Chemistry Reference
In-Depth Information
and then the constraints are corrected which is expressed in the second term of
c
ij
X
L
Þ¼
~
t
2
c
j
ð
c
i
ð
t
þ D
t
c
i
ð
t
þ D
t
Þþ
m
D
t
Þ:
(19)
2
j
c
ij
of the constraints depend now only on the elec-
tronic part. For their determination see [
3
]. Of course the nuclei are also propagated,
their positions being obtained according to (7). From these new “positions,” i.e.,
new nuclear positions and new coefficients, the forces on the nuclei F(R
I
) and those
on the electrons f
i
are obtained. Again, the “velocities” of the coefficients are
derived as
The Lagrange multipliers
L
f
i
ð
t
þ D
t
Þ
Þ¼
~
c
0
i
ð
_
t
þ D
t
c
i
ð
t
þ D
t
Þþ
D
t
:
(20)
2
m
They are corrected afterwards by determining the constraints
L
X
ij
c
0
t
ð
c
i
ð
_
t
þ D
t
Þ¼_
t
þ D
t
Þþ
m
D
t
c
j
ð
t
þ D
t
Þ:
(21)
2
j
The difference from classical force field based simulations where the forces are
calculated from pre-defined pair potentials is that the forces are derived from the
global potential energy surface of an electronic structure theory. The vastly higher
computational costs of an electronic structure calculation restrict the system size
and the length of trajectories accessible by ab initio molecular dynamics simulations.
However, it becomes clear that CPMD and AIMD are important steps towards
general predictive methods, due to their independence from parameterizations.
2.5 Generalization of the Car-Parrinello and Born-Oppenheimer
Molecular Dynamics Approaches
In order to allow for higher order symplectic or geometric integration schemes,
Anders Niklasson et al. introduced a Lagrangian generalization of the time-revers-
ible Born-Oppenheimer molecular dynamics simulations [
7
,
8
].
Integrators in molecular dynamics simulations are supposed to be accurate, i.e.,
they should enforce the exact trajectory being followed as closely as possible.
They should provide stability, meaning that the constants of motion, e.g., the total
energy in the microcanonical ensemble, are preserved. Nevertheless, the integrators
should be efficient, which means that a minimum number of force calculations are
needed in order to save computer time. The best numerical methods are based on
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