MOLECULAR PHYSICS, 1997, VOL. 91, NO. 2, 319± 332

Theoretical calculations of the hyper® ne structure in the spectra ofH+

3 and its deuterated isotopomers

By PER JENSENFB 9, Theoretische Chemie, Bergische UniversitaÈ t-Gesamthochschule Wuppertal, D-

42097 Wuppertal, Germany

IVANA PAIDAROVA , VLADIMI R SÏ PIRKOJ. Heyrovsky Institute of Physical Chemistry, Academy of Sciences of the

Czech Republic, DolejsÏ kova 3, CZ-18223 Praha 8, Czech Republic

and STEPHEN P. A. SAUERDepartment of Physical Chemistry, H. C. é rsted Institute, University of Copenhagen,

Universitetsparken 5, DK-2100 Copenhagen é , Denmark

(Received 12 December 1996; accepted 15 January 1997)

The present paper reports a purely theoretical calculation of the hyper® ne structure in therotation± vibration spectra of H+

3 , D+3 , H2D

+ and HD+2 . In our previous publication (P.

Jensen, I. Paidarova , J. Vojtik and V. SÏ pirko, 1991, J. molec. Spectrosc., 150, 137) we havealready calculated the contributions to the hyper® ne structure resulting from the quadrupolecoupling in the deuterated isotopomers of H+

3 . Here we extend this work by considering alsothe `magnetic’ e� ects, that is spin± rotation interaction and spin± spin interaction. As the initialstep in the calculation we have computed ab initio extensive sets of values for the spin± rotationconstant tensor of the molecules H+

3 , D+3 , H2D

+ and HD+2 . These ab initio values are used,

together with a spin± spin interaction tensor derived as a function of the nuclear coordinates,and vibrational wavefunctions obtained using the rotation± vibration theory developed pre-viously by us in 1985 and 1986 for X3 and XY2 equilateral triangular molecules (V. SÏ pirko, P.Jensen, P. R. Bunker and A. CÏ ejchan, 1985, J. molec. Spectrosc. , 112, 183; P. Jensen, V. SÏ pirkoand P. R. Bunker, 1986, Ibid., 115, 269) to obtain the hyper® ne splitting parameters for thevibrational states of the molecules in question. We have calculated actual hyper® ne splittingsfor selected transitions of H2D

+ and HD+2 . A possible application of the present work is the

identi® cation of H2D+ in interstellar space by observation of its hyper® ne splitting patterns.

We discuss the feasibility of such observations and conclude that they would be very di� cultbut not completely impossible.

1. IntroductionMolecular hydrogen is by far the most abundant

chemical species in the Universe (for example [1]) andthe reaction H2 + H+

2 ® H+3 + H is strongly exothermic

(by about 1.7eV). Consequently, the H+3 molecular ion

is considered to be one of the most abundant ionicspecies in the interstellar medium. Because of itschemical properties, the H+

3 ion is also thought to playa central role (as a protonating agent) in the ion± mole-cule reaction scheme which dominates gas-phase inter-stellar chemistry [2]. Hence it is not surprising that its® rst laboratory observation [3]stimulated a tremendouse� ort (a detailed description of the history of the searchis given by Tennyson [4] in his excellent review article)to observe it also in the interstellar medium. Aftervery many failed attempts, the endeavour was ® nally

crowned with success; Geballe and Oka [5]very recentlyreported the observation of three rotation± vibrationabsorption lines in the t 2 band of H+

3 in the two mole-cular clouds GL2136 and W33A.

Because the deuterium fractionation reaction H+3 +

HD ® H2D+ + H2 is slightly exothermic (140K),

present-day models of the interstellar chemistry predictthe abundance of H2D

+ in the interstellar medium to becomparable with that of H+

3 (for example [6]). In orderto verify experimentally this result, it would clearly bedesirable to observe also the H2D

+ ion in interstellarspace. This ion has a substantial permanent dipolemoment (0.6D) and so it appears as a suitable candidatefor radio-astronomical observations in the cold inter-stellar clouds. Among the allowed rotational transitions,the lines 101 000 and 110 111 appear most likely to

0026± 8976/97 $12.00 Ñ 1997 Taylor & Francis Ltd.

be detected, and their observation has indeed beenclaimed. However, subsequent experiments aimed atverifying these observations either failed completely orat least threw substantial doubt on the initial identi® ca-tions (for more details see [4]), and consequently theunambiguous identi® cation of H2D

+ in the interstellarmedium is still an unsolved problem.

In a previous study [7] we have pointed out that, atleast in principle, the identi® cation of H2D

+ in inter-stellar space could be based on the observation of thehyper® ne structures of its allowed rotational transitions.To assess the feasibility of this approach, we calculatedin [7] the hyper® ne structures originating in the quad-rupole coupling of the deuteron for these transitions.That is, we neglected the magnetic interactions. Theactual calculations were performed using an ab initioelectric ® eld gradient surface and rotation± vibrationwavefunctions obtained using the rotation± vibrationtheory developed in [8± 11]. The calculated structureshave splittings that are considerably smaller than theDoppler widths of the lines (which are rather largeowing to the very small mass of H2D

+ and to the rela-tively high frequencies of the transitions in question).Thus the ratio-astronomical identi® cation of H2D

+

from its quadrupolar splittings would seem rather prob-lematic. However, in addition to the quadrupolar split-tings, the true spectra will also exhibit splittingsoriginating in magnetic interactions neglected in [7].The H2D

+ ion is a closed-shell species, and it isknown [12] that in the electronic ground states of suchmolecules the electronic and nuclear magnetic momentstend to cancel almost exactly, so that their contributionsto the splittings are usually smaller than the contribu-tion from quadrupole coupling. For extremely lightmolecules such as H2D

+ , however, it is conceivablethat the magnetic e� ects (spin± rotation interaction andspin± spin interaction) could be at least comparable inimportance with the quadrupole coupling. Conse-quently, we have found it worthwhile to extend ourprevious study by taking into account the nuclear quad-rupole coupling, the spin± rotation interaction and thedirect spin± spin interaction. The indirect nuclear spin±nuclear spin interaction contributes much less to thetotal energy than the dipolar interactions studied inthe present work. For the equilibrium geometry ofH2D

+ , we evaluate ab initio at the coupled cluster singlesplus doubles with perturbative triples (CCSD(T)) levelof theory that the indirect spin± spin coupling constantsare J(H± H) = 189Hz and J(H± D)= 29Hz. For the equi-librium geometry of HD+

2 , we obtain J(D± D) = 4Hz.The interaction parameters obtained in the presentwork and in [7] for the nuclear quadrupole coupling,the spin± rotation interaction and the spin± spin interac-tion are all at least two orders of magnitude larger than

these J values, so that we can safely neglect the indirectinteractions.

