Brian K Elza et al- Non-adiabaticity and other aspects of polarisation in positron-molecule...

8/3/2019 Brian K Elza et al- Non-adiabaticity and other aspects of polarisation in positron-molecule scattering

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J. Phys. B: At. Mol. Opt. Phys. 22 (1989) 113-130. Printed in the UK

Non-adiabaticity and other aspects of polarisation inpositron-molecule scattering

Brian K Elzai, Thomas L Gibson$, Michael A Morrisont and BidhanC Sahatt Department of Physics and Astronomy, Universi ty of Oklahoma, Norman, Oklahoma73019, USA$ Depa rtment o f Physics , Texas Technical Universi ty, L ubbock, Texas 79409, USA

Received 7 June 1988, in final form 26 August 1988

Abstract. We have investigated various aspects of polarisation/correlation effects at lowenergies for the prototypical p osi t ron-N 2 system. Firs t , we have fo und th at , l ike posi t ron-Hi,cross sect ions, those f or this system a re highly sensi tive to terms in the (adiaba t ic) polarisa-t ion potent ial that expl ici t ly depend on the sign of the charg e of the projectile. Insightinto the nature of these s ign-d epend ent dis tort ions is afforded by three-dime nsional grap hsof polarised N, density functions. Second , we have explored an d characterised non-adiaba t ic (co rrelat ion) effects in this system using a s imple two-param eter model . We havefou nd tha t the onset of these effects in the near-target region in positron scattering is mu chmore grad ual than in electron scattering. By including in scattering calculations adiab aticand non-a diabat ic effects , we obtain theoret ical po si t ron-N 2 cross sections in good agree-ment with recent experimental data.

1. IntroductionThe last few years have seen a dramatic increase in research on low-energy positron-molecule scattering (see, for example, the review by Armour (1988)). This activity isin part due to new technology for producing improved positron beams (Canter andMills 1982). Nevertheless, there remain considerable uncertainties in measured valuesof positron-molecule cross sections at collision energies below a few eV (cf Hoffmanet a1 1982, Charlton et al 1983, Sueoka and Mori 1984), and fundamental questionsplague the calculation of theoretical positron scattering cross sections at these energies.

Below the threshold for positronium formation, the most serious of these questionsconcerns polarisation and (short-range) correlation effects. Although these effectssignificantly influence low-energy cross sections, their inclusion in computationalmethods via accurate yet tractable approximations is problematical.Two groups have recently initiated studies based on ab initio treatments of theseeffects. Armour and collaborators (Armour 1984,1985, Armour and Baker 1985, 1986,1987) have published a series of increasingly sophisticated variational calculations ofpositron-H, collisions based on the Kohn method. Tennyson (1986) and Tennysonand Morgan (1987) have applied the R-matrix method to positron scattering from H2,N2 and CO.An alternative to these ab initio theories is the use of a physically motivated modelpotential. At large distances from the target, the form of such a potential is simpleand familiar: for a projectile at position ( r , , e,) (in a body-fixed coordinate system),0953-4075/89/010113 + 18%02.50 @ 1989 IO P Publishing Ltd 113


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114 B K Elza et a1the asymptotic polarisation potential is an analytic function of the spherical andnon-spherical polarisabilities of the target, a. and a z , .e.,

