On spectroscopic properties and isotope effects of vibrationally stabilized molecules

ChemicaI Physics 83 (1984) 333-343 North-Holland Amsterdam

ON SPECTROSCOPIC OF VIBRATIONALLY

-

_

PROPERTIES STABILIiED

333.

AND ISOTOPE EFFECIS MOLECULES

_

J. MANZ Lehrstuhl ftir Theorerzsche Chemie, Technrsche UnruersrrhtMiinchen, D -8046 Garchmg. Germany

R. MEYER Laboratorwm ftir Ph~szkalrreheChemie. ETH- Zentrum. CH-8092

Zurrch. Swttzerland

E. POLLAK Chemlcaf Phystcs Department, Wei:mann Instrtute of Scrence. 76100 Rehovor, Israel

J. RdMELT Lehrstuhl fir Theoretrsche Chemre. Unwersltat Bonn, D -5300 Bonn I, Germany

and H.H.R. SCHOR *) Lehrstuhi jiir Theoretuche Chemre. Technrsche Unwersrrat Miinchen. D - 8046 Garchmg, Germany Received

3 June 1983

Results of quantum and semxlassical calculations obtained for two different potential-energy spectroscopic properties and Isotope effects of the linear IHI and IDI molecules The potenhals

surfaces

are used to dtscuss

are a purely

repulsive

LEPS

surface and a DIM-3C potential with two van der WaaIs type minima for equivaIent IH - - I and I - - - HI configurations Both systems are dommated by the effect of vtbrational bonding giving rise to some very unusual spectroscopic phenomena, which are discussed in detail. The different vtbrational frequencies and rotational constants are roughly estimated as it = 120 (100) cm-‘, ~z = 280 (210) cm-‘, us = 360 (160) cm-’ and B = 0 0194 (0 0196) cm- ’ for IHI (IDI). A detailed discussion of the dependence of P,. pz and B on vs. their sensitwity to variations of the potential-energy surface, and a comparison wrth the vibrational frequencies of I, and HI (ID) is given. It is predicted that there exists only one exctted level of the antisymmetric stretching mode. The numbers of symmetrical stretchtng and bendmg levels are fatrly constant or may even decrease upon deuteratton Stmultaneously deuteratton destabihzes the molecule These unusual phenomena are rattonaliicd by our calculattons. A set of criteria for observing infrared and Rarnan bound-to-bound and bound-to-resonance state transitions are presented for the IHI and ID1 molecule_

1. Introduction Recent theoretical work [l-8] suggests that symmetric molecules like IHI, possibly BrHBr or ‘) Permanent versidade Brazil.

address- Departamento de Quunica, ICEX. UniFederal de Minas Gerais. 30 000 Belo Horizonte,

HLH in general, where H and L denote heavy and light (groups of) atoms, should be stabilized, in part at least, due to vibrational bonding. This new kind of chemical. bonding is important, or may become even dominant, if the bond energy, vibrational frequencies or the molecular geometry are strongly affected by vibration of the light central atom in the antisymmetric stretching mode.

0301-0104/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

In this paper we investigate the spectroscopic consequences of vibrational bending. The purpose is thus to gutde the expetimentalist towards the verificatton of vlbrattonal bonding in HLH type molecules_ We have to point out from the outset. however. that this paper deals mainly with qualitattve effects and some semiquantitative spectroscop~c predrcttons. This restrictton arises aimost inevitably from the fact that prectse potential-energy surfaces. transttion moments etc., of favourable molecular candidates (which should be the basis for quantttati\e predictions) are not available We hope that the qualitative effects presented below w111nevertheless serve as useful criteria for the experimental discovery of vibrattonal bondmg tn HLH type molecules. namely characteristic (and only very few) blbrdtiondl transitions and unexpected isotope effects. As a first step towards spectroscoptc critena of vibrational bonding. we consider m section 2 the collmear IHI and IDI molecules using two potential-energy surfaces with characteristic properties. This allows us to extrapolate typtcal vibrational frequencies for the symmetric and antisymmetric stretching modes. together with rough estimates of the relevant infrared and Raman transitions. The Investigation is extended to three-dimensional IHI and ID1 in section 3, yielding additional information on the bending frequencies. In section 4 we discuss the implications with a view towards future spectroscopic experiments. A comparison of the present theoretical predicttons with the extensive results available from matnx-isolation spectroscopy of HLH systems [9-161 is in progress.

