On isotope effects in the predissociations of HD

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On isotope effects in the predissociations of HD

J. Durup

To cite this version:

J. Durup. On isotope effects in the predissociations of HD. Journal de Physique, 1978, 39 (9), pp.941-

950. �10.1051/jphys:01978003909094100�. �jpa-00208835�

941

ON ISOTOPE EFFECTS IN THE PREDISSOCIATIONS OF HD

J. DURUP

Laboratoire des Collisions Atomiques ^et Moléculaires (*),

Université de Paris-Sud, ⁹¹⁴⁰⁵ Orsay, ^France

(Reçu ^{le 28 mars} 1978, ^{révisé le 2 mai} 1978, accepté ^{le 18 mai} 1978 )

Résumé.

On considère les couplages susceptibles ^{de donner} naissance à un effet isotopique ^dans

la prédissociation d’espèces diatomiques à noyaux de même charge ^{mais de masses} différentes.

Dans le cas de HD, ^les équations couplées ^{sont résolues} numériquement pour des processus modèles traités à l’aide de valeurs publiées ^de potentiels ^{et de} couplages calculés ab initio. Les résultats montrent que (i)

les effets isotopiques ^{ne sont dus} qu’au couplage dynamique du second ordre entre états

Born-Oppenheimer ^{et non au} couplage ^{radial du} premier ordre, (ii)

^{ces effets sont} régis par la forme asymptotique de la décroissance de la séparation gerade-ungerade à la distance internucléaire où celle-ci est égale ^{à la} séparation ^{entre les canaux de sortie} isotopiques, (iii)

^en particulier, des ^effets isotopiques importants peuvent apparaitre ^sur les chemins corrélés diabatiquement à des étáts

atomiques optiquement ^liés entre eux, tandis qu’aucun ^effet isotopique important ^ne peut apparaitre

sur le chemin corrélé diabatiquement ^{à la} paire d’ions, (iv)

l’applicabilité ^{à ce} problème ^{du modèle}

de Demkov, ^tel que l’ont utilisé van Asselt, ^{Maas et} Los, ^est confirmée. Les mécanismes de pré-

dissociation de niveaux de Rydberg élevés de la molécule d’hydrogène ^sont discutés à la lumière de

ces résultats.

Abstract.

A discussion is given ^{of the} couplings which may give ^{rise to an} isotope ^{effect in} pre- dissociation of diatomics with nuclei of equal charges but different masses. In the case of HD the

coupled equations ^are numerically solved for model _processes using published potentials ^and couplings ^{from ab initio} calculations. The results show that (i) isotope ^{effects are due} only ^{to the}

second-order dynamic coupling ^between Born-Oppenheimer ^{states and not to} first-order radial

coupling, (ii) these effects are controlled by ^the asymptotic form of the decrease of the gerade-ungerade splitting ^at the internuclear distance where it equals ^the splitting between the isotopic ^exit channels, (iii) ⁱⁿ particular, strong isotope effects may ^{occur on} paths diabatically correlated with atomic states

optically ^{connected to each} other, ^{whereas no} strong isotope ^effect can occur on the path diabatically

correlated with the ion pair, (iv) the relevance of Demkov’s model to this problem, ^as developed by

van Asselt, ^{Maas and} Los, is confirmed. From these results a discussion is given ^{of the} predissociation

mechanisms of some high Rydberg ^{states of the} hydrogen ^molecule.

LE JOURNAL DE PHYSIQUE TOME 39, ^SEPTEMBRE 1978,

Classification Physics ^Abstracts

33.80G

1. Introduction.

Various experimental ^results

have been published ^{in the recent years} concerning branching ratios in the dissociation of heteroisotopic electronically symmetric diatomics. A strong isotope

effect has been reported ^{in the} photodissociation

of HD into H + ⁺ D - and H - + D + [ 1, 2], ^{the latter}

channel being favoured with respect ^to the former by

a factor which according ^to Berkowitz and Krone- busch [1] ] ^{is 1.4 to 1.5 in the observed range} (from

threshold to 0.3 eV above), ^whereas according ^to Chupka ^{et al.} [2] it is about 2 at threshold and decreases

(*) Laboratoire associé au C.N.R.S.

LE JOURNAL DE PHYSIQUE.

T. 39, ^N° 9, SEPTEMBRE 1978

with increasing energy. This photodissociation ^occurs largely through predissociation ^of Rydberg ^states

of HD. Isotope ^{effects were} also observed in the _pre- dissociation of 3He4He+ into ’He’ + 4He and

’He ⁺ 4He+ by ^van Asselt, ^{Maas and Los} [3]. ^In

contrast, ^a branching ^ratio equal ^{to 1 was found} by Thomas, Dale and Paulson [4] ^{in the} photodissociation

of HD + into H + + D and H + D + .

In such diatomics, characterized by isotopic nuclei,

dissociation after excitation into a well-defined unbound molecular state can present ^an isotope

effect

i.e. the appearance with unequal yields ^of

the two possible fragment pairs

only ^{if this} initially populated ^{unbound state is} coupled ^{with at least one}

64

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other molecular state leading ^{to the same} fragment

states but having ^the opposite parity ^{of the electronic}

wavefunction. In effect, coupling ^{between states of the}

same parity ^{leads to an exact} wavefunction also with the same g ^{or u} character and therefore with equal weights ^{on both} isotopic channels. On the other hand, coupling ^{between molecular states} connected with different fragment states cannot lead to interference between outgoing ^{waves since the} fragment ^{states are} distinguishable ; ^{thus the} parity ^{of the} wavefunction will be retained in each of the exit channels.

