Normal-ordering approximations and translational (non) invariance

Normal-ordering provides an approach to approximate three-body forces as effective two-body operators and it is therefore an important tool in many-body calculations with realistic nuclear interactions. The corresponding neglect of certain three-body terms in the normal-ordered Hamiltonian is known to influence translational invariance, although the magnitude of this effect has not yet been systematically quantified. In this work we study in particular the normal-ordering two-body approximation applied to a single harmonic-oscillator reference state. We explicate the breaking of translational invariance and demonstrate the magnitude of the approximation error as a function of model space parameters for $^4\rm{He}$ and $^{16}\rm{O}$ by performing full no-core shell-model calculations with and without three-nucleon forces. We combine two different diagnostics to better monitor the breaking of translational invariance. While the center-of-mass effect is shown to become potentially very large for $^4\rm{He}$, it is also shown to be much smaller for $^{16}\rm{O}$ although full convergence is not reached. These tools can be easily implemented in studies using other many-body frameworks and bases.


I. INTRODUCTION
The need for an effective three-nucleon force (3NF) to describe the strong nuclear interaction in atomic nuclei is well established [1]. Its origin dates back to Fujita and Miyazawa, who computed the 3NF arising from a twopion exchange diagram [2]. In the modern understanding, 3NFs arise in effective field theories (EFTs) as a consequence of integrating out degrees of freedom. More specifically, 3NFs appear in the chiral EFT (χEFT) of the strong nuclear interaction at next-to-next-to-leading order (N2LO) and above in standard power counting schemes of the chiral expansion [3][4][5][6]. Besides the EFT arguments, it has also been shown that several experimental findings are difficult to reproduce without the inclusion of a 3NF, such as certain three-nucleon scattering observables [7], the A = 3, 4 binding energies [8,9], and selected light nucleus spectroscopy [10][11][12].
Unfortunately, the full inclusion of 3NFs in quantum many-body methods is computationally demanding due to the large increase in the number of non-zero matrix elements (NZME) [13]. In fact, the escalation of memory demands and the increase in execution time often render solutions intractable when explicit 3NFs are added. This situation has initiated searches for approximation schemes that will include the most important physics of 3NFs, but at a lower computational cost.
One such approximation scheme is the singlereference normal-ordering two-body (SR-NO2B) approximation [14,15], which potentially can incorporate the dominant piece of the 3NF as an effective two-nucleon force (2NF) and therefore at significantly reduced computational cost. This is often done by approximating the ground state with a single Slater determinant (SD) and then normal ordering the 3NF relative to this reference state using Wick's theorem [16]. The expectation value of the residual three-nucleon term, acting only outside the references state, is assumed to give a much smaller contribution to the ground-state energy than the induced two-, one-and zero-body parts-and is therefore discarded.
The SR-NO2B approximation has been used with great success in ab initio nuclear structure calculationsin particular to facilitate studies of medium-mass systems [15,[17][18][19][20]. Normal-ordering approximations beyond the single reference state have also been developed [21,22]. The accuracy of the SR-NO2B has been benchmarked, e.g., in Refs. [14,15,23]. The importance of residual 3NFs was shown to be small by explicit comparison with calculations using full 3NFs. However, these benchmarks were performed at a fixed oscillator frequency and the dependence on model-space parameters has not been investigated. This is particularly important since we show that the sensitivity of SR-NO2B to the choice of basis frequency could be significant. We stress that our concern pertains to the explicit breaking of translational invariance of the underlying Hamiltonian due the normal-ordering two-body (NO2B) approximation. The ensuing center-of-mass (CM) dependence is therefore of different origin compared to the well-known problem of CM mixing as a consequence of e.g. particular truncations of the single-particle basis that are used in some many-body solvers [20,[24][25][26][27][28][29].
In this work, we have studied the SR-NO2B approximation in a harmonic oscillator (HO) basis with a no-core shell model (NCSM) total-energy truncation. We consider the closed-shell systems 4 He and 16 O-for which the single-reference approximation is appropriate-and we explore the accuracy of the NO2B approximation and the breaking of translational invariance as a function of model space parameters N max and Ω. The realistic N2LO sat interaction [30] with both 2NF and 3NF terms is used for all numerical calculations unless otherwise stated.
The full Hamiltonian, the NCSM method, and the SR-NO2B approximation is introduced in Sec. II. The CM problem is presented in Sec. III where we also introduce and benchmark the metrics that will be used in the analysis. The NO2B approximation errors for ground-state en-ergies and radii for 4 He and 16 O are analyzed in Sec. IV, while concluding remarks are given in Sec. V.

