Full correlation single-particle positron potentials for a positron and a positronium interacting with atoms
FFull correlation single-particle positron potentials for a positron and a positroniuminteracting with atoms
A. Zubiaga ∗ and F. Tuomisto Department of Applied Physics, Aalto University,P.O. Box 14100, FIN-00076 Aalto Espoo, Finland
M. J. Puska
COMP, Department of Applied Physics, Aalto University,P.O. Box 11100, FIN-00076 Aalto Espoo, Finland
In this work we define single-particle potentials for a positron and a positronium atom interactingwith light atoms (H, He, Li and Be) by inverting a single-particle Schr¨odinger equation. For thispurpose, we use accurate energies and positron densities obtained from the many-body wavefunctionof the corresponding positronic systems. The introduced potentials describe the exact correlationsfor the calculated systems including the formation of a positronium atom. We show that thescattering lengths and the low-energy s-wave phase shifts from accurate many-body calculations arewell accounted for by the introduced potential. We also calculate self-consistent two-componentdensity-functional theory positron potentials and densities for the bound positronic systems withinthe local density approximation. They are in a very good agreement with the many-body results,provided that the finite-positron-density electron-positron correlation potential is used, and they canalso describe systems comprising a positronium atom. We argue that the introduced single-particlepositron potentials defined for single molecules are transferable to the condensed phase when theinter-molecular interactions are weak. When this condition is fulfilled, the total positron potentialcan be constructed in a good approximation as the superposition of the molecular potentials.
I. INTRODUCTION
Although the chemistry of the positron in crystallinesolids and soft-condensed matter has an intrinsic interestby itself, it is mainly studied in connection of probing theelectron chemistry and the open volume of materials bypositrons. Thermalized positrons become localized insideopen volume defects such as vacancies and voids wherethe repulsion by the nucleus is minimum. When prob-ing soft matter, the positron chemistry has to be takeninto account because a positron can bind an electron andform a positronium (Ps) atom before getting trapped intoopen volume pockets [1]. The annihilation properties ofthe positron are determined by the local electronic struc-tures and the distribution of the open volume. Positronannihilation spectroscopy (PAS) exploits this property tomeasure the type and concentration of vacancies in met-als and semiconductors [2]. By measuring the lifetime ofPs, the distribution of open volume has been studied inporous SiO [3, 4], polymers [5] and biostructures [6].The interpretation of PAS experiments benefits fromthe comparison to computational predictions. However,the description of an electron-positron system embeddedin a host material requires addressing the correlations oflight particles beyond the adiabatic approximation, thequantum-mechanical delocalization and the zero-pointenergy. Regrettably, using many-body techniques fora full quantum-mechanical treatment of the problem isclearly beyond the present-day computational capacity. ∗ [email protected] Instead, in metals and semiconductors the distributionsand the annihilation properties of positrons can be cal-culated from first principles to a good accuracy withinthe two-component density functional theory (2C-DFT)and the local density approximation (LDA) for the ex-change and correlation functionals [7]. For a delocalizedpositron in a perfect lattice the scheme works particu-larly well because the electron-positron correlation en-ergy functional is known very accurately within the LDAin this limit. Moreover, the same method can be appliedalso for positrons trapped at vacancies. The calculatedpositron annihilation parameters can then be used fora quantitative analysis of the experimental results formetals and semiconductors [2]. However, the 2C-DFTscheme is considered to be unable to describe Ps andinstead semiempirical methods have been employed todescribe the matter-Ps interaction [8–10].The many-body wavefunctions of small positronic sys-tems composed by a positron interacting with a lightatom or a small molecule can be calculated to a good ac-curacy using the Quantum Monte-Carlo (QMC) [11–16]and Configuration Interaction (CI) [17] methods. In thiswork, we have obtained accurate positron energies