Development of efficient computational techniques and codes forsecond-order Møller–Plesset perturbation calculation of extended
systems
Michio Katouda 11Introduction

Electronic structure theory such as ab initio molecular orbital(MO) theory is the powerful tool in elucidating chemical
phenomena such as electronic states, molecular structures,properties, and reaction mechanisms. The high-level quantum
chemical calculations excellently reproduce the properties forsmall molecules in the same or better accuracy with theexperiments. However, the computational costs drasticallyincrease with respect to the size of molecules when the highlevel quantum chemical calculations are performed. To elucidate
chemical phenomena of nano molecules and biologicalmolecules, the development of fast and robust theories andcomputational techniques of the ab initio quantum chemistry isdesired. Especially, non-covalent interactions such as van der
Waals forces play important roles in the chemical phenomena ofsuch systems. Those interactions come from the electroncorrelations. The generally accepted density functional theory(DFT) functionals fail to describe non-covalent interactions
because they suffer from self-interaction problems and do notincorporate the long-range correlation effect.
Therefore, the electron correlation theory based on the ab initio
MO theory is indispensable for the robust reproduction of noncovalent interactions. Among these theories, second-order
Møller–Plesset perturbation theory (MP2) [1] is the simplest
method to account for the electron correlations at the ab initio
level and often used for the practical chemical applications.
However, even the computational cost of MP2 calculations
scales O(N5) with respect to the size of molecules (N), andpractical applications are limited to molecules of moderate size.
To make the MP2 calculations applicable to the extended
systems such as nano molecules and biological molecules,
development of efficient computational techniques is desired.
In this study, we present an efficient parallel resolution-ofidentity (RI) MP2 (RI-MP2) algorithm and fragmentation based
O(N) parallel RI-MP2 methods. These methods are suitable forthe massively parallel computations on the supercomputers suchas “K computer”.
2of the massively parallel computations. In order to use morelarge number of CPU cores with the efficient load balancing, wehave changed to use the virtual orbital pairs for the parallel task
distribution. Generally, the number of virtual orbitals is fourtimes larger than the number of occupied orbitals, and the loadvalancing is improved from the original algorithm. We have
implemented the modified version of parallel RI-MP2 algorithm
into GAMESS [4] program and NTChem program. In these
implementations, we have performed MPI/Open-MP hybrid
parallelization for the efficient usage of the memories and thenetwork devices in the multi-core architectures.
Figure 1 presents the speedups of parallel RI-MP2 calculations.
The speedups scale almost liner with respect to the number ofCPU cores. These results demonstrate the efficiency of ourparallel algorithm and implementations. Using the parallel RIMP2 codes developed by us, MP2 calculations of large
molecules can be performed in modest times with massively
parallel supercomputers and low-cost personal computer (PC)clusters.

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Parallel RI-MP2 Algorithm

We have developed a parallel RI-MP2 algorithm for large
molecules [2] and a RI-MP2 method for periodic systems [3].
These methods are based on the RI approximation of four-centre
Coulomb integrals, and highly reduce the computational costs.
The parallel RI-MP2 algorithm is designed for the efficientparallel calculations by reducing the I/O and network
communication overheads and considering the uniform task
distribution for the efficient load balancing.
Recently, we have modified our parallel RI-MP2 algorithm fromoriginal one. In the original algorithm, the occupied orbital pairsare distributed to processors for the calculation of four-centre
integrals. However, the number of occupied orbital pairs issmall, and this makes the load balancing problems in the cases

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Figure. 1. Speedups of parallel RI-MP2/cc-pVDZ
calculations of valinomycin and bucky-catcher C60@C60H28.
Parallel RI-MP2 calculations were performed on the Xeon
X5670 cluster connected with InfiniBand.
3O(N) RI-MP2 methods for Extended Systems

We have developed efficient implementations that are suitable
for the massively parallel computations on the supercomputers.
However, even for employing RI-MP2 method, thecomputational costs of these codes are still O(N5) with respect tothe size of molecules N, and its practical applications are limited
to molecules of moderate size. To overcome this limitation of

computational scaling, we have applied the parallel RI-MP2
codes to the fragment molecular orbital (FMO) method [5], thedivided-and-conquer (DC) method [6], and the moleculartailoring approach (MTA) [7]. In these methods, a targetmolecule is divided spatially into small fragments and then theenergies and properties are obtained by accumulating
contributions of fragmented subsystems. The application of RIMP2 to those fragment based linear scaling methods
considerably speeds up MP2 calculations because thecomputational scaling and prefactor of RI-MP2 are smaller thanthose of conventional MP2.
3.1

