**2. Methodology of FMO-MD**

FMO-MD is based on the Born-Oppenheimer approximation, in which the motion of the electrons and that of the nuclei are separated (Fig. 1). In FMO-MD, the electronic state is solved quantum mechanically by FMO using the instantaneous 3D coordinates of the nuclei (**r**) to obtain the energy (*E*) and force (**F**, minus the energy gradient) acting on each nucleus, which are then used to update **r** classical mechanically by MD. In the following subsections, software systems for FMO-MD are described, and then the FMO and MD aspects of the FMO-MD methodology are explained separately.

#### **2.1 Software systems for FMO-MD**

FMO-MD can be implemented by using a combination of two independent programs, one for FMO and the other for MD. Most of the simulations presented in this article were

Recent Advances in Fragment Molecular

**2.2 FMO** 

*O*(*N*2).

ABINIT-MP is scheduled to be completed within 2012.

Several simulations with these systems are also presented.

the history of the general fragment methods.

calculated using the following formulae:

**2.2.1 Hartree-Fock (HF)**

Orbital-Based Molecular Dynamics (FMO-MD) Simulations 5

implementing FMO-MD directly in the ABINIT-MP program. This working version of

Though not faultless, the PEACH/ABINIT-MP system has produced most of the important FMO-MD simulations performed thus far, which will be presented in this article. Besides the PEACH/ABINIT-MP system, a few FMO-MD software systems have been reported in the literature, some using ABINIT-MP (Ishimoto *et al.*, 2004, 2005; Fujita *et al.*, 2009, 2011) and others GAMESS (Fedorov *et al.*, 2004a; Nagata *et al.*, 2010, 2011c; Fujiwara *et al.*, 2010a).

FMO, the essential constituent of FMO-MD, is an approximate *ab initio* MO method (Kitaura *et al.*, 1999). FMO scales to *N*1-2, is easy to parallelize, and retains chemical accuracy during these processes. A vast number of papers have been published on the FMO methodology, but here we review mainly those closely related to FMO-MD. To be more specific, those on the FMO energy gradient, Energy Minimization (EM, or geometry optimization), and MD are preferentially selected in the reference list. Thus, those readers interested in FMO itself are referred to Fedorov & Kitaura (2007b, 2009) for comprehensive reviews of FMO. Also, one can find an extensive review of fragment methods in Gordon *et al.* (2011), where FMO is re-evaluated in the context of its place in

We describe the formulation and algorithm for the HF level calculation with 2-body expansion (FMO2), the very fundamental of the FMO methodology (Kitaura *et al.*, 1999).

First, the molecular system of interest is divided into *Nf* fragments. Second, the initial electron density, *ρI*(*r*), is estimated with a lower-level MO method, e.g., extended Hückel, for all the fragments. Third, self-consistent field (SCF) energy, *EI,* is calculated for each fragment monomer while considering the electrostatic environment. The SCF calculation is repeated until all *ρI*(*r*)'s are mutually converged. This procedure is called the self-consistent charge (SCC) loop. At the end of the SCC loop, monomer electron density *ρI*(**r**) and energy *EI* are obtained. Finally, an SCF calculation is performed once for each fragment pair to obtain dimer electron density *ρIJ*(**r**) and energy *EIJ*. Total electron density *ρ*(**r**) and energy *E* are

> ( 2) *IJ <sup>f</sup> <sup>I</sup> I J I*

In calculation of the dimer terms, electrostatic interactions between distant pairs are approximiated by simple Coulombic interactions (dimer-ES approximation, Nakano *et al.*, 2002). This approximation is mandatory to reduce the computation cost from *O*(*N*4) to

( 2) *IJ <sup>f</sup> <sup>I</sup> I J I E EN E*

 

**rr r** (1)

. (2)

*N*

Below, subscripts *I, J, K*... denote fragments, while *i*, *j*, *k*,... denote atomic nuclei.

Fig. 1. Schematics of the FMO-MD method exemplified by an ion solvation with four water molecules. The atomic nuclei are represented by black circles (the large one for the ion, medium ones for Oxygens, and small ones for Hydrogens) and the electron cloud by a grey shadow. The electronic structure is calculated by FMO to give force (**F**) and energy (*E*), which are then used to update the 3D coordinates of nuclei (**r**) by MD, i.e., by solving the classical equation of motion.

performed by the PEACH/ABINIT-MP software system composed of the PEACH MD program (Komeiji *et al*., 1997) and the ABINIT-MP 1 (F)MO program (Nakano *et al*., 2000). We have revised the system several times (Komeiji *et al*., 2004, 2009a), but here we describe the latest system, which has not yet been published. In the latest system, the PEACH program prepares the ABINIT-MP input file containing the list of fragments and 3D atomic coordinates, executes an intermediate shell script to run ABINIT-MP, receives the resultant FMO energy and force, and updates the coordinates by the velocity-Verlet integration algorithm. This procedure is repeated for a given number of time steps.

The above implementation of FMO-MD, referred to as the PEACH/ABINIT-MP system, has both advantages and disadvantages. The most important advantage is the convenience for the software developers; both FMO and MD programmers can modify their programs independently from each other. Also, if one wants to add a new function of MD, one can first write and debug the MD program against an inexpensive classical force field simulation and then transfer the function to FMO-MD, a costly *ab initio* MD. Nonetheless, the PEACH/ABINIT-MP system has several practical disadvantages as well, mostly related to the use of the systemcall command to connect the two programs. For example, frequent invoking of ABINIT-MP from PEACH sometimes causes a system error that leads to an abrupt end of simulations. Furthermore, use of the systemcall command is prohibited in many supercomputing facilities. To overcome these disadvantages, we are currently

<sup>1</sup> Our developers' version of ABINIT-MP is named ABINIT-MPX, but it is referred to as ABINIT-MP throughout this article.

implementing FMO-MD directly in the ABINIT-MP program. This working version of ABINIT-MP is scheduled to be completed within 2012.

Though not faultless, the PEACH/ABINIT-MP system has produced most of the important FMO-MD simulations performed thus far, which will be presented in this article. Besides the PEACH/ABINIT-MP system, a few FMO-MD software systems have been reported in the literature, some using ABINIT-MP (Ishimoto *et al.*, 2004, 2005; Fujita *et al.*, 2009, 2011) and others GAMESS (Fedorov *et al.*, 2004a; Nagata *et al.*, 2010, 2011c; Fujiwara *et al.*, 2010a). Several simulations with these systems are also presented.
