The two-center atomic orbital close-coupling method is a semi-classical method for describing the collisions between ions and atoms (ions) in the intermediate energy region. The approximation of straight-line trajectories is used for the nuclear motion, and the details of this method can be found in Ref. [11]. Here only a short introduction is given.

The total wave function Ψ of the collision system can be written in the form 4

where the functions Φj and χ_n are atomic orbitals centered on the target and the projectile ion, respectively. The atomic orbitals can be determined by Slater basis in the form 5 6

where C_nk is the expansion factor, Y_lm(r) is the spherical harmonic function, α and β are the variational parameters and are shown in Table 4, and N_l(ζ_k) is the normalization coefficient for the slater basis.

Table 4.

Table 4.

Table 4. The variational parameters α and β in the Slater basis for the projectile and target, respectively. . State α β Li2+ ns 0.100 1.322 np 0.088 1.324 nd 0.145 1.255 nf 0.200 1.220 ng 0.220 1.200 Li nh 0.100 1.370 ns 0.0805 1.334 np 0.090 1.334 nd 0.085 1.355 nf 0.100 1.370

Table 4. The variational parameters α and β in the Slater basis for the projectile and target, respectively. .

The total wave function Ψ satisfies the Schrödinger equation , where are the interactions between the active electron and the projectile/target, respectively. After inserting the total wave function (4) into the above Schrödinger equation, the coupled equations for the amplitudes Aj and B_n (centered on the projectile and target, respectively) can be obtained as follows: 7 8

where the dot denotes differentiation with respect to t, Sjm is the overlap matrix, H_jk and H_nm are the direct matrices, and K_jm and K_nk are the exchange matrices.

The cross section σ_in for electron capture can be obtained by solving the above coupled equation for the amplitudes under the initial condition Aj(t → ∞) = d_ij,B_n(t → ∞) = 0. If the incident channel is channel i, the probability for finding the system in the final state n is 9

here b is the impact parameter.

As mentioned in the preceding section, the interaction of the active electron with the target ion (Li⁺) has been represented by a model potential as follows: 10

In Table 1, we give the binding energies of the ground and excited states of Li²⁺ ions and Li atoms which are obtained by the diagonalization of the single-electron Hamiltonian with the Coulomb and above model potentials. Moreover, the corresponding data from NIST are also given for comparison, good agreement with the calculated energies has been obtained.

Table 1.

Table 1.

Table 1. Energies (in units of a.u.) of Li2+ ion and Li atom obtained by diagonalization of the single-electron Hamiltonian compared with the NIST data. . Vs (present) NIST Error Li2+ 1s –4.50000 –4.50200 0.00044 2s –1.12500 –1.12554 0.00048 2p –1.12500 –1.12548 0.00043 3s –0.50000 –0.50023 0.00045 3p –0.50000 –0.50021 0.00042 3d –0.50000 –0.50018 0.00037 4s –0.28122 –0.28137 0.00054 4p –0.28125 –0.28136 0.00041 4d –0.28125 –0.28135 0.00037 4f –0.28125 –0.28135 0.00035 5s –0.17972 –0.18008 0.00197 5p –0.17999 –0.18007 0.00045 5d –0.18000 –0.18007 0.00037 5f –0.18000 –0.18006 0.00035 5g –0.18000 –0.18006 0.00035 6s –0.12380 –0.12505 0.01000 6p –0.12493 –0.12505 0.00095 6d –0.12498 –0.12505 0.00052 6f –0.12500 –0.12504 0.00035 6g –0.12500 –0.12504 0.00035 6h –0.12500 –0.12504 0.00034 Li 1s22s –0.19800 –0.19822 0.00113 1s22p –0.12994 –0.13029 0.00268 1s23s –0.07428 –0.07421 0.00091 1s23p –0.05720 –0.05726 0.00104 1s23d –0.05557 –0.05563 0.00106 1s24s –0.03867 –0.03863 0.00100 1s24p –0.03197 –0.03199 0.00055 1s24d –0.03126 –0.03129 0.00085 1s24f –0.03125 –0.03126 0.00017

Table 1. Energies (in units of a.u.) of Li2+ ion and Li atom obtained by diagonalization of the single-electron Hamiltonian compared with the NIST data. .