2. Collinear IHI and ID1 In this section, we mvestrgate common and different properttes of collinear IHI and ID1 on two potential-energy surfaces. We employ the LEPS potential-energy surface A of ref. [17] and the semi-empirical DIM-3C potential of Last [l&19] *_ For convenience, they are referred to as LEPS-A potential and Last potential below, re- We use Last’s eq (10) but replacethe misprint5’)-R by gR [ISI

spectively. The LEPS-A potential has aIready been used in refs. [l-7,17,20-23]; the Last potential has been used in ref. [7]. The methods of investigation include, among others, exact classical [1,2,5] and quantum [3,4,7] evaluations of bound states as well as the diagonal corrected vibrational adiabatic hyperspherical (DIVAH) prediction of bound state and resonance energies [8,22] (see also refs. [3.7,24]). These techniques are used here directly, without repetition of computational details which have been presented previously [l-3,5,7,8,22,25]. The LEPS-A and Last potentials are plotted in the bottom panels of figs. 1 and 2 respectively, using mass-wetghted coordmates _Y= ( “‘,.,r/“‘,r)l’Z’r

xi.

J = rx,,

giving rise to skewing angles q,,, = arctan [n*x(2~t

+ frzx)/n*f]l/l,

car= 7.2” and 10.1” for X = H and D, respectively n = 1.008 amu, nzn = 2.014 an-m, nzi = 126.9 amu). Both potentials have nearly thermoneutral energy profiles along the reaction path in accord with several other LEPS potential-energy surfaces which have been suggested recently for IHI [17,26]. predicted by (The deeper well, -28.3 kJ mol-‘, the BEBO method [27] is presumably an underestimate. For example, the BEBO method predicts for ClHCl a well depth of -6.5 kJ moI-’ [27], whereas the recommended experimental value is 36 kJ mol-’ [28].) However, the LEPS-A and Last potentials differ in their details: The LEPS-A potential is minimum-free and has a saddle point located at a symmetric collinear configuration. The barrier height is 4.6 kJ mol-‘. Thus the overall Interatomic forces are repulsive, and the LEPSA potential provides a pure case of vibrational bonding [l-73. On the other hand, the Last potential also has a symmetrical barrier of 1.3 kJ mol-‘, but in addition it has two equivalent shallow minima for the collinear I - - - HI and IH - - - I confrgurations whose well depth is 3.8 kJ mol-‘. It provides a mixed case of interfering vibratronal and van der Waals type bonding [7]. These two representative potentials should enable extrapolation of some characteristic spectroscopic properties of IHI, ID1 and similar molecuIes_

J. Mam et al. /

Vrbrationaliystabdrzed molecules

Ftg. 1. LEPS-A potential [17] and quantum states for collinear IHI (a) and IDI (b). Bottom panels show eqmpotentlal contours plotted m mass-weighted coordinates: x = (mrxt/nzxt)tfl rtxt and y = rxt, X = H and D Contours are drawn at E = -268 4, -294.7, - 304.0 kJ mol-t for IHI and -279.7, -298-7. -304.0 kJ mol-’ for IDI. The first two values correspond to the dtatomic energy levels E, and E,, for HI and DI rcspecttvely, as indtcated by arrows. The positton of the saddle point is tndicated by +_ The top panels show the minimum potential V,, and effective potentials 9, versus r = (x’ + yz)‘fl = 7_981r,, for IHI and = 5 679rn for IDI. The solid and dash-dotted horizontal hnes indicate the energy levels E”, “p with gerade and ungerade parity, respectively. A shape-resonance state produced by the &t.(r) effective potenttal is indicated by the dashed honzontal hne.

The vibrational IHI and ID1 bound-state and resonance energies are very conveniently evaluated ,with the help of the DIVAH method [8,22]. Thus method has the advantages of being numerically simple, providing ilhtminating interpretations of the results (see below), and being very accurate, typically within 10m2 kJ mol-‘, see reefs.[3,7,8,22]. For example, it predicts two collinear ID1 bound states on the LEPS-A potential, see fig. 3, with energies -300.6690 and -299.3898 kJ mol-‘, whereas the exact results, obtained with the tech-

with energies - 300.6668 and -299.3815 kJ mol-‘, respectively, see fig. 3. The physical idea behind the DIVAH method is a Born-Oppenheimer type separation between the slow IX1 symmetric and the fast antisymmetric stretching modes (see ref. [6] for a detailed discussion). These two vibrations correspond to motions along (Delves) hypersphericai coordinates r and ‘p, respectively, where (ref. [29], see also refs. [6,30,31] for equivalent definitions)

nique of refs. [3,25],

r=

also yields two bound

states

(x2 +y2y

- (m,.,,/m,,yr,,,

336

J. Manz et aL /

Vtbrationalb stabdced molecules

-270

“I’

