Motivation Method Results Outlook Recursive scheme of the perturbation theory for high-precision calculations in atoms and ions Dmitry A. Glazov Department of Physics Saint-Petersburg State University 2017-05-19 logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Outline 1 Motivation 2 Method 3 Results Binding energies Nuclear recoil effect g factor 4 Outlook logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Interelectronic interaction: Breit approximation Dirac-Coulomb-Breit Hamiltonian X X H = Λ+ h(j) + VBreit (j, k) Λ+ j j<k h = α · p + βm + Vnuc (r ) MBPT and various all-order methods based on: Dirac-Fock equation CI-DFS effective potentials MCDF configuration interaction RPP coupled cluster ... logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Interelectronic interaction: QED+Breit 1/Z 1/Z 2 1/Z 3 ... Breit QED logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Interelectronic interaction: QED+Breit 1/Z 1/Z 2 1/Z 3 ... Breit QED logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Requirements to the new method When the lowest orders are treated within QED and the remainder within the Breit approximation, there are special demands: 1 provide control over low orders of PT 2 take into account higher orders of PT 3 tame an exponential growth for higher orders logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Concept of the new method 1 One-electron basis set: |ni = |nr jlMi is constructed within the DKB-splines method 2 Many-electron basis set: |Ni = |n1 , n2 , . . . , ns i consists of Slater determinants 3 Many-electron matrix elements 4 Recursive perturbation theory logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook DCB Hamiltonian with screening potential Zeroth-order Hamiltonian H0 = X h(j) j h = α · p + βm + Vnuc (r ) + Vscr (r ) Perturbation H1 = X VBreit (j, k) − j<k VBreit = α X Vscr (rj ) j α1 · α2 1 1 − − (α1 · ∇1 )(α2 · ∇2 )r12 r12 r12 2 logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Perturbation theory Zeroth-order problem H0 |Ai = EA |Ai Exact problem (H0 + H1 )|Ãi = ẼA |Ãi Perturbation expansion ẼA = ∞ X (k ) EA (0) EA = EA k =0 |Ãi = ∞ X |A(k ) i |A(0) i = |Ai k =0 logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Perturbation theory For non-degenerate state |Ai one finds ∆E (1) = hA|H1 |Ai ∆E (2) = X ′ hA|H1 |NihN|H1 |Ai EA − EN N ∆E (3) = X ′ hA|H1 |MihM|H1 |NihN|H1 |Ai M,N (EA − EM )(EA − EN ) − hA|H1 |Ai X ′ hA|H1 |NihN|H1 |Ai N (EA − EN )2 ∆E (k ) comprises (k − 1)-fold summation → N k −1 terms logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Perturbation theory: recursive scheme ẼA = ∞ X (k ) |Ãi = EA k =0 (k ) EA = X ∞ X k =0 hA|H1 |MihM|A hA|A (k ) ∞ X X |NihN|A(k ) i k =0 N (k −1) i− k −1 X (j) EA hA|A(k −j) i j=1 M hN|A(k ) i = |A(k ) i = 1 EA − EN kX −1 X (j) hN|H1 |MihM|A(k −1) i − E hN|A(k −j) i A j=1 M k −1 1 X X (j) i=− hA |MihM|A(k −j) i 2 j=1 M Dmitry A. Glazov Recursive perturbation theory in atoms and ions logo-spbu.png Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor Systems investigated to date One-determinant states: