Charge exchange in Li2+(1s) + H(ls) collisions. a molecular approach including two-electron translation factors

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Charge exchange in Li2+(1s) + H(ls) collisions. a molecular approach including two-electron translation

factors

L.F. Errea, L. Mendez, A. Riera, M. Yáñez, J. Hanssen, C. Harel, A. Salin

To cite this version:

L.F. Errea, L. Mendez, A. Riera, M. Yáñez, J. Hanssen, et al.. Charge exchange in Li2+(1s) + H(ls)

collisions. a molecular approach including two-electron translation factors. Journal de Physique, 1985,

46 (5), pp.719-726. �10.1051/jphys:01985004605071900�. �jpa-00210013�

Charge exchange ⁱⁿ Li2+(1s) ⁺ H(ls) collisions.

A molecular approach including two-electron translation factors

L. F. Errea, ^L. Mendez, ^A. Riera, ^{M. Yáñez}

Departamento ^de Quimica Fisica y Química Cuántica, Universidad Autónoma de Madrid, ²⁸⁰⁴⁹ Madrid, Spain

J. Hanssen, C. Harel and A. Salin

Laboratoire des Collisions Atomiques (*), Université de Bordeaux I, ³³⁴⁰⁵ Talence, ^France

(Reçu ^{le 16} juillet ^1984, révisé le 17 décembre 1984, accepté ^{le 4} janvier 1985)

Résumé. 2014 Nous calculons la section totale d’échange ^de charge ^{dans la collision} Li2+(1s) ⁺ H(1s) ^{en utilisant}

un développement moléculaire à huit termes incluant le facteur de translation commun à deux électrons. Aux basses vitesses (03BD 0,5 u.a.) la contribution la plus importante ^{à la section totale est due à des} transitions aux voisinages

du pseudo-croisement ^{1303A3 - 2} 303A3(R ⁹ u.a.). ^{Pour 03BD} plus grand que 0,5 ^{u.a. un mécanisme à trois états} 1 303A3- 3 3 03A3 - 1 3 03A0 devient dominant. L’accord avec les résultats expérimentaux ^est bon à basses vitesses (0,5 ^03BD 0,8 u.a.).

Abstract.

We calculate the cross section for the charge exchange process in Li2+(1s) ⁺ H(1s) collisions, using

an eight ^{term molecular} expansion that includes a two-electron common translation factor. The results are stable with respect ^{to the} variations of the parameters ^included in this translation factor. At low velocities (03BD ^0.5 a.u.)

the most important contribution to the total cross section is due to transitions in the neighbourhood ^{of the 1303A3-}

2 303A3 pseudo-crossing (for ^{R 9} a.u.). ^{For 03BD} larger ^{than 0.5 a.u. a three state 1303A3 - 3} 303A3 - 13 03A0 mechanism becomes dominant. The agreement ^with experimental ^{data is} good ^at low velocities (0.5 ^{03BD 0.8} a.u.).

Classification Physics ^Abstracts

34.70

1. Introduction.

Collisions of multiply charged lithium ions with

hydrogen ^{atoms have} recently ^{received a} great ^{deal of} attention because of the envisaged ^{use of lithium}

blankets in fusion reactors, and of fast neutral Li beams to heat the fusion plasma. However, except ^{for refe-}

rence [2] all theoretical data refer to completely stripped ^{Li3 +} ions. In the present work, ^{we treat}

the charge exchange reactions :

....

in the impact energy range 0.5-25 keV amu-’, ^{and we}

compare the results ^to the experimental ^data [3, 4] ^for

the total charge exchange process :

(*) Equipe de Recherche CNRS No 260.

To calculate the ^cross sections for reactions (1),

we use an impact parameter formalism, ^and expand

the wave function that represents ^the electronic state of the colliding system ^{in terms} of either the OEDM

or CI bases constructed in the calculations reported

in the preceding paper [1 ] (hereafter ^{referred to as} I). ^To

eliminate residual dynamical couplings ^{at infinite}

internuclear separation (see I) ^{and to} obtain results which are independent ^{of the} origin of the electronic coordinate chosen in the impact parameter equation,

we have introduced in the formalism the common

translation factor (CTF) [5] of Errea et alp [6]. ^{We shall}

first explain the characteristics of this two-electron

CTF, and its effect on the coupling matrix elements

presented ^{in I.}

In the following, ^{we shall use the same} symbols ^as

in I, ^{and atomic} units unless otherwise specified.