-2901 I

III\

I

’

I

I

-x/A

Ftg 2 Last potenttal [IS] and quantum states for colhnear IHI (a). adapted from ref [7] and IDI (b). Bottom panels show cqutpotenttal contours plotted m mass-weighted coordmates x and) Contours are drawn at E = -267 3. -294 3. -308.2. - 311.0 kJ mol- ’ for IHI and E = - 278.8 - 298 4, - 308 2. - 311.0 kJ mol- ’ for IDI They correspond to the dtatomic energy levels E, and Et, for Hi and DI (as mdtcated by arrows), the asymptottc well depth - D, and a contour in the van der Waals type potential wells, respxtr\ely_ The posmons of the saddle pomt and minima are indicated by $ and x respecttvely. Top panels show minimum potent& v IX?,”and effective potenttais U&, versus r. The sohd and dash-dotted honzontal lines mdrcate the energy levels ELISnpwtit gerade and ungerade panty honzontal hnes

respectively.

q = arctan(

Y/A-)

==OS(r,,

-Try),

-

The hypothetical

energy levels of

-_/f + q~J2,

O=zg,
One first evaluates the effective (diagonally corrected [8,22]) antisymmetric stretching mode energies U&,(r) for fixed symmetnc stretching mode coordinate r. These U,,(r) provide vibrationally adiabatic potential energy surfaces for the sym-

V,,,

(excludmg

vibrational

bondmg)

are also indtcated

by solid

metric stretching modes. Next one evaluates the energies El,, “p as solutions of the radial Schro dinger equation for the symmetric stretching mode,

using the effective potential U,,(r). The quantum numbers U, = 0, 1, 2, ___ and (np) = (Og), (Ou), (lg), (lu), enumerate the symmetric and antisymmetric bound state and resonance energy levels wrth gerade (g) or ungerade (u) parity. Note that

_-

_.. _ f_ Manz et aL / VtbrazionaIQ &&ed~molectd&

_

.

-

.- -_ 33; -_

fold. Thus- the Jdissociation -energy,-=symmetrical stretching.mode frequencyand th~number of Yr levels increase with- quantum -number-&since the well depth of the U,,(r) increases with & Simultaneously, they d_ecreaseupon changing-from-gerade (p = g) to ungerade (p = u) parity, see figs. 1 and 2.. -.,

Fig 3. LEPS-A potential V(r,cp) and the two bound-state wavefnnctions ~_Jr(‘.‘p) for collinear IDI. Equipotential contours are plotted for E = -305. -295, -285 and -275 kJ mol-‘. The saddle pomt is indicated by $. Contours of the nonnalised wavefunctions are given at &O 04, t0.12,

&O 20

and kO.28 Negative values are indicated by broken lines.

the (np) states correspond to the spectroscopic assignment u3, for example (np) = (Og), (Ou) correspond to u3 = 0, 1. The resulting effective potentials U&, (r) and the lowest energy levels for IHI and ID1 on the LEPS-A and Last potemu& are plotted in the top panels of figs. 1 and 2, respectively. We note the following results: (1) The effective potential curves U,,(r) and Q,(r) are pairwise quasi-degenerate for large values of r. Asymptotically they approach the vibrational XI(n) levels of the dissociated products I + XI(n)

At smaller values of r, the quasi-degeneracy is removed due to the interaction of the reactant and product valleys. Obviously, the interaction region shifts towards larger values of r as the IX1 energy increases. (2) The wells in the effective potentials U,,(r) support bound states with energy EU,_RPaccording to the usual rule: the deeper the minimum the larger are the IX1 --, I + XI dissociation energy, K,(XI) -Eo.np, the frequency of the symmetrical (Y,) stretching mode,

