Excited States of Positronic Lithium and Beryllium

week ending 8 NOVEMBER 2013 PHYSICAL REVIEW LETTERS PRL 111, 193401 (2013) Excited States of Positronic Lithium and Beryllium Sergiy Bubin and Oleg V. Prezhdo Department of Chemistry, University of Rochester, Rochester, New York 14627, USA (Received 31 July 2013; revised manuscript received 19 September 2013; published 4 November 2013) Using a variational method with an explicitly correlated Gaussian basis, we study the eþ -Li and eþ -Be complexes in the ground and lowest excited states with higher spin multiplicity. Our calculations provide rigorous theoretical confirmation that a positron can be attached to the excited states: 1s2s2p 4 Po and 1s2 2s2p 3 Po for eþ -Li and eþ -Be, respectively. The result is particularly notable for the eþ -Be complex, as the excited 3 Po state lies below the autoionization threshold. We report accurate binding energies, annihilation rates and structural properties of these positron-atom systems. The existence of the ground and metastable excited states with bound positron opens up a new route to the presently lacking experimental verification of stability of a positron binding to any neutral atom. DOI: 10.1103/PhysRevLett.111.193401 PACS numbers: 36.10. k, 31.15.ac, 31.15.xt Understanding the mechanisms of the interaction of low energy positrons with matter is one of the main tasks of positron physics and chemistry. In particular, of great interest is the question whether atoms and molecules can capture positrons and form bound states stable against dissociation [1–9]. Since 1997, when the first conclusive and rigorous theoretical confirmation of a possibility of attaching a positron to a neutral lithium was given [1], there have been a number of investigations claiming the dynamical stability of positron-atom complexes. At present, at least a dozen atoms are believed to be capable of binding a positron. In contrast, on the experimental side no evidence has been collected as of yet to demonstrate the existence of positronic atoms. While different experimental approaches to study positron binding to atoms have been proposed, e.g., by measuring resonant positron-atom annihilation [10,11] or by laser-assisted photorecombination [12], the existence of excited states is of crucial importance for detecting positron-atom complexes as it should allow spectroscopic measurements. Just recently such an approach has been used by Cassidy et al. [13] to confirm the production of a positronium molecule (Ps2 ). The technique employed in Ref. [13] was based on observing small yet detectable changes in the annihilation yield of dense Ps over a narrow range of wavelengths corresponding to a transition between the ground and excited states of Ps2 . The existence of a bound excited state of Ps2 , in turn, had been previously predicted by numerical calculations [14]. Only a handful of theoretical studies so far have dealt with the investigation of excited states of positronic atoms. The simplest multielectron atom, helium, has been known to attach a positron in its 1s2s 3 S state for more than a decade [15] (here and below the term symbol refers to electrons only). Recently, it has also been shown that positron attachment is possible in three doubly excited states [16–18]. Nevertheless, in its ground singlet state He does not form a bound positron-atom complex. There was also an indication based on large configuration 0031-9007=13=111(19)=193401(5) interaction (CI) calculations that Ca and Be might also bind a positron in an excited state [6,19]. These calculations, however, did not yield positive binding energy values. Instead, they relied heavily on asymptotic series analysis and extrapolation, and involved a fixed-core approximation. It is only recently that Bressanini [9] demonstrated convincingly