulations of nanoscale device chemistry generally center on modeling the electronic structure and properties of model devices [17]. Simulations of heterogeneous catalysis typically begin with electronic structure calculations on the geometries and adsorption energies of reaction intermediates on catalyst surfaces. The resulting potential energy surfaces can be applied in microkinetics models of reactions at realistic pressures, temperatures, and catalyst compositions [18–21].

Electronic structure calculations generally model a surface as either a periodic slab [22] or a finite cluster of surface atoms [23], potentially embedded in a simpler model background [24]. Cluster models are widely used for insulators, and are readily applicable to charged species. However, cluster models for metal surfaces often converge slowly with cluster size [25]. (Differences between calculations at different levels of theory can converge more rapidly [26].) Electronic structure approximations applicable to periodic slabs are desirable for treating metal, semiconductor, and insulator surfaces on an equal theoretical footing.

1.3 Choice of Electronic Structure Approximation

Electronic structure calculations on molecules, solids, and surfaces typically must approximate the many-body electron–electron interactions [27]. To be useful, an approximation should be both accurate enough to make experimentally meaningful predictions, and computationally inexpensive enough to treat the chosen system in a reasonable time using available computational resources. Several different electronic structure approximations have been applied to surfaces. One route uses first principles ab initio approximations for the many-electron wavefunction. Recent ab initio calculations on surfaces include coupled cluster [28–31] simulations of clusters of surface atoms, and quantum Monte Carlo (QMC) [26, 32–35] and random phase approximation (RPA) [36–42] simulations of periodic slabs. Unfortunately, ab initio methods typically have rather steep computational scaling [27], making them problematic for large and realistic model surfaces. A second route is to use tight-binding Hamiltonians. These can be applied to large systems, but require substantial empirical parameterization and can have limited transferability [43].

2 GGAs for Surface Chemistry

2.1 Density Functional Theory

The vast majority of electronic structure calculations on surfaces use Kohn–Sham density functional theory (DFT) [44, 45]. DFT can often provide acceptable accuracy at modest computational expense. Figure 1 illustrates DFT’s dominance in modeling surface chemistry.

Fig. 1
figure 1

Results of a Web of Science search for “DFT catalyst* surface*” illustrating published applications of DFT to heterogeneous catalysis [46]

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Kohn–Sham DFT models the ground state of an N-electron system as a reference system of N noninteracting Fermions corrected by a mean-field Hartree interaction,

$$ {E}_{\mathrm{H}}\left[\rho \right]=\frac{1}{2}\mathit{\int}{d}^3\mathbf{r}\mathit{\int }{d}^3{\mathbf{r}}^{\mathbf{\prime}}\frac{\rho \left(\mathbf{r}\right)\rho \left({\mathbf{r}}^{\mathbf{\prime}}\right)}{\left|{\mathbf{r}}^{\mathbf{\prime}}-\mathbf{r}\right|}, $$
(1)

and an exchange-correlation (XC) density functional E XC[ρ] containing all many-body effects. ρ(r) is the probability density for finding an electron at r. The Kohn–Sham reference system has the same ρ(r) as the real system by construction. The exact E XC[ρ] is a unique and variational functional of ρ(r) [44]. The Hamiltonian of the non-interacting Fermions includes local and multiplicative Hartree and XC potentials, e.g., v XC(r) = δE XC[ρ]/δρ(r). Practical calculations typically use separate ↑- and ↓-spin densities and Kohn–Sham orbitals. This work suppresses spin dependence for conciseness. All orbitals, densities, density matrices, and exchange energies are assumed to be spin polarized.

The accuracy and computational expense of a typical DFT calculation is determined by the one-electron basis set used to expand the reference system’s wave-function, by any approximate treatments of relativistic effects, reciprocal space, and core electrons, and by the choice of approximate XC functional. DFT’s success is largely attributable to the development of accurate and computationally tractable XC approximations [47]. Systematically convergent hierarchies of XC approximations (ab initio DFT) have been developed [48]. However, the vast majority of DFT calculations use an alternative “Jacob’s Ladder” of approximations [49] based on the homogeneous electron gas (HEG, ρ(r) = constant). The first “rung” of this ladder is the local spin-density approximation (LSDA) [50, 51] constructed to reproduce the numerically exact XC energy density [52, 53] of the HEG.

Generalized gradient approximations occupy the second rung of Jacob’s Ladder. GGAs model the nonunique [54] XC energy density at point r as a function of ρ(r) and its gradient:

$$ {E}_{\mathrm{XC}}^{\mathrm{GGA}}\left[\rho \right]=\mathit{\int}{d}^3\mathbf{r}\;{e}_{\mathrm{XC}}^{\mathrm{GGA}}\left(\rho \left(\mathbf{r}\right),\left|\nabla \rho \left(\mathbf{r}\right)\right|\right). $$
(2)

GGAs tend to improve upon the LSDA for total energies, atomization energies, and reaction barriers, and expand and soften bonds to (over)correct the LSDA’s overbinding [55]. Their improved accuracy and computational simplicity makes them widely applied to periodic slab models for surfaces.