A. The Hamiltonian
The general Hamiltonian that is considered in this work can be written The potential operators are expressed in second quantized form aŝ and with Greek letters representing tuples of the well-known single-particle quantum numbers (n, l, j, j z , t z ) in a HO basis. These operators, as well as the intrinsic kinetic energy,T int , depend on relative coordinates (in position and momentum space) such that the Hamiltonian (1) is translationally invariant.

B. The no-core shell model
To solve the many-body Schrödinger equation we employ the NCSM in which the Schrödinger equation is rewritten as a finite matrix eigenvalue problem by expanding the eigenstates of the HamiltonianĤ in a finite many-body basis {|Φ i } D i=1 , i.e., the NCSM eigenstate n is The SD many-body basis state |Φ i is constructed using second-quantization and is an eigenstate of a two-component A-body (A = N + Z) fermionic HO Hamiltonian with the corresponding eigenenergy where Ω is the oscillator frequency and N i is the total HO energy quantum number where n j (l j ) is the principle quantum number (orbital angular momentum) of particle j in the basis state Φ i . The dimension D of the NCSM basis is set by a total HO-energy truncation In general, there is no guarantee that the separation of intrinsic and CM excitations due to the translational invariance of the Hamiltonian is preserved when the Hilbert space is arbitrarily truncated. However, an important feature of the total-energy truncation of the NCSM basis is that it does in fact guarantee this separation due to the energy-conserving property of the HO transformation brackets [11]. This property implies that there exists a unitary mapping of a SD basis of HO singleparticle states-truncated with respect to the HO excitation energy N max Ω-onto a Jacobi-coordinate basis. Therefore, the NCSM eigenstates of (1) can formally be written as product states with the state number n = n(i, j). The lowest energy state (n = 0) will be the product of the ground state of the CM motion (j = 0) and that of the intrinsic Hamiltonian (i = 0).

C. Single-reference normal ordering
In this section we outline the major steps of singlereference normal ordering and describe the NO2B approximation. Starting from the general expression of the vacuum normal-ordered 3NFs in equation (3) and a reference state, that is a single SD -the lowest HO states for the A-body system composed of Z protons and N = A − Z neutrons.
V 3NF can then be normal-ordered relative to |ψ ref , which results in an expansion of zero-, one-, two-and three-body operators. The contribution to the groundstate energy of the residual three-nucleon operator is assumed to be small-since it acts solely outside the reference state-and is discarded. This is known as the NO2B approximation as it results in an effective Hamiltonian with at most two-body operators.
The normal-ordering relative to |ψ ref is easiest performed with Wick's theorem [31]. A product of secondquantization operators, normal-ordered relative to |ψ ref , is here written as âbĉ · · · . Such a normal-ordered operator fulfills âbĉ · · · |ψ ref = 0. Combined with the formal definition of a contraction,âb =âb − âb , it is possible to derive the contraction ruleŝ The 3NF in equation (3) can now be normal ordered relative to |ψ ref by applying Wick's theorem, where we note that W 0 is a constant whileŴ i |ψ ref = 0 for i ∈ {1, 2, 3} due to the normal-ordered secondquantization operators. The NO2B approximation of V 3NF is then defined aŝ In the NCSM, however, the Hamiltonian is not expressed relative to a reference state and we need to apply Wick's theorem backwards to transformV NO2B 3NF into vacuum normal-ordered form. With this aim, we use the following relations and arrive at In the end, the NO2B-approximated Hamiltonian that we use in the NCSM iŝ This is clearly different compared to the full Hamiltonian, H, given by Eq. (1) where the complete 3NF is retained.