The problem ^{of the} coupling ^{between a} pair ^{of states}

of opposite parity ^{connected with the same} fragment

states in a heteroisotopic diatomic molecule has been treated along ^various lines. Thorson [5] studied the state giving ^{rise to} the double edge observed in the

absorption spectrum ^{of HD. He used a} model Hamil- tonian for describing ^{the vibronic} coupling ^between

these states, ^{which were the B’} lE: ^{state and a} hypo-

thetical companion lEg+ ^{state, and he} solved the

coupled equations by the method of Green’s functions.

For a treatment of HD +, ^Hunter [6] used the Born-

Oppenheimer wavefunctions with appropriate ^scal- ing parameters. ^The predissociation by ^rotation

of 3He4He+ was treated by ^{van Asselt et al.} [3] ^{on the}

basis of Demkov’s theory [7, 8].

°

We present ^{here a treatment of the} predissociation

of Rydberg ^{states of HD} either into an ion pair ^or

into 1 s + 2po atoms, by ^numerical integration ^{of the} coupled equations for the radial nuclear wave-

functions. In the ion-pair ^{case some} comparison ^can

be made with van Asselt et al.’s treatment [3], ^since

, Demkov’s theory ^of single charge ^transfer may easily

be extended to double charge ^{transfer. In the case of}

neutral fragment ^{atoms the} analytical ^{treatment of}

Demkov is ^not applicable.

Our calculations were performed ^{with two-state}

models. The real problem ^{of the} predissociation ^{of a} Rydberg ^{state of} hydrogen ^{into an ion} pair ^{is much}

more complicated ^{since it involves at least six states} (four of which have been well studied theoretically :

the B 1 Eu+ ^state [9], ^{the B’} 1 Eu+ ^state [10] ^{and the}

double-minimum E, F 1,E 9 ^{[11] and G,} K ’,E 9 ^{[ 12]}

states) ^and possibly ^{six other ones. The} output ^{of our}

two-state model calculations will only ^{be to show}

which processes ^can be responsible ^{for an} isotope

effect and which cannot.

In the following ^{we shall} ignore the rotation of the molecule. Rotational effects of course are very impor-

tant, when ^not determinant, in threshold predissocia-

tions. An interesting ^{case which} may be studied

experimentally ^{is the} predissociation by ^rotation

of the 2 E; ^{ground state of HD + . The treatment of}

rotation by introducing ^the centrifugal potential

into the radial Schrôdinger equation ^is straightfor- ward, ^and may be a usèful extension of the present work ; ^{such a} treatment, which ignores gyroscopic

terms, holds exactly ^{in the case of 1 ’ states.}

2. Formalism.

We shall first consider the general

situation where two molecular states of a homonuclear neutral or ionic diatomics A2 ^are dissociating ^{into the}

same pair ^{of atomic states Ai +} Ai, ^{where the} super-

scripts i :0 j ^{refer to different} quantum ^or charge

states. If i and j specify all atomic quantum numbers,

the molecular states at large internuclear distance are

simply ^the gerade ^and ungerade combinations of the

atom-pair ^{states Ai + A’ and A’ + Ai.}

Turning ^{now to an} isotopically non-symmetrical

molecule which we shall name AB (where e.g. A is

hydrogen ^{and B is} deuterium), the exit channels for the corresponding ^{molecular states become Ai + Bi}

and Ai + B‘. The electronic wavefunctions of the molecular states retain their _g and u characters at short and moderate distances, ^with respect ^{to the centre of} charge ^{of the nuclei.}

Let us start with the Born-Oppenheimer ^wave- functions 1 Pg > ^and 1 Pu > of these molecular states.

They ^are eigenfunctions ^{of the} electronic Hamiltonian

only, ^which (including ^nuclear repulsion ^and ignoring spin-orbit coupling) ^reads

where the subscripts s, s’ label ^the electrons, ZA ^and ZB

are the nuclear charges, ^{and R is the} intemuclear distance. The term species of 1 (fJg > ^and 1 (fJu > ^are

identical except ^{for the} g ^{or u} character, e.g. they ^are

both Z+, ^{or both} H, ^etc. They ^are degenerate ^at

infinite internuclear distance.

From these Born-Oppenheimer ^{states we can}

define at any internuclear distance R the channel

states :

which at large R become the exit channels Ai + Bi and Ai ⁺ B‘, respectively.

The Born-Oppenheimer ^{states defined} by 1 ({Jg >

and 1 (Qu > ^are coupled by ^two operators, â/âR ^and VI

We shall refer to these couplings ^as first-order radial

coupling and second-order dynamic coupling, ^res- pectively, and discuss their roles in turn.

The first-order radial coupling ({J g 1 a / (} R 1 (pu >

occurs only because the centre of mass of the molecule differs from its centre of charge. ^{Let the molecular}

centre of mass G be the origin of coordinates with r

denoting ^the coordinates of all electrons. _{Let p} be the electron coordinates with respect ^{to the mean} point ¹

between the nuclei.

If we ignore ^the distinction between the centre of mass

943

of the nuclei and the centre of mass of ^the molecule, ^we get ^the simple ^{relation :}

since

Here, ^R

BA, ^{the M’s are} the nuclear _masses,

and ( ^{is the} projection ^of p ^on the internuclear axis.

