134
6 Interactions
Fig. 6.2 Rotation of the
distortion in the trough of the
Mexican hat. Along the Q θ
coordinate the complex is
elongated along its z-axis.
Rotation around the centre in
the direction of Q ǫ will
shorten the z-axis and
increase the x-axis. At an
angle of revolution of 120 ◦ a
tetragonally elongated
structure is again found, but
this time with the elongation
along the x-direction, and
similarly at 240 ◦ ,withthe
elongation along the y-axis
operator for a rotation in the function space (see [9]). We can construct this by analogy with Eq. (6.76), but with an important amendment: as we have argued while
discussing Fig. 6.2, the coordinates rotate twice as quickly as the wavevector, and
hence a prefactor of 1/2 is required!
S =
1
2
|EθEǫ|−|EǫEθ|
(6.78)
Only in this case does the total momentum operator J = L + S commute with the
Hamiltonian:
J , H
′
=
L, H
′
+
S, H
′
= 0
(6.79)
As the reader will have noticed, we have made use of the standard spectroscopic
symbols for orbital angular momentum, spin momentum, and total momentum.
Vibronic coupling is indeed analogous to coupling of spin and orbit momenta in
cylindrical molecules. To form the vibronic wavefunction, describing the dynamics
of the Mexican hat system, the electronic state has to be combined with nuclear
wavefunctions. If the JT effect is pronounced, the vibronic levels take the form of
a radial oscillator, describing transverse oscillations in the bottom of the through,
and pseudo-rotational levels, describing the longitudinal motion along the bottom
of the trough. The total vibronic wavefunction should of course be single-valued
after a full turn around the trough, which takes the system back to the starting point.
Hence, since the electronic part changes sign after a full turn, the vibrational part
should also show a compensating sign change. This is indeed the case: the pseudorotational levels are characterized by half-integral angular momentum [10].
6.7 Application: Pseudo-Jahn–Teller interactions
Pseudo-JT interactions (PJT) refer to the second-order vibronic coupling between
electronic states which are separated by a gap [11]. In this section we describe the
6 Interactions
Fig. 6.2 Rotation of the
distortion in the trough of the
Mexican hat. Along the Q θ
coordinate the complex is
elongated along its z-axis.
Rotation around the centre in
the direction of Q ǫ will
shorten the z-axis and
increase the x-axis. At an
angle of revolution of 120 ◦ a
tetragonally elongated
structure is again found, but
this time with the elongation
along the x-direction, and
similarly at 240 ◦ ,withthe
elongation along the y-axis
operator for a rotation in the function space (see [9]). We can construct this by analogy with Eq. (6.76), but with an important amendment: as we have argued while
discussing Fig. 6.2, the coordinates rotate twice as quickly as the wavevector, and
hence a prefactor of 1/2 is required!
S =
1
2
|EθEǫ|−|EǫEθ|
(6.78)
Only in this case does the total momentum operator J = L + S commute with the
Hamiltonian:
J , H
′
=
L, H
′
+
S, H
′
= 0
(6.79)
As the reader will have noticed, we have made use of the standard spectroscopic
symbols for orbital angular momentum, spin momentum, and total momentum.
Vibronic coupling is indeed analogous to coupling of spin and orbit momenta in
cylindrical molecules. To form the vibronic wavefunction, describing the dynamics
of the Mexican hat system, the electronic state has to be combined with nuclear
wavefunctions. If the JT effect is pronounced, the vibronic levels take the form of
a radial oscillator, describing transverse oscillations in the bottom of the through,
and pseudo-rotational levels, describing the longitudinal motion along the bottom
of the trough. The total vibronic wavefunction should of course be single-valued
after a full turn around the trough, which takes the system back to the starting point.
Hence, since the electronic part changes sign after a full turn, the vibrational part
should also show a compensating sign change. This is indeed the case: the pseudorotational levels are characterized by half-integral angular momentum [10].
6.7 Application: Pseudo-Jahn–Teller interactions
Pseudo-JT interactions (PJT) refer to the second-order vibronic coupling between
electronic states which are separated by a gap [11]. In this section we describe the