The H2D+ ion is a suitable candidate for radio-astro-

nomical observations, but of course the hyper® ne inter-actions are present in the spectra of all H+

3 isotopomers.The corresponding spectral patterns are ® ngerprints’of the corresponding transitions and, if observed(obviously, laboratory measurements would requiresub-Doppler resolution), the hyper® ne structure wouldpermit these lines to be assigned instantly. For thisreason we have made calculations not only for H2D

+

but also for the other spectroscopically interestingisotopomers H+

3 , D+3 and HD+

2 .

2. Spin± rotation interactionThe energy associated with the interaction between

the spin magnetic moment of a nucleus l and the mag-netic moments originating in the motion of the electronsand the other nuclei in a molecule depends on the so-called spin± rotation constant tensor M(l). This tensor isthe sum of a nuclear term M

(n)(l) and an electronic termM

(el)(l) [13]. The electronic term is related to the para-magnetic contribution to the nuclear magnetic shieldingtensor h

(p)(l) [14]. For a closed-shell molecule, the com-ponents of M(l) in a molecule-® xed principal axissystem x, y, z can be written as [15]

Ma b (l) = M(n)a b (l) + M(el)

a b (l)

=gl

2p Ib b

e¹0¹N

4p åk /= l

Zk

r3kl[

´ (r2kl d a b - [rk - rl]a [rk - rl]b ) +

e¹0¹N

4me p

´ ån /=0

k 0| å i(Lli) a /r3

il|nl k n| å i(Lli) b |0l

E0 - En(

+k 0| å i(Ll

i) b |nl k n| å i(Lli) a /r3

il|0lE0 - En

) ]=

gl

2p Ib b

e¹0¹N

4p åk /= l

Zk

r3kl

(´ (r2

kl d a b - [rk - rl]a [rk - rl]b ) +mehmp

s(p)a b (l)) ,

(1)where rk and rl are the position vectors of nuclei k and lrespectively and ri is the position vector of electron i, inthe x, y, z axis system. The quantities [rk - rl]a and[rk - rl]b are the a and b coordinates respectively( a , b = x,y,z) of the vector rk - rl, and rkl is thelength of this vector, that is the instantaneous value of

320 P. Jensen et al.

the distance between the nuclei k and l. Further, ril is theinstantaneous value of the distance between nucleus land electron i. The vector Ll

i represents the angularmoment operator of electron i de® ned with respect tothe position rl of nucleus l. The atomic number ofnucleus k is denoted Zk and Ib b is the e� ectivemoment of inertia associated with the b axis( b = x,y,z). By {|nl } we denote a complete set of elec-tronic states with energies {En}. In this set {|0l } (withenergy E0) corresponds to the electronic ground state.Finally, in equation (1), gl is the nuclear g factor ofnucleus l [16], e is the elementary charge, ¹0 is the per-meability of vacuum, ¹N is the nuclear magneton, me isthe electron mass and mp is the proton mass [17]. Fromthe magnetic moment values given in [17], we obtainthat for a proton the nuclear g factor gH = 5.585694 6whereas, for a deuteron, gD = 0.8574382.

The tensor elements M(el)a b (l) and s (p)

a b (l) are second-order properties which can be calculated elegantly by thepolarization propagator technique [18]. The ® rst-orderpolarization propagator or random-phase approxima-tion (RPA), which is equivalent to the coupled Har-tree± Fock method, and the second-order polarizationpropagator approximation (SOPPA) [19] were used intest calculations. It was found, however, that the corre-lation e� ects are very small, that is the RPA and SOPPAresults di� ered by less than 1%. The complete s

(p)a b (l)

surface was therefore evaluated at the level of theRPA only. The RPA calculations were done with theOdense version of the RPAC program package [20].The basis set derived by Oddershede and Sabin [12]for the calculation of the rotational g factor of H+

3was employed. It consists of 69 uncontracted CartesianGaussian functions, namely (7s3p1d) on each atom andone di� use s orbital placed at the centre of mass of theH+

3 molecule.The s

(p)a b (l) tensor elements were calculated for 220

geometries of H+3 . From these values, the M(el)

a b (l) prop-erty surfaces were obtained for the isotopic species H+

3 ,H2D

+ , HD+2 and D+

3 . For each nucleus l the s (p)a b (l)

tensor was, if necessary, ® rst transformed to a molecularprincipal axis system and the corresponding elements ofM(el)

a b (l) were then obtained using equation (1). The ® nalcalculated tensor elements are expressed in the molecule-® xed axis system x,y,z de® ned by the centre-of-massand Eckart conditions given by equations (2) ± (4) of [9]for a general triatomic XY2 molecule with molecularsymmetry group C2v [21] and an equilateral triangularequilibrium structure. This coordinate system is not aprincipal axis system when the XY2 molecule is dis-placed from its equilibrium con® guration. We label thetwo Y nuclei 1 and 2, and the X nucleus 3. This labellingconvention, the location of the molecule-® xed x,y,zaxes relative to the XY2 molecule in its equilibrium

con® guration, and the de® nition of the internucleardistances (r1, r2,r3), which we use as vibrational co-ordinates in the present work, are shown in ® gure 1.

The parameters of analytical functions in the nuclearcoordinates were adjusted to ® t the ab initio values forthe Ma a (l) tensor elements. We choose these functionsas expansions in the quantities yk = 1 - exp (- a D rk),k = 1,2,3, where a is a parameter determined from thepotential energy function, and D rk = rk - re is the dis-placement of rk ( ® gure 1) from its equilibrium value re:

Ma a (l) = M( a a ,l)0 + å

jM( a a ,l)

j yj + åj,k

M( a a ,l)jk yjyk

+ åj,k,m

M( a a ,l)jkm yjykym + å

j,k,m,nM( a a ,l)

jkmn yjykymyn.

(2)

In de® ning yk we use a = 1.5AÊ - 1 and re = 0.873 113AÊ ;these values are obtained from the ab initio potentialenergy surface calculated by Meyer et al. [22] (seetable I of [7]).

We obtain values of the parameters M( a a ,l)jkm... of equa-

tion (2) in a least-squares ® t to the ab initio values forMa a (l). We have chosen to ® t functions that transformaccording to irreducible representations of C2v [21]withappropriate constraints on the expansion coe� cients ofthe polynomial in equation (2). Consequently we ® t foreach of the four molecules H+

3 , D+3 , H2D

+ and HD+2 ,

and for a = x,y,z, the functions Ma a (1) + Ma a (2) andMa a (3) (both of A1 symmetry in C2v) and the functionMa a (1) - Ma a (2) (of B2 symmetry in C2v).

