Chemistry Reference
In-Depth Information
Fig. 6.1
e
linear JT problem. The nuclear
displacement coordinates are the tetragonal elongation,
Q
θ
, and the orthorhombic in-plane distor-
tion,
Q
The Mexican hat potential-energy surface of the
E
×
Then the secular equation of the force element matrix in Eq. (
6.69
) becomes:
ε
k
−
F
E
ρ
2
cos
2
ϕ
F
E
ρ
2
sin
2
ϕ
−
=
0
(6.71)
Two roots are found, which are independent of the angular coordinate. The corre-
sponding eigenfunctions are:
cos
ϕ
sin
ϕ
1
=
F
E
ρ
−→ |
ψ
1
=
2
|
Eθ
+
2
|
E
(6.72)
sin
ϕ
cos
ϕ
2
=−
F
E
ρ
−→ |
ψ
2
=−
2
|
Eθ
+
2
|
E
The surface consists of two sheets and exhibits rotational symmetry.
1
2
K
E
ρ
2
E
±
=
E
0
+
±
F
E
ρ
2
K
E
Q
θ
+
Q
2
±
F
E
Q
θ
+
Q
2
1
=
E
0
+
(6.73)
A cross section of this surface looks like a two-well potential, with two displaced
parabolæ. The depth of the well is called the JT stabilization energy:
F
E
2
K
E
E
JT
=−
(6.74)
In the 2D space of the active modes these parabolæ revolve around the centre, giv-
ing rise to the Mexican hat appearance. At the origin this surface has the shape
of a conical intersection, indicating that the high-symmetry point is unstable, and
will spontaneously relax to the circular trough surrounding the degeneracy [
8
]. The