2.2 Design of GGAs

Unlike the LSDA, there is no single “best” choice of a GGA [55]. This flexibility has been exploited to construct GGAs which perform well for specific aspects of surface chemistry. In practice, a GGA’s performance is often largely a function of its exchange enhancement factor F X:

$$ {E}_{\mathrm{X}}^{\mathrm{GGA}}\left[\rho \right]=\mathit{\int}{d}^3\mathbf{r}\;{C}_{\mathrm{X}}{\rho}^{4/3}\left(\mathbf{r}\right){F}_{\mathrm{X}}(s). $$
(3)

The “exchange” portion of E XC[ρ] (4) is defined in terms of the expectation value of the electron–electron interaction operator, evaluated with the wavefunction of the noninteracting Kohn–Sham reference system [56]. Coefficient \( {C}_{\mathrm{X}}=-\frac{3}{4}{\left(\frac{6}{\pi}\right)}^{1/3} \) is from the exact Kohn–Sham wavefunction of the HEG [57]. s = |∇ρ(r)|/(2(6π 2)1/3 ρ 4/3(r)) is the unitless reduced density gradient.

Figure 2 illustrates representative GGAs’ exchange enhancement factors and their performance for some properties relevant to surface chemistry. The LSDA is included for comparison. Similar to the LSDA, GGAs with small enhancement factors (PBEsol [62], AM05 [63], Wu-Cohen [64]) combine accurate lattice parameters with overestimated adsorption energies [65]. GGAs with larger enhancement factors (revPBE [66], RPBE [67], BLYP [68, 69]) improve adsorption energies [67] and molecular thermochemistry [70] at the expense of lattice parameters and geometries [71]. PBE [55] and PW91 [50, 72] provide intermediate performance [59]. None of these GGAs perform well for gas-phase reaction barrier heights, a fact central to what follows.

Fig. 2
figure 2

Top: Exchange enhancement factors of the LSDA and representative GGAs (F X, Eq. (3)). Bottom: Mean errors in the shortest interatomic distances of 30 transition metals [58]; mean error in bulk moduli of 30 transition metals [58]; predicted chemisorption energies for CO on Pt(111) [60]; root-mean-square deviation in BH76 barrier heights and weighted mean absolute deviation in the GMTKN30 database of gas-phase properties (kcal/mol) [61]. Functionals are approximately sorted in order of increasing enhancement factor

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2.3 Dispersion Corrected GGAs

The locality of Eq. (2) prevents it from treating truly nonlocal correlation effects, including the asymptotic van der Waals interaction between distant closed-shell uncharged fragments [73]. Over the last decade, substantial resources have been invested into dispersion corrections for approximate XC functionals [74–76]. Dispersion corrections generally improve GGA simulations of molecule-surface adsorption [77], particularly for larger molecules such as coronene [78] and perylene derivatives [79, 80]. Dispersion corrections can be critical for some catalytic processes [37], some dissociation barriers [81], and some reactions of adsorbed molecules [82]. Dispersion corrections to adsorption energies are also important for apparent activation barriers on surfaces [83, 84]. However, dispersion corrections generally do not correct GGAs’ systematic underestimate of gas-phase reaction barriers. This can be seen, for example, in Tables S30–S31 of Goerigk and Grimme [85], where dispersion corrections slightly degrade PBE’s performance for the BH76 benchmark set of gas-phase reaction barriers [85–87]. This assertion is supported by the widespread adoption of dispersion-corrected beyond-GGA functionals in computational chemistry [88–92]. Similar results are seen for reaction barriers of molecules on surfaces, evaluated with dispersion-corrected GGAs (Svelle et al. [93], Sect. 4). The studies reviewed below suggest that improving DFT for surface chemistry requires both dispersion corrections and beyond-GGA functional forms.

3 Successes of GGAs for Surface Chemistry

DFT calculations with generalized gradient XC functionals have been applied to an enormous array of problems in surface science. A comprehensive discussion of this literature would extend to several volumes, and is far beyond the scope of this work. This section presents a few representative successes connected to our work. Hafner [94], Greeley et al. [95], and Nørskov et al. [96] provide more extensive recent reviews.

GGAs are widely applied to determine how surface functionalization of nanostructures controls their properties. DFT calculations have mapped out the relationship between edge functionalization and electronic properties of graphene nanoribbons [4, 17, 97–99], have predicted how adsorbates open a bandgap in graphene electronics [100, 101], and have motivated studies of other quasi-1D semiconductors [102]. GGA calculations have also provided mechanistic insight into ammonia adsorption and reaction on Si(100) surfaces, a process relevant to chemical vapor deposition of silicon nitride for integrated circuits [103–107].

GGAs are also extensively applied to heterogeneous catalysis. Microkinetics models of ammonia synthesis over ruthenium nanoparticles, constructed from the RPBE and PW91 GGAs, reproduce experimental turnover frequencies to within an order of magnitude [108]. The calculated mechanisms also clarify the role of atomic steps for the rate-limiting N2 dissociation [109]. Recent GGA calculations give new evidence for H2-induced CO dissociation on Fischer–Tropsch catalysts [21, 110–114], a mechanism appearing to show significant structure sensitivity [115]. Studies of the C–C coupling step in the Fischer–Tropsch reaction point to the importance of surface carbide species [116]. Several recent simulations of the Fischer–Tropsch reaction considered the roles of adsorbed promoters which stabilize corrugated surfaces [117] and block graphitization [118], and adsorbed sulfur poisons which block CO dissociation [119]. These results have been expanded into complete microkinetics models of the Fischer–Tropsch reaction [21, 114, 120–123]. Other studies have considered methanol synthesis and the water-gas shift reaction [124] over some industrially relevant catalysts [18, 125, 126]. A recent combined computational and experimental study of methanation over nickel catalysts further illustrates the value of GGA calculations for interpreting experiment [112]. GGA calculations have also provided atomic-scale explanations for the catalytic activity of gold nanoparticles, particularly bound to defective metal oxide supports [127–131]. Newer GGAs resolve the “CO/Pt(111) puzzle,” in which standard GGAs’ over-delocalization leads to qualitatively incorrect site dependence for CO adsorption on coinage metals [132]. Perhaps most importantly, GGA calculations have yielded general insights into periodic trends in adsorption [133, 134] and reactivity [135, 136] on catalyst surfaces.

