Cite this Article
Jia Z M, Zeng Z N, Tang W T, Li R X. Electron localization of H 2 + in a dc electric field. Chinese Physics B, 2017, 26(1): 013201  
Permissions
Electron localization of H 2 + in a dc electric field
Jia Z M1, Zeng Z N1, †, Tang W T2, Li R X1
State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
Signal and Information Processing, Jingchu University of Technology, Jingmen 448000, China

 

† Corresponding author. E-mail: zhinan_zeng@mail.siom.ac.cn

Abstract

A dc electric field is utilized to steer the electron motion after the molecular ion is excited by an ultrashort ultraviolet laser pulse. The numerical simulation shows that the electron localization distribution and the dissociation control ratio are dependent on the polarization direction and amplitude of the dc electric field. Most electrons of the dissociation state move opposite to the dc electric field and stabilize at the dressed-up potential well, for the dressed-down well is occupied by the electrons of the 1s state.

PACS: 32.80.Rm;33.80.Rv;42.50.Hz;42.65.Ky
Keyword:dissociation localization;time dependent Schrödinger equation;Coulomb potential;ultraviolet laser pulse
1. Introduction

Coherent control of electrons and fragments in chemical reactions and photoelectron processes has attracted a great deal of interest. One of the main goals has been to find a way to selectively break and form molecular bonds in photochemical reactions.[1–3] As for searching for the underlying mechanism in realizing electron localization control during the dissociation process, several solution routes have been proposed, which include the mixture of the 1s and 2p states,[4] the interference between the 2p and 3s states,[5] the superposition of the 2p and 2p states,[6] and so on. In addition, the other quantum coupled equations[7–9] or laser-induced Stark shift effect[10–12] have also been used to reveal the dissociation control mechanism.

The hydrogen molecular ion is of special interest since it is the simplest molecule, consisting of three particles only. Even so, it plays an important role in the theoretical and experimental studies of electron localization in the dissociation of molecules. When the distance of the two protons of is small enough, the electron on the dissociation state can oscillate between them. With the increase of the distance, the inter-nuclear potential barrier would suppress such an oscillating behavior until the electron localization is stabilized.[13] In this paper, we utilize a dc electric field to steer the electron motion between the two protons after an ultrashort ultraviolet (UV) laser pulse is used to excite the electron wave packet onto the dissociation 2p state. Both the UV and dc electric fields are linearly polarized. The polarization direction of the UV pulse is along the molecular axis. The results show that the electron localization distribution and the electron dissociation control ratio are obviously dependent on the polarization direction and amplitude of the dc electric field. By adjusting the cross angle between the polarization axis of the dc electric field and the molecular axis, a probability as high as 98.8% to localize electrons on one of the two protons can be obtained, with a total dissociation probability of 53.9% and an ionization probability of 3.37%. Furthermore, for the electrons of the dissociation state, they move opposite to the dissociation control electric field (the dc electric field) and stabilize at the potential well which is dressed-up due to the occupation of the dressed-down potential well with the electrons of the 1s state.

This paper is organized as follows. Section 2 describes briefly the model system of and the parameters of the external electric fields we are using. In Section 3, the numerical results for the exploration of the control landscape of electron dynamics is studied. The physical mechanism underlying the localization is demonstrated in Section 4 with a simplified model. Finally, Section 5 contains a brief summary and the conclusion.

2. Simulation model

The polarization direction of the electric field component of the UV laser pulse is assumed to be parallel to the molecular axis, while the cross angle between the polarization axis of the dc electric field and the molecular axis varies from 0.0 to , therefore, we can adopt the two-dimensional non-Born–Oppenheimer time-dependent Schrödinger equation (TDSE) to do the simulation. The corresponding TDSE can be written as ( in atomic unit (a.u.), which are used throughout the paper unless otherwise stated)[6]

(1)
where is the field-free Hamiltonian of the system, stands for the soft-core Coulomb interaction, indicates the interaction of the particle with the external laser pulse, R is the relative internuclear distance, and z and x collectively represent the electronic coordinates with respect to the center of mass of the two nuclei, as shown in Fig. 1.

Fig. 1. The 2-dimensional model for molecular ion in the calculation, where R is the relative internuclear distance, and θ is the cross angle between the polarization axis of the dc electric field and the molecular axis. The electronic axes z and x are parallel and perpendicular to the molecular axis, respectively.

For our model, the kinetic energy in Eq. (1) reads

(2)
the electronic axes z and x are assumed to be parallel and perpendicular to the molecular axis, respectively. Then the Coulomb potential of the system can be expressed as
(3)
Here and are the electron and proton masses ( , respectively.

