Few-Body Syst (2009) 46: 173–181DOI 10.1007/s00601-009-0055-2

Sabyasachi Kar · Y. K. Ho

Doubly Excited 1,3 Pe Resonance Statesof the Positronium Negative Ion with Coulomband Screened Coulomb Potentials

Received: 17 April 2009 / Accepted: 27 April 2009 / Published online: 13 May 2009© Springer-Verlag 2009

Abstract We have investigated the doubly excited 1,3 Pe resonance states of positronium negative ion withCoulomb and screened Coulomb potentials using highly accurate correlated exponential wavefunctions. ForCoulomb interaction, the stabilization and the complex-rotation methods are employed to extract resonanceparameters (resonance positions and widths). We have obtained two 1 Pe resonances and three 3 Pe resonancesbelow the n = 3 Ps threshold. In addition to Feshbach resonances lying below n = 3 Ps threshold, we havecalculated one 3 Pe shape resonances lying above the Ps (n = 2) threshold. For screened Coulomb (Yukawa)interaction, we employ the stabilization method to extract resonance parameters as functions screening param-eter. The resonance energies and widths for 1,3 Pe resonance states of Ps− below the n = 3 Ps threshold fordifferent screening parameters ranging from infinity (Coulomb case) to small values are reported, along withthe Ps(3S) and Ps(3P) threshold energies. The screened Coulomb results for the 1,3 Pe resonance states arereported for the first time in the literature.

1 Introduction

There is upsurge of interest to study various properties of the positronium negative ion (Ps−) after the observa-tion of such spices and the subsequent annihilation rate in the laboratory [1,2]. The Ps− is one of the simplestthree lepton systems interacting through Coulomb forces. Recent experimental advances ([3–5], referencestherein) in positron trapping and accumulation technology have opened up access to study several propertiesof such three-body system. A great number of theoretical studies on Ps− have been performed in last fewdecades. In this work, our interest is to study P-wave even parity states of Ps− interacting with Coulomband screened Coulomb potentials. Several studies have been performed on the resonances in e−-Ps scatteringusing the theoretical methods such as complex-coordinate rotation method [6,7], the Kohn-variational method[8], adiabatic treatment in the hyper spherical coordinates [9,10], adiabatic molecular approximation [11], thehyperspherical close coupling method [12], and the stabilization method [13–15]. In contrast with H−, there isno 3 Pe bound states in Ps− below the n = 2 threshold. Absence of such a bound states in Ps− was establishedby Mills [16] as well as by Bhatia and Drachman [17,18] using a variational method and by Botero [19,20]with the use of adiabatic hyperspherical potential curves. The existence of shape resonance above the Ps

S. Kar (B)Center for Theoretical Atomic and Molecular Physics, Harbin Institute of Technology,The Academy of Fundamental and Interdisciplinary Sciences,150080 Harbin, People’s Republic of ChinaE-mail: skar@hit.edu.cn

Y. K. HoInstitute of Atomic and Molecular Sciences, Academia Sinica,PO Box 23-166, 106 Taipei, Taiwan, People’s Republic of ChinaE-mail: ykho@pub.iams.sinica.edu.tw

174 S. Kar, Y. K. Ho

(n = 2) threshold has been indicated by Botero and latter such shape resonance has been determined using thecomplex-rotation method [21]. The 3 Pe [21] and 1,3Po [22] resonance states up to the Ps (n = 6) thresholdhave been calculated in the framework of complex-rotation method using Hylleraas-type basis functions. Su-permultiplet structures of the doubly excited Ps− have been reported in the literature [23]. Recently, resonancesof Ps− in two and three dimensions have also been reported [24]. For the recent advances in the theoreticalstudies on the resonances in Ps−, readers are referred to recent reviews [25–27]. From the experimental side,there have been recent interests in the investigations of few-body systems involving positron and positronium[28,29]. Also, the positronium negative ion may play a role in the observation on 0.511 MeV gamma-ray aroundthe center of our galaxy, the Milky Way [30]. With the improved experimental technique, it is important toinvestigate 1,3 Pe states in Ps− using different correlated wavefunctions in the framework of both the powerfulvariational techniques: the stabilization method [31–33] and the complex coordinate rotation method ([34],references therein).

