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Low-lying states of cs22+
Volume 71, number 3
LOW-LYlNG
CHEMICAL
STATES
PHYSICS
LETTERS
lSApnl1980
OF Css’ *
G. DAS and R-C. RAFFENE-M’IS C3zemtstry Dnmon. Argonne Nattonal Laboratory Argonne. Illmo~s 6O-l.39. USA Recelred
20 December
1979, m final form 3 hIarch 1980
’ xi, Usmg an MC SCF CI method, warefuntions for the ground state I S’g and the excited states of the symmetnes of the Cs$+ iomc system are generated The potenbal curves for eleven r z& twelve rile and six 'Ag states are calculated Results suggest a small charge-transfer cross section for the reaction Cs* + Cs*+ Cs + Cs-•.
*l$, and I+
1_ Introduction Several heavy eons are possible candidates for use in the fast ion-beam induced fuslor. process [ I]. The Cs+ eon, because of the closed-shell structure, IS hkeiy to have low charge-changmg cross sectlons, and therefore, long beam hfetime m storage rings.. A chargechangmg cross sectlon of the order of IO-l7 cm2 is deslrab!e to ensure a beam hfetrme of =I s. The charge state of an ion may be changed either by charge transfer (Cs’ + Cs+ + Cs” + Cs”) or romzation (Cs” + cs+ + cs2+ + Cs+ + e)_ Dependmg on the electromc configuratron of ions, the charge-transfer process may have a large cross section (>l O-l5 cm?) owmg to a possrble interaction of molecular states at large mternuclear separatron. To make a rehable esttmate of the charge-transfer cross sectron at relatrve kmetrc energies of mterest (Cl 50 keV) It IS necessary to consider a molecular descnptron of the compound system and study the possrble pseudo-crossmg, rfany, of the ground state and nelghbonng excited-state curves. In case of such crossmgs a simple LandauZener-type treatment IS able to grve a semr-quantrtative estimate of the charge-transfer cross sectlon. Howewr, rf there 1s no curve crossing even at small nuclear
* work performed under the auspices of the Office of Basic Energy kences, Diwslon of ChenuuI Saences, US Department of Energy ’ Present address Apphed h¶athematics Dwwon. Argonne National Laboratory, Argonne, Ulmon 60439. USA.
198
separation, as shown to be the case for Cs? in this work. at Ieast a two-state treatment [2] $ is needed involving the utitial and final states. However, since the finaI state consistmg of a doubly-charged ion and a neutral atom is energeticauy close to many other excited states arising from a variety of combinations of the Iowest-lying states of the neutral atom as well as the SmgIy-charged ion mvoivmg the 6s, 6p and Sd shells, it 1s necessary to include ali the near-degenerate molecutar states. Since the charge-transfer operator, hke the total hamtltoman, IS symmetnc m the centers of the homopolar system under connderatton, we need only the excited states of the symmetry 1 2g’, II-Is and ‘+ We present below “MC SCF + CI” calculations on the ground and vanous exerted states beloneg to the above symmetries.
2. Basis set, configurations, calculations
and other details of the
The basis set selected consrsts of a minunal set for and double-zeta plus the core electrons (n = l-4) polanzation functions for the valence electrons (PI = 5,6). We start wrth a set of Slater-type orbitals (STOs) generated by a least-squares fit apphed to the nonrelativrstic Hartree-Fock numerical atomic orbttals * For a serm-classical treatment, see ref. 131.
