Dissociative excitation of HD D 2 and DT by electr

Cent. Eur. J. Phys. • 8(1) • 2010 • 87-94
DOI: 10.2478/s11534-009-0093-x
Central European Journal of Physics
Dissociative excitation of HD+,D+
2, and DT+by
electron impact
Research Article
Mariana Duca, Magda Fifirig∗
Chemistry Department, University of Bucharest, Bd Regina Elisabeta 4-12, R-030018 Bucharest, Romania
Received 4 April 2009; accepted 21 May 2009
Abstract: In the framework of the Multi-Channel Quantum Defect Theory (MQDT), a theoretical study of the disso-
ciative excitation is presented. Numerical results for the dissociative excitation cross sections of HD+,D+
2,
and DT+with electrons of energy between 2and 12eV are reported. The contribution of the vibrational
continua of the two lowest electronic states as explicit ionization channels has been considered. Within
aquasi-diabatic representation of the molecular electronic states, the Born expansion of second order is
done in the K-matrix evaluation.
PACS (2008): 34.80.Gs, 34.80.Ht
Keywords: dissociative excitation • hydrogen molecular ion • multi-channel quantum defect theory • wave packet dis-
cretization
©Versita Warsaw and Springer-Verlag Berlin Heidelberg.
1. Introduction
In the reactive collisions between a diatomic molecular
cationAB+intheboundvibrationallevel+
andanelec-
tronofenergyabovethedissociationthresholdofmolec-
ularcation, thedissociativeexcitation(DE)plays anim-
portantrole
AB+(+
)+−()→A+B++−(0)A++B+−(0)(1)
where 0is the energy of the expelled electron. Since
thebound vibrationallevels ofthe Rydbergstatesof the
neutralsystemarenotaccessibleatthisenergy,onlythe
states corresponding to the scattering of the electron by
themolecularcationexcitedintothevibrationalcontinuum
∗E-mail: magda.fifi[email protected]
canbeattained. Twomechanismsareresponsiblefordis-
sociativeexcitation:
i) the electron captureby the molecular cation, con-
ducting to the formation of a neutral excited
molecule which subsequently autoionizes into an
antibonding state (AB+)∗leading to the fragmen-
tationinoneneutralandonechargedatomicfrag-
ments
AB+(+
)+−()→AB∗∗ →AB+∗+−(0)
→A+B++−(0)A++B+−(0)(2)
ii) theexcitationofthemolecularcationintoarepul-
sivestate(AB+)∗conductingtooneneutralandone
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Dissociative excitation of HD+,D+
2, and DT+by electron impact
chargedatomicfragments
AB+(+
)+−()→AB+∗+−(0)
→A+B++−(0)A++B+−(0)(3)
The last mechanism occurs at any collision energy ex-
ceeding the excitation energy i.e. the energy difference
betweentheenergyoftherepulsivestateandthatofthe
groundstateattheouterclassicalturningpoint.
Figure 1. Dissociative excitation of HD+in vibrational level +
= 1with electrons of energy . Thick black solid line represents the potential
energy curve of HD+in its electronic ground state, thick grey broken line is the potential energy curve of doubly excited state HD∗∗
situated below the 2Σ+
excited state of HD+(thick grey solid line). A more complete set of potential energy curves for HD+and HD can
be found in Ref. [1]. The arrow leading to −indicates the autoionization. Thin grey horizontal line represents the dissociation energy
of HD+(approximately −04998a.u.). Panels (a) and (b) illustrate the electron capture into HD∗∗ followed by the autoionization and
dissociation, and direct excitation into repulsive state 2σfollowed by dissociation, respectively.
Fig.1schematicallydescribesthetwomechanismsacting
in the dissociative excitation of HD+in the vibrational
level+
=1. Panel(a)presentstheindirectreactionpath
involving the electron capture into HD∗∗ repulsive dou-
blyexcitedstate(asinDissociativeRecombination(DR))
whichsubsequentlyautoionizesintotheantibondingstate
and dissociates into H++Dand D++H, see Eq. (2).
Thismechanismispossibleifthecollisionenergyexceeds
the dissociation energy of HD+, which is approximately
243eVfor+
=1. Panel(b)showsthedirectexcitation
fromtheelectronic ground stateof HD+tothe repulsive
state2σ, whichleads to dissociationintoH++Dand
D++H,seeEq.(3). For+
=1thismechanismoccursat
anyinternucleardistanceRatwhichthecollisionenergy
exceeds74eV.TheexperimentperformedbyAndersenet
al.[2]emphasizes thatthetwo endchannelsH++Dand
D++Hareofequalstrength.
