Applying the Coupled-Cluster Ansatz to Solids and Surfaces in the Thermodynamic Limit

Modern electronic structure theories can predict and simulate a wealth of phenomena in surface science and solid-state physics. In order to allow for a direct comparison with experiment, such ab initio predictions have to be made in the thermodynamic limit, substantially increasing the computational cost of many-electron wave-function theories. Here, we present a method that achieves thermodynamic limit results for solids and surfaces using the “gold standard” coupled cluster ansatz of quantum chemistry with unprecedented efficiency. We study the energy difference between carbon diamond and graphite crystals, adsorption energies of water on $h$-BN, as well as the cohesive energy of the Ne solid, demonstrating the increased efficiency and accuracy of coupled cluster theory for solids and surfaces.

Popular Summary

All physical properties and chemical processes in solids, molecules, and atoms can, in principle, be calculated from first principles. However, in practice, any theory that aims at solving the complex interactions among the many electrons in such systems has to make a trade-off between accuracy and computational cost. Properties of solids and surfaces need to be calculated using models that contain more than a hundred atoms, as well as periodic boundary conditions. For this reason, coupled-cluster theories—based on an explicit description for the many-electron wave function and, hence, potentially very accurate—have not yet been fully explored for solid-state materials. Here, we report a simple yet robust method to predict properties of real materials using coupled-cluster theories with unprecedented computational efficiency.

We introduce corrections to the coupled-cluster approach that allow for calculating thermodynamic-limit properties of solids and surfaces using models that contain only a few tens of atoms. We demonstrate the increased efficiency and accuracy by studying the energy difference between carbon diamond and graphite crystals and adsorption energies of water on the hexagonal form of the crystal boron nitride ($h$-BN).

Our work paves the way for the routine use of accurate coupled-cluster theories in the field of surface science and solid-state physics. Future work will greatly benefit from the ability to produce accurate benchmark results that can be used by the entire community of electronic-structure theorists to further improve upon computationally efficient theories and to more reliably interpret experimental findings.

Figure 1

Partial derivative of correlation energy with respect to the Coulomb kernel for graphite. Slices and isosurface of $S(\mathbf{G})$ for carbon graphite in the ABC stacking. Darker colors indicate more negative values. White corresponds to zero. The blue isosurface is a hexagonally shaped torus, and it reflects the anisotropic shape for $S(\mathbf{G})$. Black dots represent the sampling points using a $4\times 4\times 4$ $k$-point mesh.

Figure 2

Convergence of the energy difference between the carbon diamond and graphite using CCSD and 2 atomic unit cells with respect to the number of natural orbitals used. The (T) correlation energy contribution has been added to the CCSD energy using 16 natural orbitals per atom.

Figure 3

Energy difference between carbon diamond and graphite phases. The dotted line represents the linear fit of the uncorrected values with shifts. (T) corrections (black) are on top of CCSD-TA-FS energy obtained using a $4\times 4\times 4$ $k$-point mesh (red). Zero-point energies are included, and they stabilize graphite compared to diamond by $9\mathrm{meV}/\mathrm{atom}$.

Figure 4

Adsorption energy of a single water molecule on an $h$-BN sheet. The energies are retrieved as a function of atoms in the sheet using MP2, $\mathrm{RPA}+\mathrm{SOSEX}$, CCSD, and CCSD(T) theory. FS indicates that finite-size corrections are included. The results have been obtained using pseudized aug-cc-pVTZ basis sets and corrected for basis-set superposition errors using counterpoise corrections.