The impact of correlations on electromagnetic strengths and polarizabilities

The impact of correlations on electromagnetic strengths and polarizabilities

M. Miorelli TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, V6T 2A3, Canada Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, V6T 1Z4, Canada    S. Bacca Institut für Kernphysik and PRISMA Cluster of Excellence, Johannes Gutenberg-Universität Mainz, 55128 Mainz, Germany TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, V6T 2A3, Canada Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada    G. Hagen Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA    T. Papenbrock Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
July 15, 2019
Abstract

We study electromagnetic sum rules, such as the bremsstrahlung sum rule and the polarizability sum rule at various approximation levels in coupled-cluster theory using interactions from chiral effective field theory. We go beyond singles-and doubles excitations by employing the CCSDT-1 approach which includes leading-order three-particle-three-hole (-) excitations for the ground state, excited states, and in the similarity transformed operator. To gauge the quality of our coupled-cluster approximations we perform several benchmarks in He with the effective interaction hyperspherical harmonics approach. We find that inclusion of - excitations in the ground state is sufficient to obtain an agreement with the hyperspherical harmonics approach to better than 1%. We employ the chiral interaction NLO [Ekström et al., Phys. Rev. C 91, 051301 (2015)] and calculate and in O and Ca. For Ca we also use the chiral interaction 1.8/2.0 (EM) [Hebeler et al., Phys. Rev. C 83, 031301 (2011)], which has been shown to give accurate description of binding energies and spectra from light to heavy nuclei. We find that the effect of - excitations in the ground state is small for 1.8/2.0 (EM) but non-negligible for NLO. This reconciles the recently reported discrepancy between coupled-cluster results based on this interaction and the experimentally determined from proton inelastic scattering in Ca [Birkhan et al., Phys. Rev. Lett. 118, 252501 (2017)]. For the computation of electromagnetic and polarizability sum rules, the inclusion of leading-order - excitations in the ground state is important, while less so for the excited states. Furthermore, we find that induced three- and higher-body terms coming from the similarity transformation of the dipole operator are found to be small, thus allowing for efficient and precise approximations.

pacs:
21.60.De, 24.10.Cn, 24.30.Cz, 25.20.-x
thanks: This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan. (http://energy.gov/downloads/doe-public-access-plan).

I Introduction

In the last decade, major advancements have been made in first principles approaches to nuclear structure Navrátil et al. (2009); Epelbaum et al. (2012); Leidemann and Orlandini (2013); Barrett et al. (2013); Hagen et al. (2014); Carlson et al. (2015); Hergert et al. (2016). Realistic descriptions of nuclei as heavy as Ni and Sn have recently been achieved based on state-of-the-art nucleon-nucleon () and three-nucleon forces (3NFs) from chiral effective field theory Hagen et al. (2016); Simonis et al. (2017); Morris et al. (2017). This advancement is based on the combination of first principles methods that scale polynomial with system size Mihaila and Heisenberg (2000); Dean and Hjorth-Jensen (2004); Dickhoff and Barbieri (2004); Hagen et al. (2008, 2010); Tsukiyama et al. (2011); Roth et al. (2012); Hergert et al. (2013); Somà et al. (2014); Binder et al. (2013); Lähde et al. (2014), an ever-increasing computational power following Moore’s law, insights from the renormalization group and effective field theory Bogner et al. (2003, 2007); Epelbaum et al. (2009); Machleidt and Entem (2011), and progress with nuclear forces Ekström et al. (2015); Carlsson et al. (2016).

Electromagnetic reactions are a crucial tool to investigate nuclear dynamics Bacca et al. (2002); Gazit et al. (2006); Bacca et al. (2007); Efros et al. (2007), see Ref. Bacca and Pastore (2014) for a recent review. Due to the perturbative nature of the process, one can clearly separate the role of the known electromagnetic probe from the less well known nuclear dynamics. Through a comparison of experimental data with theory one is then able to assess the precision and accuracy of the employed nuclear interactions and associated current operators. Photo reactions on heavier nuclei can be computed by the combination of the Lorentz integral transform Efros et al. (1994) and the coupled-cluster method Bacca et al. (2013, 2014); Hagen et al. (2016a), and by alternative means Calci et al. (2016); Lovato et al. (2016); Barbieri et al. (2017); Stumpf et al. (2017); Rocco and Barbieri (2018).

