Hybrid RPA:DFT Approach for Adsorption on Transition Metal Surfaces: Methane and Ethane on Platinum (111)

The hybrid QM:QM approach is extended to adsorption on transition metal surfaces. The random phase approximation (RPA) as the high-level method is applied to cluster models and, using the subtractive scheme, embedded in periodic models which are treated with density functional theory (DFT) that is the low-level method. The PBE functional, both without dispersion and augmented with the many-body dispersion (MBD), is employed. Adsorption of methane and ethane on the Pt(111) surface is studied. For methane in a 2 × 2 surface cell, the hybrid RPA:PBE and RPA:PBE+MBD results, −14.3 and −16.0 kJ mol–1, respectively, are in close agreement with the periodic RPA value of −13.8 kJ mol–1 at significantly reduced computational cost (factor of ∼50). For methane and ethane, the RPA:PBE results (−14.3 and −17.8 kJ mol–1, respectively) indicate underbinding relative to energies derived from experimental desorption barriers for relevant loadings (−15.6 ± 1.6 and −27.2 ± 2.9 kJ mol–1, respectively), whereas the hybrid RPA:PBE+MBD results (−16.0 and −24.9 kJ mol–1, respectively) agree with the experiment well within experimental uncertainty limits (deviation of −0.4 ± 1.5 and +2.3 ± 2.9 kJ mol–1, respectively). Finding a cluster that adequately and robustly represents the adsorbate at the bulk surface is important for the success of the RPA-based QM:QM scheme for metals.


S1 -Cluster models
The clusters used in this study are presented in Figures S1. 1, respectively.They are named in the following style: Ptn(A,B,…), where n is the number of platinum atoms in the cluster, A is the number in the top layer of the cluster, B is the number in the second layer, and so on.

S2 -Clusters under pbc
The clusters shown in Section S1 were placed into cubic cells under pbc.Table S2. 1 shows the adsorption energies for the cluster in 20 Å 3 and 25 Å 3 cells.Table S2.1.Adsorption energy ΔEads (in kJ mol -1 ) of CH4 on platinum clusters Ptn(A,B,C) for PBE+MBD in 20 3 and 25 3 Å 3 cells.Where n denotes the total number of Pt atoms, A the number in the first layer, B the number in the second layer, and C the number in the third layer.

Cluster
ΔEads / kJ mol We present the adsorption energies with and without spin-polarisation in Table S2.

S3.1 DFT
We have shown that using plane waves, i.e. a near complete basis set is appropriate for describing clusters and converges towards to pbc values.However, this corroboration does not necessarily extend smoothly to Gaussian basis sets.To investigate this, we calculated the adsorption energy for the seven different clusters using PBE with a large, Gaussian basis.The choice of state then becomes important.We chose to investigate the singlet state, as this is the physical state of bulk platinum, lacking any magnetic dipole, and the lowest energy spin state, which as commonly been used in the literature. 2 The adsorption energies are plotted against the number of Platinum atoms in the cluster in Figure S3.1.The adsorption energies were obtained using PBE/def2-QZVPP and the Basis Set Superposition Error (BSSE) was accounted for using the Counterpoise Correction (CPC).S3.1.The adsorption energies were obtained using PBE/def2-QZVPP and BSSE-corrected with the Counterpoise Correction (CPC).
The difference between the adsorption energies for the singlet and lowest spin states is not significant, varying by less than 1 kJ mol -1 with the singlet state being consistently lower in energy.
We checked the adsorption energy up to the 39-tet state for Pt19 and found that it did not change significantly with multiplet state, see Figure S3.3 and Table S3.2, with only a few minor deviations.
3][4][5][6] By using DFT, one can compensate for this and make the clusters suitable for describing adsorption on surfaces. 7,8e important exception is for the Pt18,10 cluster, where the lowest energy spin state (the triplet in this case) is significantly lower in energy.Upon investigation, it became clear that this is due to close-lying triplet states that are difficult to distinguish.Instead, we show the quintet state for this is far closer to the singlet state and does not significantly deviate, so we will use this in subsequent calculations on this cluster.
Once again, we see little difference between the 2-and 3-layered clusters, generally less than 0.5 kJ mol -1 , confirming our suspicion that there is no real benefit in the use of additional layers, so we will not consider additional layers further.The general trend of adsorption energies with respect to cluster size is similar for both the singlet and lowest spin states, showing similar curves.This is matched by the trend for those adsorption energies obtained from the plane wave basis set, which indicates that they describe the electronic structure similarly.This corroborates that either a plane wave or atom-centred, Gaussian is suitable for the description of these platinum clusters using DFT(+D).This makes it appropriate to use as the low-level method in a hybrid scheme.