Some of the ab initio values correspond to very dis-placed geometries with energies very much higher thanthe equilibrium energy, and we have found it necessary

Hyper® ne e� ects in H+3 321

Figure 1. The atom-numbering convention and molecule-® xed axes for an equilateral triangular XY2 molecule.

to assign lower weights to such points in the least-squares ® tting procedure. As in [7], the relative weightswere calculated from the expression

W =1

1 + C(y21 + y2

2 + y23)2 . (3)

The constant C was determined so that the minimumweight employed in each ® t (i.e. the weight given to thepoint with the largest value of y2

1 + y22 + y2

3) was 0.01.In all the ® ttings of ab initio Ma a (l) values carried outin the present work, the ® tted functions generally repro-duce the ab initio values to within 1± 5% of the ab initiovalue. The ab initio values of Ma a (l) for the moleculesH+

3 , D+3 , H2D

+ and HD+2 , and the values of the

parameters M( a a ,l)jkm... obtained from the least-squares ® ts,

are available from the authors on request.

3. Spin± spin interactionThe energy originating in the interaction between the

spin magnetic moments of two nuclei k and l can beshown [23] to depend on the vibrational coordinatesof the molecule through a traceless tensor R(kl) withelements

Ra b (kl) =- 3[rk - rl]a [rk - rl]b + d a b r2

kl

r5kl

. (4)

In equation (4), the symbols in principle have the samemeaning as in equation (1). However, the x,y,z labelsnow refer to the molecule-® xed axes de® ned by thecentre-of-mass and Eckart conditions (® gure 1), andnot to the instantaneous’ principal axes used in equa-tion (1). With our de® nition of the vibrational coordi-nates D rj, j = 1,2,3, we have

rkl = re + D rm, (5)where klm is a permutation of the numbers 123. This canbe inserted directly in equation (4). Furthermore, we canuse equations (1) ± (4) and (14) ± (16) of [9] to obtain thecoordinates [rk - rl]a and [rk - rl]b as linear combina-tions of the symmetrized coordinates Si, i = 1,2,3,de® ned by equations (8) ± (13) of [9]. These coordinatesare expressed in terms of the D rj through equation (23)of [9]:

Si = åj

aij D rj + 1

2 åjk

aijk D rj D rk

+ 16 å

jklai

jkl D rj D rk D rl . . . . (6)

The expansion coe� cients aijk... are given to third order

in the D rj in table 1 of [9]. Finally, the D rj can beexpressed in terms of the yk:

D rj =1a å

¥

n=1

1n

ynj . (7)

In the present work, we consider only the diagonal ele-ments of the tensor R(kl). This is the approximationcustomarily made when experimental spectra are ana-lysed for hyper® ne e� ects. By using the various relationsdescribed above, we can express the diagonal tensorelements Ra a (kl) as Taylor expansions, truncated afterthe third-order terms, in the yj functions. We have usedthe computer algebra program MAPLE [24] to deriveanalytical expressions for the ® rst to third derivatives ofRa a (kl) with respect to yj, j = 1,2,3. The MAPLEcalculation produced FORTRAN code for a programcomputing numerical values of the derivatives neededfor constructing the Taylor expansions. With these deri-vatives, we can proceed to obtain vibrational averagesof the Ra a (kl) elements. In practice, we carry out thevibrational averaging of tensor elements given in SIunits [17] of energy as

Ra a (kl) = gkgl¹0¹

2N

4pRa a (kl). (8)

4. Vibrationally averaged tensor elementsThe tensor elements Ma a (l) (equation (2)) and

Ra a (kl) (equation (8)) are now averaged over the vibra-tional wavefunctions:

Ma a (l) = k V |Ma a (l)|V l (9)and

Ra a (kl) = k V |Ra a (kl)|V l . (10)The vibrational wavefunctions |V l are generated asdescribed in [7, 9]. The potential energy surface for H+

3was taken to be given by equation (26) of [7] with theparameter values from table I of [7]. These values weredetermined from the ab initio points calculated by Meyeret al. [22]. The functions |V l are given as linear combi-nations of products of Morse oscillator functions, andwhen both Ma a (l) and Ra a (kl) are given as polynomialsin the yk, the expectation values of equations (9) and(10) can be straightforwardly calculated using Morseoscillator matrix elements from [8, 11].

In calculating the vibrationally averaged values weuse the relation

Ma a (l) = 12 [Ma a (1) + Ma a (2)] 6 1

2 [Ma a (1) - Ma a (2)],(11)

where the plus sign corresponds to l = 1, and the minussign to l = 2. As mentioned above, Ma a (1) + Ma a (2) hasA1 symmetry in C2v, and Ma a (1) - Ma a (2) has B2 sym-metry. From equation (11) we obtain

322 P. Jensen et al.

Ma a (1) = Ma a (2) = 12 k V |Ma a (1) + Ma a (2)|V l . (12)

Analogous considerations are used in calculating thevibrationally averaged values of Ra a (13) and Ra a (23),where we have

Ra a (13) = Ra a (23) = 12 k V |Ra a (13) + Ra a (23)|V l .

(13)The vibrationally averaged values of Ma a (3) andRa a (12) are obtained directly from equations (9) and(10) respectively, since these functions have A1 sym-metry in C2v.

A special case arises when we consider vibrationalstates of H+

3 and D+3 with EÂ symmetry in D3h. As dis-

cussed in [7], we de® ne the two vibrational wavefunc-tions |EÂa l and |EÂb l spanning the EÂ symmetry so that|EÂa l is unchanged under the interchange (12) of nuclei 1and 2, but |EÂb l changes its sign under this operation.That is, |EÂa l has A1 symmetry in the C2v subgroup ofD3h, and |EÂb l has B2 symmetry. Consequently, we canuse equations (9), (10), (12) and (13) to obtain theexpectation values k E Âa|Ma a (l)|E Âa l , k E Âb|Ma a (l)|E Âb l ,k E Âa|Ra a (kl)|E Âa l , k E Âb|Ra a (kl)|E Âb l exactly as done forthe true’ C2v molecules H2D

+ and HD+2 . For H+

3 andD+

3 , however, we also need matrix elements between |E Âa land |E Âb l . These will be non-vanishing for Ma a (1),Ma a (2), Ra a (13) and Ra a (23) and can be obtainedfrom the equations

k E Âa|Ma a (1)|E Âb l = - k E Âa|Ma a (2)|E Âb l= 1

2 k E Âa|Ma a (1) - Ma a (2)|E Âb l (14)and

k E Âa|Ra a (13)|E Âb l = - k E Âa|Ra a (23)|E Âb l= 1

2 k E Âa|Ra a (13) - Ra a (23)|E Âb l . (15)Table 1 gives vibrationally averaged tensor elements,

and the o� -diagonal matrix elements de® ned by equa-tions (14) and (15), for the lowest vibrational states ofH+

3 and D+3 , and table 2 presents vibrationally averaged

results for H2D+ and HD+

2 .