This form pertains to both positron (plus superscript) and electron (minus superscript)scattering; it is the asymptotic limit of the polarisation potential as calculated insecond-order perturbation theory (Castillejo et a1 1960; ch 2 of Gibson 1982). Hencethis form is independent of the sign of the charge of the projectile.But nearer the target, the simple form (1) does not accurately characterise polarisa-tion effects, the second-order perturbation approximation is invalid, and V,f,,( rq , e,)does depend o n the sign; i.e., the polarisation potential for an incident positron differsfrom that for an incident electron. In 1984, Morrison et a1 demonstrated the importanceof this dependence for positron-H, cross sections and calculated a sign-dependentpotential which is accurate beyond second order in the projectile-target interactionby treating the projectiles position adiabatically and determining via the linear vari-ational method the decrease in the system energy due to polarisation of the molecule.The importance of explicitly allowing for the sign of the projectiles charge ingenerating a polarisation potential for positron scattering calculations has been anissue because of the widespread use of (model) polarisation potentials that weredetermined not for the positron-molecule system, but for the corresponding electron-molecule system. This practice continues to the present day. Thus Jain (1986a, b) andGianturco et a1 (1987) have suggested that the sensitivity to the sign of the projectileseen by Morrison et a1 in positron-H, cross sections may be a fluke: that H,, earlyspherical and with only two bound electrons, is somewhat atypical, and that for heaviertargets a treatment of polarisation that does not distinguish between electrons andpositrons as projectiles may be approximately valid. Investigation of this question fora typical positron-molecule system, positron-N,, is one focus of the present study.Our second area of concern-non-adiabaticity-becomes important within themolecular charge cloud. Here the adiabatic approximation breaks down completelyand non-adiabatic effects become important: particularly correlation effects. Severalinvestigators have developed and tested methods, ab initio and model-based, forincluding such effects in electron scattering (see Morrison et a1 1987 and referencestherein, Schneider and Collins 1983, Huo et a1 1987). Some positron-molecule studiesuse these potentials without change (see references in table I1 of Morrison 1986).However, the importance in the near-target region of higher-order perturbationterms, which are senstive to the nature of the projectile, raises questions about the useof electron-molecule polarisation potentials in positron-molecule scattering calcula-tions. Moreover, some methods for treating non-adiabaticity, such as the non-penetrating approximation (Temkin 1957), which we have applied to the electron-H,(Gibson and Morrison 1984) and electron-N, (Morrison et a1 1987) systems, seeminappropriate on physical grounds to positron-molecule scattering.A central finding of the present study (see 4) concerns the onset of these non-adiabatic effects as the projectile nears the target. To characterise this onset, we haveused a two-parameter cut-off function that smoothly eliminates the adiabatic polarisa-tion potential as rq+ 0. By comparing our calculated cross sections with measuredvalues, we can determine optimum parameters in this function; significantly, the valueswe obtain contrast strikingly with those required for electron-N, scattering. (Thisstrategem is similar in spirit to that used in recent positron-atom studies (Nakanishi


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Positron-molecule scattering theory 11 5and SchrLder 1986a, b)). We offer in 0 4 an explanation based on the underlyingphysics of positron and electron collisions for the differences in the onset of non-adiabatic phenomena in the two systems.

2. TheoryFor the scattering processes of interest here, the fixed-nuclei approximation is valid(Temkin and V asavada 1967), so we work in a body-fixed ( B F ) reference frame (Lane1980) an d choo se the origin of a single-centre coo rdin ate system at the centre of massof the molecule and the z axis along the internuclear axis. We den ote by r m ,R , , an drq the spa tial coordinates of the molecular electrons, the target nuclei, an d the scatteringprojectile. A semi-colon connotes parametric dependence on whatever variables appearafter it.Thus for a projecti le at rq scattering from a molecule with Ne target electrons atlocations collectively denoted by rm and with N , nuclei with charge 2 , f i x e d a t R,,the Coulomb potential is

In the body-frame fixed-nuclei scattering equations, which result from projectingout of the t ime-ind epen dent Schrodinger equa tion for the positron-molecule systemthe (Born-Oppenheimer) ground electronic state of the target (see Morrison 1988),the effective positron-molecule interaction potential VL,( q;R ) is the sum of staticand polarisation terms,V L ( r q ;R )= V 2 r q ;R I + v ;o l ( r q ;R ) . (3)

Th e electrostatic term is the average (th e integral with respect to electronic coordinatesr m ) of the C oulom b interaction ( 2 ) over G 0 ( r m ; ) , he ground electronic state of thetarget at internuclear separation R, i.e.,V,:(r , ;R )= $01 v:oll , l$o). (4 )

To characterise the polarisation potential V;o l ( q ; R ) t is useful to consider threeregions of configuration space. These regions can be loosely delineated by rq , thedistance of the positron from the molecular centre of mass. In the asymptotic regionthe polarisation potential reduces to the simple sign-independent analytic form (1).In the intermediate region, at values of rq outside the molecular charge cloud but nea renough to i t that equation (1) is invalid, one can obtain an accurate polarisationpotential for low-energy collisions (im pac t energies less tha n a bo ut 10 eV), by assum ing(Morrison and Hay 1979, Truhlar et a1 1979) that the target responds adiabatically tothe proj!ctile-i.e., th at the system is accu rately represe nted by an adiab atic Ha mil-tonian %& which is the su m of the (Born-O ppenheimer) elect ronic Ham il tonian ofthe target a nd the C oulom b interaction (2 ) with the fixed radial coordinate rq treatedas a parameter .The present positron-N, study uses such a treatment. At eac h internuclear sepa r-ation R, the seven N 2 molecular orbitals ( 1ug 1U , , 2ug, u,, 3ug, vu)are al lowed t ofully relax in the presence of a point charge whose location rq is frozen. From the


