Structure and resonances of the e+-He(1s2s 3Se) system

Zhenzhong Ren,1, 2 Huili Han,1, ∗ Tingyun Shi,1 and J. Mitroy3

1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,

Wuhan Institute of Physics and Mathematics,

Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China

2 Graduate School of the Chinese Academy of Sciences,

Beijing 100049, People’s Republic of China

3 Centre for Antimatter-Matter Studies and School of Engineering,

Charles Darwin University, Darwin, Northern Territory 0909, Australia

(Dated: March 3, 2012)

We have studied the geometry and resonances of the e+He(3Se) system in the

framework of hyperspherical coordinates. A model potential proposed by us is used

to describe the interaction between the out electron with the He+ ionic core. The

calculated binding energy and expectation distance of the system are in agreement

with other calculations. In addition, two resonances below the e+-He(1s3s 3Se)

threshold and one resonance below the Ps(n=2)-He+ threshold are identified.

I. INTRODUCTION

It is known by calculations that a positron (e+) cannot bind to the ground state of atomic

helium [1, 2] but can bind itself to the triplet state He(1s2s 3Se) atom into a configuration

that is stable against dissociation into Ps(e+e−)+He+ [3–6] . These calculations were mainly

aimed at investigating the stability of the system and revealing the annihilation process.

Resonances in e+-He(3Se) scattering have not been studied.

There has been increasing interest to investigate atomic resonance phenomena involving

positrons. A series of calculations for resonances in atomic systems involving positron have

been performed, for example, e+H, e+He(1Se), e+A (A=Li, Na, K) and e+He+ systems [7–

21]. Up to now, two kinds of resonances have been widely studied. One is the type formed

∗Electronic address: huilihan@wipm.ac.cn

2

by an excited positronium moving in the field of a positively charged atomic (H+, He+, Li+,

etc.) system. These resonances are due to the degeneracies of the excited Ps states that

leads to a dipole moment for the excited Ps atom [10, 20]. Another type of resonance occurs

for hydrogen where the ns, np degeneracies result in the positron inducing a dipole moment

on the atom leading to a Feshbach resonance. Feshbach resonances associated with excited

states of other atoms have also been identified [16–18]. Recently, a low energy P -wave shape

resonance in positron-Mg scattering has been predicted by calculations [22, 23]. Therefore,

the analysis of atomic resonances phenomena involving positron is very interesting.

In this paper, resonances in the e+-He(3Se) scattering continuum are studied using the

Hyperspherical methods. This approach has been applied previously to the study of both

shape and Feshbach resonances in other three-body systems [26, 27, 33].

II. THEORETICAL METHODS

The e+-He(3Se)system is treated as a three-body system consisting of the He+(1s) core,

an electron, and a positron. We use re, rp, and rep to represent the electron-core distance,

the positron-core distance, and the electron-positron distance respectively. Atomic units

are used throughout unless otherwise stated. The angle between re and rp is denoted by

θ. The hyperradius R and the hyperangle φ are defined by re = R cosφ and rp = R sinφ

respectively.

Then the Schrodinger equation for zero total angular momentum states can be written

with the rescaled wave function Ψ(R, φ, θ) = ψ(R, φ, θ)R3

2 sinφ cosφ(

−1

2

∂

∂RR2 ∂

∂R+Had(R, φ, θ)

)

ψ(R, φ, θ) = R2Eψ(R, φ, θ), (1)

where Had is the adiabatic Hamiltonian

Had(R, φ, θ) =Λ2 − 1

4

2+R2V (R, φ, θ) (2)

with

Λ2 = −∂2

∂φ2−

1

sin2 φ cos2 φ sin θ

∂

∂θ

(

sin θ∂

∂θ

)

. (3)

The adiabatic potentials Uν(R)R2 and channel functions Φν(R, φ, θ) are defined as the solu-

tions of the adiabatic eigenvalue equation:

