Theor Chem Acc (2015) 134:128
1 3
around 24, 48 fs and so on time delays. In the presence of
external fi eld as a fi rst crude approximation, we can expect
that the non-dissociated part of the wave packet also exhibits a similar periodic motion. The individual panels of the
Figs. 4 and 5 are snapshots of the nuclear density function at the special times mentioned above and also at time
42 fs which is half way between the two neighboring ones.
Applying 10 fs pulse length the external fi eld has non-vanishing value only at the fi rst interesting time interval with
large internuclear distances. The whole dissociation process is practically completed when the nuclear wave packet
is at fi rst time in the large internuclear distance region.
Then the remaining part of the wave packet oscillates back
and force in the adiabatic lower surface exhibiting interference picture is the bound region. The nuclear wave packet
will be again at the same position after another 24 fs, but
because the intensity is very low further dissociation is no
longer going to happen (see on Fig. 4 ; t = 36 fs). On the
right sides of the panels (on Fig. 4 ), the time evolution of
the dissociated part of the wave packet is illustrated using
different scaling for the internuclear distance.
On Fig. 5 results are collected by using longer laser pulse
(50 fs). For this pulse length a completely different situation
applies. The dissociation takes place in several steps since
the laser pulse is long enough. Then the direct consequence
of this process is, that on the right hand side of the panels
(on Fig. 5 ) the pictures of the time evolution of the dissociated part of the nuclear wave packet are more structured.
Similarly, the interference pictures on the bounded region
of these panels also show rich patterns. This behavior is
the direct consequence of the laser fi eld-induced rotational
nodes, which are formed due to applying long pulses. The
nuclear wave packet density is close to zero at the rotational
nodes. But above and below these nodes, the value of the
wave packet density is markedly different from zero and socalled quantum interference picture occurs. These quantum
interference effects are then the sources of the modulations
on the angular distribution curves of the photofragments.
Applying short laser pulses, such interference patterns do
not appear as rotational nodes are not formed.
4 Conclusions
We present results for the kinetic energy (KER) spectra and
for the angular distribution of the photofragments of the D
+
2
ion. By means of two-dimensional quantum dynamical calculations, we have demonstrated that the impact of the laserinduced conical intersection for these dynamical quantities
is the largest possible, if one uses high fi eld intensity and
long pulses together. Such a laser fi eld can rotate the molecules signifi cantly, which is the heart of the LICI in diatomic
systems. Moreover, we have analyzed the time evolution of
the nuclear wave packet density, which visualizes the occurring quantum interference effect during the dissociation due
to the very strong nonadiabatic coupling between the electronic, vibrational and rotational motions.
Acknowledgments The authors acknowledge the fi nancial support
by the Deutsche Forschungsgemeinschaft (Project ID CE10/50-2).
Á.V. acknowledges the OTKA Grant No. NN103251. The authors
thank Lorenz Cederbaum for many fruitful discussions.
