Overall Comparison - What does all this mean for simulation?

As expected, the classical approximation is at its best (approaches the quantum results) at high temperatures and for oscillators of low frequency (small k and/or big m). Under these conditions, the gaps between quantum states are small relative to thermal energy, $k_B T$.

The highest energy vibration we've looked at in detail has been that of protons oscillating at f = 100/ps. This frequency was chosen because it is essentially that of oxygen-hydrogen bond stretching. It represents one of the highest frequency modes of vibration in a biomolecule and thus serves as a worst-case scenario for classical approximations in macromolecular simulations. Several aspects of this motion have been depicted; see the plots of V and eigenvalues E, of probability(x), and the green curves in the plots of U and heat capacity and mean-square fluctuation.

Indeed, the mean-square fluctuation predicted classically at this frequency is about eight times too small at 300 K. We can take consolation in the scale, though. For while the quantum value of $\langle x^2 \rangle = 0.005 \AA^2$ (rms x = 0.07 Å) is large compared to the classical result, it is still modest relative to crystallographic resolutions and the equilibrium length of the O-H bond. Furthermore, when compared to motional amplitudes measured by neutron scattering, classical simulations predict too much motion [6]. Thus, the reduced motion resulting from the neglect of quantum effects is overshadowed by other approximations made in simulations (perhaps the neglect of electronic polarizability and the assumed pairwise additivity of van der Waals forces). The overestimate of protein motion by simulations is not yet understood.

Another problem with classical dynamics is the incorrect partitioning of energy (see plots of U and heat capacity). Classically $U = k_B T$ and $C_V = k_B$, independent of frequency. In reality, high frequency motions have much more energy (larger U) and much less ability to exchange energy (smaller $C_V$) than classical mechanics predicts. High frequency motions like O-H bond stretching are energetically trapped in their quantum ground state, unexcitable to higher energy levels except at very high temperatures. Thus, the average energy U for the oscillator with f = 100/ps is very nearly temperature independent ($C_V \approx 0$) all the way up to 600 K. The frequency-dependent underestimate of U by classical mechanics complicates calculations of free energy differences $\Delta G$ when vibrational frequencies are likely to change during the process under investigation. However, these errors tend to cancel in estimates of $\Delta \Delta G$ values. In classical simulations, low frequency motions exchange energy with too many (high frequency) degrees of freedom, but this unphysical give-and-take of energy with high frequency motions tends to average out.

To summarize, classical simulations are unable to analyze the details of bond stretching and angle bending quantitatively. These motions are at frequencies too high for an accurate treatment using Newton's laws. However, we've observed that the errors in motional amplitude are relatively small, and errors in energy tend to cancel out in appropriately designed calculations, as when $\Delta \Delta G$'s are calculated rather than $\Delta G$'s. For lower frequency motions ($f \approx$ 1/ps or less), observables such as U and $\langle x^2 \rangle$ become temperature independent (as quantum effects dominate) at much lower temperatures. For these motions, classical mechanics is a good approximation at physiological temperatures.