5. Hyper® ne splittingsIn the present work, we consider the hyper® ne e� ects

in one particular rotation± vibration state characterizedby the quantum number J associated with the operator

J = F- I, (16)where F is the total angular momentum of the moleculeand I is the total nuclear spin. We assume that we canneglect the interaction between the rotation± vibrationstate under study and all other rotation± vibrationstates. We employ the following Hamiltonian fordescribing the hyper® ne e� ects:

Hhfs = HQC + HSR + HSS. (17)

In equation (17), the part HQC, which describes thequadrupole coupling, is given by equation (3) of [7]:

HQC = åN

l=1V(l)·Q(l). (18)

Here, N is the number of nuclei, Q(l) denotes thenuclear quadrupole tensor of nucleus l and V(l) denotesthe electric ® eld gradient tensor at the instantaneousposition of this nucleus. The Hamiltonian HSR describesthe interaction between the nuclear spins and the overallrotation of the molecule and is given by

HSR = åN

l=1ClIl ·J. (19)

The parameters Cl in equation (19) depend on theactual rotation± vibration state considered and Il denotesthe spin of nucleus l. The third Hamiltonian HSSaccounts for the interactions between nuclear spinsand is given by

HSS = åk<l

Skl{32 [(Ik ·J)(Il ·J) + (Il ·J)(Ik ·J)]

- (Ik ·Il)J2}. (20)

The parameters Skl of HSS depend on the rotation±vibration state considered.

The matrix elements of HQC depend on the couplingconstants eQq(l)

JJ for the rotation± vibration state in ques-tion. The calculation of these quantities is the subject of[7], and values of the coupling constants for selectedrotation± vibration states in H2D

+ and HD+2 can be

found in table V of that publication. For states notlisted in table V of [7], the quadrupole coupling constantsc (l)

xx- yy and c (l)zz from table IV of [7]can be used to obtain

vibrationally averaged values of the electric ® eld gradienttensor V(l). It holds that

eQ k V |V xx(l)|V l = 12 [c (l)

xx- yy - c (l)zz ], (21)

eQ k V |V yy(l)|V l = - 12 [c (l)

xx- yy + c (l)zz ], (22)

and

eQ k V |V zz(l)|V l = c (l)zz . (23)

In the preceding equations, eQ is the quadrupolemoment of nucleus l. When the vibrationally averagedvalues of V(l) are known, the parameter eQq(l)

JJ can beobtained from equation (3) of [23]:

eQq(l)JJ = 2eQ å

a =x,y,z

k V |V a a (l)|V l k J2a l

(J + 1)(2J + 3) . (24)

Hyper® ne e� ects in H+3 323

324 P. Jensen et al.

Tab

le1.

Vib

ratio

nally

aver

aged

tens

orel

emen

tsM

aa( l)

(equ

atio

n(9

))an

dR

aa( k

l)(e

quat

ion

(10)

)for

H+ 3

and

D+ 3

.

v 1v 2

G vib

aE

b /cm

-1M

xx( 1

) /kH

zM

yy( 1

) /kH

zM

zz( 1

) /kH

zM

xx( 3

) /kH

zM

yy( 3

) /kH

zM

zz( 3

) /kH

zR

xx( 1

2)/k

Hz

Ryy

( 12)

/kH

zR

zz( 1

2)/k

Hz

Rxx

( 13)

/kH

zR

yy( 1

3)/k

Hz

Rzz

( 13)

/kH

z

H+ 3

00

AÂ1

0.0

62. 6

74. 2

62. 0

78. 7

58. 1

62. 0

-342

. 216

9.4

172.

841

. 5-2

14. 3

172.

80

1EÂa

2524

. 260

. 571

. 356

. 079

. 657

. 059

. 8-3

22. 5

159.

816

2.6

17. 4

-186

. 716

9.3

EÂb

59. 9

75. 1

58. 5

75. 0

55. 1

54. 7

-332

. 416

0.9

171.

560

. 4-2

24. 3

164.

9EÂa/

EÂb

-1. 6

-1. 0

-2. 2

-11.

315

. 0-3

. 71

0AÂ1

3177

. 961

. 473

. 060

. 877

. 756

. 760

. 9-3

33. 1

164.

916

8.2

40. 4

-208

. 616

8.2

02

AÂ1

4801

. 657

. 773

. 053

. 477

. 253

. 653

. 4-3

12. 5

150.

316

2.2

34. 6

-196

. 816

2.2

02

EÂa

5006

. 958

. 570

. 752

. 576

. 654

. 253

. 4-3

10. 9

151.

815

9.1

30. 7

-192

. 916

2.3

EÂb

57. 0

75. 1

53. 0

74. 7

54. 0

52. 3

-316

. 515

3.2

163.

341

. 3-2

01. 4

160.

1EÂa/

EÂb

-1. 0

0.0

-0. 4

-2. 8

4.7

-1. 9

D+ 3

00

AÂ1

0.0

4.9

5.7

4.9

6.1

4.5

4.9

-8. 2

4.0

4.2

1.0

-5. 1

4.1

01

EÂa

1834

. 54.

85.

64.

56.

14.

54.

8-7

. 93.

94.

00.

6-4

. 64.

0EÂb

4.7

5.8

4.7

5.9

4.3

4.5

-8. 0

3.9

4.1

1.3

-5. 3

4.0

EÂa/

EÂb

-0. 1

-0. 1

-0. 1

-0. 2

0.3

-0. 1

10

AÂ1

2300

. 54.

85.

74.

86.

04.

54.

8-8

. 04.

04.

01.

0-5

. 04.

00

2AÂ1

3536

. 04.

65.

74.

46.

04.

34.

4-7

. 73.

74.

00.

9-4

. 83.

90

2EÂa

3651

. 34.

65.

64.

36.

04.

34.

4-7

. 73.

83.

90.

8-4

. 84.

0EÂb

4.6

5.7

4.4

5.9

4.3

4.3

-7. 7

3.8

3.9

1.0

-4. 9

3.0

EÂa/

EÂb

0.0

0.0

0.0

0.0

0.0

0.0

11

EÂa

4059

. 34.

75.

54.

56.

14.

44.

7-7

. 73.

83.

90.

6-4

. 64.

