1. Introduction

Excitation of electronic states of molecules is important in several areas [¹]. It is well known that an accurate theoretical study of description of positron collisions requires more than simply changing the sign of the projectile. For example, short-range polarization and correlation effects are different from electron scattering and the role of positronium channel (Ps) represent an special challenges (Ps channel is not present for electron case). The observation and absolute experimental data still represent a problem [²], same for targets as H₂ molecule. Specifically, for inelastic positron-H₂ scattering results (theoretical calculations and experimental) are modest when compared with inelastic electron-H₂ scattering. Sullivan et al. [³] produced experimental cross sections for the  B1∑+‍u state by positron impact and some reviews of a selection of measurements for H₂ can be found also in Surko et al. [⁴]. On the theoretical side, few calculations of cross sections for excitation of H₂ by positron impact have been reported. For example, Lino et al. [⁵] reported integral cross sections (ICS) for excitation of the  B1∑+‍u state using the Schwinger multichannel (SMC) method, Arretche et al. [⁶,⁷] reported the same  B1∑+‍u state using again the SMC method, and Lino [⁸,⁹] using the SBP approach reported cross sections for the  B1∑+‍u and C¹ Π _u states. With the advance of quantum mechanical computational methods, some very accurate ab initio calculations as the close-coupling, the R-matrix method, complex Kohn and Schwinger multichannel method were performed for positron scattering and these significant efforts were published by Surko et al. [⁴]. Many analytical formulas for positron-molecule scattering have been developed to overcome these difficulties (the merit of analytical expressions is of convenience and increased applicability when cross sections may be required in more complicated modeling situation). Kim [¹⁰] originally proposed an scaling plane wave Born (called BEf-scaling) to describe electron-molecule scattering and this method was adapted by Lino [⁸,⁹] to describe positron-molecule scattering (called SBP approach). Comparison with available theoretical calculations and experimental data showed that cross sections using the SBP approach to positron-molecule are encouraging [¹¹-¹³]. Motivated by these results we have taken up the task to investigate the numerical stability of the SBP approach for H2 molecule. Here, we repeat the prior calculation of positron-H₂ scattering to  B1∑+‍u (C¹ Π _u ) state [⁹], and extend the investigation to  B’1∑+‍u, and D¹ Π _u state. To the best of our knowledge there are no corresponding theoretical or experimental inelastic cross sections and in the absence of these results, comparisons are made with analogous electron scattering. A preliminary study using three-state of approximation to investigate the influence of open channels (energetically accessible electronic states multichannel effects) also is proposed.

In Sec. 2 we identify the SBP method for positron scattering. In Sec. 3 computational procedures and results are discussed. Conclusions are presented in Sec. 4.

2. Outline of theory

The SBP approach is given by

where

where Eps is the positronium energy, Eexc is the excitation energy, E is the energy incident positron particle, faccur is an accurate dipole oscillator strength value from experiments or from accurate wavefunctions, and fBorn is the dipole oscillator strength value from first Born approximation (FBA). The first scaling proposed by Kim [¹⁰] for electron scattering replaces the “E” energy by E+B+Eexc where B is the ionization energy of the target electron. The original qualitative justification for this scaling was that the “effective” incident energy seen by the target electron is E plus the energy of the bound electron. The denominator Eq. (1) can seen as the scaling factor to represent the correlation between the positron and electron. We have used in Eq. (1)

where 6.8 eV is the ground state binding energy of positronium. Cross sections for positron-impact and electron-impact excitation are identical when Born approximation is used but the relation Eps=B-6.8 eV is used to identify the positron as incident particle. The Eq. (1) is only a indicator of the order of magnitude of a constant shift to be added to E and we have observed that studies using the SBP show equally good results for molecules. The FBA in Eq. (1) represent the starting point and is used in the scaling because the plane wave is the correct wave function at infinity for an particle with charge colliding with a target. As observed in Eq. (2), E is increased by a constant (Eps+Eexc) and this modification has some consequences practical to the performance of the SBP method [⁸,⁹]. The f(E) factor reduces the FBA at low energies while keeping the validity of the Born approximation at high energies and the SBP approach has the effect of correcting the FBA (the f(E) factor also identify the positron as incident particle). The FBA can be written as

