2 XUV Lasers for Ultrafast Electronic Control in H 2
31
The electronic structure is obtained within the Feshbach formalism [49]. The
bound electronic states are obtained by directly solving the eigenvalue problem,
ˆ
H el (r, R)φ n (r, R) = E n (R)φ n (r, R),
(2.5)
whereas the representation of continuum and autoionizing electronic states uses the
Feshbach subspaces. Two orthogonal complementary subspaces are defined ( ˆ
Q +
ˆ
P = 1) respectively containing the resonant ( ˆ
Q) and non-resonant ( ˆ
P ) contribution
to the continuum electronic wave function at a given energy. Note that precisely the
use of two subspaces, respectively holding the electronic continuum and the doubly
excited states (DES) embedded in that continuum, makes the method suitable for a
straightforward time-resolved tracing of the DES decay. We then solve the electronic
eigenvalue problem for each subspace:
[ ˆ
Q ˆ
H el ˆ
Q]φ r (r, R) = E r (R)φ r (r, R),
(2.6)
[ ˆ
P ˆ
H el ˆ
P ]φ α,, α ,ε (r, R) = E α,ε (R)φ α,, α ,ε (r, R).
(2.7)
The eigenvalue equations for the bound states [Eq. (2.5)] and the resonant component of the continuum [Eq. (2.6)] are solved using a configuration interaction
method in a basis of H
+
2 orbitals, whereas the non-resonant continuum electronic
states, Eq. (2.7), are calculated using a multichannel L 2 close-coupling procedure
[66, 67]. The basis set of H
+
2 orbitals are written as single center expansions using
spherical harmonics for the angular part and B-spline basis functions for the radial
part [68, 69]. The results presented in following sections have been obtained using angular momenta expansions up to = 16, and up to 180 B-splines functions
of order k = 8 in a box of size 60 a.u., which is large enough to avoid unphysical
reflections. For the close-coupling procedure, angular momenta of the ejected electron are included up to l α = 7, for each α electronic state in the discretized continua
associated to each ionization threshold of the H
+
2 molecule.
Once the electronic structure is obtained in a given grid of internuclear distances,
we solve the one-dimensional Schrödinger equation to calculate the nuclear wave
functions χ v i (R):
ˆ
T (R) + E x (R)
χ v x (R) = W x,v x χ v x (R)
(2.8)
where x stands for a bound, resonant or continuum electronic state, and E x is the
corresponding potential energy curve previously obtained. For the results here presented, the basis for the nuclear wave functions had up to 300 B-splines defined in
a box of size 12 a.u.
By projecting onto the basis of vibronic states, the TDSE defined in Eq. (2.1) is
reduced to a system of coupled differential equations that can be solved using ordinary integration procedures. A sixth-order Runge-Kutta algorithm, implemented in
PETSc libraries [70], is used for the integration.
Excitation and ionization amplitudes are extracted by projecting the propagated
time-dependent wave packet into the excitation or ionization final states. Since the
31
The electronic structure is obtained within the Feshbach formalism [49]. The
bound electronic states are obtained by directly solving the eigenvalue problem,
ˆ
H el (r, R)φ n (r, R) = E n (R)φ n (r, R),
(2.5)
whereas the representation of continuum and autoionizing electronic states uses the
Feshbach subspaces. Two orthogonal complementary subspaces are defined ( ˆ
Q +
ˆ
P = 1) respectively containing the resonant ( ˆ
Q) and non-resonant ( ˆ
P ) contribution
to the continuum electronic wave function at a given energy. Note that precisely the
use of two subspaces, respectively holding the electronic continuum and the doubly
excited states (DES) embedded in that continuum, makes the method suitable for a
straightforward time-resolved tracing of the DES decay. We then solve the electronic
eigenvalue problem for each subspace:
[ ˆ
Q ˆ
H el ˆ
Q]φ r (r, R) = E r (R)φ r (r, R),
(2.6)
[ ˆ
P ˆ
H el ˆ
P ]φ α,, α ,ε (r, R) = E α,ε (R)φ α,, α ,ε (r, R).
(2.7)
The eigenvalue equations for the bound states [Eq. (2.5)] and the resonant component of the continuum [Eq. (2.6)] are solved using a configuration interaction
method in a basis of H
+
2 orbitals, whereas the non-resonant continuum electronic
states, Eq. (2.7), are calculated using a multichannel L 2 close-coupling procedure
[66, 67]. The basis set of H
+
2 orbitals are written as single center expansions using
spherical harmonics for the angular part and B-spline basis functions for the radial
part [68, 69]. The results presented in following sections have been obtained using angular momenta expansions up to = 16, and up to 180 B-splines functions
of order k = 8 in a box of size 60 a.u., which is large enough to avoid unphysical
reflections. For the close-coupling procedure, angular momenta of the ejected electron are included up to l α = 7, for each α electronic state in the discretized continua
associated to each ionization threshold of the H
+
2 molecule.
Once the electronic structure is obtained in a given grid of internuclear distances,
we solve the one-dimensional Schrödinger equation to calculate the nuclear wave
functions χ v i (R):
ˆ
T (R) + E x (R)
χ v x (R) = W x,v x χ v x (R)
(2.8)
where x stands for a bound, resonant or continuum electronic state, and E x is the
corresponding potential energy curve previously obtained. For the results here presented, the basis for the nuclear wave functions had up to 300 B-splines defined in
a box of size 12 a.u.
By projecting onto the basis of vibronic states, the TDSE defined in Eq. (2.1) is
reduced to a system of coupled differential equations that can be solved using ordinary integration procedures. A sixth-order Runge-Kutta algorithm, implemented in
PETSc libraries [70], is used for the integration.
Excitation and ionization amplitudes are extracted by projecting the propagated
time-dependent wave packet into the excitation or ionization final states. Since the