**2.2.6 Model Core Potential (MCP)**

Heavy metal ions play major roles in various biological systems and functional materials. Therefore, it is important to understand the fundamental chemical nature and dynamics of the metal ions under physiological or experimental conditions. Each heavy metal element has a large number of electrons to which relativistic effects must be taken into account, however. Hence, the heavy metal ions increase the computation cost of high-level electronic structure theories. A way to reduce the computation is the Model Core Potential (MCP; Sakai *et al.*, 1987; Miyoshi *et al.*, 2005; Osanai *et al.*, 2008ab; Mori *et al.*, 2009), where the proper nodal structures of valence shell orbitals can be maintained by the projection operator technique. In the MCP scheme, only valence electrons are considered, and core electrons are replaced with 1-electron relativistic pseudo-potentials to decrease computational costs. The MCP method has been combined with FMO and implemented in ABINIT-MP (Ishikawa *et al*., 2006), which has been used in the comparative MCP/FMO-MD simulations of hydrated *cis*-platin and *trans*-platin (see subsection 3.6). Very recently, the 4f-in-core type MCP set for trilvalent lanthanides has been developed and made available (Fujiwara *et al*., 2011).

#### **2.2.7 Periodic Boundary Condition (PBC)**

PBC was finally introduced to FMO-MD in the TINKER/ABINIT-MP system by Fujita *et al.* (2011). PBC is a standard protocol for both classical and *ab intio* MD simulations.

Recent Advances in Fragment Molecular

Presently, PEACH has four fragmentation modes, as follows:

Mode 0: Use the fragmentation data in the input file throughout the simulation.

Usually, Mode 1 is enough, but Mode 2 or 3 sometimes become necessary.

threshold parameters.

forming an H-bond.

solvent molecules.

fragment.

Orbital-Based Molecular Dynamics (FMO-MD) Simulations 9

The DF algorithm was generalized later to handle arbitrary molecular systems (Komeiji *et al.*, 2010). The algorithm requires each atom's van der Waals radius and instantaneous 3D coordinate, atomic composition and net charge of possible fragment species, and certain

Mode 1: Merge covalently connected atoms, namely, those constituting a molecule, into a

Mode 2: Fragments produced by Mode 1 are unified into a larger fragment if they are

Mode 3: Fragments produced by Mode 2 are unified if they are an ion and coordinating

The modes are further explained as follows. Heavy atoms located significantly close to each other are united as a fragment, and each H atom is assigned to its closest heavy atom (Mode 1). Then, two fragments sharing an H atom are unified (Mode 2). Finally, an ion and surrounding molecules are united (Mode 3). See Figure 2 for typical examples of DF.

Fig. 2. Typical examples of fragment species generated by the generalized DF scheme. Expected fragmentation patterns are drawn for three solute molecules, A–C. Reproduced

from Komeiji *et al.* (2010) with permission.

Nonetheless, partly due to the complexity of PBC in formulation but mostly due to its computation cost, FMO-MD simulations reported in the literature had been performed under a free boundary condition, usually with a cluster solvent model restrained by a harmonic spherical potential. This spherical boundary has the disadvantage of exposing the simulated molecular system to a vacuum condition and altering the electronic structure of the outer surface (Komeiji *et al.*, 2007). Hence, PBC is expected to avoid the disadvantage and to extend FMO-MD to simulations of bulk solvent and crystals. For PBC simulations to be practical, efficient approximations in evaluating the ESP matrix elements will need to be developed. A technique of multipole expansion may be worth considering.
