# Normal Mode Langevin

### Normal Mode Langevin (NML).

The NML coarse grained method utilizes a normal mode decomposition of the dynamical space to achieve large numerical step size. Our first paper 'Normal Mode Partitioning of Langevin Dynamics for Biomolecules' authored by Christopher R. Sweet, Paula Petrone, Vijay S. Pande and Jesus A. Izaguirre was published in the Journal of Chemical Physics. It established the basic principles and theory of the method, and the corresponding informational website can be found here. To implement an efficient realization has required further work on Hessian diagonalization and Langevin methods for large step size.

The basic idea is that the maximum timestep is dependent on the highest frequency in the system, but these high frequency vibrations contribute little to the dynamics of interest. By removing frequencies above a given threshold large timesteps can be used, and the high frequencies are 'relaxed' by minimization.

### Efficient Diagonalization.

The coarse-graining strategy to computing the frequency partitioning is to find a reduced set of normalized vectors that spans the low frequency space of interest. The quadratic product of these vectors and the Hessian, which can be done in O(N) due to their block structure, produces a small matrix which is cheap to diagonalize as seen below. This avoids the O(N^{3}) cost of traditional diagonalization, instead the cost is O(N^{9/5}).

The following graphic depicts the block structure of the Hessian that is used to allow an efficient

diagonalization method.

### Langevin Thermostats for large step-size.

Traditional Langevin thermostats give large error at large values of (time-step*damping coefficient) Methods such as Langevin Impulse alleviate many of these problems but has been found to overdamp for large time-steps. We have derived a method based on the leapfrog which exactly integrates the velocities over the half step. This has been shown to give exact results for isomerization rates for alanine dipetide with step sizes from 0.25 to 12fs.

The equations are