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116 B K Elza et a1resulting distorted molecular orbitals we construct the polarised electronic wavefunc-tion (clp(m; r q , R ) of the target. The adiab atic polarisation potential at each R is justthe difference between the expectation values of eDith respect to the polarisedand unpolarised target wavefunctions, i.e.,

In the region nearest the target, within the molecular charge cloud, the adiabaticpolarisation potential improperly correlates the projectile and the bound electrons:the a diab atic approximation overestimates correlation effects, leading to a polarisationpoten tial that is to o attractive. In this region, short-range correla tion (e.g. virtualpositronium form ation ) is im por tant, and we must allow for non -adia batic effects. (A nadiabatic treatment in this region would be more appropriate were we determining aproton-molecule interaction potential, where the dominant short-range correlationeffect is virtual hydrogen-atom formation.)Our treatment of non-adiabaticity distinguishes in two ways the present positronpolarisation potentials from our non-adiabatic electron potentials (Gibson and Mor-rison 1984, M orrison et al 1987). First, we d o not use the non-penetrating a ppr oxim a-tion of Temkin (1957) to incorporate non -ad iaba tic effects in positron scatteringpotentials; this app rox im ation , which essentially entails switching off the C oulo m binteraction between the projectile and the bound electrons whenever the former isinside the charge cloud of the latter, is inappropriate for positron collisions (seeMorrison et a1 1984). Second, we retain all multipoles in the polarisation potential.(For electron scattering, only the dipole term need be retained.)Our primary interest in this study is the physical nature of non-a diab atic correctionsto the adiaba tic positron-molecule polarisation potential. We approximate thesecorrections by simply multiplying the polarisation potential, in either its asymptoticor adia batic form (equation s (1) or (5 )), by a spherical cut-off function that de pendson a djustable parameters rc and p :

C ( r q )= 1 -exp[ - ( rq / r JP] .Several investigators have used such functions in prior positron-molecule studies (see 2.1 of Morrison et a1 (1984) and references in Morrison (1986)). This functionaccounts for non-adiabatic effects by smoothly eliminating the polarisation potentialas rq decreases from rc to zero. The power parameter p controls how sharply thispotential vanishes, or alternatively, the size of the region of non-adiabaticity and theabru ptne ss of the onset of non- adia batic effects. Thus, within this scheme, p charac-terises the coarse features of these effects.Cut-off functions like equation (6) have been used to approximate non -ad iaba ticcorrections in many electron-molecule scattering calculations (see Mo rrison 1983 an dreferences therein). In such studies, the power parameter is usually chosen as p = 6and the cut-off radius rc is adjusted to force calculated cross sections to replicatesome experimentally determined feature of the system such as a shape resonance.The choice p = 6 in electron-m olecule scattering reflects the suddenne ss of the onsetof non -ad iaba tic effects as rq decreases to zero. The principal no n-a diab atic correctionis due to a velocity dependence in the polarisation potential; this dependence ariseswhen a slow scattering electron falls into the strongly attractive, short-range static-exchange p otential well. Th e electrons local kinetic energy rapidly becomes c om par -able to that of the bound electrons, which completely invalidates the adiabatic


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Positron-molecule scattering theory 117hypothesis. The resulting correlation potential is considerably weaker near the originthan the adiabatic potential.However, for positron-molecule scattering no comparable mechanism exists tocause the sudden onset of non-adiabatic effects. Because the positron-molecule staticpotential is repulsive, velocity-dependent non-adiabatic effects are much less importantthan in the corresponding electron-molecule system. Hence the dominant non-adiabatic effect is due to virtual positronium formation, which occurs gradually as thepositron penetrates the molecular charge cloud.Thus the physics of non-adiabaticity in positron-molecule scattering argues forusing a small value of the power parameter-a value that permits a gradual onset ofnon-adiabatic effects. In our positron-N, study ( 4), we characterise the dominantnon-adiabatic effects by varying this parameter, comparing calculated cross sectionswith experimental data.