The interaction of the system with the external laser field can be presented as

(4)
Here and stand for the electric component of the UV laser field and the dc electric field adopted in the simulation, respectively. θ is the cross angle between the axis of the molecular and the polarization of the dc electric field; more details can be found in Fig. 1.

The time dependent electric field component of the UV pulse is given as . Here E 0 is the peak electric field amplitude in atomic units, T is the pulse duration, and ω is the central frequency of the UV laser pulse. The four channels of dissociation are defined as[14–16]

(5)
where , and correspond to the boundaries of the , and x axes, and is the final wave function of the system. In the simulation, is taken at 34.4 fs after the on-set of the UV pulse, when the probabilities of the electron being localized on one of the protons (left—or right +, see Fig. 2) are stable.

Fig. 2. (color online) Snapshots of the common logarithm of the electron-nuclear probability density distribution in the presentations of (z,R) and (x,R) spaces, taken at the end of the simulation: (a) only the UV pulse with an intensity of W/cm2 is applied, (b) both the UV pulse and the dc electric field with an amplitude of −0.008 a.u. are utilized. The cross angle between the polarization axis of the dc electric field and the molecular axis is .

In the simulation, the 140-nm UV pulse with an intensity of W/cm2 and a pulse duration T of 15.9 fs is used to excite the electron wave packet onto the dissociative state 2p . Then the dc electric field with an amplitude of −0.008 a.u. is used to steer the electron motion. This dc electric field does not induce any further ionization. In the calculation, we set the dissociation asymmetry parameter as .

3. Simulation results and discussion

Figure 2 depicts the snapshots of the common logarithm of the electron–nuclear probability density distribution and in the (z,R) and (x,R) spaces, taken at the end of the dissociation ( fs) without or with the dc electric field, respectively,

(6)
One can find that the electron localization starts from about R = 5.0 a.u., therefore, the on-set of the R axis is 5.0 a.u. in the definition of dissociation channels of Eq. (5). From Fig. 2(a), one can see that a single UV pulse induces a symmetric electron distribution, both in the z and x directions, with and , respectively, due to the symmetric distribution of the double-well coulomb potential of the molecular ion . The total ionization probability is I = 0.0231, i.e., , where
is the electron survival probability.

When a dc electric field, whose polarization is assumed to be parallel to the molecular axis, with an amplitude of −0.008 a.u. is utilized to steer the electron motion after the excitation of the UV pulse, the symmetric electron localization distribution along the z direction is seriously broken, which can be seen in Fig. 2(b1) (the cross angle between the polarization axis of the dc electric field and the molecular axis is ). This is because along the z direction, the energies of the double-well Coulomb potential of the molecular ion are dressed up and down by the dc electric field due to the dipole interaction term . The energies of the right potential well are effectively descended by , while for the left well, the energies are ascended by . The direction of is right, and the electric field force of the electron is in the right direction, as the charge on the electron is e = 1.0 in the simulation. While for the electrons on the dissociative state 2p , they move opposite to the electric field force and stabilize at the dressed-up potential well (the left well). When the dissociation control electric field is in the right direction (a minus electric field), the electrons of the 1s state are mainly localized at the potential well with dressed-down energies, i.e., the right potential well. While for the electrons of the dissociation 2p state, most of them are stabilized at the left potential well which is dressed up, see Fig. 3. For the electrons of the 1s state, they are stabilized at the bound states of finally, because the 1s state is bounded. While for the electrons of the 2p state, most of them are localized at the left proton at the end of the simulation, for the 2p state is the first dissociative state of . When the location probabilities are stable, one can obtain , and . Thus 98.8% electrons of all the dissociation events along the z axis are localized at the left proton.

Fig. 3. (color online) The probabilities of the electron localization of the 1s and 2p states of on the double Coulomb potential wells along the z axis. Both the UV pulse with an intensity of W/cm2 and the dc electric field with an amplitude of −0.008a.u. are applied. The cross angle between the polarization axis of the dc electric field and the molecular axis is .

For the electron localization distribution along the x direction, there lies a symmetric distribution with , as can be found in Fig. 2(b2), due to the symmetric distribution of the Coulomb potential wells along this direction.