The subscripts denote the variables kept fixed in each

partial differentiation.

The first term on the right ^{hand side of} equation (3)

vanishes on symmetry grounds. The second term is

easily expressed ^{in terms of the} dipole ^transition

moment 4(,gu ^{between the} two states and of their electronic energies Hgg ^and Huu :

It is well known in atomic collision theory ^{that the}

careless introduction of the first-order radial coupling

terms (3), ^{when as in this case} they ^are only ^{due to the} asymmetric separation ^{of the} nuclei, ^{can lead to} spurious, non-physical ^effects [13, 14]. ^For example ⁱⁿ

tast collisions where the velocities of the two nuclei do not vary significantly during ^the encounter, ^{these terms}

are entirely spurious and their careless use would lead to erroneous results. One gets ^correct result if one

takes properly ^{into account the fact that the wave-}

function of each electron of the separating system should include the translation of the nucleus to which it is attached ; this leads to the introduction of the so-called translational terms [15]. This correction in fact means avoiding ^the simplifying assumption ^which

we underlined above, namely identification of the centre of mass of the pair of nuclei with that of the whole molecule.

’

In the case under consideration in the present work, namely dissociation just ^above threshold, ^{where the}

final total kinetic _energy of the nuclei will not exceed 0.3 eV, the associated kinetic _energy of each electron is less than 10-’ eV and _may be neglected. ^Conse- quently, ^the spurious part ^{of terms} (3) ^{should not}

affect our calculations. Our results will show that this holds even at higher ^kinetic energies.

However, part ^{of the effect of these} coupling ^terms

is not spurious ^{and has to} be taken into account ; ^{it is} the role of the acceleration of the nuclei with respect

to each other. Clearly, ^{in the extreme case of a} proton receding suddenly ^{from a} heavy companion ^atom

both previously ^{at rest, an electron} initially ^shared

between them will have comparatively less chance to follow the proton than in the case of a slow separation

of the nuclei. Coupling ^terms (3) ^{also are known to}

give ^{rise to shifts in} bound-state energies [16]. ^Such

terms, ^{which are} important ⁱⁿ HD, ^{would of course} be much smaller in species with less relative ^mass

difference, ^{such as} heteroisotopic oxygen ^or nitrogen

molecules.

This discussion about first-order ^radial coupling

between the Born-Oppenheimer ^states also holds as

regards ^the channel states, since from equation (2) ^and

from the orthonormality of 1 çog > ^and l ’Pu >

Thus a check of the possible ^{role of the} first-order radial coupling ^is important ^for assessing ^the validity

of Demkov’s treatment, ^{which rests} upon the assump- tion of diabaticity of the channel states with respect ^to each other.

The second-order dynamic coupling) pg ) 1 VI I Qu >

arises from the fact that the slight nuclear motion necessary ^to cancel the electronic momentum in the centre of mass frame is dependent ^on whether any electron interacts mainly ^{with the} lighter ^{nucleus or}

with the heavier one. This VR coupling ^{term includes}

not only the second-order radial coupling

but also the terms arising ^{from the} slight swinging

motion of the nuclei about the line joining ^{the centres}

ofmass of the two atoms.

The expectation ^{values of} vi for t (Qg > ^and 1 Qu >

are equal ^at infinity. ^In other words at infinite R the channel states (2) diagonalize ^the V2 operator (1), ^and

the différence between its eigenvalues ^{in the two}

channels is the cause of the _energy splitting ^between

them. As simple algebra shows this splitting ^becomes

in the Born-Oppenheimer basis twice the coupling between t Q g > ^{and 1} Qu >.

At finite R the expectation ^{values of} V2 ^{differ for}

each Born-Oppenheimer ^state (they ^{have been cal-}

culated for the B 1 Lu+ ^and E, ^F 1 Lg+ ^{states of H2}

by Kotbs and Wôlniewicz [9, 11), and therefore the channel states do not diagonalize V2 any ^more.

Furthermore, ^{it seems} reasonable to expect ^{that at} intemuclear distances where the active electron(s) strongly interact(s) with both nuclei the expectation

values of VR for both channel states merge together,

and therefore the Born-Oppenheimer ^{states now} diagonalize ^that operator. ^{This of course is} necessarily

true at the limit of the united atom, but probably ^also

at the equilibrium internuclear distance of the molecule.

We shall now tum to the so-called adiabatic

e) ^{In cases} where the atomic states i andj ^are optically connected,

a small off-diagonal matrix element of VR between the channel

states (2) persists ^{at infinite R. It is a} typically spurious term, ^as

discussed above, ^{and has to be deleted.}

states [17], ^which diagonalize the adiabatic Hamil- tonian

where H is given by equation (1) and J1 ^{is the reduced}

mass of the nuclei. Mass-polarization ^terms (see

e.g. [18]) ^are included in Jead through ^the OR term;

they are responsible for the well-known isotope ^effect

on atom energies.

In the subspace spanned by ^the Born-Oppenheimer states 1 lfJg > ^and 1 lfJu > ^{the adiabatic states} may be constructed as

with

where AE is the splitting ^{between the values of}

-

(h’12 /l)Vi ^{for the channel states. The slight}

différence between the expectation ^{values of} VR ^for 1 (Qg > ^and 1 (fJu > ^at low R has been neglected ⁱⁿ (8) ⁱⁿ comparison with the difference in electronic energies.

