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Benchmark of the CPMD code on CRESCO HPC Facilities for  Numerical Simulations of a Magnesium Nanoparticle

Introduction

Solid-state metal hydrides are considered useful for storing hydrogen although materials suitable for practical use are still under development. Magnesium is an important candidate in this respect as it can reversibly store about 7.6 wt% hydrogen, is light weight and is a low-cost material. However, its thermodynamic parameters are not completely favorable and the reaction with hydrogen often shows sluggish kinetics. Different treatments in Mg based materials have been proposed to overcome these drawbacks. One of these is tailoring Mg nanoparticles in view of an enhancement on the reaction kinetics and thermodynamics of the Mg-MgH₂ phase transformation [1-3]. Because metallic nanoparticles often show size dependent behavior different from bulk matter, a better understanding of their physical-chemical properties is necessary. For this purpose, we want to set up a numerical model to perform first principle calculations based on the density functional theory (DFT), using CPMD code. To evaluate the suitable computational demand, we performed benchmarks of the CPMD code on the HCP ENEA CRESCO computing facilities.

Computational Details
Car-Parrinello Molecular Dynamics (CPMD) code [4,5] is an ab-initio electronic structure and molecular dynamics (MD) program which uses a plane wave/pseudopotential implementation of density functional theory [6,7]. It is mainly targeted at Car-Parrinello MD simulations but also supports geometry optimizations, Born-Oppenheimer MD, path integral MD, response functions, excited states and calculation of some electronic properties. In ab-initio molecular dynamics simulation, the forces acting on atoms are calculated from an electronic structure calculation repeated every time step (“on the fly”). Thanks to electronic structure calculation which uses density functional methods, simulations of large systems with hundreds of atoms are now standard. Originally developed by Roberto Car and Michele Parrinello for applications in solid state physics and material science, this method has also been used with great success for the study of molecular systems. Applications of ab-initio Car-Parrinello Molecular Dynamics simulations range from the thermodynamics of solids and liquids to the study of chemical reactions in solution and on metal and oxide surfaces.
For all the calculations we employed the CPMD code with Goedecker-Teter-Hutter pseudopotentials for magnesium, together with Pade approximant LDA exchange-correlation potentials [8-10]. The electronic wave functions are expanded in a plane-wave basis set with a kinetic energy cut-off equal to 80 Ry. The latter value is optimized by preliminary calculations both on simple molecules and on the crystalline structures of metallic hcp Mg (lattice constant: a_Mg = b_Mg = 3.15 Å, c_Mg = 1.62*a_Mg). Experimental observations have showed that Mg nanoparticles annealed at 300 °C show evaporation, void formation, and void growth in the Mg core both in vacuum and under a high pressure gas environment. This is mainly due to the outward diffusion and evaporation of Mg with the simultaneously inward diffusion of vacancies leading to void growth (Kirkendall effect [11]). To approach numerical studying of Mg nanoparticles, starting from the magnesium bulk, we have considered the cluster built from the Mg atoms inside a 12 Å radius sphere centered on a Mg atom. To take in consideration the Kirkendall effect, we then have removed the inner Mg atoms inside a 5.6 Å radius sphere. The system is composed of a hollow nanoparticle of 266 magnesium atoms as shown in Figure 1, where magnesium atoms are in blue, outer sphere is in gray, and inner red sphere is the void space due to the Kirkendall effect.

Figure 1: Mg nanoparticle. Magnesium atoms are in blue, outer sphere is in gray and, inner red sphere is the void space due to the Kirkendall effect.

