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A new diabatic molecular representation for triplet-triplet transitions in He+-He collisions
Volume 96, number 1
2.5 March 1983
CHEMICAL PHYSICS LETTERS
A NEW DIABATIC MOLECULAR REPRESENTATION FOR TRIPLET-TRIPLET He+-He*
TRANSITIONS IN
COLLISIONS
TX. RAI DASTIDAR + Department of Chemkt~.
University of Southern California. Los Angeles. Califomia 90089. USA
K. RAI DASTIDAR Department of Physics, University of Southem California. Los Angeles, Gzlifomia 90089, USA and
R. SEN GUPTA Department of GeneraI Physics and X-Rays, Indinn Association for the Culttiation of Science. Calcutta 700032, India Received 23 August 1982;in
final form 13 January
1983
A “frozen-orbital” diabatic basis has been constructed for an impact parameter treatment of collisions of He+ \\ith metastable He(2 3S) at 1000 eV laboratory ion ener,q_ Lcept for the two highest states used, the diabatic states correlate very well with the separated-atom energies. and the P-11 rotational couplings deviate little from the proper asymptotic behaviour (=R”). Cross sections and transition probabilities are presented for some elastic and inelastic channels.
l_ Introduction
2_ Theory
In a recent publication [l] we demonstrated the use of an approximate diabatic molecular basis in resonant charge-transfer collisions of He+ with He. In the present paper we describe the use of another diabatic basis in a close-coupled treatment of collisions of He+ ions with metastable (2 3s) helium atoms, studying the elastic and the inelastic channels 2 3S + 2 3S, 2 3S -+ 2 3P and 2 3S + 3 3S together with their charge-exchange counterparts. As in ref. [l], it is found that the present diabatic basis is reasonably free from the two well-known defects of molecular orbital expansions, namely (i) failure to correlate to the proper separated-atom energies, and (ii) asymptotically incorrect (MZ-l) variation of the rotational coupling between Z-II molecular states. Atomic units are used throughout except where , otherwise stated.
and the corresponding charge transfer channels, we expand the total wavefunction \kas in ref. [l] in a truncated molecular basis set:
* Permanent address: Department of General Physics Br X-
idc/dr=(H+Q)c;
Rays, Indian Association for the Cultivation of Science, Calcutta 700032. India.
0 009-2614/83/0000-0000/S
03.00 0 1983 North-Holland
For the collision process He+(ls) + He*(ls2s,
2 %5)--t He+ + He*(ls2s, + He+ + He*(lsZp, +
w,
2 3S), 2 ‘P)
He* + He*(lsZs, 3 3S)
Jw)) = $Jq(r) Gi/I(T,q ,
(3)
(2)
where the 4 are electronic basis functions in a molecule-fured reference frame. The coupled equations to be solved for the problem have been outlined earlier [I] and here we quote only the final equation: (3)
where c is the coefficient vector defined in (2) and H 8.5
CHEMICAL
\‘olume 96. number 1
25 March 1983
PHYSICS LETTERS
l-able 1 ~~O~CCUIX
orbit&
I[
used for constructing the configration
state functions 4 as described in the text Exponents
and Q dre squire
opt-mized at
R 2 20.0
Sls' 2.00
Qs = 0.564
Sls’ 2.00 Ils = 2.00
T2po = 0.548 r2po = 0.533
Sls = 2.00
52P,
= 0.597
51,=
c3s=
0.83
2.00
matrices: (7)
IIk_r = (Gk lH,tI~,~ . Qx,=(uh~R~)~~,I--i
aiaoi$,),
(4)
the significance
of a11 quantities being the same as in ref. 111. In this clculdtion we have restricted ourselves to SIX“X staIcs and rlvo jii states in the summation (2). The basis \ector Jr ws built up from configuration SIate functions 4 hdvinp the general form
@= Ilo,
lo,, %I + Ilo,
= (lo,,
Ij i$
lo, Xl1 .
+ IQ lo, r;;,r
= u s(Kjl#I(r.
(5)
h‘) .
Icl(-=)l Ck(-O”)
= IQ-=)I =
0
solved sub-
= 2-111,
for X-f 1 or 2.