3. Compu tational considerations3.1. Representation of the N 2 targetThe ground-state ( X E l ) N2wavefunction appears in both the static and polarisationcomponents of the positron-N, interaction potential (equation (3)) . We representedthis function in the Hartree-Fock approximation, calculating molecular orbitals usinga (10s6p2d/6s4p2d) basis of contracted nucleus-centred Cartesian Gaussian functions(Morrison and Hay 1979). This basis is an augmentation of a (9s5pld/5s3pld) basis,the additional (uncontracted) s, p and d functions providing sufficient flexibility toaccommodate distortions of the N2 charge density.The static positron-molecule potential in the region of the target is strongly repul-sive. Consequently the nature of the static and polarisation potentials outside thisregion, including their long-range functional forms, significantly influences the crosssections we want to determine. I n the asymptotic region ( r, +CO) , the static potential(4) reduces to a simple analytic form (at internuclear separation R ) whose Legendreprojections

U:( rq; R ) =-h + 1 [: V : ( r q ;R)P,(cos e,) sin 8, de,2are proportional to the permanent multipole moments c (R) , as

(7)

where the plus sign corresponds to positron and the minus to electron scattering.The values of the lowest three permanent moments, calculated from the staticpotential (based o n our (10s6p2d/6s4p2d) electronic function) at the equilibriuminternuclear separation Re= 2.068ao are compared to experimental values in table 1.We extracted the theoretical moments at r, = 10.0a0, by which point each Legendreprojection uy(rq;R) is given (to the stated precision) by its multipolar form (8) .Similarly, we determine the induced polarisabilities, cy o and a2 n the asymptotic form( l ) , rom the calculated adiabatic polarisation potential V&,(rq ;R ) at rq= 25.0~2,. thevalues of these moments also appear in table 1.


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118 B K Elza e ta1Table 1. Permanent and induced moments fo r N 2 a t R = 2.0680,.

Permanent momentsTheorya Experimentbc,(R,) = 0.91eaic4(Re) 7.49~20:c6(Re) 20.68eag

11.04/i .07eai

Induced momentsTheoryc Experimentda, (R) = 11.420; 11.74*0.0040;a2 (R ) 3.370;a Morrison et al (1987). Extracted a t rq = 10.00,.

2.96 + 0.030;

Herzberg (1950), Alms et al (1975).Present s tudy. Mo men ts extracted at rq = 25.00,.Mil ler an d Bedersen (1978) and references therein.

3.2. Parameters of the scattering calculationsIn the fixed-nuclei ap pro xim ation , the total cross section is the sum of contributionsfrom inde pen den t calculations in various electron-molecule symmetries (see Mo rrison1988 an d references the rein) . These symmetries correspond to the irreducible rep-resentations of the point g roup of the molecule, an d for N 2 (the point gro up of whichis Doah) re defined by the projection A of the projectiles orbital angu lar mom entumon the internuclear axis and by the behaviour of the scattering function u nde r inversionthro ugh the origin (gera de or ung era de) . Positron-molecule cross sections decreaserapidly with increasing A, an d for positron-N, we can converge the total cross section(at energies less than or equal to 1 0 e V ) to 1% by including contributions fromZ,, E,,nu,Ig, Ag and A, symmetries.Within each symmetry, the most impo rtan t convergence param eters are the num berof partial waves (spherical harmonics Y f ) ncluded in the single-centre expansion ofthe scattering function and the value r,,, of the radial projectile coordinate at whichwe extract the scattering matrix. To satisfy our 1% convergence critieria, we requirefive partial waves in the Z symmetries ( A = 0) and four in the II (A = 1) and A (A = 2)symmetries. In the Legendre exp ansion of the static potential (see equation (7)) weinclude (eve n-A) projections u p to A,,, = 14. We must propag ate t he scattering functionto rmax= 8 6 . 0 ~ ~ .e checked these values using the time-honoured method of includingmore partial waves (u p to I = 12) and propagating further (to r,,, = 100.0 ao). Allis well.