0EÂb

4.7

5.7

4.6

5.9

4.3

4.4

-7. 6

3.6

4.0

1.3

-5. 2

3.9

EÂa/

EÂb

-0. 1

-0. 1

-0. 1

-0. 2

0.2

0.0

20

AÂ1

4554

.24.

75.

64.

86.

04.

44.

8-7

. 93.

94.

01.

0-4

. 93.

9

aV

ibra

tiona

lsym

met

ryin

D3h

.For

stat

esw

ithEÂs

ymm

etry

,the

expe

ctat

ion

valu

esin

the

com

pone

ntst

ates

|EÂal

and|E

Âblar

ela

belle

dEÂaa

ndEÂb,

and

the

o�-d

iago

nalm

atri

xel

emen

tgiv

enby

equa

tions

(13)

and

(14)

isla

belle

dEÂa/

EÂb.

bC

alcu

late

dvi

brat

iona

lene

rgy.

Hyper® ne e� ects in H+3 325

Tab

le2.

Vib

ratio

nally

aver

aged

tens

orel

emen

tsM

aa( l)

(equ

atio

n(9

))an

dR

aa( k

l)(e

quat

ion

(10)

)for

H2D

+an

dH

D+ 2

.

v 1v 2

v 3G v

iba

Eb /c

m-1

Mxx

( 1) /k

Hz

Myy

( 1) /k

Hz

Mzz( 1

) /kH

zM

xx( 3

) /kH

zM

yy( 3

) /kH

zM

zz( 3

) /kH

zR

xx( 1

2)/k

Hz

Ryy

( 12)

/kH

zR

zz( 1

2)/k

Hz

Rxx

( 13)

/kH

zR

yy( 1

3)/k

Hz

Rzz

( 13)

/kH

z

H2D

+

00

0A

10.

042

. 173

. 949

. 98.

18.

97.

8-3

40. 6

168.

117

2.5

6.1

-32.

826

. 70

10

A1

2208

. 241

. 071

. 746

. 48.

28.

77.

6-3

26. 6

161.

116

5.5

3.4

-29.

526

. 10

01

B2

2335

. 740

. 374

. 146

. 87.

98.

46.

9-3

26. 4

156.

317

0.2

8.2

-33.

925

. 71

00

A1

2992

. 341

. 072

. 348

. 48.

18.

77.

7-3

27. 6

162.

016

5.6

5.1

-31.

326

. 20

20

A1

4301

. 739

. 570

. 743

. 48.

28.

57.

4-3

11. 8

151.

716

0.1

2.1

-27.

625

. 50

11

B2

4466

. 638

. 872

. 843

. 77.

88.

36.

7-3

15. 0

150.

316

4.7

6.1

-31.

225

. 10

02

A1

4606

. 939

. 372

. 943

. 77.

88.

06.

3-3

14. 2

148.

116

6.1

8.5

-33.

324

. 8

HD

+ 20

00

A1

0.0

8.1

5.8

6.1

65. 4

29. 6

39. 1

-8. 2

4.1

4.1

6.8

-33.

526

. 70

10

A1

1967

. 97.

95.

65.

766

. 029

. 237

. 8-7

. 83.

93.

94.

2-3

0.5

26. 3

00

1B

220

80. 1

7.8

5.9

5.9

62. 0

28. 8

35. 5

-8. 2

4.1

4.1

9.5

-35.

125

. 61

00

A1

2736

. 07.

95.

75.

964

. 528

. 938

. 0-8

. 14.

04.

16.

4-3

2.3

25. 9

02

0A

138

24. 5

7.7

5.5

5.3

66. 4

28. 8

36. 4

-7. 5

3.7

3.8

2.0

-28.

026

. 00

02

A1

4052

. 47.

55.

95.

760

. 327

. 832

. 7-8

. 03.

94.

111

. 2-3

5.8

24. 6

01

1B

240

63. 7

7.5

5.7

5.9

62. 3

28. 7

34. 3

-7. 8

3.8

4.0

7.3

-32.

525

. 21

10

A1

4649

. 97.

25.

55.

565

. 128

. 636

. 5-7

. 73.

83.

93.

8-2

9.4

25. 6

10

1B

246

76. 3

7.6

5.8

5.7

60. 7

28. 0

34. 2

-8. 0

3.9

4.1

9.3

-34.

024

. 7

aV

ibra

tiona

lsym

met

ryin

C2v

.bC

alcu

late

dvi

brat

iona

lene

rgy.

The quantities k J2a l are the average values of the square

of the components of the angular momentum J alongthe molecule-® xed axes.

The parameter Cl in equation (19) is obtained fromequation (17) of [23]:

Cl = åa =x,y,z

Ma a (l) k J2a l

J(J + 1) . (25)

Finally, the parameter Skl in equation (20) is obtainedfrom equations (9) and (10) of [23]:

Skl = 2 åa =x,y,z

Ra a (kl) k J2a l

(J + 1)(2J + 3)J(2J - 1) . (26)

Thaddeus et al. [23] construct the matrix representa-tion of the Hamiltonian Hhfs in a basis set |I1 . . . IN;JF1F2 . . . FN- 1Fl obtained by the following couplingscheme:

J + I1 = F1,F1 + I2 = F2,

..

.

FN- 1 + IN = F.

(27)

Initially, we wrote a program to construct the matrixrepresentation of Hhfs for three nuclei in the basis|I1I2I3; JF1F2Fl and to diagonalize this matrix. Theappropriate matrix elements were obtained from equa-tions (22) ± (24) of [23], which can be rather straightfor-wardly obtained from angular momentum theory, forexample by use of equations (5.71) ± (5.73) of [25].There is a fairly obvious misprint in equation (24) of[23], where the factor 2IL - 1 in the second denominatorof the second line should be 2IL - 1. The programimplementing the matrix elements of [23] indeed calcu-lated the correct hyper® ne energies. However, the trans-formation properties of the basis functions |I1I2I3;JF1F2Fl under the interchange (12) of the two identicalY nuclei (labelled 1 and 2) of the XY2 molecule are noteasily determined, and so we were faced with the prob-lem that we could not decide which of the calculatedlevels were allowed by the Pauli exclusion principle.An alternative basis set, whose symmetry propertiesunder (12) are well known, consists of the functions|I1I2I3; JI12F2Fl obtained from the coupling scheme

I1 + I2 = I12,J + I12 = F2, (28)

F2 + I3 = F.

