In the context of water potentials, V 1-b in Eq. 5 is the gas-phase water monomer, 50 V 2-b is the intrinsic interaction between two water molecules, V 3-b is the intrinsic interaction between three water molecules, and V 4-b is the intrinsic interaction between four water molecules, etc. For the recent q-AQUA, q-AQUA-pol and MB-pol(2023) This approach has also been used for heterogeneous systems, for example clathrate hydrates with different guest molecules such as H 2 , 10 CH 4 , 52 and CO 2 , 53 and notably the hydrated proton. The many-body expansion for covalent interactions for a molecule is less common in the recent literature; however, there a history using this approach for triatomic and tetraatomic molecules, as developed in the 1980s by Murrell and co-workers. A recent example of this approach, relevant to the present work on alkanes, was reported by Paesani and co-workers. 9