Photodissociation and radiative association of HeH+ in the metastable triplet state

0.3 MB·English

by  J. Loreau

Photodissociation and radiative association of HeH+ in the metastable triplet state
J. Loreau,1,2 S. Vranckx,2,3 M. Desouter-Lecomte,3 N. Vaeck,2 and A. Dalgarno1
1ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA
2Service de Chimie Quantique et Photophysique,
Universit´e Libre de Bruxelles (ULB) CP 160/09, 1050 Brussels, Belgium
3Laboratoire de Chimie Physique, UMR8000, Universit´e de Paris-Sud, Orsay 91405, France.
Weinvestigatethephotodissociation ofthemetastabletripletstateofHeH+ aswellasitsforma-
tion through the inverse process, radiative association. In models of astrophysical plasmas, HeH+
3 is assumed to be present only in the ground state, and the influence of the triplet state has not
1 been explored. It may be formed by radiative association during collisions between a proton and
0
metastablehelium,whicharepresentinsignificant concentrationsinnebulae. Thetriplet statecan
2 alsobeformedbyassociationofHe+andH,althoughthisprocessislesslikelytooccur. Wecompute

n thecrosssectionsandratecoefficientscorrespondingtothephotodissociation ofthetripletstateby
a UV photons from a central star using a wave packet method. We show that the photodissociation
J cross sections depend strongly on the initial vibrational state and that the effects of excited elec-

4 tronic states and non-adiabatic couplings cannot be neglected. We then calculate the cross section
1 and rate coefficient for theradiative association of HeH+ in the metastable triplet state.

]
h I. INTRODUCTION metastable triplet state, a 3Σ+. This state, which corre-
p
lates asymptotically to He+(1s) + H(1s), can indeed be
-
m The helium hydride ion HeH+ is one of the most ele- populated and will not decay by collisions if the plasma
densityis low. As its radiativedecayto the groundstate
e mentarymolecularionsandthe firstto forminthe early
h universe[1],throughthedirectradiativeassociationpro- is spin-forbidden, the state is metastable with a lifetime
c cess [2] of 150 s [15]. Moreover, in experimental studies on the
. dissociative recombination of HeH+ [16, 17], it was sug-
s
c He(1s2)+H+ HeH+(X 1Σ+)+hν , (1) gested that the a 3Σ+ state might be present in the ion
si −→ beam and be responsible for a part of the cross section.
y which occurs in the ground state of the molecular ion. In this work, we study the main mechanisms control-
h In addition, HeH+ can also be formed by a spontaneous ling the abundance of HeH+ in the a 3Σ+ state. The
p
[ radiative transition from the vibrational continuum of firstreactionof interestis the photodissociationprocess,
the excited A 1Σ+ state to a discrete vibrational levelof which destroys the molecular ion:

1
v the ground state [3, 4]: HeH+(a 3Σ+)+hν (HeH+)∗ He(1snl 3L)+H+
−→ −→
74 He+(1s)+H(1s) HeH+(X 1Σ+)+hν (2) −→He+(1s)+H(nl)
−→
(3)
0
3 These mechanisms of production of HeH+ were inves-
The photodissociation can occur following excitation to
. tigated along with destruction processesin order to esti-
1 3Σ+or3Πelectronicstatesandleadstoatomicfragments
0 matetheabundanceofthemolecularionintheearlyuni- He + H+ or He+ + H. We investigate this reaction with
3 verseaswellasinvariousastrophysicalenvironments. In
a time-dependent method based on the propagationof a
1 particular,HeH+ waspredictedtobeobservableinplan-
wave packet on the excited electronic states coupled by
: etary and gaseous nebulae such as NGC 7027 [2, 5–7].
v non-adiabatic interactions that gives access to the con-
i However, despite searches for vibrational or rotational
X lines, no emission from HeH+ has been detected from tribution of each excited state to the cross section.