4. ResultsTo exhibit the nature of indu ced distortions of a molecular charge cloud by an incidentprojectile, electron or positron, we compare (in 9 4.1.1) neutral and polarised one-electron N2 probability densities for various (fixed) projectile locations. These three-dimensional graphs evince the sensititivity of polarisation effects to the sign of the


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Positr on- molecule scattering theory 119charge of the projectile. Furth er insight into this response follows from exam iningindividual molecular orbitals, important examples of which appear in 4.1.2. Thedifferences seen in these graphs of probability densities translate into differences inthe polarisa tion potentials themselves, w hich a re discussed in 4.2. These disparities,in turn , influence cross sections for positron-N, scattering as calculated w ith variouspolarisatio n potentials; these scattering data are comp ared in 0 4.3, where we reconsidernon-adiabaticity, a central concern of this research.

4.1. Variously polarised probability densities4.1.1. Total electron density functions. We calculate total one-elec tron probabilitydensities of the (neu tral or polarised) molecule from the B orn-O ppen heim er electronicwavefunctions, qo(rm;R ) or (Ldp' ( r m ; q , R ) (see , for example , equation (5)) . For awavefunction polarised by a projectile at rq , this density func tion is an integral overall but one of the spatial variables of the boun d electrons,

p ( r l ) = I b"'*(r ,; rq, R )& ")( rm ; rq, R ) dr ,edrN, - l . . . d r 2 . (9)To display the polarisation distortions, we show in this section graphs of the one-electron density fun ction evaluated in the xz plan e, i.e. we plot p ( r l , e l , pl = 0) for rlfrom 0 to 5a , and for f rom 0 to T . Figure 1, for example, shows this fun ction forthe X 'Xi ground state of the neutral (undistorted) N, molecule. (To enable a clearvisualisation of these density functions, their peak s (ne ar the nitrogen nuclei at *1.034a0on the z axis) have been clipped as necessary; hence th e 'flat top s' in figure 1 an delsewhere.)

Figure 2 shows distortions of the total gro und-s tate density du e to a positron ( a )and an e lec tron ( b ) ; hese projectiles were fixed at zq = 1.5 a,, just beyond the nuclei.Th e distortions seen in these figures are not simple mirror im ages of one ano ther ; theydiffer in bo th form a nd extent. Thus, the extrusion of the positron-d istorted density

5.0- 0x1s (0,)Figure 1. Total N, electron density function (at R = 2.068 a,,) in the xz plane. In order tomore clearly sho w changes in this funct ion du e to a fixed projectile (see figure 2), its peakshave been 'cl ipped' to a value of p = 1.


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120 B K Elza et a1

5.02 - x i s fa,)Figure 2. Total N, density function polarised by ( a ) a posi t ron and ( b ) n electron fixedat zq = 1.5 ao.

in figure 2 ( a ) towards the projectile, which is shap ed by that projectiles (ra dia l) field,is quite different from its counterpart in the electron-distorted density in figure 2 ( b ) .4.1.2. Molecular orbital density functions. Because these total densities include all sevenmolecular orbitals of N2, hey obscure s om e features of the distortions they a re designedto illustrate. These features a pp ea r quite vividly, however, in grap hs of the mo st diffusemolecular orbitals.The 3ugan d 2uU rbitals clearly exhibit the natu re of these (ad iab atic) polarisationdistortions. The un distorted (neutral) densities for these orbitals app ear in f igures 3(a )and 4 ( a ) , espectively. Figures 3 ( b ) an d ( c ) show the strikingly different distortionsinduced in the 3u, density by an electron and a positron at zq = 1.5 a,. The lack ofmirror symmetry in the polarised orbitals illustrated in this comparison also appearsin the distorted 2uUorbitals in figures 4 ( b ) and ( c ) .