A key ingredient to study electromagnetic reactions, such as photo-dissociation or electron scattering, is the nuclear response function, defined as

(1)

Here is the transferred energy, while is the electromagnetic operator. It depends on the momentum-transfer of the considered probe. The nuclear response function is a dynamical observables and requires knowledge of the ground state and all excited states with corresponding energies and , respectively. As most of the excited states are in the continuum, the sum in Eq. (1) really becomes an integral, and this makes the direct computation of the response function a formidable task.

Instead, it is often easier to compute sum rules of such response functions, i.e. moments of the response intended as a distribution function and defined as

(2)

Here, is typically an integer. Employing the closure relation, we rewrite Eq. (2) as a ground-state expectation value

(3)

In practice, one often inserts completeness relations on the right of and on the left of , truncates the employed Hilbert spaces, and then increases the number of states until convergence is obtained.

While knowing the response function is equivalent to knowing all of its (existing) moments, already from a few moments one can gain useful insights into the dynamics of the nucleus. In this paper we will take this approach and focus on a few well known sum rules, namely the bremsstrahlung sum rule and the polarizability sum rule [see Eq. (6) below]. We calculate these observables within coupled-cluster theory using various approximation levels, and infer information on the nuclear dynamics from a comparison to other available theoretical computations and/or experimental data.

In the long wave-length approximation, i.e., in the limit of , the electric dipole operator reads

(4)

Here and are the coordinates of the -th nucleon and the center of mass of the nucleus, respectively, and is the third component of the isospin operator. As an example, the photo-disintegration cross section of a nucleus below the pion production threshold becomes

(5)

Here is the response function (1) of the translationally invariant dipole operator (4) for . Employing an interaction from chiral effective field theory Entem and Machleidt (2003) this reaction cross section was calculated in Refs. Bacca et al. (2013, 2014) using coupled-cluster theory with singles and doubles excitations (CCSD). While those calculations agreed with experimental data for He, O and Ca, they lacked a better understanding of uncertainties related to the underlying Hamiltonian and the applied many-body approach.

The computation of a cross section or related sum rule exhibits various theoretical uncertainties. Often, the largest uncertainty results from truncations of the employed chiral effective field theory at some given order in the power counting, while statistical uncertainties due to the optimization of the interaction are significantly smaller Ekström et al. (2015); Pérez et al. (2015); Carlsson et al. (2016). While a robust quantification of systematic uncertainties related to the employed chiral interactions is still lacking, a rough estimate can be made by employing a large set of different chiral interactions Hebeler et al. (2011); Ekström et al. (2015); Hagen et al. (2016a); Calci and Roth (2016); Hagen et al. (2016). Uncertainties from finite harmonic-oscillator basis can be estimated by varying the number of oscillator shells () and the oscillator frequency (). We found that this uncertainty is of the order of for calculations in nuclei with mass number up to  Hagen et al. (2016a). Another source of uncertainty in the coupled-cluster method comes from truncating the cluster operator at some low-order particle-hole excitation rank. Previous works employed the CCSD approximation, and neglected higher order excitations, such as - excitations. It is the purpose of this paper to investigate the role of leading-order - excitations in the ground state, excited states, and the similarity transformed operator in the calculation of the bremsstrahlung sum rule  Levinger and Bethe (1950); Brink (1957); Foldy (1957); Dellafiore and Lipparini (1982) and on the electric dipole polarizability sum rule  Friar (1975).

The polarizability is related to the inverse energy weighted sum rule as

(6)

Due to the inverse energy weight, this sum rule is more sensitive to the low-energy part of the excitation spectrum. Thus, it is interesting to study its sensitivity with respect to the coupled-cluster truncation level.