A particularly interesting set of recent studies use GGA calculations to design new heterogeneous catalysts. Such studies often begin by identifying descriptors, such as adsorption or dissociation energies, which can be correlated with a catalyst’s overall performance [137]. Typical calculations yield a “volcano plot” of catalytic activity vs descriptor, with optimal catalysts having intermediate values of the adsorption or dissociation energy [138]. GGA calculations of the descriptor on many model catalysts are then used to identify optimal candidates. GGA calculations on CO adsorption energies predicted that nickel-iron alloy catalysts could outperform more expensive pure Ni for CO methanation [139], a prediction subsequently verified by experiment [140]. GGA calculations were used to identify methylene chemisorption energies as a good descriptor for ethylene hydrogenation, and subsequently to identify novel nickel-zinc alloy catalysts [141]. GGA calculations on H2 chemisorption energies were also used to identify near-surface alloy hydrogenation catalysts [142]. Nørskov et al. [143] reviews this active field.

4 Limitations of GGAs for Surface Chemistry

The successful applications reviewed above are all the more remarkable given the limitations of the GGA form. The large errors for gas-phase reaction barriers in Fig. 2 are not a special case, but are a general property of GGAs. GGAs systematically over-delocalize electrons [144–146] and overstabilize systems such as the stretched bonds of transition states. These errors are well known in the computational chemistry literature [86, 87, 147–154]. They have led to GGAs being almost entirely superseded in the computational chemistry community. (See, for example, the discussion of Fig. 1 in Burke [47].) The XC functionals which replaced them are discussed in Sect. 5.

The aforementioned difficulties of surface chemistry mean that there are relatively few experimental or computed benchmarks available to test GGAs’ performance on surfaces. GGAs’ limitations for surface chemistry are thus less well-characterized, and arguably less appreciated, than their limitations for molecules. However, available evidence strongly suggests that GGAs’ errors for gas-phase barriers carry over to reactions of molecules on surfaces. The LSDA and the BP86 [50, 68] and BLYP GGAs underestimate QCISD(T) barriers for H2 dissociation on gas-phase silanes and a cluster model for Si(100) [155]. The PW91 GGA underestimates experimental [156] and quantum Monte Carlo [26] adsorption barriers for H2 dissociative adsorption on Si(100). The PW91, PBE, and RPBE GGAs all underestimate the QMC barrier for H2 dissociation on Mg(0001) [32]. PBE also underestimates QMC barriers for hydrogen abstraction by styrene radical on hydrogen-terminated Si(001) [33]. A series of studies by Bickelhaupt and coworkers show similar GGA errors for small organic molecules reacting with atomic Pd [157–159]. We found comparable errors in several GGAs’ predicted dissociation barriers for H2 dissociation on Au3 and Ag3 clusters [160]. The BP86 GGA severely underestimates completely renormalized coupled cluster reaction barriers for methanol oxidation on Au −8 [31]. PW91 underestimates both coupled cluster reaction barriers for water splitting on an Fe atom, and RPA barriers for water splitting on Fe(100) [41]. PBE underestimates the barrier to O2 sticking on a cluster model of Al(111) [161], and PBE and RPBE incorrectly predict barrierless O2 dissociation on Al(111) slabs [162]. (While this is attributed to spin selection rules [162], recent hybrid DFT calculations [163] suggest that the GGA’s limitations also play a role.) PBE and PW91 also incorrectly predict barrierless dissociation of H2O2 on cluster models of zirconium, titanium, and yttrium oxide surfaces [164]. Dispersion-corrected PBE calculations systematically underestimate the experimental reaction barrier to alkene methylation over a slab model zeolite catalyst [93]. We showed that the LSDA, the PBE, and revPBE GGAs underestimate the coupled cluster barrier for NH3 dissociation on a cluster model of the reconstructed Si(100) surface, with differences between GGAs and beyond-GGA functionals persisting on larger clusters and periodic slabs [165]. GGAs also incorrectly predict the relative barriers to inter- vs intra-dimer NH3 dissociation [104–106, 166, 167]. The LSDA and the PBE, PW91, and revPBE GGAs tend to underestimate diffusion Monte Carlo calculations for diffusion barriers (i.e., adsorption energies at different sites [168]) of adatoms on graphene [34, 169]. PBE also underestimates accurate reaction barriers for hydrogenation of graphene model compounds [30]. Other relevant GGA errors include overbinding of Cu on cluster models of the MgO(001) surface [170], and qualitatively incorrect spin distributions for defects on titania [171] and ceria [172] surfaces.

A particularly instructive illustration of GGAs’ strengths and limitations comes from the aforementioned careful and insightful study of ammonia synthesis over ruthenium [108]. As discussed above, microkinetics models constructed from the PW91 and RPBE GGAs both predicted overall turnover frequencies within an order of magnitude of experiment. However, PW91 predicted that the rate-limiting N2 dissociation barrier was 0.6 eV lower than the RPBE barrier. This corresponds to an enormous ten orders of magnitude discrepancy in the room-temperature Arrhenius rate constant. At least one of the predicted mechanisms thus enjoyed substantial error cancellation between inaccurate adsorption energies and reaction barriers. The authors explicitly characterized this error cancellation, stating that PW91 calculations “increased the coverage … and decreased the number of free sites for dissociation” [108]. While such error cancellations are acceptable for some applications, improvements are clearly desirable. This result illustrates the handicaps faced by computational surface scientists, and motivates the development of new approximations.