The quantum number I12 is integral by necessity. If theY nuclei are protons, we have I1 = I2 = 1

2. In this case,functions |I1I2I3; JI12F2Fl with I12 even (i.e. I12 = 0) will

be antisymmetric under the operation (12), whereasfunctions with I12 odd (i.e. I12 = 1) will be symmetric.For two deuterons (I1 = I2 = 1) the functions with I12even (I12 = 0 or 2) will be symmetric, and those withI12 odd (I12 = 1) will be antisymmetric.

The two sets of basis functions are connected by therelation (see equation (4.3) of [25])

|I1I2I3; JI12F2Fl = åF1

k JF1F2F|JI12F2Fl |I1I2I3; JF1F2Fl ,(29)

where the recoupling coe� cients’ k JF1F2F|JI12F2Fl aregiven by equation (4.8) of [25]:

k JF1F Â2F Â |JI12F2Fl = d F Â2F2 d F Â F(- 1)J+I1+I2+F2

´ (2F1 + 1)1/2(2I12 + 1)1/2

´ JI2

I1

F2

F1

I12{ }. (30)

The quantity is braces in a 6j symbol. The overlap in-tegrals in equation (30) constitute the elements of aunitary matrix which we can use in a similarity transfor-mation to obtain, from the matrix representation of theHamiltonian Hhfs in the basis |I1I2I3; JF1F2Fl , therepresentation in the basis |I1I2I3; JI12F2Fl . In thelatter basis, the representation matrix is block diagonalwith one block of even-I12 basis functions and one blockof odd-I12 basis functions. Consequently, the calculatedeigenfunctions of Hhfs are linear combinations of |I1I2I3;JI12F2Fl with a common parity of I12, and the transfor-mation properties of these eigenfunctions under (12) arestraightforwardly determined. We have extended ourprogram for the calculation of hyper® ne splittings sothat it now uses the matrix elements of equation (30)to compute the matrix representation of Hhfs in thesymmetrized basis |I1I2I3; JI12F2Fl . This matrix is thendiagonalized for the upper and lower rotation± vibrationstate respectively of the rotation± vibration transitionunder study. We can then form transitions between thecalculated hyper® ne energy levels that are allowed bythe Pauli exclusion principle. The relative intensities ofthese transitions are obtained by calculating the matrixC de® ned by equation (31) of [23] and transformingit ® rst to the |I1I2I3; JI12F2Fl basis, using the matrixelements of equation (30), and then to the basis of eigen-functions for Hhfs, using the eigenvectors obtained in theenergy calculations. The squares of the elements of thetransformed C matrix constitute the desired relativeintensities.

We have used the new program to calculate the hyper-® ne splitting patterns for a number of rotational transi-tions in H2D

+ and HD+2 . For H2D

+ , we study theseven lines 110 111, 220 221, 101 000, 212 111,

326 P. Jensen et al.

211 110, 330 331 and 321 322. The ® rst three ofthese transitions have been observed in the laboratory[26± 29], and the lines 110 111 and 101 000 have beententatively assigned in spectra of the core region of theinterstellar cloud NGC 2264 and of the IRc2 region ofthe interstellar cloud M42 [30, 31]respectively. The twolines 212 111 and 211 110 are possible candidates forinterstellar observation of H2D

+ (see section 6), whereasthe last two lines, 330 331 and 321 322, are not likelyto be of interest in an astrophysical context. We calcu-late their hyper® ne structure here because they may beobservable in the laboratory. For HD+

2 , we consider thefour transitions 220 211, 111 000, 110 101, and211 202, of which the former two have been measuredin the laboratory by Jennings et al. [29] (see table 4 ofPolyansky and McKellar [32]). For the calculationsof hyper® ne structure, the quadrupole coupling con-stants were taken from tables IV and V of [7], thevibrationally averaged tensor components Ma a (l) andRa a (kl) were taken from table 2, and the averagedvalues k J2

a l for the squares of the angular momentumcomponents were calculated in the rigid rotor approx-imation using values of the ground-state rotational con-stants from [33] for H2D

+ and from [34] for HD+2 . The

strongest transitions in the calculated hyper® ne patternsare given in table 3 for H2D

+ and in table 4 for HD+2 .

These tables list the calculated hyper® ne shift D t , that isthe displacement of the line from the centre frequencycorresponding to the energy di� erence between the twounsplit rotation± vibration levels. Further, we give therelative intensity Irel normalized so that the sum of theintensities of all hyper® ne components equals thenuclear spin statistical weight factor obtained from thePauli exclusion principle in the absence of hyper® nee� ects. Tables 3 and 4 contain transitions with relativeintensities Irel > 0.1. We label the hyper® ne componentswith the value of the good quantum number F in theupper and lower states. For HD+

2 (table 4), we also givevalues of the quantum number I12. These are obtainedfrom the basis function with the largest contribution tothe eigenfunction for Hhfs. Figure 2 shows the hyper® nesplitting patterns calculated for H2D

+ , and ® gure 3shows those calculated for HD+

2 .

6. ConclusionBy comparing ® gures 2 and 3 with ® gures 2± 4 of [7],

we see that for the studied rotation± vibration transitionsof H2D

+ and HD+2 , the hyper® ne structures are gener-

ally signi® cantly changed by the addition of the spin±rotation and spin± spin interactions of protons and deu-terons to the theoretical model. For all transitions con-sidered, the `width’ of the hyper® ne splitting pattern hasincreased as a consequence of the introduction of themagnetic e� ects. We can obviously draw the immediate

Hyper® ne e� ects in H+3 327

Table 3. Calculated hyper® ne structures for the transitions110 111, 220 221, 101 000, 212 111, 211 110,330 331 and 321 322 of H2D

+ .

F Â F Â Â D ta/kHz Irel

b F Â F Â Â D ta/kHz Irel

b

110 111 101 0000 1 - 142.34 0.17 0 1 - 59.44 0.331 2 - 103.14 0.16 2 1 3.59 1.671 1 - 90.92 0.19 1 1 13.83 1.002 3 - 89.12 0.751 1 - 71.54 0.43 220 2211 2 - 69.10 0.62 2 3 - 59.43 0.161 0 - 51.32 0.10 2 1 - 32.04 0.151 1 - 50.91 0.26 2 2 - 5.63 0.692 1 - 50.74 0.45 3 3 1.63 1.242 2 - 44.77 0.98 1 1 5.58 0.452 1 - 32.55 0.21 1 2 31.99 0.153 3 - 27.40 1.56 3 2 55.43 0.163 2 - 20.62 0.191 2 - 4.06 0.34 330 3311 1 8.16 0.41 4 5 - 292.29 0.101 1 132.88 0.14 3 4 - 121.44 0.142 2 135.49 0.18 2 3 - 99.37 0.142 1 153.69 0.29 2 2 - 6.24 0.462 2 173.06 0.23 4 4 - 6.01 1.092 1 191.26 0.18 5 5 - 4.69 1.413 2 197.22 0.59 1 1 - 4.53 0.291 0 231.56 0.18 3 3 - 4.21 0.86

