32
A. Palacios et al.
wave packet is written in terms of the stationary states of the system, the expansion
coefficients directly give the corresponding amplitudes, which allows for a straightforward analysis of the spectral components (total, energy- and angle-differential)
in the wave packet. Therefore, for a given final energy W ε α v α , and a given energy
sharing for nuclei (E v α ) and electrons (ε α ) such as W ε α v α = E v α + ε α , the ionization
probability is:
d 2 P α (E v α , ε α , T )
dE v α dε α
=
C
α
αε α v α
(t = t max )
2 ,
(2.9)
where α stands for the angular momentum contribution of the ejected electron in the α ionization threshold, and T is the pulse length. The upper time
limit in the Runge-Kutta integration of the TDSE is t max , and it should be chosen such t max > T . It is important to note that the asymptotic limit is only reached
when autoionization of significantly populated DES has occurred, which implies
t max > T + τ max , being τ max the largest lifetime of those DES. This topic is further
discussed and illustrated in Sect. 2.4.
The theoretical calculations presented in the next Sections are performed for linearly polarized light parallel to the molecular axis. According with dipole selection
rules, the only relevant molecular states are those with total symmetry Σ +
g and Σ +
u .
The potential energy curves of those states are shown in Fig. 2.1 of Sect. 2.4.
2.4 Time-Resolved Imaging of H 2 Autoionization
Several molecular-dynamics processes involve DES. Autoionization receives particular attention because of its key role in electron scattering and photoionization.
Given the complexity of accounting for all degrees of freedom to properly describe
molecular autoionization, most experimental and theoretical studies on characterization, structure and mechanisms of molecular DES have used the simplest possible
targets (H 2 , D 2 ). DES are highly correlated electronic states (both electrons are simultaneously excited by absorption of one or more photons) and decay into the
background continuum in a time scale comparable to that of nuclear motion. As it
will be shown in the present work, through the evaluation of the competing dissociative and non-dissociative ionization channels, one can obtain relevant information
on the role of the combined nuclear motion with electron correlation in the DES of
the hydrogen molecule.
The potential energy curves for H 2 are shown in Fig. 2.1. In full thick lines, there
appear the ground state of the neutral molecule (X 1 Σ +
g ), the six lowest single ionization thresholds of H
+
2 (1sσ g , 2pσ u , 2pπ u , 2sσ g , 3pσ u and 3dσ g ), and the double
ionization threshold (the 1/R potential curve corresponding to the full Coulomb
break up of the system). The lowest singly excited states of 1 Σ +
g and 1 Σ +
u symmetries are also plotted. The DES of H 2 lie above its first ionization threshold. Each
one of the higher ionization thresholds have associated series of DES (labeled as
Q n ). In Fig. 2.1, there are only shown the first two series, Q 1 and Q 2 , converging
A. Palacios et al.
wave packet is written in terms of the stationary states of the system, the expansion
coefficients directly give the corresponding amplitudes, which allows for a straightforward analysis of the spectral components (total, energy- and angle-differential)
in the wave packet. Therefore, for a given final energy W ε α v α , and a given energy
sharing for nuclei (E v α ) and electrons (ε α ) such as W ε α v α = E v α + ε α , the ionization
probability is:
d 2 P α (E v α , ε α , T )
dE v α dε α
=
C
α
αε α v α
(t = t max )
2 ,
(2.9)
where α stands for the angular momentum contribution of the ejected electron in the α ionization threshold, and T is the pulse length. The upper time
limit in the Runge-Kutta integration of the TDSE is t max , and it should be chosen such t max > T . It is important to note that the asymptotic limit is only reached
when autoionization of significantly populated DES has occurred, which implies
t max > T + τ max , being τ max the largest lifetime of those DES. This topic is further
discussed and illustrated in Sect. 2.4.
The theoretical calculations presented in the next Sections are performed for linearly polarized light parallel to the molecular axis. According with dipole selection
rules, the only relevant molecular states are those with total symmetry Σ +
g and Σ +
u .
The potential energy curves of those states are shown in Fig. 2.1 of Sect. 2.4.
2.4 Time-Resolved Imaging of H 2 Autoionization
Several molecular-dynamics processes involve DES. Autoionization receives particular attention because of its key role in electron scattering and photoionization.
Given the complexity of accounting for all degrees of freedom to properly describe
molecular autoionization, most experimental and theoretical studies on characterization, structure and mechanisms of molecular DES have used the simplest possible
targets (H 2 , D 2 ). DES are highly correlated electronic states (both electrons are simultaneously excited by absorption of one or more photons) and decay into the
background continuum in a time scale comparable to that of nuclear motion. As it
will be shown in the present work, through the evaluation of the competing dissociative and non-dissociative ionization channels, one can obtain relevant information
on the role of the combined nuclear motion with electron correlation in the DES of
the hydrogen molecule.
The potential energy curves for H 2 are shown in Fig. 2.1. In full thick lines, there
appear the ground state of the neutral molecule (X 1 Σ +
g ), the six lowest single ionization thresholds of H
+
2 (1sσ g , 2pσ u , 2pπ u , 2sσ g , 3pσ u and 3dσ g ), and the double
ionization threshold (the 1/R potential curve corresponding to the full Coulomb
break up of the system). The lowest singly excited states of 1 Σ +
g and 1 Σ +
u symmetries are also plotted. The DES of H 2 lie above its first ionization threshold. Each
one of the higher ionization thresholds have associated series of DES (labeled as
Q n ). In Fig. 2.1, there are only shown the first two series, Q 1 and Q 2 , converging