Had(R, φ, θ)Φµ(R, φ, θ) = Uµ(R)Φµ(R, φ, θ) , (4)

3

with boundary conditions

Φµ(R, 0, θ) = Φµ(R,π

2, θ) = 0 . (5)

The channel functions can be expanded in terms of B-splines of order k as follows:

Φµ(R, φ, θ) =

Nφ∑

i

Nθ∑

j

ci,jBi,k(φ)Bj,k(θ). (6)

We choose k − 1 knots to be located at φ = 0 to satisfy the boundary conditions. We also

choose k − 1 knots at φ = π2. The solution ψ(R, φ, θ) of equation (1) is expanded in terms

of point-wise discrete variable representation (DVR) basis functions πj(R)

ψ(R, φ, θ) =

NDVR∑

j=1

Nc∑

µ=1

cjµπj(R)Φµ(Rj, φ, θ) , (7)

where NDVR is the number of terms in the DVR basis set and Nc is the number of coupled

channels. The πj(R) is constructed from the associated Laguerre polynomials Lαn(R/β) [28].

Since the wave function ψ(R, φ, θ) ∼ R3

2 as R → 0, we choose α = 3. The scaling factor

is β = Rmax/Rn, where Rn is the nth zero of Lαn(R) and Rmax is the size of the system.

Equation (1) can be solved using the slow variable discretization (SVD) method [29].

The V (R, φ, θ) is the interaction among the three particles with

V (R, φ, θ) = Ve(re) + Vp(rp) + Vep(re, rp) , (8)

where Ve(re), Vp(rp) and Vep(re,rp) represent the electron-core interaction, positron-core

interaction, and the electron-positron interaction respectively. The Ve(re) has the form

Ve(r) = Vdir(r) + Vexc(r) + Vpol(r) . (9)

The direct interaction,Vdir(r), between the core and the active electron is computed from

the exact He+(1s) wave function. The expression of Vdir(r)is

Vdir(r) = e−4r

(

−1

r− 2

)

−1

r. (10)

The Vpol(r) = − αd

2r4 g2(r) is the polarization potential and g2(r) = 1− exp (− r6

ρ6 ) is a cut-off

function which prevents the potential from diverging at the origin. The polarizability, αd =

0.281 25 a30 while the cut-off parameter of ρ = 2.0 a0 was set in [3, 6] and chosen so that

model potential binding energies of the triplet states of helium were in reasonable agreement

with experiment.

4

The core-exchange potential Vexc(r) is determined empirically

Vexc(r) = (ar + br2 + c)e−β1r . (11)

The parameters a, b, c and β1 are adjusted to reproduce the binding energies of the He atom.

The values chosen are a = −0.96, b = −0.0167, c = −2.044 and β1 = 2.8. The negative

values of a, b and c mean that the exchange interaction between the core and valence electron

is attractive.

The positron-core interaction has no core-exchange potential. The core-direct interaction

has the opposite sign, while the one-body polarization potential is same.

The two-body potential between the positron and valence electron is

Vep(re, rp) = −1

rep

+ Vpol12(re, rp) , (12)

where

Vpol12(r1, r2) =αdr1 · r2

r31r

32

g(r1)g(r2) , (13)

is the polarization potential for the electron-positron case. The model potentials are used to

calculate the energies of the He ground and excited states by diagonalizing the Hamiltonian

using a B-spline basis sets, as shown in Table I. Columns 3 is the results of Ryzhikh and

Mitroy [3] using the fixed-core Hartree-Fock calculations. It can be seen from the table that

the calculated energies with present model potential are in reasonable agreement with the

experimental energies.

5

TABLE I: Binding energies (in a.u.) of the He triplet states calculated using the model potential

compared with other calculations. All energies are given with respect to the He+(1s) ground state.

