120
4 Wavepacket Dynamics and Geometrical Relaxation
|Ψ exc =
π
1/2 i
2 1/2
e
iφ
v
ϕ k χ k,v
µ kv,0u · ˜
E 0 (Δω kv,0u )
(4.1)
where µ kv,0u =
ϕ k χ k,v |µ| ϕ 0 χ 0,u
is the transition dipole moment and Δω kv,0u =
ω − (E kv − E 0u )/ is the detuning. Here we adopt the Born–Oppenheimer electrostatic approximation because we have seen in Sects. 3.10 and 3.11 that the coupling
with vibrationally excited dark states can be neglected as far as the excitation with
short light pulses is concerned. As before, we shall also replace ˜
E 0 (Δω kv,0u ) with its
value at zero detuning, although the wider range of the final energies E kv makes this
approximation less accurate. So, leaving aside the inessential factors, we get
|Ψ exc =
v
ϕ k χ k,v
µ kv,0u · ˆ
e p
(4.2)
where ˆ
e p is the polarization versor of the exciting light. This expression describes
a nuclear “wavepacket” in the electronic state ϕ k with a shape that depends on the
vibrational eigenstates χ k,v and their transition dipoles µ kv,0u , but not on the features of the radiation pulse, provided it is short enough. By wavepacket we mean a
nonstationary normalizable wavefunction and this term is most often used for welllocalized distributions in the coordinate space. A generalization of Eq. 4.2 to more
electronic states can be envisaged if their PESs are close enough. A further simplification is possible when the Franck–Condon approximation is valid, i.e., in the case
of dipole allowed electronic transitions. We assume µ kv,0u µ k,0 (R
(eq)
0 )χ k,v
χ 0,u
where R
(eq)
0
are the equilibrium coordinates in the ground state. Then
|Ψ exc = µ k,0 (R
(eq)
0 ) · ˆ
e p
v
ϕ k χ k,v
χ k,v
χ 0,u = µ k,0 (R
(eq)
0 ) · ˆ
e p
ϕ k χ 0,u
.
(4.3)
The last equality stems from the completeness of the
χ k,v
set of vibrational states as
a basis for wavefunctions of the nuclear coordinates (actually it is sufficient to assume
that the χ 0,u function can be expanded on a subset of the χ k,v with energies well within
the bandwidth of the light pulse). The excitation then generates a wavepacket with
the same shape as the initial vibrational wavefunction χ 0,u , translated into the excited
PES of state ϕ k . This is called a “Franck–Condon” excitation, and the region of the
excited PES occupied by the wavepacket is called the Franck–Condon region. Of
course in the new PES the χ 0,u wavepacket is not anymore a vibrational eigenstate,
so it will evolve in time as shown in the next section. On the contrary a long pulse
with a good frequency resolution (small FWHM ω ) allows to select a single vibrational
eigenstate in the upper potential, thus creating a stationary state (apart from the
almost ubiquitous decay to lower states, with or without photon emission).
In classical terms, it is often stated that the excitation process is so fast that the
positions and momenta of the nuclei cannot change. What happens is then a “vertical
excitation” whereby the point representing the molecule in the nuclear phase space
(coordinates and momenta) is translated from the ground-state PES to the excited
4 Wavepacket Dynamics and Geometrical Relaxation
|Ψ exc =
π
1/2 i
2 1/2
e
iφ
v
ϕ k χ k,v
µ kv,0u · ˜
E 0 (Δω kv,0u )
(4.1)
where µ kv,0u =
ϕ k χ k,v |µ| ϕ 0 χ 0,u
is the transition dipole moment and Δω kv,0u =
ω − (E kv − E 0u )/ is the detuning. Here we adopt the Born–Oppenheimer electrostatic approximation because we have seen in Sects. 3.10 and 3.11 that the coupling
with vibrationally excited dark states can be neglected as far as the excitation with
short light pulses is concerned. As before, we shall also replace ˜
E 0 (Δω kv,0u ) with its
value at zero detuning, although the wider range of the final energies E kv makes this
approximation less accurate. So, leaving aside the inessential factors, we get
|Ψ exc =
v
ϕ k χ k,v
µ kv,0u · ˆ
e p
(4.2)
where ˆ
e p is the polarization versor of the exciting light. This expression describes
a nuclear “wavepacket” in the electronic state ϕ k with a shape that depends on the
vibrational eigenstates χ k,v and their transition dipoles µ kv,0u , but not on the features of the radiation pulse, provided it is short enough. By wavepacket we mean a
nonstationary normalizable wavefunction and this term is most often used for welllocalized distributions in the coordinate space. A generalization of Eq. 4.2 to more
electronic states can be envisaged if their PESs are close enough. A further simplification is possible when the Franck–Condon approximation is valid, i.e., in the case
of dipole allowed electronic transitions. We assume µ kv,0u µ k,0 (R
(eq)
0 )χ k,v
χ 0,u
where R
(eq)
0
are the equilibrium coordinates in the ground state. Then
|Ψ exc = µ k,0 (R
(eq)
0 ) · ˆ
e p
v
ϕ k χ k,v
χ k,v
χ 0,u = µ k,0 (R
(eq)
0 ) · ˆ
e p
ϕ k χ 0,u
.
(4.3)
The last equality stems from the completeness of the
χ k,v
set of vibrational states as
a basis for wavefunctions of the nuclear coordinates (actually it is sufficient to assume
that the χ 0,u function can be expanded on a subset of the χ k,v with energies well within
the bandwidth of the light pulse). The excitation then generates a wavepacket with
the same shape as the initial vibrational wavefunction χ 0,u , translated into the excited
PES of state ϕ k . This is called a “Franck–Condon” excitation, and the region of the
excited PES occupied by the wavepacket is called the Franck–Condon region. Of
course in the new PES the χ 0,u wavepacket is not anymore a vibrational eigenstate,
so it will evolve in time as shown in the next section. On the contrary a long pulse
with a good frequency resolution (small FWHM ω ) allows to select a single vibrational
eigenstate in the upper potential, thus creating a stationary state (apart from the
almost ubiquitous decay to lower states, with or without photon emission).
In classical terms, it is often stated that the excitation process is so fast that the
positions and momenta of the nuclei cannot change. What happens is then a “vertical
excitation” whereby the point representing the molecule in the nuclear phase space
(coordinates and momenta) is translated from the ground-state PES to the excited