Author: Patryk Kozlowski

Mentor: Garnet K. Chan and Yang Gao

Editor: Hannah X. Chen

## Introduction

Due to the imminent crisis of climate change, developing sustainable methods for fuel and chemical production has become more important than ever. A significant step towards progress comes from the electrochemistry processes which use renewable energy to convert molecules from the atmosphere into higher-valued products. Catalysts are instrumental in such processes, increasing the rate, efficiency, and selectivity of the chemical transformations involved^{1}.

One notable example is iron, which catalyzes the conversion of atmospheric nitrogen into ammonia (used in artificial fertilizers) in the Haber-Bosch process. In fact, the 2007 Nobel Prize in Chemistry was awarded to physicist Gerhard Ertl “for his studies of chemical processes on solid surfaces” in relation to the Haber-Bosch process^{2}. Clearly, understanding the chemistry occurring at catalyst surfaces will be crucial for the future development of enhanced catalysts. In particular, we need to be able to accurately compute surface energies – quantities that govern the chemistry at catalyst surfaces.

### The Catalytic Mechanism

Compared to atoms that are part of the bulk material, atoms on a surface are unstable due to their relative lack of stabilizing intermolecular attractions **(Fig. 1a)**. This inherent instability can cause surface atoms of a catalyst to interact with nearby gas molecules **(Fig. 1b)**. Once adsorbed to the catalyst surface, reactants undergo a desired chemical transformation and are then desorbed from the surface, at which point the job of the catalyst is complete.

### Theoretical Approaches

The surface energy measures the instability of a surface relative to the bulk material. Since surface energies thereby give insight into the adsorption process, we would really like to be able to compute them accurately using theory. Though density functional theory (DFT), the current workhorse computational method of surface science, has proven inadequate to this end, wavefunction-based methods that treat correlations between electrons have shown promise as a viable alternative^{5}.

Coupled-cluster theory (CC), established as the “gold standard” of quantum chemistry at the molecular level, is one such framework which has been used to study materials^{6–10}. CCSD methods (like coupled cluster theory including perturbative singles and doubles) have shown increased accuracy with respect to DFT in computing the properties of various materials^{6,11}.

However, CCSD has only recently been used to study surface chemistry, which is relevant to catalysis. In this work, I compare the performance of the simpler MP2 and more robust CCSD perturbative frameworks with that of DFT for computing energies of the (111) surface of platinum, a common catalyst.

## Computational Methodology

### Achieving a Gapped Platinum System

Now, since platinum is a gapless metal, its highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are degenerate in energy. This degeneracy was problematic for the perturbative framework I wanted to use, but one can numerically introduce a twisted periodic boundary condition to artificially “open a gap” in the platinum system. Thus, in my simulations I sampled the reciprocal space of platinum’s primitive cell to identify the ideal k-point mesh yielding the largest gap. I then used this k-point mesh in all proceeding calculations.

### Lattice Spacing Optimization

We optimized the bulk platinum lattice spacings computationally, yielding fair agreement with the benchmark experimental value^{14} **(Table 1)**. The minimal gth-szv and the more robust gth-dzvp basis sets overestimated and underestimated the bulk platinum lattice constant, respectively. A basis set is a set of functions representing the electronic wavefunction in DFT so that the partial differential equations of the model can be implemented efficiently on a computer.

### Computing Surface Energies

We then calculated platinum (111) surface energies according to **Equation 1**, with *E _{bulk}* and

*E*, respectively the bulk platinum and platinum surface slab energies, computed using the Table 1 bulk-optimized lattice constants.

_{slab}*N*is the size of the simulated system.

**Results and Discussion**

### Finite Size Effects

Material surfaces are effectively infinite in size. To compute their properties, we make two approximations: the size of the k-point mesh and the number of layers perpendicular to the surface slab sampled must be finite. We then consider the effects of this finite size on computed surface energies by independently varying the number of k-points and surface slab layers sampled in calculations of platinum (111) surface energies.

