⇦Chinese Physics Letters, 2023, Vol. 40, No. 4, Article code 043101 A Time-Dependent-Density-Functional-Theory Study of Charge Transfer Processes of Li$^{2+}$ Colliding with Ar in the MeV Region Hui-Hui Zhang (张会会)1, Wan-Dong Yu (于皖东)2*, Cong-Zhang Gao (高聪章)3*, and Yi-Zhi Qu (屈一至)1* Affiliations 1School of Optoelectronics, University of Chinese Academy of Sciences, Beijing 100049, China 2State Key Laboratory for Mesoscopic Physics and Frontiers Science Center for Nano-optoelectronics, School of Physics, Peking University, Beijing 100871, China 3Institute of Applied Physics and Computational Mathematics, Beijing 100088, China Received 9 February 2023; accepted manuscript online 17 March 2023; published online 4 April 2023 *Corresponding authors. Email: wandongyu@pku.edu.cn; gao_congzhang@iapcm.ac.cn; yzqu@ucas.edu.cn Citation Text: Zhang H H, Yu W D, Gao C Z et al. 2023 Chin. Phys. Lett. 40 043101    Abstract We study charge transfer of a multi-electron collision system Li$^{2+}$ + Ar using the time-dependent density functional theory non-adiabatically coupled to the molecular dynamics. By implementing the particle number projection method, the single- and double-charge transfer cross sections are extracted at MeV energies, which are in good agreement with the experimental data available. The analysis of charge transfer probabilities shows that for energies higher than 1.0 MeV, the single-charge transfer occurs for a broader range of impact parameters, while the double-charge transfer is dominated by close collisions. To gain the population of captured electrons on the projectile, we compute the orbital projection probabilities. It is found that the electrons of the Ar atom will most possibly transfer to the $2p$ orbitals of the Li$^{2+}$, and only a small portion of captured electrons distribute on the $s$ orbitals. This work verifies the capability of the present methodology in dealing with charge transfer in dressed ion collisions at MeV energies.
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 DOI:10.1088/0256-307X/40/4/043101 © 2023 Chinese Physics Society Article Text When atoms or molecules are impacted by ion beams, the charge- and energy-transfer processes often occur. A consequence of relevant experimental capabilities is the measurements of reaction cross sections.[1] As one of the essential processes between the projectile and the target, charge transfer and its cross sections have been intensively studied both experimentally and numerically for a number of atomic or molecular systems.[2-6] A mechanism lying at the heart of the charge transfer is the role of the active electrons (either in the target or the projectile) involved in collisions. Charge transfer of fully stripped projectile was studied frequently due to their structural simplicity, i.e., the projectile is usually described as a classical point of charge. However, charge transfer of dressed projectiles has attracted significant interests in recent years, because the active electrons of the projectile play a role in specific processes,[7-9] resulting in more complicated mechanisms, such as the screening effects,[10-12] the enhancement of state-selected cross sections,[13,14] and stopping powers.[15] To theoretically describe the dressed projectile, the effect of its active electrons cannot be neglected. In general, the model potential works at high impact energies (more than several MeV), but may be incapable of giving convincing details for state-resolved properties. Using an independent particle model, Baxter et al. added a screening potential for the projectile's electrons and investigated the electron capture of He$^+$ + He with some success.[16] At high energies, the semi-classical eikonal model accounting for the electronic screening effect was used to calculate the capture cross sections of dressed projectiles, such as light dressed ions colliding with He or H atoms.[17,18] Due to the computational feasibility when dealing with many-electron systems, the time-dependent density-functional theory (TDDFT) coupling with the Ehrenfest dynamics, abbreviated to Ehrenfest-TDDFT,[19] was widely used to simulate ion collisions with atoms, molecules, and solids in a wide energy regime.[20-24] However, the standard Ehrenfest-TDDFT approach is difficult to be directly applied in dressed ion collisions, since the decoupling effect emerges and may be dominant, i.e., the active projectile electrons cannot move simultaneously with the nucleus during the collision, inducing an electric-dipole moment on the projectile which is unphysical. Recently, Yu et al. developed the Ehrenfest-TDDFT for dressed projectiles and verified the particle number projector (PNP) method in the Ne$^{2+}$ + He collision.[25] They showed that the dynamics of the projectile's active electrons can be correctly described and thus the decoupling effect is circumvented. It should be noted that the charge transfer probability extracted by the PNP method has good records in Fermi systems,[26,27] and usually gives more satisfactory descriptions for correlated charge transfer dynamics. In this Letter, we calculate the charge transfer cross sections of Li$^{2+}$ + Ar collisions. Unlike the work at the keV region,[25] for the dressed Li$^{2+}$, we here extend the collision energies to the MeV region in which the ionization of electrons is substantial, and we show the capability of reproducing charge transfer cross sections at MeV energies based on the Ehrenfest-TDDFT framework. Since the Li$^{2+}$ carries an electron, to prepare its initial states, we initially boost the Kohn–Sham orbitals (KSOs) by adding a phase factor, which avoids projectile's active electrons detaching from the nucleus. To track the time-evolution of the electronic density of the Li$^{2+}$, we employ an inverse collision framework, in which we set the reference frame in which the Ar atom is the projectile that impacts the Li$^{2+}$ target. This treatment ensures the analysis of the collision dynamics and state-selective probabilities.