In the present work, we have investigated the doubly excited 1,3 Pe resonance states for one of the simplestthree-body systems that consist of three leptons, the Ps−ion, using accurate correlated exponential basis func-tions. The stabilization method and complex-coordinate rotation method are employed to extract resonanceparameters. We have calculated the 3 Pe shape resonances lying above the n = 2 Ps threshold which is com-parable to the available results. We have obtained two 1 Pe resonances and three 3 Pe resonances below then = 3 Ps threshold. The convergence of the calculations has been examined with the increasing number ofterms in the basis functions. Beside such investigations with Coulomb interaction, we have also investigatedthe 1,3 Pe resonance states of Ps− with screened Coulomb (Yukawa) potentials in the framework of the stabil-ization method. The importance of the screened Coulomb potentials has been highlighted in the earlier works([13–15,35–37], references therein). The atomic unit (a.u.) has been used throughout the work.

2 Calculations

The non-relativistic Hamiltonian describing the positronium negative ion in the screening environment char-acterized by the parameter µ is given by

H = T + V = −1

2∇2

1 − 1

2∇2

2 − 1

2∇2

3 −[

exp(−µr31)

r31+ exp(−µr32)

r32

]+ exp(−µr12)

r12, (1)

where 1 and 2 denote two electrons and 3 denotes positron and ri j is their relative distance. When the posi-tronium negative ion is placed in vacuum, we have µ = 0. For the 1,3 Pstates of the Ps− having a parity of(−1)ε, we employ highly accurate correlated wave functions [15,38,39]

� = (1 + Spn P̂12)

N∑i=1

L∑l1=ε

Ai (−1)εY l1,l2L M (r31, r32) exp[(−αi r31 − βi r32 − γi r12)ω], (2)

with

Yl1,l2L M (r31, r32) = rl1

31rl232

∑m1,m2

C L Ml1m1,l2m2

Yl1m1(r̂31)Yl1m2(r̂32), (3)

where the functions Yl1,l2L M (r1, r2) are the bipolar harmonics or Schwartz harmonics, r̂ j = ri/r j ( j = 1, 2),

Yli mi (r̂ j ) denotes the usual spherical harmonics, C L Ml1m1,l2m2

are the Clebsch-Gordon coefficients, αi , βi , γi

are the non-linear variation parameters, Ai (i = 1, . . . ., N ) are the linear expansion coefficients, L = 1 forP-states, ε = l1 + l2 = 2, Spn = 1 indicates singlet states and Spn = −1 assigns triplet states, N is numberof basis terms, and the operator P̂12 is the permutation of the two identical particles 1 and 2. The non-linearvariational parameters αi , βi and γi are chosen from a quasi-random process [13–15,35–39]. In the presentwork, the best choice of non-linear parameters, is given by the matrix notation

{αi , βi , γi

} ={ ⟨⟨

k√

2/2⟩⟩

A,⟨⟨

k√

3/2⟩⟩

B,⟨⟨

k√

5/2⟩⟩

C}

, (4)

where k = i(i + 1), A = 0.84, B = 0.34, C = 0.1, the symbol 〈〈. . .〉〉 denotes the fractional part of a realnumber. The exponential wave functions with exponents generated by the quasi-random process are widelyused in several other works ([13–15,35–39], references therein).

Doubly Excited 1,3 Pe Resonance States of the Positronium Negative Ion 175

3 Results and Discussions

As mentioned above we have used stabilization method and complex-coordinate rotation method to extractresonance parameters. First, we discuss about the investigation using stabilization method. In the stabilizationmethod [31–33], a resonance position can easily be identified from the flat plateau of the stabilization diagram,obtained by plotting the energy level E (ω) for varying ω in the mesh size 0.001. Figs. 1a and 2a shows thestabilization characters near the energy value −0.0316, −0.0281 and −0.02781 a.u. respectively for 3 Pe reso-nance states. Figure 3a shows the stabilization character near the energy value −0.0282 and −0.02782 a.u. for1 Pe resonance states. To extract resonance parameters (Er , �) for a particular resonance state, we calculatethe density of the resonance states in the stabilized plateau for each single energy level in the stabilizationdiagram using the formula

ρn(E) =∣∣∣∣ En(ωi+1) − En(ωi−1)

ωi+1 − ωi−1

∣∣∣∣−1

E=En(ωi )

, (5)

where the index i is the i th varied ω value, i.e, ωi and the index n is for the nth resonance. Also in Eq. (5),ωi−1 and ωi+1 are respectively the (i − 1)th and (i + 1)th varied ω values next to ωi . After calculating thedensity of resonance states ρn(E) using formula (5) for all the energy level of the stabilization diagram, we fitthese ρn(E), one set at a time, to the following Lorentzian form that yields resonance energy Er and a totalwidth �, with

ρn(E) = y0 + A

π

(�/2)