Volume 7 1, number 2
CHEMICAL
Table 1 All-electron Cs Slater-type-orbital basis set OrbItal type
Orbnal exponents
IS 2s
53 90674 20.23381 25 41793 12 10265 12.15194 13 66558 6 83221 6 53744 5.74113 3.42830,2 20688 3 15936,1.91135 1 87475.0 69698 l-29761,0 77646 1 16253,O 67259
2P 3s 3P 3d 4s 4P 4d 5s 5P 5d 6s 6~
15 April 1980
PHYSICS LETTERS
of Cs+. The exponents of the atomic STOs were then opturuzed by atomic I&u-tree-Fock-Roothaa calculations Next the basis set is augmented by suitable polanzatron functions. Table 1 shows the final basis set used for the ah-electron calculatrons. The basis set for the core orbitals (having pnnc~pal quantum number <4) is a mimmal one and hence IS not flexrble enough for optutuzation as the mtemuclear separatron is varied. At all mtemuclear separations considered, therefore, we are forced to use a “frozen” core (apart from skght changes brought about by the requirements of orthogonahty between orbitals at different centers). Valence orbrtals are kept orthogonal to these frozen cores. This “frozen-core approximatron” [4] has been demonstrated in the past to produce good-quahty potential curves except at short internuclear separations where the cores overlap srgnificantly. We have also verified that wMe a frozen mmimum basis core yields as good an SCF curve for Cs; as that grven by a double-zeta relaxed core, a relaxed mimmum basis core, on the other hand, leads to a rather poor potential curve, as expected. Corresponding to the atomic orbitals Ss, Sp, 6s, 6p, and Sd, we have to deal with five MOs each of the species a9 and au, three each of nu and rrg, and one each of 6, and 5 u m the moIecu1a.r ion; these are apart from the 32 core orbitals. The predominant electron configuration for the ground state is (5so,)2(5sou)2 (5~0,)~ (Sp0,)~(5p7ru)4 (5p~,)~ (oormtting the core occupancies for brevity). The ground state described
by this configuration is first optimized, yielding the description of the above occupied orbitals, both core and valence. The higher orbitals are generated as follows. To the ground state configuration we add two configurations, Spas + $, and Spa, -+ +,, and optimaze 9, and $,, for the first excited state of the 2 Zg’ symmetry, while other orbitals already optimized for the ground state are kept frozen. The next higher solutions of the Fock equations for GcI,and Gu are seIected to represent the remaining four orbitals of G symmetry. The excited s and u orbitals are obtained in rhe same way using appropriate (l llg and I+) molecular states. We then consider all those configurations that asymptoticaliy represent atoms or ions either in their ground states or those that are singly excited to the levels 6s. 6p, or 5d. These configurations are described in excitation language in tabie 2. AU configurations belonging to the symmetry 1 IZl are included in the respective CE secular equations. However, for the states of 1 Ilg and I+ symmetries, for simplicity, we have divided the configuratmns
into two groups
for which separate CE
calculations were carried out: one group corresponds to the excitations Us_ + mg,u or 4 u + ug u and the other represents the excitations 4 u‘-6 ‘_Thetwo groups of excitation interact only Gery w%Zcly. However, for some separation the curves from one group can and do cross with those from the other.
Table 2 OrbItaloccupanaes for the variouscoutigurationsusedin the w~zftmchons for the ‘xi, ‘rXg and ‘09 states of the ~5’ system. Theseare either as given by e. = (core) (5s~g)* (SSO& (5Pug)* (Spq# theiefrom Iv+ -g @O 5pug + 6s~ g SPY, - 6sa, 5pug - 6pog 5Pau -, 6~0, 5wg - 5dog 5pau+5dau 5pnu - 6~=,, 5png - 6pmg Spn, - Sdm,,
5~“~ - 5dq
(5pn”)’
(5pQ4
or various single exdtations -
Iftg 5pag 5wu 5Pog -, 5pcru -, 5pn, -
‘d9 6pg 6prru 5dng Sdnu 6sO”
5P=g -+ 65og 5pnu-,6pou 5wcz - 6pcg Spq, - Sdq, 5mg -5&g
5POg’ 5cBg spa, - 5&, 5~” -, 5mg -, 5p1r, -c 5pwg -,
6wu 6pg 5drru 5dirg
5pnu-,5d&l 5p”g -c 5&g 199
Volume 71,
number
CHEMICALPHYSiCS LETTERS
2
15 Aprtl1980
3. IkXlIts
0
L 3
---l----
5
7 RlsOHRI
9
II
Fig. 1. Ground and some excited * 2; states of the C.$+ton. The posittons of the cu~lrzswhtch asymptotrcally co&&pond to Cs + Cs*+and (Cs?* + cs’are mdtcated
In figs. 1 and 2 we plot the potential curves for the 1 I?,:, tI& and l& states. The curves, although shown only for R G 11.5 bohr, have actuaby been calculated to R = 20 bohr. The curves, ah repulsive, show some general features of mterest. First, the molecular wavefunction for the ground state appears to consist essentially of two Cs* tons repelitng each other electrostaticaiiy over the enbre range of values of R consxdered. The excited states are, however, strongty mtxed, even at as large a separation as 10 bohr. Proper identification of the states is possible only by going to around 20 bohr. They correspond, as they should, to the various combinations of the atomrc states m question, namely Cs(5p66s), Cs(5p66p), Cs(5p%d), Cs+(Sp6), Cs+(5p56s Cs+‘(5p56p), Cs”(Sp%d), and Cs2*(5p5). The SCF energies of these states relative to the Cs(Sp66s) energy, as caIcuiated by using the basis set and the ““minimumbasis” frozen core of our molecular caIcuIattons are compared in table 3 with the corresponding experimental [s] values. The starred expe~menta~ values correspond to an averagmg over a large number of relatlvlstic states belonging to the same non-relatlvlstic assignments Sp56p and Sp55d. The fau to good agreement between experiment and theory suggests that the resuitmg potenttai curves are very probably accurate enough for charge-transfer catcuiatlons. One rather curious feature of the excited state curves IS the presence of a large number of “bumps”, parttcularly at small separattons. Predtctably, the three pomt splme-fitting used m tracmg these curves IS unsatisfactory near these bumps. However, the sharp changes m the potential curves are real and are caused by numerous pseudo-crossings.
