Constructing Mixed Density Functionals for Describing Dissociative Chemisorption on Metal Surfaces: Basic Principles

Abstract

The production of a majority of chemicals involves heterogeneous catalysis at some stage, and the rates of many heterogeneously catalyzed processes are governed by transition states for dissociative chemisorption on metals. Accurate values of barrier heights for dissociative chemisorption on metals are therefore important to benchmarking electronic structure theory in general and density functionals in particular. Such accurate barriers can be obtained using the semiempirical specific reaction parameter (SRP) approach to density functional theory. However, this approach has thus far been rather ad hoc in its choice of the generic expression of the SRP functional to be used, and there is a need for better heuristic approaches to determining the mixing parameters contained in such expressions. Here we address these two issues. We investigate the ability of several mixed, parametrized density functional expressions combining exchange at the generalized gradient approximation (GGA) level with either GGA or nonlocal correlation to reproduce barrier heights for dissociative chemisorption on metal surfaces. For this, seven expressions of such mixed density functionals are tested on a database consisting of results for 16 systems taken from a recently published slightly larger database called SBH17. Three expressions are derived that exhibit high tunability and use correlation functionals that are either of the PBE GGA form or of one of two limiting nonlocal forms also describing the attractive van der Waals interaction in an approximate way. We also find that, for mixed density functionals incorporating GGA correlation, the optimum fraction of repulsive RPBE GGA exchange obtained with a specific GGA density functional is correlated with the charge-transfer parameter, which is equal to the difference in the work function of the metal surface and the electron affinity of the molecule. However, the correlation is generally not large and not large enough to obtain accurate guesses of the mixing parameter for the systems considered, suggesting that it does not give rise to a very effective search strategy.

Figure 1

Exchange enhancement factor (F_x) as a function of the reduced density gradient (s) for the RPBE (black), PBE (blue), PBEα (with α = 0.57, magenta), and PBEsol (red) functionals. The horizontal red dashed line presents the local Lieb-Oxford bound imposed in the construction of these functionals.^,,,

Figure 2

Barrier heights E_b computed with the PBEsolc, PBE, and RPBE DFs are shown as a function of the charge transfer parameter for the 16 systems present in the SBH16 database.

Figure 3

Barrier heights E_b computed with the PBE, the PBE-vdW1, and the PBE-vdW2 DFs are shown as functions of the charge transfer parameter for the 16 systems present in the SBH16 database.

Figure 4

Barrier heights E_b computed with the RPBE, the RPBE-vdW1, and the RPBE-vdW2 DFs are shown as a function of the charge transfer parameter for the 16 systems present in the SBH16 database.

Figure 5

Barrier heights E_b computed with the PBEsolc and PBEsol-vdW2 DFs are shown as a function of the charge transfer parameter for the 16 systems present in the SBH16 database.

Figure 6

Barrier heights computed with the SRPx DF (black bars) and the SRPxsol DF (red bars) are shown as a function of the fraction of RPBE exchange x, for (A) H₂ + Cu(111) (upper panel) and (B) CH₄ + Pt(111) (lower panel). Blue horizontal lines indicate the reference value of the barrier height for these systems. The black and red dashed lines linearly interpolate the barrier height as a function of x for SRPx and SRPxsol DFs, respectively. The optimal value of x is equal to the value of x for which these lines intersect with the blue lines.

Figure 7

Optimum fraction of RPBE exchange x is shown as a function of ΔE_CT for the SRPx DF (eq 1). Values falling between the two horizontal dot-dashed black lines could be obtained by the interpolation procedure illustrated in Figure 6. The green, blue, and red symbols correspond to N₂, H₂, and CH₄ + metal surface systems, respectively. The black, blue, and red dashed lines provide the linear fits corresponding to the Pearson correlation coefficients computed for all molecules, H₂, and CH₄ + metal surface systems, respectively, without omitting systems.

Figure 8

Optimum fraction of RPBE exchange x is shown as a function of ΔE_CT for the SRPxsol DF (eq 2). Values falling between the two horizontal dot-dashed black lines could be obtained by the interpolation procedure illustrated in Figure 6. The green, blue, and red symbols correspond to N₂, H₂, and CH₄ + metal surface systems, respectively. The black, blue, and red dashed lines provide the linear fits corresponding to the Pearson correlation coefficients computed for all molecules, H₂, and CH₄ + metal surface systems, respectively, without omitting systems.

Figure 9

Optimum fraction of RPBE exchange x is shown as a function of ΔE_CT for the SRPxsol-vdW2 DF (eq 5). Values falling between the two horizontal dot-dashed black lines could be obtained by the interpolation procedure illustrated in Figure 6. The green, blue, and red symbols correspond to N₂, H₂, and CH₄ + metal surface systems, respectively. The black, blue, and red dashed lines provide the linear fits corresponding to the Pearson correlation coefficients computed for all molecules, H₂, and CH₄ + metal surface systems, respectively, without omitting systems.

Figure 10

Optimum mixing parameter x is shown as a function of ΔE_CT for the SRPx-vdW1-ext DF (eq 6a). Values falling between the two horizontal dot-dashed black lines could be obtained by the interpolation procedure illustrated in Figure 6. The green, blue, and red symbols correspond to N₂, H₂, and CH₄ + metal surface systems, respectively. The black, blue, and red dashed lines provide the linear fits corresponding to the Pearson correlation coefficients computed for all molecules, H₂, and CH₄ + metal surface systems, respectively, without omitting systems.