The importance of three-body terms in the fragment molecular orbital method

A previously proposed two-body fragment molecular orbital method based on the restricted Hartree–Fock (RHF) method was extended to include explicit three-body terms. The accuracy of the method was tested on a set of representative molecules: $(H2O)n,$ $n=16,$ 32, and 64, as well as α and β n-mers of alanine, $n=10,$ 20, and 40, using STO-3G, 3-21G, 6-31G, and $6-31++G**$ basis sets. Two- and three-body results are presented separately for assigning one and two molecules (or residues) per fragment. Total energies are found to differ from the regular RHF method by at most $ΔE2/1=0.06,$ $ΔE2/2=0.04,$ $ΔE3/1=0.02,$ and $ΔE3/2=0.003 (a.u.);$ rms energy gradients differ by at most $ΔG2/1=0.0015,$ $ΔG2/2=0.000 75,$ $ΔG3/1=0.000 20,$ and $ΔG3/2=0.000 10 (a.u./bohr),$ and rms dipole moments are reproduced with at most $δD2/1=3.7,$ $δD2/2=3.4,$ $δD3/1=2.6,$ and $δD3/2=3.1$ (%) relative error, where the subscript notation $n/m$ refers to the n-body method based on m molecules (residues) per fragment. A few of the largest three-body calculations were performed with a separated trimer approximation, which presumably somewhat lowered the accuracy of mostly dipole moments which are very sensitive to slight variations in the density distribution. The proposed method is capable of providing sufficient chemical accuracy while providing detailed information on many-body interactions.