RPA
Few post-HF methods are suitable for studying metals, due to their zero band-gap.One such method is the Random Phase Approximation (RPA).0][11] Alternatively, clusters may be used. 8As these are only mimics for the surface, they do not have an exactly zero band-gap, so other methods may also be suitable, if they were also computationally feasible.We have applied RPA to the metal clusters investigated previously and encountered additional problems.Although the band-gap is not as severe an issue as might have been expected, there is nonetheless great difficulty in finding a suitable spin state for calculating.Above we tested the singlet and the lowest energy spin states and found that they worked well for DFT.We then took these PBE orbitals and used them for RPA.We found that there is a strong dependence on the HOMO-LUMO gap, 12,13 the non-periodic analogue of the band-gap.To test this, we took a cluster that performed well, the singlet state of the Pt19 cluster and set the HOMO-LUMO gap to a set value by shifting all the energy of the virtual orbitals by the same amount, then performed RPA calculations.N.B. the PBE orbitals remained otherwise unchanged.We show the adsorption energy against the HOMO-LUMO gap for CH4/Pt19 using RPA in Figure S3.2.S3.3.The red line is the adsorption energy without any shifting of the HOMO-LUMO gap.
It is clear from Figure 3.2 that the adsorption energy is linear with respect to the HOMO-LUMO gap beyond 0.2 eV, leading us to wonder whether this could be extrapolated to a "zero-gap" value to better mimic the metal surface.This did not match the unshifted HOMO-LUMO gap value (-12.9 kJ mol -1 ), however, instead underestimating it by 2 kJ mol -1 .Additionally, it is clear that this implementation of RPA (as an approximation of ring CCD) 14 is not immune from the zero-gap issue, unlike periodic RPA.Instead, the adsorption energy becomes increasingly strong as the HOMO-LUMO gap tends towards zero, resulting in unphysically strong binding.However, so long as a small, non-zero HOMO-LUMO gap is found for the cluster, good RPA adsorption energies may still be performed.
We tested the HOMO-LUMO gaps for all our clusters and found that, with the exception of the Pt19 cluster, the singlet state resulted in a negative HOMO-LUMO gap, i.e. a non-Aufbau population, and making it inappropriate for further use.This can be amended by forcing a final diagonalisation of the Fock matrix.However, this introduces an artificially large HOMO-LUMO gap, rendering the RPA adsorption energies meaningless (cf.dependency of ΔEads in Figure S3.2 and Table S3.4).This is due to doubly-degenerate orbitals being populated preferentially before singly degenerate, resulting in a hole in the orbital population.However, the singlet state of the Pt19 cluster is suitable, due to the HOMO being singly, rather than doubly, degenerate.Additionally, triplet and other multiplet states are suitable in every case.However, due to RPA calculations becoming computationally intractable for the larger clusters, we limit ourselves to the Pt19 and Pt28 clusters.The adsorption energies for different clusters using a Gaussian basis (Figure S3.1) are presented in Table S3.1.The RPA adsorption energies for CH4/Pt19 are presented in Table S3.3.For these calculations, the HOMO-LUMO gap for Pt19, CH4/Pt19, Pt19//CH4/Pt19, and Pt19(CH4)//CH4/Pt19 were set to stated value, shifting all the virtual orbitals by the same amount as the LUMO, and then the RPA calculation was performed.The HOMO-LUMO gap for cluster calculations, with number of Pt atoms given, is shown in Tables S3. 4 for the singlet states.The HOMO-LUMO gap for cluster calculations against the number of Pt atoms is presented in   The hybrid adsorption energies for Pt28 in the quintet state are shown and broken down in Table S4.2.The carbon-platinum distance r(C-Pt) was then varied and then reoptimized on the PBE+MBD level with the C atom frozen in place.The adsorption energy using the hybrid structure for these methods is shown in Table S4.3 below.S4. 3.

Ethane
The carbon-platinum distance r(C-Pt) was then varied and then reoptimized on the PBE+MBD level with the C atom frozen in place.The adsorption energy using the hybrid structure for these methods is shown in Table S4.4 below.The potential energy curve for C2H6/Pt(111) using PBE:RPA and PBE+MBD:RPA are plotted in Figure S4.3, alongside the adsorption energies for C2H6/Pt19 using PBE, PBE+MBD, and RPA.S4. 4.
A comparison between the adsorption energies for ethane on Pt(111) using different clusters and spin states is presented in Table S4.5.-24.9 -24.7 -21.4 -20.0 The number of unoccupied bands used for the RPA calculations has, thus far, not been considered.We present these below in Table S4 a frozen orbitals for PW calculations are considered to be those described by a pseudopotential, hence why Ntotal = Nocc, active + Nvirt in these cases.