Basedon the results on the photodissociationprocess,
r these objects so far, although it has been observed in
a we examine the radiative association of HeH+ in the
laboratory plasmas for many years [8, 9]. It was shown
a 3Σ+ state, which is one of the main mechanism of for-
that the observation of the J = 1 0 rotational line
of HeH+ is hindered by the near-coin−cidence of this line mation of this state. It occurs through a spontaneous
radiative transition from the vibrational continuum of
with a CH rotational line that has a greater intensity
the excited b 3Σ+ state to a discrete vibrational level of
[10]. Other possibilities to detect HeH+ include the po-
the metastable a 3Σ+ state:
tential presence of the ion in helium-rich white dwarfs
[11, 12], metal-poor stars [13], or in supernovae [14], but He(1s2s 3S)+H+ HeH+(a 3Σ+)+hν (4)
the molecular ion has so far eluded observation. −→
The formation of HeH+ is mainly due to the radiative The b 3Σ+ molecular state corresponds to collisions be-
association between He and H+ or between He+ and H tween H+ and He in its metastable 1s2s3S state, which
(processes (1) and (2)). While it has always been sup- has a lifetime of about 8000 s [18]. Since metastable he-
posed that HeH+ is formed in its ground X 1Σ+ state, lium is easily produced in anion source,the reaction(4)
one should also consider the possible role of the first could lead to the presence of HeH+ in the a 3Σ+ state
2
in the source. Since the b 3Σ+ state has a large poten-
tial well (0.73 eV), it could also be produced [19], and 3Σ+ states
its radiative decay would result in populating the a 3Σ+ -1.8 3Π states
stateaswell,aprocessthatwasinvestigatedbyChibisov
et al. [20]. Metastable helium is also present in various
astrophysical environments [21, 22], and in particular in
planetarynebulae [23, 24], mostly due to the recombina- -1.9
tionofHe+ withfreeelectrons. Theradiativeassociation
process (4) might therefore lead to the presence in space
ofHeH+ in its triplet state. HeH+ canalsobe formedin
-2
the a 3Σ+ state by a transition from the continuum to a
bound ro-vibrational state of the same potential energy
n=3
curve. However, the rate coefficient for this process is
)
extremely small [4]. u. -2.1
a.
We start in Sec. II by summarizing previous results (
of the molecular structure of HeH+ as well as the theory gy n=2
r
e
of the photodissociation and radiative association pro- En -2.2 H+ + He(1s2s 3S)
cesses. We present the photodissociation and radiative b 3Σ+
associationcrosssectionsandratecoefficientsinSec. III,
and we discuss possible applications of the a 3Σ+ state.
-2.3
II. THEORETICAL METHODS
A. Molecular data -2.4 a 3Σ+
We consider here all the 3Σ+ and 3Π electronic states H(1s) + He+(1s)
dissociatingintoatomicstateswithn=1 3,wherenis -2.5
−
thelargestprincipalquantumnumberoftheatomicfrag-
ments. Thereisonlyonen=13Σ+state,whilethereare 0 5 10 15 20
4 3Σ+ and 2 3Π n=2 states, as well as 6 3Σ+ and 4 3Π Internuclear distance (a.u.)
n=3 states. This makes a total of 11 3Σ+ states and 6
3Π states. The adiabatic potential energy curves (PEC) FIG.1. AdiabaticPECofthen=1−3tripletstatesofHeH+
for these triplet states are shown in Fig. 1. The low- fromRef. [25]. Black,3Σ+ states;Red,3Πstates. Thea3Σ+
energy electronic spectrum of the molecular ion consists and b 3Σ+ states are shown with theirdissociation limits.
of states dissociating either into H + He+ or into H+ +
He. For example, the lowest triplet state (a 3Σ+) disso-
ciatesinto H(1s)+He+(1s),while the firstexcitedstate tween adjacent states, Fm,m±1. Eq. (5) was solved nu-
(b 3Σ+) dissociates into H+ + He(1s2s3S). We adopted merically by imposing the initial condition D( ) = I.
∞
the molecular data presented in Ref. [25]. The potential The diabatic potentialenergy curvesarethe diagonalel-
energy curves, non-adiabatic couplings and dipole tran- ements of the matrix Ud = D−1 U D, where U is the
· ·
sition moments were calculated at the complete active matrixofHelintheadiabaticrepresentation,andtheoff-
space self-consistent field (CASSCF) and configuration diagonal elements of Ud are the diabatic couplings. On
interaction(CI)levels. Sincetheexcitedelectronicstates theotherhand,weneglectedthenon-adiabaticrotational
undergo avoided crossings (see Fig. 1), the radial non- couplings,astheir effectisexpectedtobenegligible[30].