8 tends towards n/4 ^at low R and 0 at infinite R, ^{so that} the adiabatic states 1 ) and 12 > tend towards the

Born-Oppenheimer states at low R and towards the channel states at large ^R. Simple algebra ^{shows that}

the expression (8) cancels 1 1 V’ 2 ) except ^{for a} small term

so that (8) ^{is exact at low} R, ^at large R, and in the critical region ^where d8 jdR ^{is maximum} (see later).

The non-crossing rule holds for the adiabatic

states (7) ^since they diagonalize ^a well-defined Hamil- tonian operator. Arguments ^{based on adiabatic}

correlations have thus been used in discussions about

isotopic ^{effects in HD} predissociations [2,19].

In the general ^{case which we are} discussing, ^the adiabatic correlations are as depicted ⁱⁿ figure ^1.

One must be aware of the fact that, ^{due to} the very small value of the OR coupling ^terms (a ^few meV), ^the dissociating system ^{will behave} adiabatically only ^at

very small velocities ; ^at energies above the meV _range it will not follow an adiabatic path but rather remain

on the initial Born-Oppenheimer ^curve and therefore dissociate without _any isotope effect. This is clear also from the formulae derived in the case of a diatomic ion AB+ by ^van Asselt et al. [3] ^{on the basis of Dem-}

kov’s model : they ^{find for the} branching ^ratio

between the exit channels the expression

where vR ^{is the radial} velocity in the critical region

and I the ionization potential ^{of the atom A or} B, ^all quantities being expressed in atomic units. Clearly

this ratio will differ measurably ^{from 1} only ^if vR

corresponds ^to energies ^{no more than about 10-2 eV.}

In order to make more general predictions, ^{let us}

consider equations (7). They ^are identical with the familiar equations describing the rotation from the diabatic to the adiabatic basis in Stueckelberg’s

treatment of the curve-crossing problem [20] ^{as well}

as in charge ^{transfer without curve} crossing, except that in these traditional cases the starting basis is made of single-configuration wavefunctions whereas here

we start from Born-Oppenheimer wavefunctions.

Table 1 compares the three situations.

TABLE 1

Construction of ^{adiabatic states in cases} of (i) ^curve crossing, (ii) charge transfer ^{without curve} crossing, and (iii) heteroisotopic dissociation.

e) Arbitrarily ^{taken as 0.}

(b) Typical ^values (the ^{actual ones} may be slightly different).

In all three cases the most general ^{treatment of the}

collision problem ^{is in terms of the adiabatic basis} (7),

whose states are coupled only by ^a ô/âR ^matrix

element, ^here given by

945

In the semi-classical picture ^this coupling ^term

appears in the coupled equations ^{with the radial} velocity VR ^{as a} multiplying ^factor.

We shall assume

and our calculations will

verify

that, ^{in the} region of R where the adiabatic states change ^{from the} Born-Oppenheimer ^{states to}

the channel states (Fig. 1), ^{the first term in the} right

hand site of equation (10) is the dominant one.

The transition will then occur with a highest pro-

bability ^{at the} internuclear distance where (d8/dR ) x vR is maximum. This also occurs in Stueckelberg’s ^or

Demkov’s treatment when it is assumed that the

single-configuration ^states used in first-ôrder approxi-

mation are diabatic in the subspace they span.

In the present ^{case the} variation of 0 with the

parameter 1 Huu - Hgg ^l/âE (see ^eq. (8)) ^{is as shown}

in figure ^2. Clearly the transition region ^{should be} where ( Huu - Hgg ^{1 ils close to AE,} i.e. twice the VR coupling ^between Born-Oppenheimer ^{states. This}

turns out_ to be similar with Demkov’s _case, where the transition is usually considered to take place ^{at the}

internuclear distance where the energy gap between the diabatic states equals ^{twice the} coupling ^between

them [8].

FIG. 2.

Rotation of the Born-Oppenheimer ^{basis to} the adiabatic basis in heteroisotopic dissociation.

It thus _appears that, ^{in so far as the radial} coupling between Qg > ^and 1 ({Ju > ^{is not} dominant in (10), ^the

adiabatic or non-adiabatic behaviour of our system will depend ^on the values of two variables in that critical region : the radial velocity vR, and the relative rate of variation with R of the ratio Huu - Hgg ^J/AE.

Since AE is only ^{a few} meV, ^the critical internuclear distance Rc where 1 Huu - Hgg ¹

^{AE is large enough}

for the electronic wavefunctions to be concentrated close to either or both nuclei. Therefore at R = Rc

we are allowed to replace ^AE by ^{a value} I1Eoo ^corres- ponding ^{to infinite} R, but which has to be taken

on the exit channels with the same electron configu-

ration as that of the channel states in the critical region.

Thus the relevant variables are vR and the rate of variation with R of 1 Huu - Hgg ^{( about the value} I1Eoo’

Now Huu - Hgg ^{may be expressed at large R in a} time-dependent picture ^as

where viilii ^{is the rate} of oscillation of the system between the channel states ij ) and ji ).

We now consider more specifically ^the predisso-

ciation of HD either into the ion pair ^{or into} ground-

state plus excited-state atoms. The potential ^curves

of H2 ^{relevant for the} present problem ^{are shown}

in figure ^{3. We shall here} summarize the discussion

given by Chupka et ^al. [2]. ^{The states which are} diabatically correlated with the ion pair ^{are charac-}

terized by the Coulomb potential ^{well. At short R} they

FIG. 3.