Figure S1. 1 .
Figure S1.1.Cluster models of the surface.Light, middle, and dark blue indicate atoms in the first, second, and third layer of the cluster, respectively.1 2 below.As the D2 and D3 dispersion corrections are post-SCF additive and not density-dependent, they are not impacted by spin polarisation besides the change in the PBE component.

Figure S3. 1 .
Figure S3.1.Adsorption energy plots for CH4 on Ptn clusters for different sized clusters in the singlet (blue) and lowest spin (red) states using Gaussian basis sets.Circles show the total

Figure 3 . 2 .
Figure 3.2.RPA adsorption energy (in kJ mol -1 ) against the HOMO-LUMO gap (in eV) for singlet CH4/Pt19.Points are tabulated in TableS3.3.The red line is the adsorption energy without any shifting of the HOMO-LUMO gap.

Figure S4. 1 .
Figure S4.1.RPA:PBE+D (D2, D3, dDsC) adsorption energy (in kJ mol -1 ) against Pt-C distance r(C-Pt) in pm for CH4/Pt(111).Red crosses are periodic RPA values; circles/ full lines are hybrid values; triangles/ dashed lines are periodic values.The experiment is shown by a dashed black line with grey error bars to indicate the range of experimental error; chemical accuracy, ±4 kJ mol -1 is shown by dashed darker grey lines.Points are tabulated in TableS4.3.

Figure S4. 2 .
Figure S4.2.Hybrid RPA:PBE and RPA:PBE+MBD adsorption energies (kJ mol -1 ) as function of the Pt-C distance, r(C-Pt), for CH4/Pt(111), along with breakdown of cluster terms.Red crosses are periodic RPA values; circles/ full lines are hybrid values; triangles/ dashed lines are periodic values; squares are cluster components (C).The experiment is shown by a dashed black line with grey error bars to indicate the range of experimental error; chemical accuracy, ±4 kJ mol -1 is shown by dashed darker grey lines.Points are tabulated in TableS4.3.

Figure S4. 3 .
Figure S4.3.RPA:PBE and RPA:PBE+MBD adsorption energy (in kJ mol -1 ) against Pt-C distance r(C-Pt) in pm for C2H6/Pt(111), along with breakdown of cluster terms.Circles/ full lines are hybrid values; triangles/ dashed lines are periodic values; squares are cluster components (c).The experiment is shown by a dashed black line with grey error bars to indicate the range of experimental error; chemical accuracy, ±4 kJ mol -1 is shown by dashed darker grey lines.Points are tabulated in Table S4.4.

Table S2 . 3 .
Adsorption energy, ΔEads (kJ mol -1 ), of CH4 on platinum clusters Ptn(A,B,C) for PBE with dispersion corrections (D2, D3, dDsC, and MBD).Where n denotes the total number of Pt atoms, A the number in the first layer, B the number in the second layer, and C the number in the third layer.The periodic calculation is for the 3-layered (2x2) cell with lateral interactions removed, pbcno lat.

Table S3 . 1 .
Adsorption energy ΔEads (in kJ mol -1 ) for CH4 on platinum clusters Ptn(A,B,C) in the singlet states and the lowest energy spin states using PBE/def2-QZVPP, and the plane wave (PW) value.Comparison is made against the Singlet state, so the non-spin-polarised PW values are given.The lowest energy spin state multiplicity (2S+1)low, Counterpoise-corrected (CPC) adsorption energies, and the Basis Set Superposition Error (BSSE) are given.Additionally, the values for the quintet state of Pt28 are shown.The adsorption energies for different spin states of Pt19 are given in TableS3.2 and shown in FigureS3.3.

Table S4 . 6 .
.6 for both VASP and TURBOMMOLE calculations.N.B.VASP, i.e. plane wave calculations, require far more virtual orbitals to converge the energy.Number of frozen and active occupied orbitals Nocc, virtual orbital Nvirt, and total orbitals Ntotal using VASP (a plane wave PW code) and TURBOMOLE (T'MOLE, a Linear Combination of Atomic Orbitals LCAO code) for CH4 and C2H6 on the Pt(111) surface; calculations in VASP used a (2x2) cell, and TURBOMOLE used a Pt19 or Pt28 cluster in the Singlet (S) or Quintet (Qu) states, respectively.