We remind the reader that previous calculations in the CCSD approximation showed good agreement with exact calculations in He, with a difference of about 1% Bacca et al. (2013, 2014); Miorelli et al. (2016). For Miorelli et al. (2016) and similarly for Ca Hagen et al. (2016a); Birkhan et al. (2017), a fairly good agreement with experimental data was obtained for specific choices of Hamiltonians including three-nucleon forces.

This paper is organized as follows. In Section II we introduce the formalism used to calculate the response functions, and the sum rules and . We also present the nomenclature used for the various approximation schemes that we implemented in this work. In Section III we present results for and in He, O and Ca, and compare with available experimental data. We also show a comparison of running sum rules and discretized responses, which allow us to monitor how excited states move as a function of energy for the various approximation schemes. Finally, in Section IV we draw our conclusions.

Ii Theoretical settings

Coupled-cluster theory Coester (1958); Coester and Kümmel (1960); Kümmel et al. (1978); Mihaila and Heisenberg (2000); Dean and Hjorth-Jensen (2004); Włoch et al. (2005); Hagen et al. (2010); Binder et al. (2014); Hagen et al. (2014) is based on the similarity transformed Hamiltonian,

(7)

Here is normal-ordered with respect to a single-reference state (usually the Hartree-Fock state), and is an expansion in particle-hole excitations with respect to this reference. The similarity transformation decouples all particle-hole excitations from the ground state, and the reference state becomes the exact ground state of . In practice, the operator is truncated at some low rank particle-hole excitation level. The similarity transformed Hamiltonian can be evaluated using the Baker-Campbell-Hausdorff expansion, and terminates exactly at quadruply nested commutators for a normal-ordered Hamiltonian containing at most up to two-body terms. The drawback from having an exactly terminating commutator expansion, is that the similarity transformed Hamiltonian is non-Hermitian and thus requires the computation of both the left and right eigenstates in order to evaluate expectation values and transitions. The left ground state is parameterized as

(8)

where is a sum of particle-hole de-excitation operators. The ground-state energy is given by the energy-functional

(9)

Here is the Hartree-Fock reference energy. The left and right ground states are normalized according to .

For excited states we employ the equation-of-motion (EOM) coupled-cluster method Stanton and Bartlett (1993) and calculate the right and left excited state of , i.e.,

(10)

Here and are linear expansions in particle-hole excitations with

(12)

The expression for is equivalent. Here, and are creation and annihilation operators, respectively. Indices run over unoccupied orbitals, while run over occupied orbitals. The left and right excited states are normalized according to

(13)

Electromagnetic transition strengths from the ground to an excited state are evaluated in the coupled-cluster method as

Here is the similarity transformed normal-ordered operator, which in this work is taken to be the electric dipole of Eq. (4). Because it is a one-body operator, the Baker-Campbell-Hausdorff expansion terminates at doubly nested commutators

(15)

Having defined the ground and excited states, we can rewrite the response function in the coupled-cluster formalism starting from Eq. (1) as

(16)

By integrating over the energy we obtain the bremsstrahlung sum rule

(17)

and using the closure relation we obtain the equivalent expression

(18)

In practice one solves for Eq. (18) by inserting a complete set on the Fock space defined by,

(19)

where label single, double, triple, and quadruple (and so on) excited reference states, corresponding to

(20)

Here, is a - state. Inserting the completeness of Eq. (19) into Eq. (18) one obtains

where the first term is identically zero for non-scalar operators , such as the electric dipole operator considered in this work. Calculating from Eq. (II) is significantly simpler than starting from Eq. (17) since no knowledge of the excited states of is required. As a proof of principle we verified that solving Eq. (II) is equivalent to calculating the response function and integrating in using Eq. (17).

The inverse-energy-weighted-polarizability sum rule of Eq. (6) can be calculated by utilizing the Lanczos continued fraction method Miorelli et al. (2016) as

(22)

Here is the Lorentz integral transform, and is a continued fraction of the Lanczos coefficients. To calculate the Lorentz integral transform Efros et al. (1994); Bacca et al. (2013), one needs to solve an EOM with a source term and the non-symmetric Lanczos algorithm is implemented by constructing the right and left normalized pivots as (see Ref. Miorelli et al. (2016) for details)

(23)
(24)

respectively.