2. Characteristics of the common translation factor.

The common translation factor (CTF) [5] approach

substitutes the usual molecular expansion ^{of the wave}

function for the electronic state of the system by ^the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004605071900

720

expression :

where xn ^are approximate eigenfunctions of the elec- tronic Hamiltonian, ^of energies En. ^In equation (3)

one could equivalently ^use the basis functions t/J j

defined in I in terms of OEDM orbitals. In the latter

case the equations below should be modified as dis- cussed in [14]. ^{The main} advantages ^of (3) is that the introduction of the CTF does not destroy ^{the conver-}

gence properties of the usual molecular expansion,

and that it does not substantially increase the com-

putational effort. In this work, ^{we have used the CTF of}

references [6-8], slightly ^{modified :}

with

where v is the relative nuclear velocity, zi ⁼ rj8:Q ^is

defined with respect ^{to a common} origin ⁰ (called

« privileged origin » ^{in Ref} [6]), and P specifies ^the neighbourhood ^{of R = 0 where} f ^{is made} negligible.

When the origin of electronic coordinates 0’ of the

impact parameter equation ^{does not} coincide with 0, f ^{and u are} changed ^{into :}

where rj ⁼ rj

bR. Substituting (3) ^{in the} impact parameter equation yields ^the system ^of coupled

differential equations :

where,the coupling matrix elements are given by :

The real part of the matrix element (8) ^is proportional ^to v2, and should not [6, 7 ] ^be neglected. ^The imaginary part of (8) ^is proportional ^{to the} velocity, ^{and can be} expressed ^{in a more} familiar form :

where b is the impact parameter, ^and Wkj ^and T kj the modified radial and rotational couplings, respectively :

where Ly ^{is the component of the} electronic angular

momentum in the direction perpendicular ^{to the colli-}

sion plane. ^It is easy ^to prove that, ^{as R -} oo, ^one has (1) :

hence the ^« spurious » ^{behaviour of the} dynamical

(1) ^{The matrix elements} Wk j ^{in (10) would, in general,}

decrease as 1/R ^{as R tends to} infinity. ^{In the} present work, it can be checked, using simple symmetry arguments, that, for the states kept ^{in the} expansion, ^the coefficient of the terms decreasing ^as I IR ^{is zero for all} W kt

couplings ^{in the treatment without} translation factors is eliminated. Furthermore, ^as explained ^{in refe-}

rence [6], because of equation (6), ^the calculated ^cross sections are independent ^{of the} origin ^{of electronic}

coordinate chosen in the impact parameter equation.

On the other hand, the results will depend, ⁱⁿ general,

on the functional form chosen for the CTF. In the present case, these functional variations take the form of dependencies ^{on the} explicit parameter fl ^of (5)

and in the so-called « privileged origin » (see ^text following Eq. (5)). For P ^{= 0} ( f ⁼ Z/R), ^{results are} independent ^{of the} «privileged origins. However,

in the latter case, the transition amplitudes ^{for small}

Fig. ^1.

Schematic correlation diagram ^{for LiH2 + .}

internuclear distances are unreliable because of the

large anisotropies which show _up in the diagonal

terms of the couplings [8]. ^{Hence, if the} region ^{R - 0}

is important ^{for the collision} (which ^{is the case here}

because of the 2 3E - 1 3n and 3 IE - I1n coupl- ings) ^{one should choose} a non-zero value for fl. ^This

has been confirmed recently by ^the optimization of fl through ^the determination of the Euclidian norm [15].

A non-zero P, ^{on the other} hand, ^{creates a cut-off in the}

CTF for R --+ 0 which makes us encounter traces of the usual problems ^{of the p.s.s. method. In} particular,

the dependence ^{of the} couplings ^on the electronic coordinates now appear, fortunately ^{to much less}

extent, ^{as a} dependence ^{on the} privileged origin.

In _sum, for the present system, ^{the value} of P ^{should be}

chosen as a compromise ^{between two} requirements : (i) P =F ⁰

(ii) results should not depend ^{much on the «} privi- leged origin » .