He-like: (1s)2 Li-like: (1s)2 2s, (1s)2 2pJ Total energy: E[(1s)2 2s], E[(1s)2 2pj ] Ionization energy: E[(1s)2 2s/2pj ] − E[(1s)2 ] Be-like: (1s)2 (2s)2 B-like: (1s)2 (2s)2 2pJ Total energy: E[(1s)2 (2s)2 2pj ] Ionization energy: E[(1s)2 (2s)2 2pj ] − E[(1s)2 (2s)2 ] Fine structure: E[(1s)2 (2s)2 2p3/2 ] − E[(1s)2 (2s)2 2p1/2 ] Dmitry A. Glazov Recursive perturbation theory in atoms and ions logo-spbu.png Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor Lithium atom Coulomb potential CH: core-Hartree potential Z ∞ dr ′ Vscr (r ) = 0 α ρ1s (r ′ ) max(r , r ′ ) 2 2 (r )) (r ) + F1s ρ1s (r ) = 2 (G1s CH1 : 1/2 of core-Hartree potential 2 2 (r )) (r ) + F1s ρ1s (r ) = (G1s Glazov et al., NIMB (2017) MBPT: Yerokhin, Artemyev, Shabaev, PRA (2007) all-order: Yan, Drake, PRL (1998) Dmitry A. Glazov Recursive perturbation theory in atoms and ions logo-spbu.png Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor Lithium atom: (1s)2 2s ionization energy [a.u.] PT order 0 1 2 3 4 5 6 7 8 9 10 0–∞ all-order Coulomb −1.125 169 1.193 706 −0.250 663 −0.008 396 −0.004 363 −0.001 812 −0.000 741 −0.000 328 −0.000 187 −0.000 100 −0.000 029 −0.198 149 CH1 −0.572 597 0.435 989 −0.058 851 −0.000 628 −0.001 662 −0.000 280 −0.000 091 −0.000 035 −0.000 012 −0.000 004 −0.000 003 −0.198 176 −0.198 159 72 Dmitry A. Glazov CH −0.183 10 −0.021 61 0.011 59 −0.009 40 0.008 40 −0.008 15 0.008 35 −0.008 91 0.009 79 −0.011 01 0.012 62 ? Recursive perturbation theory in atoms and ions logo-spbu.png Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor Systems investigated to date One-determinant states: He-like: (1s)2 Li-like: (1s)2 2s, (1s)2 2pJ Total energy: E[(1s)2 2s], E[(1s)2 2pj ] Ionization energy: E[(1s)2 2s/2pj ] − E[(1s)2 ] Be-like: (1s)2 (2s)2 B-like: (1s)2 (2s)2 2pJ Total energy: E[(1s)2 (2s)2 2pj ] Ionization energy: E[(1s)2 (2s)2 2pj ] − E[(1s)2 (2s)2 ] Fine structure: E[(1s)2 (2s)2 2p3/2 ] − E[(1s)2 (2s)2 2p1/2 ] Dmitry A. Glazov Recursive perturbation theory in atoms and ions logo-spbu.png Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor B-like argon: (1s)2 (2s)2 2pj — ionization energies [eV] PT order 3 4 5 6 7 ... 12 13 14 15 3–∞ CI-DFS 2p1/2 LDF PZ 0.3560 0.4209 −0.5623 −0.5219 0.1644 0.1284 0.0581 0.0583 −0.0608 −0.0474 ... ... 0.0016 0.0011 −0.0005 −0.0001 −0.0003 −0.0002 0.0002 0.0001 −0.0296 0.0471 −0.0295 0.0470 2p3/2 LDF PZ 0.4706 0.5294 −0.5970 −0.5535 0.1575 0.1222 0.0710 0.0689 −0.0669 −0.0520 ... ... 0.0018 0.0013 −0.0006 −0.0002 −0.0002 −0.0002 0.0002 0.0001 0.0516 0.1249 0.0515 0.1246 logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor B-like argon: (1s)2 (2s)2 2pj — ionization energies [eV] 2p1/2 Term (0) EDirac (1) EBreit (2) EBreit (>3) EBreit (1) EQED (2) EQED Erec Etotal 2p3/2 LDF PZ −757.0075 −757.7629 LDF PZ −753.8815 −754.5449 0.1972 1.5045 −0.0477 1.1541 1.6916 1.0632 1.5335 0.9219 −0.0295 0.0470 0.0515 0.1246 −0.0041 −0.0042 0.0018 0.0018 −0.0157 0.0040 −0.0156 0.0040 −0.0160 0.0039 −0.0162 0.0039 −755.1639 −755.1640 −752.3545 −752.3546 Glazov et al., NIMB (2017); Malyshev et al., NIMB (2017) Dmitry A. Glazov Recursive perturbation theory in atoms and ions logo-spbu.png Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor Perturbation theory for matrix elements ∆EA [W ] =hÃ|W |Ãi ∞ X (k ) ∆EA [W ] = k =0 (k ) ∆EA [W ] = k X hA(j) |W |A(k −j) i j=0 = k X X hA(j) |MihM|W |NihN|A(k −j) i j=0 M,N logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor Nuclear recoil effect HM = HNMS + HSMS + HRNMS + HRSMS HNMS = 1 X 2 pj 2M j HSMS 1 X = pj · pk 2M j6=k HRNMS HRSMS " # (αj · rj )rj 1 X αZ αj + · pj =− 2 2M rj r j j " # (αj · rj )rj 1 X αZ αj + =− · pk 2 2M rj r j j6=k logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor Nuclear recoil effect: Li-like ions Li-like ions 2s–2pJ transitions: E [(1s)2 2pJ ] − E [(1s)2 2s] CI-DFS: Kozhedub et al., PRA 81, 042513 (2010) 1/Z + CI-DFS: Zubova et al., PRA 90, 062512 (2014) ∆E [HM ] = K M logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor Li-like U: (1s)2 2p1/2 –(1s)2 2s Z = 92 PT order 0 1 2 3 4 2–∞ 0–∞ Coulomb potential NMS −3629.96 −3629.93 −45.40 −45.42 11.039 −0.327 0.006 10.72 10.5 −3664.64 −3664.8 SMS −4925.25 −4925.25 299.48 299.49 −6.879 0.112 −0.002 −6.77 −6.7 −4632.54 −4632.5 [THz amu] RNMS 3930.05 3930.03 −28.38 −28.37 −7.705 0.255 −0.004 −7.46 −7.2 3894.21 3894.4 RSMS 3929.40 3929.40 −267.26 −267.27 6.896 −0.109 0.001 6.79 6.5 3668.93 3668.6 1/Z + CI-DFS: Zubova et al., PRA 90, 062512 (2014) Dmitry A. Glazov Recursive perturbation theory in atoms and ions logo-spbu.png Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor Nuclear recoil effect: B-like ions B-like ions Fine structure: E [(1s)2 (2s)2 2p3/2 ] − E [(1s)2 (2s)2 2p1/2 ] CI-DFS: Zubova et al., PRA 93, 052502 (2016) ∆E [HM ] = K M logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor B-like ions: 2p3/2 –2p1/2 fine structure Z = 20 [THz amu] KS LDF CI-DFS NMS −1.8424 −1.8350 −1.843 SMS 1.2661 1.2593 1.260 RNMS 1.4163 1.4191 1.417 Z = 92 RSMS −1.9770 −1.9806 −1.978 [THz amu] CH KS CI-DFS NMS −2967.8 −2967.8 −2968. SMS 1967.9 1967.7 1968. RNMS 2530.1 2529.4 2531. RSMS −2898.2 −2896.8 −2899. CI-DFS: Zubova et al., PRA 93, 052502 (2016) logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor g factor: positive-energy states Hmagn = µ0 HU X U= αj × rj z ∆EA = gA MA µ0 H gA [+] = j (k ) ∆gA [+] = 1 hÃ|U|Ãi MA k 1 X (j) hA |U|A(k −j) i MA j=0 = k 1 X X (j) hA |MihM|U|NihN|A(k −j) i MA j=0 M,N logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor g factor: negative-energy states gA [−] = (k ) ∆gA [−] = 2 X hp|U|ni + hâ âp Ã|H1 |Ãi MA p,n εp − εn n k −1 2 X X X hp|U|ni MA ε − εn p,n p j=0 M,N (k −j−1) i ×hA |Mihâ+ n âp M|H1 |NihN|A (j) |pi : εp > 0 |ni : εn < 0 logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor g factor of Li-like silicon Z = 14 Dirac QED ∼ α QED ∼ α2 Nuclear recoil Nuclear size e-e interaction Screened QED Total theory Experiment 1.998 254 751 0.002 324 044 −0.000 003 517 (1) 0.000 000 039 (1) 0.000 000 003 0.000 314 809 (6) −0.000 000 236 (5) 2.000 889 892 (8) 2.000 889 890 (2) Wagner et al., PRL 110, 033003 (2013) Volotka et al., PRL 112, 253004 (2014) logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor g factor of Li-like silicon Z = 14 Coulomb potential PT order 1 2 3 4 5 total 3+ CI-DFS CI ∆g × 109 321 437.2 −6 826.0 (2) 108.0 (19) −12.5 (8) −2.1 (2) 314 704.6 (21) 93.4 (21) 85. (22) 94. the convergence of the PT expansion is good the uncertainty is determined by the 3rd-order term the uncertainty is 10 times smaller than with CI-DFS Yerokhin et al., arXiv:1705.04476 Dmitry A. Glazov logo-spbu.png Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor g factor of Li-like silicon Z = 14 Local Dirac-Fock potential PT order 0 1 2 3 4 5 total 3+ CI-DFS ∆g × 109 349 635.9 −33 960.4 (1) −970.0 (2) −1.1 (11) −0.4 (3) 0.0 (2) 314 702.4 (12) −1.5 (12) −5.0 (60) the convergence of the PT expansion is even better the uncertainty is determined by the 3rd-order term the uncertainty is 5 times smaller than with CI-DFS the uncertainty is 2 times smaller than with Coulomb potential logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor g factor of Li-like silicon Z = 14 Local Dirac-Fock potential PT order 0 1 2 3 4 5 total ∆g × 109 349 635.9 −33 960.4 (1) −970.0 (2) −1.1 (11) −0.4 (3) 0.0 (2) 314 702.4 (12) QED +113.7 −4.2 ±2.4 314 813.5 (12)(24) logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor g factor of Li-like silicon Z = 14 Dirac QED ∼ α QED ∼ α2 Nuclear recoil Nuclear size e-e interaction Screened QED Total theory Experiment 1.998 254 751 0.002 324 044 −0.000 003 517 (1) 0.000 000 039 (1) 0.000 000 003 0.000 314 809 (6) 0.000 314 809 (1) 0.000 314 813 (3) −0.000 000 370 (7) 2.000 889 892 (8) 2.000 889 892 (6) 2.000 889 896 (6) 2.000 889 890 (2) Volotka et al., PRL (2014) Yerokhin et al., arXiv:1705.04476 this work Volotka et al., PRL (2014) Yerokhin et al., arXiv:1705.04476 this work Wagner et al., PRL (2013) logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor g factor of Li-like calcium Z = 20 Dirac QED ∼ α QED ∼ α2 Nuclear recoil Nuclear size e-e interaction Screened QED Total theory Experiment 1.996 426 011 0.002 325 555 (5) −0.000 003 520 (2) 0.000 000 062 0.000 000 014 0.000 454 290 (9) 0.000 454 296 (4) −0.000 000 370 (7) 1.999 202 042 (13) 1.999 202 048 (10) 1.999 202 041 (1) Volotka et al., PRL (2014) this work Volotka et al., PRL (2014) this work Koehler et al., NC (2016) logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Binding energies Nuclear recoil effect g factor g factor of Li-like ions: conclusion For low-Z Li-like ions: accuracy of ∆gint within the Breit approximation is improved by order of magnitude accuracy of ∆gint is now determined by the 3-photon QED contribution accuracy of g is now mostly determined by the screened QED contribution logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions Motivation Method Results Outlook Outlook Further generalizations of the presented method (quasi-)degenerate states axially symmetric systems: DKB → ADKB hÃ|W |Ãi → X hÃ|W1 |ÑihÑ|W2 |Ãi N ẼA − ẼN logo-spbu.png Dmitry A. Glazov Recursive perturbation theory in atoms and ions
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