Bound states of positrons with atoms and molecules: Theory

Bound states of positrons with atoms and molecules: Theory

D.M. Schrader 1

Department of Chemistry, Marquette University, P.O. Box 1881, Milwaukee, WI 53201-1881, USA

Received 31 October 1997; received in revised form 30 March 1998

Abstract

Calculations of the binding energies and annihilation rates of bound atomic and molecular systems which contain a

positron are reviewed. Emphasis is placed on methods of calculation and the quality of the numerical results. In this

article we limit our attention to positrons interacting with atoms, diatomic molecules, and their ions. Ó 1998 Elsevier

Science B.V. All rights reserved.

Keywords: Positron compounds; Positronium compounds; Antimatter compounds; Bound states; Annihilation rate

1. Introduction

Bound states have been on our collective mindsfrom the beginning [1±3]. They have been postulat-ed many times to explain observations made in thegas phase [4±7] as well as in complicated environ-ments such as solids [8±11] and solutions contain-ing the chemical residue from the slowing down ofpositrons from radioisotope sources [12,13]. Allatomic and molecular bound systems which con-tain a positron have lifetimes on the order of hun-dreds of picoseconds, and in that sense they areunstable. In this article we consider another kindof stability, namely, chemical stability. A systemis considered to be chemically stable if its energy

is below all dissociation thresholds; such a systemwill hang together mechanically until its inevitableannihilation. At least three quantities are interest-ing for each such system: its binding energy, itslifetime, and the momentum distribution of the an-nihilation photons. Each of these quantities is inprinciple calculable and measurable. Both the cal-culations as well as the experiments are challengingand interesting. Evidently the calculations are lessdi�cult, because most of our knowledge of thesesystems comes from calculations.

In this article we brie¯y review our knowledge(up to September 1997) of chemically stable mixedpositron±electron systems, and compare theoreti-cal with experimental ®ndings whenever possible.The interactions responsible for binding, togetherwith various computational aspects, have beenpreviously reviewed [14±17]. Here we limit our-selves to a short discussion of the role of the at-traction of atoms for electrons, as evidenced by

Nuclear Instruments and Methods in Physics Research B 143 (1998) 209±217

1 Tel.: +1 414 288 3332; fax: +1 414 288 7066; e-mail:

[email protected].

0168-583X/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 2 7 7 - 8

the electron a�nity and the ionization potential, inthe binding of positrons and positronium. We alsocomment on the electron±positron correlationproblem, in particular the recent discovery of im-portant three-body e�ects. We use symbolismand nomenclature in this work which re¯ect ourown chemical background, and sometimes we bor-row from our friends who work with muons. Weare not systematic; we use names and symbols suchas PsH, positronium hydride; PsCl, positroniumchloride; LiPs, lithium positride; and CH3Ps,methyl positride or positromethane. We call e�Csand e�H2O positronic cesium and positronic wa-ter, resp. The ``binding energy'' of a compoundPsA and the ``positronium a�nity'' of A are oneand the same, and are denoted by the symbolBE(PsA). The electron a�nity and positrona�nity of a species A are denoted by EA(A)and PA(A), resp., and its ionization potential isIP(A).

It is important to understand that PsH is an at-om in spite of its diatomic-like symbol and name.Similarly, PsOH, positronium hydroxide, is a di-atomic molecule. This is so because the positronis so light that structurally it is more like an elec-tron than a proton.

2. Why do some atoms and molecules (and not

others) bind positrons and positronium?

Positronium binding to a neutral atom or mol-ecule might result if the electron a�nity of the at-om or molecule is large, or if its electric dipolepolarizability is large. The ®rst is a strong short-range e�ect while the latter is a much weakerlong-range e�ect and is not examined here. In ad-dition, molecules (but not atoms) might bind pos-itrons by virtue of a large permanent electricdipole moment.

We note that a few atoms unambiguously donot bind positrons: H, He, Ne, and N. That e�Hhas no bound state was proven by establishing[18] that this system fails to satisfy a necessary con-dition for binding. Similarly for e�He [19], e�Ne,and e�N [20]. Interestingly, e�He was found tobe bound in two published calculations [21,22]which are, of course, erroneous.

