Chemistry Reference
In-Depth Information
conserved quantity in this ensemble. The total momentum of the particles in
the system is also a conserved quantity. For an isolated collection of particles,
the total angular momentum is also conserved, but this quantity is not con-
served in systems with periodic boundary conditions.
If we are going to relate the properties of our system to a physical situation,
we need to be able to characterize the system's temperature,
T
. In a macro-
scopic collection of atoms that is in equilibrium at temperature
T
, the velocities
of the atoms are distributed according to the Maxwell-Boltzmann distri-
bution. One of the key properties of this distribution is that the average kinetic
energy of each degree of freedom is
1
2
m
(
v
k
B
T
2
2
)
¼
:
:
(9
6)
In molecular dynamics, this relationship is turned around to
define
tempera-
ture by
6
N
X
3
N
k
B
T
MD
2
;
1
2
i
m
i
v
:
(9
:
7)
i
1
Notice that because the kinetic energy,
K
, is not conserved by the equations of
motion above, the temperature observed in a microcanonical ensemble MD
simulation must fluctuate with time.
In essentially all systems that are physically interesting, the equations of
motion above are far too complicated to solve in closed form. It is therefore
important to be able to integrate these equations numerically in order to
follow the dynamics of the atoms. A simple way to do this is to use the
Taylor expansion:
d
2
r
i
(
t
)
d
3
r
i
(
t
)
dr
i
(
t
)
dt
1
2
1
6
dt
2
Dt
2
dt
3
Dt
3
þ O
(
Dt
4
)
r
i
(
t þ Dt
)
¼ r
i
(
t
)
þ
Dt þ
þ
:
:
(9
8)
Rewriting this expansion by assigning the names of the derivatives on the
right-hand side
gives
d
3
r
i
(
t
)
1
1
6
2
a
i
(
t
)
Dt
2
dt
3
Dt
3
þ O
(
Dt
4
)
:
r
i
(
t þ Dt
)
¼ r
i
(
t
)
þ
v
i
(
t
)
Dt þ
þ
(9
:
9)
The third derivative term (which is also the first time derivative of acceleration) also has a
name: the jerk.
