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All-Order Methods for Relativistic Atomic Structure Calculations 1 M. S. Safronova a,2 W. R. Johnson b,3 a Department of Physics and Astronomy, 223 Sharp Lab, University of Delaware, Newark, Delaware 19716

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All-Order Methods for Relativistic Atomic Structure Calculations 1 M. S. Safronova a,2 W. R. Johnson b,3 a Department of Physics and Astronomy, 223 Sharp Lab, University of Delaware, Newark, Delaware b Department of Physics, University of Notre Dame, Notre Dame, IN Abstract All-order extensions of relativistic atomic many-body perturbation theory are described and applied to predict properties of heavy atoms. Limitations of relativistic many-body perturbation theory are first discussed and the need for all-order calculations is established. An account is then given of relativistic all-order calculations based on a linearized version of the coupled-cluster expansion. This account is followed by a review of applications to energies, transition matrix elements, and hyperfine constants. The need for extensions of the linearized coupled-cluster method is discussed in light of accuracy limits, the availability of new computational resources, and precise modern experiments. For monovalent atoms, calculations that include nonlinear terms and triple excitations in the coupled-cluster expansion are described. For divalent atoms, results from second- and third-order perturbation theory calculations are given, along with results from configuration-interaction calculations and mixed configuration interaction many-body perturbation theory calculations. Finally, applications of all-order methods to atomic parity nonconservation, polarizabilities, C 3 and C 6 coefficients, and isotope shifts are given. Key words: relativistic atomic structure, many-body perturbation theory, coupled-cluster theory, electron correlation calculations, isotope shift, polarizability, hyperfine structure, oscillator strengths, lifetimes, transition moments, weak-interaction effects in atoms 1 AMO review paper: version 7 12/02/ Preprint submitted to Elsevier Preprint 7 December 2006 1 Introduction and Overview The nonperturbative treatment of relativity in atomic many-body calculations can be traced back to the formulation of relativistic self-consistent field (SCF) equations with exchange by Swirles [1] in The SCF equations, also referred to as Dirac-Hartree-Fock (DHF) equations, are based on a manyelectron Hamiltonian in which the electron kinetic and rest energies are from the Dirac equation and the electron-electron interaction is approximated by the Coulomb potential. Numerical solutions of the DHF equations without exchange were obtained during the years by Williams [2], Mayers [3], and Cohen [4]. The formulation of relativistic SCF theory by Swirles was reexamined by Grant [5] in 1961 and the DHF equations were brought into a compact and easily used form. Numerical solutions to the DHF equations with exchange were published in 1963 by Coulthard [6], by Kim [7], and Smith and Johnson [8]. The Breit interaction was included in the later two calculations [7, 8]. In 1973, Desclaux [9] published complete DHF studies of atoms with Z = and Mann and Waber [10] published DHF studies of the lanthanides, including effects of the Breit interaction. The DHF equations remain as the starting point for relativistic many-body studies of atoms and versatile multiconfiguration DHF codes are publically available; notably the codes of Desclaux [11] and Grant et al. [12]. Extensions of the DHF approximation have been developed over the past three decades driven by advances in several areas of experimental atomic physics. Of particular importance in this regard are the precise measurements of energy levels and transition moments for highly-charged ions produced in beam-foil experiments, electron beam ion trap (EBIT) experiments, tokamak plasmas, and astrophysical plasmas [13]. These measurements have reached such a high level of precision that it has become possible to detect two-loop Lamb-shift corrections to levels in lithiumlike U [14], putting very tight constraints on the accuracy of the underlying atomic structure calculations. An equally important motivating factor in the development of extensions of the DHF approximation are measurements of parity nonconserving (PNC) amplitudes in heavy atoms, especially those designed to test the standard model of the electroweak interaction and to set limits on its possible extensions [15]. For the case of cesium, measurements of PNC amplitudes have reached an accuracy of 0.4% [16]. To make meaningful tests of the standard model, calculations of the amplitudes must be carried out for heavy neutral atoms to a similar level of accuracy. One systematic extension of the DHF approximation is relativistic many-body perturbation theory (MBPT). Relativistic MBPT studies of atomic structure start from a lowest-order approximation in which the electron-electron interaction is the frozen core DHF potential and