anddensities for positronic atoms including a positron (e + H,e + He, e + Li and e + Be) and Ps (HPs and LiPs) by an ex-act diagonalization stochastic variational method (SVM)using an explicitly correlated Gaussian (ECG) functionbasis set.In e + Li an electron from Li forms a Ps atom with thepositron and becomes bound to the Li + ion. In e + Bethe polarized electron cloud binds the positron. In HPsand LiPs the unpaired atom electrons form a chemicalbond that binds the Ps strongly to the atom. On the a r X i v : . [ phy s i c s . a t m - c l u s ] M a y other hand, the positron is not bound to the atom in thee + H and e + He systems. ECG-SVM accounts for morecorrelation energy for the bound states [18–20] and theresulting binding energies are larger than for QMC andCI.On the basis of our many-body results we propose asingle-particle potential for the positron and we deriveit for all the positronic systems by inverting a single-particle Schr¨odinger equation. We check the accuracy bycomparing the ensuing scattering lengths and the s-wavephase shifts to the corresponding many-body values. Wealso compare the many-body densities and the intro-duced single-particle potentials to the corresponding re-sults of 2C-DFT within LDA for bound e + Li, e + Be, HPsand LiPs. The agreement seen predicts that 2C-DFTand LDA can be the starting point to describe positronbound states including systems in which Ps is formed.Finally, we discuss the utility of the single-particle ef-fective potentials to approach a practical and predictivedescription of positron and Ps states in condensed mat-ter.The organization of the present paper is as follows.The many-body ECG-SVM as well as the 2C-DFTschemes are shortly described in Chapter II. Chapter IIIpresents and discusses the effective single-particle po-tentials, the elastic scattering parameters and the self-consistent 2C-DFT-LDA results. Chapter IV is devotedto discuss the utility of the introduced potentials to de-scribe positron and Ps states in condensed matter andchapter V presents our conclusions.
II. COMPUTATIONAL METHODSA. ECG-SVM
ECG-SVM [21] is an all-particle quantum ab-initiomethod used to calculate the many-body wavefunctionof N particles (electrons, positrons and nuclei) interact-ing through the Coulomb interaction. The Hamiltonianof the system with the kinetic energy of the center-of-mass (CM) subtracted is H = (cid:88) i p i m i − T CM + (cid:88) i
50 a.u..
Table I. Main properties of the calculated systems. The first four columns give the name of the system, the size of the basisused, the total energy, and the mean positron-nucleus distance (cid:104) r p (cid:105) . The next three columns give the asymptotic state, thetotal energy of the corresponding atom or ion and the interaction energy.System Basis size Energy (cid:104) r p (cid:105) Asymptotic Atom/Ion E e + int /E Psint (a.u.) (a.u.) state Energy (a.u.) (a.u.)e + H 200 -0.49974 67.47 e + -0.5 (H) 0.262 × − e + He 1000 -2.90332 56.98 e + -2.9036937 (He) 0.372 × − e + Be 2000 -14.6694 10.972 e + -14.6670283 (Be) -2.33 × − -14.3246131 (Be + )e + Li 1000 -7.53226 9.928 Ps -7.2798377 (Li + ) -2.42 × − HPs 1000 -0.78919 3.662 Ps -0.25 (Ps) -39.187 × − LiPs 2000 -7.73898 6.432 Ps -7.4779733 (Li) -11.011 × − The confinement radius is chosen large enough (100 a.u.)so that the shape of the wavefunction is not affected bythe confinement potential in the interaction region of thepositron or Ps with the atom. The resulting (cid:104) r p (cid:105) is largeand the interaction energy is small.We defined the asymptotic non-interacting state of theatom-positron and atom-Ps systems when the positron isfar from the atom. For unbound systems, the asymptoticstate is the main scattering channel, i.e., the positronscatters off the neutral atom, and for bound systems it isthe main dissociation channel, i.e., the positron (e + Be)or Ps (e + Li, HPs and LiPs) detaches leaving behind anatom or an ion. e + Be splits into a neutral atom anda positron, e + Li splits into a Li + ion and a Ps atomand both HPs and LiPs dissociate into neutral atomsand Ps. We define the positron interaction energy as E e + int = E e + X − E X , i.e., the difference between the en-ergy of the interacting positronic system, E e + X , and theatom without the positron, E X . The Ps interaction en-ergy, E P sint = E e + X/XP s − E X + /X − E P s is the differencebetween the energy of the interacting system, E e + X/XP s ,and the sum of the total energies of the positive ion orthe atom, E X + /X , and Ps, E P s , after Ps has dissociated.