2.1. The role of the ionization potential; the bindingof e� to neutral atoms

One view of the binding of a positron to a neu-tral atom holds that the electron is a resonant mix-ture of two structures which might be representedas e�A and PsA� [23]. If the ionization potential ofA is less than 6.8 eV, the structure PsA� dominatesthe mixture, otherwise the reverse. In this reso-nance picture, a large binding e�ect occurs if theionization potential of A is close to 6.8 eV. Thereare several atoms whose ionization potentials arewithin 0.1 eV of 6.8 eV: Ti, V, Cr, Zr, Nb, andHf. In other words, the two thresholds e� + Aand Ps + A� are close for these atoms. These acci-dental near-degeneracies do not imply extremelydeeply bound levels for positrons as some havesuggested [24], but instead only a more or lessequal mixing of the two resonance structures. Inthe resonance picture, this produces two levelswith the lower one being below the lower thresholdso that a bound state for the system exists. This ar-gument would seem to guarantee the existence ofone bound state for every positron±atom system,but in fact several atoms are known not to binda positron, as noted above, so the simple reso-nance picture sketched above evidently is an over-simpli®cation. For example, the positron±cationrepulsion is clearly part of any complete theory.

In general, neutral atoms o�er an inhospitableenvironment to a positron. The static potential isstrong and repulsive everywhere. There is a weaklong-range polarization potential varying asÿa=2r4 for large atom±positron distances r, anda short-range correlation potential that is also at-tractive. The interplay of these opposing e�ects de-

Table 1

Positron a�nities and other parameters from a many-body per-

turbation calculation a

Atom Mg Cd Zn Hg

e� a�nity (eV) 0.87 0.35 0.23 0.045

Ionization potential (eV) 7.65 8.99 9.39 10.44

Polarizability (�A3) 10.6 7.2 6.4 5.4

a From Ref. [24].

210 D.M. Schrader / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 209±217

termines binding, and only quantum calculationscan sort out the balance.

Just this was recently attempted by Dzuba andcoworkers [24], who performed extensive many-body perturbation calculations on the atoms Mg,Zn, Cd, and Hg, and found all of them to bebound. Their binding energies and other parame-ters are given in Table 1. It can be seen that thecalculated binding energies fall as the ionizationpotential increases beyond 6.8 eV (in accordancewith the resonance model discussed above), andas the polarizability decreases. 2 It is di�cult to as-sess this work. The method does not providebounds on calculated energies, but it works wellfor some purely electronic properties such as theenergy levels of alkali metal atoms [25,26] andatomic dipole polarizabilities [24]. Regarding pos-itron binding, the authors warn [24] that there maybe some ``double counting'' from using Ps interme-diate states, and that ``the accuracy of the Ps-for-mation potential is not as well established'' as isthe polarization potential in purely electronic at-oms.

2.2. The role of the electron a�nity; the binding ofPs to neutral atoms and molecules

The binding energy of the compound PsA canbe written as

BE�PsA� � EA�A� � PA�Aÿ� ÿ 6:80 eV: �1�Of course, there are other relationships one can

write for BE�PsA�, but we consider only Eq. (1)here for the limited purposes of our discussion.Readers interested in a fuller discussion shouldconsult the older reviews [14±17]. If an atom ormolecule with a small electron a�nity adds anelectron, the anion is large and has a di�use outerelectron density. Such an electron is free to corre-late strongly with an added positron, but the pos-itron is simultaneously repelled by theincompletely shielded nucleus. One conjecturesthat the nucleus must be light in order for the at-traction to the added electron to dominate the re-pulsion by the nucleus. PsH is bound by 1.0661 eV

[27] but two calculations on LiPs [28,29] fail toshow binding. Perhaps the limitation on the sizeof the nucleus for atoms with small electron a�n-ities to bind positronium is a strong one, so that ofsuch atoms, only H can bind a positronium atom.

If an atom or molecule has a large electron af-®nity, then the added electron is more tightlybound, and the resulting anion more closely resem-bles a heavy negative point charge. 3 A positron in-teracting with such an anion must, in order tobind, ®nd that the long range Coulomb attractionof the anion is strong enough to overcome themuch larger but shorter range repulsion from theincompletely shielded nucleus.

The relationship between the electron a�nity ofA and the positron a�nity of Aÿ for stable PsAspecies is shown in Table 2. PsH is an exampleof Ps binding to a species with a low electron a�n-ity; the positronium halides show binding of Ps tospecies with large electron a�nities. PsOH is an in-termediate case. The PA(Aÿ) values have a nearlylinear dependence on the electron a�nities of A.The e�ect of increasing nuclear charge for speciesA with nearly the same electron a�nities is appar-ent for the halogens.