Dissociativeexcitation(DE)haslimitedtheoreticalinves-
tigations[3–6]. InliteraturetherewerereportedDEmea-
surementsat heavy-ion storagering [2,7,8], DE experi-
mentsperformedbytheanimatedcrossedbeammethod[9]
andbymergedbeammethod[10]. Mosttheoreticalstudies
dedicatedto the electron-collisionwith hydrogenmolec-
ularions,haveinvestigatedelectronenergyregionbelow
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Mariana Duca, Magda Fifirig
1eV[11–16],situationinwhichDRisthedominantprocess
anddissociativeexcitationvanishes. WementiontheDR
measurements for vibrationally cold hydrogen molecular
ions performed in the ion storage ring experiments [17–
26].
Here we continue our previous investigations regarding
theDEofHD+initiallyinthevibrationallevel+
ofthe
electronicgroundstate, withelectrons ofenergy ranging
from2eVto12eV[6]andtheDRandDEindirectmech-
anism,describedbyEq.(2),forH+
2,HD+andDT+[27].
Inthispaper dedicated to the DEprocess for HD+,D+
2,
andDT+we haveappliedour MQDTprocedure[6]con-
sisting of the inclusion of the vibrational continua of the
two lowest electronic states of hydrogen molecular ion
as explicit ionization channels. Sec. 2is devoted to the
presentation of our theoretical approach amenable to a
proper description of the reactive collisions between di-
atomic molecular ions and electrons of energy above the
dissociation threshold of the molecular ion. In order to
simplifyourcomputationshere,wehaveneglectedthero-
tationaleffects,takingintoaccountonlyvibrationalstruc-
ture and interactions. In Sec. 3we extend our model to
HD+,D+
2, and DT+DE cross sections evaluating them
for+
=0and+ =1. Moreexcited states builton the
2σioniccoreareincludedinthiswork. Ananalysisof
thetwoDEmechanismsisalsocarriedout.
2. Basic equations
In the framework of the MQDT [28,29] adapted to DR
processes[30–32] we have performed calculations for the
DEcrosssectionstakingintoaccountthecontributionof
the first excited core of the molecular cation involved in
the electron reactive collision. Here we have achieved a
nonrotationaltreatmentwithcurvecrossing(thepotential
energycurveofadoublyexcitedstatecrossesthepoten-
tialenergycurveoftheionelectronicgroundstateinthe
Franck-Condonregion).
Present theoretical approach consists of the inclusion of
thevibrationcontinuaofthetwolowestelectronicstates
of the molecular cation as explicit ionization channels.
This inclusion yields a good description of the electron-
molecularcationreactivecollisionabovethedissociation
thresholdof the molecular cationground state. Theion-
izationchannels associatedwith thevibrationalcontinua
oftheelectronicstatesofthemolecularcationcorrespond
tostatesobtainedbythediscretizationofthesecontinua.
Eachoftheseionizationchannelsiscorrelatedtoawave
packetχ(R)constructedoftheexactwavefunctionsofthe
continuumspectrum
χ(R)= 1
√δZE+δ/2
E−δ/2χ(RE)E  ≥1(4)
E =E0< E1<  < E<  < EN(5)
wheretheindexdenotestheionizationchannelsassoci-
atedwithvibrationalcontinuaandthepositivequantityδ
is smaller than the energy gap ∆between two consecu-
tivediscretizedlevels(∆=E1−E0= =EN−EN−1),
E beingthedissociationlimitofthecationgroundstate
andENthe superior limit of the energygrid used in the
vibrationalcontinuumdiscretization. Theexactwavefunc-
tions χ(RE)are energy normalized, while the functions
definedby(4)areorthonormalized. Thewavepacket (4)
is practically the eigendifferential [33] of the radial part
χ(RE)ofthecontinuumfunction.
A similar continuum discretization scheme was used by
TakagiintheDEstudyofhydrogenmolecularion[19,20]
andintheDRcomputationofHeH+below1eV[34]. Tak-
agi treated the channels associated with the discretized
levels as dissociation ones. The wave-packet continuum
discretizationmethodwasalsoused forsolvingthescat-
teringofacompositeparticleonatargetnucleus[35,36].