B. Two-component DFT
Within LDA of the 2C-DFT the total energy functionalof a positronic atom is E [ n − ( r ) , n + ( r )] = F [ n − ] + F [ n + ] + E epC [ n − , n + ] + E epcorr [ n − , n + ] , (4)where E epC [ n − , n + ] is the attractive mean-field Coulombinteraction between the electrons and the positron and E epcorr [ n − , n + ] is the electron-positron correlation energyfunctional. F [ n ] is the usual one-component densityfunctional F [ n ] = E kin [ n ] + E ext [ n ] + E H [ n ] + E xc [ n ] , (5) where E kin [ n ] is the Kohn-Sham kinetic energy and E ext [ n ], E H [ n ], and E xc [ n ] are the electron(positron)-nucleus interaction, the Hartree energy functional andthe exchange-correlation energy functional, respectively.For the last one, we have used the parametrization byPerdew and Zunger [23]. The self-interaction corrected(SIC) density functional for a single positron F [ n ] is F [ n ] = E kin [ n ] + E ext [ n ] . (6)The asymmetric treatment of the electron and positronself-interactions for positron states in solids has beenshown to give results in a quantitative agreement withexperiments [24, 25]. The resulting Kohn-Sham equa-tions for the electron φ − i and positron φ + orbitals are (cid:20) − ∇ − Zr + (cid:90) n − ( x ) − n + ( x ) | (cid:126)r − (cid:126)x | d(cid:126)x + δE xc [ n − ] δn − + δE epcorr [ n + , n − ] δn − (cid:21) φ − i = (cid:15) − i φ − i (7) (cid:20) − ∇ Zr − (cid:90) n − ( x ) | (cid:126)r − (cid:126)x | d(cid:126)x + δE epcorr [ n + , n − ] δn + (cid:21) φ + = (cid:15) + φ + , (8)where Z is the atomic number of the nucleus and (cid:15) − i and (cid:15) + are the electron and positron energy eigenval-ues, respectively. Equations 7 and 8 are solved self-consistently with a DFT code that solves the all-electronand positron radial Kohn-Sham equations [26]. Themean-field Coulomb potential plotted in figure 2 is com-posed by the second and third terms of equation 8.In our LDA energy functional we use a two-component electron-positron correlation energy func-tional E ep [ n + , n − ]. To build up this functional, thereis only data for a homogeneous electron-positron plasmain the metallic regime calculated by Lantto [27]. TheLDA parametrization of reference [25] describes correctly -4 -3 -2 -1 -4 -3 -2 -1 R ad i a l den s i t y ( n x π r a . u . - ) -4 -3 -2 -1 e + Li e + Be HPs e + He Li He H Be H Li LiPs e + H Figure 1. (Color online) Electron and positron densities of the calculated systems. The electron densities of the isolated atoms(filled blue curves), and the interacting positron-atom systems (red broken curves), as well as the positron densities (black fullcurves) are given. E ep [ n + , n − ] and V ep [ n + , n − ]= δE ep [ n + , n − ]/ δn + for theelectron densities typical in metals and semiconductors r s = (3 / /π/n ) / ∼ E ep [ n + , n − ] is not knownaccurately at medium electron and positron densities, be-yond r s > E ep [ n + , n − ] when one or both densities are small but fi-nite, i.e. 8 < r s <
20 a.u. The asymptote of E ep [ n + , n − ]is { / ( n e V BNep [ n p ]) + 1 / ( n p V BNep [ n e ]) } − where V BNep isthe parametrization given by Boronski and Nieminen [24]for the single positron or electron limit. The interpola-tion and its first functional derivatives are continuouseverywhere. Finally, we cut E ep [ n + , n − ] when both theelectron and positron densities are vanishingly small, be-yond r s >
20 a.u.
