Proton-molecule collisional interactions

CHEMICAL. PHYSICS LE-ITERS

Volume 71, number 1

PROTON-MOLECULE VIBRATIONAL

COLLISIONAL

1 April 1980

INTERACTIONS.

THRESHOLD EFFECTS

IN COMPUTED CROSS SECTIONS FOR CO TARGETS

F-A. GIANTURCO * l~¶ax-Ptanck-Instltut fir Stromungsforschu~zg.

Giithngen. FRG

and

U-T. LAMANNA and M. ATTIMONELLI Ishtuto dt Chm~ca Fmca dell’Umvemt& Bari. Italy Received 20 December 1979

The vlbrationally meIastic mtegral cross section for the IO) -c I 1) process m H* coksions with CO<’ X ) molecules is exammed m the energy regon spannmg the successwe openmgs of the higher vlbratlonal wet channeIs (Eco~ < 1.0 eV). The spherical component of a prevtously computed ab m&o potenttal energy surface is used as interaction and the vibrational coupI& IS obtamed via an approwmation presented before. The effects on computed cross secttons origmatmg from the strongly attractive potential with a deep weII at R J 2.7 au are analysed and discussed.

1. Introduction The energy transfer processes that have been observed in simple molecules colhdmg with collimated proton beams [l-3] have opened up a new area of investigation m the domam of molecular subreactive, inelastic colhsions. The special nature of these ion-molecule encounters, in fact, IS immediately perceived by sunply observing the dfferent behamour of H+ projectiles with respect to that of the well studed lithium ions on sinular targets (e.g., see ref. [4]). The lack of core electrons in the former partners allows much stronger interactions unth the molecular electronic cloud smce no e--erepulsion terms contnbute at smaller relative distances, as they Instead do for the latter Ions, and therefore the relatively more marked alterations of electron densities withm the bonchng regions are in turn expected to cause drfferences in the scattering observables. As an example, the existence of interference effects * Von-HumboldtStitung

Seruor Fellow 1979-1980. Permanent address- Istltuto dI Chunica Fwca deIl’Uruversi& CitG Umversltana. Rome, Italy.

that give rise to glory-like undulations in the forward scattering direction has been detected in the measured vibratlonally inelastic differential cross sections of Ha on H2 [S ] and H+ on CO [6]. The latter behaviour has also been confiied by our recent model calculations over a wide range of colhsion energies [7] that employed a previously computed [S ] potential energy surface (PES) for that system. It therefore becomes interesting to try and extend the theoretical analysis of the scattering behaviour for thus class of molecular encounters to other energy domains where the nature of the interaction plays an even more transparent role in affecting dynamical observabks hke integral cross sections, eIastic and inelastic, or the corresponding differential cross sections in the rainbow angles and beyond. In the present work we are examining the low-energy behaviour of the vibratlonally inelastic integral cross sectlon, associated to the (u = 0 + u’= 1) excitation of CO targets colliding with H+ projectiles, by carrying out a preliminary model study that preserves however the main features of the PES. We have in fact employed the spherical component V&, R) obtained From a multipolar expansion of the ab initio PES already discussed 49

Volume

71, number

CHEWCAL

1

PHYSICS

m a prevrous paper f8j. The former IS a strongly attractive potential wtth a weII depth of 3.2 eV at a distance R from the molecular centrz of mass of K!.7 au. The strong chemrcal forces that play a dominant role m this region, the protonation “belt” around the CO targ2t, extznd further out to rather large R \rdues and then gave rise to the dominant terms of the standard asymptotic form of the full multipolar interaction (m au) @Ii ‘W )fga

= - I3 347f2R-t

- (,.396/‘Rq

- (0 OU/R”)P,

+ 1 859/R3)Pz(cos

(cos 6)

0) + 0(P7)

- (1)

The mam physrcal effects of ttus potentral on colbsion dynamics can aheady be studied by startmg with only The latter turns out to be the Y. contribution able to predict some g2neral qualitative features that ..ue hkely to apply to a wider class of molecular targets colhding with slow protons

