Essentials of Computational Chemistry

78 3 SIMULATIONS OF MOLECULAR ENSEMBLES
(not third) about t + t/2 are employed, in particular
and
q
q
t + 1
2
t + 1
2
1
t +
2 t
= q t + 1
2 t
+ v t + 1
2 t
1 1
t +
2 2! a
t + 1
2 t
1
2 t
2 (3.21)
1
t −
2 t
= q t + 1
2 t
− v t + 1
2 t
1 1
t +
2 2! a
t + 1
2 t
1
2 t
2 When Eq. (3.22) is subtracted from Eq. (3.21) one obtains
Similar expansions for v give
(3.22)
q(t + t) = q(t) + v t + 1
2 t
t (3.23)
v t + 1
2 t
= v t − 1
2 t
+ a(t)t (3.24)
Note that in the leapfrog method, position depends on the velocities as computed one-half
time step out of phase, thus, scaling of the velocities can be accomplished to control temperature.
Note also that no force-field calculations actually take place for the fractional time
steps. Forces (and thus accelerations) in Eq. (3.24) are computed at integral time steps, halftime-step-forward
velocities are computed therefrom, and these are then used in Eq. (3.23)
to update the particle positions. The drawbacks of the leapfrog algorithm include ignoring
third-order terms in the Taylor expansions and the half-time-step displacements of the position
and velocity vectors – both of these features can contribute to decreased stability in
numerical integration of the trajectory.
Considerably more stable numerical integration schemes are known for arbitrary trajectories,
e.g., Runge–Kutta (Press et al. 1986) and Gear predictor-corrector (Gear 1971) methods.
In Runge–Kutta methods, the gradient of a function is evaluated at a number of different
intermediate points, determined iteratively from the gradient at the current point, prior to
taking a step to a new trajectory point on the path; the ‘order’ of the method refers to
the number of such intermediate evaluations. In Gear predictor-corrector algorithms, higher
order terms in the Taylor expansion are used to predict steps along the trajectory, and then
the actual particle accelerations computed for those points are compared to those that were
predicted by the Taylor expansion. The differences between the actual and predicted values
are used to correct the position of the point on the trajectory. While Runge–Kutta and Gear
predictor-corrector algorithms enjoy very high stability, they find only limited use in MD
simulations because of the high computational cost associated with computing multiple first
derivatives, or higher-order derivatives, for every step along the trajectory.
A different method of increasing the time step without decreasing the numerical stability is
to remove from the system those degrees of freedom having the highest frequency (assuming,