In-medium similarity renormalization group with three-body operators
IIn-medium similarity renormalization group with three-body operators
M. Heinz,
1, 2, ∗∗ A. Tichai,
3, 1, 2, †† J. Hoppe,
1, 2, ‡‡ K. Hebeler,
1, 2, §§ and A. Schwenk
1, 2, 3, ¶¶ Technische Universität Darmstadt, Department of Physics, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
Over the last decade the in-medium similarity renormalization group (IMSRG) approach has proven to beone of the most powerful and versatile ab initio many-body methods for studying medium-mass nuclei. So far,the IMSRG was limited to the approximation in which only up to two-body operators are incorporated in therenormalization group flow, referred to as the IMSRG(2). In this work, we extend the IMSRG(2) approach tofully include three-body operators yielding the IMSRG(3) approximation. We use a perturbative scaling anal-ysis to estimate the importance of individual terms in this approximation and introduce truncations that aim toapproximate the IMSRG(3) at a lower computational cost. The IMSRG(3) is systematically benchmarked fordi ff erent nuclear Hamiltonians for He and O in small model spaces. The IMSRG(3) systematically improvesover the IMSRG(2) relative to exact results. Approximate IMSRG(3) truncations constructed based on compu-tational cost are able to reproduce much of the systematic improvement o ff ered by the full IMSRG(3). We alsofind that the approximate IMSRG(3) truncations behave consistently with expectations from our perturbativeanalysis, indicating that this strategy may also be used to systematically approximate the IMSRG(3). I. INTRODUCTION
A key challenge in nuclear structure theory is the calcula-tion of the properties of atomic nuclei with predictive powerextending to unmeasured, exotic systems targeted by modernrare-isotope facilities.
Ab initio many-body approaches seekto accomplish this by solving the many-body Schrödingerequation in a systematically improvable manner using two-and three-body nuclear interactions as input. The rapid growthof the range of systems within reach of ab initio many-bodymethods over the past two decades [11–33] can be understood interms of improvements in the input interactions [44–1111] andimprovements in many-body approaches for medium-massnuclei.To access mid-mass and heavier systems, the many-bodyapproaches used in ab initio calculations start from an A -bodyreference state upon which corrections are systematically con-structed. These methods scale polynomially in the size of thecomputational basis N rather than exponentially in the num-ber of particles A , as is the case for the exact solution of the A -body Schrödinger equation. Examples of such methods arecoupled-cluster (CC) theory [1212, 1313], the in-medium similar-ity renormalization group (IMSRG) [1414–1616], self-consistentGreen’s function (SCGF) theory [1717, 1818], many-body pertur-bation theory (MBPT) [1919–2222], and nuclear lattice simula-tions [2323]. These methods all share a many-body truncationthat can be systematically relaxed and in the limit of no many-body truncation recover the exact results.In this work, we focus on the systematic improvementof the IMSRG, which is currently truncated at the normal-ordered two-body level, the IMSRG(2) approximation. In ∗ [email protected]@theorie.ikp.physik.tu-darmstadt.de † [email protected]@physik.tu-darmstadt.de ‡ [email protected]@theorie.ikp.physik.tu-darmstadt.de § [email protected]@physik.tu-darmstadt.de ¶ [email protected]@physik.tu-darmstadt.de coupled-cluster theory, many di ff erent methods have been de-veloped to approximately and exactly handle three-body ef-fects in many-body calculations [1212, 2424–2929]. These e ff ectshave been shown to be important for the reproduction of arange of observables, such as 2 + excited-state energies atclosed shells [3030, 3131], dipole polarizabilities [3232, 3333], and nu-clear β -decay matrix elements [3434]. In these cases, the IM-SRG(2) performance is deficient relative to methods that areable to treat three-body e ff ects. For the IMSRG, truncationsthat include induced three-body e ff ects have been applied toshell-model diagonalizations using universal shell-model in-teractions [3535]. In quantum chemistry, the driven similarityrenormalization group, a similar many-body method to theIMSRG, has been extended to approximately include three-body e ff ects in ways designed to reproduce the success ofcoupled-cluster