2. Description of the mode1 For a sphertcal interactton I’(r, R), where r describes the internal molecular coordinate, a standard expansion of th2 tota! wsvzfunctron over vrbrattonal target states ‘I’@. R) = 2

,

x” (r)F;

(R)Pl(d+)

(2)

,

leads to the well-known couplttd equattons for each of the park11 wave contrtbuttons F/‘(R) that needs to be detcrmmed. [d’/dR’ = 2~9

- I(l+I)fR’

+k;lF;(R)


,

(3)

with k$ = 9,&z, - er,p+ k$L?pCr), where the mittal target state has a:bitrarrIy been assigned ths subscript v. The vlbratronaI ~~upl.rnng matru elements provrde tn turn the interaction between asymptotrcaby different channels associated to each of the final ~bratlon~ states lu’) of the target moIecuIe. In the present case they have bsen obtained within an approxtmate model [7] that wntes. <@‘I I’lx”‘> = v(r,,, The coupling 50

Rn7”u’ (Rcta)

mtegral

.

is now obtamed

(4) analytically:

1

LETTERS

April 1980

~;k&) =~Wx$WI * X exe b&d wrth xhi bemg internal modes. depsnds on the stder2d cokron the Iogarithmic The various then given by CT v-_u(&,~~)

x~&9y

(r - r,,)l

(5)

Morse oscllfator descriptions of the target Now the “stretching” parameter o(Rcla) classical turning point Rcla at the conenergy Ecott and is berng evahrated from denvattv2 of I’&,_, R) at Rcla [7] _ vrbratronahy mehutic cross sections are

= (n/k:)

7

x IS”,“. - s~,U~(~co,*)l’

(=+

I)

-

(6)

When the proton energy is sufficrent to exiteN t~brattonal levels of the unperturbed target, the corresponding couplzd equations (3) wrll generate a set of matrix elements appearmg m eq. (6&S{, 5’4, SAtl for each partial wave. Above the threshold for the highest of such channeh these sets of matnx elements have terms proporttonal to exp (I~IR), exp (lkZR), exp (~k~,,+~l?) respectively, whde below the threshold of, say, the (*Vi- f)st channel the latter becomes the first closed charm21 and the descrtptron of its scatterrng waveFunction must change. The associated wavenumber becomes m fact complex and the soiutron goes into a deexp (- Ik;v+I Ur). caying exponential For sphericai mteracttons with long-range tad behavtour, the potential scattenng solutton Just above threshold e_tibtts the weII known kit:: energy dependtnce. When one IS treating its mult~ch~eI extensron as in the present modeI, however, such threshold dependence IS not clearly estabhshed in such a simple analytical form. Moreover. umtarrty constderatrons on the totalSmatrix suggest that in gong through a threshold some effect IS hkriy to appear on the lower-Iymg excitation channeh and it wdI depend on the nature of the rnteractron potential [9]. These socalled Wigner cusps have been studred m atomic electronic excitations [ 10.1 I J, observed r.n vlbratiomd excitation of polar moIecuIes by electron scattering [ 121 and computed rn rotatronal excitation of I-IF by electron impact 1131. They essentially describe the effect of strong, long-range potentials on the stow outgomg particle above threshold, whereby the structure of the S-matrrx element associated to the observed