III. THE CENTER-OF-MASS PROBLEM
The translational symmetry of the NO2Bapproximated Hamiltonian (20) is explicitly broken since we neglect the residual 3NF and this renders the CM dependence of the reference state manifest. Indeed, retaining the residual 3NF restores translational symmetry since the normal ordering in Eq. 15 is an exact relation. With the NO2B approximation it is therefore no longer guaranteed that the ground state Ψ NO2B gs NCSM ofĤ NO2B is factorized into a product of CM and intrinsic states as in equation (9). Instead we must expect a linear superposition of product states, In this more general situation the intrinsic and CM states are no longer pure quantum states and must be expressed with density matricesρ int andρ CM . This mixing of CM and intrinsic degrees of freedom can potentially have a huge effect on various observables and is here labeled as the center-of-mass problem. It is therefore crucial to quantify the CM mixing. In the following we will introduce two metrics that have exactly this purpose.

A. Introducing center-of-mass metrics
The mixing of CM and intrinsic states is a known problem in many-body physics. It might occur also when using fully translational-invariant Hamiltonians as a consequence of approximations used in the many-body solver. In particular, Galilean invariance is broken explicitly when employing lattice methods [32] and CM mixing can occur in basis-expansion methods when imposing a basis truncation at the single-particle level [20,[24][25][26][27][28][29]. A very common approach to diagnose the problem in basis expansion methods is to evaluate the (energy-shifted) HO CM Hamiltonian or the corresponding CM number operator with expectation value N CM (ω). Small expectation values of these operators,Ĥ CM (Ω) and N CM (Ω), evaluated at the basis frequency Ω, are then used as evidence for satisfactory CM factorization. Large expectation values, on the other hand, indicate problematic mixing. However, it might be too assertive to claim proper CM separation based on this single observable. We argue here that additional metrics are needed. In addition, there are claims [25] that the factorization does occur but that the CM state is not necessarily a ground state of the HamiltonianĤ CM (Ω) (22) constructed using the basis frequency Ω.

The ξCM metric
Consider an eigenstate of a translationally invariant Hamiltonian that factorizes into a product of an intrinsic state |Ψ int and a CM state Ψ ω ξ gs CM , where the latter corresponds to the ground state ofĤ CM (ω ξ )-the Hamiltonian (22) constructed with an oscillator frequency ω ξ which does not necessarily correspond to the basis frequency Ω. In this situation we would obtain the expectation values and with the oscillator length b = b(ω ξ ) = /Amω ξ , and the expectation values are with respect to the full ground state. This fact was utilized by Parzuchowski et al. [27] in their study of transition operators within the in-medium SRG framework. They introduced the quantity which will evaluate to ξ CM = 0 if Ψ ω ξ gs CM is a HO ground state, regardless of the frequency ω ξ , while ξ CM > 0 if it is not. Note, however, that a HO eigenstate with one frequency, ω, cannot be exactly represented in a truncated HO basis with a different basis frequency Ω = ω. This is illustrated in Fig. 1 and further discussed in Appendix A. In the case when ξ CM ≈ 0 it is possible to identify the corresponding frequency of the underlying HO Hamilto-nianĤ CM (ω ξ ) by whereT CM is the CM kinetic energy, and the expectation value is with respect to the ground state. When the NCSM Hamiltonian is translationally invariant the frequency ω ξ will equal the basis frequency in the NCSM method. However, ω ξ = Ω indicates a broken symmetry. Note that we might still have a product state (9) in this situation-such that CM mixing is not problematic-and that we can measure this with ξ CM .

The NCM metric
In applications of the coupled-cluster (CC) method the computed ground state |Ψ gs CC is assumed to be separable such that the CM state is the ground state of a HO Hamiltonian with a frequency ω N that not necessarily equals the underlying HO basis frequency Ω. The frequency ω N is obtained by evaluating [25] and identifying We can then define the operatorN CM (ω N ) analogous to Eq. (23) and evaluate its expectation value N CM (ω N ) which will be small if the wave function factorizes.