adiabatic couplings must be taken into account. Due to
thelargenumberofsuchcouplings,wedonotshowthem
hereandreferthereadertoRefs. [25,26]wheretheyare B. Photodissociation cross sections and rates
discussed. The matrix of the radial non-adiabatic cou-
plings in the basis of the adiabatic electronic functions
The photodissociation cross section from a given rovi-
ζm , Fmm′ = ζm ∂R ζm′ , can be used to build the brational state v,J is computed with a time-dependent
{ } h | | i
diabatic representation [27]. The adiabatic-to-diabatic
wavepacket method that has been described in detail in
transformation matrix D(R) is found by solving the dif-
Refs. [28, 29], so we only briefly outline its main fea-
ferential matrix equation
tures. The first step is to calculate the rovibrationalen-
∂ D(R)+F(R) D(R)=0 (5) ergiesEvJ andwavefunctionsψvJ(R)ofthea3Σ+ state,
R
· whichisachievedusingaB-splinemethod. Thepotential
As in previous studies involving the excited states of supportssixvibrationalstatesforJ =0,aswasreported
HeH+ [28, 29], we have only retained the couplings be- in Ref. [15]. The ground vibrational state is bound by
3
664.8 cm−1 while the v = 5 state is bound by less than as the wavefunction for this weakly bound levelextends
2 cm−1, and there is a total of 51 bound rovibrational to large internuclear distances.
states.
Theratecoefficientforthe photodissociationfollowing
The wave packet at t = 0, Φf (R,t =0) is defined on
vJ the absorption of a photon emitted by a central star of
each excited state f as the product of the wave function
radius R with blackbody temperature T at a radial
⋆ ⋆
of the rovibrational state by the dipole matrix element
distance R is given by [2]
µ (R)= ζ dζ betweentheinitialstateiandthefinal
if i f
h | | i
state f:
2π R 2 E2
⋆ ph
ΦfvJ(R,t=0)=µif(R)ψvJ(R) . (6) k(T⋆)= h3c2(cid:18) R (cid:19) Z eEph/kBT⋆ 1σ(Eph) dEph ,
−
(9)
In this case, the initial state i is the a 3Σ+ state while Wehavechosenthevalue(R /R)2 =10−13,assuggested
⋆
the final state f can be any excited 3Σ+ or 3Π state.
in Ref. [2] .
The transition to Σ states occurs for a parallel orien-
tation of the field with respect to the molecular axis,
while the transition to Π states occurs for a perpendic-
ular orientation. The wave packet subsequently evolves
C. Radiative association
on the coupled potential energy curves, and the time-
propagationisperformedwiththesplit-operatormethod
in the diabatic representation [31, 32]. The total pho- The cross section for the radiative association process
todissociation cross section can then be computed from canbe obtainedfromthe photodissociationcrosssection
the Fourier transform of the autocorrelation function using the detailed balance principle. The radiative asso-
C(t) = Φf (R,0)Φf (R,t) [33]. The propagated ciationcrosssectionfor the formationofthe a 3Σ+ state
fh vJ | vJ i
wavepacPketthereforecontainstheinformationaboutthe into a level v,J along the potential of any excited state
cross section for all energies. can be calculated as [35]
However,inthisworkwearenotinterestedinthetotal
cross section but rather in the partial photodissociation
E2
cross sections for each of the excited electronic states. σa (E )= ph σd (E ) . (10)
We wish to understand which states form the dominant vJ k µc2Ek vJ ph
contribution to the cross section, and the partial cross
sections are also necessary if one wants to compute the In this equation, σd (E ) is the partial cross section
vJ ph
radiativeassociationcrosssection(seebelow). Thecom- corresponding to the photodissociation of the v,J level
putation of the partial cross sections is realized using of the a 3Σ+ state via an excited state as a function of
a method based on the Fourier transform of the wave the photon energy,while σa (E ) is the cross section for
vJ k
packet at a point R∞ located in the asymptotic region radiative association into the v,J level as a function of
[34]. The partialcrosssectionforeachfinalexcitedstate the relative kinetic energy of the incident particles. It is
f is given by related to the photon energy by
σf (E )= 4π2αa20kfE Af 2 , (7)
vJ ph µ ph vJ Ek =Eph ∆E , (11)
(cid:12) (cid:12) −
(cid:12) (cid:12)
where µ is the reduced mass of the system, E = hν
ph where ∆E is the difference between the dissociation en-
isthe photonenergy,k = 2µ(E +E Ef )isthe ergyoftheelectronicstatealongwhichthecollisiontakes
f vJ ph as
wavenumberintheelectronqicstatef withan−asymptotic place and the energy of the rovibrational level of the
energy Ef , and a 3Σ+ state in which the association is realized. As
as
the association can occur into any bound rovibrational
1 level, the cross section for radiative association must be
Af = Φf (R∞,t)ei(EvJ+Eph)tdt . (8)
vJ √2π Z vJ summed over all possible values of v and J. It should
be noted that, by construction, the radiative association
The calculations were performed using a spatial grid crosssectionfromaparticularelectronicstatecalculated
of 213 points from 0.5 to 100au. The point at which the using Eq. (10) includes non-adiabatic effects.