Volume 71. number 1

CHEMICAL

PHYSICS

excitation is altered when the co!hsron energy goes through the threshold associated wrth the next higher excited states of the target [l I]. Smce the present system contains long-range charge-dipole mteractron and, in its spherical PES component, a strong polarrzabrhty coefficrent, cyG,we should expect that even “vo only” calculations wi!! etibit such discontmurties on the first derivative of, say, the o,,_~(E,_& mtegral cross sections when gomg throug!-r the opening of the higher CO vrbrational channels. Athough they m&t str!! be too narrow for present-day experimental resolutions, such effects could help to shed some light on the special mteraction of proton beams with polar, highly polansable, molecular targets. Another set of rmportant phenomena can also take place when the proton colhdes with the CO molecule and excites its internal vrbrational states, for the proJectlle separates wrth a hnetrc energy duninished by the amount of energy transferred mto mtema! excrtatron and the target can sometime be so !ughly excited that, at certam energies, the proton remains bound to the molecule in the attractive we!! of the mteractron potentral. Of course, this compound state has a fmrte hfetrme and the excrtatron wdl eventually decay thereby letting the partners separate. Such temporary associations are usually call Feshbach or “core-exerted” resonances [ 141 or “quasi-bound” resonances [ 151. Since the molecule’s excited state IS a closed channel at the Ecou considered and there IS not enough energy in the system for rt to separate without the target being de-excited, they are also referred to as closedchannel resonances [ 161. For generally attractive potent& wrth short-range repulsions also open channel resonances are possble rn which the centnfugal barrier traps the incommg light proton temporardy because of resonant couphng wrt!rin a narrow energy band of translational states associated with a gtven orbital angular momentum value in eq. (3) and the quasrbound vrbratrona! states of the (CO-H+) system temporardy formed during collision. The latter phenomena are called “orbiting resonances” [ 171 and are we!! known in electron-molecule scattering and nuclear physics where they are frequently cal!ed “shape resonances”. These metastable states with nonzero angular momentum quantum number of the pseudo-molecule formed are obviously related to the long-range nature of the attractive mterachon. They are of great importance in several fields of chemical physics

LETTERS

1 April 1980

or for the understanding of recombination phenomena m the gas phase. in the present, exploratory study on the (CO-H)+ system we will show thar strong strucLures mdeed appear, at colhsion energies above the asymptotic target u = 1 threshold, when computing the vrbratronally inelastrc integral cross section and also *-hat they can be related to the deeply attractive nature of the potential and to its long-range part of the interaction.

3. Results and discussion Smce studying the resonances in any collision problem is srmrlar to studying the bound and pseudo-bound states of the compound system, it IS useful to have some idea on the number and distribution of such Ievek For a spherical potential it has been shown [l&j that the number of bound states depends essential!y on a drmensionless quantity B = 2peR&/f$, where ~_r, E and Rbf are the reduced mass, the well depth and the minimum position for the whole system interaction. in the present case B is very large (2: 2.9 X 103 au), due to the large well depth (~3.2 eV) for the Vo contrbution and therefore a rather large number of bound states is supported by the potential employed here: a numerical solution of the ccrrespondmg Schrodinger equation yietds m fact 19 vrbrationa! levels with zero angular momentum * and a very large number of distinct (2C+ I)-fold degenerate energy levels for each effective potential relevant at the Eco,t values considered (j < 70). Since the V, contnbution to the multipolar expansion is very large in the region of the well of vt, (-2.0 eV [EtJ), the drfferences between diagonal potentials in the case of the full mteraction are considerably larger than the Morse oscdlator (MO) levels employed here to describe the isolated target lower-lying vibrational levels (Aeol e 269 meV). Hence there should be little correlation between the bound or quasrbound states of the different diagonal potentials and meaningful tests can be produced by starting computations wrth only Vo [19J _ Fig. 1 presents the interaction potential V. for different asymptotic values of the target vibrational statesA few bound levels, with zero angular momentum. near to the dissociation limit for the compound state, are * We thank Dr. J. Schaefer for performing this caJcdatia~_

CHEMICAL

Volume 7 1, number 1

PHYSICS LE-ITERS

cOtvI*H* v, only

7

Fig. 1 SphencaI mteractlons (I = 0) for the CO plus proton case wth the target molecule m the U’ = 0. 1, Z and 3 vibrational states The honzontal dashed lines represent the hghest five and the bottom two bound vlbratlonal Ierels for the (CO-H)* system.