where V is the Coulombic interaction between the incident positron and the molecular target where Φi and Φf are initial and final electronic states of the target, respectively. Qualitatively, the FBA does not account distortion of plane wave in the vicinity of the target and is apply only to integrated excitation cross sections. When dealing with dipole transitions the long-range character of the dipolar coupling requires a larger number of partial waves and because of it, higher partial waves are not well described for several sophisticates ab initio methods [⁵]. This consideration is very important and we have observed that the SBP approach can provide realistic excitation cross sections for many targets, which are not only difficult to measure but also cannot easily be calculated with existing theories. The present study has several goals: first, as cited before to the best of our knowledge, no theoretical study using the SBP approach has been observed to B'1∑+‍u, D¹ Π _u , and also E,  F1∑+‍u states. Second, in the present study we reexamine our previous calculations to  B1∑+‍u, and C¹ Π _u , states [⁹], and third, a study preliminary on a indirect multichannel effects (using E,  F1∑+‍u state) is discussed. As we will see, our results for positron-H₂ scattering are highly encouraging.

3. Computational procedures and results

Many articles have been published on positron-H₂ scattering using sophisticate basis set and here we have used the basis set presented in Table I (see Ref. [¹⁴]).

Whit this basis set in Table II we shows vertical excitation energies using the improved virtual orbital (IVO) [¹⁴].

Many theoretical papers were published on the generalized oscillator strengths (GOSs) for these transitions in H₂, from which integrated Born excitation cross sections can be derived. The integrated Born cross section σ _Born for the excitation of an atom or molecule in terms of the GOS is given by

where R is the Rydberg energy and Q the momentum transfer and its minimum and maximum values. For a dipole and spin-allowed excitation, the GOS can be calculated using correlated wave functions fitted accurately [¹²]. With the fitted analytic function for the GOS, the integration of GOS in σ _Born can be carried out analytically for arbitrary “E”. The appropriate f _acc and f _Born value to use in the SBP approach is given in the Table III (see Ref. [¹²]).

As a first step we will show results obtained using FBA for all states. These examples were chosen to allow assessment of the numerical quadrature technique using the FBA [⁵]. In Figs. 1-5 we show the FBA integral cross section for the  B1∑+‍u, C¹ Π _u , B'1∑+‍u, D¹ Π _u , and E,  F1∑+‍u states compared with other FBA [¹⁴].

Figure 1 Integral cross sections (ICS) for positron-H₂ (B 1∑+‍, state) using FBA. Solid line, our FBA; dashed line, FBA [¹⁴]. 

Figure 2 Same as Fig. 1. C 1∑+‍, state. 

Figure 3 Same as Figure 1. B'1∑+‍, state. 

Figure 4 Same as Fig. 1. D¹ Π _u , state. 

Figure 5 Same as Fig. 1. E,  1F∑+g, state. 

As observed in Figs. 1-5 our FBA results predicts equal cross sections with other FBA [¹⁴]. The results suggest a good convergence of cross sections in order to better describe the SBP approach. When dealing with dipole allowed transition, such as  B1∑+‍g, the long-range character of the dipolar coupling requires a larger number of partial waves and the SBP approach, in principle, can be used to include these contributions. In Fig. 6 we show the SBP method for 15, 20, 25, and 30 eV compared with the sophisticate ab initio SMC method [⁶], and experimental data [³]. As observed the SBP approach provides cross sections with the same order of magnitude as the experimental data [³] at low energy. Similarity (shape) between SBP approach and SMC method [⁶] also can be observed (as the SMC method is expected to produce better cross sections, the present SBP cross sections may be very encouraging).

Figure 6 Integral cross sections for positron-H₂ ( B1∑+‍g, state). Circle, experimental data[³]; solid line, SBP approach; dott line, SMC method [⁶]. 

As a next test to verify the stability of the X1∑+‍g→B1∑+‍g cross section, we have compared in Table IV the SBP method with the sophisticated convergence close coupling (CCC) for positron impact [¹⁵] at low energies (the comparison of our two-state ICS with CCC method shows that the SBP approach is encouraging).

As observed in Table II the agreement between our SBP and CCC [¹⁵] is reasonable, giving confidence that the SBP approach is promise (for positron scattering the CCC is expected to produce better cross sections [¹⁵]). In Table V we shows cross sections for the C¹ Π _u state using the SBP compared with CCC [¹⁵] method. Again the similar behavior between the two methods can be observed.