III. RESULTS
The ionization energies of H and He are 0.5 a.u. and0.90369 a.u., respectively. They are well above the bind-ing energy of Ps (0.25 a.u.), therefore the electrons, asshown in figure 1, remain tightly bound to the nucleiwithout an appreciable polarization. The positron iscompletely delocalized in these systems. The ionizationenergy, 0.34242 a.u., of the closed 2s orbital of Be is onlyslightly larger than the Ps binding energy so that theatom becomes polarized and the positron is bound bythe induced (dynamic) dipole of Be. On the other hand,the Li ionization energy of 0.198 a.u. is lower than thebinding energy of Ps and the positron forms a Ps clusterwith the Li 2s electron [30]. In HPs and LiPs, the electronin Ps forms a strong chemical bond with the unpaired selectron of the atom, keeping Ps as a distinguishable unit.The formation of a Ps cluster is manifested at r > + Li, HPs and LiPs.
A. Effective potentials
We introduce now an effective single-particle potential V eff using our many-body results and in Chapter IV wewill propose it as the starting point to describe positronand Ps states in condensed matter. We invert a single-particle Schr¨odinger equation using the positron densitiesof the interacting systems and obtain V eff ( r ) = E eff + 12 M eff ∇ (cid:112) n + ( r ) (cid:112) n + ( r ) . (9)The effective energy E eff is the interaction energy of theasymptotic state in table I. For e + Li, HPs and LiPs theeffective mass M eff is the mass of Ps (2 m e ) and for theother systems it is the mass of the positron ( m e ). V eff is a single-particle potential for the positron even in sys-tems where Ps is formed. We remark when the effectivepotential describes a system comprising a Ps atom bynaming it as the Ps V eff . The introduced potential isequivalent to the exact Kohn-Sham potential for a sin-gle positron with effective mass M eff . E eff ensures thatits asymptotic value far from the nucleus is zero also forsystems including Ps.We also define a single-particle mass-normalized Ps ef-fective potential V P seff (cid:48) = 2 E P sint + ∇ ( √ n + ) / (2 √ n + ) withthe effective mass m e . The densities obtained by solvingthe Schr¨odinger equation with the mass-normalized po-tential are the same as those of V eff and the energies aremultiplied by a factor of 2. In the present work, we usethe mass-normalized potential to compare the Ps V eff tothe corresponding positron DFT potential.According to figure 2, when r < ∼ V eff becomes attractive at larger separations, whenthe electron-positron correlation is comparable to thepositron-nucleus Coulomb repulsion. The repulsive coreof V eff range from r < ∼ + H and e + He and r < ∼ + Li and e + Be. Although the attractive V eff well is slightly deeper for e + H than for e + He, dueto the larger polarizability of H, it is very shallow forboth unbound systems. For bound e + Be the minimumof V eff is deeper, -84.58 × − a.u. at r=3.18 a.u. Theminimum value of the Ps V eff potential of e + Li is shal-lower, -24.57 × − a.u. at r=4.62 a.u., but the potentialwell extends longer distances. Finally, the attractive Ps V eff wells of the strongly bound HPs and LiPs are deep,-0.280 a.u. and -0.102 a.u., respectively.To show that V eff can predict the correct positrondensity and interaction energy, we have calculated thepositron (Ps) binding energy to Be (Li + ) by solving nu-merically the radial single-particle Schr¨odinger equation.For the ground state it reduces to the one-dimension problem − M eff d Udr + V eff U = EU, (10)where U = r Ψ and Ψ is the s-type wavefunction. Theboundary conditions for U are U ( r = 0)=0 and U ( r →∞ )=0. For e + Li, HPs and LiPs the effective potentialsare the Ps V eff potentials and M eff =2 m e . The resultingbinding energy and (cid:104) r p (cid:105) given by equation 10 are, E b = 2.414 × − a.u. and (cid:104) r p (cid:105) =10.213 a.u. for e + Li, E b = 2.33 × − a.u. and (cid:104) r p (cid:105) = 11.104 a.u. for e + Be, E b =39.210 × − a.u. and (cid:104) r p (cid:105) = 3.673 a.u. for HPs,and E b = 10.394 a.u. and (cid:104) r p (cid:105) = 6.457 a.u. for LiPs. B. Scattering lengths
