Efficient Sampling of Quantum Systems Using Path Integral Molecular Dynamics: Application to Weakly Bound Systems r θ Christopher Ing, Konrad Hinsen*, Jing Yang, Tao Zeng, Hui Li, Pierre-Nicholas Roy Department of Chemistry, University of Waterloo *Centre de Biophysique Moleculaire, CNRS Hi, I’m Jing and I’m presenting some research from a Masters student in my lab who couldn’t make it to this conference because he lost his passport! I only contributed a small part to this project, the development of potentials, but I will try my best to walk you through this cool research.

Our Goals We developed our own code to do path integral molecular dynamics (PIMD) in an open-source package, Molecular Modelling Toolkit (http://dirac.cnrs-orleans.fr/MMTK/). We decided to test this code by simulating a familiar system in our group, the doped helium cluster, (He)N-CO2 at low temperatures. The goals of this research were to: 1) test our path integral molecular dynamics code developed in our lab 2) learn a little bit about the challenges of simulating doped helium clusters at low temperatures. We chose the He-CO2 system because we have studied it before in our group so we have lots of data on it. I’ll briefly review what path integral MD is in a few slides, but first let me show you a classical simulation of He-CO2 using normal molecular dynamics.

He-CO2 Clusters using MD 1) On the left I’ve shown a 2D potential energy surface for a He-CO2 potential aligned about the x-axis. 2) On the right is a path integral MD simulation using classical molecular dynamics, newton’s equations, H. Li and R. J. Le Roy, Phys Chem Chem Phys 10, 4128 (2008). H. Li, N. Blinov, P.-N. Roy, and R. J. L. Roy, J. Chem. Phys. 130, 144305 (2009).

Reformulate that using Path Integrals! Low Temp. Requires PIMD Interaction Between Two Classical Atoms Reformulate that using Path Integrals! rCO2-He 1) In the last slide I showed a movie of the whole CO2 molecule but in our simulations we treat it as a point CO2 particle which is free to translate. 2) Using smooth analytic potential forms, we can define our helium-CO2 interactions based solely on pair distance. 3) However, at low temperatures, we have quantum effects and classical mechanics fails, 4) we can use path integrals to obtain a more accurate simulation.

Interaction Between Two Classical Atoms Path Integral MD Interaction Between Two Classical Atoms Interaction Between Two Quantum Atoms 1) By replicating our system, P times, where Im showing P=3 in this slide, we can describe the interaction between two QUANTUM ATOMS, which is exact in the limit of P=infinite beads. 2) As I’ve shown on the right, this requires us to replicate our system P times and adding spring terms between each replicas. These terms have a spring constant which is a function of P*T^2*mass of the atom divided by hbar. All the mathematics is described well by Chandler and Wolynes in JCP 74. 3) This formulation give us an EFFECTIVE potential that can be simulated using a set of normal classical equations of motion, but that gives us a quantum average when we run our simulation long enough. D. Chandler, P.G. Wolynes, J. Chem. Phys. 74, 4078-4095 (1981).

P=64 Path Integral MD of HeCO2 1) Here’s a visualization of a path integral simulation with only helium represented at P=64, or as we call it it, 64 beads. 2) In practice both the CO2 and the helium are represented with 64 beads. 3) But how do you know how many beads to use to get good quantum averages? You have to do a few trial simulations first… (switch to next slide)

HeCO2 Structural Convergence 1) Using these simulations, we can study what happens to something like the radial distribution (the pair distance from CO2 to helium) as we increase the number of path integral beads. 2) You can see that 512 beads in green is sufficiently close to the exact basis set calculation for this system.

He-CO2 Energy Convergence 1) We can also study what happens to an averaged property like the energy of our helium co2 system as we increase P. 2) Here I’ve plotted energy as a function of 1/(kB*T*P), so, energy versus inverse number of beads. 3) The point at 0.003 represents 896 beads. 4) We know mathematically that the energy gets better quadratically, so we can fit it and extrapolate what a PIMD simulation would be like with thousands and thousands of beads. 5) We see that all our simulations using MD are within error bars of the basis set calculation. (White noise langevin equation is blue, Path integral langevin equation is black (just a better langevin method by manolopolous published in 2010), path integral monte carlo is red)

(He)N-CO2 Radial Distribution We can also include a helium-helium potential and make our clusters larger. Here we show the radial distribution as we increase the number of helium atoms.

Connect to Experiment But to confirm these structural distributions shown in the last slide, we can actually connect to experiment The shift in the asymmetric stretch of CO2 can be measured using spectoscoptic techniques. We can compute a band origin shift with the difference in potential between the ground state and first excited state for a summation and normalization of radial and angular pairs H. Li, N. Blinov, P.-N. Roy, and R. J. L. Roy, J. Chem. Phys. 130, 144305 (2009).

(He)N-CO2 Vibrational Shifts Here we see the vibrational shift in wavenumbers as we add heliums to our cluster. (N=5 heliums form a donut around the center of CO2) Our results in red match PIMC, but differ by a nearly constant factor. This difference is due to the lack of rotation (a centrifugal effect) and exchange (quantum statistics of bosons) in PIMD. We confirmed by PIMC calculations with the worm algorithm in green, which matches the blue.

Conclusion Path Integral Molecular Dynamics is a viable method for quantum sampling neglecting rotation and exchange. PIMD is useful for understanding quantum effects at low temperature in He-CO2 systems. PIMD works for He-CO2 simulations We learned how energetic and structural properties converged We learned about the importance of exchange and rotation

Future Studies New and exciting applications, He-CO2 dynamics using RPMD, Centroid MD Quantifying rotational and exchange effects Confined hydrogen in water cages Sugars with explicit water on GPU Path Integral Ground State MD simulations, Comparison to Path Integral Ground State Monte Carlo Numerous exciting applications of this code within our group 1) Contact me after the presentation if you’d like to get a copy of the code, which will be released on our group website soon.