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 +, D2+, 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 dissociative excitation is presented. Numerical results for the dissociative excitation cross sections of HD + , D2+ ,
and DT + with electrons of energy between 2 and 12 eV are reported. The contribution of the vibrational
continua of the two lowest electronic states as explicit ionization channels has been considered. Within
a quasi-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 discretization
© Versita Warsaw and Springer-Verlag Berlin Heidelberg.
1.
Introduction
can be attained. Two mechanisms are responsible for dissociative excitation:
In the reactive collisions between a diatomic molecular
cation AB+ in the bound vibrational level vi+ and an electron of energy ε above the dissociation threshold of molecular cation, the dissociative excitation (DE) plays an important role
AB+ (vi+ ) + e− (ε) → A + B+ + e− (ε 0 ), A+ + B + e− (ε 0 ), (1)
where ε 0 is the energy of the expelled electron. Since
the bound vibrational levels of the Rydberg states of the
neutral system are not accessible at this energy, only the
states corresponding to the scattering of the electron by
the molecular cation excited into the vibrational continuum
∗
E-mail: [email protected]
i) the electron capture by the molecular cation, conducting to the formation of a neutral excited
molecule which subsequently autoionizes into an
antibonding state (AB+ )∗ leading to the fragmentation in one neutral and one charged atomic fragments
AB+ (vi+ ) + e− (ε) → AB∗∗ → AB+
∗
+ e− (ε 0 )
→ A + B+ + e− (ε 0 ), A+ + B + e− (ε 0 ).
(2)
ii) the excitation of the molecular cation into a repulsive state (AB+ )∗ conducting to one neutral and one
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Dissociative excitation of HD + , D2+ , and DT + by electron impact
charged atomic fragments
AB+ (vi+ ) + e− (ε) → AB+
∗
+ e− (ε 0 )
→ A + B+ + e− (ε 0 ), A+ + B + e− (ε 0 ).
Figure 1.
The last mechanism occurs at any collision energy exceeding the excitation energy i.e. the energy difference
between the energy of the repulsive state and that of the
ground state at the outer classical turning point.
(3)
Dissociative excitation of HD + in vibrational level vi+ = 1 with electrons of energy ε. 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 2 Σ+
u 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 e− indicates the autoionization. Thin grey horizontal line represents the dissociation energy
of HD + (approximately −0.4998 a.u.). Panels (a) and (b) illustrate the electron capture into HD ∗∗ followed by the autoionization and
dissociation, and direct excitation into repulsive state 2pσu followed by dissociation, respectively.
Fig. 1 schematically describes the two mechanisms acting
in the dissociative excitation of HD + in the vibrational
level vi+ = 1. Panel (a) presents the indirect reaction path
involving the electron capture into HD ∗∗ repulsive doubly excited state (as in Dissociative Recombination (DR))
which subsequently autoionizes into the antibonding state
and dissociates into H + + D and D + + H, see Eq. (2).
This mechanism is possible if the collision energy exceeds
the dissociation energy of HD + , which is approximately
2.43 eV for vi+ = 1. Panel (b) shows the direct excitation
from the electronic ground state of HD + to the repulsive
state 2pσu , which leads to dissociation into H + + D and
D + + H, see Eq. (3). For vi+ = 1 this mechanism occurs at
any internuclear distance R at which the collision energy
exceeds 7.4 eV. The experiment performed by Andersen et
al. [2] emphasizes that the two endchannels H + + D and
D + + H are of equal strength.
Dissociative excitation (DE) has limited theoretical investigations [3–6]. In literature there were reported DE measurements at heavy-ion storage ring [2, 7, 8], DE experiments performed by the animated crossed beam method [9]
and by merged beam method [10]. Most theoretical studies
dedicated to the electron-collision with hydrogen molecular ions, have investigated electron energy region below
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Mariana Duca, Magda Fifirig
1 eV [11–16], situation in which DR is the dominant process
and dissociative excitation vanishes. We mention the DR
measurements for vibrationally cold hydrogen molecular
ions performed in the ion storage ring experiments [17–
26].
continuum spectrum
Here we continue our previous investigations regarding
the DE of HD + initially in the vibrational level vi+ of the
electronic ground state, with electrons of energy ranging
from 2 eV to 12 eV [6] and the DR and DE indirect mechanism, described by Eq. (2), for H2+ , HD + and DT + [27].