Fouriertransform(8)isevaluatedwaschosenasR∞ =75
The radiative association rate constant is calculated
au. To avoid unphysical reflexions of the wave packet at
assuming a Maxwell-Boltzmann distribution of incident
the end of the grid, an absorbing potential starting at
particles,
R = 80 au was introduced. The time step used in the
propagation was 1 au, and tests with shorter steps were
∞
performedtoassesstheconvergenceofthecrosssections. k(T)= 2 3/2 1 E e−Ek/kBTσ(E ) dE .
k k k
For the highest vibrationallevel, v =5, we extended the (cid:16)kBT(cid:17) √πµZ0
grid to 150 au and took R∞ =100 au. This is necessary (12)
4
III. RESULTS AND DISCUSSION 0.7
v=0, 3Σ+
v=0, 3Π
A. Photodissociation 0.6 v=0, total
2)
m 0.5
Thephotodissociationcrosssectionfortheinitialrovi- 6 c
1
brational level v = 0,J = 0 is presented in Fig. 2. The -10 0.4
total cross section, as well as the contributions of the n (
o
3Σ+ and 3Π states, are shown. In the case of dissocia- cti 0.3
e
tion through 3Σ+ states, the cross section is composed s s
of two peaks centered around 9.5 eV and 11.6 eV. The os 0.2
Cr
first peak is due to the b state and the second to the
0.1
three remaining n=2 states. The smallstructure in the
range 15 17 eV is due to the n = 3 states. We note
− 0
that the cross section starts abruptly at 8.92 eV, which 8 10 12 14 16 18
corresponds to the energy difference between the initial Photon energy (eV)
state (i.e., the v = 0 level of the a 3Σ+ state) and the
dissociation energy of the b 3Σ+ state. The b 3Σ+ state FIG. 2. Photodissociation cross section for the initial state
v = 0,J = 0. Blue dashed lines: contribution of the 3Σ+
hasadeeppotentialwellofabout0.73eV,ascanbeseen
states. Reddottedlines: contributionofthe3Πstates. Black
from Fig. 1, with an equilibrium position R = 7.75a
e 0 full line: total cross section.
[25]. Acomponentoftheinitialwavepacket(6)willhave
an energy below the dissociation energy of the b 3Σ+
state, and this component will oscillate in the potential 1 v=1, 3Σ+
well. Therefore, it never reaches the asymptotic region 0.9 v=1, 3Π
and makes no contribution to the cross section of the v=1, total
0.8
b 3Σ+ state. On the other hand, the component of the 2)
m
wave packet with sufficient energy will traveltowardthe 6 c 0.7
1
asymptoticregion,whichexplainsthethresholdbehavior -0 0.6
1
of the cross section. As in the case of the photodissocia- n ( 0.5
o
tion from the ground X1Σ+ state of HeH+ [28], we ob- ecti 0.4
soenrvtehde aphsottroondgissioncfliuaetinocnecorfostshesencotino-nasd,iadbuaetitcoctohuepllainrgges oss s 0.3
Cr
number of avoided crossings affecting the excited states 0.2
PEC (see Fig. 1). 0.1
Thecrosssectionfordissociationvia3Πstatesisdom-
0
inatedbyanarrowpeakcenteredat10.3eVandawider 8 10 12 14 16 18
peak around 13 eV, both due to the two n = 2 states. Photon energy (eV)
The shape of the cross section can be explained using
FIG. 3. Photodissociation cross section for the initial state
the reflection principle, which states that the cross sec-
v = 1,J = 0. Blue dashed lines: contribution of the 3Σ+
tion reflects the spatial distribution of the wave function
states. Reddottedlines: contributionofthe3Πstates. Black
of the initial state, and that its width is proportional to
full line: total cross section.