Several groups have attempted to remedy these issues with new GGAs. Some explore parameterized GGA forms similar to those pioneered in Becke [173]. The BEEF-vdW dispersion-corrected GGA incorporates 31 empirical parameters fitted using Bayesian error estimation [174]. Calculations using this functional accurately treat a wide variety of properties, from small-molecule heats of formation and noncovalent interactions to lattice constants, bulk moduli, and chemisorption energies [174]. Applications to metal–carbon bond formation from hydrogenation of supported graphene [175] and chemisorption on zeolites [176] leverage its strengths for noncovalent interactions. A study of CO2 hydrogenation even points to improvements over RPBE for some reaction barriers on surfaces [177]. However, BEEF-vdW still gives a 0.26 eV mean absolute error in representative gas-phase reaction barriers [150], comparable to RPBE (0.27 eV), and significantly larger than the B3LYP global hybrid (0.17 eV) [174]. This is especially noteworthy given that B3LYP is, for a hybrid functional, not particularly accurate for reaction barriers. (To illustrate, the HISS screened hybrid discussed in Sect. 5.3 gives mean absolute errors of 1.7 and 1.8 kcal mol–1 in the HTBH38/04 and NHTBH38/04 test sets of gas-phase reaction barriers [86, 87], significantly smaller than B3LYP errors 4.23 and 4.34 kcal mol–1 obtained with a different computational setup [87].) The SOGGA11 [178] and non-separable N12 [179] GGAs, which respectively incorporate 18 and 24 empirical parameters, also give fairly large errors in these test sets [180]. These errors are dramatically reduced by the empirical meta-GGAs discussed in Sect. 5.4 [180].

“Specific reaction parameter” GGAs interpolating between, for example, PW91 and RPBE have also been proposed [181]. The interpolations are not guaranteed to be transferable, and require fitting to known experimental values. Interpolations can also be problematic where the known experimental value is not bracketed by two different GGAs [182]. GGAs’ systematic underestimate of reaction barriers appears to make this circumstance rather common. Indeed, specific reaction parameter functionals originally tuned a hybrid functional’s GGA term and fraction of exact exchange [183], exploiting the effects discussed in Sect. 5.1. Unfortunately, it appears that this “Procrustean dilemma” (Perdew et al. [62]) is an inherent limitation of the GGA form. This fact motivates exploration of methods beyond the GGA.

5 Beyond the GGA

DFT calculations on small and medium-sized molecules almost exclusively use beyond-GGA approximations for the exchange-correlation functional [47]. Extension of these methods to surface chemistry offers a potential solution to the dilemma presented in Fig. 2. This section focuses on screened hybrids and meta-GGAs, two beyond-GGA approximations that show particular promise for surface chemistry.

5.1 Hybrid XC Functionals

DFT’s widespread adoption for computational chemistry [47] is arguably directly attributed to the ability of fourth-rung “hybrid” XC approximations to outperform Hartree–Fock theory and second-order many-body perturbation theory for chemical bond breaking. Hybrid functionals [56] incorporate a fraction of exact exchange:

$$ {E}_{\mathrm{X}}^{\mathrm{ex}}\left[\rho \right]=-\frac{1}{2}\mathit{\int}{d}^3\mathbf{r}\mathit{\int }{d}^3{\mathbf{r}}^{\mathbf{\prime}}\frac{\left|\gamma \left(\mathbf{r},{\mathbf{r}}^{\mathbf{\prime}}\right)\right|{}^2}{\left|\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime}}\right|}. $$
(4)

Here, γ(r, r′) is the nonlocal one-particle density matrix of the noninteracting Kohn–Sham wavefunction, constructed from the occupied orbitals of the Kohn–Sham wavefunction {ϕ i (r)} as γ(r, r′) = Σ i ϕ i (r)ϕ i  * (r′). These orbitals and density matrices are thus implicit density functionals. Equation (4) provides the exact XC functional for one-electron systems, where it exactly cancels the Hartree interaction.

Admixture of a fraction of Eq. (4) to a GGA is justified by an adiabatic connection between the real system and the noninteracting Kohn–Sham reference [56, 184]. Such admixture corrects GGAs’ over-delocalization, improving the prediction of a variety of properties including reaction barrier heights [146]. GGA calculations with more localized Hartree–Fock orbitals [153, 185], self-interaction corrections [186], and explicit constraints on localization [187] all give related improvements.

Hybrid functionals’ success can be rationalized in terms of GGAs’ simulation of nondynamical correlation [188]. Briefly, the many-body correction to the Hartree interaction may be modeled as a “hole” h XC[ρ](r, r′) in the electron density about an electron at point r [145]. The total Hartree + XC energy becomes

$$ {E}_{\mathrm{HXC}}\left[\rho \right]=\frac{1}{2}\mathit{\int}{d}^3\mathbf{r}\rho \left(\mathbf{r}\right)\mathit{\int}{d}^3{\mathbf{r}}^{\mathbf{\prime}}\frac{\rho \left({\mathbf{r}}^{\mathbf{\prime}}\right)+{h}_{\mathrm{XC}}\left[\rho \right]\left(\mathbf{r},{\mathbf{r}}^{\mathbf{\prime}}\right)}{\left|\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime}}\right|}. $$
(5)

The XC hole is delocalized in systems such as stretched H +2 , where h XC[ρ](r, r′) = – ρ(r′). Nondynamical correlation in stretched covalent bonds localizes the XC hole about r, such that (for example) an electron on the left atom in stretched singlet symmetric H2 pushes the other electron to the right atom. GGA exchange functionals use localized exchange holes by construction. Thus, GGA “exchange” in practice models both exchange and nondynamical correlation [189].