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Fig. 1. The initial schematic diagram of the Li$^{2+}$ + Ar collisions in the inverse collision framework. The distance $b$ is the impact parameter, and ${\boldsymbol v}$ denotes the incident direction. The whole space is divided into two parts, i.e., the spherical region $V_{\rm T}$ centered on the Li$^{2+}$ target and its complementary region $V_{\rm P}$.
Theoretical Model. We model the ion-atom collision using the Ehrenfest-TDDFT model,[19] in which the electronic wavefunction is described by the TDDFT, and the nuclei are described as classical point particles. Specifically, we deal with the Li$^{2+}$ + Ar collision using the inverse collision method.[25] Different from the normal collision in which the Li$^{2+}$ is the projectile, as shown in Fig. 1, the inverse collision framework is implemented using the strategy of the projectile-target role reversal, i.e., the Li$^{2+}$ is treated as the target, and the Ar is the projectile. The inverse framework has certain advantages. (a) One can track the time evolution of single-electron wavefunctions surrounding the Li$^{2+}$, which makes the analysis of charge transfer dynamics possible and the extraction of capture probabilities more accurate. (b) The TDDFT with the local-density-approximation (LDA) functional preserves the Galilean invariance, which is an exact property of the many-electron system.[28] (c) The Ar atom carries eight electrons and its initial states can be easily implemented in the orbital-dependent TDDFT using the electron translation factor.[29] (d) All valence electrons of the collision system described on the same quantum-mechanical footing, together with the ionic motion. All calculations are carried out in a development version of the OCTOPUS program.[30] We implemented the PNP method in OCTOPUS. All numerical quantities are discretized in a rectangular box ($94 \times 80 \times 70$ a.u.) with a uniform grid spacing (0.38 a.u.). Figure 1 shows the inverse collision diagram, in which the boosted Ar is initially located at ($-20 a_0$, $-5.0+b a_0$, 0) and the ground-state Li$^{2+}$ is initially placed at ($0$, $-5.0 a_0$, 0). The initial impact velocity ${\boldsymbol v}$ is in the direction of the positive $x$ axis. We employ the norm-conserving pseudopotentials[31] for the core-electron effects and an adiabatic LDA[32] for the electronic exchange-correlation effects. The equations of the nuclear motion are solved by the velocity Verlet algorithm.[33] A mask absorbing boundary[34] is set on each edge of the simulation box with a width of 15.0 a.u., which is used to prevent the reflection of the scattered electronic wavefunction. The total simulation time is 100 fs, and the time step is fixed to $4.83 \times 10^{-4}$ fs. To extract converged cross sections, the size of the target space $V_{\rm T}$ covers the whole simulation box, which is large enough to ensure the results' convergence. To extract the charge transfer probability, we employ the particle number projection (PNP) method, based on the idea of a geometrical division of the whole space.[26,27,35,36] The application of the PNP method depends on the spatial division of the whole space. As shown in Fig. 1, the spherical region $V_{\rm T}$ centers on the Li$^{2+}$, and its complementary region $V_{\rm P}$ contains the Ar projectile. The $n$-charge transfer probability is defined as the probability that $n+1$ electrons remain in the $V_{\rm T}$ region. For instance, the single- and double-charge transfer probabilities are defined as the probabilities that 2 and 3 electrons stay in the $V_{\rm T}$, respectively. Note that the Li$^{2+}$ is less probable to capture more than 2 electrons. In the Kohn–Sham TDDFT, the probability that $m$ electrons remain in the $V_{\rm T}$ is computed by \begin{align} P_m(t) = \sum_{\{ \boldsymbol{\varOmega}\} \in S_m} \mathrm{{\det}}\{\langle \psi_i(t)|\psi_j(t) \rangle_{\varOmega_i}\}, \tag {1} \end{align} where $\psi_j$ represents the KSOs. The set $\{ \boldsymbol{\varOmega} \}=(\tau_1,\dots,\tau_N)$ belongs to $S_m$, and $S_m$ is the combination that the $V_{\rm T}$ appears $m$ times and $V_{\rm P}$ appears $N-m$ times. The probability $P_m$ runs over all possible elements $\varOmega_i$ in $\{ \boldsymbol{\varOmega} \}$. Therefore, for the Li$^{2+}$ + Ar collisions, the $n$-charge transfer cross section is defined by the integral over all the impact parameters (i.e., $b$), \begin{align} \sigma_n = 2\pi\int P_{n+1}(b)bdb. \tag {2} \end{align} To extract converged cross sections, the size of the target space $V_{\rm T}$ covers the whole simulation box, which is large enough to ensure the results' convergence.