include an order-by-order 2 perturbation expansion (in powers of the residual interaction) of energies and wave functions. Relativistic MBPT was used to predict properties of alkalimetal atoms from Li to Cs in Ref. [17], where energy levels for the ground state and the first few excited states were calculated to second order. In [17], electric-dipole matrix elements for the principal transitions and hyperfine constants were calculated through second order and included dominant third-order corrections. Although accurate values for energies, transition matrix elements, and hyperfine constants were obtained for Li, results for heavier alkali-metal atoms were significantly less accurate. The ground-state energy for Cs was accurate to 1.5%, while the Cs transition and hyperfine matrix elements were accurate to about 5% as determined by comparisons with precise experimental data. Later, complete third-order calculations of electric-dipole matrix elements, including all third-order terms were carried out in Ref. [18] for alkali-metal atoms and for Li-like and Na-like ions. The agreement with available experiments was very good for lighter atoms (within experimental precision for Li and Na), but decreased significantly for Cs and Fr. To achieve the accuracy required for tests of the standard model in heavy atoms, it is imperative to include contributions beyond third order in MBPT. Although extensions to fourth order represent one possibility, the resulting calculations are formidable; for each first-order matrix element there are four terms in second order, 60 terms in third order, and 3072 terms in fourth order [19]. Owing to this very rapid increase in computational effort with MBPT order, one seeks alternatives to MBPT beyond third order. One such alternative is the coupled-cluster singles-doubles (CCSD) method in which single and double excitations of the DHF ground state are included to all orders of perturbation theory. A nonrelativistic version of this method was used to calculate precise values of energies and hyperfine constants of 2s and 2p states of Li by Lindgren [20]. A linearized, but relativistic, version of the coupled-cluster method was later used to obtain energy levels, finestructure intervals, and dipole matrix elements in Li and Be + in Ref. [21]. These all-order calculations substantially improved the accuracy of energies and matrix elements compared to older MBPT results [17]. A nonrelativistic CCSD calculation for Na was reported in [22], where energies and hyperfine constants of 3s and 3p states and the 3s 3p electric-dipole matrix elements were calculated. Partial contributions to the 3s energy and hyperfine constant from triple excitations were also included in [22]; the resulting 3s energy was accurate to 0.01% and the 3s hyperfine constant to 0.2%. A relativistic version of the CCSD method was applied to calculate energy levels of alkali-metal atoms in [23] and excellent agreement with experiment was found. A linearized version of the coupled-cluster formalism, including single, double, and partial triple excitations (SDpT) was used to determine atomic properties of Cs in Ref. [24], where removal energies agreed with experiment to 0.5% and matrix elements agreed with measurements to better than 1%. Properties of Na-like 3 ions (Z = 11 16), such as energies, transition matrix elements, and hyperfine constants were studied using the linearized CCSD method in Ref. [25], and similar studies of alkali-metal atoms including polarizabilities were reported in [26]. Although we concentrate on relativistic all-order coupled-cluster methods in this review, it should be noted that perturbation theory in the screened Coulomb interaction (PTSI) developed by Dzuba et al. [27, 28], in which important classes of MBPT corrections are summed to all-orders, is an alternative method that has been successfully applied to atomic structure calculations for heavy neutral atoms. Moreover, for atoms with more than one valence electron, relativistic configuration-interaction (CI) calculations in an effective Hamiltonian extracted from the linearized SD theory, which has been developed and applied to small systems by Kozlov [29], is a promising alternative to CCSD methods for large systems. 2 Relativistic Many-Body Perturbation Theory In the simplest picture of a relativistic many-electron atom, each electron moves independently in a central potential U(r) produced by the remaining electrons. The one-electron orbitals φ a (r) describing the motion of an electron with quantum numbers a = (n a, κ a, m a ) satisfy the one-electron Dirac equation where h(r)φ a (r) = ɛ a φ a (r), (1) h(r) = c α p + βmc 2 Z r + U(r). (2) The quantities α and β in Eq. (2) are 4 4 Dirac matrices. The Dirac eigenvalues ɛ a range through values: ɛ a mc 2 for electron scattering states, mc 2 ɛ a 0 for electron bound states, and mc 2 ɛ a for positron states. The point of departure for our discussions of many-electron atoms is the no-pair Hamiltonian obtained from QED by Brown and Ravenhall [30] and illuminated in Refs. [31 34]. In