3. The electron±positron correlation problem

That Schr�odinger wave mechanics is applicablefor calculating wave functions and properties ofmixed electron±positron systems was recognizeda very long time ago [31]. This discovery is crucial,for it provides validity for practically all the theo-retical work done so far on such systems.

One can learn useful lessons from simple con-siderations using Schr�odinger wave mechanics.For example, the annihilation of a free positron

2 This correlation does not imply a causal relationship.

3 One should not carry the obvious analogy with antihy-

drogen too far, for even a very compact anion is much larger

than an antiproton, and furthermore the threshold closest to

PsA is (not considering dissociation of a molecule A) invariably

Ps + A, not e� + Aÿ. Nevertheless, one should expect the

positron to have an in®nite number of levels converging on the

e� + Aÿ threshold, but only those below the lower Ps + A

threshold are bound states. So far, no excited bound states of

PsA have been found.

D.M. Schrader / Nucl. Instr. and Meth. in Phys. Res. B 143 (1998) 209±217 211

embedded in an atomic or molecular environmentproceeds from virtual positronium. ``Virtual'' inthis sense refers to a feature of the wave function[32], namely a well-developed enhancement of thewave function at short electron±positron distanceswhich mimics the wave function of positronium,including the cusp at coalescence [33]. The cuspvalue, the logarithmic derivative of the wave func-tion at coalescence, has a known value for the ex-act wave function (ÿ1=2), so its value for anapproximate wave function is of interest as ameans of assaying its accuracy at coalescence[34]. This is particularly important for the calcula-tion of annihilation rates because the coalescencepoint is the only point in the relative space of theannihilating particles which contributes to thatquantity. More generally, one concludes that, foraccuracy in calculations, the wave function for aone-positron, many-electron system must some-how include the e�ects of the highly correlated mo-tion of each electron±positron pair in such a wayas to substantially reproduce positronium embed-ded in the electron sea. The con®guration interac-tion method, applied so successfully to purelyelectronic systems, has been demonstrated to bemuch more slowly convergent for mixed elec-tron±positron systems owing to the e�ect of virtualpositronium in the wave function [35,36], and ithas never been successfully applied to a calculationof a positronium binding energy or an annihilationrate for a positronic compound.

3.1. Insights from cluster expansion calculations

Many-electron wave functions can be expandedin a series of electron cluster functions each of

which serves to correlate the motions of a certainnumber of electrons. In its simplest form [37],

w�x1; x2; . . . ; xn� � wHF �Xn

i

fi �Xn

i<j

Uij

�Xn

i<j<k

Uijk � � � � � U123...n; �2�

where x � �r; r� denotes the space and spin coordi-nates of the particle indicated by the subscript, andwHF is the Hartree±Fock wave function, a Slaterdeterminant of spin-orbitals. Each of the n-elec-tron cluster functions fi and the U 's is orthogonalto the terms which precede it on the right hand sideabove and is properly antisymmetrized and nor-malized. Each of these has the e�ect of correlatingthe motions of the subscripted electrons beyondthat provided by lower order terms. A more com-plete description is given by Sinano�glu [37].

In approximate calculations, the series is trun-cated. The ``one-body cluster'' term

Pfi is zero

unless higher order terms are included in the trun-cated expansion. The calculation of the pair clusterfunction Uij has been and continues to be muchstudied [38±40]. The three-body function Uijk isthought to be a less important part of the wavefunction for electronic systems because the e�ectsof both Coulomb repulsion and exclusion serveto make the wave function small in those regionsof many-electron space where three electrons areclose. The higher order terms are perforce still lessimportant. For many-electron systems which con-tain a positron, the importance of two-body clus-ters which consist of the positron and an electronhave long been recognized and discussed in thecontext of electron±positron correlation [34,41].

Table 2

Electron a�nities of A, and positron a�nities of Aÿ for stable compounds PsA a

PsH PsOH PsF PsCl PsBr

EA(A) 0.754 1.828 3.40 3.62 3.37

PA(Aÿ) 7.112 5:58� 0:16 5:38� 0:17 5:09� 0:09 4:57� 0:11

BE(PsA) b 1.0661 0:61� 0:16 1:98� 0:17 1:91� 0:16 1:14� 0:11

a Entries are arranged in order of increasing nuclear charge. Energies are in electron volts. Electron a�nities are from Ref. [30].