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Fig. 2. Single- and double-charge transfer cross sections as a function of collision energy. Our results (stars and circles) are calculated by choosing the $V_{\rm T}$ to be the whole computational box. The other scattered symbols are the experimental data of Losqui et al. in Ref. [37] and Dmitriev et al. in Ref. [38].
Results. Figure 2 shows the single- and double-charge transfer cross sections from 0.5 MeV to 2.5 MeV. On the whole, both single-charge transfer (SCT) and double-charge transfer (DCT) cross sections decrease monotonously with the increase of the impact energy. Regarding SCT cross sections, our results agree well with the experimental data[37,38] from 0.5 to 1.0 MeV. For the region of 1.5–2.5 MeV, while the present results underestimate the experimental data roughly 20% $\sim$ $50$%, they still give the right tend. The DCT cross sections are generally lower than the SCT ones by more than one order of magnitude. Since the DCT probability is relatively weak, it must be investigated both experimentally and theoretically. Our results reproduce the right trend but systematically overestimate the experimental data. The discrepancy between our results and experiments may come from the impurity of incident ions in the experiments. In early works, the ion beam may be polluted by excited ions, and the mixing of metastable ions may bring variations of the cross sections. Imai et al.[39] provided theoretical evidence that metastable ions play a role in measurement of charge transfer cross sections. Discussion on this issue is beyond the scope of the present study. In short, for SCT cross sections, the TDDFT calculations reproduce experimental data satisfactorily at a few MeV energies, but are still applicable to a more extended energy range. For the DCT cross sections, the TDDFT results are almost two times higher than the experimental data at MeV energies.
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Fig. 3. Charge transfer probabilities as a function of impact parameter: (a) SCT probabilities, (b) DCT probabilities.
To learn more about the dependence of cross sections on the impact parameter and the collision energy, we examine the details about charge transfer probabilities. In Fig. 3, we plot the probability $2\pi b P$ as a function of impact parameter for SCT and DCT cross sections. As shown in Fig. 3(a), for each impact energy, the curve of SCT probabilities is unimodal. As the impact energy increases from 0.5 to 2.5 MeV, the peak value drops monotonously, and the peak position moves towards low impact parameters. This feature indicates that, as the impact energy increases, the charge transfer processes prefer the spatial region close to the nuclei. The DCT probabilities are shown in Fig. 3(b). The DCT probabilities are roughly an order of magnitude lower than the SCT ones. The unimodal distribution is preserved for all the impact energies. Similar to the SCT case, as the energy increases, the peak position concentrates towards the low region of the impact parameter and its magnitude drops monotonously. For energies more than 1.0 MeV, the DCT is dominated by close collisions ($b\le3$ a.u.), while the SCT can take place on a wider range of impact parameters. Moreover, one can see that the computed DCT cross sections are much higher than the experimental data. This discrepancy may be attributed to two aspects. On the one hand, the wavefunction used in the PNP method may not be well approximated by the TDDFT orbitals. On the other hand, the LDA functional does not catch some basic properties of the exact exchange-correlation functional (e.g., the asymptotic behavior, the functional discontinuity, and the free of the electronic self-interaction), which may lead to the overestimated charge transfer cross sections.
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Fig. 4. Snapshots of the electronic density distribution inside the simulation box for the 0.5 MeV collision at $b = 3.0$ a.u. for the sequence of time instants.
To illustrate the charge transfer, Fig. 4 shows the time evolution of total electronic density. Except for the initial configuration in Fig. 4(a), the incident Ar is isolated from the Li ion due to the long-distance separation. For the 0.5 MeV collision at $b = 3$ a.u., we find a violent ionization before the first 10 fs. Then, the electrons gradually get out of the simulation box, and this process takes roughly 70 fs until the final density does not show any significant changes. In addition, for all the impact energies, we verify that the convergence of SCT and DCT cross sections depends on the ionization of $\sim$  $10^{-7}$ magnitude of the electronic density (see yellow isosurfaces in Fig. 4). In other words, unless such $\sim$ $10^{-7}$ electronic density becomes stable, SCT and DCT cross sections do not converge. After the collision, both Ar and Li are excited. In the inverse framework, the Ar is scattered after $\sim$ $10$ fs, and the Li stays in the simulation box. To get insight into the electronic excitation of the charge transfer process, we define the following state-selected projection probability: \begin{align} P_{\rm{orb}} = \sum_j |\langle \psi^{\rm{g.s.}}_{\rm{orb}}(t) | \psi_{j}(t) \rangle|^2, \tag {3} \end{align} where $\psi_j(t)$ is the $j$th KSO and $\psi^{\rm{g.s.}}_{\rm{orb}}(t)$ is the DFT ground-state KSO of the Li$^{2+}$ ion. It is emphasized that the $\psi^{\rm{g.s.}}_{\rm{orb}}(t)$ is computed at each time $t$, for the reason why the positions of the Li ion are different from its initial ones. In practice, we project the KSOs of Ar (i.e., the KSOs initially located on the Ar atom) onto the DFT KSOs of the Li$^{2+}$. Therefore, the projection probability $P_{\rm{orb}}$ can provide us more details about the excitation dynamics of the residual Li$^{2+}$.