In order to study the adequacy of V eff to modelpositron and Ps states, we consider a positron or Ps scat-tering off light atoms. Many-body calculations of the s-wave phase shifts ( δ ) and scattering lengths ( A ) existfor e + H, e + He and e + Be. Zhang et al. [31] used the sta-bilized ECG-SVM to calculate the positron A of H andHe. Houston et al. [32] applied Hylleraas wavefunctionsand the Kohn variational method to positrons scatteringoff H and Bromley et al. [33] studied positron scatteringoff Be using polarized orbital wavefunctions. Ps scatter-ing off Li + ion has been studied by Mitroy and Ivanov [34]using the stabilized ECG-SVM.Here we calculate δ and A using the corresponding V eff or Ps V eff and compare them to the many-bodyvalues in the literature. For a positron scattering off Li,the Ps formation channel is open at all energies [35, 36]and therefore the Ps V eff A and δ are compared to themany-body values of Ps scattering off Li + . We obtainthe s-wave scattering wavefunction for a positron withthe energy E = k / M eff by solving equation 10. Atlarge distances from the nucleus the wavefunction hasthe form lim r −→∞ ψ = sin ( kr + δ ) kr . (11)The wavefunction calculated numerically is fitted to thisasymptote to obtain δ as a function of k . A is then cal-culated at the low-energy limit from k cot δ = − /a + O ( k ).The calculated δ ( k ) are plotted in figure 3. Theyshow a good agreement with the many-body values for k < ∼ − which suggests that (Ps) V eff will re-main valid to describe quasi-thermalized positrons atroom temperature. For larger momenta the dynamicalcorrelation becomes important and our values are sys-tematically slightly lower. For Ps scattering off Li + thedifference is the largest, 0.3-0.4 radians, because both thetarget and the projectile are deformed. For a positronscattering off Be the agreement is very good, consideringthat the positron binding energy to Be is 0.8 × − a.u. P o t en t i a l ( a . u . ) M a ss N o r m . V e ff ( a . u . ) e + Li e + Be e + H e + He HPs LiPs
Figure 2. (Color online) V eff (black curves) of all the calculated systems and the mean-field Coulomb potentials (green dottedcurves) of the systems composed by an atom and a positron. For e + Li, HPs and LiPs the Ps V eff have been plotted. The insetin the e + Li panel compares the mass normalized V eff and the mean-field Coulomb potentials. smaller (26%) than the best many-body value [19] and0.5 × − a.u. smaller (16%) than the binding energyby Bromley et al. [33] However the A value, see ta-ble II, shows the largest mismatch with the referencemany-body value. For H, He and Li the present A arecomparable to the many-body values. C. Two-component DFT + Li, HPs, andLiPs allows us to draw conclusions also about systemsincluding a Ps cluster. Using the vanishing positron-density limit for the electron-positron correlation energy
Table II. Positron (e + H, e + He and e + Be) and Ps (e + Li) scat-tering lengths. The first column shows the values computedusing V eff and the last column are many-body calculationsfrom the literature. All the values are given in a.u.e + H -1.86 -2.094 [31], -2.10278 [32]e + He -0.55 -0.474 [31]e + Be 18.76 16 [33]Ps-Li + and potential [28, 29] the LDA 2C-DFT doesn’t pre-dict the binding of positrons to atoms. We use instead E ep [ n + , n − ], a LDA functional that depends on the elec-tron and positron densities and it predicts the formation P ha s e s h i ft (r ad i an ) -1 ) 321 P ha s e s h i ft (r ad i an ) -1 ) Figure 3. (Color online) s-wave phase shifts for positrons scattering off H (black curve), He (red curve) and Be (green curve)and Ps scattering off Li (blue curve). The many-body values obtained by Zhang et al. [31] for H (black circles) and He (redsquares), Mitroy et al. [34] for Li (blue diamonds) and Bromley et al. [33] for Be (green triangles) are also shown. of bound atom-positron states.Overall, the LDA 2C-DFT predicts accurate positrondensities and potentials comparable to the many-bodyresults. Figure 4 compares the LDA and the many-body electron and positron densities of e + Li and e + Be.The LDA positron density of e + Be matches the many-body density whereas the LDA electron density is slightlymore delocalized than the many-body density. The LDApositron density of e + Li is also accurate, however, theLDA electrons are more tightly bound to the nucleusthan in the many-body calculation. The potential wellsof the LDA positron potentials match V eff of e + Be andthe mass-normalized V P seff (cid:48) of e + Li. Close to the Li nu-cleus the LDA positron potential is less repulsive than V eff but the effect on the positron density is minor. Al-though V ep [ n + , n − ] is small compared to the mean-fieldCoulomb potential, it is necessary to obtain a bound statefor e + Li and e + Be.In the case of HPs and LiPs the mean-field Coulombpotential alone is able to predict the formation of a boundstate but including V ep [ n + , n − ] increases the accuracy ofthe positron density. For both systems the LDA positronpotential wells are deeper than the corresponding V eff but their widths are similar up to distances, r ∼
