PHYSICAL REVIEW A VOLUME 33, NUMBER 5 MAY 1986

Binding energy of positronium molecules

Y. K. HoDepartment of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803

(Received 6 January 1986)

The binding energy of positronium molecules is calculated variationally by using extensive Hylleraas-typewave functions in which all six interparticle coordinates are used. The binding energy against dissociationinto two positronium atoms is determined as 0.0302 Ry. The present result is consistent with the value of0.0303+ 0.0005 Ry, obtained by Lee using the Green's-function Monte Carlo method.

This work reports a variational calculation of the bindingenergy for positronium molecules, Ps2, a system consistingof two positronium atoms (or a four-particle system consist-ing of two electrons and two positrons, and interacted viaCoulomb forces). There have been continuous studies offew-body atomic problems involving positrons. For exam-ple, recently there were intense theoretical studies' of vari-ous properties of positronium negative ions, a three-bodysystem consisting of two electrons and one positron. Thisthree-body system has been observed experimentally. ' Thepositron annihilation rate for this system has also been mea-sured. ' For positronium molecules, the first calculation thatshowed such a system does form a bound system was byHylleraas and Ore. They showed that the binding energyfor Ps2 is 0.116 eV. Over the years, several theoretical stud-ies have appeared for this system. ' Possible experimentalstudies of positronium molecules have been discussed re-cently by Mills.

This work begins an extensive study of positroniummolecules. The goal of these studies is to achieve an accu-rate variational value for binding energy for this system and

I

to study its various properties. The Hamiltonian for thissystem is

2cos(e„";k)

iWjk mi

j&k

with

fij + fik fjk.2 2 — 2

~ij,ik2f(j fl'k

where m; and Z; are the mass and charge of the particle i,respectively. Atomic units are used in this work with ener-

gy expressed in Rydbergs. Figure 1 shows the coordinatesystem where a and b denote the positrons, and 1 and 2 theelectrons. The interparticle coordinate rl, represents thedistance between the electron 1 and positron a. A generalform of wave function for positronium molecules is

g CkmnijI [flaf2afabf2bfibf12 eXP( Clfla C2f2a C3 "ab C4I'2b C3flb —Cbf12)k m n i '

I

+ flaf2afabf Ibflbf12 eXp( —C2fla Cl f2 C3f b C3f2b C4I'lb Cbf12)

+flbf2bf,"bf2. fjl ff2 eXp( —Clflb C2f2b C3fab C4f2 C3f] Cbfl2)

+ I'lbf2bf hfdf "1 f12 exP( —c2flb —clf2b —c3f b cSf2 c4f1 cbf12) ]

1 1X I (Olp2 —plI22) I- (Ika pb paCIb )

v'2 42

where Ik and p are the spin-up and -down wave functions,respectively. The wave function in the form of Eq. (2) is

antisymmetric with respect to interchange of the two elec-trons or of the two positrons. To solve the necessary in-

tegrals involved in this extensive form of wave functions isnot any easy task. Some simplications are hence made here.First, we omit the symmetry of the two positrons, i.e., thelast two terms in Eq. (2) are dropped. The implications ofsuch an approximation will be discussed later in the text.Secondly, we let C4= C~= C6=0. The omission of the ex-plicit exponential factors involving rl~, r2~, and rl2 will becompensated by the use of extensive terms involving po~erseries of such interparticle coordinates. Under these ap-proxirnations, the wave function becomes

e (1) e+(b )

e'(a ) r, e (2)FIG. 1. Coordinate systems for the positronium molecules; parti-

cles a and b are for positrons and 1 and 2 for electrons.

33 3584 1986 The American Physical Society

33 BINDING ENERGY OF POSITRONIUM MOLECULES 3585

TABLE I. Energies and cusp values for positronium molecules: ~& ——k+m+n+i +j+l, with m =0=i,cu2 = k + m + n + i +j, with m = 0.

z (By)

(et=3, %=35ru&

——4, N =70rv) =5, % =126

—0.981 08—1.006 32—1.016 17

—0.444—0.490—0.482

0.1820.2890.386

0.4330.3750.481

—0.333—0.398—0.441

cv2= 5,cll2 = 5,QJ2 = 5b

co2= 6,

l =0, N =126l =1, N =252l=2, %=322l ~4, % =400

—1.017 16—1.025 29—1.027 36—1.030 21

—0.485—0.491—0,491—0.498

0.00.1950,4580.485

0.5110.5370.5240.509

—0.441—0.449—0.447—0.479

Exact —0.5 0,5 0.5 —0.5

g Ckmaijl ( flaf2afabf2bf ibfl2 eXp( Cl fla C2f2a C3fab ) + flaf2afabf jbfIbfl2 eXP( —C2f2a Clfla Clfab ) ~ ~ (C4li82 pla2) .