References
1. Born M, Oppenheimer JR (1927) Ann Phys 84:457
2. Köppel H, Domcke W, Cederbaum LS (1984) Adv Chem Phys
57:59–246
3. Baer M (2002) Phys Rep 358:75–142
4. Worth GA, Cederbaum LS (2004) Annu Rev Phys Chem
55:127–158
5. Domcke W, Yarkony DR, Köppel H (2004) Conical intersections: electronic structure, dynamics and spectroscopy. World
Scientifi c, Singapore
6. Baer M (2006) Beyond born oppenheimer: electronic non-adiabatic coupling terms and conical intersections. Wiley, New York
7. Matsika S (2007) Rev Comput Chem 23:83–124
8. Althorpe SC, Stecher T, Bouakline FJ (2008) Chem Phys
129:214117
9. Bouakline F (2014) Chem Phys 442:31–40
10. Truhlar DG, Mead A (2003) Phys Rev A 68:032501
11. Tishchenko O, Li R, Truhlar DG (2010) PNAS 107:19139–19145
12. Moiseyev N, Sindelka M, Cederbaum LS (2008) J Phys B
41:221001–221006
13. Sindelka M, Moiseyev N, Cederbaum LS (2011) J Phys B
44:045603–045606
14. Halász GJ, Vibók Á, Sindelka M, Moiseyev N, Cederbaum LS
(2011) J Phys B 44:175102–175112
15. Halász GJ, Sindelka M, Moiseyev N, Cederbaum LS, Vibók Á
(2012) J Phys Chem A 116:2636–2643
16. Halász GJ, Vibók Á, Sindelka M, Cederbaum LS, Moiseyev N
(2012) Chem Phys 399:146–150
17. Halász GJ, Vibók Á, Moiseyev N, Cederbaum LS (2012) J Phys
B 45:135101–135110
18. Halász GJ, Vibók Á, Meyer HD, Cederbaum LS (2013) J Phys
Chem A 117:8528–8535
19. Halász GJ, Vibók Á, Moiseyev N, Cederbaum LS (2013) Phys
Rev A 88:043413-6
20. Halász GJ, Csehi A, Vibók Á, Cederbaum LS (2014) J Phys
Chem A 118:11908–11915
21. Halász GJ, Vibók Á, Cederbaum LS (2015) J Phys Chem Lett
6:348–354
22. Cederbaum LS, Chiang YC, Demekhin PV, Moiseyev N (2011)
Phys Rev Lett 106:123001–123005
23. Demekhin PV, Chiang YC, Cederbaum LS (2011) Phys Rev A
84:033417
24. Demekhin PV, Cederbaum LS (2013) J Phys B 46:164008
25. Estrada H, Cederbaum LS, Domcke W (1986) J Chem Phys 84:152
26. Feuerbacher S, Sommerfeld T, Cederbaum LS (2004) J Chem
Phys 120:3201
27. Demekhin PV, Cederbaum LS (2013) J Chem Phys
139:154314–154315
28. Natan A, Ware MR, Bucksbaum PH (2015) Book of ultrafast
phenomena XIX Springer proceedings in physics, vol 162, pp
122–125
172
Reprinted from the journal
1 3
around 24, 48 fs and so on time delays. In the presence of
external fi eld as a fi rst crude approximation, we can expect
that the non-dissociated part of the wave packet also exhibits a similar periodic motion. The individual panels of the
Figs. 4 and 5 are snapshots of the nuclear density function at the special times mentioned above and also at time
42 fs which is half way between the two neighboring ones.
Applying 10 fs pulse length the external fi eld has non-vanishing value only at the fi rst interesting time interval with
large internuclear distances. The whole dissociation process is practically completed when the nuclear wave packet
is at fi rst time in the large internuclear distance region.
Then the remaining part of the wave packet oscillates back
and force in the adiabatic lower surface exhibiting interference picture is the bound region. The nuclear wave packet
will be again at the same position after another 24 fs, but
because the intensity is very low further dissociation is no
longer going to happen (see on Fig. 4 ; t = 36 fs). On the
right sides of the panels (on Fig. 4 ), the time evolution of
the dissociated part of the wave packet is illustrated using
different scaling for the internuclear distance.
On Fig. 5 results are collected by using longer laser pulse
(50 fs). For this pulse length a completely different situation
applies. The dissociation takes place in several steps since
the laser pulse is long enough. Then the direct consequence
of this process is, that on the right hand side of the panels
(on Fig. 5 ) the pictures of the time evolution of the dissociated part of the nuclear wave packet are more structured.
Similarly, the interference pictures on the bounded region
of these panels also show rich patterns. This behavior is
the direct consequence of the laser fi eld-induced rotational
nodes, which are formed due to applying long pulses. The
nuclear wave packet density is close to zero at the rotational
nodes. But above and below these nodes, the value of the
wave packet density is markedly different from zero and socalled quantum interference picture occurs. These quantum
interference effects are then the sources of the modulations
on the angular distribution curves of the photofragments.
Applying short laser pulses, such interference patterns do
not appear as rotational nodes are not formed.
