Quantum Charge Transfer Study of Triply Charged Ions in the Adiabatic Representation : the ( BHe ) 3 + System

O estudo quântico do processo B + He → B + He foi investigado para energias de colisão variando de 1-102 eV usando um potencial de interação ab-initio. Um método novo foi usado para resolver a equação de Schrödinger na base adiabática, onde os acoplamentos rotacional e radial foram considerados. Além disso, uma discussão sobre os acoplamentos de diferentes simetrias é apresentada. Usando o modelo de Landau-Zenner, mostramos que o modelo de dois estados não deve ser usado para este sistema. Este estudo poderia indicar que tal modelo deve ser usado cuidadosamente para outros sistemas onde o processo de transferência de carga seja considerado. Finalmente, as seções de choque total quântica foram comparadas com trabalhos publicados anteriormente por Gargaud e co-autores e uma concordância razoável foi observada.


Introduction
Charge transfer in atomic and molecular collisions has been intensively studied in the past few decades [1][2][3][4][5] .The importance of this process arises in different branches of science such as astrophysics, chemistry, laser research, material science and many others 5 .
Theoretical or experimental works in this field have grown up rapidly due to new techniques that have become 6 available, especially in the case of high resolution molecular beam experiments 7 .
From the theoretical point of view the mechanisms to elucidate charge transfer processes need accurate potential energy functions and therefore exact quantum calculations have to be carried out.However, most charge transfer processes studied have been mainly concerned with only single charge transfer systems 5 .In this case only covalent dissociating states are involved and hence the quantum scattering calculations are simpler.On the other hand, for systems with multiple charge transfer such calculations are more difficult to perform due to covalent and ionic states being involved.The first quantum theoretical work where double charges were considered was that of Braga et al. 8 In the latter work, the Ar 2+ /He system was analysed by quantum mechanics and an intermolecular potential model was proposed based on ab-initio calculations.The quantum scattering calculations were carried out in the diabatic basis and no rotational to radial couplings were included.Nevertheless satisfactory agreement was achieved compared with the experimental measurements of Friedrich and collaborators 9 .The recent research review of Herman 10 has updated the existing literature on charge transfer in atomic and molecular collisions in the experimental and theoretical pictures.As shown by Herman, the experimental apparatus have shown better resolution and consequently lower relative errors can be obtained for the experimental data.
For the charge transfer process with doubly charged molecules, Herman and co-workers [10][11] have shown that theory and experiment are well developed for resolving excited electronic states.For example, Herman and collaborators 11 have studied the CO 2+ +Ne system by using the potential energy function calculated by Larsson et al. 12 for this single charge transfer process.This latter work showed a quantitative agreement with recent experiments of cross molecular beams perform by Hamdan and Brenton 13 , and has confirmed that the intermolecular potential of Larsson et al. 12 , for CO 2+ /Ne collision is able to describe the energy separation of the minima and the equilibrium distance for the 1 3 Π and 1 3 Σ states.However, due to the resolution of the energy, about 0.3 eV, the vibrational states were not fully resolved although the important features of the population for the energy spectra were analysed.For polyatomic molecules, such as NH3 and H2S colliding with He 2+ , the theoretical and experimental work of Fárník and co-workers 14 are the most recent advance in multiple charge transfer processes.By using the crossed molecular beam technique with resolution of about 120 meV, they were able to study, in detail, the contribution of the energy transfer between all quantum state.The relaxation into each rotational, vibrational and electronic states has provided important results such as the Franck-Condon factors.
According to Herman 10 , simple models have been applied to dication-molecule and cation-molecule chemical reactions and charge transfer processes.In addition, charge transfer processes with multiple charge are one of the research areas that have recently turned out to be of considerable interest.
Very recently Boyd and collaborators 15 have proposed an inversion of the diabatic coupling potentials for multiple charge transfer on the C 4+ /He system.Experimental inelastic cross sections were used as the input data.By using a general inversion algorithm based on functional sensitive analysis, the diabatic coupling potential for such a system was derived.This inversion was accurate especially in the region where accurate ab-initio calculations are difficult to perform.
The present work will analyse the charge transfer of the B 3+ /He system, which is isoelectronic to C 4+ /He, and where ionic and covalent states are also involved.López-Castillo and Ornellas 16,17 have recently produced a potential energy surface where all the important couplings (rotational and radial) for the dynamics were well defined.The reactions presented in Eq. 1 were calculated with an accurate basis set 17 .
Although the above authors have produced quality potentials and couplings, their ab-initio calculations have been tested using the classical path method 16 for colinear trajectories and for very high collision energy (1 keV up to 50 keV).Nevertheless, their calculations are in fair agreement with experimental cross sections, but some experimental data 18,19 are overestimated.The agreement is almost qualitative, but it is quite consistent with the subsequent experimental results of the Iwai et al. 20 This paper aims to calculate the state-to-state total cross sections for the above process, in the collision energy ranging from 1 eV to 102 eV and using the renormalized Fox-Goodwin method as given by Braga and Belchior 21 .The latter method provides a powerfull tool to propagate the radial wavefunction with the intermolecular potential in an adiabatic basis.This combination of low collision energy and quantum analysis will provide a better understanding of the quality of the potential energy surfaces given by López-Castillo and Ornellas 16 .In addition, the well known two state Landau-Zener (LZ) model will be applied to analyse the validity of approaching a problem of several states by a two state model.Such a method is well detailed elsewhere 8 and will not be repeated here.Since there is no experimental data in this collision energy range, a comparison with previous theoretical work will be done 22 .Actually, the low collision energy used in the present calculations could be achieved by experimentalists for carrying out measurements in this energy range.