It is crucial to use a large expansion basis in the TCAOCC method for ensuring the convergence of the cross section results. In Tables 2 and 3, we compare the nl-partial cross sections for the dominant (n = 4) and sub-dominant (n = 3,5,6) capture channels at the collision energies of 1 keV/u and 10 keV/u. For avoiding the problem of linear correlation of the expansion basis sets centered on the projectile and target, different expansion basis sizes are used in the present calculation for different impact energy regions. In the energy range 0.1–2 keV/u, the projectile contains all bound states of n ≤ 8 (l ≤ 5), and the target contains all bound states of n ≤ 4. For energies above 3 keV/u, the projectile contains all n ≤ 13 (l ≤ 5) bound states and 68 quasi-continuum pseudostates, while the target contains only all bound states of n ≤ 4.

Table 2.

Table 2.

Table 2. The nl-partial cross sections (in units of 10−16 cm2) for the dominant and sub-dominant capture channels in Li2+–Li(1s22s) collisions at 1 keV/u with different basis sets. . nl Li2+(n ≤ 7)/Li(n ≤ 4) Li2+(n ≤ 8)/Li(n ≤ 4) Li2+(n ≤ 8)/Li(n ≤ 5) Li2+(n ≤ 9)/Li(n ≤ 2) 3s 0.74786 0.74525 0.74207 0.67981 3p 2.78162 2.79866 2.79958 2.82756 3d 1.26894 1.27982 1.28445 1.29247 4s 12.78756 12.88953 12.87795 12.07162 4p 32.57727 32.65314 32.72342 30.67455 4d 49.70066 49.59046 49.60763 46.50709 4f 82.62856 82.66100 82.64863 80.08355 5s 0.72093 0.79581 0.82376 0.81078 5p 2.29183 2.28852 2.23026 3.02780 5d 3.95315 4.02875 4.13100 4.73969 5f 3.90052 3.87196 3.91839 4.38944 5g 1.65605 1.63617 1.61871 2.14910 6s 0.23295 0.26380 0.25269 0.38003 6p 0.90188 0.87596 0.90841 1.97223 6d 1.02486 1.04176 0.97867 2.50951 6f 0.87311 0.92330 0.91127 2.14721 6g 0.79499 0.79785 0.78122 1.77867 6h 0.34124 0.29657 0.32079 0.75191

Table 2. The nl-partial cross sections (in units of 10−16 cm2) for the dominant and sub-dominant capture channels in Li2+–Li(1s22s) collisions at 1 keV/u with different basis sets. .

Table 3.

Table 3.

Table 3. The nl-partial cross sections (in units of 10−16 cm2) for the dominant and sub-dominant capture channels in Li2+–Li(1s22s) collisions at energies of 10 keV/u and 100 keV/u with different basis sets. . nl E = 10 keV/u E = 100 keV/u Li2+(n ≤ 13)/Li(n ≤ 4) Li2+(n ≤ 15)/Li(n ≤ 6) Li2+(n ≤ 13)/Li(n ≤ 4) Li2+(n ≤ 15)/Li(n ≤ 6) 3s 0.64262 0.66742 0.00546 0.00435 3p 2.87482 2.91122 0.05494 0.05645 3d 5.69733 5.73932 0.04826 0.05352 4s 0.85291 0.90728 0.00371 0.00294 4p 4.16113 4.29256 0.02472 0.02627 4d 12.82695 13.11016 0.02206 0.02246 4f 50.18307 50.40761 0.00697 0.00880 5s 0.74181 0.72777 0.00274 0.00193 5p 2.66397 2.63733 0.01372 0.01439 5d 7.31421 7.35082 0.01019 0.01065 5f 14.34866 14.77341 0.00402 0.00556 5g 14.44446 14.47175 0.00134 0.00150 6s 0.52141 0.55310 0.00231 0.00130 6p 1.73196 1.61667 0.00921 0.00993 6d 4.03045 3.91881 0.00560 0.00540 6f 5.16323 5.07694 0.00218 0.00288 6g 3.72710 3.82662 0.00098 0.00117 6h 1.65248 1.63444 0.00020 0.00021

Table 3. The nl-partial cross sections (in units of 10−16 cm2) for the dominant and sub-dominant capture channels in Li2+–Li(1s22s) collisions at energies of 10 keV/u and 100 keV/u with different basis sets. .