where the index i denotes the ionization channels associated with vibrational continua and the positive quantity δ
is smaller than the energy gap ∆ between two consecutive discretized levels (∆ = E1 − E0 = . . . = EN − EN−1 ),
Eas being the dissociation limit of the cation ground state
and EN the superior limit of the energy grid used in the
vibrational continuum discretization. The exact wavefunctions χ(R, E) are energy normalized, while the functions
defined by (4) are orthonormalized. The wave packet (4)
is practically the eigendifferential [33] of the radial part
χ(R, E) of the continuum function.
A similar continuum discretization scheme was used by
Takagi in the DE study of hydrogen molecular ion [19, 20]
and in the DR computation of HeH+ below 1 eV [34]. Takagi treated the channels associated with the discretized
levels as dissociation ones. The wave-packet continuum
discretization method was also used for solving the scattering of a composite particle on a target nucleus [35, 36].
In this approach, two different electronic couplings are accomplished. They are connected with the interactions between the channels characterizing the electron-molecular
cation reactive collision, the ionization channels and the
dissociation channels. The ionization channels built on
the ground ionic core, noted here by c1 , are labeled by the
pair (v, l), with v the vibrational quantum number of the
ground electronic state and l the angular quantum number of the incoming (or outgoing) electron, while the dissociation channels are labeled by d, the quantum number
associated with the electronic dissociative state. At slow
energies of the colliding electron, the dominant interchannel interactions are the electronic couplings between the
ionization channels and the dissociation channels. These
interactions permit the electron capture into dissociative
channels with competing autoionization back to open ionization channels. The electronic coupling between the
dissociation channel d and the ionization channel (v, l) is
given by
In this paper dedicated to the DE process for HD + , D2+ ,
and DT + we have applied our MQDT procedure [6] consisting of the inclusion of the vibrational continua of the
two lowest electronic states of hydrogen molecular ion
as explicit ionization channels. Sec. 2 is devoted to the
presentation of our theoretical approach amenable to a
proper description of the reactive collisions between diatomic molecular ions and electrons of energy above the
dissociation threshold of the molecular ion. In order to
simplify our computations here, we have neglected the rotational effects, taking into account only vibrational structure and interactions. In Sec. 3 we extend our model to
HD + , D2+ , and DT + DE cross sections evaluating them
for vi+ = 0 and vi+ = 1. More excited states built on the
2pσu ionic core are included in this work. An analysis of
the two DE mechanisms is also carried out.
2.
Basic equations
In the framework of the MQDT [28, 29] adapted to DR
processes [30–32] we have performed calculations for the
DE cross sections taking into account the contribution of
the first excited core of the molecular cation involved in
the electron reactive collision. Here we have achieved a
nonrotational treatment with curve crossing (the potential
energy curve of a doubly excited state crosses the potential energy curve of the ion electronic ground state in the
Franck-Condon region).
Present theoretical approach consists of the inclusion of
the vibration continua of the two lowest electronic states
of the molecular cation as explicit ionization channels.
This inclusion yields a good description of the electronmolecular cation reactive collision above the dissociation
threshold of the molecular cation ground state. The ionization channels associated with the vibrational continua
of the electronic states of the molecular cation correspond
to states obtained by the discretization of these continua.
Each of these ionization channels is correlated to a wave
packet χi (R) constructed of the exact wavefunctions of the
Z Ei +δ/2
1
χ(R, E)dE, i ≥ 1
χi (R) = √
δ Ei −δ/2
Eas = E0 < E1 < . . . < Ei < . . . < EN ,
(4)
(5)
+
l
(el)
Vd,lc1 (R) = hA Φ+
| Φd (q, R)i,
c1 (q , R)φε (q, R) | H
(6)
where H (el) is the electronic Hamiltonian, A the antisymmetrization operator, ε the energy of the external electron,
and R the internuclear distance. The integration is performed over electronic coordinates, which are denoted collectively by q for the neutral molecule and by q+ for the
molecular ion. Φ+
c1 and Φd are the electronic wavefunctions of the molecular ion state associated with the ionic
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Dissociative excitation of HD + , D2+ , and DT + by electron impact
core c1 and of neutral molecule dissociative state, respectively, and φε is the radial wavefunction of the external
electron.