the steepness of the potential energy curves in the ex-
citation region [36]. As can be seen in Fig. 1, in the
excitation region the PEC of the lowest 3Π state is al- b 3Σ+ state. We reach similar conclusions for the higher
most flat, leading to a very narrow cross section, while vibrational levels v 2. On the other hand, the cross
thatofthesecond3Πstateissteeper,resultinginawider section is almost ind≥ependent of J and is insensitive to
cross section. The diffuse structure in the cross section rotationalexcitationastheinitialwavefunctiondepends
between14and17eVisduetothen=3states. Asthese only weakly on J.
statesinteractstronglythroughnon-adiabaticcouplings, The photodissociation rate coefficient, given by Eq.
thecrosssectionismuchlessstructuredthanforthetwo (9), is presented in Tab. I as a function of the star tem-
n=2 states. perature T and for three different value of the matter
⋆
The photodissociation cross section is strongly depen- temperature T . The matter temperature T governs
m m
dentonthe initialvibrationallevel. This isillustratedin therovibrationaldistributionoftheinitialstateofHeH+.
Fig. 3, which shows the cross sectionfor the initial state Assuming equilibrium, the population of each state is
v = 1,J = 0. It has a more complicated structure than given by
for the v = 0 case, reflecting the oscillatory behavior of
the initial wave function, but we still observe the domi- p (T)=(2J +1)exp( E /k T )/Z(T ) (13)
vJ vJ B m m
nance of the 3Π states as well as a threshold behavior at −
anenergycorrespondingtothedissociationenergyofthe where k is the the Boltzmann constant and Z(T ) is
B m
5
the partition function. The photodissociation cross sec-
tionsandratecoefficientsmusttakethisdependenceinto
10-20
account. WegiveinTab. Ithecontributionoftheb3Σ+
state to the rate coefficient (denoted by k ), as well as
the total contribution of the 3Σ+ and 3Π stbates (kΣ and 2m) 10-22
tkhΠa,treevspenecatitvleolwy)taenmdpethraetutoret,alitraisteno(tktsout)ffi.ciWenetotbosceorvne- on (c 10-24
sider only the b 3Σ+ state to obtain an accurate value ecti
s
of the photodissociation rate coefficient. Other excited ss 10-26
electronic states, and the 3Π states in particular, must Cro
also be taken into account. For example, at T = 104 K
the contribution of the b 3Σ+ state to the tota⋆l rate co- 10-28
efficient is less than 50% while at T =5 104 K it does
⋆
not exceed 20%. This is in stark contra×st to the pho- 10-30
todissociation from the ground state of HeH+, in which 10-3 10-2 10-1 100 101
Energy (eV)
case the rate is dominated by dissociation into the first
excited state [37]. FIG. 4. Radiative association cross section for the radiative
association along theb 3Σ+ state, process (4).
B. Radiative association
10
Theradiativeassociationcrosssectionalongtheb3Σ+
state, calculated using Eq. (10), is presented in Fig. 4 -1s) 8
3
as a function of the collision energy. The cross section is m
c
dominatedbytransitionstothelevelswithv =0−2,and -160 6
itisseveralordersofmagnitudelargerthanthecrosssec- 1
tion for radiative association along the a 3Σ+ state that nt (
e
was studied Kraemer et al. [4]. In that case, the as- effici 4
sociation takes place via a radiative transition from the o
c
continuum to a bound level of the same potential, lead- ate 2
R
ing to a small cross section. The cross section shown
in Fig. 4 decreases sharply for energies over 2.77 eV,
which corresponds to the threshold energy above which 0
10 100 1000 10000
the transition probability from the b 3Σ+ state to the
Temperature (K)
a 3Σ+ state becomes strongly suppressed. It should be
notedthatthecrosssectionpresentedinFig. 4isonlythe FIG. 5. Rate constant for the radiative association process
non-resonantpartofthecrosssectionandthatthecontri- b 3Σ+ →a 3Σ++hν.