In Table 2, the cross sections calculated with different basis sets are compared for the impact energy of 1 keV/u. From this table, we can see that the nl-state-selective cross sections for the basis set Li²⁺(n ≤ 8)/Li(n ≤ 4) are very closed to those of basis set Li²⁺(n ≤ 8)/Li(n ≤ 5). It indicates that the basis placed on the target is reasonable. However, when compared with the results of basis set Li²⁺(n ≤ 9)/Li(n ≤ 2), we can see that there exist obvious differences in the nl-state-selective cross sections for both dominant and sub-dominant reaction channels. This means that the number of expansion basis set on the target is not enough. Further, by comparing the results of basis set Li²⁺(n ≤ 8)/Li(n ≤ 4) with those of basis set Li²⁺(n ≤ 7)/Li(n ≤ 4), it can be seen that the cross sections from these two basis sets are comparable. It means that the basis sets placed on the incident particles are reasonable.

In Table 3, the cross sections calculated with different basis sets are compared for the impact energies of 10 keV/u and 100 keV/u. The results are reliable for their convergence under the selected basis sets. A similar procedure is also performed below for the collision systems of Li³⁺–Li(1s²2p₀) and Li³⁺–Li(1s²2p₁).

The present total charge transfer cross sections for the collision of Li³⁺–Li(1s²2s) in the energy range 0.1–300 keV/u are shown in Fig. 1. In the same figure, the previous AOCC calculations by Schweinzer et al.^[9] are also shown for comparison, and no other available data can be used for this collision system until now. It can be seen from this figure that the previous AOCC results of Schweinzer et al.^[9] have the same energy-dependent behavior with our present results, but the magnitude of the previous AOCC cross section is obviously larger than the present result in the overlapping energy range. The difference between the results of different AOCC calculations maybe come from both the different model potentials used to represent the electron–ion interaction and the difference in the size of the AO expansion basis used in the dynamic calculations. It is noted that the present total charge exchange cross sections show a flat energy behavior for the collision energy below 10 keV/u. Above this energy, the present AOCC cross sections start to decrease sharply. The near flat energy behavior of the total charge exchange cross section in the energy region below 10 keV/u is a result of the different energy behavior of the nl-state-selective electron capture cross section in the same energy region.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. Total charge exchange cross sections for Li3+–Li(1s22s) collisions. Schweinzer refer to the results of Ref. [9].

Figure 2 shows our AOCC cross sections for electron capture to n = 3, 4, 5, 6, 7 shells of Li²⁺ ion in the Li³⁺–Li(1s²2s) collisions. The theoretical CTMC results of Otranto et al.^[10] in the same energy region are also shown for comparison. No other theoretical data have been reported for these n-partial cross sections. It can be observed that the electron capture to the n = 4 shell dominates in the energy region below ∼ 15 keV/u for this collision system, but for higher energies, the cross sections for capture to n = 3, 5, 6, 7 shells become comparable. In the overlapping energy region, the present AOCC results for capture to n = 4, 6, 7 shells of Li²⁺ ion are in general agreement with the CTMC results of Otranto et al.^[10] For the n = 5 shell, the present AOCC cross sections agree with the CTMC calculations by Otranto et al.^[10] for the collision energies above 1.5 keV/u. While the present AOCC results for capture to the n = 3 shell agree poor with those of CTMC calculations^[10] in the considered energy range. The significant disagreement between the present TC-AOCC results and CTMC calculations^[10] in the low energy region can be attributed to a fact that CTMC is a full classical method and it is not suitable to describe the collision processes at the low collision energies. Furthermore, the CTMC method can not give the n- and nl-state-selective electron capture cross sections directly. Within that method, the determination of quantum numbers (n,l,m) of the captured electron is based on the quantum–classical correspondence principle, which is appropriate only for high n values. However, in the present TC-AOCC method, the use of suitable model potentials and very large expansion basis (including pseudostates) leads to an accurate description of the collision dynamics to generate convergent and accurate state-selective electron-capture cross sections in the intermediate collision energy region. It should be noted that all the n-partial electron capture cross sections (including the total charge transfer cross section in Fig. 1) decrease sharply for the collision energies above ∼10 keV/u. This decrease for the energy region above 10 keV/u is mainly due to a fact that the excitation and ionization processes become more important at the high collision energies.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. Cross sections for electron capture to n = 3–7 shells of Li2+ ion in Li3+–Li(1s22s) collisions. CTMC data are from Ref. [10].