that lithium can attach a positron in the excited 1s2s2p 4 Po state. While an encouraging fact by itself, it is somewhat unfortunate that this quartet state of Li lies far above the autoionization threshold. Its lifetime on the order of microseconds [20] is primarily determined by the relatively quick autoionization process. The question of the existence of positron-atom complexes in metastable states, whose energies are below the autoionization threshold, is much more intriguing from the practical point of view. In this Letter, we report a rigorous confirmation of such possibility for the 1s2 2s2p 3 Po state of a Be atom (see Fig. 1). The lowest excited state of Be, it can only decay radiatively to the ground 1 S state by emitting two or more photons. Its lifetime exceeds 2 s [21]. Our calculations allowed us to estimate also the lifetimes of both eþ -Li FIG. 1 (color online). Energy levels of Be and Beþ (solid black lines) and eþ -Be (dashed blue lines). 193401-1 Ó 2013 American Physical Society week ending 8 NOVEMBER 2013 PHYSICAL REVIEW LETTERS PRL 111, 193401 (2013) and eþ -Be complexes against electron-positron annihilation. An accurate description of positronic systems is challenging even in their ground states. The difficulties stem from the presence of a different kind of light particle and weak binding energies. Moreover, due to strong repulsion from nuclei, the positron transforms the system into a clusterlike structure, which may result in poor convergence of traditional quantum-chemical approaches [22,23]. Explicitly correlated Gaussian (ECG) basis sets that depend on all interparticle distances provide effective means for overcoming these obstacles for small systems. For the case of L ¼ 0; 1 considered in this work, where L is the total orbital angular momentum, a suitable form of the spatial part of ECGs is 
X X 2 ; 2 ðL¼0Þ r (1) r ¼ exp ijk ij ik i k i<j i positron-atom state can be generally viewed as a mixture of two major configurations involving different binding mechanisms, namely, a positron interacting with a polarized atom and a polarized Ps atom interacting with an atomic ion. The convergence of the total nonrelativistic energies for the eþ -Li and eþ -Be positronic complexes and relevant states of the atoms and ions is shown in Table II. We performed calculations using finite masses for the atomic nuclei: Mð7 LiÞ ¼ 12 786:3933me and Mð9 BeÞ ¼ 16 424:2037me , where me is the electron mass. To enable direct comparison with energies from published works we also recalculated all quantities by setting the nuclear mass to infinity. TABLE II. Convergence of total nonrelativistic energies and positron binding energies (BE). All values are in hartree. Basis size Nucleus ðL¼1;M¼0Þ ¼ zpk exp k
X 2 ik ri i X i<j 2 ijk rij  ; (2) where ri are particle coordinates, rij ¼ jri rj j, and ik , ijk , and pk are variational parameters. The explicit use of the spin part of the wave function can be avoided by employing Young projection operators within spin-free formalism [24]. Variational expansions in terms of ECGs have demonstrated an exceptional performance in calculations of various quantum few-body systems, including those containing positrons [25–28]. It has been known that the condition and likelihood of positron binding to an atom A depends on the ionization potential (IP) of the atomic state [5]. When the IP is greater than 0.25 hartree (binding energy of Ps), the threshold for the total energy is set by dissociation channel eþ þ A. In this case the key property affecting the existence of a bound state is the atomic polarizability. When the IP is smaller than 0.25, the dissociation channel Ps þ Aþ becomes more competitive. It has been observed that a positron is more likely to form a bound state with an atom when the IP value is not far from 0.25. The