RI-MP2 with FMO Method (FMO-RI-MP2)

The FMO method is efficient approach for the rapid ab initio
quantum chemical calculations of large biological moleculesand nanomolecules [8,9]. FMO reduces the computational costs
considerably by dividing the target system into fragmented
subsystems and performing ab initio quantum chemical
calculations for each monomer and dimer (and trimer, ifnecessary) of fragments. FMO can treat the electron correlation
successfully in the MP2 level (FMO-MP2). This opens up many
practical chemical applications of FMO to biological molecules
such as protein–ligand bindings [8,9].
We have developed a new implementation of FMO-RI-MP2
based on the GAMESS version of parallel RI-MP2 program
mentioned in Section 2. In this implementation, we haveextended FMO-RI-MP2 to treat the three-body FMO method(FMO3-RI-MP2) as well as the two-body FMO method (FMO2RI-MP2).
FMO-RI-MP2 considerably speeds up FMO-MP2 calculationsof biological molecules without sacrificing chemical accuracy.
Table 1 presents the computation times and errors of totalenergy of Trp-cage protein (PDB ID: 1L2Y). FMO2-RI-MP2
and FMO3-RI-MP2 calculations of 1L2Y protein were finishedwithin 0.5 and 8.2 h, respectively, whereas FMO2-MP2 andFMO3-MP2 calculations took 1.1 and 23.9 h, respectively. Theerrors of FMO2-RI-MP2 and FMO3-RI-MP2 are at most 3,745
and 0.104 mHartree, respectively.
Recently, we have ported our FMO-RI-MP2 code to K computerand have performed MPI/Open-MP hybrid parallelization. The
code was tested for an FMO-RI-MP2 calculation of real protein
on K computer up to 86016 CPU cores. The FMO-RI-MP2 code
has been supplied as a library program in K computer [10].
Table 1. MP2 energies, errors of MP2 energy, and wall times of FMOMP2/6-31G* and MP2/6-31G* calculations of 1L2Y protein.
Energy [H]Error [mH]Time [h]FMO2RI-MP2-7461.560235
3.745
0.5
MP2-7461.560450
3.530
1.1
FMO3
RI-MP2-7461.563877
0.104
8.2
MP2-7461.564100-0.12023.9
Full
RI-MP2-7461.563758
0.222
33.8
MP2-7461.563980
59.2*cc-pVTZ auxiliary basis set was employed for RI-MP2 calculations.
Calculations were performed on 32 nodes of Pentium 4 640 PC cluster
connected with the gigabit ethernet.

3.2

RI-MP2 with DC Method (DC-RI-MP2)

The MP2 method based on the DC approach (DC-MP2) is oneof the efficient methods for the calculation of large
nanomolecules [11]. In DC-MP2, the total system is divided into
small nonoverlapping subsystems called central regions. Some
atoms that are located around a central region are added to thesubsystems to account for the environmental effects that are
missing by the division of the total system. The MP2 correlationenergy of a total system is obtained as a sum of the MP2correlation energies of subsystems, which are evaluatedexploiting the localized molecular orbitals provided by the DC
Hartree–Fock (HF) calculations.
Recently, we have developed a new implementation of DC-RIMP2 code based on the GAMESS version of parallel RI-MP2
program mentioned in Section 2. In this implementation, we
also performed multilevel parallelization of DC-MP2, which issuitable for the calculations of extended systems on themassively parallel supercomputers [6]. The parallelization
scheme is a combination of the coarse-grain parallelization
assigning each subsystem to a group of processors and the finegrain parallelization where the computational tasks forevaluating the MP2 correlation energy of the assigned
subsystem are distributed among processors in the group. TheDC-RI-MP2 code has been ported to K computer and supplied
as a library program in K computer [10].
4Summary

We have developed an efficient parallel RI-MP2 algorithm. We
also have developed efficient computational techniques andcodes of RI-MP2 based on molecular fragmentation based O(N)quantum chemical methods such as FMO-RI-MP2 and DC-RIMP2. These methods and codes are suitable for the massively
parallel computations of extended systems on thesupercomputers. The codes developed in this study have been
supplied to users of K computer as a library program [10].
Acknowledgements
This work was supported by the Nanoscience Program in theNext Generation Super Computing Project of the MEXT. Part ofcalculations were performed at K computer, Kobe, Japan andthe Research Center for Computational Science, Okazaki, Japan.
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