Indian J. Phys. 65B (3 ), 217-225 (1991)
Ab-initio molecular treatment o f single electron capture process for the 0 ‘ + + He collision
K Amezian and M C Bacchus-Montabonel
Laboratoire de Spectrom<5trie lonique et Molt^culaire (associe au C.N.R.S. n*171) University LYON L Bat. 205,43 Bd du 11 Nov. 1918
69622 Villeurbanne Cedex, Franco
Received 15 May 1990, accepted 16 January 1991
A b stra ct: An ab-initio molecular expansion method with a semi-classical collisional treatment have been developed for the interpretation of single electron capture process in the O^'^+He collision. Radial and rotational coupling matrix elements calculated from ab-initio Cl wavefunctions are presented. Calculated partial cross sections for the electronic capture on the n — 3 levels of the 0 “+(3') + He^ system are seen to be in fairly good agreement both with experiment and previous theoretical results in the 20-100 keV energy range.
Keyw ords : Ab-initio molecular expansion method, single electron capture process, O " ' + H e collision.
PA CS N o : 34.70. I e
I . Introduction
Recently, we have developed a full ab-initio treatment for the interpretation of the single electron capture process in the collision of the multiply charged ion on a He target (Bacchus-Montabonel 1987, 1989). This procedure involves a configuration interaction calculation of the adiabatic potential energies and of the corresponding wavefunctions by means of the CIPSI (Configuration Interaction by perturbation of a Multiconfiguration wavefnnction Selected Iteratively algorithm (Huron et al 1973) followed by a semi-classical treatment of the collisional problem (Aubert and LeSech 1976). The partial cross sections of single electron capture on the n = 3 levels of ^(31)-!'He+, which has been shown experimentally to be the main process in the f He collision (Cotte et al 1985), were fairly well reproduced by our calculations. So we have performed the same theoretical treatment for the single electron capture process in the collision of the isoelectronic ion 0^'^ on a He target which has been shown experimentally to present a different behaviour than the N®^ + He system in the 10-100 keV energy range.
The 0®"*^ + He collisional system has been extensively studied in the past few years. From an experimental point of view. It has been Investigated by a wide variety of techniques : V U V spectroscopy (Gordeev et al 1983, Dijkkamp et at
1 217
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K Atnezlan and M C Bacchus-Montabonel
1985, Liu et al 1989), time of flight techniques (Okuno et al 1983, Roncin et al 1989), electron spectroscopy (Bordenave-Montesquieu et al 1988). Recent theoretical calculations are also available which use an atomic orbital expansion method (Fritsch and Lin 1986) as well as a molecular orbital expansion (Shimakura et al 1987). They reproduce the experimental measurements w ith a good accuracy.
2. Calculations and results
We present in this paper, calculated values of the partial cross sections regarding the collision reaction 0 ® '^(1s®) + He (Is ® )-> 0**(1s®, 3l) + He+(1s). The adiabatic potential energy curves have been displayed in a previous paper (Amezian and Bacchus-Montabonel 1989) for the entry channel ^£+(0"+(1s*)-|-He (1s®)5 and for all the and ^17 states dissociating to the (0*+(1s*, 31) f He^(1s)|
configuration.
The resolution of the collisional coupled equations involves the evaluation of the non-adiabatic couplings between molecular states. The radial coupling matrix elements = < i/'r
I
41
fAii > b®tween all the pairs of states of the same' dR 1
symmetry have been calculated by means of the finite difference technique (Cimiraglia et al 1982)
gKt(R) = lim ^ 1 y
with d = 0.0012 a.u.
The origin of the electronic coordinates has been taken on the O nucleus in order to eliminate the non vanishing coupling terms at long range.
A three-point numerical differentiation with calculations at R + d and R - A shows a precision of about lO ” ** a.u. for the radial coupling terms except for very sharp peaks where It Is lowered to 5.10“ at the top of tha peak.
The diagonal gjrjr(R) matrix elements are zero up to the fifth figure according to the antisymmetry of the d/dR operator.
The values of the g^^fR) matrix elements between the states are displayed in Table 1 and Figures 1('a)-(c), Calculated values for the radial couplings between the states are very weak, so they have been neglected in the present investigation.