Unitarity was conserved to within (see refs. 12, 31 for computational
2-3 parts in 1000 details)_
3. Results and discussion
Table 1 gives the hl0 configurations of the ei$t CSFs used in this work. As in ref. [ 11, the MOs were chosen ds linrdr combinations of minima! Slater-type atomic orbirals: the orbrtals were variationally adjusted to minimize the configuration ener_gies at large internuclear sepArarion H and thereafter “frozen”. Finally the dlabatrc basis \LKIS constructed in the fonn \tV.R,
The coupled equations (3) were therefore ject to the boundary condition
(6)
\rhere S(K) is the Schmidt ortho~iormalizatioIi matrix and U = U(R > 20.0) is the “frozen” unitary lnatrh Jiagxmlizing S(R)+at large R. An elementary anal) sis yields the separated-atom limits of the MO conl$ratrons; thus states 1 and 2 can be shown to dissociate to the limit
The basis set was constructed and the l-i and Q were set up in a Burroughs 6700 computer with 48-bit-single-precision arithmetic; these matrices were input in an IBM 370/168 computer with 64-bit double precision to solve the coupled equations (3) The CSFs, 4 are eigenfunctions of Sz and S’ with eigenvalues S, = l/2 and S = 312. The motivation behind the s&p of freezing the non-linear expounded as well as the linear configuration coefficients (Le. the U-matrix elements) has already been expounded in earlier papers [ 1,2] ; this makes the radial momentum coupling negligible and hence the basis approaches Smith’s [4] criterion for diabaticity. Table 2 shows the separated-atom limits and the asymptotic energies of the configurations (HII) after diagonaiization at large R. Except for the highest-lying states 7 and 8 which require much more elaborate configuration matrices
25 hlarch 1983
CHEMICAL PHYSICS LETTERS
Volume 96, number 1
Table 2 Diagonal elements of the H matrix The experimental energy values are from ref. 151 I
Separated-atom confmration of 01
HllCR 2 20.0)
Wpt
1,2 3-4
He+@) + He(ls2s)3S He*&) + He(ls2po)sP
-4.1761 -4.1321
-4.1753
5.6 798
He+(k) + He(ls2p+)3P
-4.1326
He+(ls) i- He(ls3s13S
-3.4805
interaction, the correlation to the experimental energies [S] is seen to be quite satisfactory. Fig. 1 shows the variation of the diagonal terms of the diabatic H matrices with R. We feel that because of the “frozen” nature of our orbitals as well as the neglect of hi-her configurations, no particular significance can be attached to any comparison between our diabatic HII values and any adiabatic calculations II
I
-4.1332
A.0687
at intermediate and small internuclear separations. State 4 contains the MO 4fo,, and as in ref. [l] the behaviour of N44 is peculiar (see ref. 161 in this context). The angular momentum couplings between the pairs (l--5), (3-9, (7-5) and (Z-6), (4-6), (8-6) of states are shown in fig. 2. Their long-range behaviour almost duplicates that of the rotational coupling elements in ref. [l] _ Of course, only the terms (7-5) and (8-6) go to the proper limit, whereas the two other pairs lie somewhat above and below the correct lmrit; but their R variation is quite close to the proper asymptotic behaviour (R-‘), unlike many other molecular basis expansions which show a spurious R-l variation at long range [7]_ From the solutions of the coupled equations we have computed the different channel transition probabilities and the cross sections. Table 3 gives the cross sections at a laboratory ion energy of 1 keV (i-e. 500 eV in cm. system) for the channels given in (1) (except elastic scattering) together with their charge-
0 I
20
I
40
5
60
*
80
I
106
t
200
3
Fig. 1. Diagonal potential matrix elements for the eight states. Full lines, x states; broken lines, II states
0i
Fis 2. X-n rotational coup&
terms.
87
CHEMICAL
Volume 96, number 1 Table 3 Cross sections (X2> for the various scatwing 1000 eV laboralor)‘ ion energy ---_ - --_ Channel
Cross section
Z%-
:!3P
13s....
231’
54.@8 6.53
(chnrzx c\changecI 735 ;&.j3s
3.71 181‘ ==5 0 191.
6.48 3.59
33s
[clwgc
channels at
Other theoretical results (see text)
--_,‘S2% (charge c+.cb.mge)
PHYSICS LETTERS
3.43 _I
e\cJ~ang) ---_-
counterp,wts. To our knowledge there are no parallel calculations in the literature 10 compare our results with; but we give in the table the results for 1?%G- 2 3P excitation in neutral He--He*{2 3S) collision given by Evans et a1. [S] and by Lenamon et al. [91_ (The latter group obtained their results at this exchange
25 March 1983
energy by extrapolating lower-energy calculations by a procedure due to Russek [IO].) It should be noted that Evans et al. did not consider the X-Il coupling, but Lenamon et al. did_ Fig_ 3 shows a sample of ihe 2 3S + 2 3S elastic and charge-transfer channel probabilities in a region where their behaviour can be seen without getting lost in a labyrinth of detail, pIotted against the orbital anguIar momentum I which is related to the impact parameter b (semiclassical equivalence) as b = (I f $)fk, k = p_@z. It wiIl be seen that in general for large impact parameters the direct and the charge exchange probabilities are out of phase with each other, but towards low b this situation no longer holds. This is of course to be expected on account of the many curve crossings as well as rotational couplings_ Unlike ref. [ l] we have not assigned here any particular potential to the classical nuclear trajectory and hence do not give any simple relation between the impact parameter and the scattering angle.