Unfortunately, this rather crude model tends to overestimate nondynamical correlation and overbind. It is especially problematic in stretched H +2 and other odd-electron bonds [190]. (Notions of self-interaction error [144, 191, 192], delocalization error [193], exact constraints on the XC hole [194], and symmetry breaking [195] provide additional insights into these effects.) Exact exchange admixture tunes this model, providing a surprisingly effective treatment of chemical bonding.

It is worth noting another useful aspect of hybrid functionals. Generalized Kohn–Sham calculations with a nonlocal XC potential v XC(r, r′) = δE XC/δγ(r, r′) approximate the exact functional’s derivative discontinuity, allowing occupied-virtual orbital energy gaps to better approximate fundamental band gaps better [196–198]. Kohn–Sham calculations with hybrid functionals require the local and multiplicative effective potential δE exX [ρ]/δρ(r), typically constructed with optimized effective potential methods [199–201] or variants thereof [202]. Cohen et al. [197] provides a particularly useful illustration of the differences between Kohn–Sham and generalized Kohn–Sham calculations.

5.2 Hybrid Functionals’ Limitations

Hybrid functionals have three limitations which are particularly relevant to surface chemistry. First, the optimal fraction of exact exchange is not known a priori, but depends on the role of nondynamical correlation in the system or property of interest. In practice, simple global hybrids typically require ~20% exact exchange for accurate thermochemistry [203, 204], ~40–50% exact exchange for kinetics [205], 100% exact exchange far from finite systems where Eq. (4) is the exact XC functional [206–208], and relatively small exact exchange admixtures for many organometallic properties [209, 210]. Figure 3 illustrates how this is problematic for reactions on metal surfaces. Other implications for surface chemistry are discussed in Hafner [211]. (The “+U” method, where the Kohn–Sham reference system includes a Hubbard repulsion on special sites [212], has similar issues centered on choosing the magnitude of U [213].) There has been substantial interest in overcoming this limitation through system-dependent [214–219] or position-dependent [220–222] exact exchange admixture, or through more sophisticated mean-field models of the XC hole [223, 224]. However, existing position-dependent “local hybrids” are not unambiguously more accurate than global hybrids [222, 225], and system-dependent exact exchange admixture can introduce size consistency issues [217].

Fig. 3
figure 3

Schematic of the optimal admixture of exact exchange in various regions of a reaction on a metal surface

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Other limitations for surface chemistry arise from the long-range piece of exact exchange (large |r − r′| in Eq. (4)) Evaluating this long-range contribution is computationally expensive in metallic systems where the Kohn–Sham γ(r, r′) delocalizes over a large range of |r − r′| [226]. (More sophisticated treatments of this term have been proposed [227, 228].) Additionally, long-range exact exchange is exactly cancelled by higher order electron correlation effects in the HEG, and approximately cancelled in metals [229, 230]. There are thus relatively few global hybrid DFT calculations on periodic metal slabs [161, 163, 231–233].

5.3 Screened Hybrids

Screened hybrid functionals [234, 235] rigorously [236] cut off the problematic long range of Eq. (4), sacrificing [237] an exact treatment of density tails [206–208] to enable facile treatments of periodic systems. The HSE06 screened hybrid [235, 238, 239] includes 25% of the error-function-screened exact exchange:

$$ {E}_{\mathrm{X}}^{\mathrm{SR}-\mathrm{ex}}\left[\rho \right]=-\frac{1}{2}{\displaystyle \int {d}^3}\mathbf{r}{\displaystyle \int {d}^3}{\mathbf{r}}^{\mathbf{\prime}}\frac{\left|\gamma \left(\mathbf{r},{\mathbf{r}}^{\mathbf{\prime}}\right)\right|{}^2}{\left|\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime}}\right|}\mathrm{erfc}\left(\omega \left|\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime}}\right|\right). $$
(6)

This is combined with 75% screened PBE exchange and 100% long-range PBE exchange and PBE correlation. Screened PBE exchange is constructed from explicit models [240, 241] of the angle- and system-averaged PBE exchange hole of Eq. (5):

$$ {E}_{\mathrm{X}}^{\mathrm{SR}-\mathrm{P}\mathrm{B}\mathrm{E}}\left[\rho \right]={\displaystyle \int {d}^3}\mathbf{r}{e}_{\mathrm{X}}^{\mathrm{SR}-\mathrm{P}\mathrm{B}\mathrm{E}}\left(\rho \left(\mathbf{r}\right),\nabla \rho \left(\mathbf{r}\right)\right), $$
(7)
$$ {e}_{\mathrm{X}}^{\mathrm{SR}-\mathrm{P}\mathrm{B}\mathrm{E}}\left(\rho \left(\mathbf{r}\right),\nabla \rho \left(\mathbf{r}\right)\right)=-\frac{1}{2}\rho \left(\mathbf{r}\right){\displaystyle \int {d}^3{\mathbf{r}}^{\mathbf{\prime}}}\frac{h_{\mathrm{X}}^{\mathrm{PBE}}\left(\rho \left(\mathbf{r}\right),\nabla \rho \left(\mathbf{r}\right),\left|\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime}}\right|\right)}{\left|\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime}}\right|}\mathrm{erfc}\left(\omega \left|\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime}}\right|\right). $$
(8)

Exchange screening accelerates hybrid DFT calculations on periodic systems. Calculations with atom-centered basis functions can integrate |r – r′| over a reduced number of replica cells [242]. Calculations with plane-wave basis functions can downsample the k-space mesh for reciprocal space integration [226]. HSE06 is implemented in standard codes [243–245] and has been extensively applied to semiconductors [246, 247].