3 3 - 4.02 0.69321 322 3 3 8.33 0.81

3 3 - 11.17 0.84 4 4 10.57 1.014 4 3.81 1.21 2 2 10.93 0.512 2 8.77 0.63 3 2 89.11 0.14

4 3 127.99 0.155 4 298.17 0.10

212 111 211 1101 1 - 185.15 0.12 1 2 - 238.78 0.120 1 - 184.86 0.10 2 2 - 218.15 0.283 3 - 180.33 0.32 2 1 - 197.37 0.162 2 - 169.56 0.23 1 1 - 53.22 0.202 1 - 165.60 0.58 0 1 - 36.91 0.131 1 - 143.52 0.11 1 1 - 34.19 0.371 1 - 124.14 0.17 2 1 - 21.38 0.712 1 8.36 0.86 3 3 - 15.45 0.461 1 19.29 0.28 2 1 - 13.57 0.472 2 20.61 0.32 2 2 - 5.82 0.113 2 44.28 1.04 2 2 31.76 0.293 3 46.97 0.14 4 3 34.68 1.802 2 48.31 0.18 3 2 36.07 1.384 3 48.53 1.80 3 2 46.31 0.943 2 53.71 1.26 2 1 52.54 0.502 1 66.53 0.55 2 1 77.78 0.191 0 79.82 0.29 1 0 97.41 0.25

a Hyper® ne shift.b Relative intensity. Only transitions with Irel > 0.1 are

given.

conclusion that these e� ects are just as important for thehyper® ne structures as the quadrupole coupling of thedeuterons, which was the only e� ect considered in [7].One exception to this rule is the 220 221 transition ofH2D

+ ( ® gure 2) whose hyper® ne structure is largelyunchanged by spin± rotation and spin± spin interaction.This is because for this transition the Pauli exclusionprinciple requires that the spin functions be antisym-

metric under the interchange of the two protons.Hence these functions have I12 = 0 (see section 5). Asdiscussed generally by Thaddeus et al. [23]and exempli-® ed by Shostak et al. [35] for the case of the watermolecule, in such a rotation± vibration state the spin±rotation interaction of the protons, the spin± spin inter-action between them, and their spin± spin interactionwith the deuteron do not contribute to the hyper® ne

328 P. Jensen et al.

Table 4. Calculated hyper® ne structures for the transitions 111 000, 211 202, 110 101, and 220 211 of HD+2 .

I Â12 F Â I Â Â12 F Â Â D ta/kHz Irel

b I Â12 F Â I Â Â12 F Â Â D ta/kHz Irel

b

111 000 110 1012 5/2 2 3/2 - 66.42 2.00 1 3/2 1 5/2 - 62.70 0.240 1/2 2 3/2 - 56.99 0.67 1 1/2 1 3/2 - 60.17 0.472 3/2 0 1/2 - 51.63 1.33 1 3/2 1 3/2 - 37.80 0.472 1/2 2 3/2 - 15.11 0.67 1 1/2 1 1/2 - 31.69 0.302 3/2 0 1/2 20.18 1.33 1 5/2 1 5/2 - 4.38 1.400 3/2 2 3/2 29.87 1.33 1 3/2 1 3/2 - 2.30 0.262 5/2 2 3/2 34.44 2.00 1 3/2 1 1/2 5.79 0.352 7/2 2 5/2 42.79 2.67 1 1/2 1 3/2 6.00 0.17

1 3/2 1 5/2 18.58 0.36211 202 1 3/2 1 1/2 20.35 0.362 5/2 2 5/2 - 84.15 0.18 1 3/2 1 3/2 43.48 0.540 3/2 2 3/2 - 68.96 0.11 1 5/2 1 3/2 56.01 0.542 7/2 2 9/2 - 60.21 0.16 1 1/2 1 1/2 64.15 0.142 1/2 2 3/2 - 56.51 0.102 3/2 2 5/2 - 52.76 0.35 220 2112 5/2 0 5/2 - 51.60 0.59 2 7/2 2 9/2 - 103.65 0.212 3/2 0 5/2 - 41.09 0.16 2 5/2 2 7/2 - 97.70 0.352 5/2 2 7/2 - 40.99 0.27 2 3/2 0 3/2 - 78.98 0.192 7/2 0 5/2 - 39.93 0.16 2 3/2 2 5/2 - 58.99 0.282 5/2 2 7/2 - 29.39 0.33 2 7/2 2 7/2 - 35.86 1.002 7/2 2 7/2 - 29.32 0.94 2 5/2 2 5/2 - 33.10 0.350 5/2 0 5/2 - 25.85 0.24 2 3/2 2 1/2 - 28.10 0.192 5/2 2 5/2 - 25.70 0.21 2 5/2 2 7/2 - 25.67 0.212 7/2 2 9/2 - 22.27 0.21 2 5/2 2 3/2 - 24.51 0.232 1/2 2 3/2 - 8.27 0.19 2 7/2 2 9/2 - 18.41 0.160 3/2 2 3/2 - 5.63 0.32 2 5/2 2 5/2 - 14.01 0.622 9/2 2 7/2 0.52 0.24 0 3/2 0 3/2 - 7.04 0.472 9/2 2 9/2 7.58 1.63 2 5/2 2 3/2 - 6.04 0.402 3/2 2 1/2 9.60 0.24 2 1/2 2 3/2 3.42 0.272 3/2 2 3/2 12.42 0.19 2 3/2 2 1/2 5.66 0.202 3/2 2 5/2 17.22 0.14 2 3/2 2 3/2 11.26 0.232 1/2 2 3/2 18.02 0.16 2 7/2 2 7/2 11.44 1.002 5/2 2 3/2 22.62 0.24 0 3/2 2 5/2 12.95 0.110 5/2 2 5/2 32.46 0.63 2 1/2 2 1/2 14.59 0.162 7/2 2 7/2 35.21 1.08 2 1/2 2 3/2 20.19 0.202 7/2 2 5/2 38.89 0.29 2 3/2 2 5/2 21.76 0.260 3/2 2 3/2 42.61 0.26 2 7/2 2 5/2 23.11 0.262 3/2 2 1/2 56.99 0.16 2 7/2 2 5/2 28.74 0.382 1/2 2 1/2 62.59 0.19 2 9/2 2 9/2 33.38 1.632 5/2 2 3/2 65.24 0.21 0 5/2 0 5/2 35.23 0.902 7/2 2 5/2 76.83 0.15 0 3/2 2 1/2 43.84 0.100 5/2 2 5/2 90.91 0.11 0 5/2 2 7/2 49.30 0.112 9/2 2 7/2 103.00 0.13 2 9/2 2 7/2 63.23 0.21

2 9/2 2 7/2 101.17 0.16a Hyper® ne shift.b Relative intensity. Only transitions with Irel > 0.1 are given.