(3) The shapes of U,,(r) are a direct, consequence of the potential profiles V(r,cp) along ‘p at a given value of r. At large values of r, the profile is a symmetrical double well, separated by a very large barrier (the three-atom dissociation energy) hence the quasi-degenerated asymptotic behaviour of U,,(r) and Q”(r). For smaller values of r the quasi-degeneracy of the U&(r) and U,,(r) is removed, and near the saddle point of the potential surface V(r,v) the double well merges to form a broad single well. Therefore the zero-point energy of the antisymmetric IX1 stretching mode is reduced. Of course, with further decrease of r the single well becomes narrow and the yj frequency increases again. The lowering of the IX1 zero-point energy in the transition region is the origin of “ vibrational bonding’! [l-7]. When the zero-point energy function is superimposed on the variation of the minimum potential energy defined as Gm(‘)

= og*V

6,

V(r~q),

one finds a well in the vibrationally adiabatic potential U&Jr). A similar behaviour is found for the excited potentials U,,, Uts, U,, shown in figs. 1 and 2. Note that figs. 1 and 2 are scaled such that the curves for Vmm(r) coincide for IHI and ‘IDI. The findings (l)-(3) are characteristic of arbitrary HLH system. We now turn to a more detailed investigation of IHI and IDI on the LEPS-A and Last potentials. (4) For the LEPS-A potential, the minima of v,,(r) are exclusively .due to vibrational bonding smce Vm,,,(r) is a purely repulsive curve. For the also arise Last potential, the minima of U,,(r) mainly from vibrational bsnding, but now they are enhanced by the van der Waals type minimum of I&,(r)_ Comparison of both potentials indicates that the well depth of U&(r) can be seen as a sum

and the number of ZJ~levels for a given np mani-

of two contributions [7].-One, the result of vibra-

33s

J Man: et at /

Vzbratronat~stabked

tlonal bondmg, which is = -9 kJ mol-’ for IHI [7] and = -6 kJ mol-’ for ID1 independent of the exact details of the potential surface. The other is the well m the q,,,,,(r) curve, found at r = r*. where I-* denotes the iocation of the minimum of u&(r). We stress this point since it gives us an esumate of the magnitude ol’ vibrational bonding. see section 3. (5) Especially Important is the effect of the wells In I’,,,,(r) on V&,(r). which is always the most shallow effective potential curve_ Obviously. the repulsive V,,,(r) for the pure case of IHI \lbr,ttlonal bonding on the LETS-A potential provldes J. Itmitmg case of L/,,(r): its mmlmum IS extremely shallow and so it supports resonance levels but no truly bound quantum states. Howe\er, the true IHI potential should have at least a van der Waals minimum. dnd the Last potential mdlcates that some levels should then exist In the O;,(r) potentral. This is of conslderable spectroscoptc relevance smce it implies that the allowed Infrared l!3 = 0 --, 05 = 1 transitions may be observed Otherwise. only 0 -+ 11~2 3 IR bound-stateto-resonance transItIons \vlth much larger frequencies would be observable. (6) Comparison of the L&,(r) surfaces for IHI .md ID1 yields .m unusual isotope effect: the IHI bond is destabihzed upon deuteration. This effect IS contrary to the well-known stabilization of an ordmary R-H bond upon deuteratlon. It may be understood as d consequence of the smaller c~nrra:IOIIF of antisymmetric V~ ID1 zero-point energies. In comparison with IHI. due to the heavier mass of D m comparison with H. The rule is thus- the the hghtsr the mass r7z,_ of d HLH molecule. stronger IS the vibrational bonding. A rather extreme example of this rule has already been discovered m ref. [S]. the molecule ClHCl apparently does not eylst (in the gas phase) but if hydrogen is replaced by muomum. the isotopic molecule ClMuCl IS vtbratlonally stabilized. tnlh,,, = nrH/9; T&i”= 2.2 lls). (7) The second remarkable isotope effect 1s that the number of vlbrationally stablhzed Y, levels m a given V~ manifold may decrease if IHI is replaced by IDi. Usunliy deuteration mcreases the number of bound states by a factor = (nro/nzH)“’ in an ordinary RH bond.

motecxdes

(8) The

antisymmetric stretching frequencies are sensitive to deuteration. The ho, = Eo.oll - % Qg ratio fiw,(IHI)/ttw,(IDI) may exceed the value (n~,/m,)‘~, depending on the depths of the G&(r) and Q,(r) wells (see, e.g., fig. 2). symmetric stretching frequencies (9) The Aw,(G’~) are only weakly isotope dependent. For example, tro,(Og)= 1.48 (1.25) kJ mol-’ and fiti, = 0.87 (0.81) kJ mol-’ for IHI (IDI) on the Last potential. The case of pure vibrational bondmg, represented by the LEPS-A potential, apparently provides a lower limit for Ao,(Og), i.e. tiw,(Og)~l

3kJmol-‘==llOcm-‘.