Our values of the radial couplings for the states are seen to be rather different from those of Shimakura et fli (1987) around 4.5 a.u. In this region, the calculations of Shimakura et ol show a peak (0.4-0.8 a.u. high) for the coupling matrix elements between the entry channel (6 S ) and the states of the S0*^(1s®, 3l)-FHe+(1s)l configuration (3S, 4S, 5S) while our values of the corres
ponding radial coupling matrix elements ( g , « , g*^, g«e ) are a bit lower (0.25- 0.45 a.u.). A s shown by Shimakura, peaks of the radial coupling matrix elements
Ab-fnitio molecular treatment etc
are observable in the 2-3 a.u. internuclear distance range. In order to interpret
2 1 9
these peaks we have determined quasi-diabatic potential energy curves by means of a method recently proposed by Cimiraglia et al (1985). The details of the calculation have been given in a previous paper (Bacchus-Montabonel and Amezian
Ta b le t. Radial coupling matrix elements b e t w e e n S t a t e s (in a.u.) (Labels are defined in Figure 2).
R (a.u.) Ss 4 5 A Ear Sa (1
1.5 0.549 0.137 0.290 0.208 0.145 0.265
1.9 0.133 0.502 0.527 0.456 0.200 1.482
2.0 0.368 4.283 1.731 0.768 1.801 1.771
2.1 12.224 1.366 4.115
2.2 0.830 14.800 2.774 4.425 1.431
2.3 0.959 7.570 2.021 0.053 1.101 1.349
2.5 0.511 1.073 1.531 0.031 1.960 1.458
2.8 1.420 0.95'j 0.375 2.165 5.289
3.0 1.884 0.227 0.120 0.715 0.104 0.363
3.5 0.055 0.064 0.126 0.064 0.108 0.054
4.0 0.332 0.299 0.086 0.118 0.016 0.369
4.25 0.106 0.158 0.002
4.5 0.634 0.062 0.252 0.175 0.349 0.251
4.75 0 213 0.450 0.256
5.0 0.365 0.032 0.195 0.185 0.396 0.167
5.5 0.186 0.016 0.097 0.164 0.190 0,057
6.0 0.120 0.003 0 037 0.140 0.072 0.027
10.0 0.043 0.013 0.0 0.043 0.001 0.0
1990) but for seek of clarity the curves are displayed in Figure 2. From these quasi-diabatic potential energy curves, the peaks presented by the radial coupling matrix elements seem to correspond (i) to a crossing between the states dissociating to |0'‘+ (3 s)-m e + (1 s)} (3 ) and to }0*^(3p)-FH e+ (I s )} (4 ) at about 3.0 a.u., (ii) to a crossing between the entry channel and both the (3 ) and (4) states at about 2.2 a.u. It is worth noting that the ggg radial coupling matrix element between the ’^i:+,'0®'^(3s) + He+(1s)! and the entry channel for R lower than 4 a.u. is particularly sharp (14.80 a.u. high and about 0.30 a.u. wide at half height) corresponding probably to some 4s character (Shimakura et al 1987) for the (3) state at short internuclear distances.
The rotational coupling matrix elements <
I
| y between the ^27+ and states have been evaluated analytically from the Cl wavefunctions by means of the L+ and L_ operators. They are presented In Table 2 and Figure 3(a) and (b) and are in an overall good accordance w ith those of Shimakura et of (1987). They are very weak for large internuclear distances except for and states correlated to the same configuration.220 K Amezlan and A4 C Bacchus-Montabonel
E (a.u.)
Ab-lnitio molecular treatment etc
22 1F ig u re 2. Quasi-diabatic potential energy curves for the states of OHe®*^
3 refers to the state dissociating to (0 '‘*(3s) + He^(1s)}
4 refers to the state dissociating to { 0 “+(3p) + H e*^(1 s)>
5 refers to the state dissociating to { 0 ‘‘^(3cf) + He^(1s)}
6 refers to the state dissociating to He(ls^)}
A detail of the curves in the region of pseudo-crossings is given in the right top of the figure.
T a b l e ! . Rotational coupling matrix elements between (labelled 3, 4, 5, 6) and ' n (labelled 7 , 8) states of OHe'’^ (in a.u.)
(Labels 3, 4 , 5, 6 are defined in Figure 2).
(Labels 7 and 8 refer respectively to the states dissociating to vO^^(3p) + He^}
and l He^).