B. The relation between different metrics
There is an interesting connection between ξ CM and N CM (ω ξ ), i.e. the two different metrics expressed in terms of the same CM oscillator frequency ω ξ . We pick ω ξ since the two frequencies ω ξ and ω N are equal in the limit N max → ∞ if the state is separable as in equation (9). Consider a factorized state with i.e., it is a HO state with frequency ω ξ , radial quantum number N and orbital angular momentum L. Then we have and Using the definitions of the two metrics we find that they become equal in this scenario Now consider the possibility that |Ψ CM is a linear superposition of HO states |φ NiLi = a † NiLi | . To simplify further calculations we introduce and where A, B are real and A ≥ 3/2. Then we find and which gives It is clear from equations (38) and (41) that where we have assumed that ε ≡ B 2 /A 2 1. Therefore, the difference between these two metrics can be used as a measure of how much of the CM state is in higher It turns out that a similar relation can be derived if we have CM mixing such that the CM state is not a pure quantum state. Then we find that ξ CM = N CM if there does not exist any HO basis in which the CM density matrix is diagonal. In this situation the coefficients A and B in Eqs. (38) and (41) are given by In conclusion, when finding that ξ CM , N CM > 0 we cannot know if the CM state is a pure quantum state or a mixed one. Only the situation ξ CM = N CM = 0 assures a proper separation of the intrinsic and CM parts of the eigenstate as in Eq. (9).

C. Benchmark of center-of-mass metrics
To benchmark the CM analysis metrics we consider an interacting many-body system in an external HO trap with Hamiltonian where we use the 2NF part of N2LO sat as a realistic interactionV 2NF . Then we compute the NCSM ground state of 4 He for different basis frequencies Ω ∈ {8, 12, . . . , 32, 36} MeV while keeping the trapping potential frequency fixed at ω trap = 20 MeV. The metrics described in Sec. III A are then evaluated for the ground state. While this Hamiltonian depends on the CM coordinate-such that translational invariance is explicitly broken-it is still block-diagonal in a CM-part and an intrinsic part. Therefore, it is possible to precisely control the CM part of the ground state and this property makes it a suitable benchmark of the CM metrics. However, it is not equivalent to the non block-diagonal CM coupling of the NO2B-approximated Hamiltonian.
We compute the expectation values R 2 CM , P 2 CM and Ĥ CM (ω) for each NCSM model space (N max , Ω). This allows us to extract the optimal decoupling frequencies ω ξ and ω N and to test the decoupling by evaluating the metrics ξ CM and N CM (ω N ). In addition, the standard CMdecoupling metric N CM (Ω) can be evaluated-although it is expected to fail when ω trap = Ω. All of these quantities are plotted in Fig. 2.
We numerically confirm that all three metrics, shown in the top row of Fig. 2, become equal to zero when the basis frequency Ω is equal to the trap frequency. However, while N CM (Ω) in panel (c) fails to reveal the decoupling for other basis frequencies, the two metrics N CM (ω N ) and ξ CM , shown in panels (a) and (b), respectively, do indicate decoupling by exhibiting small values.
The fact that both N CM (ω N ) and ξ CM are larger for small basis frequencies indicates that a superposition of excited HO states is needed to describe the CM ground state in this truncated space (see Appendix A). However, for Ω > ω trap the metrics are very small already at modest N max indicating a good CM state representation with these basis frequencies.
The corresponding optimal frequencies, ω N and ω ξ shown in panels (d) and (e), do approach the trap frequency as N max increases. For N max = 0, where there is a single SD basis state, this analysis will always return the basis frequency as the optimal one, as shown by the diagonal, straight line.
Finally, the differences between the optimal frequencies found via the ξ CM and N CM methods are shown in panel (f), while the difference between the two metrics are displayed in panels (h) and (i). As a general conclusion we find that the two analysis methods provide basically identical results, but that the ξ CM metric is easier to implement and compute. Furthermore, the problem of representing a HO state of another frequency than that of the truncated HO basis hampers the analysis at small basis frequencies (see Appendix A).