Potential curves of some relevant states of H2. ^{Sources :} Sharp [30] ^{for the B, E, F, and B’ states at 1.2-7 a.u., and the B"}

state at 1.2-4 _{a.u. ;} Glover and Weinhold ^{[12a] for the G, K state}

at 1-5.75 a.u. ; Kolos [11] ^{for the B",} B state at 4-12 a.u. ; Wôlniewicz and Dressler [12b] ^{for the H state.} Else : crude interpolation by ^the present ^{author. More recent data on the} B, B’ and B" states have been given by Jungen and Atabek [31], ^{and a recent} calculation of the

G, ^K state also by Wôlniewicz and Dressler [12b].

are the B ’2:’ ^{state and the F part of the} E,F 03A3’

state ; ^at large ^R they ^{cross the} Rydberg ^{states diaba-} tically correlated with H(1 s) ^{+ H} (n

2, 3, 4). ^The

first set of crossings, ^{which occurs at R}

^11.1 a.u., becomes strongly ^{avoided when. the} diagonalization

of the electronic Hamiltonian is performed [21].

The second set of crossings ^{at R = 35.6 a.u.} may still permit transitions between Rydberg ^{states dia-} batically correlated with H( 1 s) ^{+ H} (n ⁼ 3) and ^ion- pair ^states, although ^{with a small} probability (about

4 percent, ^{from the} best values quoted by ^Janev [21]).

The third set of crossings ^{at 270 a.u.} may be ignored.

Bearing in mind this description, ^{we see that for a} given exit channel the occurrence of an isotope ^effect

in HD predissociation ^will depend ^{on the} path ^{of the}

system, ^{which at each} crossing may be either adiabatic

or diabatic as regards ^the electronic Hamiltonian.

Once this path ^{is known, if the} first-order radial coupl- ing ^between Born-Oppenheimer ^{states is of minor} importance ⁱⁿ (10), ^the eventuality ^{of an} isotope ^effect

will depend ^{on the} electronic configuration ^{at the}

critical internuclear distance Rc where the splitting 1 Huu - Hgg ¹ equals Des ^as earlier defined.

If the electronic configuration is of the ion-pair type, V,iii ⁱⁿ equation (11) ^{is the rate of} exchange ^of

the whole electron pair between the proton ^{and the} deuteron, which falls off exponentially ^{with R in a}

very fast _way. Then d0/dR is very large ^at Rc. ^Besides,

to be able to dissociate the system necessarily passes Rc with a kinetic _energy at least equal ^to 1/Rc (Coulombic potential). From these two statements one may expect the behaviour of the system ^{to be} essentially ^non- adiabatic, ^{in the sense} that it will not evolve smoothly

from a Born-Oppenheimer ^{state to a channel state but}

will rather stay ^{in its initial} Born-Oppenheimer ^state.

Then no isotope ^{effect is} expected, ^{i.e. the} branching

ratio between exit channels will be close to unity.

If in contrast the electronic configuration ^at Rc ^is

that of a Rydberg state, Vij,ji ⁱⁿ equation (11) ^{is the}

rate of excitation transfer between ground-state ^and

excited-state atoms, which decreases with increasing R according ^{to an inverse} power law in R - 3 if atomic states i and j ^are optically ^connected by ^a dipolar

transition [22]. The decrease of Vij,ji will be much faster if the transition between i and j ^is dipole- forbidden, e.g. it will be as R - 5 for a quadrupolar transition, and will be an exponential decrease if the transition is spin-forbidden.

Thus, ^{if in} particular ^the system ^{is at} Rc ^{on a} segment of path diabatically correlated with the atoms in states 1 s + np (n ⁼ 2, 3, 4), ^{the rate} of decrease of Huu - Hgg ¹

with increasing R, and therefore also d0/dR, ^{will be} comparatively low. If in addition the dissociation

occurs near threshold on such a path, the radial

velocity vR ^at Rc will be low too. From these two statements one may expect the behaviour of the system ^{to be} partially adiabatic, giving ^{rise to an} isotope effect in favour of the exit channel adiaba-

tically correlated with the Born-Oppenheimer ^state

of the system ^below Rc.

These are the general expectations which will now

be checked by ^numerical computations.

Whereas the preceding discussion put ^the emphasis

on a description ^{of the} system ^{in terms} of the adiabatic basis, ^for practical ^{reasons the} computations ^were performed ^{in the} Born-Oppenheimer ^basis.

3. Numerical computation ^{of the} isotope ^{effect :}

model and method used.

We assume the HD mole- cule to be excited from its ground ^{state to a} given

rovibrational level of a bound Rydberg ^{state with}

ungerade electronic wavefunction. The predissociation

is supposed ^{to lead to the B} 1 03A3u+ ^{state, whose} potential

curve at low R is not far from the left part of those of the bound Rydberg ^states. Regarding ^the dissociation

path ^{two model cases were} considered :

(i) ^the system always ^{evolves on the Coulombic} potential ^{curve and} dissociates into the ion pair ;

(ii) ^the system ^{follows at the first} set of crossings

the electronically ^adiabatic path, ^which is assumed to lead to the 1 s + 2pnr ^{states of the atoms.}

In both cases only ^the couplings ^{with the} companion

state leading ^{to the same atomic states are} considered.

At low R this state is taken to be the E, F 11 9 + ^state.