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Fig. 5. State-selected projection probabilities for Li$^{2+}$ ion at (a) $b =1.0$, (b) 2.5, (c) 6.5 a.u. for the impact energy of 0.5 MeV.
As shown in Fig. 5, we compute the state-selected projection probability for 0.5 MeV collisions at three different impact parameters (i.e., $b = 1.0, 2.5, 6.5$ a.u.). Note that the curves in Fig. 5 illustrate how electrons of the Ar atom finally populate to the KSOs of the Li$^{2+}$. It is found that most of the excited electrons are populated on 4 KSOs, i.e., the $1s$, $2s$, $2p$, and $3s$ of the Li$^{2+}$. For the close collision at $b = 1.0$ a.u., the electron transfer to the $2p$ is the highest. The $2s$ population is roughly three times higher than the $1s$ case. In addition, it is found that the electrons are seldom populated to the $3s$ orbital. However, the electronic excitation is quite different at larger impact parameters. For the collision at $b = 2.5$ a.u., the $2p$ population is the highest, and the other three orbital projections are much smaller. At $b = 6.5$ a.u., the interaction between the projectile and target is relatively weak, and therefore the projection probabilities go down roughly an order of magnitude. Similarly, the $2p$ projection is the foremost. The dominant $2p$ projection probabilities may be explained as an orbital-matching effect, i.e., the $2p$ orbitals are dumbbell shape and usually distribute more widely than the $1s$ and the $2s$ orbital. In addition, the $3s$ orbital of the Li$^{2+}$ has the highest energy, about 0.35 a.u. higher than its $2p$ orbital, so electrons of the Ar may not prefer to populate on the $3s$ orbital of the Li$^{2+}$. Further, by analyzing the orbital ionization and computing the energy differences for the possible matching orbitals between the Ar and the Li$^{2+}$, it is found that the $3p$ orbital of the Ar contributes most of the captured electrons and its orbital energy is more close to the $3s$ orbital energy of the Li$^{2+}$ rather than the $2p$ orbital energy of the Li$^{2+}$. This feature indicates that the orbital-matching effects can not be simply reduced to the energy-matching effects between these involved orbitals. For impact energies higher than 0.5 MeV, we observed similar charge distributions and the electronic population to the $2p$ orbital is still the highest. In summary, we have studied the charge transfer dynamics as well as cross sections for the Li$^{2+}$ + Ar collision at MeV energies. For SCT cross sections, we find that the Ehrenfest-TDDFT works well at $\sim$ $1$ MeV, and underestimate experimental data above 1.5 MeV. For DCT cross sections, our results yield the right tend for available experimental data but quantitatively overestimate them with $\sim$ $2$ times. For the collisions at 0.5 MeV, we illustrate the time-resolved electronic density surrounding the Li ion. We find that the convergence of the final cross sections is related to the decay of the electronic density within the first $\sim$ $80$ fs. The state-selected probability is computed by projecting the time-dependent KSOs onto the instantaneous ground state DFT orbitals of the Li$^{2+}$. Note that, in the Kohn–Sham TDDFT, the only real observable is the total electronic density and the many-electron wavefunction is unknown, so it is emphasized that Eq. (3) is an approximation to the exact state-selected probabilities. However, this approximation does not obstruct our quantitative discussions on the state-selected probabilities because Eq. (3) is still able to describe the electronic excitation on a single-particle level. Moreover, the LDA approximation is lack of self-interaction corrections, which may lead to the inadequate description on the scattering dynamics[40] and higher electronic capture probabilities[41] for low-energy collisions. This work shows the capability of the Ehrenfest-TDDFT in dressed Li$^{2+}$ ion collisions at MeV energies, which may have implications for the study of the charge transfer in more complex systems in the future. Acknowledgement. This work was financially supported by the National Key Research and Development Program of China (Grant No. 2017YFA0402300), and the National Natural Science Foundation of China (Grant Nos. 11774344, 11704039, and 12104019). References Roadmap on photonic, electronic and atomic collision physics: III. 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