R (a.u.) ^ 7 8 ?7 4 ^ 7 a Stt» S.4 Sv a
2 .0 0.632 0.169 0.296 1.278 0.540 1.111 0.143 0.263
3.0 0.770 0.454 0.105 - 0.097 0.437 1.306 0.247
3.5 0.941 0.585 0.217 0.777 0.204 0.373 1.360 0.349
4.75 0.590 0.992 0.204 0.465 0.093 0.401 1.462 0.330
5.5 0.365 1 . 1 1 0 0.257 0.107 0.035 0.365 1.554 0.089
7.0 0 .2 0 1 1.075 0.199 0.003 0.017 0.294 1.635 0.005
8.5 0.130 1.043 0.144 0.006 0.0 1 1 0.234 1.678 0 .00 1
9.5 0 . 1 0 2 1.030 0.118 0.006 0.009 0.198 1.695 0.001
1 1 .0 0.074 1.018 0.089 0.004 0.007 0.155 1.710 0.00 1
13.0 0.052 1.009 0.065 0.001 0.004 0.114 1.720 0 .0
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K Amezlan and
A1 CBacchus-Montabonel
The ab-initio potential energy curves and the radial and rotational coupling matrix elements have been involved in a six-channel (4 channels and 2 channels ^JT) collisional treatment of the single electron capture process within a semi-classical formalism (Aubert and LeSech 1976). The state has not been
Figures 3 (a ) & ( b ) . Rotational co up lin g matrix elements of the O H e"* system (in a .u .).
considered in the calculation. The strong g^s radial coupling has been fitted by a Lorenzian shape function while the other couplings and the potentials which show less sharp variations have been fitted by spline cubic functions.
The values of the g » i(R ) radial coupling terms are seen to decrease as R increases from 5 a.u. Their fitted values are thus almost zero (lower than 10"® a.u.) at R = 1 5 .0 a.u. which is the upper limit used during the integration of the collisional coupled equations.
The partial cross sections of single electron capture are presented in Table 3 and Figure 4 and compared to the experimental results of Dijkkamp et al (1985). They are in an overall good agreement with experiment especially for the
Ab-initio molecular treatment etc 2 2 3
partial cross sections on the 3p and 3d levels. Our calculated values of the partial cross section on the 3s level are too high for collision energies lower thanT a b le 3. Partial cross sections of single electron capture on the n - 3 levels.
Energy (k e V )
Calculations ( l O " ^ ’ cm *)
Experiment ( l O - ' " c m »)
9 2.4 0.46
1.7 4.5
3 .9 3.0
18 2.1 0.8
4 .3 6.0
3.1 2.8
48 1.0 1.8
7 .0 6.4
2.5 2.7
72 2.6 2.3
5.1 5.0
2.7 3.2
t1 4 2.6 2.7
CTsp 4.2 3.9
3.7 3.6
20 keV (see in Figure 4 the two crosses corresponding to a collisional energy of about 1 keV/a.m.u.). They are about the same order of magnitude as the cross
F ig u re 4. Partial cross sections of s in g le electron capture on the n - 3 levels for the 0 " * 4 He system (in 1 0 '”*'' cm ).
---our c a lc u la tio n ,... D ijkkam p et al (1 9 8 5 ).
2 2 4 K
Amezian and M C Bacchus-Montabonel
section on the n = 4 levels measured by Dijkkamp c t al ( 1 9 8 5 ) and might bo due to
a 4s c h a r a c t e r fo r t h a s t a t e dissociating to {0*+(3s) + He+}. This hypothesis may be supported by the sharp radial coupling matrix element which could be c o m p a r e d t o t h e 6 s - 7 s c o u p l i n g p r e s e n t e d by Shimakura e t al ( 1 9 8 7 ) .
Nevertheless, the value of the cross section on t h e n = 4 levels is still controverted (Roncin et al 1 9 8 9 ) and further calculations should be performed.
Our results are also in an overall good agreement w ith tha theoretical curves of Fritsch and Lin (1986) and of Shimakura et al (1987). We obtain a slightly better agreement w ith experiment for the partial cross section on the 3d level than that obtained by Shimakura et al. This could be due to the introduction of all the electrons in the molecular calculation instead of taking a pseudo-potential on the O nucleus.
The agreement between our results and previous theoretical and experimental ones enables us to assume that inclusion of electron translation factors which has not been done in the present work, should not affect much the numerical values of the partial cross sections on the n -- 3 levels (Gargaud et al 1981).
The values of the partial cross sections of single electron capture on the ‘ 27+
and ‘ /7 states are presented in Table 4. This is a measure of the relative importance T a b le 4. Values of the single electron capture partial cross sections for the and ^11 states (in 10 c m ”') -
3s
Energy (k e V ) 3p 3d
‘ 11 ‘ n
9 2.4 1.2 0 .4 3.6 0.4
18 2.1 2.3 2 .0 1.1 2 .0
48 1.0 3 .8 3.2 1.7 0.8
72 2.6 1.7 3.4 1.2 1.5
114 2.6 2.9 1.3 2.7 1 .0
of the radial and rotational couplings. From our results, both of them have to be taken into account to have a fair evaluation of the cross sections.
3. Conclusion
These results show that our complete ab-initio treatment is suitable for electron capture processes. Such a procedure could be extended to reactions involving metastable ions.
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