In Fig. 3, we show the n-distribution (n = 2–6) of the charge transfer cross sections in Li³⁺–Li(1s²2s) collisions at impact energies of 1 keV/u, 10 keV/u, 25 keV/u, and 50 keV/u, respectively. The results of the present AOCC calculations are compared with the CTMC calculations of Otranto et al.^[10] It can observed that for all considered impact energies, our n-distribution of the charge transfer cross sections is in general agreement with that of CTMC calculations,^[10] especially for the low collision energies. It should be noted that for E = 1 keV/u, the electron capture cross sections increase sharply with the increase of n, reach a maximum at n = 4, and then decrease sharply. This means that n = 4 is the dominant capture channel at this collision energy. We further note that with increasing collision energy, the charge transfer cross sections still first increase up to n = 4 and then decrease. But the magnitudes of the cross sections for capture to different n shells become comparable, especially for the high energies. It is interesting to note that for E = 50 keV/u, the charge transfer cross sections reach the maximum at n = 3, and then decrease slowly. It means that with increasing collision energy, the captured electron will have a broad distribution on the different excited states of the projectile ions. It also demonstrates that only a large expansion basis can lead to the accurate nl-state-selective electron capture cross sections, especially for the high energies.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. The n-distributions of charge transfer cross sections for Li3+–Li(1s22s) collisions at impact energies of (a) 1 keV/u, (b) 10 keV/u, (c) 25 keV/u, and (d) 50 keV/u.

In Fig. 4, we present our AOCC state-selective cross sections for electron capture to the 3l, 4l, 5l, and 6l states of Li²⁺ ion in Li³⁺–Li(1s²2s) collisions, respectively. No other available data of state-selective cross sections has been reported for this collision system until now. It should be noted that for the weak electron capture channels (3l, 5l, 6l), there exist some oscillatory structures in the state-selective cross sections in the energy region below ∼ 1.5 keV/u, these structures are from the more complex population dynamics at low collision energies for these nl states.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. Cross sections for electron capture to (a) 3l, (b) 4l, (c) 5l, and (d) 6l states of Li2+ ion as a function of the impact energy in Li3+–Li(1s22s) collisions.