We took the Born-Oppenheimer energies H.

and Hgg ^{from Kolos and} Wôlniewicz’ calculations of states B 1 Eu+ [9] ^and E, ^F 1 03A3g+ [11], given ^{between 1}

and 12 a.u. Within this range ^we used a spline ^inter- polation ; ^{at lower R we} supposed ^the potential ^curves

to parallel ^{that of} H2 , 1 sa g’

For case (i) ^we used these B and E, ^F potentials only

up ^to 6 a.u. At higher R ^we took the Coulomb law for (Hgg ^{+ Huu)/2 and we} extrapolated (Hg - H u)/2

above 6.5 a.u. according ^{to an} exp( - 2 IH ⁺ ÏH

law, ^where IH ^{is the} ionization potential ^{of H and} IH _

its electron affinity. ^This expression generalizes ^that

used in single charge ^transfer [8].

For case (ü) ^we kept ^{the B and} E, ^F potentials up to 12 _{ao. At} higher R ^we extrapolated (Hg. ^{+ Huu)/2} according ^to an R -6 law, ^and (Hgg - ^{Huu)/2 according}

to an R - 3 law.

With such potentials ^{the critical} point ^where Hgg - ^{Huu =} AE. ^is at ,Rc

^{6.6 a.u. in case} (i) ^and

at Re

^{19.0 a.u. in case} (ii).

We took the dipole transition moment Mgu ^from

Wôlniewicz’ calculation of transition

given between 0 and 12 ^a.u.

For case (i) ^we used these values only up ^to 3.6 a.u.

and thereafter joined ^them smoothly ^to the theoretical

expression ^{for the} ion-pair ^states, namely MgU

^R.

For case (ii) ^we used the whole _range of values from Wôlniewicz [23] plus ^{his value at} infinite R (taken

as 20 a.u.), ^{with a} spline interpolation.

We assumed the VR ipatrix ^between Born-Oppen-

heimer states to keep ^at any R ^{its value at} infinity. ^As

discussed above, ^this certainly ^{holds in the critical} region where the V’ couplings ^are operating. ^At

lower R the off-diagonal matrix elements actually

tend to zero whereas the diagonal ^{ones can be deduced} approximately ^{for HD from Kolos and} Wôlniewicz’

data for H2 [9,11], ^{but these} corrections are immaterial in the present problem.

For case (i) ^{we took the} off-diagonal matrix element of V 2 ^{from the} expérimental gap between the exit channels [24]. ^The diagonal ^{matrix element was}

obtained by multiplying ^the off-diagonal ^one by

(MD ⁺ MH2)/(MD2 - MH) [25]. ^We might ^{have made}

947

use of the theoretical values [26, 27] : the whole matrix would have been raised by ^{a factor of 1.015.}

For case (ii) both matrix elements were obtained from simple theory.

Table II displays ^{the set} of values used in both

cases (2).

TABLE II

The V2 ^matrices used between the [ Qg ) and 1 Qu > ^states (*)

Here Gu(R ) ⁼ RFu(R ) ^and Gg(R) ⁼ RF,(R),

where Fu ^and Fg ^{are the} radial wavefunctions asso-

ciated with the 1 Qu > and 1 Qg > ^{states ;}

Ebeing ^{the total} energy of the system ;

4l(R) ^{is the source term} describing ^the feeding ^of

the B 1 Lu+ ^state by predissociation ^{of the} initially

formed Rydberg ^level.

If 03A6(R) ^is given, ^{the most} general solution of

equation (12) vanishing ^{at low R includes two} arbitrary parameters. ^{The two} conditions _necessary for deter-

(2) Very recently ^{a numerical} computation ^{of the} vi coupling

between the B and E, F states in the range of 1 ^to 12 a.u. was per- formed by Alemar-Rivera and Ford [35]. ^The general ^{trend is as} described above (section 2) ^{and the} coupling extrapolates ^well to its value for case (ii) (see ^Table II).

The resolution of the two-state problem ^was performed ^{in the} Born-Oppenheimer basis, ^with

account of the first-order radial coupling ^{and the}

second-order dynamic coupling ^{as discussed in sec-}

tion 2.

The familiar coupled equations ^read (see ^Ref. [28]) :

mining ^thèse parameters ^are that boti1 G. ^and Gg ^be

pure outgoing ^{waves at infinite R- Thé} desired branch-

ing ratio between the isotopic ^exit channels is given by

where the k’s are the wave numbers for exit channels

1 ij > and ji ), respectively. With definition (2) ^{of the}

channel states, p is the ratio Pij/Pji ^{of the yields of}

these exit channels.

From the structure of equation (12) it results that the asymptotically ^correct (Gu, Gg ^{solutions are} linearly dependent ^{on the source function} O(R).

Therefore if the numerical calculation yield ^{the same} p from various functions «R) ^{it will still be found for}

any linear combination of them. Consequently, ^{if we}

solve equation (12) ^{with as} 4l(R) various delta func- tions â(R - Ro) scanning ^the expected range of the

source function, ^{and if we find} always ^{thé same value}

of _p, it will be the correct p for the true 4l(R). ^This

turned out to be the case in our present ^{work : in a case} where a strong isotope ^{effect was observed} (case ⁽ⁱⁱ⁾

at 9.23 meV above the higher ^exit channel), ^{p varied in}

an oscillatory way between 8.00 and 8.16 in the _range of 0.9 to 2.1 _{a.u. ;} the standard deviation was no more

than 0.7 percent ^{of the} average value. This satisfactory

result ^was expectable ^{since no} significant coupling

between the Born-Oppenheimer ^{states was} expected

to occur at low R, ^{where the} potential curves are far apart from each other.