(E − Er )2 + (�/2)2 , (6)

where y0 is the baseline offset, A is the total area under the curve, Er is the center of the peak, and � denotesthe full width of the peak of the curve at half height. The fit that gives the best fit (with the least chi-square, χ2

and with the best value of the square of the correlation coefficient, r2) to the Lorentzian form is consideredas the desired results for that particular resonance. Details of the stabilization method can be found from theearlier works ([13–15,31–33,36,37], references therein). Figures 1b,c, 2b, 3b shows the best fittings of thedensity of resonance states for 1,3 Pe resonance states in Ps− indicated inside the figures. The stabilizationdiagrams and the fittings presented here are calculated using 700-term basis functions for 3 Pe and 600-termbasis functions for 1 Pe states. The resonance parameters (Er , �) obtained from the Lorentzian fittings arepresented in Table 1. Table 1 also shows the convergence with the increasing number of terms in the wavefunctions 2. The resonance width for the 1 Pe(2) resonance states are narrow and it appears from the fittingsthat the resonance width would be of order than 10−6. We prefer not to present the resonance widths for such1 Pe (2) states.

Next we discuss the present study using complex rotation method. In the complex-coordinate rotationmethod [34], the radial coordinate ri j are rotated through an angle θ by the transformation ri j → ri j exp(iθ),where θ is real and positive, and the Hamiltonian H takes the form

H = T exp(−2iθ) + V exp(−iθ). (7)

Resonance poles can be identified by observing the complex energy levels E (θ, ω). A complex resonanceeigenvalue W = Er − i�/2, where Er and � denote the resonance energy and width respectively, that exhibitsstationary behavior for small change of θ, and the minimum of ∂W/∂θ ≈ minimum will yield the resonanceposition and width (see Table 2) [15,39]. For the details of the complex rotation method, readers are referred tothe review articles [34]. Once the resonance poles are identified by observing the energies in the complex energyplane, then near a particular resonance pole, the plots of E(ϑ, ω) show that the rotational paths (see Figs. 4, 5and 6) are slowed down and all the energies for a range of ω values in the complex plane will approximatelymeet at a point, and from which the resonance parameters can be determined. Figure 4 shows the rotationalpath of the 3 Pe shape resonance above the Ps (n = 2) threshold. The rotational paths of the 1 Pe(1) and 3 Pe(3)Feshbach resonances below the Ps (n = 3) threshold are presented in Figs. 5 and 6 respectively. We nextcompute the minimum of changes in complex energy with respect to the change of ϑ, |W (ϑ + �ϑ) − W (ϑ)|,around the resonance pole for different ω values. Among all the minima for different ω values, the minimumvalue of the relative minima (∂W/∂ϑ ≈min.) will yield the resonance position and width for that particularresonance. Table 2 shows such behavior for the lowest 3 Pe resonance states.

The final results for the 3 Pe resonance parameters obtained from the present work using complex rotationmethod are reported in Table 3. In Table 3, we compare the results using complex rotation method with those

176 S. Kar, Y. K. Ho

(a)

(b) (c)

Fig. 1 (Color online) a Stabilization diagram for the 3 Pe(1) states of screened Ps. b The best fittings (solid line) of the calculateddensity of states (circles) corresponding to 23th energy level in the stabilization diagram (a) for the lowest 3 Pe states. c The bestfittings (solid line) of the calculated density of states (circles) corresponding to 23th energy level in the stabilization diagram forthe 3Pe(2)states, respectively

(a) (b)

Fig. 2 (Color online) a Stabilization diagram for the 3 Pe(3) resonance state of screened Ps−. b The best fittings (solid line) ofthe calculated density of states (circles) corresponding to 26th energy level in the stabilization diagram (a) for the 3 Pe(3) states

obtained using stabilization method. The comparison is also made with the reported 3 Pe resonance statesbelow n = 3 threshold [21]. We have found the third 3 Pe resonance below the n = 3 threshold using expo-nential basis functions and this resonance state has not been reported before. The 1,3 Pe resonance energiesand widths obtained using stabilization method are comparable with those obtained using complex-rotationmethod. As for the width of the second 1 Pe resonance, we conclude that such a resonance has a very narrowwidth, and we can only estimate its value as 2 × 10−7a.u.< �< 4.0 × 10−5 a.u. The present results in Table 3using complex-rotation method are calculated using 700-term wavefunctions.