bution from shape resonances due to the centrifugal po-
tential,aswellasresonancescausedbythenon-adiabatic
interactions,should also be included. Unfortunately, the the rate coefficient for association in the ground state
accurate determination of these resonances requires im- [3, 4]. However,the value of the rate is probably slightly
practically long propagation times [19]. Taking shape larger than the one presented in this work due to the
resonancesintoaccountresultsinanenhancementofthe contribution of shape resonances that we neglected.
radiativeassociationratecoefficientandcanbeevaluated As we have calculated the partial photodissociation
in a separate calculation using the Breit-Wigner theory cross sections into all the n = 1 3 excited states, we
−
[38, 39]. The contribution of resonances on the radiative can easily evaluate the cross section for radiative asso-
associationrate can be quite large in cold environments, ciation in a discrete level of the a 3Σ+ state following
butitdecreasesquicklywithincreasingtemperature[40]. a spontaneous transition from the continuum of any of
Inaddition,sincethea3Σ+ statehasasmallbindingen- the excited electronic states. In Fig. 6, we show the
ergyandsupportsonly51boundrovibrationalstates,we rate coefficients for the dominant channels of formation
canreasonablyexpectthenumberofshaperesonancesto of HeH+ in the a 3Σ+ state. It should be noted that
be small and their contribution to the rate coefficient to it is necessary to include the effect of the non-adiabatic
be negligible except at very low temperature, similarly couplings to obtain accurate rate coefficients, due to the
to what was recently shown for the radiative association strong interaction between the excited electronic states
of LiHe+ [41]. of HeH+. In the 3Σ+ symmetry, the radiative associa-
The rate coefficient, calculated using expression (12), tion process is dominated by formation along the b 3Σ+
is shown in Fig. 5. It presents a maximum at 10 K state. The electronic channels that give the next largest
and decreases at higher temperature. It is smaller than ratecoefficientsfortheradiativeassociationofthea3Σ+
6
TABLEI.Photodissociation ratecoefficients(s−1). Tm(K)isthemattertemperatureandT⋆(K)isthetemperatureofthestar.
kb denotes the contribution of the b 3Σ+ state to the rate coefficient, kΣ is the contribution of all the 3Σ+ states, kΠ is the
contribution of the3Π states, and ktot =kΣ+kΠ is thetotal rate.
Tm(K) T⋆(K) kb kΣ kΠ ktot
50 1×104 1.69×10−10 1.84×10−10 1.61×10−10 3.44×10−10
2×104 4.60×10−8 6.02×10−8 8.05×10−8 1.41×10−7
5×104 1.54×10−6 2.62×10−6 4.59×10−6 7.21×10−6
1×105 6.35×10−6 1.22×10−5 2.32×10−5 3.54×10−5
2×105 1.76×10−5 3.57×10−5 7.06×10−5 1.06×10−4
5×105 5.29×10−5 1.11×10−4 2.23×10−4 3.34×10−4
500 1×104 1.11×10−10 1.31×10−10 1.43×10−10 2.74×10−10
2×104 3.00×10−8 4.67×10−8 7.34×10−8 1.20×10−7
5×104 1.01×10−6 2.20×10−6 4.18×10−6 6.38×10−6
1×105 4.19×10−6 1.05×10−5 2.10×10−5 3.15×10−5
2×105 1.16×10−5 3.11×10−5 6.38×10−5 9.48×10−5
5×105 3.50×10−5 9.68×10−5 2.01×10−4 2.98×10−4
5000 1×104 9.93×10−11 1.22×10−10 1.41×10−10 2.63×10−10
2×104 2.69×10−8 4.44×10−8 7.25×10−8 1.17×10−7
5×104 9.08×10−7 2.12×10−6 4.10×10−6 6.22×10−6
1×105 3.77×10−6 1.01×10−5 2.05×10−5 3.06×10−5
2×105 1.05×10−5 3.01×10−5 6.21×10−5 9.22×10−5
5×105 3.15×10−5 9.39×10−5 1.96×10−4 2.90×10−4
state are the second and fourth n = 2 states (see Fig.
