## Comparison Of Classical And Quantum Results

Until recently, QMC had not been applied to many body chemical systems and direct comparison of classical and quantum simulations was not possible. Recently, it was found that simple diffusion QMC (no importance sampling) performed well on polyethylene chains with 50 or fewer atoms [20]. Beyond this number, the method failed due to the controllability issue discussed above.

ICQMC was first applied to united atom model polyethylene chains with 100 monomer units; typical results are shown in Fig. 1 [21]. The ground state energy is slightly below the normal modes result. In addition, as expected for the ground state, wavefunctions have a nearly Gaussian shape. As shown by the end to end distance distribution, the ground state is quite rigid; the width of the end to end distance distribution is of the same order as those of the individual bond lengths and is centered near the equilibrium value. It should be emphasized that the trial wavefunction constrains only internal coordinates; no restrictions are placed on the end to end distance, unlike Cartesian trial functions.

So far, we have calculated ground states for polyethylene chains with up to 400 atoms. For 100 or fewer atoms, it takes several thousand time steps to converge. As the system gets larger, the number of iterations required for convergence increases sharply. For 400 atom chains, it took upwards of 100,000 iterations to converge. We are attempting to address this issue by improving the quality of the trial functions (by including local coupling effects, for instance).

These results are in sharp contrast with classical MD simulation results. Fig. 2 shows end to end distance profile for a polyethylene chain at (a) 100% and (b) 25% of the zero point energy. At these energies, the polymer chain undergoes either significant coiling or large amplitude motion. This occurs because energy which should be locked in placed quantum mechanically flows freely in classical simulations, subject only to the conservation of energy. Any vibrational energy that enters low-frequency, highamplitude modes may cause unrealistically large amplitude motion. This phenomenon is known as the zero point energy problem or adiabatic leak. The significance of these results is underscored by a previous study which reported high-dimensional chaos in MD simulations of polyethylene chains even at temperatures below 2K [22]. In few-body systems, it is possible to construct classical trajectories that are quasi-periodic and that mimic the behavior of quantum states; no such trajectories have yet been found for polyethylene chains.

Admittedly, these results pertain to a worst case scenario: a one-dimensional bond network with no external constraints. Similar comparisons have been made for carbon nanotubes [23], which are much more highly connected. In this case, the agreement is much better in that the nanotube keeps its shape in the MD simulations; however, the magnitude of longitudinal motion and ring breathing is still far in excess of the quantum result.

Essentially, the problem of adiabatic leak is mitigated by the presence of constraints, whether in the form of steric hindrance, external forces, or in reduction of dimensionality (for example, simulations in torsion space). In simulations of proposed nanomachines, for example, rigid body dynamics is sometimes applied to components or substructures in order to make the entire nanostructure more rigid. All of this suggests that the prediction of failure modes in some nanomachine designs due strictly to vibrational motion is overly pessimistic. However, this does not by any means minimize concerns such as the difficulty of building desired nanostructures, chemical reactivity, and so on.

It should be noted that we are comparing the results of classical, constant energy simulations with quantum results. In many kinds of simulations, zero point energy is far less of a concern. For example, classical Monte Carlo calculations are used, among other applications, to calculate equilibrium structures of polymer micelles and other formations; in this case, the iterations do not correspond to a progression of time ,steps.

Several attempts have been made to correct the zero point energy problem; however, these have been applied so far only to few-dimensional systems such as Henon-Heiles and an idealized model of water [24-27]. In several of these schemes, energies of single vibrational modes are prevented from falling below their zero point values. This means not entering an elliptical region of (p, x) space; the correction schemes differ according to how this is done. For example, Bowman and co-workers [23,24] reverse the momentum of a mode when the ellipse is reached; McCormack and Lim [25,26] slide around the ellipse and move away on the "other side". To adapt these or other schemes to many-body systems will very likely require accounting for specific features of the problem.

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