Previous: Parallel ab-initio molecular dynamics
The term ab-initio molecular dynamics is used to refer to a class of methods for studying the dynamical motion of atoms, where a huge amount of computational work is spent in solving, as exactly as is required, the entire quantum mechanical electronic structure problem. When the electronic wavefunctions are reliably known, it will be possible to derive the forces on the atomic nuclei using the Hellmann-Feynman theorem[1]. The forces may then be used to move the atoms, as in standard molecular dynamics.
The most widely used theory for studying the quantum mechanical electronic structure problem of solids and larger molecular systems is the density-functional theory of Hohenberg and Kohn[2] in the local-density approximation[2] (LDA). The selfconsistent Schrödinger equation (or more precisely, the Kohn-Sham equations[2]) for single-electron states is solved for the solid-state or molecular system, usually in a finite basis-set of analytical functions. The electronic ground state and its total energy is thus obtained. One widely used basis set is ``plane waves'', or simply the Fourier components of the numerical wavefunction with a kinetic energy less than some cutoff value. Such basis sets can only be used reliably for atomic potentials whose bound states aren't too localized, and hence plane waves are almost always used in conjunction with pseudo-potentials[3] that effectively represent the atomic cores as relatively smooth static effective potentials in which the valence electrons are treated.
Car and Parrinello's method[4] is based upon the LDA, and uses pseudopotentials and plane wave basis sets, but they added the concept of updating iteratively the electronic wavefunctions simultaneously with the motion of atomic nuclei (electron and nucleus dynamics are coupled). This is implemented in a standard molecular dynamics paradigm, associating dynamical degrees of freedom with each electronic Fourier component (with a small but finite mass). The efficiency of this iteration scheme has opened up not only for the mentioned pseudopotential based molecular dynamics studies, but also for static calculations for far larger systems than had previously been accessible. Part of this improvement is due to the fact that some terms of the Kohn-Sham Hamiltonian can be efficiently represented in real-space, other terms in Fourier space, and that Fast Fourier Transforms (FFT) can be used to quickly transform from one representation to the other.
Since the original paper by Car and Parrinello[4], a number of modifications[5,6] have been presented that improve significantly on the efficiency of the iterative solution of the Kohn-Sham equations. The modifications include the introduction of the conjugate gradients method[5,6,7] and a direct minimization of the total energy[6].
The present work is based upon the solution of the Kohn-Sham equations using the conjugate gradients method. We use Gillan's all-bands minimization method[7] for simultaneously updating all eigenstates, which is important when treating metallic systems with a Fermi-surface.
The Car-Parrinello code (written in Fortran-77) employed by us has been used for a number of years, and has been optimized for vector supercomputers and workstations. On a single CPU of a Cray C90 the code performs at about 350-400 MFLOPS (out of 952 MFLOPS peak), mainly bound by the performance of Cray's complex 3D FFT library routine. On a single node of a Fujitsu VPP-500/32 at the JRCAT computer center in Tsukuba, Japan the code achieves about 500 MFLOPS (out of 1600 MFLOPS peak).
Previous: Parallel ab-initio molecular dynamics