As the distance of the (fixed) projectile fro m the molecule increases, the distortionsof these densities decrease; ultimately, they regain their neutral shapes. In 9 4.2 we


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:i.8

Figure 3. ( a ) The one-particle density function evaluated in the xz plane for th e 3 ~ 7 ~molecular orbital of neutral N,. Also shown are the densi ty funct ions that resul t whenthis orbital is polarised by ( b ) posi t ron and ( c ) an electron at z , = 1.5 a,,.

will discuss the effect on the polarisation potential of varying the projectiles location.The effect of this variation on the energies of the 3a, and 2a, orbitals is shown infigure 5. As expected, as the incident (negatively charged) electron enters the fringesof the molecular charge cloud, it enhances screening of the outermost electrons fromthe nucleus, causing their binding energies to decrease; precisely the opposite behaviourresults from the incursion of an incident (positively charged) positron.In any case, the way in which the positron- and electron-distorted orbitals correlateto the appropriate neutral energies (as r4+a) s strongly dependent on the sign ofthe charge. These comparisons reflect again the inadequacy of a second-order perturba-tive treatment of polarisation, an inherently sign-independent treatment, in the non-asymptotic region.4.2. Adiaba tic and non-adiabatic polarisation potentialsThe energy lowerings due to the electron- and positron-distorted density functions infigure 2-and their counterparts for other values of r4-are the adiabatic polarisation


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122 B K Elza et a1

0.8'OI I !1

2d 0.6z!: .4e+- 0.2

0.0

z1VI

U

.-sWP5E+YW

1.0

0.8

0.6

0.4

0.2

0.0 D

''OI.8

.O

Figure 4. ( a ) The one-particle density function evaluated in the xz plane for the 2uumolecular orbital of neutral N,. Also shown are the density functions that result whenthis orbital is polarised by ( b ) posi tron an d ( c ) an electron at zq = 1.5 a,.


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Positron -molecule scattering theory 123

1

Figure 5. Energies of N2 molecular orbitals polarised by an incident electron (- - -) a ndby an incident positron (-) as a function of the distance of the (fixed) projectile alongthe z axis. Sho wn are these 'correlation diagrams' for polarised orbitals that correlate (asrq+ CO) t o t he ( a ) 3ug nd ( b )2u, orbitals of the neutral.

potentials (see equation (6)). We have designated these potentials by the acronymsADELEC (for an incident electron) and ADPOS (for a positron). The two lowest-orderLegendre projections (cf equation (7)) of these potentials are compared in figures 6 ( a )( A =0, the spherical projection) and 6(b) ( A = 2). Note that as r q + q the ADELECand ADPOS potentials become identical: both reduce to the asymptotic form (1).The similarities between the projections of these purely adiabatic potentials aredeceptive; quite different potentials result when non-adiabatic effects are taken intoaccount. To illustrate this point, we also show in figure 6 the projections of twonon -adiab atic polarisation potentials: the 'better tha n adiabatic dipole' (BTAD) potentialdetermined by Morrison et a1 (1987) for electron-N, scattering an d a cut-off ADPOSpotential. Th e latter produces to tal positron-N, cross sections that agree closely withmeasured values (see figure 10(b)) ; we shall discuss this ADPOS potential in the nextsection.These no n-a dia ba tic potentials highlight an important difference between positron-and electron-induced distortions of N, . This can m ost clearly be seen in the diferencesbetween the purely adia bat ic and n on-adiabatic potentials: within the approxim aterepresentation of non-adiabaticity inherent in both the BTAD and cut-off ADPOS models,these differences are 'the non-a diab atic pola risation p otentials' d u e to, respectively,an electron and a positron. The spherical and A = 2 projections of these differencesin figure 7 show that non-adiabatic effects are stronger and more extensive for adistorting electron than for a positron.4.3. Positron-N2 cross sections4.3.1. Total integrated cross sections. Our procedure for approximating the effects ofnon-adiabaticity-multiplying the adi aba tic (o r asymp totic) polarisation potential byth e sph erical cut-off func tion (6)-facilitates stud y of the importa nce of these effectsto cross sections, by simply varying the adjustable parameters in C ( r q ) : he powerparameter p and the cut-off radius r , . Figure 8, for example, shows the sensitivity ofthe total positron-N, cross section as calculated with the ADPOS (figure 8 ( a ) ) o r ADELEC(figure 8 ( b) ) polarisation potentials to variations in the power parameter p . Figure 9shows their sensitivity to variation of the cut-off radius r , .