The total and partial cross sections for electron capture to n = 3–6 shells of Li²⁺ ion in Li³⁺–Li(1s²2p) and Li³⁺–Li(1s²2p₁) collisions from the present AOCC calculations are displayed in Fig. 5. To the best of our knowledge, no other available data have been reported for these collision systems so far. From Fig. 5(a), we can see that for the Li³⁺–Li(1s²2p) system, the electron capture to the n = 5 shell dominates in the energy below 30 keV/u, but it coincides with the case of capture to the n = 4 shell for energies above 2 keV/u. In the energy region 3–5 keV/u, the magnitude of the n = 6 captured cross section becomes comparable with those of n = 5 and n = 4 shells. In the energy region above 50 keV/u, the n = 3 shell becomes the dominant reaction channel due to the momentum transfer. From Fig. 5(b), we can see that the electron capture to the n = 5 shell of Li²⁺ ion dominates up to 5 keV/u. For the energy region 5–20 keV/u, the cross sections for electron capture to n = 4–6 shells become comparable. By comparing Fig. 5(a) with Fig. 5(b), it can be seen that the energy behavior and magnitude of the n-partial electron capture cross sections have significant difference for the collision energy below ∼ 20 keV/u. This difference demonstrates that the n-partial electron capture cross sections are sensitive to the initial p-state charge cloud alignment, especially for the low collision energies. In the energy region above 20 keV/u, we can see that the magnitude and energy behavior of the n-partial cross sections from different initial target states are very similar.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. Total and partial cross sections for electron capture to n = 3–6 shells of Li2+ ion in (a) Li3+–Li(1s22p0) and (b) Li3+–Li(1s22p1) collisions.

Meanwhile, comparing Fig. 5 with the results from the initial Li(1s²2s) state (shown in Fig. 2), we can see that the dominant electron capture channel and the magnitudes of the different n-partial electron capture cross sections all have significant changes for the case of Li(1s²2p_0,1) target.

The cross sections for charge transfer to the 3l,4l,5l, and 6l states of Li²⁺ ion in Li³⁺–Li(1s²2p₀) and Li³⁺–Li(1s²2p₁) collision systems are displayed in Figs. 6 and 7, respectively. No other theoretical or experimental data have been reported for these subshell-selective cross sections until now. It can be observed that for the dominant reaction channel, the magnitudes of the 5l-capture cross sections are significantly larger than those of the subdominant channels in the energy region below about 1 keV/u for both collision systems. With increasing impact energy, for energies above 50 keV/u, the magnitudes of different subshell-selective cross sections become comparable. The more complex population dynamics of the nl states at low energies result in some oscillatory energy behavior of the nl cross sections in the low energy region. Comparing the partial charge exchange cross sections of the Li³⁺–Li(1s²2p₀) system with those of the Li³⁺–Li(1s²2p₁) system, we can see that there exist obviously differences in the magnitude and energy behavior of the nl-state-selective cross sections due to the strong alignment dependence on the initial targets.

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. State selective cross sections for electron capture to (a) 3l, (b) 4l, (c) 5l, and (d) 6l states of Li2+ ion from the initial Li(1s22p0) state.

Fig. 7.

	Figure Option ViewDownloadNew Window
	Fig. 7. State selective cross sections for electron capture to (a) 3l, (b) 4l, (c) 5l, and (d) 6l states of Li2+ ion from the initial Li(1s22p1) state.

In Fig. 8, we display the n-distribution of the charge exchange cross sections for the Li³⁺–Li(1s²2s) and Li³⁺–Li(1s²2p_0,1) collision systems at impact energies of 1 keV/u, 10 keV/u, 50 keV/u, and 100 keV/u, respectively. It can be seen from this figure that for E = 1 keV/u, the dominant electron capture channel is the n = 4 shell for the Li³⁺–Li(1s²2s) system, while for the Li³⁺–Li(1s²2p_0,1) collision systems, n = 5 is the dominant reaction channel. We further note that the magnitudes of the cross sections for these dominant reaction channels are significantly larger that those of the subdominant channels. With the increase of the collision energy, we can see that for E = 50 keV/u, the dominant electron capture channels for the Li³⁺–Li(1s²2s) and Li³⁺–Li(1s²2p_0,1) systems become the n = 3 and n = 4 shells, respectively. For E = 100 keV/u, the dominant reaction channels for these three collision systems have all become the n = 3 shell, and the magnitudes of the cross sections for the dominant and subdominant channels become comparable.

Fig. 8.

	Figure Option ViewDownloadNew Window
	Fig. 8. The n distributions of electron capture cross sections in Li3+–Li(1s22s) and Li3+–Li(1s22p0, 1) collision systems at impact energies of (a) 1 keV/u, (b) 10 keV/u, (c) 25 keV/u, and (d) 50 keV/u.