State This Work Ryzhikh and Mitroy [3] Experiment [30]

1s2s 3Se -0.175 228 -0.175 386 -0.175 367

1s3s 3Se -0.068 824 -0.068 718 -0.068 836

1s4s 3Se -0.036 580 -0.036 523 -0.036 662

1s2p 3Po -0.133 170 -0.132 514 -0.133 308

1s3p 3Po -0.058 082 -0.057 872 -0.058 230

1s4p 3Po -0.032 326 -0.032 235 -0.032 475

1s3d 3De -0.055 669 -0.055 640 -0.055 783

1s4d 3De -0.031 307 -0.031 291 -0.031 439

1s5d 3De -0.020 031 -0.020 022 -0.020 173

III. RESULTS

A. The e+He(3Se) bound state

The accuracy of the hyperspherical potentials is important to the final results, therefore

a convergence test for hyperspherical potentials at fixed hyperradius R = 25, R = 200 and

R = 261 a0 has been performed, as shown in Table II. From the table, it can be seen that

when the basis set size is Nθ = 35, Nφ = 86, the lowest 7 potentials are converged to at least

six significant digits with respect to further increase in the basis. Therefore, in the present

calculation, Nθ = 35, Nφ = 86 are used to compute the whole hyperspherical potentials.

The convergence of the hyperspherical potentials as a function of radius are also shown

in Table II. Most of the potentials are about 10−5 a.u. away from their asymptotic values

at R = 261 a0. The exception being the Ps(n=2) states which have values 10−4 a.u. away

from their asymptotic values. The relatively poor convergence could be the result of the

hyperspherical boundary conditions distorting the shape of any Ps(n=2) state near the

boundary. Further, the potential curve asymptotic to the Ps(2p) state will have an L = 1

centrifugal potential which could also act to slow convergence.

6

TABLE II: Convergence study of the hyperspherical adiabatic potentials Uµ(R)/R2 (in a.u.) at the

hyperradius R = 25, 200 and 261 a0 with increasing basis sizes for the e+He(3Se). The threshold

energies are from the present model potential.

(Nφ, Nθ) µ = 1 µ = 2 µ = 3 µ = 4 µ = 5 µ = 6 µ = 7

R = 25 a0

( 78, 30) -0.250 795 06 -0.177 067 91 -0.133 202 49 -0.088 685 44 -0.062 607 97 -0.050 849 03 -0.045 110 61

( 86, 30) -0.250 795 06 -0.177 067 91 -0.133 202 49 -0.088 685 44 -0.062 607 97 -0.050 849 03 -0.045 110 61

( 86, 35) -0.250 795 09 -0.177 067 91 -0.133 202 49 -0.088 685 44 -0.062 607 97 -0.050 849 03 -0.045 110 61

(102, 50) -0.250 795 09 -0.177 067 91 -0.133 202 49 -0.088 685 44 -0.062 607 97 -0.050 849 04 -0.045 110 61

R = 200 a0

( 78, 30) -0.250 0089 -0.175 2486 -0.133 1609 -0.068 8754 -0.062 8121 -0.062 2096 -0.058 1152

( 86, 30) -0.250 0090 -0.175 2494 -0.133 1610 -0.068 8755 -0.062 8121 -0.062 2096 -0.058 1152

( 86, 35) -0.250 0093 -0.175 2494 -0.133 1610 -0.068 8755 -0.062 8123 -0.062 2098 -0.058 1152

(102, 50) -0.250 0094 -0.175 2495 -0.133 1610 -0.068 8756 -0.062 8123 -0.062 2098 -0.058 1152

R = 261 a0

( 78, 30) -0.250 0046 -0.175 2336 -0.133 1645 -0.068 8521 -0.062 6825 -0.062 3291 -0.058 0992

( 86, 30) -0.250 0047 -0.175 2381 -0.133 1646 -0.068 8532 -0.062 6825 -0.062 3291 -0.058 0993

( 86, 35) -0.250 0052 -0.175 2381 -0.133 1646 -0.068 8532 -0.062 6828 -0.062 3293 -0.058 0993