Table 3
R(BOHR1 Eig 2 Some excited i I’$ and ‘A8 state.s curves. The pos~hons of the cur+= whtch asymptottcaliy correspond to locally excited and charge-transfer states are mdtcated. The ground state curve is tncluded for comparison
200
State
Calculated enermes fiartree)
Experimental energies Ourtree)
5~~6~ Sp%d SP6 SP56s 5~~6~ 5p55d 5PS
005 0.07 013 0 65 0.74 0.75 0.98
0.05 0.07 0.14 0.64 o-68* 0.68 * 105
Volume 71, number: 2
CHEMKM.
PHYSICS
Smce there IS no ‘%urve crossing” (in the d~aba~ic sense) of any of the exctted states with the ground state, we expect that the charge-transfer cross section is small <
LETTERS
References [I 1 A_ FaRens, E Hayer, D. Keefe and L.3. iaslen. PEO[2] I3 j
AcknowIedgement Our thanks are due to Dr. Y.-K. K&I for suggesting the problem and for many helpful discussions_
[4] 15]
atedzqp of the Heavy Ion Fusion Wo&shop held at Argonne National tiboratory, ANL-7941 (1978) p. 38.N F. hiott and H SW. Massey, Theory of atomic coUi&ons, 3rd Ed. @arendon Press, Oxford, 196.565) p- 347D.R. Bates and D S-F. Crother~, Rot- Roy- Sot- A31S (1970) 465. W.H. Fmk, J. Chem. pfiys. 57 (1972) 1822; E 5. Sachs and J. Smut, J. Chem. Fhys. 62 (197513393. C.E. Moore, Atomic Energy Levels. Vol. 3, National Bureau of Stzmdasls, US t3fzptnment of Commerce (29.58).
201
LOW-LYlNG
CHEMICAL
STATES
PHYSICS
LETTERS
lSApnl1980
OF Css’ *
G. DAS and R-C. RAFFENE-M’IS C3zemtstry Dnmon. Argonne Nattonal Laboratory Argonne. Illmo~s 6O-l.39. USA Recelred
20 December
1979, m final form 3 hIarch 1980
’ xi, Usmg an MC SCF CI method, warefuntions for the ground state I S’g and the excited states of the symmetnes of the Cs$+ iomc system are generated The potenbal curves for eleven r z& twelve rile and six 'Ag states are calculated Results suggest a small charge-transfer cross section for the reaction Cs* + Cs*+ Cs + Cs-•.
*l$, and I+
1_ Introduction Several heavy eons are possible candidates for use in the fast ion-beam induced fuslor. process [ I]. The Cs+ eon, because of the closed-shell structure, IS hkeiy to have low charge-changmg cross sectlons, and therefore, long beam hfetime m storage rings.. A chargechangmg cross sectlon of the order of IO-l7 cm2 is deslrab!e to ensure a beam hfetrme of =I s. The charge state of an ion may be changed either by charge transfer (Cs’ + Cs+ + Cs” + Cs”) or romzation (Cs” + cs+ + cs2+ + Cs+ + e)_ Dependmg on the electromc configuratron of ions, the charge-transfer process may have a large cross section (>l O-l5 cm?) owmg to a possrble interaction of molecular states at large mternuclear separatron. To make a rehable esttmate of the charge-transfer cross sectron at relatrve kmetrc energies of mterest (Cl 50 keV) It IS necessary to consider a molecular descnptron of the compound system and study the possrble pseudo-crossmg, rfany, of the ground state and nelghbonng excited-state curves. In case of such crossmgs a simple LandauZener-type treatment IS able to grve a semr-quantrtative estimate of the charge-transfer cross sectlon. Howewr, rf there 1s no curve crossing even at small nuclear
* work performed under the auspices of the Office of Basic Energy kences, Diwslon of ChenuuI Saences, US Department of Energy ’ Present address Apphed h¶athematics Dwwon. Argonne National Laboratory, Argonne, Ulmon 60439. USA.