ground and excited states of Li and Be considered in this Letter satisfy that condition fairly well, as demonstrated in Table I. Regardless of the dissociation threshold, a bound TABLE I. Atomic ionization potentials (in hartree, from Ref. [29]) and dissociation thresholds for positron-atom complexes. eþ -A(state) eþ -Lið1s2 2s 2 SÞ eþ -Lið1s2s2p 4 Po Þ eþ -Beð1s2 2s2 1 SÞ eþ -Beð1s2 2s2p 3 Po Þ IP[A(state)] Dissociation threshold 0.198 0.255 0.343 0.242 Ps þ Liþ ð1s2 1 SÞ þ e þ Lið1s2s2p 4 Po Þ eþ þ Beð1s2 2s2 1 SÞ Ps þ Beþ ð1s2 2s 2 SÞ 500 1000 1500 2000 2500 3000 3000 Ref. [30]a 500 1000 1500 2000 2500 3000 3000 Ref. [9]b 500 1000 1500 2000 2500 2500 Ref. [31]c 500 1000 1500 2000 2500 2500 a 7 Li 7 Li 7 Li 7 Li Li 7 Li 1 Li 1 Li 7 7 Li 7 Li 7 Li 7 Li 7 Li 7 Li 1 1 Li Li 9 Be 9 Be 9 Be Be 9 Be 1 Be 1 Be 9 9 Be 9 Be 9 Be 9 Be 9 Be 1 Be BE Liþ ð1 SÞ 7:279 321 52 7:279 321 52 7:279 321 52 7:279 321 52 7:279 321 52 7:279 321 52 7:279 913 41 7:279 913 4 eþ -Lið2 SÞ 7:531 731 45 7:531 802 42 7:531 811 66 7:531 814 32 7:531 815 32 7:531 815 73 7:532 410 48 7:532 395 5 0.002 410 0.002 481 0.002490 0.002 493 0.002494 0.002 494 0.002 497 0.002 482 Lið4 Po Þ 5:367 605 51 5:367 605 79 5:367 605 81 5:367 605 82 5:3676 058 2 5:367 605 82 5:368 010 15 5:367 33ð3Þ eþ -Lið4 Po Þ 5:373 199 50 5:373 386 70 5:373 417 78 5:373 427 57 5:373 431 52 5:373 433 43 5:373 835 48 5:3710ð2Þ 0.005 594 0.005 781 0.005 812 0.005 822 0.005 826 0.005 828 0.005 825 0.0037(2) Beð1 SÞ 14:666 428 42 14:666 434 60 14:666 435 25 14:666 435 37 14:666 435 47 14:667 356 45 14:667 338 eþ -Beð1 SÞ 14:669 153 13 14:669 553 55 14:669 633 47 14:669 662 10 14:669 674 34 14:670 592 86 14:670 519 0.002 725 0.003 119 0.003 198 0.003 227 0.003 239 0.003 236 0.003 181 Beþ ð2 SÞ 14:323 863 15 14:323 863 47 14:323 863 49 14:323 863 49 14:323 863 49 14:324 763 18 eþ -Beð3 Po Þ 14:572 774 52 14:574 263 28 14:574 629 84 14:574 784 65 14:574 863 80 14:575 764 95 ECG, 1200 basis functions. Fixed-node diffusion Monte Carlo simulations. c ECG, 2200 basis functions. b 193401-2 0:001 089 0.000 400 0.000 766 0.000 921 0.001 000 0.001 002 week ending 8 NOVEMBER 2013 PHYSICAL REVIEW LETTERS PRL 111, 193401 (2013) The data in Table II exhibit excellent agreement with and marginal improvement over the best previous calculations of the ground states of eþ -Li and eþ -Be by Mitroy et al. [30,31]. We have confirmed the stability of the excited 1s2s2p 4 Po state of positronic lithium, and significantly increased the accuracy of the binding energy from 0.0037 hartree reported in Ref. [9] to 0.005 827 6(10) hartree obtained in this work. Finally, our efforts to compute the 1s2 2s2p 3 Po state of eþ -Be have provided conclusive evidence that beryllium can attach a positron in an excited triplet state. The positron binding energy in that state is rather small and equal to just 0.001 000 3(400) hartree or 0.027 22(10) eV. Because of tighter energy convergence in the case of bare atoms in comparison with the corresponding positron-atom complexes, our calculated binding energies are undoubtedly bounds from below to the exact nonrelativistic values. The uncertainties are estimated by extrapolating to the limit of the infinite basis size. Given the sufficiently high convergence of the total energies and no approximations involved in the calculations (at the nonrelativistic level), the prediction of the existence of the bound excited state is rigorous. The inclusion of relativistic corrections would not alter this conclusion, as they cancel out to a large extent in weakly bound systems when the energy difference (e.g., eþ -A vs Ps þ Aþ ) is evaluated. This happens because the largest contribution to the relativistic correction is due to core electrons, which remain essentially undistorted upon binding a positron. Rough estimates suggest that