in fig. 1 for the u’=O case. Also the three lowestlying vrbrational levels supported by the potential are reported in fig. 1_ One clearly sees that the higher levels are rather closely spaced and wdl be pushed rnto the posttive energy region by adding the centnfugal terms of the effective interactron appearmg rn eq (3) Our calculations showed that partial waves up to I = 70 contnbuted to the inelastic cross sectrons of eq. (6) III the

shown

energy domam exammed, yreldmg barrier herghts up to =250 meV. Accordmg to the classical picture orbiting can only occur for effectrve potentials that exhrbrt maxima. The usual LJ(12,6) potential yields maxrma, (Veff)mau, for Emu G 0.8~ thus one might expect that the present system would show orbrtmg effects for colhsion up to X2.5 eV or, at any rate, up to unuSually large values of E, above the excitation thresholds. Moreover, the width of each resonance wrll depend on the strength of the coupling that controls the overall barrier shape and that determines the rate of tu’hnelhng 52

1 April 1980

through the barrier for encounters at E,, < ( Veff),,,ay_ The present, strong potential v. IS therefore hkely to produce fairly narrow resonances in the range of colhsronal states just above the u’= 1 threshold. Moreover, the many closely spaced quasibound states generated by the effective potential wdl also be very often degenerate with a corresponding kmetic energy value and therefore hkely to produce a large number of resonances for the present system. Frg. 2 summarizes the results of our calculatrons for the vibrationally inelastic integral cross section withm the collisron energy range of 1 .O eV, i.e. for kmetic energy values up to e-73 eV above the IJ’= 1 viirational threshold of the target. The various target thresholds for vibrational excitation are also reported for clarity, being numbered wuh the reference level u’=O corresponding to Eii). In fig. 2a the region of collision energies up to the u’= 2 threshold IS shown, spannmg about 0 25 eV of kinetic energy. The corresponding melastic cross section rises very rapidly and shows a broad maximum at about 0.1 eV above threshold: the contnbuting opacity functions d,, exhrbrt maxima for 18 < I< 30 that correspond to Veff barners up to @.I2 eV. The closely spaced pseudo-bound states of the compound triatomrcs are very hkely to be often degenerate with the corresponding E,, and therefore strongly contnbute to vlbratronal melasttctty via several, narrow orbiting resonances. This mechamsm becomes increasingly less effective as one approaches the Et:) of the u’ = 2 state of the target.

The openmg of this new channel, however, affects the structure of the S-matrix because of the strong, charge-polansabihty mteraction with the slow outgomg projectile and therefore a marked mflectron on u,-,+ 1 appears as a result of Wrgnercusp behavrour [9j. Fig. 2b shows, in fact the rapid mcrease of the latter cross section across the new threshold_ As a consequence of the addrtronal coupling with this new excited target vrbrational state, located at ~266 meV above the previous u’= 1 target level, open-channel virtual excitations of the CO molecule become however possible during the observed pnmary process u = 0 + u’= 1. The associated cross section therefore “delays” its expected goes rapid return to revious values once the E,, beyond the Et,C3pthreshold 113 ] . A qualitative explanation for thrs effect can be provided by the renewed existence of orbiting following the above vutual excita-

Volume 7 1, number I

CHEhf [CAL PHYSICS LETTERS

I April i980

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40

T

35

-

FIN 2. VIbrationally inelasuc rntegral cross sechon o,-,+l (au) as a function of collrsron energies. (a) Ecou values between the v’= I! h, for the target asymptotic channels (b) Colhsion energies between the ~‘2 2 and the threshold, E$,! and the U*= 2 threshold, E(3) fouiowmg vlbrauonal threshold E@wrth the CO molecule in its v’= 3 state. (e) E cog values up to 1.0 eV above the motecuk zero pomt energy.