Figure 4 shows the dissociation asymmetry parameter A, the total ionization probability I, and the probabilities of the electron being localized on the left (or right) proton (or as functions of θ, the cross angle between the polarization axis of the dc electric field and the molecular axis. The amplitude of the dc electric field is a.u. The intensity of the 140-nm UV laser pulse is W/cm2. From this figure, one can find that the polarization direction of the dc electric field plays a very important role in the electron localization distribution. The largest dissociation control ratio can be obtained when the polarization is parallel to the molecular axis, i.e., when the cross angles are and π, the dissociation asymmetry parameters are and 0.972, respectively. and electrons of all the dissociation events can be steered onto the left or right proton, respectively, due to the dressing effect of the dc electric field. When the cross angle is , the left potential well is dressed-up. When , the dressed-up potential well is the right one. For the electrons of the dissociation state 2p , most would move opposite to the dc electric field and stabilize at the ascended potential well at the end of the simulation, as mentioned above.

Fig. 4. (color online) The dissociation asymmetry parameter A, the total ionization probability I, and as functions of θ, the cross angle between the polarization axis of the dc electric field and the molecular axis. The amplitude of the dc electric field is −0.008 a.u. The intensity of the 140-nm UV pulse is W/cm2.

The dissociation control ratio decreases with the polarization of the dc electric field changing from the parallel to the perpendicular direction of the molecular axis due to the weakening of the dressing effect of the dc electric field. When and , a symmetric electron localization distribution can be obtained with and , respectively, as a result of the symmetric distribution of the Coulomb potential wells (the amplitude of the dc electric is 0.0).

Figure 5 shows the dissociation asymmetry parameter A, the total ionization probability I, and the probabilities of the electron being localized on the left (or right) proton (or as functions of , the amplitude of the dc electric field. The cross angle between the polarization axis of the dc electric field and the molecular axis is . The intensity of the 140-nm UV laser pulse is W/cm2. When the amplitude of the dc electric field is between −0.004375 a.u. and −0.03275 a.u., , more than electrons of the dissociation 2p state can be steered onto the left proton and the total dissociation probability is larger than . When is less than 0.004375 a.u., it is not strong enough to ensure a good electron-direction selection. If is higher than 0.03275 a.u., more electrons are excited onto the higher 3sσ state and the asymmetry parameter A decreases. Furthermore, more electrons are ionized with the increase of the absolute amplitude of the dc electric field. When a.u., the total ionization probability is I = 0.02426. When a.u., I = 0.1113. Therefore, the dc electric field with an absolute amplitude between 0.004375 a.u. and 0.03275 a.u. can be utilized to control the electron localization.

Fig. 5. (color online) The dissociation asymmetry parameter A, the total ionization probability I, and as functions of , the amplitude of the dc electric field. The cross angle between the polarization axis of the dc electric field and the molecular axis is . The intensity of the 140-nm UV pulse is W/cm2.
4. Simulation with analytical solution

In this section, the electron localization of the 1s and 2p states of a double-well Coulomb potential model system in an external dc electric field is demonstrated in detail with an analytical solution. The corresponding potential curve of the Coulomb potential can be expressed as[17]

(7)
where C and S are constants, and the barrier height of the double-well potential is .

The trial wave packet can be written as

(8)
where and are the wave packets of the left and right potential wells, respectively, and . The wave packets of the 1s (denoted as and 2p ( states are
(9)
By inserting Eqs. (7) and (8) into the Schrödinger equation and making use of , the Schrödinger equation can be given as
(10)
with
(11)
where and represent the overlap integral and the tunneling effect between the two potential wells, respectively, and and are the energy levels of these two potential wells. When
(12)
one can obtain
(13)
where F is the external dc electric field, whose polarization direction is along the molecular axis. Then we can rewrite Eq. (10) with the representation as
(14)
the electron localization of the left and right potential wells are
(15)
where and . . and are the electron localization ratios of the 1s and 2p states on the left and right potential wells, respectively. The double-well potential is dependent on S. A large S will lead to a single potential well, so in the calculation. When , and . The electron localization ratios of the 1s and 2p states on the right potential well can be expressed as
(16)

When the double-well potential is definite, B is a constant. The localization ratios are dependent on and .

When the amplitude of the external dc electric field is F = 0.0, , the electron distribution of the 1s and 2p states on the left and right potential wells is symmetric. With the increasing of decreases, the electron localization ratio of the 1s state on the right potential well decreases. For the electron localization ratio of the 2p state, with the increase of increases too. While F increases faster than and increases with the increase of F. The electron localization ratio of the 2p state on the right potential well increases, as can be seen from part 1 in Fig. 6.

Fig. 6. (color online) Electron localization ratios of the 1s and 2p states on the left and right potential wells as functions of the amplitude of the external dc electric field.