this Hamiltonian, the electron kinetic and rest energies are from the Dirac equation and the potential energy is the sum of Coulomb and Breit interactions. Contributions from negative-energy (positron) states are projected out of this Hamiltonian. The no-pair Hamilto- 4 nian can be written in second-quantized form as H = H 0 + V, where H 0 = i ɛ i [a ia i ], (3) V = 1 (g ijkl + b ijkl ) [a 2 ia ja l a k ] ijkl + (V HF + B HF U) ij [a ia j ] + 1 (V HF + B HF 2 U) aa. (4) ij 2 a In Eqs. (3-4), a i and a i are creation and annihilation operators for an electron state i, and the summation indices range over electron bound and scattering states only, since, as mentioned above, contributions from negative energy states are absent in the no-pair Hamiltonian. Products of operators enclosed in brackets, such as [a ia ja l a k ], designate normal products with respect to a closed core. The summation index a in the last term in (4) ranges over states in the closed core. The quantity ɛ i in Eq. (3) is the eigenvalue of the Dirac equation (1). The quantities g ijkl and b ijkl in Eq. (4) are two-electron Coulomb and Breit matrix elements, respectively 1 g ijkl = ij b ijkl = ij r kl 12, (5) α 1 α 2 + (α 1 ˆr 12 )(α 2 ˆr 12 ) 2r 12 kl. (6) In Eq. (4), the core DHF potential is designated by V HF and its Breit counterpart is designated by B HF ; thus, (V HF ) ij = b (B HF ) ij = b [g ibjb g ibbj ], (7) [b ibjb b ibbj ], (8) where b ranges over core states. For neutral atoms, the Breit interaction is often a small perturbation that can be ignored compared to the Coulomb interaction. In such cases, it is particularly convenient to choose the starting potential U(r) to be the core DHF potential U = V HF, since with this choice, the second term in Eq. (4) vanishes. The third term in (4) is, of course, a c-number and provides an additive constant to the energy of the atom. It should be noted that, although the no-pair Hamiltonian is a useful starting point for relativistic many-body calculations, certain small contributions to wave functions and energies, including frequency-dependent corrections to the Breit interaction, self-energy and vacuum-polarization corrections, and corrections from crossed-ladder diagrams, are omitted in this approach. Perturbation theory based directly on the Furry representation of QED includes all such omitted effects [35]. In calculations based on the no-pair Hamiltonian, 5 contributions from these omitted terms are usually estimated and added as an afterthought. Recently, however, an energy-dependent formulation of MBPT that includes QED corrections completely has been developed by Lindgren et al. [36] and applied to heliumlike ions. Let us return to MBPT and concentrate on the simplest atoms, those with a single valence electron. For monovalent atoms, we write the lowest-order state vector as Ψ (0) v = a v 0 c, (9) where 0 c = a aa b a n 0 is the state vector for the closed core, 0 being the vacuum state vector and a v being a valence-state creation operator. If we ignore the Breit interaction and start our calculation using DHF wave functions for one-electron states (U = V HF ), then the lowest-order energy of the atom, obtained from H 0 Ψ (0) v = E (0) Ψ (0) v, is E (0) = ɛ v + a ɛ a, (10) and the first-order energy is E (1) = Ψ (0) v V Ψ (0) v = 1 2 (V HF ) aa. (11) a We see that through first order, the energy separates into a core contribution and a valence contribution, with E (0+1) core = a ɛ a 1 (V HF ) aa = (h 0 ) aa + 1 (g abab g abba ), (12) 2 a a 2 E (0+1) v = ɛ v. (13) The summation indices a and b in Eqs. (11) and (12) range over core states. The quantity (h 0 ) aa is the matrix element in state a of the sum of the kinetic energy and nuclear potential terms in the Dirac Hamiltonian (2). The sum of zeroth- plus first-order energies in (12) is precisely the DHF energy of the core. The energy of a one-electron atom splits order-by-order into core and valence contributions E (k) = E core (k) + E v (k). Since the core contribution is the same for each valence states, it is sufficient to consider valence contributions when studying excitation or ionization energies of one-electron atoms using MBPT. The second-order contribution to the valence energy is found to be [37] E (2) v = g abvn g vnab nab ɛ v + ɛ n ɛ a ɛ b mnb ab g vbmn g mnvb ɛ m + ɛ n ɛ v ɛ b. (14) Here and in the following sections, we adopt the convention that letters near the start of the alphabet (a, b, c, ) designate core states, letters in the middle 6 of the alphabet (m, n, o, ) designate virtual states, and letters near the end of the alphabet (v, w, x, ) designate valence states. We let the letters (i, j, k, ) designate either core or virtual (general) states. In Eq. (14), we have also used the notation g ijkl = g ijkl g ijlk to designate anti-symmetrized two-particle matrix elements. The much longer expression for the third-order contribution to the valence energy for a monovalent atom E v (3) is given in Ref. [37] and will not be repeated here. To evaluate the expressions for second- and third-order