The other interchannel interactions refer to the interactions between two ionization channels belonging to dif-
ferent ionic cores. The electronic coupling between an
ionization channel built on the ground ionic core c1 and
an ionization channel built on the excited core c2, is given
by
+
l
(el)
+
l0
Vlc1,l0 c2 (R) = hA Φ+
| A Φ+
c1 (q , R)φε (q, R) | H
c2 (q , R)φε 0 (q, R) i.
The present computation considers the case in which all
the above couplings (6) and (7) are functions of the internuclear distance R only.
The elements of the interaction operator V associated with
the couplings between an ionization channel (v, l) and a
dissociation channel d are
Z
Vdc1 = Vc1d =
0 ≤ v ≤ Nc1 ,
1 ≤ d ≤ Nd ,
(8)
where Fd,k is the regular solution of the nuclear
Schrödinger equation
in the repulsive molecular potenp
tial and k = 2M(E − Ed )/~2 with M the reduced mass
of the two constituent atoms, Ed the asymptotic energy
of the electronic dissociative state, and E the total energy of the system E = Evi + ε, Evi being the energy of
the initial vibrational state of the molecular cation. Nd is
the number of dissociation channels and Nc1 is the total
number of ionization channels associated with the ground
ionic core c1. In the above equation the wavefunction
χc1,v designates both the eigenfunction of the vibrational
level Ev and the wavefunction (4) associated with a discretised level of the ground state vibrational continuum.
The index v labels both the vibrational bound levels and
discretized levels of the ground state vibrational continuum. The matrix elements given by Eq. (8) are dependent
on the total energy of the system, through the regular
eigenfunction Fd,k (R).
Another non-null elements of the interaction operator V
correspond to the couplings between the ionization channels associated with two different electronic cores
Z
Vc1c2 = Vc2c1 =
to the discretised level w of the excited core vibrational
continuum.
In the chosen quasi-diabatic representation of the molecular states, the interaction operator V has no matrix elements between channels associated with the same ionic
core
Vdd = 0,
Fd,k (R)Vd,lc1 (R)χc1,v (R)dR,
Vdc2 = 0,
0 ≤ w ≤ Nc2 ,
Vc1c1 = 0,
Vc2c2 = 0.
(10)
Starting from V one can build the reaction matrix K
by solving the Lippmann-Schwinger integro-differential
equation
1
K,
(11)
K =V +V
E − H0
where H0 is the Hamiltonian operator excluding the electronic interaction V . In the case of weak interactions
the Lippmann-Schwinger equation can be solved perturbatively. In the first-order Born expansion, K matrix coincides with the matrix associated with the interaction operator V


0 Vdc1
0


(12)
K (1) =  Vc1d 0 Vc1c2  .
0 Vc2c1 0
Assuming the energy-independent electronic interactions
in Eq. (11), the reaction matrix K (2) (an N ×N matrix, with
N = Nd + Nc1 + Nc2 the total number of channels) can be
constructed from blocks of the type Kdd (an Nd × Nd submatrix of K ), Kdc1 (an Nd × Nc1 submatrix of K ), Kdc2 (an
Nd × Nc2 submatrix of K ), Kc1 c1 (an Nc1 × Nc1 submatrix
of K ), Kc2 c2 (an Nc2 × Nc2 submatrix of K ), and Kc1c2 (an
Nc1 × Nc2 submatrix of K ) corresponding to an arrangement of the ionization channels after their affiliation to the
electronic cores. So, K matrix has the structure
χc1,v (R)Vlc1,lc2 (R)χc2,w (R)dR,
0 ≤ v ≤ Nc1 ,
(7)

(9)
where Nc2 is the total number of ionization channels associated with the electronic core c2. In Eq. (9) the wavefunction χc2,w designates the wavefunction (4) corresponding
K (2)

Kdd Kdc1 Kdc2


=  Kc1d Kc1c1 Kc1c2  .