Doubly Excited 1,3 Pe Resonance States of the Positronium Negative Ion 177

(a) (b)

Fig. 3 (Color online) a Stabilization diagram for the 1 Pe resonance states of screened Ps− using 600-term wave functions. b Thebest fittings (solid line) of the calculated density of states (circles) corresponding to 20th energy level in the stabilization diagram(a) for the lowest 1 Pe states

Table 1 Convergence of the 1,3 Pe states of Ps− using the stabilization method with the increasing number of term (N ) in thewavefunctions

1 Pe 3 Pe

N = 500 N = 600 N = 600 N = 700

Er −0.0282030261 −0.0282030259 −0.0316287483 −0.031630266� 0.750 [−8] 0.736 [−8] 0.0001795 0.0001793Er −0.027821 −0.027821 −0.028104421 −0.028104442� 0.00002754 0.00002786Er −0.02781151 −0.02781144� 0.0000034 0.0000033

Table 2 Stabilized behavior for the lowest 3 Pe resonance of Ps− below the n = 3 Ps threshold

θ Re[E] (10−2) Im[E] (10−9) |W (θ + �θ) − W (θ)| Re[E] (10−2) Im[E] (10−9) |W (θ + �θ) − W (θ)|ω = 0.55, N = 600 ω = 0.60, N = 600

0.25 −3.163073266 −8.9530 −3.163067675 −8.95630.30 −3.163067504 −8.9546 5.8 [−8] −3.163067429 −8.9551 2.5 [−9]0.35 −3.163067492 −8.9551 1.2 [−10]* −3.163067337 −8.9547 9.2 [−10]*0.40 −3.163067191 −8.9560 3.0 [−9] −3.163067541 −8.9544 2.0 [−9]0.45 −3.163065068 −8.9597 2.1 [−8] −3.163067954 −8.9504 4.1 [−9]

ω = 0.55, N = 700 ω = 0.60, N = 7000.25 −3.163067564 −8.9530 −3.163067765 −8.9553

0.30 −3.163067569 −8.9550 5.0 [−11] ∗⊥ −3.163067574 −8.9550 1.9 [−9]0.35 −3.163067597 −8.9551 2.8 [−10] −3.163067528 −8.9550 4.6 [−10]*0.40 −3.163067766 −8.9554 1.7 [−9] −3.163067366 −8.9547 1.6 [−9]0.45 −3.163068676 −8.9571 9.1 [−9] −3.163066512 −8.9530 8.5 [−8]

* indicates ∂W/∂θ=min. ⊥ Minimum value among all the minimaThe numbers in the square brackets denote the powers of ten

In the similar way stated above, we extract the resonance parameters as function of the screening parametersin the framework of the stabilization method using 600-term wave functions. Figure 7 shows the stabilizationdiagram and the best fitting of the density of states for the lowest 3 Pe states of screened Ps− for the screenedparameters 1/µ = 50. The resonance energy and width for the 1 Pe(1),3 Pe(1) and 3 Pe(2) states of Ps− asfunctions of screening parameter µ are presented in Table 4 and Fig. 8. We have estimated the Ps(3S) andPs(3P) threshold energies by diagonalizing atomic Hamiltonian with standard Slater-type orbitals and ourcalculated energies are presented in Table 4. For the Ps atom, 3S-3P degeneracy is destroyed with the 3S statelying lower than 3P states. The resonance energies for each resonance states decrease and ultimately close ton = 3 Ps threshold with increasing screening parameter µ and the corresponding resonance widths decreasewith increasing screening parameters. Underlying reason behind the decreasing resonance widths have beenexplained in our previous works [13,14,37].

178 S. Kar, Y. K. Ho

-0.062205 -0.062202 -0.062199 -0.062196-1.395

-1.390

-1.385

-1.380

-1.375

-1.370

-1.365

-1.360

ω=0.50 ω=0.55

ω=0.35 ω=0.40 ω=0.45

Im (

10−4

a.u

.)

Re[E] (a.u.)

θ=0.35 (0.05) 0.65

Fig. 4 (Color online) Rotational path of the 3 Pe shape resonance of Ps− lying above the Ps(n = 2) threshold, in the complexplane for five different values of the scaling factor, ω using 700 basis functions

-282.0302604 -282.0302597 -282.0302590-3.74

-3.72

-3.70

-3.68

-3.66

-3.64

-3.62

-3.60

-3.58

ω=0.30 ω=0.35 ω=0.40 ω=0.45 ω=0.50

Im[E

] (10

−9 a

.u.)

Re[E] (10−4 a.u.)

θ=0.10(0.05)0.35

Fig. 5 (Color online) Rotational path of the 1 Pe (1) resonance state of Ps− in the complex plane for five different values of thescaling factor, ω using 700 basis functions

-0.0278122 -0.0278120 -0.0278118 -0.0278116-0.24

-0.22

-0.20

-0.18

-0.16

-0.14

-0.12

ω=0.325 ω=0.350

ω=0.250 ω=0.275 ω=0.300

Im[E

] (10

−5 a

.u.)

