Environmental Engineering Reference
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throughout the dynamics. However, the reported CPMD studies were completely
inadequate because of their short simulation time (due to high computational cost)
and their unrealistic initial model geometries. Based on researchers' limited simu-
lation results, researchers suggested that the segregation of carbon linear chains
and atomic rings on the surface of a liquid-like cobalt-carbide particle represents
the first stage of the cap nucleation process. In the many papers, researchers, also
simulated the incorporation of ve carbon atoms into a preassembled half-fullerene
cap attached to a cobalt metal surface for 15 ps. CPMD is an ab initio molecular
dynamics method which is the combination of first principles electronic structure
methods with MD based on Newton's equations of motion. Grand-state electronic
structures were described according to DFT within plane-wave pseudo potential
framework. The use of electronic structure methods to calculate the interaction
potential between atoms overcomes the main shortcomings of the otherwise high-
ly successful pair potential approach. There have been plenty of excellent refer-
ence topics on MD and DFT, and some simulation tricks can be found. Here, some
more details about CPMD, which are different from the traditional classical MD
and DFT will be discussed. In CPMD, considering the parameters {wi}, {RI},
{av} in energy function are supposed to be time-dependent, the Lagrangean:
1
1
1
{ } { } { }
∑ ∑∑
2
∫
3
2
2
L
=
m ψ +
dr
MR
+ ma − ψ
E
,
R
,
a
(96)
i
II
vv
i
I
v
2
2
2
i
I
v
Ù
1
1
1
2
{ } { } { }
∑ ∑∑
∫
3
2
2
L
=
m ψ +
dr
MR
+ ma − ψ
E
,
R
,
a
(97)
i
II
vv
i
I
v
2
2
2
i
I
v
Ù
was introduced, where the wi
i
are subject to the holonomic constraints:
∑
∫
(
)
(
)
3
*
dr
ψ
rt
,
ψ
rt
,
=d
i
j
ij
(98)
i
Ù
In Eqs. (96) and (97), w
i
are orbitals for electrons, RI indicate the nuclear coordi-
nates and a
v
are all the possible external constraints imposed on the system, w*(r)
is the complex conjugate of wave function w(r), h is the reduced Planck constant,
m is the mass of electron and n(r) = Ri |wi(r)|
2
is the electron density; the dot in-
dicates time derivative, M
I
are the physical ionic masses. Then, the equations of
motion can be written as:
d
E
(
)
+
∑
(
)
(99)
m
ψ
rt
,
=−
Λψ
rt
,
(
)
i
ik
K
d
ψ
*
rt
,
i
I
(100)
MR
= −∇
E
I
I
RI
�
�
∂
E
(101)
ma =−
€
‚
νν
∂a
ƒ
„
ν
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