So far we have not introduced any approximations in the coupled-cluster formulations for ground and excited states. The most commonly used approximation for the ground state is CCSD (i.e. , and ), which typically amounts for about of the full correlation energy in systems with well defined single-reference character Bartlett and Musiał (2007). In the following we will denote the CCSD approximation in short with D. We will also go beyond the CCSD level by including leading-order - excitations using the CCSDT-1 approach Lee et al. (1984). CCSDT-1 is a good approximation to the full CCSDT approach and accounts for about of the correlation energy. In brief, CCSDT-1 is an iterative approach that includes the leading-order contribution (here the index denotes that only connected terms contribute Shavitt and Bartlett (2009)) to the amplitudes with an energy denominator given by the Hartree-Fock single-particle energies, while all contributions to the and amplitudes are fully included. We will also solve for the corresponding left ground state in the CCSDT-1 following Ref. Watts and Bartlett (1995). To simplify the notation, in the following we will label the CCSDT-1 approximation with T-1. The corresponding approximations we will employ for excited states given in Eq. (II) includes up to - in the EOM-CCSD approach, and leading-order - excitations in the EOM-CCSDT-1 approach Watts and Bartlett (1995); Jansen et al. (2016). Because the calculation of and requires a particle-hole expansion of the ground state ( and ) and one for excited states [ and for or Eq. (II) for ], we need to label both of them appropriately. In this work we will investigate different approximation levels in both the ground and excited states. In order to keep the notation concise we therefore denote each scheme with a pair of labels (separated by a ‘’ symbol), with the largest order of correlation included in the ground state on the left, and the largest order of correlation included in the excited states on the right as shown in Table 1. In the previous work on dipole strengths and polarizabilities Bacca et al. (2013, 2014); Hagen et al. (2016a); Miorelli et al. (2016); Birkhan et al. (2017) both ground and excited states were approximated at the CCSD level, an approximation we label by D/D in this work.

ground state EOM label
D S D/S
D D D/D
T-1 S T-1/S
T-1 D T-1/D
T-1 T-1 T-1/T-1
Table 1: List of labels to denote the various coupled-cluster expansions used for the ground state (left of ‘’) and for the excited states (right of ‘’). The symbol (S for singles, D for doubles, T for triples and T-1 for linearized triples) represents the highest order of correlation considered (with all lower orders always fully included).

Examining the similarity transformed one-body operator of Eq. (15) reveals that for , the expansions of Eqs. (II) and (22) terminate at triply excited determinants (). If one includes the expansions terminate at quadruply excited determinants. Thus, the inclusion of - excitations in the ground and excited states requires the implementation of a number of new coupled-cluster diagrams. As usual, we checked such diagrams by comparing the -coupling and -coupling schemes (see, e.g., Ref. Miorelli (2017)).

Iii Results

The goal of this paper is to study and for He, O and Ca with the various approximation schemes outlined in Table 1. In all the results presented below He is calculated with the the chiral interaction at next-to-next-to-next-to-leading order (NLO) from Ref. Entem and Machleidt (2003), and O is calculated using the NLO interaction Ekström et al. (2015), respectively. The choice of interaction for He is motivated by the fact that we want to benchmark the various approximation schemes against virtually exact results from the effective hyperspherical harmonics approach Barnea et al. (2001) which cannot easily employ the NLO due to the non-locality of the 3NF. For O, the inclusion of 3NFs is necessary for a realistic description of the charge radius and observables correlated with it, such as and Miorelli et al. (2016); Hagen et al. (2016a); Birkhan et al. (2017). This makes the interaction NLO a good choice. For Ca, we will use two established interactions, namely NLO and 1.8/2.0 (EM) from Hebeler et al. (2011). The 1.8/2.0 (EM) interaction is constructed following Ref. Nogga et al. (2004). It starts from the chiral interaction at NLO Entem and Machleidt (2003) and “softens” it with the similarity renormalization group Bogner et al. (2007) at a cutoff/resolution scale . The 3NF is taken as the leading chiral 3NF using a non-local regulator and cutoff , with short-ranged coefficients and adjusted to nuclei. The “1.8/2.0 (EM)” interaction reproduces binding energies and spectra in medium-mass and heavy nuclei Hagen et al. (2016); Simonis et al. (2017); Morris et al. (2017).

