Chemistry Reference
In-Depth Information
methods. Both of these ideas are illustrated by examining the Verlet algorithm
for constant energy MD, Eq. (9.10).
First, consider the computational effort involved in taking a single time step
with this algorithm. During every time step, the force acting on each degree of
freedom,
F
i
(t)
, must be determined. This is (roughly) equivalent to calculating
the total potential energy,
U U
(
r
1
,
,
r
3
N
), at every time step. Even when
this potential energy is a very simple function of the coordinates, for example,
a sum of terms that depends only on the distance between atoms, the work
involved in this calculation grows rapidly with system size. Because evaluat-
ing the forces dominates the computational cost of any MD simulation, great
care must be taken to evaluate the forces in a numerically efficient way.
Second, the derivation of the Verlet algorithm from a Taylor expansion
highlights the fact that this algorithm is only accurate for sufficiently small
time steps,
Dt
. To estimate how small these time steps should be, we can
note that in almost any atomic or molecular system, there is some characteristic
vibration with a frequency on the order of 10
13
s
2
1
. This means that one
vibration takes
100 fs. If we want our numerically generated MD trajectories
to accurately describe motions of this type, we must take a large number of
time steps in this period. This simple reasoning indicates that a typical MD
simulation should use a time step no larger than
10 fs. A trajectory following
the dynamics of a set of atoms for just one nanosecond using a time step of 10
fs requires 10
5
MD steps, a significant investment of computational resources.
...
9.2
AB INITIO
MOLECULAR DYNAMICS
The explanation of classical MD given above was meant in part to emphasize
that the dynamics of atoms can be described provided that the potential energy
of the atoms,
U U
(
r
1
,
,
r
3
N
), is known as a function of the atomic
coordinates. It has probably already occurred to you that a natural use of DFT
calculations might be to perform molecular dynamics by calculating
U
U
(
r
1
,
...
,
r
3
N
) with DFT. That is, the potential energy of the system of interest
can be calculated “on the fly” using quantum mechanics. This is the basic
concept of
ab initio
MD. The Lagrangian for this approach can be written as
...
2
X
3
N
1
2
i
L ¼ K U ¼
m
i
v
E
[
w
(
r
1
,
...
,
r
3
N
)],
(9
:
16)
i
1
where
w
(
r
1
,
,
r
3
N
) represents the full set of Kohn-Sham one-electron wave
functions for the electronic ground state of the system. (See Section 1.4 for
details.) This Lagrangian suggests that calculations be done in a sequential
...