(102, 50) -0.250 0054 -0.175 2404 -0.133 1647 -0.068 8538 -0.062 6828 -0.062 3293 -0.058 0993

Threshold -0.250 0000 -0.175 2279 -0.133 1704 -0.068 8243 -0.062 5000 -0.062 5000 -0.058 0824

Asymptotic State Ps(n=1) He(1s2s) He(1s2p) He(1s3s) Ps(n=2) Ps(n=2) He(1s3p)

The structure of the e+He(3Se) has been identified as a Ps atom weakly bound to a

He+(1s) ion, therefore, the box size is an important parameter. Table III tests the con-

vergence of the ground state energy and expectation values with respect to the size of box

and basis set size. When the box size is 250 a0 and the basis set size is NDV R = 70, Nc =

30, the ground state energy is converged to 10−5 and expectation values have at least three

significant digits.

Computed energy and expectation values for the e+He(3Se) system obtained in the present

work and comparison with other results are shown in Table IV. There are two fully ab-initio

calculations of this system. Both are variational calculations using explicitly correlated

Gaussian (ECG) basis sets, One calculation was performed using the stochastic variational

method (SVM) [4, 35]. The other calculation, is for all practical purposes an SVM calcu-

lation, but is identified in the table in the row labeled as Frolov [5]. The fixed-core SVM

method (FCSVM) with the He+(1s) core fixed used a model potential very similar in design

to that used in the present calculation. The present ground state energy is only 2 × 10−4

a.u. away from the FCSVM energy [6]. Other expectation values agree to 1-3% with the

7

FCSVM [6] values. Note that it is not possible to directly compare 〈re〉 calculated with the

model potential to that obtained used the exact Hamiltonian.

TABLE III: Behavior of the e+He(3Se) energy and expectation values (in a.u.) with increasing box

size and basis set size. The basis set size for the Rmax convergence check is NDV R = 70, Nc = 10.

The radius for the basis set convergence test is Rmax = 250 a0.

Rmax Energy 〈re〉 〈rp〉 〈rep〉 〈r2e〉 〈r2

p〉 〈r2ep〉

150 -0.250 600 15.289 15.733 3.1497 347.46 355.02 13.532

200 -0.250 600 15.308 15.752 3.1496 349.36 356.93 13.531

250 -0.250 600 15.310 15.753 3.1493 349.47 357.04 13.531

(NDV R, Nc) Energy 〈re〉 〈rp〉 〈rep〉 〈r2e〉 〈r2

p〉 〈r2ep〉

(60, 5) -0.250 585 15.471 15.916 3.1421 357.64 365.34 13.453

(70, 5) -0.250 585 15.471 15.916 3.1421 357.64 365.34 13.453

(70, 10) -0.250 600 15.310 15.753 3.1493 349.47 357.04 13.531

(70, 20) -0.250 604 15.270 15.714 3.1504 347.43 355.01 13.540

(70, 30) -0.250 605 15.261 15.705 3.1506 346.97 354.55 13.542

TABLE IV: Computed energy and expectation values (in a.u.) for the e+He(3Se) system obtained

in the present work and comparison with other results.

Energy 〈re〉 〈rp〉 〈rep〉 〈r2e〉 〈r2

p〉 〈r2ep〉

Present -0.250 605 15.261 15.705 3.1506 346.97 354.55 13.542

FCSVM [6] -0.250 5863 15.4524 15.8902 3.148 72 356.266 363.781 13.5235

SVM [6] -2.250 595 08 15.805 61 359.518

Frolov [5] -2.250 593 72 15.749 59 353.868

B. Stabilization calculations

The stabilization method is an easy method to apply to determination of resonance

parameters. It has been used previously to extract resonance parameters in the e+Li and

e+Na systems [17, 18]. We first repeatedly diagonalizes the Hamiltonian in a hyperspherical

8

box of size Rmax to obtain the stabilization diagrams of the system with basis set size NDV R