iii.1 Model space truncation for - excitations

Going beyond the D/D approximation and including - excitations in the T-1/S, T-1/D, and T-1/T-1 approaches, the computational cost grows significantly both in terms of number of computational cycles and memory associated with storage of the amplitudes. The computational cost associated with the most computational expensive term in the D/D approximation is given by , while in the T-1/S, T-1/D, and T-1/T-1 approaches it is . Here is the number of occupied orbitals in the reference state and is the number of unoccupied orbitals. Clearly, the computational load grows rapidly with the mass of the nucleus () and the model-space size (). In order to overcome this computational hurdle in the T-1/S, T-1/D, and T-1/T-1 approaches, we introduce an energy cut on the allowed - excitations. The truncated space that we employ for the - excitations are thus given by and with being the harmonic-oscillator shell and the harmonic-oscillator shell at the Fermi surface. In Fig. 1 we show the convergence with respect to for in O (a) and Ca (b) calculated within the T-1/D scheme. One observes that, e.g., by truncating to 14, calculations are converged at the 1-level for O and at the 0.5-level for Ca. Unless stated otherwise, in the remainder of this work we will present results that are converged with respect to .

Figure 1: (Color online) Convergence of for O (a) and Ca (b) calculated in the T-1/D scheme as a function of the number of configurations for the . The adopted model space is . Each point corresponds to a jump of one unit (two units) in for ) (Ca) and selected values of are highlighted along the curves.

iii.2 The similarity transformed transition operator

In the T-1/S, T-1/D, and T-1/T-1 approaches, contributions are included in the one- and two-body parts of the similarity transformed Hamiltonian, while three-body parts from are only included via (here is the normal ordered one-body Fock matrix). By treating the similarity transformation of a normal-ordered one-body operator consistently with the similarity-transformed Hamiltonian, one has

(25)
(26)
(27)

where the index again denotes connected diagrams Shavitt and Bartlett (2009) and is the similarity-transformed operator in the D approximation. Due to the hierarchy among correlations, one can expect that the terms in Eq. (25) that contain are sub-leading with respect to the term. These terms are also computationally much more demanding since they involve calculating and storing configurations (see Ref. Miorelli (2017) for full expressions). Thus, it is convenient to explore their relevance with respect to using Eq. (26), or even just using Eq. (27), where the operator is similarity transformed as in the D approximation.

In this paper we compute observables by including - excitations in the ground and excited states as well as in the similarity transformed operator, and benchmark the various approximations for in He and O, as shown in Table 2. We see that for both and , the additional terms in Eqs. (25) and (26) have a negligible contribution with respect to Eq. (27), amounting to a sub-percent effect of about 0.2 and 0.7%, respectively. This finding is important in the light of performing computations of heavy nuclei, where calculations with Eq. (27) are more tractable, while using Eq. (25) would be substantially more computationally demanding. Consequently, when calculating Ca, we will use Eq. (27).

  He    O scheme
Table 2: Effect of contributions of - correlations in a T-1/T-1 computation, when is truncated as in Eqs. (25) or (26), with respect to the full expression of Eq. (27). Both and are computed with (a) MeV and and (b) MeV, and .

Note that when using Eq. (27) in the calculation of in the T-1/T-1 approximation, - excitations enter only in the ground state [see Eq. (18)], and therefore this corresponds to the T-1/D approximation. On the contrary, triples would enter both in the ground and excited states in a calculation of , for which T-1/T-1 and T-1/D are different.

iii.3 Sum rules in various approximation schemes

We now explore the convergence in terms of the model space size for two approximation schemes that includes - excitations, namely T-1/T-1 and T-1/D, and compare it to the D/D approximation.