Positron a�nities of the anions are calculated from data in Table 4 and Eq. (1).b From Table 4.


Recently Saito and Sasaki applied the clusterexpansion formalism to positronium hydride,PsH [42]. This system has only three light particles,so the cluster expansion of its wave function iscomplete with the three-body term. They ignoredthe one- and three-body terms and computed thetwo-body term in several approximations at di�er-ent levels of accuracy. We consider here only theleast and most accurate of these, which are the in-dependent pair approximation (IPA), and thecomplete coupled pair (CCP) procedure, respec-tively. The reader should consult the original pub-lication for more details, but new insight of greatvalue is given by considering only the IPA andCCP results. These are summarized in Table 3.

The coupled cluster methods do not providebounds on the total energies. As Saito and Sasakipoint out [42], the IPA method is known to over-estimate correlation energies and thereby tends togive calculated total energies that are too low.The neglect of the one- and three-body clusterterms give a positive error in the total energy,and evidently these two errors fortuitously cancelin the case of the IPA calculation on PsH [42].By improving the calculation to the CCP level, thiscancellation is at least partly removed, thus givingless accurate results.

It is not hard to understand the importance ofone-positron, two-electron clusters in retrospectof the work of Saito and Sasaki: The electronsare attracted to the positron by Coulomb forces,and for electrons with opposite spin, exclusion

does not play a role. Therefore, one should expectthree-body clusters which contain one positron tobe a more important part of the wave functionthan are three-body clusters with only electrons.The conclusion is that three-body clusters whichcontain a positron must be accounted for some-how in accurate calculations of energies and anni-hilation rates of mixed electron-positron systems.

Since positrons o�er a stringent test of wavefunctions on account of the importance of three-body clusters, it follows that positrons will providea valuable testing ground for quantum chemists af-ter they solve the two-body cluster problem totheir satisfaction and turn to the three-body clusterproblem. One might expect to see reports in the fu-ture of calculations on many-positron systems likeCPs4 (substituted methane) and C6Ps6 (substitutedbenzene)!

3.2. Annihilation rates and momentum distributions

It has been shown [31,44] that the annihilationrate is the expectation value of a one-electron op-erator. A wave function which provides accuratelyfor electron±positron correlation is required, elseerrors of up to an order of magnitude will result.The simplest wave function which provides satis-factory electron±positron correlation is [41]

~W � wp�xp�AY

lwl�xl; rp�: �3�

A is the electron antisymmetrizer and renormaliz-er, and the spin orbitals w are:

Table 3

Results of PsH calculations of the total energy, the binding energy, the two-gamma annihilation rate, and the electron±positron cusp

value

Method ÿEtot (a.u.) BE (eV) k2c �nsÿ1� Cusp value (aÿ10 ) References

Hartree±Fock 0.66695 )2.260 0.22 0.0 [29,34]

IPA 0.78899 1.065 2.088 )0.652 a [42]

CCP 0.77471 0.672 1.972 )0.664 a [42]

Accurate 0.789178 b 1.0661 b 2.436 0.0 c [27]

Exact )0.5 [33]

a Calculated following Ref. [43].b These ®gures (and all the others in this table) are for a proton with in®nite mass. For a proton with its correct mass, the total energy,

binding energy, and annihilation rate are )0.788853 a.u., 1.0647 eV, and 2.436 nsÿ1, resp.c Gaussians have zero cusp values but can be the basis of accurate calculations nonetheless.


wp�xp� � up�rp� � �a�p� or b�p��;

wl�xl; rp� � ui�rl; rp� �a�l� for l odd;

b�l� for l even:

��4�

Each electronic orbital ui is doubly occupied anddepends dynamically on the position of the posi-tron as well as, of course, that of the electron.For this approximate wave function, the annihila-tion rate takes a particularly simple form

k2c � 2pr20c

a30

Xi

Zjup�r�ui�r; r�j2 dr: �5�

Similarly, the momentum distribution of the anni-hilation gammas is:

q2�k� /X

i

Zjup�r�ui�r; r�eik�rj2 dr: �6�

In this case, interestingly, electron±electron corre-lation is not required for reasonable accuracy. Thisis because experiments sample only the momentumof the center of mass of the annihilating particles.Each one is attracted to and accelerated by theother, and this produces all the correlation e�ectsincluding the cusp of the wave function at coales-cence. However, the substantial relative momentaexperienced by the particles from this e�ect adds

to zero and is therefore not carried away by the an-nihilation gammas. For this reason, simple Har-tree±Fock wave functions provide accuratemomentum distributions.