IV. NO2B RESULTS
In this section we present a numerical study of the SR-NO2B approximation in the HO SD basis applied to the doubly-magic systems 4 He and 16 O. All results shown here are obtained with the realistic nuclear interaction model N2LO sat [30]. Throughout this study we will compare results obtained with full and with NO2Bapproximated 3NFs.
We employ the Jacobi-coordinate version of the NCSM [33,34] to compute the ground state of 4 He with full 3NF. The normal ordering is performed in the Mscheme SD basis and we employ the NCSM code pAntoine [35,36] to perform the diagonalization. Unfortunately, the huge number of 3NF matrix elements in the M -scheme SD basis limits our studies to model spaces N max ≤ 10. Specifically, with N max = 10 we have 5.4·10 9 elements while N max = 12 would require 66.5 · 10 9 .
For 16 O we are limited by the size and the number of non-zero elements of the Hamiltonian matrix. With NO2B-approximated interactions we use pAntoine and reach model spaces N max ≤ 8 with up to D = 6·10 8 basis states. With full 3NFs we use the NCSD code [37] and are able to reach model spaces N max ≤ 6 corresponding to D = 1.6 · 10 6 .
The direct comparison between results obtained with full and NO2B-truncated 3NFs allows us to focus on the size of the approximation error as a function of the mass number and model space parameters. The origin of the approximation error will here be analyzed in terms of possible CM mixing. In this context it is important to point out that all calculations in the M -scheme SD basis are performed without a Lawson projection term acting on the CM coordinates. Instead, we will employ the metrics presented in Sec. III A as diagnostic tools.

A. Ground state energy of 4 He
We first compute the ground-state energy of 4 He at the fixed basis frequency Ω = 20 MeV, which is close to the position of the variational minimum for this system with the N2LO sat interaction. Results are shown in Fig. 3 as a function of increasing NCSM truncation N max and compared to the converged result E gs = −28.43 MeV [30].
At this basis frequency, we find that the NO2B approximation captures the N max behavior of the results obtained with the full Hamiltonian to within 1%. This means that the approximation error in the total binding energy is smaller than 250 keV. We can also observe the importance of the 3NF since a full removal of this part of the Hamiltonian (green dashed line in Fig. 3) leads to approximately 2 MeV underbinding.
However, the magnitude of the NO2B-approximation error turns out to be highly sensitive to the choice of basis frequency. This finding is highlighted in Fig. 4 where the binding energy per nucleon is computed for Ω ∈ {8, 12, . . . , 32, 36} MeV. The solid lines in the upper panel correspond to E gs /A computed with the full Hamiltonian, while the dashed ones correspond to the NO2B-approximated 3NF. The difference between these two results is shown in the lower panel as a function of the basis frequency.
There seem to be an optimal frequency Ω ≈ 20 MeV for which the approximation error is very small as we transition from under-to overbinding with the NO2B truncation. For higher frequencies there is an increasing difference between the NO2B and the full-3NF results. Note also that the NO2B truncation at N max = 0 is identical to the full Hamiltonian as the single reference state is the only basis state.
The main hypothesis of this paper is that the explicitly broken translational symmetry of the NO2B Hamiltonian can become the origin of a strong Ω-dependence of the approximation error. Consequently, the NCSM eigenstates might not necessarily separate into a product of CM and intrinsic states.
To test this hypothesis we evaluate the CM metrics, ξ CM and N CM -defined in Sec. III A-and the corresponding CM oscillator-state frequencies ω ξ and ω N . These results are shown in Fig. 5. We observe that ω ξ Ω for large basis frequencies. However, the ξ CM metric clearly indicates that there is no CM separation in this scenario so the value of ω ξ does not really have any significance.
In contrast, for small basis frequencies we have a clear factorization of the eigenstate, as indicated by both metrics, and we also find that the extracted frequencies are very similar and very close to the basis frequency. There is a transition region around Ω ≈ 20 MeV where the metrics indicate CM separation at a frequency that is slightly larger than the basis one.
The finding that CM mixing is less of a concern for small basis frequencies also indicates that the NO2Bapproximation error of 500 keV/A in this region is due to the neglected, residual 3NF. As shown in the lower panels of Fig. 5  indicating a separation between the CM and intrinsic parts of 4 He ground state. However, as the basis frequency increases beyond 20 MeV both measures increase drastically, suggesting that there is no longer any separation. The frequencies ω N and ω ξ start to differ visibly from the basis frequency already at 16 MeV which is below the observed optimal frequency. This indicates that the NO2B approximation does affect the CM state, albeit very weakly.

