Chemistry Reference
In-Depth Information
S
xy
(
R
j
−
R
i
)
=
cos (
α
x
) cos (
α
y
)S
pp
σ
(R
ij
)
−
cos (
α
x
) cos (
α
y
)S
pp
π
(R
ij
)
(2.15)
S
xz
(
R
j
−
R
i
)
=
cos (
α
x
) cos (
α
z
)S
pp
σ
(R
ij
)
−
cos (
α
x
) cos (
α
z
)S
pp
π
(R
ij
)
(2.16)
for basis functions centred on different atoms and if the two basis functions are
centred on the same atom, S
ss
=
S
xx
=
S
yy
=
S
zz
=
1if
μ
=
ν
and S
μν
=
0 for
μ
ν
.
The E
rep
repulsive par potential is also given in ref. (Porezag et al.
1995
) with the
help of Chebyshev polynomials.
The atomic force
=
n
i
C
i
μ
C
i
ν
∂
H
μν
m
ε
i
∂
S
μν
∂
R
k
∂
E
rep
∂
R
k
F
k
=−
∂
R
k
−
−
(2.17)
i
μ
,
ν
is calculated from the Hellmann-Feynman theorem.
In the canonical ensemble molecular dynamics calculations the constant environ-
mental temperature was controlled with the help of Nosé-Hoover thermostat (Nosé
1984
; Hoover
1985
; Allen and Tildesley
1996
; Frenkel and Smit
1996
). In this ther-
mostat the environmental temperature is T
env
, and the force acting on the atom in the
position R
k
is determined as
n
i
C
i
μ
C
i
ν
∂
H
μν
∂
R
k
−
m
ε
i
∂
S
μν
∂
R
k
∂
E
rep
∂
R
k
−
F
NH
k
=−
−
ξ
P
k
(2.18)
i
μ
,
ν
The friction coefficient
ξ
is given by the first-order differential equation
f
Q
k
B
(T
·
ξ
=
−
T
env
)
(2.19)
where k
B
is the Boltzmann constant, Q is a thermal inertial parameter, f
=
3N is the
number of degrees of freedom and the kinetic temperature is
N
P
k
2M
k
1
3Nk
B
T
=
(2.20)
k
=
1
The thermal inertial parameter determines the time scale of the kinetic temperature
oscillation and strength of interaction with the environment. The detailed nature of
the dynamics depends on the value of Q chosen but the average properties are less
sensitive to its value, thus it can be arbitrary.
The equation of motion was solved with the help of the Verlet algorithm (Verlet
1967
)
R
k
(
t
+
t)
−
R
k
(t
−
t)
V
k
(t)
=
(2.21)
2
t
P
k
(t)
=
M
k
V
k
(t)
(2.22)