10 a.u.(HPs) or ∼
12 a.u. (LiPs), where the positron densitiesof the bound states are already negligible. Figure 5 showsthat the LDA electron and positron densities are slightlymore localized than the many-body densities in both sys-tems. The kinks in the LDA positron potentials of HPsand LiPs are caused by the cut-off imposed to the po-tential at low densities. Without the cut-off, the poten-tials have a long-range attractive tails which make thepositron densities too delocalized. Many-body calcula- tions at the low-density range of the electron-positronplasma would be required to obtain an electron-positroncorrelation potential which is accurate beyond the metal-lic density regime.The asymmetric behavior of the LDA electron andpositron densities with respect to the many-body calcula-tions reflects the means of DFT to describe correlations inthe interacting many-body system [37]. The electron self-interaction causes the 2s orbital of e + Li to be poorly de-scribed in DFT. Moreover, 2C-DFT within LDA cannotdescribe accurately strongly-correlated systems like Ps.Accordingly, the LDA densities of e + Li don’t show theformation of Ps. However, in HPs and LiPs at long sepa-rations the electron and positron densities overlap as ex-pected when Ps forms. Overall, figures 4 and 5 show con-vincingly that the electron-positron correlation potentialderived from the energy of an electron-positron plasmayields surprisingly accurate positron densities in boundpositronic atoms, including systems where Ps forms.The positron binding energies of all the studied sys-tems are only qualitative, reflecting the general inade-quacy of LDA to accurately describe binding betweenatoms. Moreover, it is a well known problem that DFTwithin LDA is not able to describe dispersion interac-tions [38].
IV. V eff FOR POSITRON AND Ps STATES INCONDENSED MATTER
It is well established that the LDA 2C-DFT yields re-liable densities for positrons trapped at vacancies insidemetals and semiconductors [25]. To simplify the calcu- R ad i a l den s i t y ( n x π r a . u . - ) P o t en t i a l ( a . u . ) P o t en t i a l ( a . u . ) e + Li e + Be Li Be Figure 4. (Color online) Many-body (full curves) and LDA (broken curves) densities and positron potentials for e + Li ande + Be. The radial electron densities of the isolated atoms (filled blue curves), and the interacting positron-atom systems (redcurves), as well as the positron densities (black curves) are represented in the main panel. The insets compare the single-particlepositron potentials. For e + Li the mass normalized potential has been plotted. R ad i a l den s i t y ( n x π r a . u . - ) -0.4-0.20.00.20.4 P o t en t i a l ( a . u . ) P o t en t i a l ( a . u . ) HPs
LiPs H Li Figure 5. (Color online) Many-body (full curves) and LDA (broken curves) densities and potentials for HPs and LiPs. Theradial electron densities of the isolated atoms (filled blue curves), and the interacting positron-atom systems (red curves), aswell as the positron densities (black curves) are represented in the main panels. The insets compare the LDA positron potentialand the mass-normalized V Pseff (cid:48) . lations or to compare different approaches, it would bedesirable to calculate the positron potentials also as su-perpositions of atomic or molecular V eff in condensedmatter. However, the transferability of V eff deducedfrom single positronic atoms or molecules is of concern.The trapping of positrons in vacancies inside metals andsemiconductors occurs partly because the valence elec-trons relax into the vacancy as attracted by the positronincreasing the binding energy and the degree of local-ization of the positron. In the 2C-DFT this is taken into account through the electron-positron correlationfunctional which lowers the energy of the positron insidethe vacancy. However, in V eff obtained from an atom-positron system the valence