B. Ground state energy of 16 O
We will now study the NO2B approximation when performing NCSM calculations of the 16 O nucleus. For this system we are limited to N max ≤ 8 for Hamiltonians including only 2NFs, and N max ≤ 6 when using the Hamiltonian with full 3NFs. Such differences in computational limits are the main reason for using the NO2B approximation in the first place. In this work, NCSM computations with the full 3NF for 16 O are only performed at a few basis frequencies: Ω = 16, 20, 24, 36 MeV.
In addition, it is well known that CM effects are suppressed in heavier systems since the excitation of CM motion is energetically costly. Accordingly, in Fig. 6 we find that the NO2B approximation captures the N max dependence of the ground-state energy results rather well for a wide frequency range. Note, however, that we are relatively far from convergence at N max = 8 when using large basis frequencies. For comparison, we also show the converged result from CC calculations E gs /A = −7.78 MeV [30].
The lower panel of Fig. 6 shows that the NO2Bapproximation error is on the order of 100 keV/A, corresponding to ∼ 1.5 MeV in the total binding energy (just over 1%).
The evaluated CM metrics are shown in Fig. 7, confirming the satisfactory factorization of the eigenstate. In fact, both N CM and ξ CM are orders of magnitude smaller for 16 O compared to 4 He. Moreover, the HO frequency of the CM state is very close to the one for the basis across the frequency range we explore. for ground-state energies-there does not seem to exist an optimal frequency where the error is at a minimum.
For 16 O we find a good agreement between results with NO2B approximated and the full Hamiltonian, even though the point-proton radius shows a much slower convergence rate for very high basis frequencies. The approximation error is 0.01 fm. When comparing the radius predictions for 4 He and 16 O we again find that the NO2B-approximation error and ensuing CM contamination decreases with increasing mass number, as expected. It is also interesting to note that the maximal NO2Bapproximation error for the total energy per particle and point-proton radius are both reduced by roughly a factor of 10 when going from A = 4 to 16. At least for the range of oscillator frequencies that we explore here.

