Made the timestep calculation example more explicit and added another example

Author: EfremBraun

paper/basic_training.tex

@@ -1007,7 +1007,8 @@ \subsubsection{Choosing an appropriate timestep}
 For the commonly used second order integrators such as the Verlet and Leapfrog algorithms, the velocities and accelerations should be approximately constant over the timestep.
 Thus, the timestep is limited by the highest frequency motion present in the system, which for all-atom simulations is usually bond vibrations.
 It is commonly found that using a timestep that is one tenth of this vibration's characteristic period is sufficient to conserve energy in the microcanonical ensemble.
-For example, if hydrogen molecules are present in the simulation box and the H-H bond vibration is the highest-frequency motion in the system, one can determine that with a force field harmonic force constant of approximately 500 N/m, the oscillation period will be 8 fs; thus, a 0.5 fs timestep can be used.
+For example, if hydrogen molecules are present in the simulation box and the H-H bond vibration is the highest-frequency motion in the system with its force field harmonic force constant set to 500 N/m, the oscillation period can be calculated using the equation for simple harmonic motion ($T=2\pi\sqrt{\frac{\mu}{k}}$, where $\mu$ is the reduced mass and $k$ is the force constant) to be 8 fs; thus, a 0.5 fs timestep can be used.
+As another example, if an ab initio MD simulation is being conducted in which C-H bond vibrations are known to be the highest-frequency motion, infrared spectra can be consulted to find that this bond vibration frequency will be approximately 3000 cm$^{-1}$, which is 11 fs; thus, either a 0.5 or 1.0 fs timestep would be recommended.
 For all-atom simulations with constraints on the high-frequency bonds, timesteps can be commonly increased to 2 fs; coarse-grained simulations with particles of higher mass and smaller force constants can have much larger timesteps.
 After choosing a timestep, a test simulation should be run in the microcanonical ensemble to ensure that the choice of timestep yields dynamics that conserve energy.
 Methods also exist to increase the timestep beyond the limit imposed by the system's highest-frequency motion.