tmn whereby the residual ktnetic energy becomes of the order of WI 1 eV and several narrow resonances can agam take place each tnne Ecou is degenerate wrth one of the many vibratronal levels of the compound (CO-H)+ thrs tune wrth the %uget” molecufe CO temporanly in its n’=Z exerted vrbrationai state from which It however decays to u’= 1 before the system separates + _ When the collision energy goes through the next target channel threshoid, E$), the long-range tad of the potential causes once more the sharp discontmuity in the first derivative of UO_,.~(E,~)_ This rs due to the appearance of a new and higher accessrble internal state of CO, in the same way as discussed before for the u’ = 2 state. The change ISnow however m the opposite dtrectron (fig. 2~). Once beyond this threshold other vntual excitations, 10) + 12) and I 1) + I2), become possible but thts tune only over a very narrow range of&In values (W.05 eV), since the strength of the duect coupling with the levels involved m the primary * The slow, outgoing partners experience m fact a long interaction me during whxh energy redrstrtbu&on between modescan occur [2O].

process IS markedly smaller and thus the additional presence of target-excited orbiting resonances contributes less than before to the enhancement of the crtil cross sections. Computa~ons made at higher Ecou did not show any of the strong features discussed above, but rather they begun to present the broad osciUations caused by entirely drfferent effects [6,7]. in conclusion, the model cross section caIcu&ions performed here for energies above the fust viiratio& threshold of the CO target appear already to indicate the presence of strong effects in proton-molecule collrsions and manage to attribute them to the particular nature of the PES involved. The strongly attractive long-range tail and the deep well exhibited by the qherical component of the PES multipolar expansion are held responsrble for the presence of several narrow shape resonances, of marked Wiier cusps at each higher threshold opening and of strong vibrational coupe that leads to open-channel vutual excitations at low E oon. The use of a more realistic, full potential is obviously going to yield different quantitative answers, although we suspect that the strong features presented by our model are not likely to be ~~t~~~~y ~&~ed. 53

Volume 7 1, number

1

CHEMICAL PIiYSICS LETTERS

One of us (FAG) thanks Professor JP. Toenrues for the warm hospitality m his laboratory. Fruitful dlscussions with Professor Don Secrest and Dr. D J. Kourl are also thankfully acknowledged. FmalIy, we gratefully appreciate Dr. Gert Barg’s careful readmg of the manuscnpt

References K Rudolph and J P Toenmes, J. Chem Phys 65 (1967) 4483 H. Schmxdt, V. Hermann and F. Lmder, Chem. Phys Letters41 (1976) 365. J. Krutem and F. Lmder, J Phys B 7 (1977) 1363 hi Faubel and J P Tornntcs. Advan At Mel Phls 13 (1977) 2’9 J. Krutem. G Blschof, F Lmder and R Schmhe, J Phys. B I2 (1979) 57

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1 Aprti 1980

f6] J. Krutein and F. Ltnder, Chem. Phys. Letters 5 1 (1977) 597. 171 F A Gtanturco, U.T Lamarma and M_ Attimoneiit, Chem Phys , to be pubhshed. [S] F.A. Granturco, U.T. Lamanna and D. ignazz~, Chem Phys , to be published [9] E P. Wagner, Phys Rev. 73 (1948) 1002. [lOJ A.I. Baz. Zh. Ekspertm iTeor- Ftz. 11(1?58) 709. [ll] A. Herzenberg and D. Ton-That, J. Phys. B 8 (1975) 426. [12] K Rohr and F Ltnder, J. Phys B 9 (1976) 2521. 1131 F-A. Gianturco and N K Rahman, J. Phys B 10 (1977) L219 [ 141 H. Feshbach. Ann. P&s 5 (1958) 357. [IS] J T. lCfu=ckerman and R.B. Bemstem. Chem. Phys Letters 4 (1969) 183. [ 161 L N. Smith, D J Mahk and D. Secrest, J. Chem Phys , to be pubhshed. [17] J-P. Toennies, CommentsAt Mel Phys 8 (1979) 137 [18J A-S. Dtcktnson and R B. Bemstem, h¶ol Phys. 18 (1970) 305. 1191 A S. Dtckmson, D J. Koun, C A. Wells and W.A. Lester Jr , J. Chem. Phys 65 (1976) 1501. f20] H S. Kook rtld E Yavlonontch, Phys. Rev. Letters 41 (1978) 745