4.2 Vibrational Wavepacket Dynamics
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factor exp(−iωt/2), the system is back to the initial wavefunction at times that are
integer multiples of T : χ(Q, nT ) = χ(Q, 0). Moreover, at half-integer multiples of
T all coefficients with even v take their initial values, while those with odd v change
in sign, so the wavefunction converts to the specular image of the initial one with
respect to the equilibrium point Q = 0, i.e., χ(Q, T /2 + nT ) = χ(−Q, 0).
Putting together the concepts of Franck–Condon excitation, Ehrenfest’s theorem
and the harmonic approximation, we can sketch the short-time dynamics as follows:
1. Excitation by an ultrashort pulse creates a localized nuclear wavepacket in an
excited state PES. The wavepacket resembles the starting vibrational state in the
ground-state PES.
2. Radiationless transitions to other states can be neglected, unless they are very
fast, as in the presence of conical intersections (see next chapter). The dynamics
is therefore “adiabatic.”
3. The center of the wavepacket moves in the excited PES according to classical
mechanics, as far as the wavepacket is sufficiently well localized in the coordinate
space.
4. If the excited PES is approximately harmonic and the wavepacket can be decomposed as a product of factors, each depending on one normal coordinate, it will
oscillate along each coordinate according to the relative frequency.
Note that the harmonic approximation is usually a good one for the lowest vibrational levels in the ground-state PES and yields wavefunctions that are factorized
in the normal coordinates Q
(gs) : χ v (Q
(gs)
) =
r χ
(r )
v r
(Q
(gs)
r ). In the Franck–Condon
approximation, this wavepacket is translated into the excited state PES without modification. However, independent oscillations along each normal coordinate, as referred
to in the last point of the above list, only occur if the wavepacket is factorized as
a function of the excited state normal coordinates Q
(ex) . Unfortunately the Q
(ex)
are linear combinations of the Cartesian coordinates of the nuclei just as the Q
(gs) ,
but with different coefficients. The factorization of χ v in the Q
(ex) coordinates is
therefore approximately true only if one can neglect the difference between the two
normal coordinates systems, i.e., the so-called Duschinsky effect.
Figures 4.1, 4.2, and 4.3 illustrate the relationship between the excitation process
and the adiabatic dynamics for one-dimensional potentials, i.e., for a model diatomic
molecule in which we consider two electronic states. The transition dipole moment
is assumed to be independent on the internuclear distance and the radiation pulse
has a Gaussian shape as in Eq. (3.76). Its amplitude has been kept sufficiently low as
to be in the first-order TDPT regime, the excitation probability being less than 1%
in all cases: however, the results illustrated in this section are based on numerically
exact calculations. The carrier frequency ω is tuned to the vibronic transition with the
largest Franck–Condon factor, i.e., to χ e,24 for the harmonic potentials and to χ e,40
for the Morse ones (see Fig. 4.1). Figures 4.2 and 4.3 show the average internuclear
distance R in the excited state as a function of time.
In the harmonic model R undergoes perfectly periodic oscillations between two
turning points. With an ultrashort pulse (τ = 10 fs) the requirements for a Franck–
Condon excitation are fulfilled; thus the inner turning point is close to the equilibrium
123
factor exp(−iωt/2), the system is back to the initial wavefunction at times that are
integer multiples of T : χ(Q, nT ) = χ(Q, 0). Moreover, at half-integer multiples of
T all coefficients with even v take their initial values, while those with odd v change
in sign, so the wavefunction converts to the specular image of the initial one with
respect to the equilibrium point Q = 0, i.e., χ(Q, T /2 + nT ) = χ(−Q, 0).
Putting together the concepts of Franck–Condon excitation, Ehrenfest’s theorem
and the harmonic approximation, we can sketch the short-time dynamics as follows:
1. Excitation by an ultrashort pulse creates a localized nuclear wavepacket in an
excited state PES. The wavepacket resembles the starting vibrational state in the
ground-state PES.
2. Radiationless transitions to other states can be neglected, unless they are very
fast, as in the presence of conical intersections (see next chapter). The dynamics
is therefore “adiabatic.”
3. The center of the wavepacket moves in the excited PES according to classical
mechanics, as far as the wavepacket is sufficiently well localized in the coordinate
space.
4. If the excited PES is approximately harmonic and the wavepacket can be decomposed as a product of factors, each depending on one normal coordinate, it will
oscillate along each coordinate according to the relative frequency.
Note that the harmonic approximation is usually a good one for the lowest vibrational levels in the ground-state PES and yields wavefunctions that are factorized
in the normal coordinates Q
(gs) : χ v (Q
(gs)
) =
r χ
(r )
v r
(Q
(gs)
r ). In the Franck–Condon
approximation, this wavepacket is translated into the excited state PES without modification. However, independent oscillations along each normal coordinate, as referred
to in the last point of the above list, only occur if the wavepacket is factorized as
a function of the excited state normal coordinates Q
(ex) . Unfortunately the Q
(ex)
are linear combinations of the Cartesian coordinates of the nuclei just as the Q
(gs) ,
but with different coefficients. The factorization of χ v in the Q
(ex) coordinates is
therefore approximately true only if one can neglect the difference between the two
normal coordinates systems, i.e., the so-called Duschinsky effect.
Figures 4.1, 4.2, and 4.3 illustrate the relationship between the excitation process
and the adiabatic dynamics for one-dimensional potentials, i.e., for a model diatomic
molecule in which we consider two electronic states. The transition dipole moment
is assumed to be independent on the internuclear distance and the radiation pulse
has a Gaussian shape as in Eq. (3.76). Its amplitude has been kept sufficiently low as
to be in the first-order TDPT regime, the excitation probability being less than 1%
in all cases: however, the results illustrated in this section are based on numerically
exact calculations. The carrier frequency ω is tuned to the vibronic transition with the
largest Franck–Condon factor, i.e., to χ e,24 for the harmonic potentials and to χ e,40
for the Morse ones (see Fig. 4.1). Figures 4.2 and 4.3 show the average internuclear
distance R in the excited state as a function of time.
In the harmonic model R undergoes perfectly periodic oscillations between two
turning points. With an ultrashort pulse (τ = 10 fs) the requirements for a Franck–
Condon excitation are fulfilled; thus the inner turning point is close to the equilibrium