V. DISCUSSION
There is a dilemma between the need to include 3NFs in nuclear calculations to achieve increased physical accuracy and precision, and the significant increase in computational complexity caused solely by the inclusion of 3NFs in ab initio methods. In this paper we have studied the SR-NO2B approximation of 3NFs, that aims to reduce the computational complexity to that of 2NFs while still capturing the most important effects of the 3NF physics. Our study is performed in the framework of the NCSM method.
The SR-NO2B approximation utilizes Wick's theorem to expand the 3NF potential in a sum of a constant, one-, two-and three-body operators that are normal-ordered relative to a non-vacuum reference state |Ψ ref , taken to be a single SD. In this work the reference state is constructed in the HO basis and we explore the sensitivity of computed observables to the choice of the basis frequency. If the reference state is a good approximation to the ground state of the nucleus, then the normal-ordered three-body term can be discarded as it is legitimate to assume that it will have a negligible contribution to the ground-state energy.
A problem with the SR-NO2B approximation is that it breaks the translational symmetry of the underlying Hamiltonian. In this work we have focused on the con- sequences of this symmetry breaking by introducing CM metrics and studying the NO2B-approximation error for energies and radii of 4 He and 16 O.
The main findings and conclusions of this study are: • Translational invariance is explicitly broken in the NO2B approximation.
The truncation of the normal-ordered Hamiltonian operator introduces a CM dependence of the reference state, which can lead to CM mixing even if a total-energy truncated NCSM basis is used.
• Metrics are important for assessing the CM mixing in eigenstates obtained with the SR-NO2B approximation. We have found that the previously introduced ξ CM and N CM metrics are useful for this purpose.
• The comparison of different CM metrics can reveal more information about the details of the CM factorization. ξ CM = N CM (ω ξ ) = 0 imply proper CM factorization with the CM part in its ground state. However, non-zero metrics do not help us determine whether we have a mixed state or a linear superposition.
• The ability of the NO2B approximation to describe the 4 He ground-state energy depends strongly on the NCSM basis frequency Ω. The NO2Bapproximation error is the smallest for Ω = 20 MeV, but it increases significantly for larger basis frequencies. A very weak dependence is observed for smaller frequencies. Both CM metrics indicate negligible CM mixing at small frequencies, and strongly increasing mixing at large ones.
• The CM problem is much less significant for the ground state of 16 O. For this system the difference between the NO2B-approximated ground-state energies and the full-3NF ones do not exhibit any significant basis-frequency dependency. Furthermore, there seems to be no significant CM mixing, since both ξ CM and N CM are small.
• We recommend further investigations of the CM problem in the NO2B approximation also when using other basis functions. In this study we have focused on the SR-NO2B approximation with a HO basis. However, reference states constructed from other single-particle bases might yield better results. In particular the Hartree-Fock and the natural orbit bases are being used in some many-body solvers and results could be analyzed in a similar fashion as in this work.
• Expectation values of other observables than ground-state energies can be strongly affected by the use of the NO2B-approximation. Expectation values are computed with respect to the eigenstates, and might therefore exhibit a stronger CM-mixing effect. While we did compute the approximation error for point-proton radii-and found that it was particularly large for 4 He-the general effects of the NO2B approximation on other observables were not fully analyzed in this work.

VI. ACKNOWLEDGMENT
We thank S. R. Stroberg for useful discussions and suggestions. We thank P. Navrátil for useful discussions and for support in the use of the NCSD code. It is not possible to fully represent a HO-ground state with a frequency ω N in a truncated HO basis with frequency Ω = ω N . Therefore, it is possible for the metrics N CM (ω N ) and ξ CM to be non-zero even if the eigenstate is factorized Ψ NO2B gs NCSM = |Ψ gs int ⊗ |Ψ gs CM (A1) where |Ψ gs CM is a HO-ground state with frequency ω N .
Here we will study eigenstates of the HO CM Hamiltonian (22). In particular, N CM (ω N )-which is the smallest eigenvalue toN CM (ω N ) in the current, truncated NCSM basis-is shown in Fig. 10 as a function of ω N /Ω. Note that the horizontal axis is logarithmic. It is obvious that this metric is not necessarily zero even if we have a factorized product state.
We also observe in Fig. 10 that N CM (ω N ) is invariant under the transformation ω N Ω → Ω ω N . Here we will demonstrate this algebraically. The analytical expression for the matrix elementsĤ CM (ω N ) in the CM-coordinate HO basis |N , L with frequency Ω is The diagonal is invariant under the transformation ω N Ω → Ω ω N , but the off-diagonal terms change sign. However, since the matrix is symmetric and tridiagonal, the offdiagonal terms will be squared in the characteristic equation, eliminating the sign change. Thus the characteristic equation is invariant for ω N Ω → Ω ω N . It follows then that the eigenvalues must be invariant too. Since N CM (ω N ) is the lowest eigenvalue this demonstrates that it also must be invariant under ω N Ω → Ω ω N in accordance with Fig. 10.
If N CM (ω N ) ≈ N CM (ω N ) then Ψ NO2B gs NCSM is separated in a HO-CM ground state and some intrinsic state. If, on the other hand, N CM (ω N ) N CM (ω N ) then the CM state is not a HO-ground state.