Chemistry Reference
In-Depth Information
The kinetic energy is then given by
m
i
d
S
i
d
t
2
21
1
2
T
=
i
=
1
d
S
i
d
t
2
d
S
i
d
t
2
3
21
M
2
m
2
=
+
(4.92)
i
=
1
i
=
4
Here,
M
is the atomic mass of uranium, and
m
is the atomic mass of fluorine. The
kinetic energy can be reduced to a uniform scalar product by mass weighting the
coordinates, i.e., by multiplying the
S
coordinates with the square root of the atomic
m
ass
of the displaced atom. We shall denote these as the vector
Q
. Hence,
Q
i
=
√
m
i
S
i
:
d
Q
i
d
t
2
2
i
2
i
1
1
Q
i
T
=
=
(4.93)
where the dot over
Q
denotes the time derivative. The potential energy will be ap-
proximated by second-order derivatives of the potential energy surface
V(
Q
)
in the
mass-weighted coordinates:
∂
2
V
∂Q
i
∂Q
j
V
ij
=
(4.94)
These derivatives are the elements of the Hessian matrix,
, which is symmetric
about the diagonal. The potential minimum coincides with the octahedral geometry.
The resulting potential energy is
V
2
i,j
1
V
=
V
ij
Q
i
Q
j
(4.95)
The kinetic and potential energies are combined to form the Lagrangian,
L
=
T
−
V
.
The equation of motion is given by
∂L
∂Q
k
=
d
d
t
∂L
∂ Q
k
(4.96)
The partial derivatives in this equation are given by
2
i
∂L
∂Q
k
=−
∂V
∂Q
k
=−
1
V
kk
Q
k
−
(V
ik
+
V
ki
)Q
i
=
k
=−
V
kk
Q
k
−
V
ki
Q
i
=−
V
ki
Q
i
(4.97)
i
=
k
i
d
d
t
∂L
∂ Q
k
=
d
d
t
∂T
∂ Q
k
=
d
d
t
Q
k
=
Q
k
(4.98)