ENEA CRESCO computing facilities
CPMD code runs on many different computer architectures and allows good scalability up to a large number of processors depending on the system size. We use the CPMD compiled with Intel Fortran Compiler, MKL (Math Kernel Library), ACML (AMD Core Math Library) and MPI (Message Passing Interface) parallelization on the high performance ENEA CRESCO computing facilities. The benchmark results demonstrate the high performance of the CPMD parallelization on CRESCO architectures and the good scalability up to hundreds of cores.
CRESCO (Computational Research Center for Complex Systems [12]) is an ENEA (Italian National Agency for New Technologies, Energy and Sustainable Economic Development [13]) Project, co-funded by the Italian Ministry of University and Research. The performance for the CRESCO HPC system ranked 125 in June 2008 top500 list with Rmax= 17,14 Tflops in the HPL benchmark. Please find below same of the hardware and software specifics.
The ENEA CRESCO computing facilities are based on the multi-core x86_64 architecture and is made up of various clusters: we tested the CRESCO cluster located in Portci, Casaccia and, Frascati  ENEA centers. The CRESCO cluster in Portici consists of two main sections: section POR_1 for high memory request and moderate parallel scalability; section POR_2 for limited memory and high scalability. We are interested in the analysis of the POR_2 section performance. This section has 340 compute nodes that are equipped with three types of Intel Xeon Quad-Core processors. The CRESCO clusters in Casaccia (CAS) and in Frascati (FRA) are equipped with AMD Opteron processors. Hardware details of the clusters are reported in Table 1. The Operating System (OS) for the three clusters is the Red Hat Enterprise Linux version 2.6.18-194.26.1.el5. We compiled the CPMD version 3.15.1 using the same version of the mathematical library MKL. On CRESCO clusters POR_2 and CAS we used ifort compiler version 11 and OpenMPI version 1.2.8; and on CRESCO cluster FRA we used ifort version 12 and OpenMPI version 1.4.2. Finally we performed a further test on the CRESCO cluster FRA choosing as mathematical library the ACML version 5.1.0. Table 1 summarizes some of the hardware and software characteristics. Further details are reported in Ref. [14].

Table 1: Some of the hardware and software characteristics for the CRESCO clusters.

Results and Discussions
The Kohn-Sham method of DFT simplifies calculations of the electronic density and energy of a system of N_e electrons in an external field without solving the Schroedinger equations with 3N_e degrees of freedom, but it takes into consideration the electronic density as the fundamental quantity (with only 3 degrees of freedom). The total ground-state energy of the system can be obtained as the minimum of the Kohn-Sham energy which is an  explicit functional of the electronic density. This leads to a set of  equations (Kohn-Sham equations) that has to be solved self-consistently in order to yield density and the total energy of the electronic ground-state. In the calculations, starting from an initial guess for the electronic density, the Kohn-Sham equations are solved iteratively until convergence is reached. To find the better computational demand, we calculated the average time for iteration in the resolution of the  Kohn-Sham equations.
In Table 2 and Figure 2, we reported the average time for iteration in function of the number of cores for the three CRESCO clusters. At the beginning, for 24 cores case, we observed that the times for the cluster POR_2 are much longer than the times for cluster CAS and FRA. This is due to the peak memory requirement (PMR) for core that is 2697.0 Mbytes, whereas clusters POR_2a, POR_2b and, POR_2c have 2048 Mbytes for cores. For the others cases the cluster POR_2c, which is equipped with processors Intel Xeon Quad-Core Westmere E5620, has the fastest times, except for the 192 cores case, where as the cluster  FRA has the best performance. However, the times for the clusters POR_2b, POR_2c and, FRA are very similar. This is also highlighted in the Figure 2, where the blue, green and red lines are overlapped. Finally, using ACML library, we observed the performance improvement of the cluster FRA which has AMD Opteron Magny Cours 6174MS processors. The times in the last column (FRA_acml) in Table 2 are reduced in the range of 4.5-6.0 % compared to those in column FRA_mkl in which MKL library were used. It is worth pointing out that we have used the Intel compiler even on AMD processors. In Figure 3 is shown the speed up. For each cluster, the speed up was calculated with respect to the 32 cores time. We observed a good scalability of the code on all the clusters, with the best linear scaling for the CRESCO cluster FRA with both MKL and ACML libraries. From the speed up data we also calculated efficiency that is shown in Figure 4. This figure confirms the CRESCO cluster FRA as the most efficient. In fact, it has an efficiency higher than 75%.