Adiabatic quantum scattering theory
The matrix representation of the Schrödinger equation in the diabatic basis, k 2 being the channel wave number, l the angular momentum, µ the system reduced mass and V(R) the intermolecular potential, is not always convenient to use in the study of charge transfer process.The non-commutability of the rotational to radial operators prevents the above representation from always be useful.In a standard procedure rotational to radial couplings are neglected and the diabatic representation is used for studying charge transfer dynamics.Concerned with the above problem Braga and Belchior 21 have recently developed an algorithm to integrate the Schrödinger equation in the adiabatic representation, which is defined in such a way that, in this new representation, the diabatic potential becomes diagonal.Certainly this is performed with the expense of making the kinetic operator non-diagonal in this adiabatic representation.Nevertheless the two representations should coincide for the large scattering coordinate where the transformation between these two representations, denoted by U(R), becomes R independent and goes to a constant equals to the identity transformation.
In this new representation the Schrödinger equation is then written as, The algorithm to solve the above equation 21 , which combines efficiency and simplicity, is based on a three point recurrence relation established by Fox and Goodwin in 1949 23 .Defining the quantities, and, the adiabatic Schrödinger equation is then transformed 21 into the recurrence relation, where R(R) = X(R+h)X -1 (R).This renormalization gives the necessary stability for the closed channels as discussed in detail by Braga and Belchior 21 .The matrices U(R) and D(R) are provided by the electronic calculations.A bicubic spline interpolator was implemented with the quantum scaterring code and hence no analytical fitting of these matrices were necessary.For details about the electronic calculation see references 16,17 .
In the asymptotic region the Schrödinger equation, either in the adiabatic or the diabatic representation, is decoupled.Nevertheless the ionic and covalent channels have to be matched with different boundary conditions due to the tail of the coulomb potential that decreases slower than the centrifugal term.These equations are then given, for the large scattering coordinate, as where ρi = kiR and For the covalent channel this quantity has to be taken equal to zero.
The Riccati-Bessel functions were generated by using forward and backward recurrence relations 24 whereas the Coulomb functions were generated using Steed's algorithm in complex form 25 .Denoting the diagonal matrices that contain the Riccati-Bessel and Coulomb functions of the first kind and second kind respectively by J(R) and N(R) the K matrix can then be obtained by, Scattering matrices and cross sections are then obtained in the usual way 8 , i.e, an average over all partial waves are given by σi→j = π where Sij l is the scattering matrix for each partial wave and for the transition i to j.

The interaction potential
To solve the Hartree-Fock-Roothaan equation, initially one has to select a set of atomic basis functions for the molecular expansion.These functions are normally ex-pressed as a linear combination of cartesian Gaussians, which are easier to work in numerical calculations.
The basis sets selected for the [BHe] 3+ system are of dupla-zeta type and are described in Refs.26 and 27.They also contain p-type and d-type polarization functions for the He and B atoms respectively.They are represented by (11s,6p)/[5s,4p] for boron, with three d-type polarization function with coefficients 1.11, 0.402 and 0.145 26 and (6s,1p)/[4s,1p] for helium 27 .The electronic structure calculation was performed using the MELD program 28 and the potential energy curves were then calculated in the adiabatic approximation for the quasi-molecule [BHe] 3+ in the configuration interaction (CI) level.Single and double excitations relative to the Hartree-Fock configuration (B 3+ (1s 2 ) + He 0 (1s 2 )) were therefore generated.For more details see Ref. 16.
The radial and rotational couplings can be formally written as and where Ψn is a CI wave function, Ci n are the CI coefficients, Pαβ mn represent the element αβ of the transition density matrix between the Born-Oppenheimer states ψm and ψn in the molecular base {αβ}, ap α are the SCF coefficients and ϕ's are the atomic orbitals.These couplings were solved by the central field method (CFM) 16,29 where the dependence with the origin was eliminated.

