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symplectic and time-reversible integrators. For integrator algorithms see Sect.
2.2,
in which the Str
€
omer-Verlet integrator was introduced. However, in general, long
time stability is more important than short time accuracy.
For ab initio molecular dynamics simulations it is important to understand how
the errors in the forces affect the long term MD stability of the simulations. While
the error in the energy [min
f
E
[{
f
}, R
I
], see (10)] is, due to the variational
principle, of second order in the error
df
of the wave functions, the error in the
force (
dE
[{
. This suggests that MD stability can
only be achieved with numerically highly accurate wavefunctions.
In the following the indices for the nuclei and the electrons will be omitted.
Applying the extended Lagrangian method introduced by Niklasson [
7
,
8
] a general
expression for the AIMD Lagrangian can be written as
f
}, R
I
]/
d
R
I
) is of first order in
df
1
2
M
1
2
m _
q
2
x
2
Lð
q
; _
q
;
x
; _
x
Þ¼
_
þ
E
ð
q
;
y
Þþ
k
m
G
ðjj
x
y
jjÞ:
(22)
q and x are now generalized coordinates of the nuclei and electrons, respec-
tively. The vector y expresses the wave function after complete or partial
optimization:
y
¼
F
ð
q
;
x
Þ:
(23)
) is a retention potential that ensures that the propagated
wavefunction x stays close to the optimized wavefunction y and
G
(
k
x
y
k
m
is a mass
2
is the force constant of the retention potential.
From the generalized Lagrangian follows the equations of motion
parameter,
k
¼ o
¼
@
E
q
@
E
y
@
F
m
@
G
y
@
F
M
€
q
q
þ
k
(24)
@
@
@
@
@
q
and
¼
@
E
y
@
F
m
@
G
q
þ
@
G
@
y
@
F
m
€
x
x
þ
k
:
(25)
@
@
@
@
x
In this notation the Car-Parrinello molecular dynamics scheme is obtained with
the condition
y
¼
x
)
G
ðjj
x
y
jjÞ ¼
0
:
(26)
This leads directly to the Car-Parrinello Lagrangian [see (11)]
1
2
M
1
2
m _
q
2
x
2
Lð
q
; _
q
;
x
; _
x
Þ¼
_
þ
E
ð
q
;
x
Þ
(27)
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