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124 B K Elza et a1

t - - 1I I1.0 2.0 3.0 4.0 5.0Rdius (sol

I11.0 2.0 3.0 4.0 5.0Rdius ( a o )

Figure 6. Adiabatic and non-adiabatic polarisation potentials for an incident electron andfor an incident positron: ( a ) the spherical Legendre projection, and ( b ) the A = 2 projection.For each type of projectile, two curves are shown: purely adiabatic (long broken curve foran incident electron, short broken curve for a positron) and non-adiabatic (dotted curvefor an electron, full curve for a positron). The latter potentials were chosen as those thatmost closely reproduce experimental cross section data for scattering from N, : the non-adiabatic electron polarisation potential is the BTAD model of Morrison et a1 (1987); thenon-adiabatic positron potential is the cut-off A D W S model (with r, = 1.59~1, and p = 1)discussed in 0 4.

Two points des erve emphasis. First, cross sections calculated using a positron-induced polarisation potential differ qualitatively from those determined with anelectron-induced p otential. Fo r example, the cross sections in figure 8(b), calculatedusing the cut-off ADELEC potential, exhibit structure (Ramsauer minimum) not seenin the id entically cut-off ADPOS results of figure 8 (a ) no r in experimental data ). Figure9(a ) shows, however, tha t we can force such a minimum into the ADPOS cross sectionby choosing an unrealistically large cut-off radius, r,> 2ao .Th is first observation confirms th at the findings of Morrison et a1 (1984) regardingpos itron-H 2 scattering als o pertain to this typical heavier sy stem: i.e., tha t the differencesin variously polarised electron densities ( 0 4.1) and in the polarisation potentialscalculated from them (0 4.2) transla te into significant differences in scattering qu antities.Second, the total cross section is especially sensitive to the sharpness of onset ofnon-adiabatic efects as controlled by the power parameter p. In order to determine


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Positron- molecule scattering theory

1.0 2.0 3.0 4.0 50Radius (a,)

0.0 ,,\\

-0.5 ,Radius (a,)

I0.0 1.0 2.0 3.0 4.0 5.0

Figure 7. Model non-adabatic polarisation potentials fo r an incident electron (brokencurves) and an incident positron (full curves): ( a ) he spherical ( A =0) and ( b ) A = 2Legendre projections. In each case, these are the diferences between the purely adiabaticpotentials in figure 6 and their counterparts including non-adiabatic effects: the BTAD fo ran incident electron, the cut-off ADPOS for an incident positron.

Figure 8. Total positron-N, cross sections calculated using variously cut-off ( a ) ADPOSand ( b ) ADELEC polarisation potentials: -, p = 1.0; - - -, p = 2.0; - - - -, = 6.0. Forthese comparisons, the cut-off radius was fixed at r, = 1.59 a, and the power parameter pwas varied. See also figure 9.


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Positron -molecule scattering theory 127Table 2. Total a nd part ial cross sect ions (in a : ) and eigenphase sums ( in rad ians , shownin parentheses) fo r positron-N, scattering.

0.25 3 1.689 4.026 0.140 0.899 36.7710.50 11.693 5.964 0.238 1.898 19.8270.75 4.739 7.017 0.340 2.651 14.8111 o 1.955 7.426 0.446 3.156 13.0821.5 0.703 7.094 0.663 3.590 12.23 22.0 1.328 6.093 0.874 3.536 12.1043.0 3.546 3.924 1.224 2.851 11.9955.0 6.868 1.296 1.522 1.449 11.84 77.0 8.344 0.414 1.419 0.756 11.7659.0 8.774 0.332 1.151 0.550 11.658

10.0 8.777 0.428 1.007 0.547 11.594

(0.232) (0.083) (0.013) (0.031)(0.212) (0.143) (0.023) (0.062)(0.178) (0.191) (0.034) (0.089)(0.144) (0.228) (0.045) (0.112)(0.081) (0.278) (0.066) (0.148)(0.027) (0.304) (0.087) (0.174)

(-0.063) (0.313) (0.126) (0.201)(-0.204) (0.267) (0.185) (0.209)(-0.322) (0.197) (0.221 ) (0.192)(-0.430) (0.125) (0.237) (0.167)(-0.481) (0.089) (0.240) (0 .153)

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4.0 6.0 8.0.0 I0.0 2.0 Energy ( e V 1

t-- I"5 .01 - .POLea" o f fman et 01 119821t**tt Sumko and Mori (19841-A Chorlton e t a / 119831'i1JI, 1