aa1nr),edwmHhuiecc+hh(sc1mosr)arle+llearHtfeo(a2rssty)h,merspeetsosptteaiccttaeilsvl.yelOtyo.nHtHheoe(w1oset2vhpeer3r,Phtaoh)ned+r,aHtthe+es 1) 10-14 He+(1s) + H(2p) b33ΣΠ 3+ Σs s+tatatetes
yHapieep(ld1rossa2rcpaht3ePalcooo)neg+ffitchHiee+nttwasontdlhoawHteeas+rt(e31Πms)us+tcahtHleas(r2(gdpe)irs,strohecasipnaettichntegivrienalyttoe) 3-nt (cms 10-15 He(1s2p 3P) + H+
e
corresponding to the transition b 3Σ+ a 3Σ+. This ci
tshheowmsotshtaetffircaideinattiwveayasosfopciraotdiouncinalgonHgeHt→h+eibn3tΣh+e ais3nΣo+t ate coeffi 10-16 He(1s2p 3P) + H+
state. R
He+(1s) + H(2s)
10-17
C. Applications 10 100 1000 10000
Temperature (K)
Metastable helium can be present in significant con-
FIG. 6. Rate constants for radiative association of HeH+ in
centration in astrophysical environments such as plan-
the a 3Σ+ state along various electronic states. Black full
etary nebulae due to recombination of He+ with elec- line: association along the b 3Σ+ state; dotted blue line: as-
trons [42], and it is conceivable that it might influence sociation along other 3Σ+ state; dashed red line: association
the abundances of various atomic and molecular species. along 3Π states.
Roberge and Dalgarno [2] already considered the role of
metastable helium on the production of HeH+ in the
ground state following collisions between He(1s2s 3S) the b 3Σ+ state, we have
andatomicormolecularhydrogenandshowedthatunder
n(HeH+)kd =n(23S)n(H+)ka (14)
some circumstances,these reactionscanbe a substantial
source of HeH+. where n(HeH+), n(23S), and n(H+) are the density of
Based on the rates presented in Sec. III, we can pro- HeH+, He(1s2s3S) and H+, respectively, while kd and
vide aroughestimate ofthe abundanceofHeH+(a 3Σ+) ka are the rates for the photodissociation and radiative
in planetary nebulae. If we assume equilibrium be- association processes. We consider here a typical neb-
tween photodissociation and radiative association along ula with parameters n(H+) = 104 cm−3, n(He+) = 103
7
cm−3, T = 5 104 K and T = 104 K [23]. A formula curves, non-adiabatic couplings and dipole moments of
⋆ e
givingthe abun×danceofmetastableheliuminnebulaeas the molecular ion HeH+ to calculate the cross section
a function of the He+ density and electron density and and rate coefficient for this process using a wave packet
temperature was derived by Clegg [24]. Using the pa- approach. Wefoundthatthephotodissociationcrosssec-
rameters above, we get a density of metastable helium tionsandratecoefficientsaredominatedbythecontribu-
n(23S) 4 10−3 cm−3. Combining with the results tionoftheexcitedstates,andinparticularthe3Πstates.
≈ ×
for the rates kd and ka given respectively in Tab. I and Thecrosssectionandratecoefficientsfortheinversepro-
Fig. 5, we get n(HeH+) = 3 10−10 cm−3. The den- cess, radiative association, were computed on the basis
×
sity of HeH+ in its triplet state is much smaller than of the photodissociation cross sections. HeH+ can be
in the ground state [7] but it can nonetheless influence formed in its triplet state by association of metastable
theabundanceofotherspeciesthroughreactions(3)and helium and a proton, a process which is likely to occur
(4). The radiative association process (4) could also be in various astrophysical environments. Based on these
of importance in the chemistry of the early universe [1]. results, we estimated the abundance of this triplet state
Neutral helium was first produced following recombina- and found it to be much lower than HeH+ in its ground
tion of He+, therefore populating the 23S state. While state.
metastableheliumwasprobablyre-ionizedduetoitslow
binding energy,it could still form HeH+ by radiative as-
sociation with H+. ACKNOWLEDGMENTS
This work was supported by the U.S. Department of
IV. CONCLUSIONS Energy and by the Communaut´e franc¸aise of Belgium
(Action de Recherche Concert´ee) and the Belgian Na-
WehaveinvestigatedthephotodissociationofHeH+ in tionalFundforScientificResearch(FRFC/IISNConven-
its metastable triplet state by means of time-dependent tion). S. Vranckx would like to thank the FRIA-FNRS
methods. We used previously reported potential energy for financial support.
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