Fig. 2 shows the convergence of and in He with respect to for  MeV. Note that for He it was not necessary to employ any energy cut on the allowed - excitations. Calculations in the T-1/T-1 and T-1/D scheme were performed with Eq. (26) and Eq. (27), respectively. The convergence with respect to is of similar quality both for (a) and (b).

Figure 2: (Color online) Convergence of (a) and (b) for He with respect to the model-space size for MeV. Two triples approximations schemes T-1/T-1 (green diamonds) and T-1/D (red circles) are compared to the D/D case (blue squares). The similarity transformed operator is implemented with Eq. (26) and (27), in T-1/T-1 and T-1/D, respectively.

We see that for , the results obtained within the T-1/D approximation are close to those obtained in T-1/T-1 approximation. The slight difference stems from the fact that the T-1/T-1 calculations are performed with the similarity transformed operator given in Eq. (26), while T-1/D results are obtained with Eq. (27). We implement these two different equations to graphically show that our findings presented in Table 2 are consistent in various model spaces and approximations. For we observe a slightly larger difference between the T-1/T-1 and T-1/D approximations, due to the fact that - excitations enter in the calculations of excited states as well. Overall, the effects of - excitations in He are small, amounting to about 1.5.

Figure 3: (Color online) Convergence of (a) and (b) for O with respect to the model-space size for MeV and . Two triples approximations schemes T-1/T-1 (green diamonds) and T-1/D (red circles) are compared to the D/D case (blue squares). The similarity transformed operator is implemented with Eq. (26) and (27), in T-1/T-1 and T-1/D, respectively.

In Fig. 3 we show convergence plots for O, adopting  MeV and . The situation is similar as for He, with the convergence rate being qualitatively similar in the T-1/T-1 and T-1/D approaches as in the D/D approach. At the largest model space size the residual dependence amounts to about 1.5. Also in this case, for , the T-1/T-1 and T-1/D calculations almost coincide, while some difference is observed in . While in O the effect of triples is slightly larger than in He, the overall effect of - excitations is small and amounts to 4 and 6 for and , respectively. Both for and the inclusion of - excitations in the T-1/T-1 and T-1/D approaches reduce their magnitude as compared to the results obtained in the D/D approach.

Figure 4: (Color online) Comparison of (a) and (b) in the D/D (blue/left), the T-1/D (red/central) and the T-1/T-1 (green/right) approximations against hyperspherical harmonics results (black line) in He.

It is now interesting to compare the various coupled-cluster results with respect to the hyperspherical harmonics benchmark values Barnea et al. (2001); Bacca et al. (2013) for He. Figure 4 shows (a) and (b) obtained for various approximations in coupled-cluster theory: D/D (blue/left), the T-1/D (red/central) and the T-1/T-1 (green/right). The widths of the bands reflect the residual dependence for the largest model space . The black line is the virtually exact calculation from hyperspherical harmonics expansions using the same interaction. The D/D calculations already get close to the hyperspherical harmonics result, and the addition of - correlations in both the T-1/D and T-1/T-1 approaches further improves the agreement for . For the T-1/D calculation agrees better with the hyperspherical harmonics result than the T-1/T-1 approach. The overall effect of - excitations is small. This benchmark with hyperspherical harmonics suggests that the T-1/D scheme is to be preferred for electromagnetic and polarizability sum rules, and that the inclusion of - correlations in the ground state plays a more significant role than the corresponding one in the excited states.

Figure 5: (Color online) Comparison of (a) and (b) in the D/D (blue/left), the T-1/D (red/central) and the T-1/T-1 (green/right) approximations against the integrated experimental data by Ahrens et al. (1975) in O.

In Fig. 5 we compare (a) and (b) for O obtained in the D/D (blue/right) scheme with the T-1/D (red/central) and the T-1/T-1 (green/right) approximations. The D/D value is obtained at and MeV. Due to the large number of - configurations, for O we are able to calculate only up to a maximum model space of and for T-1/T-1. For consistent results we adopt the same truncation for in T-1/D. The bands in Fig. 5 are obtained by assigning a uncertainty, accounting for the combined uncertainty from the cut and the residual -dependence.