energies, we first sum over magnetic quantum numbers analytically to obtain expressions involving radial Dirac wave functions and angular momentum coupling coefficients, then we sum over the remaining principal and angular quantum numbers numerically. To aid in the numerical work, we replace the spectrum of the radial Dirac equation, which consists of bound states, a positive-energy continuum of scattering states, and a negative-energy continuum of positron states, by a finite pseudospectrum. For the calculations discussed in this review, the pseudospectrum was constructed from B-splines confined to a large but finite cavity, as described in Ref. [38]. In Table 1, we give a breakdown of the zeroth-order, second-order, and thirdorder MBPT contributions to ionization energies of alkali-metal atoms and compare the sum with various all-order calculations and with experiment. Differences between third-order MBPT calculations and experiment range from fractions of 1% for Li and Na to about 3% for Cs. Moreover, for Cs, including third-order corrections actually worsens the agreement with measured energies found in second order, emphasizing the need for all-order methods. 3 Relativistic SD All-Order Method As an introduction to relativistic all-order calculations, we briefly describe the relativistic singles-doubles (SD) method, a linearized version of coupledcluster theory; a more detailed description can be found in [21, 25]. In the coupled-cluster theory, the exact many-body wave function is represented in the form [39] Ψ = exp(s) Ψ (0), (15) where Ψ (0) is the lowest-order atomic state vector. The operator S for an N-electron atom consists of cluster contributions from one-electron, twoelectron,, N-electron excitations of the lowest-order state vector Ψ (0) : S = S 1 + S S N. (16) 7 The exponential in Eq. (15), when expanded in terms of the n-body excitations S n, becomes Ψ = { 1 + S 1 + S 2 + S S2 1 + S 1 S } 2 S2 2 + Ψ (0). (17) In the linearized coupled-cluster method, all non-linear terms are omitted and the wave function takes the form Ψ = {1 + S 1 + S 2 + S S N } Ψ (0). (18) The SD method is the linearized coupled-cluster method restricted to single and double excitations only. The all-order singles-doubles-partial triples (SDpT) method is an extension of the SD method in which the dominant part of S 3 is treated perturbatively. A detailed description of the SDpT method is given in Refs. [24, 26]. Inclusion of the non-linear terms in the relativistic SD formalism and a more complete treatment of the triple excitations is given in [40, 41] and will be considered later. Restricting the sum in Eq. (18) to single and double excitations yields the following expansion for the SD state vector of a monovalent atom in state v: Ψ v = [ 1 + ma ρ ma a ma a mnab ρ mnab a ma na b a a + + ρ mv a ma v + ρ mnva a ma na a a v Ψ (0) v, (19) m v mna where Ψ (0) v is the lowest-order atomic state vector given in Eq. (9). In Eq. (19), the indices m and n range over all possible virtual states while indices a and b range over all occupied core states. The quantities ρ ma, ρ mv are singleexcitation coefficients for core and valence electrons and ρ mnab and ρ mnva are double-excitation coefficients for core and valence electrons, respectively. It should be noted that the operator products in Eq. (19) are normally ordered as they stand. To derive equations for the excitation coefficients, the state vector Ψ v is substituted into the many-body Schrödinger equation H Ψ v = E Ψ v, and terms on the left- and right-hand sides are matched, based on the number and type of operators they contain, leading to the following equations for the single and double valence excitation coefficients: 8 (ɛ v ɛ m + δe v )ρ mv = g mbvn ρ nb + g mbnr ρ nrvb g bcvn ρ mnbc, (20) bn bnr bcn (ɛ vb ɛ mn + δe v )ρ mnvb = g mnvb + g cdvb ρ mncd + g mnrs ρ rsvb cd rs [ + g mnrb ρ rv g cnvb ρ mc + ] g cnrb ρ mrvc + v b, (21) r c rc m n where δe v = E v ɛ v, the correlation correction to the energy of the state v, is given in terms of the excitation coefficients by δe v = ma g vavm ρ ma + g abvm ρ mvab + g vbmn ρ mnvb. (22) mab mna In Eq. (21), we use the abbreviation ɛ ij = ɛ i + ɛ j, and in Eq. (22), we use the notation ρ mnvb = ρ mnvb ρ nmvb. Equations for core excitation coefficients ρ ma and ρ mnab are obtained from the above equations by removing δe v from the left-hand side of the equations and replacing the valence index v by a core index a. The core correlation energy is given by δe c = 1 2 mnab g abmn ρ mnab. (23) After removing the dependence on magnetic quantum numbers, Eqs. (20) and (21) are solved iteratively. To this end, states a, b, m, and n are represented in a finite B-spline basis, identical to that used in the MBPT calculations discussed in Sec. 2. As a first step, equations for the core single- and doubleexcitation coefficients ρ ma and ρ mnab are solved iteratively; the core excitation coefficients are stored after the core correlation energy has converged to a specified accuracy.

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