(3)

The advantage of using this wave function is that all thenecessary integrals have already been done in a relatedproblem of binding-energy calculation of positronium hy-dride, PsH. ' Following the investigation of PsH, we firstlet m =i = 0 in Eq. (3). The wave functions with &a = k + m

+n+i +j+I=1,2, 3, 4, and 5 correspond to a total ofIV=5, 15, 35, 70, and 126 terms, respectively. The non-linear parameters are optimized at N =70 terms. They aredetermined as C~=0.385, C2=0.48, and C3=0.37. For1V = 70 terms we obtain the lowest eigenvalue of —1.00632Ry (see Table I). It is seen that even with this somewhatextensive %=70 basis set, the ground-state energy liesabove that of Hylleraas and Ore (see Table II). This is be-cause we do not include the explicit exponential factors in-volving r~~ and r~~. The use of po~er series for such fac-tors leads to slow convergence. %e are, nevertheless, ableto obtain a better energy when we extend the calculation upto eo = 5, a total of N = 126 terms. The ground-state energyfor Ps2 is obtained as —1.01617 Ry. The binding energy ishence 0.22 eV, about a 10% improvement over the value of0.2 eV obtained in Ref. 7 (see Table II).

Since the convergence in energy using the wave functionsdescribed above [Eq. (3)] is slow, an alternative form ofwave function is desirable. The slow convergence is partlydue to the fact that we omit the last two terms of the wave

I

function. This indicates that the products of the exponen-tial factors of exp( —C3flb) exp( —C4f2b) are important. Inorder to take into account these factors, the condition i = 0in Eq. (3) is relaxed. In other words, products of fib andr2b are explicitly included. With &F2=5, l =0, ~here cu2 isdefined as cu2= k+ m +n+i +j, we obtain an energy ofE= —1.01716 Ry with this N =126 term wave function.This is not a bad result considering that we have not includ-ed the r~2 terms, the explicit correlation factor between thetwo electrons. Such a factor has been proven to play an im-portant role in accurate calculations of the related systemPsH. Next, when we include the r~2 factor, i.e., let l =1and ia2= 5 (N = 252), we obtain E = —1.025 29 Ry, a signi-ficant improvement of binding energy. Next, we add 70terms of l =2 to the wave function. With a total of N =322terms, the ground-state energy is determined as —1.02736Ry. The most extensive set of wave function used in thiswork is N =400. This wave function includes terms withl ~ 4 and some terms with ~2 = 6. %e stop here at N = 400because of practical reason. Kith N =400 terms we obtainthe ground-state energy of —1.03021 Ry. This result isconsistent with the value of —1.0303+0.0005 Ry obtainedin Ref. 8 with the Green's-function Monte Carlo method.It should be mentioned that the uncertainty of + 0.0005 Rylisted in Ref. 8 is due to statistical error.

TABLE II. Calculations of binding energies of positronium molecules.

Authors (Ry}Binding energy

(eV)

Hylleraas and Ore 1947 (Ref. 4)Ore 1947 {Ref. 5)Akimoto and Hanamura 1972 (Ref. 6)Brinkman, Rice, and Bell 1973 {Ref. 7)Lee, Vashishta, and Kalia 1983 (Ref. 8)Lee 1985 (Ref. 8)Present calculation

0.00850.0090.01350.01450.03 + 0.0020.0303 + 0.00050.0302

0.1160.1220.1840.1970.408 + 0.0270.412 + 0.0070.411

Y. K. HO 33

TABLE III. Energy spectra for Ps with ~-2 wave functions

[see Eq. {4)]. For g -0, 1, and —1 correspond to N = 10, 7, and 3,respectively.

TABLE IV. Average distances (in ao) between various pairs ofcharged particles.

g ~0(~l ) (~2 ) (rl2)

5.93

(r,b) (fib) (f2b)

-0.521 546—0.402435-0.347 328-0.065 143

0.0466730.1274550.1549350.6021690.968 3441.280 176

—0.521 546

—0.347 328

0.0466730.127 4550.154935

0.968 3441.280 176

—0.402 435

—0.065 143

0.602 169

+grl rfrl2 exp( —c2rl —clr2) ] (4)

For the purpose of discussion, we insert a constant factor gin front of the second term in Eq. (4). When g- I and

k ~ m, the wave function represents the singlet states of thesystem. When g —1 and k & m, the wave functionrepresents the triplets. The case when g -0 is in analog tothe present calculation for positronium molecules with thelast two terms missing in Eq. (2). For pl k+m+n 2

and c~ c2 0.362, the total numbers of terms for g =0, 1,and -1 are N -10, 7, and 3, respectively. The energy spec-tra for all these three cases are sho~n in Table III. It is

seen that for the g 0 case the energy spectrum containsenergies for both the singlet and triplet states of Ps . Sincethe lowest energy state of Ps is singlet, we can assume thatthe lowest energy state for g -0 is also singlet. In the caseof Ps2 we can make the same assumption. (In vigoroussense, we cannot tell what spin state for positrons that thewave function represents. ) The example discussed abovealso shows why the convergence is slow when we let g -0.In the Ps case, the 10-term (g 0) wave function canonly produce the same singlet state results for the N = 7

term (g-1) wave functions.To test the qualities of the wave functions, we have calcu-

lated the electron-electron and electron-positron cusp val-

ues. For a system interacting through Coulomb forces, theaverage value of the cusp condition between particles i and jis given by"