= 70, Nc = 15, and then calculate the density of resonance states ρ(E) for two energy levels

at the avoided crossing with the help of the following formula [36]:

ρ(E) =1

∆R

∑

n

∣

∣

∣

∣

En(Ri+1) − En(Ri−1)

Ri+1 −Ri−1

∣

∣

∣

∣

−1

En(Ri)=Ei

, (14)

where the index i indicates the ith value of R and n indicates the nth energy level. Resonance

parameters can be obtained by fitting the ρ(E) to the following Lorentzian form that involves

the resonance energy Er and the width Γ:

ρ(E) = y0 +A

π

Γ/2

(E − Er)2 + (Γ/2)2, (15)

where y0 is the baseline offset, A is the total area under the curve from the base line, Er is

the center of the peak, and Γ is the full width of the peak of the curve at half height.

Figure 1 show stabilization diagram of e+He(3Se) system. From these figures, it appears

that there are stabilization lines near −0.079 a.u. and −0.068 a.u..

230 235 240 245 250 255 260-0.082

-0.080

-0.078

-0.076

-0.074

-0.072

-0.070

-0.068

E (a

.u.)

Rmax (a0)

FIG. 1: Stabilization diagram for the S-wave e+-He(3Se) system for the energy region between

−0.082 and −0.068 a.u.

9

230 235 240 245 250 255 260-0.0650

-0.0645

-0.0640

-0.0635

-0.0630

-0.0625

-0.0620

E

(a.u

.)

Rmax (a0)

FIG. 2: Stabilization diagram for the S-wave e+-He(3Se) system near −0.0688 a.u.

10

230 235 240 245 250 255 260-0.0700

-0.0696

-0.0692

-0.0688

-0.0684

-0.0680

E

(a.u

.)

Rmax (a0)

FIG. 3: Stabilization diagram for the S-wave e+-He(3Se) system near −0.0625 a.u.

Before interpreting states as resonances it is worth taking the impact of the finite size cav-

ity into account. Any particle in a finite range cavity has a zero-point energy. A reasonable

estimate can be made from the known hyperspherical radius. Scattering states consisting

of a positron and a He target state will have a delocalized positron and a tightly bound

electron, e.g. the valence electron radius for the He(1s3s 3Se) state is about 〈re〉 ≈ 6 a0.

The hyperspherical box radius can be regarded as roughly equivalent to the maximum radius

for the positron. Setting sin(k0rp) = 0 at rp ≈ 261 a0, gives k0 ≈ π/261 = 0.012 04 a−10 , and

E0 = 0.012 042/2 = 7.2 × 10−5 a.u.

Figure 4, 5 and 6 show the densities of the features near −0.079, −0.068 and −0.062 8 a.u.

respectively. The solid squares are the present calculated values, and the solid line is the

fitted Lorentzian to the corresponding density. From the fitting of densities, one resonance

state with an energy of Er = −0.079 22 a.u. and a width of Γ = 0.000 23 a.u. is identified.

The state densities used in the fit were taken from stabilization curves in the range from

11

230 to 261 a0

More uncertainty is associated with the possible resonance of near −0.068 8 a.u. The

density of states analysis in Fig. 5 gives resonance parameters of Er = −0.068 88 and Γ =

0.000 03 a.u. The energy difference of the resonance from the He(1s3s) energy threshold at

Rmax = 261 a0 is only 2 × 10−5 a.u. This is less than the zero point energy of the positron

inside the hyperspherical surface. The width of the resonance, 0.000 03 a.u. also exceeds the

energy difference from the He(1s3s) threshold.

There is a relatively flat stabilization like curve in figure 2 just above −0.068 8 a.u. Fitting

a Lorentzian to the first feature above −0.068 8 gave Er = −0.068 67 a.u. which is above

the e+He(1s3s) threshold. This feature arises from the first continuum state associated with

the e+He(1s3s) threshold and is not a resonance. The small change in the zero-point energy

when Rmax changes from 230 to 261 a0 leads to the relatively flat curve in Fig. 2.