The resolution of equation (12) ^with

is relatively easy. We made use of Numerov’s algo- rithm ; ^two linearly independent ^{solutions are pro-} pagated, starting from low R far in the classically-

forbidden region. The detailed prescriptions ^{for the} procedure ^{in the} simple ^{two-state collision} problem

were given by ^{Evans et al.} [29], ^{and we} independently

checked their validity in earm calculations. Now in the présent problem ^the irruption ^of b(R - Ro)

as 4l(R) ⁱⁿ équation (12) ^{means that at R =} Ro ^the

first derivative of the Gu ^function suddenly ^increases.

by ^some finite amount, ^as compared with its value in the absence of the source term. Since in Numerov’s

algorithm ^{at each} point ^{the functions are determined}

from their values at the two preceding points, ^this

sudden change ^{in the first} derivative of Gu ^{at a} given point, ^{of index n, is} equivalent ^{to a} change ^{in the}

value of Gu ^at point (n - 1). Since in Numerov’s

algorithm ^{the solution at} point (n ⁺ 1) ^{is a linear}

function of its values at (n - 1) ^and n, it turns out that the correct solution will be a linear combination of the solution without a source term and of a solution

starting ^{with the} input ^data (1, 0) ^for Gu ^and (0, 0)

for Gg ^at points (n - ^1, n). ^In other words one has to

propagate ^two solutions with usual input ^{data and one}

extra solution with the above-indicated input, ^{and to}

determine the linear combination of these three solutions which will yield ^{the correct} asymptotic

behaviour.

Actually ^Numerov’s algorithm ^{cannot be} applied directly ^to equation (12), because of the _presence of the d/dR ^{terms. To} avoid this difficulty ^{we made use of}

the transformations :

Thus coupled differential equations ^{in Y and Z are}

obtained which include no first derivatives.

4. Results and discussion.

Table III displays ^the

values of the isotopic branching ratio which we

obtained by ^the procédure described in section 3.

In the model where the dissociation of the B and E, ^F

states would lead to the ion pair (case (i)), ^the isotope

effect is weak, except very ^near threshold, ^{where it is}

dominated by ^{the term} kij/kji ⁱⁿ equation (15). ^{In a}

classical picture ^{this term describes} the fact that in _any time interval the system ^moves by ^a larger path length

if on the lower-energy channel than on the higher-

energy one, and therefore has a larger ^{chance to} escape

on the former channel out of the region ^{where the}

electron pair ^is likely ^{to be} exchanged. ^{The classical} description fails in the present situation, where the k’s

begin ^{to differ} significantly from each other only ^{at so} large ^{distances that the electron} pair ^{cannot be trans-}

ferred _any more. Thus the kij/kji ^{effect has to be}

considered in a purely quantum-mechanical picture,

where the motion of the system ^{cannot be viewed as} unidirectional.

In the model where the dissociation of the B and E, ^F

states would lead to the 1 s + 2pa ^states (case (ii)), ^a large isotope effect is observed near threshold.

These findings ^{are in} agreement ^{with the} qualitative

discussion in section 2.

We tested the relative roles of the first-order and second-order couplings ^between Born-Oppenheimer

states in case (ii), by suppressing artificially ^{either of}

them. We did not apply ^{this test in case} (i), ^{since a} significant isotope ^{effect occurs} only just ^above threshold, ^{where an} extremely large integration range is required for the numerical procedure ^to converge.

TABLE III

Computed isotopic branching ^ratios

(a) ^{Above the} higher channel, ^i.e. H- + D+.

(b) ^{Accurate to the last} figure ^indicated.

(’) Using-eq. (5) of van Asselt et al. [3].

(d) ^{Above the} higher channel, i.e. H(ls) ⁺ D(2p).

(e) ^1.009 if only first-order radial coupling ^between (pg ^and (pu is included ; ^8.12 if only second-order dynamic coupling is included.

(f ) ^{1.002 if} only first-order radial coupling ^between Qg ^{and Qu} is included.

949

As shown in table III, ^notes (e) ^and (f), ^{the iso-}

tope effect is practically entirely ^{due to second-}

order dynamic coupling. This remains true far from

threshold, ^{with as much as} 2 eV relative kinetic energy.

Thus the effect of first-order radial coupling ^between Born-Oppenheimer states turns out to be negligible

in the present problem. ^{This removes} any doubt on

the validity of using ^Demkov’s treatment, ^{as was done} by ^van Asselt et al. in the case of 3He4He+ [3]. ^We

further checked this point by comparing ^{our results}

in case (i) ^{with van} Asselt et al’s formulae (5) (eq. ^{(9) of}

the present ^{paper) where we replaced} J2 ^{I by the term} 2 IH ^{+ IH _ which we} had used in our calculation as

relative rate of decrease of Hgg - ^{Huu as a function}

of R. The comparison, which is shown in table III, gives ^{an excellent} agreement except very ^near threshold. Thus the applicability of Demkov’s treat- ment is strongly supported by ^the present ^results.

No such comparison could be made in case (ii),

where the decrease of Hgg - Hu" ^{is not} exponential,

and Demkov’s theory ^{does not} apply.

We shall now turn to the discussion of the actual

problem ^{of the} isotope ^{effect in HD} predissociation.

PREDISSOCIATION INTO ION PAIRS.