In Fig. 7 we shows cross sections for the B'1∑+‍u state compared with the sophisticated ab initio SMC method [¹⁴]. As observed the SBP approach is similar with SMC method [¹⁴].

Figure 7 Integral cross sections for positron-H₂ (B'1∑+‍u, state). Solid line, SBP approach, dashed line, SMC method [¹⁴]. 

In Fig. 8 we shows cross sections using the SBP approach for the  D1∑+‍g state compared now with the BEf-scaling for electron case [¹⁶]. As expected at high energies the cross sections by positron impact and electron impact tend to get closer.

Figure 8 Integral cross sections for positron-H₂ ( D1∑+‍g, state). Solid line, SBP approach, dashed line, BEf-scaling for electron [¹⁶]. 

As a possible criteria to verify again the stability of the X1∑+‍g→B1∑+‍g cross section, we have carried out calculations in three-state of approximation to investigate the influence of open channels (energetically accessible electronic states multichannel effects). Evidently, the FBA does not account multichannel effects and here we have used a alternative mechanism, i.e., a indirect study to verify the contribution (∑‍ Born) n=states and (∑‍Eexc) in Eq. (1). The three states was done considering the ground X¹, B¹ and E states.

As knowledge is expected that the “E ₁ state” as a open channel does not disturb the X1∑+‍g→B1∑+‍g cross section, which should not be a surprise [⁶]. We can easily see that the Fig. 9 satisfy this criterion, giving confidence that our preliminary study is consistent.

Figure 9 Integral cross sections for positron-H₂ using tree states. Dashed line, X1∑+‍g→B1∑+‍g using SBP approach. Solid line, X1∑+‍g→B1∑+‍g + E,  1F∑+g states using SBP approach. 

In the Table VI integral cross sections for the  B1∑+‍u, C¹ Π _u , B'1∑+‍u, and D¹ Π _u electronic state using SBP approach are listed for future reference (units ao2)

When positrons annihilate on many-electron molecules, the annihilation cross sections is traditionally written as [¹⁷-²⁰]

where

is the effective number of electrons that contribute to the annihilation process, and ψ(r1,…,rzr) is the total wavefunction of the Z electron and one positron coordenates. If substitutes the asymptotic, plane wave function into Eq. (6), one obtains Zeff=Z (classical). We have noted that the SBP approach can determine the Zeff classical based on the Born approximation, i.e., the Zeff(SBP) represent also a important criteria to check the quality of SBP approach using the FBA. The accuracy of the Zeff Born approximation can be given by

In Fig. 10, we present Zeff(SBP) for the H₂ using, for example, B1∑+‍g state. As observed the Zeff shows aspects of convergence with Z classical (Z = 2) and indicate that the SBP approach is significant and suggested that Eq. (7) can be used as a strategies to treat convergence characteristic of the SBP approach.

Figure 10  Zeff using the SBP approach. Solid line, SBP approach. 

4. Conclusion

Electronic excitation by positron impact is very challenging for theoreticians and experimentalists and as observed in our study inelastic cross sections for e+- H₂ scattering still are scarce. Analysis of the results indicated that electronic excitation of H₂ by positron impact using the SBP approach illustrate good convergence characteristics of the procedure when compared with ab initio method at low energies for the  B1∑+‍u state. A second observation is that the  B1∑+‍u, and C1π cross sections are practically equal and these results can motive experiments. In the absence of experimental or theoretical data for C1π, and D 1∑+‍ electronic states the SBP approach was compared with electron scattering case and for energies lower the cross sections for electrons being smaller in magnitude compared with positron scattering (this is expected). A preliminary study on effects multichannel showed that a indirect description multichannel can be improved to estimate cross sections, as for example the presence of the E,  F1∑+‍g state as an indirect open channel not disturb the X1∑+‍g→B1∑+‍g cross section, which is not surprise but the test using the SBP approach is specially relevant. Finally, the SBP approach for positron-H₂ suggests that the method is economical in the sense that relatively small basis set are capable of providing reliable cross sections with small effort computational. We also conclude that Z classical parameter is consistently good, giving confidence that the SBP method is significant for investigate positron-H₂ scattering. We believe that the cross sections using the SBP method may stimulate other experimental measurements be done to confirm our studies.