Symplectic molecular dynamics integration using normal mode analysis

Abstract

ABSTRACT:The split integration symplectic method (SISM) for molecular dynamics (MD) integration using normal mode analysis based on a factorization of the Liouville propagator is presented. This approach is quite distinct from others that use fractional-step methods, owing to the analytical treatment of high-frequency motions. The method involves splitting the total Hamiltonian of the system into a harmonic part and the remaining part. Then the Hamilton equations are solved using a second-order generalized leapfrog integration scheme in which the purely harmonic Hamiltonian (which represents the main contribution of the chemical bonds and angles) is treated analytically, i.e., independent of the step size of integration, by a normal mode analysis that is carried out only once, at the beginning of calculation. The whole integration step combines analytical evolution of the harmonic part of the Hamiltonian with a correction arising from the remaining part. The proposed algorithm requires only one force evaluation per integration step. The algorithm was tested on a simple system of linear chain molecules. Results demonstrate the method makes possible the integration of the MD equations over larger time steps without loss of stability while being economical in computer time.

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“…Many thanks to M. Praprotnik [18], A. Kodre, and G. Veble. This work was supported by the Slovenian Office of Science (Program P1-0099) and U.S.-Slovene NSF Joint Found (Grant No.…”

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“…Schematically the SISM reads as follows 10: At the outset of calculation vibrational frequencies and normal modes, represented by normal coordinates P , Q , of H 0 are determined. Rotate the normal coordinates, P , Q , in the phase space by the corresponding vibrational frequency ω j for Δ t /2: For ω j ≠ 0 vibrations are taken into account and for ω j = 0 translations and rotations. Equations describe translations and rotations in normal coordinates. Coordinate transformation from normal coordinates P , Q to Cartesian displacement coordinates p , q . Numerical integration of momenta (one force calculation per integration step) Back transformation from Cartesian displacement coordinates p , q to normal coordinates P , Q . Again, the rotation of normal coordinates in the phase space by the corresponding vibrational frequency ω j for Δ t /2: which concludes one SISM integration step. Go to 2 until the desired number of integration steps is reached. …”

“…To determine the parallel efficiency of the SISM the system of linear molecules was used as a model system. For this model system the MD Hamiltonian 10 is where i and j run over all atoms, m i is the mass of the i ‐atom, b 0 and ϑ 0 are reference values for bond lengths and angles, respectively, k b and k ϑ are corresponding force constants, e i denotes the charge on the i ‐atom, r ij is the distance between atoms i and j , and ε ij and σ ij are the corresponding constants of the Lennard–Jones potential.…”

“…The results of comparison of the sequential and parallel LFV and SISM are presented in Figure 5. The integration steps of 1 fs for LFV and 4.5 fs for SISM were used to compare computational performances for the same level of accuracy for both methods 10. Total execution times are shown for a simulation length of 1 ns.…”

“…We recently introduced an efficient split integration symplectic method (SISM) 10–12 for MD integration. The method allows the use of a time‐step up to an order of magnitude larger than the commonly used LFV algorithm, which is of the same order and computational complexity as SISM.…”