-0.0800 -0.0796 -0.0792 -0.0788 -0.0784

2.20x104

4.40x104

6.60x104

Den

sity

of s

tate

E (a.u.)

FIG. 4: The density of resonance states fit to the Lorentzian form in e+-He(3Se). The solid

squares are the calculated values and the solid line is the fit function. The resonance parameters

are determined to be Er = −0.079 22 a.u. and Γ = 0.000 23 a.u.

12

-0.06900 -0.06894 -0.06888 -0.06882 -0.068760.0

3.0x105

6.0x105

D

ensi

ty o

f sta

te

E (a.u.)

FIG. 5: The density of resonance states fit to the Lorentzian form in e+-He(3Se). The solid

squares are the calculated values and the solid line is the fit function. The resonance parameters

are determined to be Er = −0.068 88 a.u. and Γ = 0.000 03 a.u.

13

-0.0630 -0.0629 -0.0628 -0.0627 -0.0626 -0.0625

1x105

2x105

3x105

D

ensi

ty o

f sta

te

E (a.u.)

FIG. 6: The density of resonance states fit to the Lorentzian form in e+-He(3Se). The solid

squares are the calculated values and the solid line is the fit function. The resonance parameters

are determined to be Er = −0.062 79 a.u. and Γ = 0.000 08 a.u.

There is another structure just below the Ps(n=2) thresholds at −0.062 5 a.u. The

stabilization diagram for energies near the Ps(n=2) threshold is shown in figure 3. There

are indications of a resonance near −0.062 5 a.u. but the evidence is not conclusive. The

density of states shown in figure 6 is also not conclusive. The identification of this resonance

is impeded by the finite size of the hyperspherical cavity. The finite size splits the degeneracy

of the Ps(2s) and Ps(2p) states (refer to Table II) which are crucial to the formation of these

resonances. The density of states and the Lorentzian fit is shown in figure 6. The density

of states had some irregularities. The resonance energy is −0.062 79 a.u.

14

TABLE V: Resonance energies (Er) and widths (Γ) (in a.u.) as calculated with different methods.

Resonance EAA UAA Stabilization calculation

Er Er Er Γ

1 −0.080 74 −0.078 92 −0.079 22 0.000 23

2 −0.069 84 −0.068 79 −0.068 88 0.000 03

3 −0.062 83 −0.062 69 −0.062 79 0.000 08

C. The adiabatic hyperspherical calculations

The hyperspherical Schrodinger equation in the adiabatic approximation is[

−1

2

d2

dR2+Wµµ(R) +

Uµ(R)

R2− E

]

Fµ(R) = 0 , (16)

where

Wµµ(R) = −1

2

⟨

Φµ(R, θ, φ)

∣

∣

∣

∣

d2

dR2

∣

∣

∣

∣

Φµ(R, θ, φ)

⟩

(17)

is the diagonal coupling term. The ground state energy obtained by solving Eq. (16) is an

upper bound (UAA) to the true ground state energy and the energy is a lower bound (EAA)

without Wµµ(R) [34]. The properties of bound states or scattering states for each channel

µ are related directly to the shape of the potential Vµ(R) = Wµµ(R) + Uµ(R)/R2 [33]. The

Feshbach resonances can be identified with the hyperspherical adiabatic potential curves,

Vµ(R), having an attractive well. A shape resonance can be produced if the potential

possesses a repulsive barrier at large distances, R, outside of an attractive well at small R

[33]. The 1Po shape resonance above the H(n=2) threshold in H− [27] and a shape resonance

in Ps− [34] are both caused by potentials having this form.