Kc2d Kc2c1 Kc2c2
(13)
Taking into account the spectral representation of the
Green operator [31], the expression of the elements Kdi dj
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(with 1 ≤ di , dj ≤ Nd ) of the block Kdd , corresponding to
the interaction between the dissociation channels di and
dj , is
X
Vdi v Vvdj I(Ev ),
(14)
Kdi dj =
v
where
Z
I(Ev ) = P
εb
dε
E − (Ev + εa )
, (15)
= ln
E − (Ev + ε)
E − (Ev + εb )
εa
Kvi vj =
X
Vwi v Vvwj I(Ev ),
1 ≤ wi , wj ≤ Nc2 .
(17)
v
(16)
The summation in Eq. (20) is taken over all the ionization
channels built on the ground ionic core.
Since the ionization channel thresholds are the successive
vibrational levels or discretized levels, which are ranked
with respect to their energy, the last step in the construction of K matrix consists of the permutation of its lines
and columns. The lines and columns of the new K matrix
correspond to the natural arrangement of the ionization
channels according to their energy.
The frame transformation from the interaction zone to the
asymptotic one is characterized by the matrices C and S.
The elements of the matrix C are expressed in terms of the
eigenvalues tan ηα and the corresponding eigenvectors Uα
of the reaction matrix K
Cv + α =
X
Ulvα hχc1,v + | cos(πµc1,l + ηα ) | χc1,v i,
l,v
In (17) the summation is taken over all the ionization channels built on the ground ionic core.
The matrix elements Kvd and Kdv of the blocks Kc1d and
Kdc1 (associated with the interaction between the dissociation channels and the ionization channels built on the
ground ionic core) and the elements Kvw and Kwv of the
blocks Kc1c2 and Kc2c1 (corresponding to the interaction
between two ionization channels associated with the two
ionic cores, c1 and c2 ), are given by
Kvd = Kdv = Vdv ,
1 ≤ v ≤ Nc1 ,
1 ≤ d ≤ Nd , (18)
Kwv = Kvw = Vvw ,
1 ≤ v ≤ Nc1 ,
1 ≤ w ≤ Nc2 . (19)
Finally, the matrix elements Kwd and Kdw of the blocks
Kc2d and Kdc2 are constructed by elements defined by
Kwd = Kdw =
Similarly, the elements Kvi vj (with 1 ≤ vi , vj ≤ Nc1 ) of the
block Kc1c1 corresponding to the interaction between the
ionization channels vi and vj built on the ground ionic core
c1 (which is 1sσg in the case of the hydrogen molecular
cations), are
X 1 Z Z
X
Vvi w Vwvj I(Ew ),
χc1,vi (R)Vvi d (R)Fd,k (R< )Gd,k (R> )Vdvj (R 0 )χc1,vj (R 0 )dRdR 0 +
W
w
d
where W is the Wronskian of Fd,k and Gd,k , with Gd,k the
irregular solution of the nuclear Schrödinger equation in
the dissociative potential, lagging in phase by π/2 relative
to Fd,k . R< and R> denote the lesser and the greater,
respectively, of R and R 0 . In Eq. (16) v indexes all the
ionization channels built on the c1 ionic core, associated
with the bound and discretized vibrational levels. The
other index w labels all the ionization channels built on
the c2 ionic core. In (16) the summation is taken over all
the ionization channels built on the excited core.
The elements Kwi wj of the block Kc2c2 corresponding to
the interaction between the ionization channels wi and wj
associated with the excited core c2 (which is the 2pσu core
in the case of the hydrogen molecular cations), are
Kwi wj =
where P is the principal part integral. In (14) the summation is taken over all the ionization channels built on the
ground ionic core.
X
Vdv Vvw I(Ev ),
0 ≤ v ≤ Nc1 ,
Cw + α =
X
(21)
Ulwα hχc2,w + | cos(πµc2,l + ηα ) | χc2,w i,
l,w
Cdα =Udα cos ηα ,
0 ≤ w ≤ Nc2 ,
(22)
1 ≤ d ≤ Nd ,
(23)
where α runs from 1 to N, the total number of channels.
In the case of the hydrogen molecular cations µc1,l (R) and
µc2,l are the quantum defects for Rydberg states built on
1sσg ionic core (along with the continuum lying above)
and those at the 2 Σu threshold, respectively. The blocks
Sv + α , Sw + α and Sdα of S matrix are obtained by replacing
the cosine function by the sine function in Cv + α , Cw + α and
Cdα .