Correlations arising from - excitations reduce the size of these observables by a few percent, with effects being slightly larger on than for . Similar to the He case, we find that results for obtained in the T-1/D and T-1/T-1 approaches almost coincide, while for they slightly differ. This is expected, because is calculated as a ground-state expectation value, while also requires the solution of the excited states from Eq. (II). We use the difference between the T-1/D and T-1/T-1 results as an estimate of neglected higher order correlations. This amounts to a 4 effect for and an even smaller effect of 0.4 for .

In Fig. 5 the theoretical results are also compared with the experimental value obtained by integrating the data from Ahrens et al. (1975) (grey bands). We see that the addition of triples leads to a deviation of and with respect to the experimental data, which agreed better in the D/D approximation using the NLO interaction. We note that NLO was constrained to reproduce the charge radius of O using coupled-cluster theory in the D approximation. As shown in Refs. Hagen et al. (2016b); Miorelli et al. (2016), the charge radius is correlated with , and it would therefore be interesting to quantify the effect of - excitations in the T-1 approach on charge radii as well. We also note that the extraction of these sum rules from photo-absorption data may be prone to larger systematic uncertainties than those quoted because it is not possible to estimate the role of multipoles beyond the dipole.

Figure 6: (Color online) Comparison of (a) and (b) in the D/D (blue/left) and the T-1/D (red/right) approximations in Ca. Results for the NLO (leftmost bands) and (EM) (rightmost bands) interactions are shown. For the polarizability the results are compared against experimental data by Birkhan et al. (2017).

We now turn our attention to Ca. In the recent experiment using proton inelastic scattering off a Ca target, the electric dipole strength was disentangled from other multipoles contributions resulting in  fm Birkhan et al. (2017). Calculations based on interactions that reproduced the charge radius yielded results for in the D/D approximation that were somewhat larger than the measured value. Figure 6 shows results obtained in the D/D and T-1/D approaches using the NLO (leftmost bands) and (EM) (rightmost bands) interactions, for and in the top and bottom plots, respectively. For NLO we find that the addition of - excitations in the ground state using the T-1 approach improves the agreement with the data. As calculations of excited states in the T-1 approach are computationally demanding and currently not feasible at sufficiently large cut, we did not employ the T-1/T-1 approach for Ca. However, based on our studies for He and O one may expect that the T-1/D and T-1/T-1 approaches would yield similar results. We observe that both for and the T-1 approximation for the ground state leads to a reduction of the strength, bringing theory in agreement with the recent data from Birkhan et al. (2017), and reducing the Hamiltonian model dependence. The effect of the triples on amounts to 15 for the NLO and 6 for the (EM) interaction, which is consistent with the latter being a much “softer” interaction. The triples effect on is smaller, amounting to about 5 for NLO and 2 for the (EM) interaction.

iii.4 Running polarizability sum rule

Figure 7: (Color online) Discretized response with MeV (a) and running sum of (b) for O with various coupled-cluster approximations compared with experimental results from Ahrens et al. (1975).
Figure 8: (Color online) Discretized response with MeV (a) and running sum of (b) for Ca with various coupled-cluster approximations compared with experimental results from Birkhan et al. (2017)

At this point it is also interesting to study the running of the sum rules as a function of the maximum integration limit. If one solves for the excited states in Eq. (II) using the Lanczos technique, it is possible to define the sum rule from the integral of the dipole response function as

(28)

and study its running as a function of . A discretized response function can also be obtained by a calculation of the Lorentz integral transform  Efros et al. (1994); Bacca et al. (2013) for a very small width parameter as (see Ref. Miorelli et al. (2016) for details)

(29)

The discretized response consists of smeared peaks and does not properly take the continuum into account. However, it will allow us to see how excited states and their corresponding strengths change within various approximation schemes, thus affecting the running sums.