Note that the corresponding Raman frequency for the u, = 0 --, u, = 1 transition should thus be considerably lower than the vibrational frequency of I?. Aw = 2.6 kJ mol-’ = 214.5 cm-‘. On the other hand. tio,(Og) is stdl considerably larger (by more than a factor two!) than the hypothetical value for IHI “wIthout vlbrational bonding”, as represented by V,,,(r) instead of 0&(r) as effective potential for the symmetric stretch. see fig. 2. Note that V”,,,,(r) represents the hypothetical case of exclu-

sive variations of potential energy along r. without changing the potential profile along q. i.e. without mbrational bonding. This comparison confirms the dommance of vibrational bonding on the Last potential (see item (4) above and ref. [7]). (10) The LEPS-A and Last potentials indicate that excited levels are unstable for u3 > 2 unless the respective curves I/,,(r), Y”(r) etc. ‘Ldlve*’ below .&,(X1). Apparently, ths would require a conslderably deeper mimmum of the potential-energy surface_ most favourably in the equilateral IHI configuration. If tlus happens at all, It should be more probably realized for IDI than for IHI. Otherwise, uj = 0 ---, u, > 3 bound-state to resonance-state transition may still be observable_ The width of these transition lines should correspond to the width of the EL,,,,, etc. resonances, which is unknown but could be even smaller than the = 0.01 kJ mol-’ = 1 cm-’ resonance widths of the C! + HCl - ClH + Cl reactlon probabilities [S]. (11) The curves U&, (r ) also determine the range of the wavefunctions. compare figs. 1 and 3, which m turn affect the transition amplitudes for IR and

J. Manz et aL /

Vtbratroncl&stabbed

Raman transitions. In the case of the Last potential (fig. 2), the amplitudes of allowed ground-state to O,,CJ,= 1 transitions should be small for ui = 0 and increase with (small) quantum number u,. This is, of course, a consequence of the shift between the minima of U,,(r) and U,,(r), which in turn is basically a consequence of the location of the van der Waals type potential wells. If the true IHI potential should have a smgle minimum at a symmetric configuration, then the shift between the U&(r), U,,(r) minima should be re‘duced, implying larger transition amplitudes for ground-state to u, = 0, 1, - - - , uj = 1 transitions. (12) The minima of the curves U&,(r) at r = r* provide estimates of the IX1 bond length. For example, for IHI in the ground state, one obtains 1/2r* = 3.70 A and for IDI: r; = G = (~“I/~,H,) 3.68 A and thus B = A’/m,r*f, = 0.232 kJ mol-’ = 0.0194 cm-’ for IHI and B = 0.234 kJ mol-’ = 0.0196 cm-’ for IDI. A systematic investigation of the rotational constants is included in section 3.

3. Three-dimensional IHI and IDI The symmetric and antisymmetric stretching modes of collinear symmetric HLH molecules (section 2) are complemented in three dimensions by the degenerate bending modes. Clearly the bendmg modes will tend to destabilize any symmetric ABA molecule since they involve a possible decay channel not accounted for in the collinear case. The simplest estimate of the energy levels E,,,-J,, may be obtained by adding bending mode energes AK:+ to the collinear results, E, ,+, = EL,,,,,, (3.1) The notation used in eqs. (3.1) implies that the bending energies, like the symmetric stretching energies (cf. section 2) depend primarily on the fast antisymmetric stretching mode ZJ~.In the following we will see that the bending energies contribute substantially to a very large isotope effect. A number of techniques are available to evaluate directly the three dimensional energy levels &,,,I,, [4-6,251. Certainly an exact quantum evaluatton (see ref. [4]) would be most desirable_ However,