We have found that, ^{for 1} p ^{3, results} vary by

less than 1 %. ^{This can} be related to a much smaller variation of the couplings ^{with the} «privileged origin » (Figs. ^{2 and} 3) than in the case of the calcula- tions without CTF (see Fig. ^{2 of I).}

For a value 1 p ^{3 we} present ⁱⁿ figure ^{2 the}

modified radial coupling Wkj between the states

1 1,3 E, ^{21,3 E and 3 1,3E of the LiH 2+} quasimolecule

and in figure ³ the modified rotational couplings Tkj between these states and 1 1,3n.

For internuclear distances R > 2 _a.u., the matrix elements W kj ^{are very close to} the radial couplings (Figs. ^{2-4 of} I) ^whose physical origin ^{was discussed in}

detail in I, except for those that were strongly origin dependent ^{such as the constant residual} couplings

which are eliminated according ^to equation (12). ^In particular, the Stark and Lorentzian avoided crossing

contributions are unaffected by ^{the inclusion of the}

translation factor in the molecular basis. Analogously,

Fig. ^2.

Modified radial couplings Wj (Eq. (10)) ^between

the molecular states of the LiH2+ quasimolecule. a) Singlet subsystem. b) Triplet subsystem. Privileged origin

on Nuclear Centre of Charges ; - - - - Privileged origin ^on

the H nucleus.

722

Fig. ^3.

Moditied rotational couplings Tti (Eq. (11))

between the molecular states of the LiH2+ quasimolecule.

a) Singlet subsystem. b) Triplet subsystem.

^Privi- leged origin ^on the Nuclear Centre of Charge ; - - - - ^Privi- leged origin ^{on the H nucleus.}

the new matrix elements T kj ^{are close to} the rotational

couplings ^{described in I} (Figs. 5, 6) except ^{for the} cancellation of contributions in these couplings ^that

increased linearly ^{with R. The} limiting ^behaviour

lim 3 11; liLy 11 In > ^{= lim} 2 31; liLy 11 3 n > ⁼¹

R--m R-oo

is not affected by the CTF since it is due to the dege-

neracy of the corresponding ^{3 11; - 1} l II, 2 3 E - 13Il molecular functions in this limit

It _may be worth mentioning ^{that the} only reported

calculations we are aware of involving two-electron translation factors (common ^or not) ^{are those of}

Errea et al. [8] and of Kimura et al. [9]. However, ^{in the} latter case, the V-n expansion ^{of the} couplings ^is

truncated after the first term. This is equivalent ^to neglecting ^the origin dependent part ^{of the} dynamical couplings (for ^a discussion of this procedure ^see [10]).

3. Results and discussion.

In the absence of effective spin-orbit forces, ^the pro-

cesses for the singlet ^and triplet subsystems, ^corres- ponding ^{to the 2 1 I and 3 3 I entrance} channels, respectively, ^can be studied independently. ^The system of coupled differential equations (7) is then solved

separately ^{for the} singlet ^and triplet ^{states and the} charge exchange probabilities ^are given by (see Fig. ¹

of I) :

When transitions to all other molecular channels can be neglected, ^{the total} charge exchange probability

is given by :

and because of these statistical weights, the processes

occurring ^{for the} triplet subsystem ^{will dominate the}

outcome of (1).

When using ^{a molecular} expansion without transla-

tion factors (setting ^{U = 0 in} (3))

^{such as the}

OEDM or CI approaches ^{of I}

^the probabilities (13)

oscillate indefinitely ^as functions of time, ^{unless the}

origin of electronic coordinates is placed ^{on the Li}

nucleus, because of the residual couplings ^{that exist}

between the I ’Z - 3 1 E’and 1 3E - 2 3Z molecular states. Then, ^{one can} only ^evaluate P(lS) ⁺ P(iP) ^and p(3S) ⁺ P(3P), ^{which do not oscillate.}

The numerical solution of the coupled equations corresponding ^{to the OEDM, CI and CI + CTF bases}

was carried out using the program PAMPA [1 ], conveniently ^{modified to} include the new coupling

terms. The charge exchange ^{cross sections are then}

obtained by integrating ^the corresponding ^transition probabilities ^over all nuclear trajectories. ^{For the}

CI + CTF basis set, ^we present ⁱⁿ figure ^{4 the values}

of our calculated total charge exchange ^{cross section,} together ^{with the} experimental results of Seim et al. [3]

and Shah et al. [4]; ^we include the reported ^uncer-

tainties of these experimental data. We also present in figure ⁴ the results of using the OEDM and CI molecular bases, without translation factors; ^{these two}

basis sets yield practically ^{identical results, and we}

have performed calculations using ^two choices for the origin of electronic coordinates : the centre of nuclear charge ^{and the} position ^{of the} proton.

