Efficient implementation of the three-dimensional reference interaction site model method in the fragment molecular orbital method

The three-dimensional reference interaction site model (3D-RISM) method was efficiently implemented in the fragment molecular orbital (FMO) method. The method is referred to as the FMO/3D-RISM method, and allows us to treat electronic structure of the whole of a macromolecule, such as a protein, as well as the solvent distribution around a solute macromolecule. The formalism of the FMO/3D-RISM method, for the computationally available form and variational expressions, are proposed in detail. A major concern leading to the implementation of the method was decreasing the computational costs involved in calculating the electrostatic potential, because the electrostatic potential is calculated on numerous grid points in three-dimensional real space in the 3D-RISM method. In this article, we propose a procedure for decreasing the computational costs involved in calculating the electrostatic potential in the FMO method framework. The strategy involved in this procedure is to evaluate the electrostatic potential and the solvated Fock matrix in different manners, depending on the distance between the solute and the solvent. The electrostatic potential is evaluated directly in the vicinity of the solute molecule by integrating the molecular orbitals of monomer fragments of the solute molecule, whereas the electrostatic potential is described as the sum of multipole interactions when an analog of the fast multipole method is used. The efficiency of our method was demonstrated by applying it to a water trimer system and three biomolecular systems. The FMO/3D-RISM calculation can be performed within a reasonable computational time, retaining the accuracy of some physical properties.