As for global hybrids, screened hybrids’ optimum fraction of exact exchange and optimum screening parameter ω are not known a priori. The HSE06 exact exchange admixture comes from perturbation theory arguments [248], and the screening parameter is chosen empirically [235, 238]. Both parameters significantly affect the functional’s computational cost in solids, and its performance for many properties [238, 249]. For example, while the HSE06 screening parameters appear nearly optimal for semiconductor bandgaps [250], they underestimate the bandgaps of large-gap insulators [218] and overestimate metals’ energy bandwidths [226]. (Marques et al. [218] gives a particularly enlightening perspective on this effect, based on a relation between generalized Kohn–Sham equations with a screened hybrid functional and many-body GW calculations [251] with an effective static dielectric constant.) This has led to the exploration of several other screened hybrid forms [153, 252–257]. The “middle-range” screened hybrid HISS [152, 258] shows particular promise for surface chemistry. HISS uses a second screening function to include additional exact exchange at moderate |r – r′|. HISS accurately treats semiconductor bandgaps and lattice parameters [259], as well as some reactions on surfaces [165, 169]. Its more aggressive screening reduces its computational cost relative to HSE06 [259].

5.4 Meta-GGAs

A second route to fixing GGAs’ limitations is third-rung functionals incorporating the noninteracting kinetic energy:

$$ \tau \left(\mathbf{r}\right)=\frac{1}{2}{\displaystyle \sum_i{\left|\nabla {\phi}_i\left(\mathbf{r}\right)\right|}^2}=\frac{1}{2}\underset{{\mathbf{r}}^{\mathbf{\prime}}\to \mathbf{r}}{ \lim }{\nabla}_{\mathbf{r}}\cdot {\nabla}_{{\mathbf{r}}^{\mathbf{\prime}}}\gamma \left(\mathbf{r},{\mathbf{r}}^{\mathbf{\prime}}\right), $$
(9)
$$ {E}_{\mathrm{XC}}^{\mathrm{mGGA}}\left[\rho \right]=\mathit{\int}{d}^3\mathbf{r}\;{e}_{\mathrm{XC}}^{\mathrm{mGGA}}\left(\rho \left(\mathbf{r}\right),\left|\nabla \rho \left(\mathbf{r}\right)\right|,\tau \left(\mathbf{r}\right)\right). $$
(10)

Meta-GGAs may also use the density Laplacian, which incorporates information similar to τ [260]. Meta-GGA calculations are not much more expensive than calculations with GGAs [209, 261]. This makes meta-GGAs particularly attractive for calculations on solids and surfaces. Early meta-GGAs [262, 263] showed promise for properties such as surface energies [264] and molecular thermochemistry [261, 265]. However, their adoption was limited by the fact that they are only comparable to GGAs for lattice parameters [263] and gas-phase kinetics [266]. Modifications improving lattice parameters [267] do not improve reaction barriers [180]. This performance was somewhat disappointing, given that Eq. (9) should contain some information about the short-range nonlocal one-particle density matrix of Eq. (6), which clearly improves hybrid functionals’ performance.

A breakthrough came with the demonstration that the M06-L meta-GGA containing 37 empirical parameters can treat many properties, from lattice parameters to molecular thermochemistry to reaction barriers, with accuracy approaching hybrid functionals [268, 269]. Reverse-engineering M06-L [270] showed that part of its success comes from the inclusion of t – 1 = τ/τ HEG. Here τ HEG = (3/10)(6π 2)2/3 ρ 5/3 equals τ in the HEG. t –1 can differentiate covalent vs non-covalent interactions [270]. The success of M06-L motivated subsequent development of minimally empirical meta-GGAs based on α = (τ–τ W)/τ HEG, where τ W = |∇ρ|2/(8ρ) ≤ τ, equals τ in one-electron systems [271–273]. Viewed from this perspective, M06-L becomes a meta-GGA form fit to the exact functional (which would presumably give zero error in the fitting set), whose fitting coefficients revealed meta-GGAs’ previously unrecognized possibilities. Other empirical meta-GGAs have also been explored [180, 274]. These new meta-GGAs are designed in part to overcome the oscillatory behavior of M06-L [275].

6 Beyond the GGA for Surface Chemistry

The methods introduced in Sect. 5 have begun to be applied to chemistry on periodic slab model surfaces. Results to date indicate that these new methods have a great deal of potential, and point to some remaining limitations which motivate further development.

6.1 Recent Applications

One important series of studies applies the M06-L meta-GGA to molecule-surface adsorption. Hammer and coworkers showed that the M06-L meta-GGA accurately treats “medium-range” noncovalent interactions for adsorbates on graphene [270, 276, 277], despite its lack of true long-range correlation [278]. The authors have applied M06-L in subsequent studies of hydrogenation [279] and CO intercalation [280] of supported graphene, and of RS-Au-SR “staple” motifs [281] in alkanethiol monolayers on Au(111) [282].