Inthisapproach,twodifferentelectroniccouplingsareac-
complished. Theyareconnectedwiththeinteractionsbe-
tweenthechannelscharacterizingtheelectron-molecular
cation reactive collision, the ionization channels and the
dissociation channels. The ionization channels built on
thegroundioniccore,notedhereby1,arelabeledbythe
pair (), with the vibrational quantum number of the
ground electronic state and the angular quantum num-
beroftheincoming(oroutgoing)electron,whilethedis-
sociationchannelsarelabeledby,thequantumnumber
associatedwith theelectronicdissociativestate. At slow
energiesofthecollidingelectron,thedominantinterchan-
nelinteractionsaretheelectroniccouplingsbetweenthe
ionizationchannelsandthedissociationchannels. These
interactions permit the electron capture into dissociative
channelswithcompetingautoionizationbacktoopenion-
ization channels. The electronic coupling between the
dissociationchannelandtheionizationchannel()is
givenby
V1(R)=hAΦ+
1(+R)φ (R)|H()|Φ(R)i(6)
whereH()istheelectronicHamiltonian,Atheantisym-
metrizationoperator,theenergyoftheexternalelectron,
and Rthe internuclear distance. The integration is per-
formedoverelectroniccoordinates,whicharedenotedcol-
lectivelybyfortheneutralmoleculeandby+forthe
molecular ion. Φ+
1and Φare the electronic wavefunc-
tionsofthemolecularionstate associatedwiththe ionic
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Dissociative excitation of HD+,D+
2, and DT+by electron impact
core1andofneutralmoleculedissociativestate,respec-
tively, and φis the radial wavefunction of the external
electron.
The other interchannel interactions refer to the interac-
tions between two ionization channels belonging to dif-
ferent ionic cores. The electronic coupling between an
ionization channel built on the ground ionic core 1and
anionizationchannelbuiltontheexcitedcore2,isgiven
by
V102(R)=hAΦ+
1(+R)φ (R)|H()|AΦ+ 2(+R)φ0
0(R)i(7)
The present computation considers the case in which all
theabovecouplings(6)and(7)arefunctionsoftheinter-
nucleardistanceRonly.
TheelementsoftheinteractionoperatorVassociatedwith
the couplings between an ionization channel () and a
dissociationchannelare
V1=V1=ZF(R)V1(R)χ1(R)R
0≤≤N11≤≤N(8)
where F is the regular solution of the nuclear
Schrödinger equation in the repulsive molecular poten-
tialand=p2M(E−E)/~2withMthe reducedmass
of the two constituent atoms, Ethe asymptotic energy
of the electronic dissociative state, and Ethe total en-
ergy of the system E=E+,Ebeing the energy of
theinitialvibrationalstateofthemolecularcation. Nis
the number of dissociation channels and N1is the total
numberofionizationchannelsassociatedwiththeground
ionic core 1. In the above equation the wavefunction
χ1 designatesboth the eigenfunction ofthe vibrational
level Eand the wavefunction (4) associated with a dis-
cretised level of the ground state vibrational continuum.
Theindexlabelsboth thevibrationalbound levels and
discretized levels of the ground state vibrational contin-
uum. ThematrixelementsgivenbyEq.(8)aredependent
on the total energy of the system, through the regular
eigenfunctionF(R).
Another non-null elements of the interaction operator V
correspondtothecouplingsbetweentheionizationchan-
nelsassociatedwithtwodifferentelectroniccores
V12=V21=Zχ1(R)V12(R)χ2(R)R
0≤≤N10≤≤N2(9)
whereN2isthetotalnumberofionizationchannelsasso-
ciatedwiththeelectroniccore2. InEq.(9)thewavefunc-
tion χ2 designates the wavefunction (4) corresponding
to the discretised level of the excited core vibrational
continuum.