Since the coupling matrix elements (8) ^{are unsensi-}

tive to changes ^{in the} parameters ^{of the} CTF, ^{it is not} surprising ^{that the cross} sections calculated with the CI + CTF basis set turn out to be stable with respect

to those changes; ⁱⁿ particular they ^are independent ^of

the so-called [6] privileged origin ^{for 1} j8 ^{3; this}

should be compared ^{with the} origin dependence ^{of the}

results for the molecular bases without translation factors (Fig. 4). ^{We also} present ^{in table} I, for illustra- tive _purpose, the values of the partial ^{cross sections} O’(1,3S), O’(1,3p) and the total cross section _0’, for the molecular basis with translation factors.

Using the molecular basis with the CTF, ^{we see}

from figure 4, ^{that our} results for the cross section of reaction (2) fall within the estimated error of the _expe- rimental data of Seim et al. [3], ^whereas they ^lie higher

than those of Shah et al. [4] ^{for v} > 0.8 a.u. (E ^>

16 keV amu - 1 ). ^{We have} explicitly ^{checked that,}

for the whole range of nuclear velocities considered in the present work, ^the first excited 2 1,3 n states do not

give ^an appreciable contribution to the charge exchange ^cross section of (2)

^{in contrast with the} findings ^{for the} He+(ls) ⁺ H(ls)’collision process [12],

because the 1 n - 2 17 energy curves are here much

more separated. Then, the difference between our

results and those of reference [4] ^{in the} higher energy range considered may be due. to small contributions to the charge exchange ^process (2) ^{from a} series of other excited states, ^{as well as the onset} of the ionization process :

whose cross section, ^{for v = 1.3 a.u.} (E ^{= 42 keV} amu-’), ^coincides [12] ^with that for the charge exchange process (2). Obviously, ^{none of our mole-}

Fig. ^4.

^Total charge ^{transfer cross section for reaction} (2) : 0 - Experimental results of reference [43] ; ^{A -} Experi-

mental results of reference [4] ; ^{Our results} using translation factors ; - - - - ^Our results without translation factors with origin ^of electronic coordinates at the Nuclear Centre of Charge;

Our results without translation factors with the origin of electronic coordinates at the H

nucleus; ^{- x - Partial} charge exchange ^{cross secticn} corresponding ^{to the} triplet subsystem using translation

factors; ^-o- Partial charge exchange ^{cross section cor-} responding ^{to the} singlet subsystem using translation factors.

Table 1.

Charge transfer ^{cross section} for ^the

molecular basis with translation factors (cm2.10-16).

0’(1,3S) corresponds ^{to the} first ^reaction (1); 0’(1,3p) corresponds ^{to the second reaction} (1); Qtoca, corresponds

to reaction (2).

cular bases allows for transitions to the ionization continuum.

We shall now briefly study ^the mechanisms involved in the _processes (1). ^The sharp ^avoided crossings

between the molecular energy ^curves which _appear

for R > 15 a.u. (see Fig. 1) ^{are traversed} diabatically;

724

Fig. ^5.

^{Plot of b. P vs.} impact parameter b for transition

probabilities ^{defmed in} equation (13) ^{of text}

hence, for R = 15 _a.u., only ^{the I II and 2 3I states}

are populated ^{in the} ingoing part of the nuclear tra-

jectory. ^{As mentioned} above, ^{we can} consider sepa-

rately ^the mechanisms corresponding ^{to the} triplet

and singlet subsystems.

