Chemistry Reference
In-Depth Information
equations of motion. The interaction potentials have been explicated in connection
with the MC approach. The most time-consuming step is the evaluation of energies
and, when needed, forces [
21
,
23
]. The Newtonian equations of motion are solved
in proper time steps
t
. The following considerations influence the choice of
algorithm: (1) the time reversibility of Newton's equations of motion should be
conserved and (2) the generated trajectories should conserve volume in phase
space; that means the integrator should be symplectic (see p. 381 in [
127
]). This
is important to conserve equilibrium distributions in phase space, because deviation
from symplectic behaviour will produce time-dependent weight factors in phase
space [
128
]. The computing time for MD simulations is dominated by force calcu-
lations. Therefore, approaches that use only one force evaluation per time step are
preferred.
A more rigorous derivation of integration schemes, which leads to the possibility
of splitting the propagator of the phase space trajectory into several time scales, is
based on the phase space description of a classical system. The time evolution of a
point in the 6N dimensional phase space is given by the Liouville equation
D
e
iLt
G
ð
t
Þ¼
G
ð
0
Þ;
(22)
where
G
¼
(q,p) is the 6N dimensional vector of generalised coordinates, q
¼
q
1
,
...
, q
N
, and momenta p
¼
p
1
,
...
, p
N
.
L
is the Liouville operator, defined as
!
X
N
@
q
j
@
q
j
þ
@
p
j
@
@
@
@
@
iL
¼ :::;
f
H
g ¼
:
(23)
t
t
p
j
j¼
1
Equation (22) is the starting point for the derivation of numerical integration
schemes. In order to construct a discrete time-step integrator, the Liouville operator
is split into two parts,
L
1
þ
L
2
, and a Trotter expansion [
129
,
130
] is performed:
e
iLdt
e
iðL
1
þL
2
Þdt
e
iLdt
e
iL
1
dt=
2
e
iL
2
dt
e
iL
1
dt=
2
¼
¼
¼
:
(24)
The exact formula is the so-called Baker-Campbell-Hausdorff formula [
131
].
Any partial operators can be chosen to act only on positions or momenta.
Assuming Cartesian coordinates for a system of N free particles, this can be written as
X
X
N
N
F
j
@
@
v
j
@
@
iL
1
¼
;
iL
2
¼
r
j
:
(25)
p
j
j¼
1
j¼
1
Applying (24) to the phase space vector
and using the property exp(
a
/
x
)
G
∂
∂
f
(
x
)
¼
f
(
x
þ
a
) for any function
f
, where
a
is independent of
x
, gives
v
i
ð
t
þ d
t
=
2
Þ¼
v
ð
t
Þþð
F
i
ð
t
Þ=
m
Þd
t
=
2
;
(26)
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