moiecules

339

since the exact potential-energy surface is unknown, we consider it quite appropriate - and in f,jct most economic - to use the- semiclassical evaluation [1,2,5,32,33]. Note that previous work [h-6,32,33] implies that the semiclassic-al results agree well with exact and approximate DIVAHtype ones. For simplicity, the discussion is restricted mainly to the LEPS-A potential in this section. The semiclassical method used has been described in detail previously, here we give a brief review of the essential ideas. Collinearly, the bound and resonance states of IHI and ID1 are formed as a result of a well in the vibrationally adiabatic potential-energy surface U”,,(r), cf. section 2. As we have shown elsewhere [2,23] the adiabatic wells of the averaged adtabatic potential-energy surface U, = OS(U,, + U”/,,) may be found semiclassically by quantising resonant periodic orbits (RPOs) with a (2n + 1)h action condition. Both the location and energies of the quantised RPOs are in excellent agreement with the location and energy of the wells of U;(r). Thus the first ingredient we use is an evaluation of the adiabatic wells via quantised RPOs. To generahse to 3D, one should look for the 3D analogues of the collinear adiabatic wells. Here, we utilise the periodic reduction method devised in ref. [34]. A general u-i-atomic molecule undergoes rotational, bending and stretching motion_ Surely for a system such as linear IHI, the overall rotation is much slower than the vibrational modes because of the very large moment of inertia of the molecule. Furthermore, it is often the case that bending motion is much slower than the antisymmetric stretching motion. Thus we use a Bom-Oppenheimer type separation, in which we first freeze the rotational and bending angles, then find the RPO at each set of fiied angles, and finally, quantise it. Each RPO corresponds automatically to a near!y frozen symmetric stretching motion, in accord with the Born-Oppenheimer scheme. Thus we obtain a bend angle y dependent adiabatic well U,(y), where n corresponds to the previous quantum number of the asymmetric stretch (section 2), and the corresponding quantum u,( = 0) for the frozen symmetric stretch is omitted. We now average the hamiltonian over the (fast)

340

J Mm: et al / Vlbratlonal!v stabbed

antisymmetric stretching motion of the RPO. This provides an effective hamiltonian for the bending motion of the n th adiabattc well:

Here B,( y ) is the angle-dependent bending coefficient correspondin, 0 to the inverse of a r-dependent reduced mass, and P, is the bending momentum. We defme the bending angle usmg the Natanson-Smith-Radau coordinate system [35. 361. More details on the exact defmition of B,(y) are gtven in refs. [5.34]. The t?rth bending level E,,IG of the Z antisymmetric stretching manifold of coplanar III1 or ID1 may now be found via semiclassical quantisation of ff,-_ in order to evaluate the rotational motion_ one averages the hamiltonian over the bending motton. One can show [34] that for collinear-type potential-energy surfaces the hamtltonian H,,,; assoctated with the nzth bending fith anti-symmetric stretching level is Just

where J IS the total angular momentum of the system and B,,,, - the rotational constant - is just half the inverse moment of inertia of the moiecule, averaged over the vibrational motions. In this fashion one has generated the semiclassical 3D properties of the (averaged) adiabatic wells L&(Y) keepmg the symmetric stretch motion frozen. Past experience [4-61 has shown that it is reasonable to assume that the bending mode energies of a given pair of adiabatic wells U,s, Q,,, are well approximated by the energies of the averaged well U,. Finally. we assume that the symmetric stretching and bending modes are separable. This assump-

tion is probably not too good for excited symmetnc stretching states. however, our main purpose here IS to give a qualitative guide to 3D effects and as such it should be reasonable_ This assumption of separability enables us to evaluate the 3D levels simply by adding the bending energies AE,,Q,, to the collinear u,u3 results [cf. eq. (3-l)]. In fig. 4 we show the effective bending potentials Q(y) at ri= 0, 1 for the IHI and ID1 systems on the LEPS-A surface Note that Q(y) IS defined

t\l

176

molecules

’ ’ ’ I/t

I

180

r

184

’

’

I

1

IDI

IHI

176

180

181

r

Ag 4. Effectwe bending potentials U,( y ) versus bending angle y at n = 0, 1 on the LEPS-A potenttal(17j for IHI (a) and ID1

(b) The antisymmetric stretching quantum numbers fi = 0 and 1 correspond to us = 0, 1 and u3 = 2, 3. respectively. The honzontal hnes represent the semiclassical coplanar bending states.