Figure 7 shows a number of the lowest hyperspherical potential curves for the e+He(3Se)

system. The potential curve associated with the He(1s3s 3Se) threshold has a potential well

which could support the resonances at −0.079 22 and −0.068 88 a.u. Eq.( 16) is solved using

the e+-He(1s3s 3Se) curve as potential and the calculated EAA and UAA energies are shown

in Table V. From the table we can see that the two resonances are indeed related with the

potential curve approaching to the He(1s3s 3Se) threshold.

Figure 7 also shows that the potential curve from the e+-He(1s3s 3Se) and the Ps(n=2)-

He+ threshold interacts with each other strongly with an avoided crossing at R ≈ 45 a0. The

15

strong interaction between the two channels leads to a barrier in the potential curve from the

He(1s3s 3Se) threshold when including the Wµµ(R) term. The potential curve associated

with the Ps(n=2)-He+ threshold is expected to support many bound states due to the

degenerate Ps(n=2) states. However, the spectral behavior of the Ps(n=2)-He+ channel is

modified greatly by the the pronounced avoided crossing. We only find one stabilization line

near −0.062 8 a.u. The calculated EAA and UAA energies of the last resonance are also

shown in table V.

FIG. 7: The hyperspherical potential curves Uµ(R)/R2 (solid lines) for the e+-He(3Se) system are

shown as functions of the hyperspherical radius (the horizontal axis is a logarithmic scale). The

positions of resonance and bound states are labelled by dotted lines in the potentials. Number 0

indicates the bound state and 1, 2, 3 indicate the possible resonances. The inset is the magnification

of a part of the curve including the diagonal coupling term. In the inset, the dashed line represents

Vµ(R) = Wµµ(R) + Uµ(R)/R2, the solid horizontal lines indicate the He(1s3s 3Se) and Ps(n = 2 )

thresholds.

16

D. Perturbing the interaction

In order to elucidate more information about the structure of the resonances, the effec-

tive Hamiltonian is perturbed and all calculations are repeated. The attractive exchange

interaction (Eq.(11)) is multiplied by a factor of two and this results in a larger energy sepa-

ration between the e+-He(1s3s 3Se) and Ps(n=2)-He+(1s) thresholds. When this is done the

energy of the He(1s3s 3Se) state shifts to −0.073 70 a.u. and the He(1s3p 3Po) state shifts

to −0.060 89 a.u.

The lowest energy resonance becomes −0.081 79 a.u. The next lowest resonance state

associated with the e+-He(1s3s 3Se) that could conceivably identified as a resonance shifts

to −0.073 62 a.u., i.e. above the threshold energy.

Strengthening the attractive interaction between the electron and the core results in the

Ps(n= 2) resonance position becoming more deeply bound and the stabilization plots are

more clearly those of a resonance. There is a resonance at −0.063 92 a.u. The stronger

attractive interaction between core and the electron makes the resonance formation less

dependent on achieving exact degeneracy between the Ps(2s) and Ps(2p) levels.

IV. CONCLUSION

Low energy resonances of the e+He(3Se) system have been investigated in a three-body

model using the hyperspherical coordinate method. The model potential describing the

interaction between the valence electron and positron and the He+ ionic core is validated

by a calculation of the properties of the e+He(3Se) ground state. One resonance below the

He(1s3s) has been clearly identified. While the stabilization structure at −0.068 88 a.u.

has a Lorentzian state density it is so close to the e+He(3Se) threshold that its unequivocal

identification as a resonance is not possible. The splitting of the Ps(2s) and Ps(2p) thresholds

caused by the imposition of finite dimension hyperspherical boundary conditions disrupts

the formation of the expected Ps(n=2)-He+ resonances. Nevertheless, there does appear to

be a resonance just below the Ps(n=2)-He+(1s) threshold at −0.062 8 a.u.

17

V. ACKNOWLEDGMENTS

This work was supported by the National Basic Research Program of China under Grant

No. 2012CB821305 and the NSFC under Grant No. 11004225. This work was partially

supported under the Australian Research Council’s Discovery Program (# 0665020).

18

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