The matrices C and S are the building blocks of the ”generalized” scattering matrix
v
1 ≤ w ≤ Nc2 ,
1 ≤ d ≤ Nd .
(20)
X=
C + iS
.
C − iS
(24)
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Dissociative excitation of HD + , D2+ , and DT + by electron impact
Finally, we build the scattering matrix S dimensioned only
to open channels [28]
Soo = Xoo − Xoc
1
Xco ,
Xcc − e−2iπν
(25)
where o and c denote the open and closed channel, respectively. The matrix ν is a c × c diagonal
matrix having
p
(in atomic units) the elements νv = 1/ 2(Ev − E) running
only over the closed ionization channels for which Ev > E.
In order to determine the DE cross section we have to
consider the vibrational excitation in the ground state vibrational continuum and the vibrational excitation in the
excited state vibrational continuum. The expression (in
atomic units) of the partial cross section for vibrational
excitation in a state characterized by the vibrational quantum number v + is
sym
σv + ←lv + (ε) =
i
π
sym
ρ | Sv + ←lv + |2 ,
i
4ε
Ev + > Evi+ .
The ratio (DE cross section)/(DE cross section+DR cross
section) for HD + initially in vibrational level vi+ = 0. Black
solid line: present numerical results, and black full circles:
experimental results from ASTRID [2].
(26)
The addition over the parameters l and sym leads to the
total cross sections for vibrational excitation from the vibrational level vi+ to the vibrational level v + . So, the partial cross section for vibrational excitation in the ground
state vibrational continuum is obtained by adding all the
contributions of the type (26) associated with all the discretized levels of the ground state vibrational continuum,
and the partial cross section for vibrational excitation in
the excited state vibrational continuum is given by the
summation of terms of the type (26) over all the discretized
levels of the excited state. Finally, the total DE cross
section σDE,vi+ is the sum of two terms, one of them corresponds to the vibrational excitation in the ground state
vibrational continuum and the other to the vibrational excitation in the excited state vibrational continuum.
This MQDT approach enables a proper description of the
reactive collision between a diatomic molecular cation
in its elecctronic ground state and an electron of energy above the dissociation threshold of molecular cation
ground state can be adapted to account other excited cores
of the molecular cation.
3.
Figure 2.
Results and comments
Here we present numerical results for the DE cross sections of HD + , D2+ , and DT + in the ground state vibrational
levels vi+ = 0 and vi+ = 1 with electrons of energies between 2 and 12 eV. The energies are measured from the
initial vibrational level vi+ of the electronic ground state
of the hydrogen molecular cations.
Some of the input quantities of the MQDT computations:
the potential energy curves of the electronic states and the
dissociative states, the electronic couplings and the quantum defects for the Rydberg states built on the 1sσg ionic
core are the same data as for the previous investigations
regarding DR of hydrogen molecular ions [13, 27]. The
other input data: the electronic couplings characterizing
the interactions between the ionization channels associated with the two ionic cores and the quantum defects at
the 2 Σu threshold are extracted from the numerical results
reported by Tennyson [37].
The present computations were performed with seven
1,3
doubly-excited dissociative states 1 Σ+
Πg ,1,3 Πu ,1,3 Σu .
g,
The Rydberg states with the 1sσg ground ion core,
involved in this calculation are (1sσg )(εsσg ) and
(1sσg )(εdσg ). The angular momentum quatum number l
of the external electron implicated in the autoionization
into the antibonding state 2 Σ+
u has two values: 1 and 3
1,3
for 1 Σ+
Πg states, 0 and 2 for 1,3 Σu states, and 2 and
g and
4 for 1,3 Πu states. The discretized levels situated below
the total energy are taken into account in the computation. The energy limit EN in (5) is taken in such a manner
that the values of the DR and DE cross sections remain
unchanged. The numerical results reported here are obtained for EN = 13.26 eV and the number of discretized
levels of 200, when the results are almost convergent.