The top of Figures 7 and 8 show the Lorentz integral transform calculated for MeV using various coupled-cluster approximations for O and Ca, respectively. For both figures, the bottom plots show the running of the polarizability sum rule as a function of . The various approximation schemes are shown in comparison with experimental data (gray bands). Besides the D/D calculation, we present three different schemes with increasing correlation order, namely D/S, T-1/D and T-1/T-1. For O, the D/S scheme coincides with the T-1/D calculation, and is also very close to the most expensive T-1/T-1 calculation, deviating from the D/D approximation by a few per cent.

Method    [fm
Ref. Barbieri et al. (2017) 0.50
D/S 0.503
T-1/D 0.508
T-1/T-1 0.528
Table 3: Comparison of coupled-cluster results with self-consistent Green’s function data from Ref. Barbieri et al. (2017) in O using the NLO interaction.

In Table 3 we compare our results for with those of Barbieri et al. (2017). We see that the results obtained with the D/S approach agree with calculations from the self-consistent Green’s functions method that used a random phase approximation approach for the excited states. We find that the inclusion of - correlations in the ground state and in the excited states decreases the polarizability with respect to a D/D calculation, making this more sophisticated calculation to coincidentally agree with simpler schemes, such as the D/S approximation.

For Ca we see that for the D/S approximation the strength is dominated by a single state at around 19 MeV of excitation energy, while higher-order correlations included in D/D and T-1/D shift and fragment the strength significantly. The inclusion of - excitations via the T-1 approach in the ground state, makes the strength more fragmented and also increases some of the strength at higher energy, which leads to a reduction of due to the inverse energy weight in the polarizability sum rule. Coincidentally, in this case again the computationally least expensive D/S approximation agrees with D/D and not with T-1/D.

Iv Conclusions

We studied the role of correlations in coupled-cluster theory on electromagnetic sum rules such as the total dipole strength and the electric dipole polarizability. Leading order - excitations were implemented in the CCSDT-1 approximation for the ground state, excited states and similarity transformed dipole operator. The effect of - excitations on the latter was found to be negligible. This finding is important as it allows for precise and efficient approximations in heavy nuclei. We showed that the inclusion of - excitations in the ground state via CCSDT-1 improves the agreement with the hyperspherical harmonics approach for He. The effect of triples is quite small in He, and becomes larger in O and Ca. In O we find the effect of triples excitations to be at the order of 4-6%, and reduces the size of and . In the case of Ca, the effect of triples excitations is of the order of 15% for the NLO interaction, and reconciles the recently reported disagreement between coupled-cluster calculations and the experimental result for obtained from inelastic proton scattering. Finally, we observed that some simpler approximations may occasionally coincide with more sophisticated computations using the CCSDT-1 method both for the ground and excited states. We conclude that the effect of - excitations in the ground state is more important than their effect in excited states for the electromagnetic sum rules studied in this work. The inclusion of triples excitations in the excited states, while possible for light nuclei such as He and O, will require further developments in order to overcome the hurdles associated with the increase in computational cost for heavier nuclei.

Acknowledgements.
This work was supported in parts by the Natural Sciences and Engineering Research Council (NSERC), the National Research Council of Canada, by the Deutsche Forschungsgemeinschaft DFG through the Collaborative Research Center [The Low-Energy Frontier of the Standard Model (SFB 1044)], and through the Cluster of Excellence [Precision Physics, Fundamental Interactions and Structure of Matter (PRISMA)], by the Office of Nuclear Physics, U.S. Department of Energy under Grants Nos. DE-FG02-96ER40963 (University of Tennessee) DE-SC0008499 (SciDAC-3 NUCLEI), DE-SC0018223 (SciDAC-4 NUCLEI), and the Field Work Proposals ERKBP57 and ERKBP72 at Oak Ridge National Laboratory. Computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Oak Ridge Leadership Computing Facility located in the Oak Ridge National Laboratory, supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725, and computational resources of the National Center for Computational Sciences, the National Institute for Computational Sciences, and TRIUMF.

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