4 Conclusions
We present results for the kinetic energy (KER) spectra and
for the angular distribution of the photofragments of the D
+
2
ion. By means of two-dimensional quantum dynamical calculations, we have demonstrated that the impact of the laserinduced conical intersection for these dynamical quantities
is the largest possible, if one uses high fi eld intensity and
long pulses together. Such a laser fi eld can rotate the molecules signifi cantly, which is the heart of the LICI in diatomic
systems. Moreover, we have analyzed the time evolution of
the nuclear wave packet density, which visualizes the occurring quantum interference effect during the dissociation due
to the very strong nonadiabatic coupling between the electronic, vibrational and rotational motions.
Acknowledgments The authors acknowledge the fi nancial support
by the Deutsche Forschungsgemeinschaft (Project ID CE10/50-2).
Á.V. acknowledges the OTKA Grant No. NN103251. The authors
thank Lorenz Cederbaum for many fruitful discussions.
References
1. Born M, Oppenheimer JR (1927) Ann Phys 84:457
2. Köppel H, Domcke W, Cederbaum LS (1984) Adv Chem Phys
57:59–246
3. Baer M (2002) Phys Rep 358:75–142
4. Worth GA, Cederbaum LS (2004) Annu Rev Phys Chem
55:127–158
5. Domcke W, Yarkony DR, Köppel H (2004) Conical intersections: electronic structure, dynamics and spectroscopy. World
Scientifi c, Singapore
6. Baer M (2006) Beyond born oppenheimer: electronic non-adiabatic coupling terms and conical intersections. Wiley, New York
7. Matsika S (2007) Rev Comput Chem 23:83–124
8. Althorpe SC, Stecher T, Bouakline FJ (2008) Chem Phys
129:214117
9. Bouakline F (2014) Chem Phys 442:31–40
10. Truhlar DG, Mead A (2003) Phys Rev A 68:032501
11. Tishchenko O, Li R, Truhlar DG (2010) PNAS 107:19139–19145
12. Moiseyev N, Sindelka M, Cederbaum LS (2008) J Phys B
41:221001–221006
13. Sindelka M, Moiseyev N, Cederbaum LS (2011) J Phys B
44:045603–045606
14. Halász GJ, Vibók Á, Sindelka M, Moiseyev N, Cederbaum LS
(2011) J Phys B 44:175102–175112
15. Halász GJ, Sindelka M, Moiseyev N, Cederbaum LS, Vibók Á
(2012) J Phys Chem A 116:2636–2643
16. Halász GJ, Vibók Á, Sindelka M, Cederbaum LS, Moiseyev N
(2012) Chem Phys 399:146–150
17. Halász GJ, Vibók Á, Moiseyev N, Cederbaum LS (2012) J Phys
B 45:135101–135110
18. Halász GJ, Vibók Á, Meyer HD, Cederbaum LS (2013) J Phys
Chem A 117:8528–8535
19. Halász GJ, Vibók Á, Moiseyev N, Cederbaum LS (2013) Phys
Rev A 88:043413-6
20. Halász GJ, Csehi A, Vibók Á, Cederbaum LS (2014) J Phys
Chem A 118:11908–11915
21. Halász GJ, Vibók Á, Cederbaum LS (2015) J Phys Chem Lett
6:348–354
22. Cederbaum LS, Chiang YC, Demekhin PV, Moiseyev N (2011)
Phys Rev Lett 106:123001–123005
23. Demekhin PV, Chiang YC, Cederbaum LS (2011) Phys Rev A
84:033417
24. Demekhin PV, Cederbaum LS (2013) J Phys B 46:164008
25. Estrada H, Cederbaum LS, Domcke W (1986) J Chem Phys 84:152
26. Feuerbacher S, Sommerfeld T, Cederbaum LS (2004) J Chem
Phys 120:3201
27. Demekhin PV, Cederbaum LS (2013) J Chem Phys
139:154314–154315
28. Natan A, Ware MR, Bucksbaum PH (2015) Book of ultrafast
phenomena XIX Springer proceedings in physics, vol 162, pp
122–125
172
Reprinted from the journal