6.6 Application: The Jahn–Teller Effect
133
distorted system in the trough orbits around the origin. This motion is a pseudorotation, i.e. it is not a rotation of the molecular frame, but a gradual redistribution
of the distortions between the Cartesian directions. We illustrate this in Fig. 6.2.
The starting point is at ϕ = 0, in the direction of the Q θ mode. In this mode the
z-axis is elongated, and the xy-plane is contracted. A counterclockwise rotation activates the orthorhombic Q ǫ mode, while the tetragonal Q θ mode is receding. This
introduces a difference between the x- and y-axes: the distortion in the x-direction
becomes more pronounced, while the y-axis contracts further. At the same time the
elongation of the z-axis diminishes. At an angle of 60 ◦ the x- and z-axes are both
elongated to an equal extent, giving rise to a weak xz-plane and a strong y-axis.
At the 90 ◦ point the Q θ contribution vanishes, and the distortion is orthorhombic,
withashorty-axis, an elongated x-axis, and an undistorted z-axis. At an angle of
120 ◦ we reach the point where the x-axis is weak, and the perpendicular yz-plane is
strong. We thus have regained an elongated tetragonal configuration, but the elongation has been rotated from the z-axis to the x-axis. Continuing now at 240 ◦ we
shall have travelled another third of the trough and reoriented the tetragonal axis
along the y-direction. We can also follow the wavevector along the trough. If it is
assumed that F E < 0, the lower eigenfunction will be the |ψ 1 eigenfunction of Eq.
(6.72). In the starting elongated tetragonal configuration the ground state coincides
with the |Eθ basis function. By the time we have reached the orthorhombic configuration at ϕ = 90 ◦ , the ground state has rotated by only half that angle and equals
1/
√
2(|Eθ+|Eǫ). At 180 ◦ , we reach a structure which is tetragonally compressed
along the z-direction. Accordingly, the Q θ mode has changed sign, in contrast to the
eigenfunction, where |Eθ is replaced by |Eǫ. The observation of rotational symmetry is an unexpected feature, which is not related to the point group, but which
stems from the limitation of the JT Hamiltonian to linear terms. To describe this
symmetry we first reformulate the force element Hamiltonian in Eq. (6.69)inDirac
notation:
H
′ = F E
Q θ
|EθEθ|−|EǫEǫ|
+ Q ǫ
|EθEǫ|+|EǫEθ|
(6.75)
The angular momentum operator, corresponding to a rotation in coordinate space,
is given by:
L =
∂
∂ϕ
=
∂Q ǫ
∂ϕ
∂
∂Q ǫ
+
∂Q θ
∂ϕ
∂
∂Q θ
= Q θ
∂
∂Q ǫ
− Q ǫ
∂
∂Q θ
(6.76)
The partial derivatives were obtained from Eq. (6.70). The commutator of this operator with the Hamiltonian is:
L, H
′
= F E
−Q ǫ
|EθEθ|−|EǫEǫ|
+ Q θ
|EθEǫ|+|EǫEθ|
(6.77)
Surprisingly, this commutator does not vanish. This is an important observation,
which directly points to the vibrational-electronic or vibronic coupling between the
distortion modes and the electronic wavevector. When the system rotates around the
origin in coordinate space, not only are the coordinates changing, but the wavevector is also rotating simultaneously, so we must also provide an angular momentum
133
distorted system in the trough orbits around the origin. This motion is a pseudorotation, i.e. it is not a rotation of the molecular frame, but a gradual redistribution
of the distortions between the Cartesian directions. We illustrate this in Fig. 6.2.
The starting point is at ϕ = 0, in the direction of the Q θ mode. In this mode the
z-axis is elongated, and the xy-plane is contracted. A counterclockwise rotation activates the orthorhombic Q ǫ mode, while the tetragonal Q θ mode is receding. This
introduces a difference between the x- and y-axes: the distortion in the x-direction
becomes more pronounced, while the y-axis contracts further. At the same time the
elongation of the z-axis diminishes. At an angle of 60 ◦ the x- and z-axes are both
elongated to an equal extent, giving rise to a weak xz-plane and a strong y-axis.
At the 90 ◦ point the Q θ contribution vanishes, and the distortion is orthorhombic,
withashorty-axis, an elongated x-axis, and an undistorted z-axis. At an angle of
120 ◦ we reach the point where the x-axis is weak, and the perpendicular yz-plane is
strong. We thus have regained an elongated tetragonal configuration, but the elongation has been rotated from the z-axis to the x-axis. Continuing now at 240 ◦ we
shall have travelled another third of the trough and reoriented the tetragonal axis
along the y-direction. We can also follow the wavevector along the trough. If it is
assumed that F E < 0, the lower eigenfunction will be the |ψ 1 eigenfunction of Eq.
(6.72). In the starting elongated tetragonal configuration the ground state coincides
with the |Eθ basis function. By the time we have reached the orthorhombic configuration at ϕ = 90 ◦ , the ground state has rotated by only half that angle and equals
1/
√
2(|Eθ+|Eǫ). At 180 ◦ , we reach a structure which is tetragonally compressed
along the z-direction. Accordingly, the Q θ mode has changed sign, in contrast to the
eigenfunction, where |Eθ is replaced by |Eǫ. The observation of rotational symmetry is an unexpected feature, which is not related to the point group, but which
stems from the limitation of the JT Hamiltonian to linear terms. To describe this
symmetry we first reformulate the force element Hamiltonian in Eq. (6.69)inDirac
notation:
H
′ = F E
Q θ
|EθEθ|−|EǫEǫ|
+ Q ǫ
|EθEǫ|+|EǫEθ|
(6.75)
The angular momentum operator, corresponding to a rotation in coordinate space,
is given by:
L =
∂
∂ϕ
=
∂Q ǫ
∂ϕ
∂
∂Q ǫ
+
∂Q θ
∂ϕ
∂
∂Q θ
= Q θ
∂
∂Q ǫ
− Q ǫ
∂
∂Q θ
(6.76)
The partial derivatives were obtained from Eq. (6.70). The commutator of this operator with the Hamiltonian is:
L, H
′
= F E
−Q ǫ
|EθEθ|−|EǫEǫ|
+ Q θ
|EθEǫ|+|EǫEθ|
(6.77)
Surprisingly, this commutator does not vanish. This is an important observation,
which directly points to the vibrational-electronic or vibronic coupling between the
distortion modes and the electronic wavevector. When the system rotates around the
origin in coordinate space, not only are the coordinates changing, but the wavevector is also rotating simultaneously, so we must also provide an angular momentum