4. Present status of our knowledge of bound systems

There has been a great deal of speculation thatbound states of various kinds exist; they have beeninvoked repeatedly to explain laboratory data, es-pecially lifetime spectra in condensed phases. In asmall number of cases, experiments in condensedphases shed light on annihilation rates and mo-mentum distributions. There is a modest literatureof calculations, but all except for a handful ofthese are very approximate and do not materiallycontribute to our understanding of these systems.Published work on gas phase experimental deter-mination of binding energies is limited to justtwo reports. In short, we are just beginning to un-derstand bound states.

4.1. From calculations

All our high-quality knowledge of the bindingenergies and annihilation rates of gas-phase mixed

Table 4

Totality of our high quality knowledge about bound states as of September 1997

System Binding energy

(eV)

Dissociation

products

Annihilation

rate (nsÿ1)

Method Year, Ref.

Ps 6.8029 e� � eÿ 2.003 a Elementary atomic theory 1945 [46]

Psÿ 0.3267 eÿ � Ps 2.086 Gaussian variational expansion 1990, 1993 [47,48]

Ps2 0.4354 Ps + Ps 1.111 b Same as above 1996, 1997, [27]

PsH 1.0661 c Ps + H 2.436 c Same as above 1997 [27]

PsF 1:98� 0:17 Ps + F Di�usional quantum Monte Carlo, ®xed nodes,

model core potential

1993 [49]

PsCl 1:91� 0:16 Ps + Cl Same as above 1992 [50]

PsBr 1:14� 0:11 Ps + Br Same as above 1993 [49]

PsOH 0:61� 0:16 d Ps + OH Di�usional quantum Monte Carlo, ®xed nodes 1996 [51]

a This, the spin-averaged rate, is the only experimental number in this table. The annihilation rate of p-Ps is 7.991 nsÿ1 [52], and of o-

Ps, 7.088 lsÿ1 [53].b This is one-fourth the ®gure reported in Ref. [27], where the spin part of the wave function is assumed to be solely (p-Ps)±(p-Ps) in

character. Actually, the spin structure of the wave function for the ground state of Ps2 is only one-fourth (p-Ps)±(p-Ps) character

[54,55], and only that part can annihilate with the production of two gammas.c These (and all other quantities given above) are for an in®nite nuclear mass. Finite mass values are given in Table 3, footnote b.d 0.7 eV is reported in Ref. [56], but this is fortuitous because the authors make two serious cancelling errors [57].


positron±electron bound systems comes from cal-culations, and can be put into a small table, whichis Table 4. There are a number of other calcula-tions, but these reveal more about the shortcom-ings of the methods used than the systemsthemselves. That part of the literature which pre-dates 1995 is reviewed in Ref. [15]. In addition tothe perturbation calculations on PsH [29] alreadydiscussed above, LiPs, PsF, and PsCl were alsotreated in the same paper. The newer literature in-cludes quantum Monte Carlo (QMC) calculationson PsH and LiPs [28], in which the binding energyof PsH is found to be 1:048� 0:014 eV, and LiPs isfound to be unbound. A recent higher-qualityQMC calculation [45] gives the binding energy as1:064� 0:002 eV, and the annihilation rate as2:43� 0:11 nsÿ1. These results are as accurate(but not as precise) as the recent results of Frolovand Smith listed in Table 4.

It is evident from Table 4 that three methods ofcalculation provide all the high-quality informa-tion known today: elementary quantum theory,large scale Ritz variational expansions in correlat-ed Gaussian basis functions, and the QMC meth-od. The ®rst is applicable only to Ps. The secondmethod has been applied to systems with onlytwo and three particles 4 because the di�culty toexpress a trial wave function for a larger systemis prohibitive. Gaussian basis functions have anadvantage over functions with linear exponentialarguments in that the required evaluation of ma-trix elements by integration is simpler, thus permit-ting longer and more accurate expansions. TheQMC has no limitations of applicability exceptits voracious appetite for computer time. QMC isthe only method that has been successfully appliedto both large [49±51] and small [45] systems, butimprovements in cluster expansion methods willeventually change this assessment.