only for a finrte range of angles. For angles larger than the ranges shown in fig. 4 one no longer finds RPOs with the Dresctibed value of the action. At these large angles the saddle point energy rises too steeply and the RPOs cease to exist. We also show in fig. 4, as straight solid hnes, all semiclassical bending states supported by the bending potentials. Thus for coplanar mechanics IHI has two ti = 0 bending levels and three 5 = 1 bending levels while ID1 supports two states and four states, respectively_ As expected, substitution of H by D increases the number of bending states. Note also. that as n Increases, the bendmg potential becomes much shallower. At higher energies, the potential-energy surfaces “open up” and the system is no ionger so severeIy constrained

to the collinear region. This opemng up causes a curious isotope effect with regard to the bending IeveI spacing in the 5= 1 resonance states. Since the ii = 1 surface for IDI occurs at a lower energy than that of IHI, the bending potential at ff = 1 for IDI is stiffer at the bottom than that of IHI at ti= 1. The net result is that for the excited state (5 = 1) the bending spacing of ID1 is of comparable magnitude to that of IHI. In contrast, for the ground state (% = 0) the ratio of IHI and IDI

J. Manz et ai. / Vtbrationolly stabdizedmolecules Table 1 3D spectroscopic constants of iHI ‘1 I z

A

:b

0 0 0 1 1 1

22 3r 40

1 1 1

:b o”

AK+,

(W mol-‘)

3.35 6.79 0.22 1.51 3.50 5.49 5.93 792 10.35

341

Table 2 3D spectroscopic constants of IDI a) &,G., (cm

-1

)

0.01% 0.0199 0.0202 0.0178 0.0182 0 0185 0 0185 0.0188 0.0191

n) The molecule is in its rotational and symmetric stretchmg mode ground state, J= 0 and u, = 0. The antisymmetric stretching mode quantum number R= 0 or 1 represents us = 0, 1 or 2, 3, respechvely.

IJ:

- n

00

0

:i

A E”;,,, (kJ mol-

’)

Bu2Iv3(cm -‘I

0

2.52 7.67 5.10

0.0199 0.0202 0 0201

o”

I

1.63

0.0185

:b 22

1 1

5.21 3.42 540

0 0190 0.0187

::

1

7.55 720

00189 0.0191 0.0192

,“z

1

9.34 9.18

0 0.0194 0194

z

1

0 0196 0198

1347 1131

-’ Notation as m table 1.

bending spacings is - (m,-,/mt.t)1’2, as expected. For the 3D system, the bending motion is of course doubly degenerate. Thus if in the coplanar tz manifold we find A4 bending states, then in 3D the nth manifold will support M2 states. In 3D the bending level energies AJ??,,,~~~~ with respect to the bottom of the bending potentials l&(y) with m, quanta in one degenerate bending mode and m2 in the other will be

AE t?l,l?*z” -= E,,,,,+E,_R-2Un(y=~) = AEt,;(np, = AC+,, where the connection between the coplanar and 3D quantum numbers is u2 = ml + m2 and I = irn, - mJ_ Similarly the rotational constant B,_+ has contributions from each bending mode, to first order

In tables 1 and 2 we present the bending energies and rotational constants associated with each i? manifold for IHI and IDI, respectively. The semiclassical Boo,-,for the LEPS-A potential agree well with the collinear quantum estimates B for the Last potential, see section 2. This is a consequence of the similar ground state IHI, ID1 geometries which are dominated by vibrational bonding for both potentials, not by the van der Waals minima

of the Last potential which in fact would have implied a large I-I distance, hence a smaller value of B. Note that for the LEPS-A surface addition of the bending energy to the collinear bound-state energies leaves only one bound 3D, J = 0 IHI state 14-61 while for IDI we predict that in 3D no truly bound states exist, the bending energy will dissociate the two collinear bound states. Here we have an example of an anomalous isotope effect, substitution

of

destruction

hydrogen

by

deuterium

causes

the

of the molecule instead of its stabiliza-

tion. Let LEPS-A

us now

use the bending

potential

energies

of

the

as a rough guide to assess the

number of bound 3D states one would find on the Last potential energy surface. From eq. (3.1) one finds that for IHI the bound states would be E ulooo, ut = O-5; Eu,,,ol, u, = O-2; E,,,I~, u1 = O-2 or altogether fifteen bound states at J= 0 (of which three are doubly degenerate). For IDI one would find the bound states Eu,o~o. u, = O-5, E u,o~l, D, = O-3; E,,po, u1 = 0, 1; Eopl or altogether sixteen bound states (of which three are doubly degenerate). Thus, in the Last potential, because of the deeper adiabatic well in IHI (see section 2 and fig. 2) and the smaller bending frequency of IDI one finds approximately the same number of J = 0 bound IHI and IDI states. Note though, that usually one would expect D

342

3. Man:

et al. /

VtbrarronalJv srabtked

substitution to increase the number of states by a factor of 2”‘. The quantitative bending frequencies derived in , this section are of course sensitive to variations of the potential-energy surface. however the qualitative trends should be Independent of the potential as long as vlbrational bonding is donunant.