Another important milestone concerns treatments of CO on noble metal surfaces [132]. The HSE03 screened hybrid and the M06-L and revTPSS meta-GGAs all improve the binding site preference. The meta-GGAs correctly predict that CO preferentially adsorbs atop a single Pt atom on Pt(111), along with encouraging accuracy for lattice constants, surface formation energies, and adsorption energies [283, 284]. HSE03 provides the correct site preference for CO on Cu(111) and Rh(111) surfaces. While it still fails for Pt(111), it reduces the top-fcc energy difference relative to PBE [231]. HSE03 shows similar trends for CO adsorption on the terraces of stepped Rh(553) [59].

A third application is to adsorbates and defects on metal oxide surfaces. These studies build upon screened hybrids’ successes for modeling electrons localized in bulk defects [246, 285, 286]. HSE06 and the B3LYP global hybrid were used to analyze Au adatoms on ceria defects [172], a system where GGAs’ over-delocalization leads to qualitatively incorrect results [213, 287]. HSE was used to check PBE + U calculations on ceria-supported vanadia catalysts [288], to confirm conclusions about the reactivity of oxygen vacancies on titania surfaces [289], and to check the spin polarization of graphite surface defects [290]. Pacchioni [291] reviews some other relevant studies.

There have been relatively few applications of meta-GGAs and screened hybrids to reaction barriers of molecules on surfaces. However, results to date suggest that these functionals’ improvements for gas-phase barriers carry over to surfaces. Unlike the PBE and RPBE GGAs [162], the HSE06 screened hybrid and PBE0 global hybrid predict a substantial barrier to O2 dissociation on Al(111) slabs [163]. (As discussed above, this failure of GGAs was previously attributed entirely to spin selection rules for triplet O2 dissociation [162].) M06-L predicts reasonable binding energies for molecules on zeolite catalysts [83], and has been applied to heterogeneous catalysis by zeolites and metal-oxide frameworks [292, 293]. We showed that the HSE06 and HISS screened hybrids, and the M06-L meta-GGA, improved upon GGAs’ underestimate of the dissociation barrier for ammonia dissociation on a cluster model of Si(100). Similar trends were seen for calculations on Si slabs [165]. We also showed that HSE06, HISS, and M06-L improve GGAs’ underestimated diffusion barriers for adatoms on graphene [169] and H2 dissociation on gold and silver clusters [160]. Interestingly, the TPSS meta-GGA increases the too-low PBE barriers for H2 dissociation on reconstructed Si(001) surfaces, despite the two functionals’ similarity for gas-phase barriers [266].

6.2 Limitations

Some studies have identified limitations of existing beyond-GGA functionals for surface chemistry. Lousada et al. [164] shows that, similar to GGAs, M06-L predicts a qualitatively incorrect barrierless dissociation of H2O2 on metal oxides. Valero et al. [294, 295] shows that M06-L is problematic for the frequency shifts of CO and NO on nickel and magnesium oxides. These errors are mitigated by global hybrids [294, 295]. The M06-L and TPSS meta-GGAs do not improve upon the dispersion-corrected B97-D GGA for the aforementioned problem of methanol oxidation over Au −8 , giving mean unsigned errors in reaction barrier heights of 10.1, 9.2, and 7.4 kcal mol–1, respectively [31]. However, the B3LYP and M06 [269] global hybrids improve upon B97-D, with errors of 3.6 and 3.9 kcal mol–1. HSE06’s overestimated metal bandwidths [226] appear to contribute to its aforementioned problems for CO on Pt(111) [231]. The B3LYP global hybrid has other problems for metals, which arise because its GGA for correlation [69] does not recover the correct HEG behavior [296].

6.3 Systematic Trends

Our recent extension [169] of diffusion Monte Carlo studies on adatom adsorption and diffusion over graphene [34] provides a systematic illustration of how screened exchange affects the Procrustean dilemma [62] faced by GGAs for surface chemistry. Part of that study considered systematic modification of the PBE GGA by both rescaling of the exchange enhancement factor (F X = βF PBEX , see (3)), and admixture of a fraction α of screened exact exchange. (Put another way, changing β is similar to specific reaction parameter GGAs used for surface chemistry [181, 182]. Changing α is similar to the original specific reaction parameter global hybrids [183]). Figure 4 illustrates how changing β and α affect representative surface and molecule properties. The left panel shows calculated adsorption energies and diffusion barriers for O atom on graphene, compared to the diffusion Monte Carlo results of Hsing et al. [34]. The right panel shows errors in standard sets of gas-phase molecular thermochemistry and kinetics [297]. Computational details are in Barone et al. [169].

Fig. 4
figure 4

Systematic variations in GGA enhancement factor F X = βF PBEX (dotted lines) and screened exchange admixture α (solid lines). β = 1, α = 0 is the PBE GGA, β = 1, α = 0.25 is the HSE06 screened hybrid. Left: Diffusion barrier vs adsorption energy for O on graphene. Right: Mean signed errors in gas-phase BH6 [297] kinetics vs AE6 thermochemistry. “Ref.” are diffusion Monte Carlo from Ren et al. [34] (left) and zero mean signed error (right). Adapted with permission from Barone et al. [169]. Copyright 2013 American Chemical Society

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Figure 4 shows that GGA rescaling β simultaneously changes both adsorption energies and reaction barriers, and that no value of β can treat both properties. Figure 3 of Barone et al. [169] shows that a simple dispersion correction increased the GGA chemisorption energies, but did not affect reaction barriers. This is consistent with the results of Fig. 2 and with the limitations of empirical GGAs discussed in Sect. 4. In contrast, screened exchange admixture α increases both surface diffusion barriers and molecule reaction barriers, while maintaining reasonable thermochemistry and atomization energies. The results suggest that screened hybrids have the potential to improve reaction barriers on surfaces, just as global hybrids improve gas-phase reaction barriers.