As expected, we observed that surface energies steadily approach a thermodynamic limit value as both more k-points **(Fig. 4ab, 6)** and more slab layers **(Fig. 5ab)** are sampled. We see this from the observed oscillatory convergence in the former case and monotonic convergence in the latter. Furthermore, as expected, a comparison of Figures 4a and 4b shows that surface energies approached in the thermodynamic limit by the more robust gth-dzvp basis are more accurate than those approached by the minimal gth-szv basis.

### DFT Performance

It is worth mentioning that we corroborated a trend observed in the literature of DFT’s systemic overestimation of surface stability with respect to experiment^{5}. That is, **Fig. 4ab** show that DFT consistently yields lower surface energies with respect to experiment when approaching the thermodynamic limit of more k-points sampled.

### MP2 Performance

The perturbative MP2 method struggled with the still near energetic degeneracy of platinum’s HOMO and LUMO discussed previously. This is because the 2^{nd} order energetic perturbative correction, the component of the MP2 energy aiming to capture correlations between electrons, contains the difference between molecular orbital energies *ε _{i}*,

*ε*,

_{j}*ε*, and

_{a}*ε*in its denominator

_{b}**(Equation 2)**.

The near energetic degeneracy of the HOMO and LUMO makes this denominator go to zero, causing the MP2 energy to go to infinity in a physically unrealistic manner.

We speculate that this degeneracy caused MP2’s poor surface energies relative to DFT, evidenced by a comparison of the results from **Fig. 6** and **4b**, respectively, at the robust basis set (gth-dzvp) and the largest comparable reciprocal space sampling (9 k-points).

### CCSD Performance

The CCSD method performed poorly in computing platinum surface energies. Even an assessment of the accuracy of CCSD surface energies was inhibited by trouble with getting CCSD calculations, which rely on an iterative method, to converge numerically. CCSD, which treats correlations between electrons based on the perturbative CC ansatz, struggled to converge numerically due to the still near energetic degeneracy of platinum’s HOMO and LUMO.

We obtained CCSD convergence only using a minimal 1 k-point sampling, though results from such a minute k-point sampling are untrustworthy, customarily yielding surface energies over 10 J/m^{2} off the experimental value **(Fig. 5, 7)**. Additionally, the CCSD method has a steep scaling in computational cost of O(N^{6}), where N is the size of the simulated system. Thus, computations performed at the robust basis set (gth-dzvp) and modest reciprocal space samplings (16 k-points or more) required for obtaining accurate platinum surface energies proved to be prohibitively expensive.

## Conclusions

This study confirmed the inadequacy of DFT for computing accurate surface energies as identified in the literature. Furthermore, we found that even for a maximally gapped platinum system, the perturbative MP2 and CCSD frameworks are unable to compute accurate surface energies. Platinum’s metallic character causes the former method to yield surface energies less accurate than DFT and plagues the latter method with numerical convergence issues. Ultimately, these results show the ineffectiveness of perturbative methods for computing surface energies of metals, like platinum.

A natural direction for future work will be to assess the performance of perturbative methods, including CCSD, for computing surface energies of non-metal catalysts, in which sufficiently large HOMO/LUMO gaps exist. We would also seek to compute other quantities that help elucidate catalytic mechanisms, such as the adsorption energy.

**Acknowledgements**

**Acknowledgements**

I would like to thank Yang Gao for guiding me through my project on a daily basis in this difficult remote setting and Professor Chan for finding time to discuss my project each week throughout the summer. Lastly, I would like to sincerely thank the John Stauffer Charitable Trust for funding my SURF this summer, enabling me to get deeply involved in research I am passionate about.

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## Further Reading

[1] A brief conceptual introduction to catalysis can be found here.

[2] An introduction to basic electronic structure theory of materials can be found in Chapters 1-4 of this reference.

[3] An introduction to coupled-cluster theory and a review of its recent application to materials science is found here.