4. P- scussion The results of sections 2 and 3 imply the followmg spectroscopic criteria for infrared and Kaman bound-state-to-bound-state transitions of vibratlonally bonded IHI, ID1 and similar HLH molecules: (1) The HLH symmetric stretching frequency should be smaller than the corresponding vibrational frequency of the H-H molecule. Its minimum value is obtained In the case of pure vibrational bonding. For example. tto,( u3 = 0) > 1.3 kJ mol-’ = 110 cm-’ for IHI. If the antisymmerric stretching mode is excited to uli = 1, tio,( ug = 1) may be considerably smaller, depending on the potential. The general rule is: the smaller tto,, the stronger the influence of vIbrationa bonding. Deuteratlon may decrease w, by as much as 26%. The number of symmetrical stretching (v,) levels in a given V~ manifold may also decrease upon deuteratlon. (2) The bendmg frequency trwz was found in the range 250 cm- ’ = 3 kJ mol-’ which is larger than the symmetnc stretching frequency, tto,( u3 = 0) Deuteratton should lower kw,(o, = 0) by the usual factor (mH/~zD)1/2. As d rule, there exist only very few bending levels for u3 = 0. e.g., u2 5 1 for IHI or IDI, if vlbrational bonding is domin‘ant, simply because bendmg tends to dissociate the weak Mbrational bond. (3) Only very few antisymmetric stretching levels should exist, typically only u3 = 0, I for IHI and perhaps I+ = 2 for IDI. Conversely, spectroscopic detection of infrared LJ~= 0 ---, o; = 3 transitions would clearly indicate that vibrational bonding IS no longer dominant_ The u3 = 0 --) u; = 1 transltion frequency is very sensitive to the potential-energy surface. Deuteration should reduce this

molecules

frequency by a factor larger than (mD/mH)lfl. (4) All HLH bound-to-bound-state transition frequencies should be smaller than the zero-point energy of the dissociated LH molecule. At larger frequencies, one expects bound-state-to-resonance transitions, e.g., u3 = 0 + u; = 3 whose width will depend on the lifetimes of the resonances. The resonance energies have been discussed in sections 2 and 3. The dissociation energies for HLH ---, H + LH have a contribution from vibrational bonding which apparently does not depend very much on the potential [7]. This contribution is due to bonding variations of the antisymmetric stretching zero-point energies, section 2. This bonding contribution should be stronger than the antibonding [6] variations of the bending zero-point energies, section 3. For IHI the-sum of both contributions is = 9 - 3 = 6 kJ mol-‘, and a 50% smaller value results for IDI. namely =6-3=3kJmol-‘.Asa consequence, vibrahonal bonding will be dominant if the true potential well depth does not exceed 6 (3) kJ mol-’ for IHI (IDI). Simultaneously, one should observe destabihzatlon upon deuteration. This isotope effect should be remarkable in temperature dependent spectroscopic studies, i.e. upon heating the hnes of ID1 should vanish more rapidly than those of IHI. These spectroscopic criteria and isotope effects include a number of observable parameters which are very sensitive to the potential-energy surface, i.e. they may be used to mvert spectroscopic into potential-energetic information_

Acknowledgement Helpful correspondence with Dr. B.S. Ault and fmancial support of the Deutsche Forschungsgemeinschaft, the Minervastiftung, the Fonds der Chenuschen Industrie, as well as grants of the US - Israel Bi-national Science Foundation and the C.N.Pq. (Brazil) are gratefully acknowledged. The computations have been carried out on the CYBER 175 of the Bayerische Akademie der Wissenschaften, at the ETH Zdrich, at the Weizmann Institute and at the RHRZ of the Universitgt BOM_ JM and HHRS

would also like to thank for

_ J. Matu et al. /

yibrationaky stabdried mcifecades

the kind hospitality experienced during their visits of the Chemical Physics Department of the Weizmann Institute at Rehovot and of the Lehrstuhl fiir Theoretische Chemie of the Technische Universitgt Miinchen, respectively.

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