the net effect of relativistic corrections on the positron binding energy in the 1s2 2s2p 3 Po state of eþ -Be is at least 1–2 orders of magnitude smaller than the binding energy itself. We have computed the change in the binding energy due to the inclusion of the two largest corrective terms, the mass velocity and Darwin terms, and obtained a shift of only 0.000 007 hartree. Table III lists expectation values, which allow us to compare the structure of the excited states of positronic lithium and beryllium with that of the ground states as well as with the related atoms and ions. As a result of weak binding, positronic atoms typically have a very large spatial extent and possess halolike properties [32]. Remarkably, for both eþ -Li and eþ -Be, the size of the system as a whole in the excited state is smaller than in the ground state. The value of the average nucleus-positron distance, hrneþ i, drops notably in the exited states. This fact can be rationalized in the case of Li by the almost twice higher binding energy in the 1s2s2p 4 Po state compared to the 1s2 2s 2 S state. For Be, however, where binding in the excited state is extremely weak, the exact reason for such behavior is not immediately clear. The atomic 3 Po state, i.e., when no positron is attached, is predictably larger than the ground 1 S state. The near equality of hrneþ i with hreþ e i leads to the conclusion that eþ -Be (1s2 2s2 1 S) fits well into the picture of a bare positron interacting with a distant atom. This conclusion becomes even more evident when comparing nucleus-electron and electron-electron pair correlation functions, gij ðrÞ ¼ hðrij rÞi, shown in Fig. 2. The change in these two distributions upon going from the Be atom to the eþ -Be complex is tiny. In contrast, the same two distributions, gne ðrÞ and ge e ðrÞ, exhibit noticeable difference between Li (1s2 2s 2 S) and eþ -Li (1s2 2s 2 S). The ratio between hrneþ i and hreþ e i for eþ -Li (1s2 2s 2 S) suggests that this system can be mainly viewed as a weakly distorted Ps-Aþ complex. The geometric structure interpretation for the excited P states of positronic atoms is somewhat less straightforward. The value of hrneþ i is larger for eþ -Be compared to eþ -Li. At the same time the average positron-electron distance, hreþ e i, is smaller. While the ratio between the hrneþ i and hreþ e i in the 1s2 2s2p 3 Po state of eþ -Be is not exactly consistent with a Ps-Aþ configuration, the positron-electron correlation function shown in Fig. 3 has a pronounced maximum at small positron-electron separation, which suggests that this configuration dominates. TABLE III. Expectation values (in atomic units) for the ground and excited states of positronic complexes and relevant atoms and ions. For convenience, we use n, eþ , and e as indices, to denote nucleus, positron, and electron respectively.
is the spin-averaged annihilation rate (in 109 s 1 ). All quantities were computed using the largest generated ECG basis. Values in parentheses show estimated remaining uncertainty due to finite size of the basis. System Ps Liþ ð1 SÞ Lið2 SÞ eþ -Lið2 SÞ Lið4 PÞ eþ -Lið4 PÞ Beþ ð2 SÞ Beð1 SÞ eþ -Beð1 SÞ Beð3 PÞ eþ -Beð3 PÞ hrne i 3.000 000 0.572 821(0) 1.663 312(0) 3.4112(4) 2.034 578(0) 2.272 48(8) 1.033 863(0) 1.493 194(0) 1.535 73(2) 1.556 996(1) 2.326(6) hre e i hrneþ i hreþ e i hðrne Þi hðre e Þi hðrneþ Þi hðreþ e Þi 0.039 789 0.862 373(0) 2.889 697(0) 6.3635(8) 3.495 662(0) 3.913 73(15) 1.755 787(0) 2.545 443(0) 2.606 26(4) 2.693 011(2) 4.148(10) 6.850 35(3) 4.613 04(5) 9.9476(11) 7.7773(8) 4.5706(15) 2.978 053(50) 9.0956(8) 8.8956(8) 2.9780(20) 11.699 22(30) 8.839 48(15) 9.9998(35) 9.9575(35) 8.819(15) 8.7136(15) 9.55(8) 8.78(3) 8.710(10) 193401-3 0.533 608(6) 0.181 404(10) 0.178 43(10) 0.000 0018(2) 0(0) 0(0) 0.000 0019(3) 0.526 780(40) 0.267 56(4) 0.2676(4) 0.000 002 5(10) 0.261 22(10) 0.2633(5) 0.000 0016(5)