7 New Frontiers

Meta-GGAs and screened hybrids are not yet standard methods for simulating heterogeneous catalysis or surface chemistry. Additional work is needed to understand better their strengths and limitations, and to develop more accurate extensions. This section briefly reviews selected recent work along those lines, focusing on results from the author and his collaborators.

7.1 Dispersion-Corrected and Empirical Screened Hybrids

Barone et al. [169] suggests that GGAs’ limitations for dispersion interactions and reaction barriers are largely orthogonal. Dispersion-corrected screened hybrids could in principle combine the aforementioned successes of dispersion corrections for adsorption energies and hybrid exchange for reaction barriers. A dispersion-corrected screened hybrid was recently benchmarked for rare-gas solids [255], and applied to C–H bond cleavage in crystalline polyethylene [298] and Au adatom adsorption on defective CeO2(111) [172] mimicking ceria-supported gold catalysts [299]. Dispersion-corrected PBE and HSE calculations gave similar barriers to tetrachloropyrazine chemisorption on Pt(111) [300]. Dispersion-corrected global hybrids have also been applied to some molecular crystals [301], crystalline polymers [302], and surface chemistry [303].

It is interesting to consider whether empirical functional forms [173, 174, 179, 180] could benefit from screened exchange admixture. Perverati and Truhlar [257] proposed screened hybrids built upon the parameterizations of [179, 180]. These functionals improve upon HSE06 for gas-phase reaction barriers and some lattice constants [257]. They show modest promise for binding and relative energies of water clusters, properties which appear to be improved by dispersion corrections [304]. However, they have not yet been extensively tested for surface chemistry.

7.2 Rung 3.5 Functionals

We have proposed a new class of approximate functionals constructed to be intermediate between third-rung meta-GGAs and fourth-rung screened hybrids. Rung 3.5 functionals replace one of the one-particle density matrices in (4) with a GGA model density matrix γGGA:

$$ {E}_{\mathrm{X}}^{\prod}\left[\rho \right]=-\frac{1}{2}{\displaystyle \int {d}^3}\mathbf{r}{\displaystyle \int {d}^3{\mathbf{r}}^{\mathbf{\prime}}}\frac{\gamma \left(\mathbf{r},{\mathbf{r}}^{\mathbf{\prime}}\right){\gamma}_{\mathrm{GGA}}\left(\rho \left(\mathbf{r}\right),\nabla \rho \left(\mathbf{r}\right),\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime}}\right)}{\left|\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime}}\right|}. $$
(11)

The integrand of (11) is symmetrized in r, r′ before use. The GGA model density matrix is implicit in the construction of the GGA exchange hole of Eq. (8):

$$ {h}_{\mathrm{X}}^{\mathrm{GGA}}\left(\rho \left(\mathbf{r}\right),\nabla \rho \left(\mathbf{r}\right),\left|\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime}}\right|\right)=-\frac{1}{2}{\rho}^{-1}\left(\mathbf{r}\right){\left\langle {\left|{\gamma}_{\mathrm{GGA}}\left(\rho \left(\mathbf{r}\right),\nabla \rho \left(\mathrm{r}\right),\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime}}\right)\right|}^2\right\rangle}_{\varOmega }, $$
(12)

where 〈 … 〉 Ω denotes angle averaging. Most existing exchange hole models treat the angle- and system-averaged hole [240, 241], while γGGA is explicitly not angle averaged [305]. γGGA, similar to h GGAX , decays rapidly in |r – r′| by construction, aiding evaluation of Eq. (11) in metals. (Recall from Sect. 5.1 that this localization is central to the GGA “exchange” functionals’ model of nondynamical correlation.) γGGA also tunes the amount of nonlocal information incorporated at each point, potentially providing a route to simultaneously treating several of the regions in Fig. 3. (Note that Eq. (11) cannot include 100% long-range exact exchange, and is not exact for one-electron regions.) Rung 3.5 functionals thus have the potential to address all three limitations of exact exchange admixture discussed in Sect. 5.1. Janesko [306] reviews our applications of Rung 3.5 functionals. Benchmarks for molecular thermochemistry and kinetics show that they can provide accuracy intermediate between standard GGAs and screened hybrids.

Table 1 presents previously unpublished results applying the Rung 3.5 functional Π 1PBE [305] to ammonia dissociation on Si(100). Π 1PBE admixes 25% of Eq. (11) to the PBE GGA, using a model density matrix γPBE constructed to reproduce the PBE exchange enhancement factor. Calculations use a small cluster model (nine Si atoms) from Sniatynsky et al. [165]. The Rung 3.5 reaction barrier is between the third-rung TPSS meta-GGA and the fourth-rung HSE06 screened hybrid, indicating that the functional lives up to its name. (Results in Table 1 differ by ~0.02 eV from Sniatynsky et al. [165] because of a different basis set.) We are currently exploring more extensive applications of Rung 3.5 functionals to surface chemistry.

Table 1 Adsorption and dissociation of NH3 on Si(100)
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8 Conclusions

The successes of GGAs for surface chemistry are particularly remarkable, given their underlying limitations. Density functional approximations beyond the GGA, largely developed in the computational chemistry community, show promise for ameliorating these limitations in simulations of surface chemistry. Recent calculations illustrate these new methods’ potential and point to remaining issues. It is hoped that these promising preliminary results motivate density functional developers to consider further the applications to surfaces, and motivate surface scientists to test beyond-GGA approximations on new systems. More accurate, computationally tractable methods including beyond-GGA DFT will help build upon GGAs’ successes for surface chemistry.