In the QMC method, the time-dependentSchr�odinger equation is transformed into a classi-cal di�usion equation by the simple substitutionit ! s, which permits its solution by random walk

techniques developed many years ago. In essence,the Schr�odinger equation is solved exactly by ran-dom walk simulations of the corresponding di�u-sion problem but for the necessary limitation ofusing ®nite time steps in calculations. The QMCequation for the wave function W may be written

oWos� �Dr2 ÿ V � E�W; �7�

where the di�usion coe�cient D � �h=2m, and Vand E are the quantum mechanical potential andeigenvalue, resp., in units of �h. The space in whichthe di�usion takes place has dimensionality 3nwhere n is the number of positrons + electrons,plus the number of nuclei if the Born±Oppenhei-mer approximation is not invoked. Each di�usingparticle, or walker, is a particular con®guration ofeach of the 3n real particles which constitute thesystem. To initiate a calculation, a large numberof walkers are placed in their 3n-dimensional spaceby random selection of coordinate values. As thewalkers move around in their space, their distribu-tion approaches that of W itself. Precision and ac-curacy are in principle unlimited. In practice,realistic goals for precision and accuracy areachieved by balancing computational resourceswith the optimal combination of a large numberof walkers moving in short steps for a long time.Inadequate precision and accuracy result in the ab-sence of proper care. In a recent calculation onPsCH3 [58], the calculated binding energy was re-ported to be 0.2 eV, but the uncertainty of this re-sult, which is inherent in QMC methods but wasnot reported by the authors, is considerably larger.

The important thing to realize about QMCmethods is that all the subtleties of electron±posi-tron correlation, including the e�ects of three-bodyclusters discussed in Section 3.1, are correctlytreated automatically. There are several featuresof QMC methods that require careful technicaltreatment. For example, spin coordinates are nota part of QMC methods, so the spin structure ofthe wave function must be treated in an ad hocmanner. Wave functions, unlike solutions of classi-cal di�usion equations, have nodes and negativeparts. These features require the use of a trial wavefunction for each system which is not provided byQMC. The trial wave function should have

4 When the center of mass is transformed out of the

quantum mechanical problem, Psÿ becomes a two-particle

problem, and Ps2, a three-particle problem.


accurate nodes, which in general are surfaces in�3nÿ 3�-dimensional space, and should satisfy allcusp conditions. Solutions for these and otherproblems have been largely achieved [59], andQMC methods are today successfully used for cal-culations on atoms and molecules [60].

Because of its ability to correctly treat the elec-tron±positron correlation problem correctly,QMC methods are likely to be the methods ofchoice in the near future for calculating bindingenergies and annihilation rates of many-electron±positron systems. Con®guration interaction meth-ods converge too slowly, and techniques relying onexpansions in correlated basis functions appear tobe intractible for more than four light particles.

4.2. From experiments

There are only two reported experimental de-terminations of the binding energy of a chemicallystable positronic compound in the gas phase, al-though one of these is very old and is based on in-direct measurements and questionable inferences[61]. In this study, the annihilation rates in Cl2

gas and gaseous mixtures of Cl2 and Ar were mea-sured, and a resonance in the annihilation rate, es-timated to occur at 0.5 eV, was attributed to thereaction

Ps� Cl2 ! PsCl� Cl: �8�Since the bond energy of Cl2 is 2.5 eV, the bindingenergy of PsCl must be about 2.0 eV, according tothese authors. The remarkable agreement with themodern value shown in Table 4 is probably fortu-itous. Their conclusions cannot be independentlyassessed because their report is not complete [61].

A more modern and direct determination of abinding energy, on PsH, was recently reported[62]. The measurement gives 1:1� 0:2 eV, in agree-ment with the much more precise theoretical resultshown in Table 4. This experiment consists of de-tecting the threshold for the dissociative attach-ment of positrons to methane

e� � CH4 ! PsH� CH�3 : �9�The binding energy of PsH is deduced from know-ledge of the energy of the positron beam E�e�� andrelevant bond energies and ionization potentials,

assuming that all molecular species are in theirground states:

BE�PsH� � BE�H±CH3� � IP�CH3�ÿ 6:80 eVÿ E�e��: �10�

A measurement by the Oak Ridge group of thebinding energy of PsF in the gas phase by a similarexperiment on CH3F is in progress and is reportedelsewhere in these proceedings [63].

Acknowledgements

The authors are indebted to Mr. Nan Jiang forhelpful discussions, and to the National ScienceFoundation for support (PHY-9600416).

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Documents

Bound states of positrons with atoms and molecules: Theory