For the triplet subsystem, ^{the most} important

mechanism occurs through ^the I 3 I - 2 3 I modified radial coupling, especially ^{in the} neighbourhood ^of

its peak (at ^{R = 9} a.u.) where the corresponding

energy ^curves have a narrow pseudo-crossing. ^For

v 0.5 _a.u., the charge exchange process takes place mainly through partial transitions in the pseudo- crossing region

unlike the case for the singlet subsystem, since the pseudo-crossing for the latter

subsystem ^{occurs at R = 20 a.u.} and is therefore crossed diabatically. ^Hence r(?S) Q(3S ⁺ ’P) ^>

a(IS ⁺ ’P) (see Fig. ^{1 and Table} I). ^{For v = 0.2 a.u.}

(v ^{= 4.375 x 10’} cm.s-1), ^we illustrate this behaviour

by plotting ^{b.P vs. the} impact parameter ^{b for the}

transition probabilities defined in equation (13), ⁱⁿ

figures ^{4a, b. We have} further checked this interpreta-

tion of the low energy mechanism by performing

the calculations with the 1 3E and 2 3 L states alone, which yields ^cross sections of the same order as those

presented in table I.

As v > 0.5 increases, ^{the I 3Z - 2 3E transition}

becomes more diabatic and therefore contributes less to the cross section of (1 ) ; simultaneously, ^{the 13E -} 33 _F (and ^{to a lesser extent the} 2 3 L - .3 3E) ^radial couplings begin ^to be effective. When the 3 3L mole- cular state is populated through ^these couplings, ^the

rotational 3 3 L - 1 3 n matrix element (11) ^almost entirely transfers this population ^{to the 1 3 n state.}

The effectiveness of this rotational coupling, ^which

for R > 6 a.u. is practically equal ^to bvR-2, ^{is due to}

the fact that for this _range of internuclear distances the

corresponding ^{3 3 Land 1 3 n} energy curves are prac-

tically degenerate (see 1) ^{and to the fact} that, ^unlike

radial couplings, rotational matrix elements have the

same sign ^{for ± t.} The total effect of these couplings ^is that, ^{as v} > 0.5 a.u. increases, ^{the cross section} U(3p)

becomes comparable ^to Q(3S). ^We illustrate the pre- vious discussion by plotting ⁱⁿ figures ^{5c, d, bP(b)}

as a function of the impact parameter b, ^{for a nuclear} velocity v ^{= 0.8 a.u.} (v ^{= 1.75 x 108 cm.} S-l). ^For

v ⁼ 0.5 a.u. and b ⁼ 5 _a.u., we show in figure ^{6 the}

« history » of the collision by drawing the values of

I C13 12, 1 C2 3_1 12, (1 ^{C3 3Z} 12 + 1 C13n 12 ) ^{as functions of}

time; notice that even at this high velocity ^{the beha-}

viour of the quasimolecule ^{at the} pseudocrossing ^is

not completely ^diabatic.

For the singlet subsystem, ^as explained above, ^the 1 lE - 2 ’Z pseudo-crossing ^occurs for R = 20 a.u.

(see I) ^{and is so} sharp that it is crossed diabatically

for the impact energy range considered here. Apart

from this fact, which lowers the charge exchange ^cross

section considerably below that for the triplet ^sub- system ^{at low} velocities, ^{the mechanism is the same as}

for the latter and will not be repeated ^here (see ^illus-

tration in Figs. 5a, c).

4. Conclusion.

Finally, it may be worth mentioning ^{that our results}

constitute a good illustration of the applicability ^of

the common translation factor to a two-electron _pro-

Fig. ^6.

^Plot of PiZ) = 1 Cj(Z) 12 ^{(see Eq. (7)) as function}

blem. It is significant ^{that the cross} section obtained

using ^Common Translation Factors does not coincide with those of the standard molecular approach,

without translation factors, ^{when the} origin ^{of elec-}

tronic coordinates is placed ^{either on} nuclei, ^{or on the}

centre of nuclear charge, ^or of nuclear mass, and for the whole _range of impact energies ^considered

Our results are in _very good agreement ^{with the} first group of experimental ^{data at} low velocities. So,

the most important mechanisms which explain ^the charge exchange process ^are those introduced in ^our

analysis.

At larger velocities, ^our results exceed the experi-

mental _ones, because in ^our calculations we neglect

channels which _compete with charge exchange ^{such as}

ionization or, perhaps, excitation of the hydrogen

target.

726

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