- , ,2 0 4 0 6.0 8.0Energy ( e V I

Figure 10. Total positron-N, cross sections. ( a )Compar i son of identically cut-off modelpolarisation potentials; fo r each calculation, the parameters in the cut-off function ofequat ion (6 ) were chosen as p = 1 and r, = 1.59 a, . Also shown are experimental data ofHoffman e l a l (1982) and the R-matrix theoret ical resul ts of Tennyson (1986). ( b ) C o m -par i son of our best cut-off ADPOS results with existing experimental data.

tion potentials. For perspective, we also show in this figure the R-matrix results ofTennyson (1986) and the experimental data of Hoffman et a1 (1982). Additionalexperimental data (Charlton et a1 1983, Suoeka and Mori 1984) are compared withour optimised cut-off ADPOS results in figure 10(b ) . In spite of the uncertainty in thevarious experimental results below -3 eV, these data support the veracity of the ADPOS


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128 B K Elza et a1

Angle (deg)

0.0; ' ' i o ' ' 60 ' ' 90 ' 120 ' 1$ ' 1oAngle (deg)

Figure 11. Tota l differential positron -N, cross sections at impact energies of ( a ) 1.5 eVa nd ( b ) 10 eV as calculated using identically cut-off A D E L E C (- - -) an d ADPOS (-)polarisation potentials. The parameters in the cut-off function in these calculations werer , = 1 . 5 9 a O a n d p = l .

potential using a very mild cut-offunction. No set of cut-off parameters rc and p couldprodu ce such agreement when either the ADELEC or asymptotic forms were used.4.3.2. Diferential cross sections. Not surprisingly, differential positron-N2 cross sec-tions calculated using (identically cut-off) ADPOS and ADELEC polarisation potentialsare different. Interestingly, they differ primarily in the forw ard directio n, as show n infigure 1 1 at a low (1.5 eV) and a high (10.0 eV) energy.Apart from their intrinsic interest, the energy and angle dependences of the differen-tial cross section provide useful g uidance in the absolute norm alisation of experimentalpositron scattering data (see, for example, Hoffman et a1 1982). This information iscompactly represented by the three-dimensional graph in figure 12 , which presentsdata calculated with the cut-off ADPOS potential that gives the total cross sections infigure 10.5. ConclusionsThe graphs of total and orbital electron densities for polarised states of N2 ( 8 4.1)illustrate the nature of adiabatic distortions induced by an incident positron and


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Positron- molecule scattering theory 129

- Ih

Figure 12 . Total differential posit ron-N , cross sectio ns fo r impact energies less than orequa l to 10 eV as calculated using the cut-off ADPOS polarisation potential of f igure 11.The cor respond ing in tegra ted cross sec t ions a re co mpared wi th exper iment in figure 10(b ) .

demonstrate how different these distortions are from those induced by an electron.The observation that the resulting adiabatic polarisation potentials (figure 6) dependon the nature of the projectile extends to this prototypical positron-molecule systemconclusions of our earlier research on scattering from H 2 . This aspect of positronscattering is to be expected from the high sensitivity of positron scattering cross sectionsto polarisation effects in general. As the potential and cross section graphs in 4 attest,non-adiabatic effects exacerbate this dependence, especially at low impact energies.Indeed, both adiabatic and non-adiabatic effects appear to be essential to a physi-cally realistic model of polarisation in positron scattering: only by including both couldwe generate cross sections in agreement with experimental data. Significantly, to doso we had to adopt a far softer cut-off than that used in electron scattering calculations.This suggests that the nature of non-adiabaticity in positron scattering is essentiallydifferent from that in electron scattering. In the latter case, velocity-dependent correc-tions to adiabatic polarisation effects are of crucial importance near the target; but,in positron scattering, correlation effects (which give rise to virtual positronium forma-tion) appear gradually as the projectile nears the target, where they become dominant.We hope that this study will contribute to the formulation of an ab initio (parameter-free) model polarisation potential for positron scattering from molecules that accuratelyreflects the essential physics of their interaction.

AcknowledgmentThis research was supported by NSF grant PHY-8505438.

ReferencesAlms G R, B ur nha m A K an d F lygare W H 1975 J. Chem. Phys. 63 3321-32Armour E A G 1984 J . Phys. E : At . Mol. Phys. 17 L375-82


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