Density Functional Theory (DFT) is the prevalent method for materials simulations, but its accuracy is limited by the choice of exchange-correlation functional. Coupled cluster theory, particularly the single, double, and perturbative triple excitation variant (CCSD(T)), is considered the gold standard in quantum chemistry due to its high accuracy but suffers from extremely high computational cost, making its application to materials simulations, especially at finite temperatures, extremely challenging. Finite-temperature properties necessitate molecular dynamics (MD) simulations or Monte Carlo (MC) sampling, greatly amplifying the computational burden. For instance, a direct CCSD(T) calculation of the enthalpy of adsorption in the system studied here would take an impractical amount of time. Machine learning (ML) offers a solution by fitting interatomic potentials to data generated from fewer, high-accuracy calculations. Previous ML-assisted coupled cluster MD simulations have been limited to small molecular systems. This work overcomes these limitations by combining advanced coupled cluster implementations with ML and thermodynamic perturbation theory (TPT) to enable accurate finite-temperature CCSD(T) simulations of periodic solid materials. The specific application chosen is the calculation of CO2 adsorption enthalpy in protonated chabazite zeolite, a system of significant importance in various applications such as depollution and catalysis.

The authors reviewed existing literature on coupled cluster theory for periodic systems, highlighting the computational challenges associated with its application at finite temperatures. They discuss previous work on combining ML with MP2 and coupled cluster methods for molecular systems and condensed phases (such as liquid water), noting the scarcity of applications to periodic solids at finite temperatures. The authors also reference existing work on machine learning potentials, emphasizing their efficiency in accelerating molecular dynamics simulations and the use of techniques like the smooth overlap of atomic positions (SOAP) kernel for efficient representation of atomic environments. The literature review underscores the novelty of their approach in applying CCSD(T) to large periodic systems at finite temperatures and the need for efficient methods to address the high computational cost.

The methodology combines three key components: 1) an efficient periodic coupled cluster implementation that incorporates techniques to accelerate convergence; 2) a machine learning model based on kernel ridge regression with a SOAP kernel; and 3) thermodynamic perturbation theory (TPT) or, alternatively, Monte Carlo (MC) sampling. The process begins with ab initio molecular dynamics (AIMD) simulations at a lower-cost DFT level (PBE + D2), generating a trajectory of configurations. A subset of these configurations is then used for high-level (MP2 and CCSD(T)) single-point energy calculations. These calculations are used to train the ML model, which then predicts energies for the remaining configurations in the trajectory. The TPT approach reweights the DFT-level statistics to obtain the target (MP2 or CCSD(T)) level statistics. An alternative approach, MLMC, uses the trained ML model to directly perform MC sampling at the CCSD(T) level, avoiding the reliance on the DFT-level trajectory. This eliminates potential bias from insufficient overlap between the DFT and CCSD(T) configurational spaces. The overlap is analyzed using the Iw index, which assesses the similarity between the DFT and post-HF configurational spaces. Radial distribution functions are computed to further evaluate the reliability of the TPT approach, focusing on crucial structural parameters like Si-O distances in the zeolite framework. The adsorption enthalpy is then calculated using standard thermodynamic relations. The machine learning model is designed to predict the energy difference between the high-level method and the lower-level DFT, making training more efficient. The authors also discussed the limitations of a simplified static correction approach compared to their MLPT and MLMC approaches.

The study successfully computed the enthalpy of adsorption of CO2 in protonated chabazite using CCSD(T), achieving an accuracy comparable to experimental values. Table 1 presents the enthalpy of adsorption calculated using different methods and sampling techniques. The MLPT approach, using CCSD(T) as the target method, yields an enthalpy of adsorption of -8.32 ± 0.28 kcal/mol, which agrees well with the experimental value of -8.41 kcal/mol. The MLMC approach, which avoids the biases of TPT, gives a very similar result of -8.09 ± 0.71 kcal/mol. In contrast, the DFT (PBE + D2) method gives a significantly different value (-9.72 ± 0.27 kcal/mol), showcasing the improvement achieved using the ML-assisted CCSD(T) method. The analysis of the Iw index and radial distribution functions confirms the reliability of the employed TPT and MC approaches and the suitability of the DFT trajectory for reweighting. Figure 2 shows a t-SNE visualization of the configurational spaces sampled by DFT and CCSD(T), demonstrating a reasonable overlap. Figure 3 compares radial distribution functions for Si-O pairs, indicating similar structural properties at different theoretical levels, further supporting the validity of the perturbation theory approach. The authors demonstrated that a simplified static energy correction approach, while sometimes yielding reasonable results, fails to capture the full complexity of the energy surface and can lead to misleading conclusions. The CCSD(T) calculations were performed using the CC4s code interfaced with VASP.

The results demonstrate the feasibility and accuracy of using ML-accelerated CCSD(T) methods to calculate finite-temperature properties of periodic materials. The excellent agreement between the calculated and experimental enthalpy of adsorption validates the methodology, demonstrating its capability to predict accurate thermodynamic observables. The use of ML and TPT/MC significantly reduces the computational cost, making high-level calculations for complex periodic systems computationally tractable. The analysis of configurational space overlap and radial distribution functions confirms the reliability of the perturbation theory and MC sampling approaches. The paper highlights the limitations of simpler, static correction approaches, emphasizing the importance of the sophisticated reweighting schemes employed. The success of this work paves the way for applying highly accurate post-Hartree-Fock methods to a wider range of problems in materials science.

This study successfully demonstrates the application of CCSD(T) to compute the enthalpy of adsorption in a periodic material at finite temperatures using a combination of efficient coupled cluster methods and machine learning techniques. The excellent agreement with experimental data showcases the accuracy and computational efficiency of the developed methodology. Future research can expand the application of this approach to other challenging problems, such as calculating free energies of activation for catalytic reactions, which is crucial for understanding reaction mechanisms and designing new catalysts.

The current methodology is still computationally more expensive than standard DFT calculations. While significantly reducing the computational cost of CCSD(T) calculations, the approach still requires a considerable number of high-level calculations for training the machine learning model. The accuracy of the ML model relies on the quality of the DFT-level trajectory, and potential biases could arise if the DFT and CCSD(T) configurational spaces have insufficient overlap. However, the authors address this issue by employing both MLPT and MLMC methods and by analyzing the overlap using the Iw index and radial distribution functions.