Hyper® ne e� ects in H+3 329

Figure 2. Calculated hyper® ne structures for the transitions 110 111 (observed centre frequency, 372421.34MHz [26, 27]),220 221 (observed centre frequency, 155987.185MHz [29]), 101 000 (observed centre frequency, 1370146MHz [30]),212 111 (centre frequency estimated to be 2363GHz from the term values given in [33]), 211 110 (centre frequency estimatedto be 3102GHz from the term values given in [33]), 330 331 (centre frequency estimated to be 44.6GHz from the term valuesgiven in [33]), and 321 322 (centre frequency estimated to be 857.9GHz from the term values given in [33]) of H2D

+ .

330 P. Jensen et al.

Figure 3. Calculated hyper® ne structures for the transitions 110 101 (centre frequency estimated to be 691.9GHz from the termvalues given in [34]), 211 202 (centre frequency estimated to be 1039.0GHz from the term values given in [34]), 220 211(observed centre frequency, 1370051.6MHz [29]) and 111 000 (observed centre frequency, 1476605.5MHz [29]) of HD+

2 .

Figure 4. The spectral pro® le obtainedfor the 110 111 transition of H2D

+

for the absolute temperaturesT = 1K (´´´ ), T = 2.7K (± . . ± ),T = 5K (± . ± ) and T = 10K (± ±± ), where it is assumed that thebroadening of the lines is causedentirely by the Doppler e� ect (seethe text): ( Ð Ð ), the spectral pro® lefor the unsplit transition (i.e. for onetransition at D t = 0 with an inten-sity equal to the sum of the relativeintensities of the hyper® ne com-ponents) obtained for T = 10K.

structure. Hence this structure is caused entirely by thequadrupole coupling and spin± rotation interaction ofthe deuteron. From table IV of [7], the values ofeQ k V |V xx(3)|V l , eQ k V |Vyy(3)|V l and eQ k V |V zz(3)|V lfor the vibrational ground state of H2D

+ are obtainedas 17.3, 87.1 and - 104.4kHz respectively. The corre-sponding values of Ma a (3) (table 2) are all less than10kHz. Consequently, the hyper® ne structure is deter-mined essentially by the quadrupole coupling of thedeuteron, and a structure very similar to that in ® gure2 of [7] is obtained in the present work.

The deuterium fractionation reaction H+3 + HD ®

H2D+ + H2 mentioned in section 1 produces a high

abundance of H2D+ only for kinetic temperaturesbelow 30K (for example [30]). Consequently, if theH2D

+ ions were in thermal equilibrium at such tempera-tures, essentially all of them would populate the rota-tional ground state (note that the lowest ortho state,JKaKc = 111, is separated from the ground state by anenergy di� erence D E for which D E /k < 90K, where kis the Boltzmann constant). This would of course makeit virtually impossible to observe any emission of H2D

+ ,and the only means of detecting the ion would be the101 000 absorption line. However, according to moresophisticated models (for example [36]) for the inter-stellar chemistry, at temperatures below 20K theH2D

+ ortho-to-para ratio will deviate from that corre-sponding to thermal equilibrium, and the ortho form willbecome most abundant. The lowest ortho state, 111, isseparated from the next ortho state, 110, by approxi-mately 12.4cm- 1 [33] corresponding to D E /k < 18K.Thus there would seem to be a good chance of observingH2D

+ by means of the 111 110 emission line. Further,a high abundance of ortho H2D

+ would favour theabsorption lines 110 111 (372421.34MHz), 212 111(2363GHz) and 211 110 (3102 GHz). For each ofthese lines, we give the centre frequency in parentheses.The frequency of the 110 111 transition has beenobtained experimentally [26, 27], whereas for the othertwo lines we have estimated the frequencies from theterm values of [33]. The calculated hyper® ne splittingpatterns of the three lines are given in ® gure 2. The® gures show that for the three transitions, the hyper® nesplitting patterns have comparable `widths’. However,the transition 110 111 occurs at a considerably lowerfrequency than the other two lines, and so it wouldappear that, only for this transition, could one hopethat the hyper® ne splitting pattern will not be smearedout completely by Doppler broadening.

To investigate whether the hyper® ne splitting patternof the rotational transition 110 111 can be used toidentify H2D

+ in the interstellar medium we have eval-uated the spectral pro® le of this line at di� erent tem-peratures. We assumed that the line broadening is

caused entirely by the Doppler e� ect, so that we useda line-shape function depending on the frequency t as[37]

FD( t - t 0) =2

D t D

ln 2p( ) 1/2

exp - 4 ln 2t - t 0

D t D( ) 2[ ],(31)

where t 0 is the central frequency of the line and the fullwidth of half height (FWHH) D t D is given by

D t D = 2t 0

c2kT ln 2

M( ) 1/2

. (32)

In equation (32), T is the absolute temperature, M isthe total mass of the molecule and c is the velocity oflight in vacuum. Figure 4 shows the spectral pro® le ofthe 110 111 transition of H2D+ for T = 1K (dottedcurve), T = 2.7K (dot± dot± dash curve), T = 5K (dot±dash curve) and T = 10K (broken curve). The full curveis the spectral pro® le for the unsplit transition (i.e. forone transition at D t = 0 with an intensity equal to thesum of the relative intensities of the hyper® ne compo-nents) obtained for T = 10K. It is evident from the® gure that, in order for the hyper® ne structure to bere¯ ected in the spectral pro® le, the temperature mustbe extremely low. There exist interstellar objects withsuch low temperatures. For example, Downes et al.[38] have determined the excitation temperature of for-maldehyde in the dust cloud L134 to be 1.4K. Conse-quently, even though the identi® cation of interstellarH2D

+ by means of its hyper® ne structure requires theions to be located in a part of interstellar space withrather unusual properties, it is not downright impos-sible.

We are grateful to J. Tennyson for helpful commentson an earlier version of this paper. We acknowledgesupport from the Grant Agency of the Academy ofSciences of the Czech Republic (grant No. A4040501),from the Grant Agency of the Czech Republic (grantNo. 203/96/0947), from the Danish Natural ScienceResearch Council (grant No. 11-0924 and grantNo. 9313314), and from the Fonds der ChemischenIndustrie.

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332 Hyper® ne e� ects in H+3