Results and Discussion
The state-to-state charge transfer quantum cross sections were calculated by using a new method to solve the Schrödinger equation 21 where the radial and rotational couplings can be taken into account.In this case, all couplings were included except the one with unphysical behavior.
The state-to-state charge transfer quantum cross sections and the LZ results are shown in table 1 for several collision energies.Because in the LZ calculation only two states (2 1 Σ with 3 1 Σ) are taken into account, these cross sections are to be considered as the total cross section.For an analysis of these results, the quantum cross sections were split out for each allowed inelastic transition.The coupling between the 2 1 Σ and 1 1 Π states was neglected since in the electronic calculations this coupling generally does not have the correct asymptotic limit.This kind of procedure was also adopted recently by Boyd and co-workers 15 .Therefore, the coupling between these states are inferred from other couplings.As can be observed in Table 1, there is a weak coupling between the lowest (1 1 Σ) and highest (3 1 Σ) states used in our calculations and this is expected because such states have very weak radial coupling and no rotational coupling et al.The contribution due to the ionic and covalent channels, i.e., the 3 1 Ζ + state corresponding to the covalent and to the second ionic channel, were used in the LZ calculations.As can be seen in table 1 the LZ model describes correctly the general behaviour of the cross sec-Table 1. Cross sections in Å 2 .The LZ results are the total cross sections for the process under study.Subscripts 1,2 and 3 refer to the Σ states and subscript 4 refers to the Π state.

T/eV
σT LZ σ13 q σ23 q σ34 q 1.0 83.70 0.0  tion in this collision energy regime.For higher energies, corrections for this model have certainly to be taken into account.Moreover, it can also be observed from Table 1 that the charge transfer processes, for high energy collisions is not a two state problem.Figure 1 compares the total cross section out of the covalent state with the calculations of Gargaud and co-workers 22 .When the energy increases above 51 eV the agreement becomes poorer and the main reason for this is certainly due to the absence of rotational coupling which is important for higher collision energies.Such a deviation can also be attributed to the difference between the intermolecular potential of Gargaud el al. 22 and the ab-initio potential of Lópes-Castillo and Ornellas 16 .This can be analysed by checking the potential parameters used in the two calculations as shown in Table 2 and compared with other calculations.The present results are equivalent to those of Fraija and co-workers and the greatest disagreement might be due to the small ∆E of the outer crossing of the model potential 22 .In this case our ∆E is almost twice that of the Gargaud results.
The important region for a charge transfer process is near the avoided crossing; although Gargaud et al. was more concerned with the difference of energy in the asymptotic region.It is recognized that the CI calculation is more reliable at intermediary R than at the asymptotic limit.In the former case the potential model was parametrized for the asymptotic energies which have provided an error less than 0.0001%.Therefore one can expect that the potential used in the present work can be more reliable for describing the charge transfer process than that reported by Gargaud and collaborators 22 , although they have good agreement for the collision energy range used in this work.
If the results of Fraija and co-workers 30 , and those reported by López-Castillo and Ornellas 16 do not describe correctly the outer crossing, then the results of Shipsey et al. 31 also should not properly describe the system, since the three calculations employ the same method for the calculation.The agreement between Gargaud et al. calculations 22 with those of Shipsey and collaborators 31 , for the outer crossing, is likely to be incorrect.In addition, since the Shipsey et al. calculation is less accurate than those previouly reported 16,30 , it might be concluded that the results of Gargaud and co-workers are not quantitatively correct when compared with the calculations of Fraija et al.Moreover, Shipsey et al. have not used the translation factor and from their results 16 (Fig. 6) it can be observed that the use of the central field method (or the use of the translation factor) will increase the absolute value of the total cross section 16 .Therefore, a comparison between the results of Gargaud and those of Shipsey will certainly disagree if such a factor is taken into account.
The comparison of the total quantum cross section against the results of Gargaud and co-workers 22 in Fig. 1 shows that our results increase faster than those of Gargaud and co-workers.An extrapolation of these results for higher energies can show that the ab-initio cross section calculated in the present work will match the experimental results of Iwai and co-workers 20 .This can also be observed in the work of López-Castillo and Ornellas 16 where they used a colinear classical path method.Actually, a simple extrapolation of the semi-classical total cross section presented by López-Castillo and Ornellas 16 (Fig. 5) to lower energies indeed confirms this point.

Figure 1 .
Figure 1.Comparison of total cross sections between our results (boxes)and Gardaud and co-workers22 cross sections (triangles).

Table 2 .
Comparison of the potential parameters for some ab-initio calculations and the potential model where Rc is the crossing point and ∆E is the difference between the adiabatic potentials at the crossing.