Let us consider the various possible paths ^{and see} which of them can

give ^{rise to a} strong isotope ^{effect as} reported ⁱⁿ

references [1, 2].

Any path such that the critical distance where

1 Hgg - Huu 1 = AEoo ^{occurs on} the Coulombic part of the path ^{is ruled out from the} results of our cal- culation for case (i). ^{This is not the case for the} path proposed by Chupka et ^al. [2], ^{which starts from}

the 4 fu 1 Lu+ ^{state, now described} by ^Kolos [11] ^as

the B", B state. As apparent ⁱⁿ figure ^{3 the} potential

curves of this state and of the companion ^H 103A3 g + ^state

come close to each other in the region ^{where the domi-}

nant configurations ^are 1 s 4fQu ^and 1 s 3du g’ ^both diabatically connected with fragments ^{1 s +} 2pQ.

However, neither has the H 1 L g+ ^{state been calculated}

with high accuracy [12b], nor is ^{the dipole transition}

moment between H and B", B known. Therefore no

numerical computation ^{of the} isotope ^effect along ^the path proposed by Chupka et ^al. [2] ^can presently ^be performed.

Another possibility is that the ion pairs ^{are formed} through ^a Rydberg ^state diabatically correlated with

H(ls) ^{+ H} (n ⁼ 3) and arise from the small part ^of the wavefunction which follows the electronically

adiabatic path ^{at the second} crossing, i.e. within the

Rydberg-type part ^{of the} path, ^{and a} significant

isotope ^{effect should} appear. This isotope effect will be in the right ^direction only if the relevant 103A3 u + ^state

is above its companion 1 Lg+ ^{state, as was indeed the}

case in the path proposed by Chupka et ^al. [2]. ^{For the}

states correlated with H(l s) ^{+ H} (n

3) ^the potential

curves are not well known [12b]. ^Some support ^{for the} present hypothesis ^comes from studies of Balmer-series

fluorescence, which show that the H(ls) ^{+ H} (n ⁼ 3)

channel is important ^{in the} predissociation ^of Rydberg

states of H2 ^and D2 ^{in the} energy region which includes the ion-pair ^threshold [32, 33]. ^From the work of

Borrell, Guyon ^and Glass-Maujean [33] ⁱⁿ particular

it _appears clearly that the main peaks associated with

ion-pair ^formation [2, 24] ^{feed the} H(ls) ^{+ H} (n

4)

channel and to a lower extent the H( l s) ^{+ H} (n ⁼ 3)

channel. Thus, ^{if the} Rydberg ^levels which are observed

to lead to the ion pair ^do predissociate into lower

Rydberg ^states diabatically correlated with H(ls) ^{+ H} (n ⁼ 3) ^and H(ls) ⁺ H (n

4), ^the Balmer-a and

Balmer-P yields should be much larger ^{than the} ion-pair yield. ^{If in contrast the} initially ^formed Rydberg ^levels predissociate ^{into the} B", B 103A3u+ state, the ion-pair yield should be much larger ^{than the}

Balmer yields. ^Current experimental ^{evidence rather} supports the former assumption, ^{with an ion} yield

of 2 x 10-3 [1] ^{and a} Balmer-(a + fJ) yield ^of

ca 8 x 10 - 2 [33]. ^{A third} possibility, ^{however, is that}

the initially formed ^{Rydberg levels} predissociate ^both

to the B", ^{B state and to states} diabatically ^correlated

with H(ls) ^{+ H} (n ⁼ 3, 4), ^{with a} branching ^ratio favouring ^{the latter states over the former one.}

A strong isotope ^effect, favouring ^{H* + D with}

respect ^{to D* + H} by ^{a factor of 2, was observed} by

Camahan and Zipf [34] ^{for the} production ^{of fast} long-lived high-Rydberg ^{H* or D* atoms} by impact

of 100-eV electrons ^on HD. The explanation proposed by ^these authors is essentially ^{as follows. Fast H*}

(or D*) ^atoms arise from dissociation of doubly

excited HD molecules (2p6u n/À), ^which may also

undergo autoionization. The n/À molecular Rydberg

orbital during the dissociation _process interacts for a

longer time with the D than with ^{the H nucleus}

because of the dissymmetry ^of the dissociation. There-

fore, according ^to whether the inner (2pau) ^electron

chooses either of the nuclei the probability ^{of the} Rydberg ^{electron to attach on the other} nucleus rather than autodetaching ^{will be different.}

This mechanism is an exemple of the effect of first- order radial coupling associated with a sudden _sepa- ration of the nuclei, ^as discussed in section 2 of the present paper. Besides, ^{such a mechanism} requires

that autoionization still be energetically ^{allowed as,}

internuclear distances large enough ^{for the choice of}

the inner electron _among the nuclei to be completed, namely ^{in the} region where the Isug-2pau ^splitting

is less than the reciprocal ^{of the} dissociation time.

This requirement, which is fulfilled in the case of the process studied by Carnahan and Zipf, ^{does not seem}

to be satisfied in our case : the various paths leading ^to

the ion pair ^{as well as to 1 s +} 2pu ^{atoms leave the} region where autoionization _may occur at internuclear distances where the _g-u splitting ^{is still} large (of ^the

order of an eV) ^as regards ^the potential ^{curves known}

from theory ^{or from} spectroscopy. However, ^a precise

knowledge ^{of the states} leading ^to H(ls) ^{+ H} (n ⁼ 3),

along ^with experimental ^{evidence on the branching}