Re[E] (a.u.)

θ=0.10 (0.05) 0.30

Fig. 6 (Color online) Rotational path of the 3 Pe (3) resonance state of Ps− in the complex plane for five different values of thescaling factor, ω using 700 basis functions

Doubly Excited 1,3 Pe Resonance States of the Positronium Negative Ion 179

Table 3 Comparison of the 1,3 Pe resonance parameters (in a.u.) of Ps− obtained using stabilization method with those obtainedusing complex-rotation method (CRM), along with the available results of the 3 Pe states using CRM [21]

1 Pe 3 Pe

Present work Present work Ho and Bhatia [21]Stabilization CRM Stabilization CRM

Ps(n = 2) threshold : −0.0625Er −0.062201* −0.06220*� 0.000276 0.000270

Ps(n = 3) threshold : −0.0277777778Er −0.028203026 −0.028203026 −0.0316303 −0.03163068 −0.031630675� 0.7[−8] 0.73[−8] 0.0001795 0.0001791 0.0001792Er −0.027821 −0.027821 −0.0281044 −0.02810462 −0.02810474

� < 4 × 10−5 0.000028 0.00002734 0.000027(3)Er −0.027811 −0.027812� 0.000003 0.000003

* This is a shape resonance lying above the Ps(n = 2) thresholdThe numbers in the square brackets denote the powers of ten

(a)(b)

Fig. 7 (Color online) a Stabilization diagram for the 3 Pe resonance states of screened Ps− for the screening parameter, µ=0.02using 600-term wave functions. b The best fittings (solid line) of the calculated density of states (circles) corresponding to 21thenergy level in the stabilization diagram (a) for the lowest 3 Pe states

Table 4 The 1,3 Pe Feshbach resonances of the screened Ps− lying below the Ps(n = 3) thresholds, using 600-term wavefunctions,along with Ps(3S) and Ps(3P) threshold energies

1/µ 3 Pe(1) 3 Pe(2) 1 Pe(1) Ps(3S) Ps(3P)

∞ −Er 0.0316287 0.0281044 0.0282030259 0.0277777778 0.0277777778� 0.0001795 0.2754[−4] 0.7356[−8]

200 −Er 0.0267854 0.0233022 0.0233971007 0.0230994289 0.0230765524� 0.0001788 0.2584[−4] 0.7152[−8]

100 −Er 0.0222833 0.0189707 0.0190439258 0.0190100072 0.0189261946� 0.0001746 0.1745[−4] 0.5723[−8]

60 −Er 0.0168594 0.0143607953 0.0141540781� 0.0001623

50 −Er 0.0144083 0.0123461336 0.0120661768� 0.0001530

40 −Er 0.0110616 0.0096762774 0.0092788759� 0.0001339

30 −Er 0.0063956 0.0060788930 0.0054866033� 0.0000926

The numbers in the square brackets denote the powers of ten

180 S. Kar, Y. K. Ho

(a)

(b)

Fig. 8 (Color online) The lowest 3 Pe resonance energy [in (a)] and width [in (b)] of the Ps− as functions of the screeningparameters µ, along with the Ps(3S) and Ps(3P) thresholds

4 Conclusions

In the present work, we have investigated 1,3 Pe resonance states of Ps− below the n = 3 Ps threshold usingcorrelated exponential basis functions. We have employed both the stabilization and the complex-coordinaterotation method to extract resonance parameters. We have obtained two 1 Pe resonances and three 3 Pe res-onances below the n = 3 Ps threshold. The third 3 Pe Feshbach resonance has not been reported before.In addition to Feshbach resonances lying below n = 3 Ps threshold, we have also calculated a 3 Pe shaperesonance lying above the n = 2 Ps threshold. Beside such investigation for Coulomb interaction, we havemade first investigation on the 1,3 Pe resonance states of Ps− with screened Coulomb (Yukawa) potentials. The1,3 Pe resonance parameters of Ps− below the n = 3 Ps threshold are estimated as functions of the screeningparameters. The present work is also a first investigation of P-wave even parity resonance states using highlycorrelated exponential basis functions. With the improved experimental technique and with the importance ofthe screened Coulomb potentials on the different branches of physics and chemistry, we believe our findingscan serve as a benchmark reference for further studies on such three-body systems.

Acknowledgments S. Kar wishes to thank Natural Scientific Research Innovation Foundation in Harbin Institute of Technologyfor financial support (Grant No.HIT.NSRIF.2008.01). Y. K. Ho is supported by NSC of Taiwan.

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