2.0028 0.011 582(20) 1.7484(30) 0.003 941(10) 0.595 11(150) 0.002 140(10) 0.430 95(200) 0.005 05(7) 1.0170(150) PRL 111, 193401 (2013) PHYSICAL REVIEW LETTERS week ending 8 NOVEMBER 2013 FIG. 3 (color online). Nucleus-positron and positron-electron pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi correlation functions, 2gðrÞ [r  ð; zÞ,  ¼ x2 þ y2 ], for the excited P states of positronic Li and Be. FIG. 2 (color online). Pair correlation functions for the ground S states of positronic lithium and beryllium. Dashed lines correspond to isolated atoms with no positron. Interestingly, the positron-electron correlation function for the P state of positronic lithium also has a noticeable (albeit shorter) peak at small distances. Therefore, there should be certain admixture of Ps-Aþ configuration in this state as well. Nucleus-positron correlation functions for the P states of both eþ -Li and eþ -Be presented in Fig. 3 do not have a node (only a concavity) at z ¼ 0. The absence of the node suggests that the positron is in a mixture of states with zero and nonzero orbital angular momenta. In Table III we also show the expectation values of the two-particle delta functions. Knowledge of the average electron-positron densities at coalescence points allows us to predict the lifetimes against annihilation. The total spin-averaged particle-antiparticle annihilation rate (see Ref. [33] for details) is given by    4c
19 17
¼ 1þ 2 2 ln Neþ Ne heþ e i; a0 12  (3) where is the fine structure constant, c is the speed of light, a0 is the Bohr radius, and Neþ and Ne are the number of positrons and electrons in the system. The terms following the unity in the square brackets of Eq. (3) represent the leading radiative corrections [34,35]. It is important to note that the annihilation rates in the ground and excited states of both eþ -Li and eþ -Be are significantly different. In the case of lithium,
is about 3 times smaller in the excited 4 Po state than in the ground 2 S state, 1:7484ð30Þ  109 s 1 vs 0:595 11ð150Þ  109 s 1 , respectively. For beryllium, the situation is opposite. The ground state survives for roughly twice longer than the excited state:
ð1 SÞ ¼ 0:430 95ð200Þ  109 s 1 , while
ð3 Po Þ ¼ 1:0170ð150Þ  109 s 1 . The lower electronpositron annihilation rate in eþ -Lið4 Po Þ and eþ -Beð1 SÞ should be attributed to the fact that in these states the positron is alone and only weakly interacts with the atom. Hence, the electron and positron components of the wave function overlap little. In contrast, the most significant configuration in eþ -Lið2 SÞ and eþ -Beð3 Po Þ is Ps-A. The electron in the Ps atom is the one which most likely annihilates with the positron. A substantial difference in
of the ground and excited states should be a welcome fact for possible spectroscopic studies that detect optically induced changes in the annihilation yield. To summarize, we have investigated positron-atom complexes in excited states eþ -Li (1s2s2p 4 Po ) and eþ -Be (1s2 2s2p 3 Po ). Based on the results of variational calculations with all-particle correlated basis sets, we have obtained a rigorous confirmation that both of these states should be dynamically stable. The 1s2 2s2p 3 Po state has the advantage of being the lowest excited state of Be and having a long (exceeding seconds) lifetime against radiative decay. Theoretical evidence for the capability of twoto four-electron atoms to form excited states with positrons obtained in Refs. [9,15,16] as well as in this work raise the question as to whether positron binding in excited states should be of common occurrence. Indeed, while the ground state of an atom may not always possess the properties 193401-4 PRL 111, 193401 (2013) PHYSICAL REVIEW LETTERS necessary for