Figure 2: Average time (s) for iteration in function of the number of cores for the different processors on the
CRESCO clusters.





Figure 3: Speed up for the different processors on the CRESCO clusters. The speed up was calculated with
respect to the 32 cores time.





Figure 4: Efficiency for the different processors on the CRESCO clusters.

Summary
Magnesium is seen as one of the most important candidates in the solid media hydrogen storage. However, to reach this aim same disadvantages have to be overcome. In this context Mg nanoparticles could solve some drawbacks of the bulk material. In view of an extensive characterization of Mg nanoparticles by numerical simulations, and to evaluate the suitable computational demand, we performed benchmarks of the CPMD code on the HCP ENEA CRESCO computing facilities. The benchmark results demonstrate the high performance of the CPMD parallelization on CRESCO clusters architectures and the good scalability up to hundreds of cores.
 
Acknowledgment
The computing resources and the related technical support used for this work have been provided by CRESCO-ENEAGRID High Performance Computing infrastructure and its staff; see Ref. [12] for information. CRESCO-ENEAGRID High Performance Computing infrastructure is funded by ENEA, the “Italian National Agency for New Technologies, Energy and Sustainable Economic Development” and by national and European research programs.

References

[1] PASQUINI L., CALLINI E., PISCOPIELLO E., MONTONE A., VITTORI ANTISARI M. “Metal-hydride transformation kinetics in Mg nanoparticles” Appl. Phys. Lett. 94 (2009) 041918.

[2] KRISHNAN G., KOOI B. J., PALASANTZAS G., PIVAK Y,. DAM B. “Thermal stability of gas phase magnesium nanoparticles”  J. Appl. Phys. 107 (2010) 053504.

[3] PASQUINI L., CALLINI E., BRIGHI M., BOSCHERINI F., MONTONE A., JENSEN T. R., MAURIZIO C., VITTORI ANTISARI M., BONETTI E. “Magnesium nanoparticle with transition metal decoration for hydrogen storage ” J. Nanopart. Res. 13 (2011) pp. 5727-5737.

[4] CAR R., PARRINELLO M. “Unified approach for Molecular Dynamics and Density-Functional Theory” Phys. Rev. Lett. 55 (1985) pp. 2471-2474.

[5] CPMD V3.15.1 Copyright IBM Corp 1990-2011, Copyright MPI fuer Festkoerperforschung Stuttgart 1997-2001.

[6] HOHENBERG P., KOHN W. “Inhomogeneous electron gas” Phys. Rev. 136 (1964) pp. B864-B871.

[7] KOHN W., SHAM L. J. “Self-consistent equations including exchange and correlation effects” Phys. Rev. 140 (1965) pp. A1133-A1138.

[8] GOEDECKER S., TETER M., HUTTER J. “Separable dual-space Gaussian pseudopotentials” Phys. Rev. B 54 (1996) pp. 1703-1710.

[9] HARTWIGSEN C., GOEDECKER S., HUTTER J. “Relativistic separable dual-space Gaussian pseudopotentials from H to Rn” Phys. Rev. B 58 (1998) pp. 3641-3662.

[10] KRACK M. “Pseudopotentials for H to Kr optimized for gradient-corrected exchange-correlation functionals” Theor. Chem. Acc. 114 (2005) pp. 145-152.

[11] SMIGELSKAS A. D., KIRKENDALL E. O. “Zinc Diffusion in Alpha Brass” Trans. AIME 171 (1947) pp. 130-142.

[12] www.cresco.enea.it

[13] www.enea.it

[14] www.afs.enea.it/project/eneagrid/Resources/CRESCO_documents/index.html

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last update: 2012-05-08| simone giusepponi