Acknowledgement mo]
I
I
I
I
I
Part of the work has been supported by a grant from the Indian Space Research Organisation. Department of Space, Government of India. The authors are indebted to Professor KS. Taylor of the University of Southern California, Los Angeles for grant of computer time under NSF grant CHE-7910583 for the rest of the work.
i
References
f I] TX [ 21 { 31 141 [ 51 [ 61
1250
r500
2000
3oco
4000
5oco
E
0
Fig. 3. Sdmplc tr.msirion probabilities plotted against orbital anwlar momentum !. Full line, 2s - 2s (elastic); broken line. 2s - 2s (charge e~cbange). Note the change of scale to the Iefr off = 2000 and further change to the left ofl= 1500.
88
171 [8l
I91
ftol
Rai Dastidar and K. Rai Dastidar, Chem. Phys. Letters 85 (1982) 229. TX Rai Dastidar. Ii. Rai Dastidar and 51. Bose, Chem. Phys. 43 (1979) 183. TX. Rai Dastidar and S.S. Bhartacharya, Indian _I. Phys. 50 (1976) 731. ET. Smith, Phyr Rev- 179 (1969) IllC-E. Moore, Atomic energy levels, NBS Circular 467 (Natl. Bur. Std., Washington. 1949). J-P_ Gauyacq, in: Electronic and atomic collisions, ed. G. Watel (North-Ho~and, Amsterdam, 1978). LB. Delos, Comments At. Mol. Phys. 11 (1981) 21 I. S.A. Evans. J.S. Cohen and N.F. Lane, Phyr Rev_ A4 (1971) 2235. L. Lenamon,J.C. Browne and RE. Olson. Phys. Rev. A8 (1973) 2380. A. Russek, Phys. Rev. A4 (1971) 1918
2.5 March 1983
CHEMICAL PHYSICS LETTERS
A NEW DIABATIC MOLECULAR REPRESENTATION FOR TRIPLET-TRIPLET He+-He*
TRANSITIONS IN
COLLISIONS
TX. RAI DASTIDAR + Department of Chemkt~.
University of Southern California. Los Angeles. Califomia 90089. USA
K. RAI DASTIDAR Department of Physics, University of Southem California. Los Angeles, Gzlifomia 90089, USA and
R. SEN GUPTA Department of GeneraI Physics and X-Rays, Indinn Association for the Culttiation of Science. Calcutta 700032, India Received 23 August 1982;in
final form 13 January
1983
A “frozen-orbital” diabatic basis has been constructed for an impact parameter treatment of collisions of He+ \\ith metastable He(2 3S) at 1000 eV laboratory ion ener,q_ Lcept for the two highest states used, the diabatic states correlate very well with the separated-atom energies. and the P-11 rotational couplings deviate little from the proper asymptotic behaviour (=R”). Cross sections and transition probabilities are presented for some elastic and inelastic channels.
l_ Introduction
2_ Theory
In a recent publication [l] we demonstrated the use of an approximate diabatic molecular basis in resonant charge-transfer collisions of He+ with He. In the present paper we describe the use of another diabatic basis in a close-coupled treatment of collisions of He+ ions with metastable (2 3s) helium atoms, studying the elastic and the inelastic channels 2 3S + 2 3S, 2 3S -+ 2 3P and 2 3S + 3 3S together with their charge-exchange counterparts. As in ref. [l], it is found that the present diabatic basis is reasonably free from the two well-known defects of molecular orbital expansions, namely (i) failure to correlate to the proper separated-atom energies, and (ii) asymptotically incorrect (MZ-l) variation of the rotational coupling between Z-II molecular states. Atomic units are used throughout except where , otherwise stated.
and the corresponding charge transfer channels, we expand the total wavefunction \kas in ref. [l] in a truncated molecular basis set:
* Permanent address: Department of General Physics Br X-
idc/dr=(H+Q)c;
Rays, Indian Association for the Cultivation of Science, Calcutta 700032. India.
