6.6 Application: The Jahn–Teller Effect
131
The second-order expressions then are reduced to a diagonal matrix element:
Γ a Γ 0 Γ a
Γ 0 |ΓγΓ
′ γ
′
Γ a γ b |Γ 0 Γ a γ b
=
1
√
dim(Γ )
Γ a Γ 0 Γ a δ ΓΓ ′ δ γγ ′ δ γ a γ b = K Γ δ ΓΓ ′ δ γγ ′ δ γ a γ b (6.65)
K Γ in this equation is the harmonic force-constant. It gives rise to a constant diagonal term which provides an attractive potential around the minimum and keeps
the surface bound at larger distances from the origin. The general expression for the
potential-energy surface then becomes:
E k (Q) = E 0 +
Γ
1
2
K Γ
γ
Q
2
Γγ
+ ε k (Q)
(6.66)
Here, ε k (Q) represents the kth root of the Hamiltonian matrix. This equation describes a surface with multiple sheets, one for each root, which cross in the highsymmetry origin. In its simplest form the Hamiltonian can be restricted to the linear
terms only. In the second-order approximation non-totally-symmetric second-order
terms will also be included.
The prototype of the JT surface is the celebrated Mexican hat potential, which
describes the effect of the twofold-degenerate cubic or trigonal E state. A typical
example is the 2 E g ground state of octahedral Cu
2+ complexes, with (t 2g ) 6 (e g ) 3
configuration. The JT-active mode in this case is restricted to an e g mode, corresponding to the symmetrized square.
[E g × E g ]−A 1g = E g
(6.67)
This distortion mode consists of the tetragonal and orthorhombic stretchings, which
we already encountered as vibrational modes of UF 6 , and are depicted in Fig. 6.1.
By use of the appropriate Ei|Ej E k coupling coefficients the JT matrix can easily
be derived. The force element is defined as:
F E =
Eθ
∂H
∂Q θ
Eθ
(6.68)
The matrix then becomes:
H =
E 0 +
1
2
K E
Q
2
θ + Q
2
ǫ
10
01
+ F E
Q θ
Q ǫ
Q ǫ −Q θ
(6.69)
To diagonalize this Hamiltonian, it is convenient to transform to cylindrical coordinates {ρ,ϕ}:
Q θ = ρ cos ϕ
Q ǫ = ρ sin ϕ
(6.70)
131
The second-order expressions then are reduced to a diagonal matrix element:
Γ a Γ 0 Γ a
Γ 0 |ΓγΓ
′ γ
′
Γ a γ b |Γ 0 Γ a γ b
=
1
√
dim(Γ )
Γ a Γ 0 Γ a δ ΓΓ ′ δ γγ ′ δ γ a γ b = K Γ δ ΓΓ ′ δ γγ ′ δ γ a γ b (6.65)
K Γ in this equation is the harmonic force-constant. It gives rise to a constant diagonal term which provides an attractive potential around the minimum and keeps
the surface bound at larger distances from the origin. The general expression for the
potential-energy surface then becomes:
E k (Q) = E 0 +
Γ
1
2
K Γ
γ
Q
2
Γγ
+ ε k (Q)
(6.66)
Here, ε k (Q) represents the kth root of the Hamiltonian matrix. This equation describes a surface with multiple sheets, one for each root, which cross in the highsymmetry origin. In its simplest form the Hamiltonian can be restricted to the linear
terms only. In the second-order approximation non-totally-symmetric second-order
terms will also be included.
The prototype of the JT surface is the celebrated Mexican hat potential, which
describes the effect of the twofold-degenerate cubic or trigonal E state. A typical
example is the 2 E g ground state of octahedral Cu
2+ complexes, with (t 2g ) 6 (e g ) 3
configuration. The JT-active mode in this case is restricted to an e g mode, corresponding to the symmetrized square.
[E g × E g ]−A 1g = E g
(6.67)
This distortion mode consists of the tetragonal and orthorhombic stretchings, which
we already encountered as vibrational modes of UF 6 , and are depicted in Fig. 6.1.
By use of the appropriate Ei|Ej E k coupling coefficients the JT matrix can easily
be derived. The force element is defined as:
F E =
Eθ
∂H
∂Q θ
Eθ
(6.68)
The matrix then becomes:
H =
E 0 +
1
2
K E
Q
2
θ + Q
2
ǫ
10
01
+ F E
Q θ
Q ǫ
Q ǫ −Q θ
(6.69)
To diagonalize this Hamiltonian, it is convenient to transform to cylindrical coordinates {ρ,ϕ}:
Q θ = ρ cos ϕ
Q ǫ = ρ sin ϕ
(6.70)