Fig. 2 shows the comparison between our numerical results for σDE /(σDE + σDR ), the ratio between the dissociative excitation cross section and the dissociative excitation cross section plus dissociative recombination cross
section (which illustrates very well the competition of the
two destruction processes of molecular cations, the dissociative excitation and the dissociative recombination) for
HD + initially in the vibrational level vi+ = 0 and those
carried out by Andersen et al. [2]. In a large range of the
electron energy, the comparison is good. We note that
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Figure 3.
(a) DE cross sections of HD + in the vibrational level
vi+ = 0. Solid line: DE process, broken line: direct DE
mechanism, chain line: indirect DE mechanism. (b) same
as (a) but for D2+ . (c) same as (a) but for DT + .
Figure 4.
Same as Fig. 3 but for vi+ = 1.
DE process dominates the DR process above 4 eV. The
broad peak located at approximatively 6 eV is due to the
indirect DE mechanism (see Eq. (2)) while the next growth
of the value of the ratio σDE /(σDE + σDR ) is owing to the
direct DE mechanism (see Eq. (3)) involving the repulsive
state 2 Σ+
u.
In Fig. 3 we have plotted the DE cross section as a function of the electron energy for HD + in panel (a), D2+ in
panel (b), and DT + in panel (c). The DE cross sections
for all the above mentioned molecular cations are obtained
using the same approach and the same electronic data. In
Fig. 3, all the hydrogen molecular cations are considered
in the vibrational level vi+ = 0. The other case investigated in this work, vi+ = 1 is shown in Fig. 4. In order to illustrate the contributions of the two DE reaction
channels we have depicted them here. Hence, the broken line represents the indirect DE mechanism while the
chain line the direct DE mechanism. Since, the excitation
energy (the energy difference between the energy of the
repulsive state and that of the initial vibrational level at
the outer classical turning point) and the DE threshold
(the energy difference between the dissociation threshold
of molecular cation and the energy of the initial vibrational level) are expressed with respect to the energy of
the initial vibrational level, we expect their dependence
on the initial vibrational quantum number vi+ . The energies of the vibrational levels of the three cations being
different we wait for the isotopic effects on DE threshold.
We illustrate these facts presenting the values of the incident electron energy for which the DE process opens
(DE threshold) and of the excitation energy for the three
cations investigated in this work. So, the DE threshold
is ≈ 2.67 eV for vi+ = 0, ≈ 2.43 eV for vi+ = 1 for HD +
cation, ≈ 2.69 eV for vi+ = 0, ≈ 2.49 eV for vi+ = 1 for
D2+ cation, and ≈ 2.7 eV for vi+ = 0, ≈ 2.52 eV for vi+ = 1
for DT + cation. The direct DE mechanism starts to work
at ≈ 9 eV for vi+ = 0, ≈ 7.4 eV for vi+ = 1 for HD +
cation, at ≈ 9.5 eV for vi+ = 0, ≈ 7.8 eV for vi+ = 1 for
D2+ cation, and at ≈ 9.6 eV for vi+ = 0, ≈ 8 eV for vi+ = 1
for DT + cation. We note the increasing of the magnitude of the DE cross section for vi+ = 1. We expect such
behavior because the Frank-Condon factor for vi+ = 1 is
larger than that for vi+ = 0. From the panels of Figs. 3
and 4 we stress a significant isotopic effect on DE cross
section for the colliding electron energies below the direct DE threshold. It can be explained by the cumulative
contributions of the dissociative Rydberg states with the
2pσu core (denoted Q1 states in [1]) on DE cross sections
through indirect mechanism depicted by the Eq. (2) and
by the panel (a) of Fig. 1.
4.
Concluding remarks
In summary, we report theoretical cross sections for the
dissociative excitation of HD + , D2+ , and DT + in their electronic ground state vi+ = 0 or 1 for the incident electron
energy range 2 − 12 eV. In the framework of the MultiChannel Quantum Defect Theory, our computations were
performed in the second order of Born expansion of Kmatrix, neglecting rotational interactions. The comparison between our theoretical results and the experimental
ones performed by Andersen et al. [2], for HD + in the
vibrational level vi+ = 0 reveals a good agreement. Significant isotopic effect on DE cross section is emphasized
for the colliding electron energies below the direct DE
threshold.
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Acknowledgments
The authors acknowledge the financial support of UEFISCSU through the grant ID 967 under the Contract
143/2007. They are grateful to I F Schneider for useful
discussions.
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