0 009-2614/83/0000-0000/S
03.00 0 1983 North-Holland
For the collision process He+(ls) + He*(ls2s,
2 %5)--t He+ + He*(ls2s, + He+ + He*(lsZp, +
w,
2 3S), 2 ‘P)
He* + He*(lsZs, 3 3S)
Jw)) = $Jq(r) Gi/I(T,q ,
(3)
(2)
where the 4 are electronic basis functions in a molecule-fured reference frame. The coupled equations to be solved for the problem have been outlined earlier [I] and here we quote only the final equation: (3)
where c is the coefficient vector defined in (2) and H 8.5
CHEMICAL
\‘olume 96. number 1
25 March 1983
PHYSICS LETTERS
l-able 1 ~~O~CCUIX
orbit&
I[
used for constructing the configration
state functions 4 as described in the text Exponents
and Q dre squire
opt-mized at
R 2 20.0
Sls' 2.00
Qs = 0.564
Sls’ 2.00 Ils = 2.00
T2po = 0.548 r2po = 0.533
Sls = 2.00
52P,
= 0.597
51,=
c3s=
0.83
2.00
matrices: (7)
IIk_r = (Gk lH,tI~,~ . Qx,=(uh~R~)~~,I--i
aiaoi$,),
(4)
the significance
of a11 quantities being the same as in ref. 111. In this clculdtion we have restricted ourselves to SIX“X staIcs and rlvo jii states in the summation (2). The basis \ector Jr ws built up from configuration SIate functions 4 hdvinp the general form
@= Ilo,
lo,, %I + Ilo,
= (lo,,
Ij i$
lo, Xl1 .
+ IQ lo, r;;,r
= u s(Kjl#I(r.
(5)
h‘) .
Icl(-=)l Ck(-O”)
= IQ-=)I =
0
solved sub-
= 2-111,
for X-f 1 or 2.
Unitarity was conserved to within (see refs. 12, 31 for computational
2-3 parts in 1000 details)_
3. Results and discussion
Table 1 gives the hl0 configurations of the ei$t CSFs used in this work. As in ref. [ 11, the MOs were chosen ds linrdr combinations of minima! Slater-type atomic orbirals: the orbrtals were variationally adjusted to minimize the configuration ener_gies at large internuclear sepArarion H and thereafter “frozen”. Finally the dlabatrc basis \LKIS constructed in the fonn \tV.R,
The coupled equations (3) were therefore ject to the boundary condition
(6)
\rhere S(K) is the Schmidt ortho~iormalizatioIi matrix and U = U(R > 20.0) is the “frozen” unitary lnatrh Jiagxmlizing S(R)+at large R. An elementary anal) sis yields the separated-atom limits of the MO conl$ratrons; thus states 1 and 2 can be shown to dissociate to the limit
The basis set was constructed and the l-i and Q were set up in a Burroughs 6700 computer with 48-bit-single-precision arithmetic; these matrices were input in an IBM 370/168 computer with 64-bit double precision to solve the coupled equations (3) The CSFs, 4 are eigenfunctions of Sz and S’ with eigenvalues S, = l/2 and S = 312. The motivation behind the s&p of freezing the non-linear expounded as well as the linear configuration coefficients (Le. the U-matrix elements) has already been expounded in earlier papers [ 1,2] ; this makes the radial momentum coupling negligible and hence the basis approaches Smith’s [4] criterion for diabaticity. Table 2 shows the separated-atom limits and the asymptotic energies of the configurations (HII) after diagonaiization at large R. Except for the highest-lying states 7 and 8 which require much more elaborate configuration matrices
25 hlarch 1983
CHEMICAL PHYSICS LETTERS
Volume 96, number 1
Table 2 Diagonal elements of the H matrix The experimental energy values are from ref. 151 I
Separated-atom confmration of 01
HllCR 2 20.0)
Wpt
1,2 3-4
He+@) + He(ls2s)3S He*&) + He(ls2po)sP
-4.1761 -4.1321
-4.1753
5.6 798
He+(k) + He(ls2p+)3P
-4.1326
He+(ls) i- He(ls3s13S
-3.4805
interaction, the correlation to the experimental energies [S] is seen to be quite satisfactory. Fig. 1 shows the variation of the diagonal terms of the diabatic H matrices with R. We feel that because of the “frozen” nature of our orbitals as well as the neglect of hi-her configurations, no particular significance can be attached to any comparison between our diabatic HII values and any adiabatic calculations II
I
-4.1332
A.0687
at intermediate and small internuclear separations. State 4 contains the MO 4fo,, and as in ref. [l] the behaviour of N44 is peculiar (see ref. 161 in this context). The angular momentum couplings between the pairs (l--5), (3-9, (7-5) and (Z-6), (4-6), (8-6) of states are shown in fig. 2. Their long-range behaviour almost duplicates that of the rotational coupling elements in ref. [l] _ Of course, only the terms (7-5) and (8-6) go to the proper limit, whereas the two other pairs lie somewhat above and below the correct lmrit; but their R variation is quite close to the proper asymptotic behaviour (R-‘), unlike many other molecular basis expansions which show a spurious R-l variation at long range [7]_ From the solutions of the coupled equations we have computed the different channel transition probabilities and the cross sections. Table 3 gives the cross sections at a laboratory ion energy of 1 keV (i-e. 500 eV in cm. system) for the channels given in (1) (except elastic scattering) together with their charge-
0 I
20
I
40
5
60
*
80
I
106
t
200
3
Fig. 1. Diagonal potential matrix elements for the eight states. Full lines, x states; broken lines, II states
0i
Fis 2. X-n rotational coup&
terms.
