Chemistry Reference
In-Depth Information
F
I
ð
t
þ D
t
Þþ
F
I
ð
t
Þ
R
I
ð
Þ¼
R
I
ð
t
þ D
t
t
Þþ
Dð
t
Þ:
(8)
2
M
I
t
) can be calculated from the
current positions R
I
(
t
), velocities R
I
(
t
), and forces F
I
(
t
). Similarly, the new
velocities can be obtained from knowledge of current velocities and forces F
I
(
t
)
as well as from the new forces which are available as soon as the new positions (7)
are calculated. An overview over how integration algorithms are derived is
provided in [
10
].
In many molecular systems it is desirable to freeze fast degrees of freedom. This
can be necessary in order to allow the integration of the slower motions using larger
time steps. Or the freezing of fast degrees of freedom might be necessary if the
quantum nature of such degrees of freedom (e.g., bond stretch vibrations including
hydrogen atoms) are important. A technique developed [
10
,
15
] to handle properly
such constraints to the molecular structure in molecular dynamics simulation is
based on undetermined multipliers. The constraint conditions with the undeter-
mined multipliers are added to the Lagrangian of (1). The constraint condition then
gives rise to additional (constraint) forces G
I
in the equation of motion
It is apparent how the new positions R
I
(
t
+
D
M
I
R
I
¼
F
I
þ
G
I
:
(9)
The constraint forces depend linearly on the multipliers which have to be
determined in accordance with the numerical integration scheme. This usually
leads to nonlinear equations which can in special cases be solved directly. However,
the most common algorithm, called SHAKE [
15
], solves the equations iteratively,
until self consistency between input and output multipliers is achieved.
In order to avoid surface effects for condensed phase simulations, periodic
boundary conditions are applied. The central computational box is replicated
infinitely in all dimensions. A detailed description can be found in the textbooks
of Allen and Tildesley [
10
] as well as of Frenkel and Smit [
11
].
2.3 Born-Oppenheimer Molecular Dynamics Simulations
Instead of using a pre-parameterized potential, the potential can be calculated on
the fly using electronic structure theory within the Born-Oppenheimer approxima-
tion. In such calculations the potentials are obtained by solving a time-independent
quantum chemical electronic structure problem:
U
ð
R
I
Þ¼
min
fCg
E
½fCg;
R
I
:
(10)
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