When is expanded in , one can obtain

(17)
Then and . For the 1s state, no electron localizes at the right potential well. For the 2p state, all the electrons are stabilized at the right potential well, as shown in part 2 in Fig. 6.

When the amplitude of the external dc electric field is , with the increase of decreases, the electron localization ratio of the 2p state on the right potential well decreases, and increases because increases faster than and . The electron localization ratio of the 1s state on the right potential well increases. When , one can obtain The electron localization ratios of the 1s and 2p states on the right potential well are and . All the electrons of the 1s state and no electron of the 2p state are localized at the right potential well. With the same way, one can easily obtain that all the electrons of the 2p state are stabilized at the left potential well. The same conclusion can be obtained as the above TDSE.

5. Summary

We have shown the electron localization in the dissociation of H molecular ion in a dc electric field after the excitation of a UV laser pulse. A high localization probability of 98.8% has been demonstrated when the polarization direction of the dc electric field is assumed to be parallel to the molecular axis. The total dissociation and ionization probabilities are 53.9% and 3.37%, respectively. For the electrons of the dissociation state 2p , they move opposite to the dissociation control electric field force and stabilize at the dressed-up potential well due to the occupation of the dressed-down potential well with the electrons of the 1s state. The electron localization distribution and the dissociation control ratio are dependent on the polarization direction and amplitude of the dc electric field. The largest dissociation control ratio can be obtained when the polarization of the dc electric field is parallel to the molecular axis. When the polarization direction is perpendicular to the molecular axis, a symmetric electron distribution can be found due to the symmetric distribution of the Coulomb potential wells. A dc electric field, whose polarization is parallel to the molecular axis, with an absolute amplitude between 0.004375 a.u. and 0.03275 a.u. can be utilized to control the electron localization. When the amplitude of the dc electric field is less than 0.004375 a.u., it is not strong enough to ensure a good electron-direction selection. If is higher than 0.03275 a.u., more electrons of the ground state are excited onto the higher 3sσ state, and the dissociation control ratio decreases.

Reference
1 Gaire B McKenna J Zohrabi M Carnes K D Esry B D Ben-Itzhak I 2000 Phys. Rev. A 85 023405
2 Roudnev V Esry B D 2007 Phys. Rev. A 76 023403
3 Vladimir R Esry Ben-Itzhak I B D Ben-Itzhak I 2004 Phys. Rev. Lett. 93 163601
4 McKenna J Anis F Sayler A M Gaire B Johnson Nora G Parke E Carnes K D Esry B D Ben-Itzhak I 2010 Phys. Rev. A 85 023405
5 Fischer B Kremer M Pfeifer T Feuerstein B Sharma V Thumm U Schröter C D Moshammer R Ullrich J 2010 Phys. Rev. Lett. 105 223001
6 He F 2012 Phys. Rev. A 86 063415
7 Kling M F Siedschlag C Znakovskaya I Verhoef A J Zherebtsov S Krausz F Lezius M Vrakking M J J 2008 Mole. Phys. 106 455
8 Kling M F Siedschlag C Verhoef A J Khan J I Schultze M Uphues T Ni Y Uiberacker M Drescher M Krausz F Vrakking M J J 2006 Science 312 246
9 Sansone G Kelkensberg F Pérez-Torres J F Morales F Kling M F Siu W Ghafur O Johnsson Swoboda P M Benedetti E Ferrari Lépine F F Sanz-Vicario J L Zherebtsov S Znakovskaya I L’Huillier A Ivanov M Y Nisoli M Martín F Vrakking M J J 2010 Nature 465 763
10 Dietrich P Ivanov M Y Ilkov F A Corkum P B 1996 Phys. Rev. Lett. 77 4150
11 Niikura H Corkum P B Villeneuve D M 2003 Phys. Rev. Lett. 90 203601
12 Niikura H Villeneuve D M Corkum P B 2004 Phys. Rev. Lett. 92 133002
13 He F Ruiz C Becker A 2007 Phys. Rev. Lett. 99 083002
14 Jia Z Zeng Z Li R Xu Z Deng Y 2014 Phys. Rev. A 89 023419
15 Jia Z Zeng Z Li R Xu Z 2014 Chin. Phys. B 23 083201
16 Jia Z Zeng Z Li R Xu Z Deng Y 2015 Chin. Phys. B 24 013204
17 Cao D Li Y Bi D Wang L Cheng Y 2004 Acta Phys. Sin. 53 2444 in Chinese