198
separation, as shown to be the case for Cs? in this work. at Ieast a two-state treatment [2] $ is needed involving the utitial and final states. However, since the finaI state consistmg of a doubly-charged ion and a neutral atom is energeticauy close to many other excited states arising from a variety of combinations of the Iowest-lying states of the neutral atom as well as the SmgIy-charged ion mvoivmg the 6s, 6p and Sd shells, it 1s necessary to include ali the near-degenerate molecutar states. Since the charge-transfer operator, hke the total hamtltoman, IS symmetnc m the centers of the homopolar system under connderatton, we need only the excited states of the symmetry 1 2g’, II-Is and ‘+ We present below “MC SCF + CI” calculations on the ground and vanous exerted states beloneg to the above symmetries.
2. Basis set, configurations, calculations
and other details of the
The basis set selected consrsts of a minunal set for and double-zeta plus the core electrons (n = l-4) polanzation functions for the valence electrons (PI = 5,6). We start wrth a set of Slater-type orbitals (STOs) generated by a least-squares fit apphed to the nonrelativrstic Hartree-Fock numerical atomic orbttals * For a serm-classical treatment, see ref. 131.
Volume 7 1, number 2
CHEMICAL
Table 1 All-electron Cs Slater-type-orbital basis set OrbItal type
Orbnal exponents
IS 2s
53 90674 20.23381 25 41793 12 10265 12.15194 13 66558 6 83221 6 53744 5.74113 3.42830,2 20688 3 15936,1.91135 1 87475.0 69698 l-29761,0 77646 1 16253,O 67259
2P 3s 3P 3d 4s 4P 4d 5s 5P 5d 6s 6~
15 April 1980
PHYSICS LETTERS
of Cs+. The exponents of the atomic STOs were then opturuzed by atomic I&u-tree-Fock-Roothaa calculations Next the basis set is augmented by suitable polanzatron functions. Table 1 shows the final basis set used for the ah-electron calculatrons. The basis set for the core orbitals (having pnnc~pal quantum number <4) is a mimmal one and hence IS not flexrble enough for optutuzation as the mtemuclear separatron is varied. At all mtemuclear separations considered, therefore, we are forced to use a “frozen” core (apart from skght changes brought about by the requirements of orthogonahty between orbitals at different centers). Valence orbrtals are kept orthogonal to these frozen cores. This “frozen-core approximatron” [4] has been demonstrated in the past to produce good-quahty potential curves except at short internuclear separations where the cores overlap srgnificantly. We have also verified that wMe a frozen mmimum basis core yields as good an SCF curve for Cs; as that grven by a double-zeta relaxed core, a relaxed mimmum basis core, on the other hand, leads to a rather poor potential curve, as expected. Corresponding to the atomic orbitals Ss, Sp, 6s, 6p, and Sd, we have to deal with five MOs each of the species a9 and au, three each of nu and rrg, and one each of 6, and 5 u m the moIecu1a.r ion; these are apart from the 32 core orbitals. The predominant electron configuration for the ground state is (5so,)2(5sou)2 (5~0,)~ (Sp0,)~(5p7ru)4 (5p~,)~ (oormtting the core occupancies for brevity). The ground state described
by this configuration is first optimized, yielding the description of the above occupied orbitals, both core and valence. The higher orbitals are generated as follows. To the ground state configuration we add two configurations, Spas + $, and Spa, -+ +,, and optimaze 9, and $,, for the first excited state of the 2 Zg’ symmetry, while other orbitals already optimized for the ground state are kept frozen. The next higher solutions of the Fock equations for GcI,and Gu are seIected to represent the remaining four orbitals of G symmetry. The excited s and u orbitals are obtained in rhe same way using appropriate (l llg and I+) molecular states. We then consider all those configurations that asymptoticaliy represent atoms or ions either in their ground states or those that are singly excited to the levels 6s. 6p, or 5d. These configurations are described in excitation language in tabie 2. AU configurations belonging to the symmetry 1 IZl are included in the respective CE secular equations. However, for the states of 1 Ilg and I+ symmetries, for simplicity, we have divided the configuratmns
into two groups
for which separate CE
calculations were carried out: one group corresponds to the excitations Us_ + mg,u or 4 u + ug u and the other represents the excitations 4 u‘-6 ‘_Thetwo groups of excitation interact only Gery w%Zcly. However, for some separation the curves from one group can and do cross with those from the other.