87
CHEMICAL
Volume 96, number 1 Table 3 Cross sections (X2> for the various scatwing 1000 eV laboralor)‘ ion energy ---_ - --_ Channel
Cross section
Z%-
:!3P
13s....
231’
54.@8 6.53
(chnrzx c\changecI 735 ;&.j3s
3.71 181‘ ==5 0 191.
6.48 3.59
33s
[clwgc
channels at
Other theoretical results (see text)
--_,‘S2% (charge c+.cb.mge)
PHYSICS LETTERS
3.43 _I
e\cJ~ang) ---_-
counterp,wts. To our knowledge there are no parallel calculations in the literature 10 compare our results with; but we give in the table the results for 1?%G- 2 3P excitation in neutral He--He*{2 3S) collision given by Evans et a1. [S] and by Lenamon et al. [91_ (The latter group obtained their results at this exchange
25 March 1983
energy by extrapolating lower-energy calculations by a procedure due to Russek [IO].) It should be noted that Evans et al. did not consider the X-Il coupling, but Lenamon et al. did_ Fig_ 3 shows a sample of ihe 2 3S + 2 3S elastic and charge-transfer channel probabilities in a region where their behaviour can be seen without getting lost in a labyrinth of detail, pIotted against the orbital anguIar momentum I which is related to the impact parameter b (semiclassical equivalence) as b = (I f $)fk, k = p_@z. It wiIl be seen that in general for large impact parameters the direct and the charge exchange probabilities are out of phase with each other, but towards low b this situation no longer holds. This is of course to be expected on account of the many curve crossings as well as rotational couplings_ Unlike ref. [ l] we have not assigned here any particular potential to the classical nuclear trajectory and hence do not give any simple relation between the impact parameter and the scattering angle.
Acknowledgement mo]
I
I
I
I
I
Part of the work has been supported by a grant from the Indian Space Research Organisation. Department of Space, Government of India. The authors are indebted to Professor KS. Taylor of the University of Southern California, Los Angeles for grant of computer time under NSF grant CHE-7910583 for the rest of the work.
i
References
f I] TX [ 21 { 31 141 [ 51 [ 61
1250
r500
2000
3oco
4000
5oco
E
0
Fig. 3. Sdmplc tr.msirion probabilities plotted against orbital anwlar momentum !. Full line, 2s - 2s (elastic); broken line. 2s - 2s (charge e~cbange). Note the change of scale to the Iefr off = 2000 and further change to the left ofl= 1500.
88
171 [8l
I91
ftol
Rai Dastidar and K. Rai Dastidar, Chem. Phys. Letters 85 (1982) 229. TX Rai Dastidar. Ii. Rai Dastidar and 51. Bose, Chem. Phys. 43 (1979) 183. TX. Rai Dastidar and S.S. Bhartacharya, Indian _I. Phys. 50 (1976) 731. ET. Smith, Phyr Rev- 179 (1969) IllC-E. Moore, Atomic energy levels, NBS Circular 467 (Natl. Bur. Std., Washington. 1949). J-P_ Gauyacq, in: Electronic and atomic collisions, ed. G. Watel (North-Ho~and, Amsterdam, 1978). LB. Delos, Comments At. Mol. Phys. 11 (1981) 21 I. S.A. Evans. J.S. Cohen and N.F. Lane, Phyr Rev_ A4 (1971) 2235. L. Lenamon,J.C. Browne and RE. Olson. Phys. Rev. A8 (1973) 2380. A. Russek, Phys. Rev. A4 (1971) 1918