Inthechosenquasi-diabaticrepresentationofthemolec-
ular states, the interaction operator Vhas no matrix el-
ementsbetweenchannelsassociatedwiththesameionic
coreV =0 V2=0 V11=0 V22=0(10)
Starting from Vone can build the reaction matrix K
by solving the Lippmann-Schwinger integro-differential
equation K=V+V1
E−H0K (11)
whereH0istheHamiltonianoperatorexcludingtheelec-
tronic interaction V. In the case of weak interactions
the Lippmann-Schwinger equation can be solved pertur-
batively. Inthefirst-orderBornexpansion,Kmatrixcoin-
cideswith the matrixassociated with the interaction op-
eratorV
K(1) =
0V10
V10V12
0V210
(12)
Assumingtheenergy-independentelectronicinteractions
inEq.(11),thereactionmatrixK(2) (anN×Nmatrix,with
N=N+N1+N2thetotalnumberofchannels)canbe
constructedfromblocksofthetypeK (anN×Nsub-
matrixofK),K1(anN×N1submatrixofK),K2(an
N×N2submatrixofK),K11(anN1×N1submatrix
ofK),K22(anN2×N2submatrixofK),andK12(an
N1×N2submatrix of K) corresponding to an arrange-
mentoftheionizationchannelsaftertheiraffiliationtothe
electroniccores. So,Kmatrixhasthestructure
K(2) =
K K1K2
K1K11K12
K2K21K22
(13)
Taking into account the spectral representation of the
Greenoperator[31], the expression ofthe elements K
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Mariana Duca, Magda Fifirig
(with1≤≤N)oftheblockK,correspondingto
theinteractionbetweenthedissociationchannelsand
,is K=X
VVI(E)(14)
where
I(E)=PZε
ε
ε
E−(E+ε)=lnE−(E+ε) E−(E+ε)(15)
wherePistheprincipalpartintegral. In(14)thesumma-
tionistakenoveralltheionizationchannelsbuiltonthe
groundioniccore.
Similarly,theelementsK(with1≤≤N1)ofthe
blockK11corresponding to theinteraction between the
ionizationchannelsandbuiltonthegroundioniccore
1(which is 1σin the case of the hydrogen molecular
cations),are
K=X
1
WZ Z χ1(R)V(R)F(R<)G(R>)V(R0)χ1(R0)RR0+X
VVI(E)(16)
whereWistheWronskianofF andG,withG the
irregularsolutionof thenuclear Schrödingerequationin
thedissociativepotential,lagginginphasebyπ/2relative
to F.R<and R>denote the lesser and the greater,
respectively, of Rand R0. In Eq. (16)indexes all the
ionizationchannels built onthe 1ioniccore, associated
with the bound and discretized vibrational levels. The
other index labels all the ionization channels built on
the2ioniccore. In(16)thesummation istakenoverall
theionizationchannelsbuiltontheexcitedcore.
The elements Kof the block K22corresponding to
theinteractionbetweentheionizationchannelsand
associatedwiththeexcitedcore2(whichisthe2σcore
inthecaseofthehydrogenmolecularcations),are
K=X
VVI(E)1≤≤N2(17)
In(17)thesummationistakenoveralltheionizationchan-
nelsbuiltonthegroundioniccore.
The matrix elements K and K of the blocks K1and
K1(associated with the interaction between the disso-
ciationchannelsandtheionizationchannelsbuiltonthe
ground ionic core) and the elements K and K of the
blocks K12and K21(corresponding to the interaction
betweentwoionizationchannelsassociatedwiththetwo
ioniccores,1and2),aregivenby
K =K =V1≤≤N11≤≤N(18)
K =K =V1≤≤N11≤≤N2(19)
Finally, the matrix elements K and K of the blocks
K2andK2areconstructedbyelementsdefinedby
K =K =X
VVI(E)
1≤≤N21≤≤N(20)
ThesummationinEq.(20)istakenoveralltheionization
channelsbuiltonthegroundioniccore.
Sincetheionizationchannelthresholdsarethesuccessive
vibrational levels or discretized levels, which are ranked
withrespecttotheirenergy,thelaststepintheconstruc-
tion of Kmatrix consists of the permutation of its lines
andcolumns. ThelinesandcolumnsofthenewKmatrix
correspond to the natural arrangement of the ionization
channelsaccordingtotheirenergy.
Theframetransformationfromtheinteractionzonetothe
asymptoticoneischaracterizedbythematricesCandS.
TheelementsofthematrixCareexpressedintermsofthe
eigenvaluestanηαandthecorrespondingeigenvectorsUα
ofthereactionmatrixK
C+α=X
 Uαhχ1+|cos(πµ1 +ηα)|χ1i
0≤≤N1(21)
C+α=X
 Uαhχ2+|cos(πµ2 +ηα)|χ2i
0≤≤N2(22)
Cα =Uα cosηα1≤≤N(23)
where αruns from 1to N, the total number of channels.
Inthecaseofthehydrogenmolecularcationsµ1(R)and
µ2 arethe quantum defectsfor Rydbergstates built on
1σionic core (along with the continuum lying above)
and those at the 2Σthreshold, respectively. The blocks
S+α,S+αandSα ofSmatrixareobtainedbyreplacing
thecosinefunctionbythesinefunctioninC+α,C+αand
Cα.
ThematricesCandSarethebuildingblocksofthe”gen-
eralized”scatteringmatrix
X=C+S
C − S(24)
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