electrons remain bound tothe atom and its atomic superposition cannot predict thepositron trapping inside vacancies of crystalline solids.The utility of V eff will not be limited by this problemin condensed matter systems where the electronic struc-tures of the constituent atoms or molecules remain nearlyundisturbed like in molecular soft condensed matter andliquids where inter-molecular interactions are weak.The superposition of molecular V eff potentials is par-ticularly interesting from the point of view of studying Psembedded in molecular materials like polymers, liquids orbiostructures. Typically, the exchange repulsion betweenthe Ps and the HOMO-LUMO gap ( ∼ V eff can be derived forthe molecules forming the material. The calculation of V eff requires high quality many-body positron densities,which is computationally demanding with the presentcomputing capacity. Smaller systems like HePs can bestudied [39, 40], instead. He does not bind Ps due to itsclosed shell structure and its low polarizability. It pos-sesses a HOMO-LUMO gap in a spin-compensated elec-tron structure similarly to molecular matter and thus itprovides a good model system to study the interactionof Ps. The knowledge gained studying model systemswould allow building V eff when ab-initio methods cannotbe used. Moreover, our notion that the computationallyefficient 2C-DFT within LDA reproduces accurately themany-body single-particles potentials for systems withPs, raises the expectation that it could be used to con-struct Ps V eff . V. CONCLUSIONS
We have calculated the ECG-SVM many-body wave-functions for positronic systems including a light atom(H, He, Li and Be) and a positron or Ps. Basedon these results we have proposed an effective single-particle positron potential by inverting the single-particle Schr¨odinger equation arising from the many-body positron density. V eff is a single-particle potentialfor the positron interacting with an atom which includesthe full many-body correlations and it also describes apositron inside a Ps atom. The many-body positron den-sities and binding energies are, by construction, predictedby the introduced potential. The scattering lengths areconsistent with the many-body values in the literature and the s-wave phase shifts show also good agreementfor moments k < ∼ . − . The low-energy correla-tions are well described by V eff up to energies larger thanthat of quasi-thermalized positrons and Ps at room tem-perature. The success of V eff to describe the positronwhen a Ps complex forms, suggests that the potentialcan be also a valid single-particle description for the low-energy (quasi-thermalized) positron forming Ps withoutsolving the Schr¨odinger equation for the many-body sys-tem. This possibility should be further studied in con-nection with Ps interacting with molecular systems. Thesuperposition of atomic or molecular V eff to calculatethe positron potentials and the ensuing positron distri-butions in molecular condensed matter can be a valid de-scription of the positron in Ps when the inter-molecularinteractions are weak and the transferability is not ofconcern.We have also shown that the positron densities arewell described within the LDA 2C-DFT for bound e + Li,e + Be, HPs and LiPs when the finite positron-densityfunctional is used for the electron-positron correlationenergy. The self-consistent LDA 2C-DFT positron po-tentials reproduce the binding potential well of V eff ac-curately and predict the many-body positron densities.Although LDA 2C-DFT is less consistent predicting theelectron densities, our results indicate that it yields goodpositron distributions also for Ps bound to atoms. Thisresult opens the possibility to use 2C-DFT also to de-scribe Ps interacting with extended systems. However,the need for accurate treatment of the correlations forlow-density electron-positron plasmas calls for furthermany-body studies. ACKNOWLEDGMENTS
This work was supported by the Academy of Finlandthrough the individual fellowships and the centre of ex-cellence program. We acknowledge the computationalresources provided by Aalto Science-IT project. Thanksare due to K. Varga for providing us the ECG-SVM codeused in this work, to I. Makkonen for insightful discus-sions and to the referees for the valuable comments thatimproved the manuscript. [1] O. E. Mogensen, in
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