2.1. Hybrid QM:QM Calculations with Counterpoise Correction

Our hybrid QM:QM approach uses the subtractive scheme³⁸⁻⁴⁰ with periodic boundary conditions (pbc).³⁹ The hybrid energy E_HL:LL(pbc) is

where E_LL(pbc) is the total energy per cell of the periodic system calculated using the low-level method, and E_LL(C) and E_HL(C) are the low- and high-level energies of the finite cluster, respectively. The hybrid energy may be conceived as a high-level correction ΔHL(C) to the energy of the periodic system using a low-level method:

Alternatively, but entirely equivalently, the hybrid energy may be conceived as adding a low-level, long-range correction ΔLR(pbc, C) to the high-level energy of the cluster:

Since we employ an atomic orbital basis set for cluster calculations, there is a need to correct for the BSSE.⁴¹ The interaction energy between two monomers A and B is defined as

We correct for the BSSE by using the counterpoise correction (CPC) scheme:⁴¹

where “//” denotes “at the structure of”, meaning that the energies are computed at the structure of the A·B complex. The E(A{B}//A·B) and E({A}B//A·B) refer to the A and B entities, respectively, in the full basis of the A·B complex. The BSSE correction is then obtained by

The pbc calculations in our approach are carried out using a plane wave basis set, eliminating the need for BSSE corrections.