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J Chem Phys. 2008 Jul 14;129(2):024105. doi: 10.1063/1.2953308.

An efficient ring polymer contraction scheme for imaginary time path integral simulations.

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Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom.


A quantum simulation of an imaginary time path integral typically requires around n times more computational effort than the corresponding classical simulation, where n is the number of ring polymer beads (or imaginary time slices) used in the calculation. However, this estimate neglects the fact that the potential energies of many systems can be decomposed into a sum of rapidly varying short-range and slowly varying long-range contributions. For such systems, the computational effort of the path integral simulation can be reduced considerably by evaluating the long-range forces on a contracted ring polymer with fewer beads than are needed to evaluate the short-range forces. This idea is developed and then illustrated with an application to a flexible model of liquid water in which the intramolecular forces are evaluated with 32 beads, the oxygen-oxygen Lennard-Jones forces with seven, and the intermolecular electrostatic forces with just five. The resulting static and dynamic properties are within a few percent of those of a full 32-bead calculation, and yet they are obtained with a computational effort less than six times (rather than 32 times) that of a classical simulation. We hope that this development will encourage future studies of quantum mechanical fluctuations in liquid water and aqueous solutions and in many other systems with similar interaction potentials.


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