attaching a positron, it is more likely that suitable candidates (e.g., those with higher polarizability and ionization potential close to 0.25 hartree) can be found among the multitude of excited states. Routine search for weakly bound positron-atom states, however, remains a difficult task and will require the development of new, reliable, and computationally inexpensive approaches in the future. We hope that the conclusive evidence for the existence of a weakly bound excited state of positronic beryllium may stimulate future experimental efforts to perform spectroscopic measurements that involve the ground to excited state transition in this and other similar systems. The authors gratefully acknowledge financial support from the NSF, Grant No. CHE-1300118. [1] G. G. Ryzhikh and J. Mitroy, Phys. Rev. Lett. 79, 4124 (1997). [2] G. G. Ryzhikh, J. Mitroy, and K. Varga, J. Phys. B 31, 3965 (1998). [3] K. Varga, Phys. Rev. Lett. 83, 5471 (1999). [4] Z.-C. Yan and Y. K. Ho, Phys. Rev. A 59, 2697 (1999). [5] J. Mitroy, M. W. J. Bromley, and G. G. Ryzhikh, J. Phys. B 35, R81 (2002). [6] M. W. J. Bromley and J. Mitroy, Phys. Rev. Lett. 97, 183402 (2006). [7] D. M. Schrader, in Physics with Many Positrons, Proceedings of the International School of Physics ‘‘Enrico Fermi’’ Vol. 174, edited by A. Dupasquier, A. Mills, and R. Brusa (IOS Press, Amsterdam, Netherlands, 2010), p. 77. [8] X. Cheng, D. Babikov, and D. M. Schrader, Phys. Rev. A 83, 032504 (2011). [9] D. Bressanini, Phys. Rev. Lett. 109, 223401 (2012). [10] V. A. Dzuba, V. V. Flambaum, and G. F. Gribakin, Phys. Rev. Lett. 105, 203401 (2010). [11] G. F. Gribakin, J. A. Young, and C. M. Surko, Rev. Mod. Phys. 82, 2557 (2010). [12] C. M. Surko, J. R. Danielson, G. F. Gribakin, and R. E. Continetti, New J. Phys. 14, 065004 (2012). week ending 8 NOVEMBER 2013 [13] D. B. Cassidy, T. H. Hisakado, H. W. K. Tom, and A. P. Mills, Phys. Rev. Lett. 108, 133402 (2012). [14] K. Varga, J. Usukura, and Y. Suzuki, Phys. Rev. Lett. 80, 1876 (1998). [15] G. Ryzhikh and J. Mitroy, J. Phys. B 31, 3465 (1998). [16] M. W. J. Bromley, J. Mitroy, and K. Varga, Phys. Rev. Lett. 109, 063201 (2012). [17] J. Mitroy and J. Grineviciute, Phys. Rev. A 88, 022710 (2013). [18] Y. K. Ho, Phys. Rev. A 41, 68 (1990). [19] M. W. J. Bromley and J. Mitroy, Phys. Rev. A 75, 042506 (2007). [20] M. Levitt, R. Novick, and P. D. Feldman, Phys. Rev. A 3, 130 (1971). [21] C. F. Fischer and G. Tachiev, At. Data Nucl. Data Tables 87, 1 (2004). [22] M. W. J. Bromley and J. Mitroy, Phys. Rev. A 66, 062504 (2002). [23] K. Strasburger and H. Chojnacki, in Explicitly Correlated Wave Functions in Chemistry and Physics, Progress in Theoretical Chemical Physics Vol. 13, edited by J. Rychlewski (Springer, Netherlands, 2003), p. 439. [24] M. Hamermesh, Group Theory and Its Application to Physical Problems (Addison-Wesley, Reading, MA, 1962). [25] Y. Suzuki and K. Varga, Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems, Lecture Notes in Physics (Springer, Berlin, 1998). [26] S. Bubin, M. Pavanello, W.-C. Tung, K. L. Sharkey, and L. Adamowicz, Chem. Rev. 113, 36 (2013). [27] J. Mitroy, S. Bubin, W. Horiuchi, Y. Suzuki, L. Adamowicz, W. Cencek, K. Szalewicz, J. Komasa, D. Blume, and K. Varga, Rev. Mod. Phys. 85, 693 (2013). [28] S. Bubin, O. V. Prezhdo, and K. Varga, Phys. Rev. A 87, 054501 (2013). [29] A. E. Kramida, Yu. Ralchenko, J. Reader, and NIST ASD Team, NIST Atomic Spectra Database 5.1, http:// physics.nist.gov/asd (2013). [30] J. Mitroy, Phys. Rev. A 70, 024502 (2004). [31] J. Mitroy, J. At. Mol. Sci. 1, 275 (2010). [32] J. Mitroy, Phys. Rev. Lett. 94, 033402 (2005). [33] S. Bubin and K. Varga, Phys. Rev. A 84, 012509 (2011). [34] I. Harris and L. M. Brown, Phys. Rev. 105, 1656 (1957). [35] I. Khriplovich and A. Yelkhovsky, Phys. Lett. B 246, 520 (1990). 193401-5