Table 2 OrbItaloccupanaes for the variouscoutigurationsusedin the w~zftmchons for the ‘xi, ‘rXg and ‘09 states of the ~5’ system. Theseare either as given by e. = (core) (5s~g)* (SSO& (5Pug)* (Spq# theiefrom Iv+ -g @O 5pug + 6s~ g SPY, - 6sa, 5pug - 6pog 5Pau -, 6~0, 5wg - 5dog 5pau+5dau 5pnu - 6~=,, 5png - 6pmg Spn, - Sdm,,
5~“~ - 5dq
(5pn”)’
(5pQ4
or various single exdtations -
Iftg 5pag 5wu 5Pog -, 5pcru -, 5pn, -
‘d9 6pg 6prru 5dng Sdnu 6sO”
5P=g -+ 65og 5pnu-,6pou 5wcz - 6pcg Spq, - Sdq, 5mg -5&g
5POg’ 5cBg spa, - 5&, 5~” -, 5mg -, 5p1r, -c 5pwg -,
6wu 6pg 5drru 5dirg
5pnu-,5d&l 5p”g -c 5&g 199
Volume 71,
number
CHEMICALPHYSiCS LETTERS
2
15 Aprtl1980
3. IkXlIts
0
L 3
---l----
5
7 RlsOHRI
9
II
Fig. 1. Ground and some excited * 2; states of the C.$+ton. The posittons of the cu~lrzswhtch asymptotrcally co&&pond to Cs + Cs*+and (Cs?* + cs’are mdtcated
In figs. 1 and 2 we plot the potential curves for the 1 I?,:, tI& and l& states. The curves, although shown only for R G 11.5 bohr, have actuaby been calculated to R = 20 bohr. The curves, ah repulsive, show some general features of mterest. First, the molecular wavefunction for the ground state appears to consist essentially of two Cs* tons repelitng each other electrostaticaiiy over the enbre range of values of R consxdered. The excited states are, however, strongty mtxed, even at as large a separation as 10 bohr. Proper identification of the states is possible only by going to around 20 bohr. They correspond, as they should, to the various combinations of the atomrc states m question, namely Cs(5p66s), Cs(5p66p), Cs(5p%d), Cs+(Sp6), Cs+(5p56s Cs+‘(5p56p), Cs”(Sp%d), and Cs2*(5p5). The SCF energies of these states relative to the Cs(Sp66s) energy, as caIcuiated by using the basis set and the ““minimumbasis” frozen core of our molecular caIcuIattons are compared in table 3 with the corresponding experimental [s] values. The starred expe~menta~ values correspond to an averagmg over a large number of relatlvlstic states belonging to the same non-relatlvlstic assignments Sp56p and Sp55d. The fau to good agreement between experiment and theory suggests that the resuitmg potenttai curves are very probably accurate enough for charge-transfer catcuiatlons. One rather curious feature of the excited state curves IS the presence of a large number of “bumps”, parttcularly at small separattons. Predtctably, the three pomt splme-fitting used m tracmg these curves IS unsatisfactory near these bumps. However, the sharp changes m the potential curves are real and are caused by numerous pseudo-crossings.
Table 3
R(BOHR1 Eig 2 Some excited i I’$ and ‘A8 state.s curves. The pos~hons of the cur+= whtch asymptottcaliy correspond to locally excited and charge-transfer states are mdtcated. The ground state curve is tncluded for comparison
200
State
Calculated enermes fiartree)
Experimental energies Ourtree)
5~~6~ Sp%d SP6 SP56s 5~~6~ 5p55d 5PS
005 0.07 013 0 65 0.74 0.75 0.98
0.05 0.07 0.14 0.64 o-68* 0.68 * 105
Volume 71, number: 2
CHEMKM.
PHYSICS
Smce there IS no ‘%urve crossing” (in the d~aba~ic sense) of any of the exctted states with the ground state, we expect that the charge-transfer cross section is small <
LETTERS
References [I 1 A_ FaRens, E Hayer, D. Keefe and L.3. iaslen. PEO[2] I3 j
AcknowIedgement Our thanks are due to Dr. Y.-K. K&I for suggesting the problem and for many helpful discussions_
[4] 15]
atedzqp of the Heavy Ion Fusion Wo&shop held at Argonne National tiboratory, ANL-7941 (1978) p. 38.N F. hiott and H SW. Massey, Theory of atomic coUi&ons, 3rd Ed. @arendon Press, Oxford, 196.565) p- 347D.R. Bates and D S-F. Crother~, Rot- Roy- Sot- A31S (1970) 465. W.H. Fmk, J. Chem. pfiys. 57 (1972) 1822; E 5. Sachs and J. Smut, J. Chem. Fhys. 62 (197513393. C.E. Moore, Atomic Energy Levels. Vol. 3, National Bureau of Stzmdasls, US t3fzptnment of Commerce (29.58).
201