(4'[ig(rb)(d/Brj) (0)(q I &(r&) I+)

We now discuss the implication of the fact that the last

two terms in Eq. (2) were dropped. In other words, the ra-

dial wave function in Eq. (2) is not symmetric with respectto the interchange of the two positrons. In order to under-

stand more of the problem, let us consider the case of bind-

ing energy of Ps . The generalized Hylleraas-type wave

function is used,

X Cb „[rblr2rl2 exp( —clrl —c2r2)

and the exact value for v;, is

pg = Zl Zgp, gg

~here Z; is the charge for the particle i and p,„ is the re-duced mass for the particle i and j. The exact values for theelectron-electron and electron-positron conditions are hence+0.5ao ' and —0.5ao ', respectively. The cusp values arealso shown in Table I. For the wave function that providesthe lowest energy, it is seen that the cusp values are close tothe exact ones, with at most a discrepancy of about 4'10.

Since the cusp values are related to the first-order error inthe wave function, and energy is related to the second or-der, we estimate the error in the ground-state energy isabout 0.16%. Such an error in the ground-state energy of1.0302 Ry is equivalent to about 6'/o in the binding energyof 0.0302 Ry. We should, however, mention that the errorestimate is for qualitative references only, and is not rig-orous since there are no bound principles in the calculationsof cusp values.

We have also calculated all the six averaged interparticledistances, (r&). For economic reasons, we only use the126-term wave function (see row 3 in Table I) in this work.The results are shown in Table IV. In principle, (rl, )should equal (rib). The reason that they are not equai isbecause the last two terms in Eq. (2) are omitted. On theother hand, they are close to each other that this implies thewave function used in this work [Eq. (3)] is a goodrepresentation of the system. One of the interesting resultsshown in Table IV is that all the six interparticle distancesseem to have the same value of 5.9ap (if we assign a 10%uncertainty to (rib) ). This suggests that on average, thefour particles form a triangular pyramid, with the two elec-trons occupying any two of the four vertices, and the twopositron occupying the other two. All the six edges havethe same length of 5.9ao. In this arrangement, the systemis symmetric with respect to the interchange of the two elec-trons, or of the two positrons, as well as to the interchangeof the two positronium atoms.

In summary, the binding energy of positronium moleculesis calculated as 0.0302 Ry. The present result agrees verywell with a Green's-function Monte Carlo calculation of0.0303+0.0005 Ry. Also, all the six averaged interparticledistances are estimated to have the same value of 5.9ao,suggesting that these four particles form a triangularpyramid.

I thank M. A. Lee and R. J. Drachman for helpful con-versations.

33 BINDING ENERGY OF POSITRONIUM MOLECULES

'Y. K. Ho, Phys. Rev. A 19, 2347 (1979); J. Phys. B 16, 1530(1983}; Phys. Lett. 102A, 348 (1984); A. K Bhatia and R. J.Drachman, Phys. Rev. A 2$, 2523 (1983); A. P. Mills, Jr. , ibid.

24, 3242 (1981); S. J. %ard, J. %. Humberston, and M. R. C.McDowell, J. Phys. B 18, L525 (1985); J. Botero and C. H.Greene, Phys. Rev. A 32, 1249 (1985); Y. K. Ho, ibid. 32, 2501(1985).

~A. P. Mills, Jr., Phys. Rev. Lett. 46, 717 (1981).~A. P. Mills, Jr., Phys. Rev. Lett. 50, 671 (1983).4E. A. Hylleraas and A. Ore, Phys. Rev. 71, 493 (1947}.5A. Ore, Phys. Rev. 71, 913 (1947).t'O. Akimoto and E. Hanamura, Solid State Commun. 10, 253

(1972).

~%. F. Brinkman, T. M. Rice, and B. Bell, Phys. Rev. B 8, 1570(1973).

~M. A. Lee, P. Vashishta, and R. K. Kalia, Phys. Rev. Lett. 51,2422 (1983); M. A. Lee (private communications).

9A, P. Mills, Jr. , in Posi(ron Scattering in Gases, edited by J. %'.Humberston and M. R. C. McDowell (Plenum, New York, 1984),p. 121.

'~Y. K. Ho, Phys. Rev. A 17, 1675 (1978}."T.Kato, Commun. Pure Appl. Math. 10, 151 (1957); D. P. Chong

and D. M. Schrader, Mol. Phys. 16, 137 (1969).