theory in electronic systems [3636]. For the IM-SRG, however, studies of the role of three-body operators fornuclear systems have not yet been performed.To systematically study three-body operators in the IM-SRG, we extend the many-body truncation to the normal-ordered three-body level, defined as the IMSRG(3) approx-imation, and construct various di ff erent approximate IM-SRG(3) truncation schemes with reduced computational cost.We apply these truncation schemes to closed-shell systems insmall model spaces and analyze their properties in detail us-ing perturbative tools. We study how they compare to exactresults obtained from full diagonalizations, analyze the sys-tematics of the many-body expansion in these systems, andinvestigate how full IMSRG(3) results can be approximatedat a lower computational cost.In Section IIII, we give an overview of the IMSRG formal-ism. Section IIIIII discusses the IMSRG(3) truncation, pro-vides the fundamental commutators for the truncation, givesan overview of the perturbative analysis to understand theirrelative importance, and introduces approximate IMSRG(3)truncation schemes. In Section IVIV, we apply the IMSRG(3)and our approximate truncation schemes to the closed-shellnuclei He and O. Finally, we summarize our results in Sec-tion VV. a r X i v : . [ nu c l - t h ] F e b II. MANY-BODY FORMALISMA. Operator representation
In this work, an A -body operator O = O (0) + · · · + O ( A ) (1)is composed of zero- through A -body parts, given in second-quantized form by O ( A ) = A !) (cid:88) p ,..., p A O p ··· p A a † p · · · a † p A a p A · · · a p A + (2)with the antisymmetrized matrix elements O p ··· p A and thefermion creation (annihilation) operators a † p ( a p ), which create(annihilate) a particle in the single-particle state | p (cid:105) .Normal-ordering techniques can be used to exactly rear-range O into normal-ordered zero- through A -body parts, O = ˜ O (0) + · · · + ˜ O ( A ) , (3)where the normal ordering is performed with respect to a ref-erence state that is a good starting approximation for the tar-geted ground or excited state. The normal-ordered A -bodyparts are given by˜ O ( A ) = A !) (cid:88) p ,..., p A ˜ O p ··· p A : a † p · · · a † p A a p A · · · a p A + : . (4)In Eqs. (33) and (44), the tilde distinguishes the normal-orderedoperator and its normal-ordered matrix elements from theirfree-space equivalents in Eqs. (11) and (22). The normal or-dering of the string of creation and annihilation operators isindicated by the surrounding colons, : · · · :. In the follow-ing, we work exclusively with normal-ordered operators andmatrix elements and leave the tilde o ff to simplify notation.We focus on the case where the reference state to describean A -body system is a single A -particle Slater determinant: | Φ (cid:105) = A (cid:89) i = a † p i | (cid:105) , (5)where | (cid:105) is the vacuum, the state where no particles arepresent. For a single-particle state | p (cid:105) , if it is occupied inthe reference state, it has occupation number n p = n p = A -body Hilbert space is spanned by the reference state and itselementary excitations | Φ a ··· a B i ··· i B (cid:105) = a † a B · · · a † a a i B · · · a i | Φ (cid:105) , (6)which can be constructed by exciting the fermions in the holestates | i (cid:105) through | i B (cid:105) into the particle states | a (cid:105) through | a B (cid:105) .This state is a B -particle B -hole ( BpBh ) excited state, where B ≤ A .A conventional notation for the normal-ordered Hamilto-nian is H = E + f + Γ + W (7) = E + (cid:88) pq f pq : a † p a q : + (cid:88) pqrs Γ pqrs : a † p a † q a s a r : + (cid:88) pqrstu W pqrstu : a † p a † q a † r a u a t a s : , (8)where E is the reference-state expectation value of the Hamil-tonian, (cid:104) Φ | H | Φ (cid:105) , and f , Γ , and W are the normal-ordered one-,two-, and three-body parts of the Hamiltonian. For example,for a Hartree-Fock (HF) reference state, E is the Hartree-Fockenergy, and f is the Fock operator, which is diagonal in theeigenbasis of the HF one-body density matrix. The physicalground state of the system is not a single Slater determinantbut some linear combination of | Φ (cid:105) and its elementary exci-tations, leading to an energy lower than the reference-stateexpectation value. In the IMSRG and other many-body meth-ods, the task is to calculate the remaining correlation energybeyond the Hartree-Fock level to obtain the exact ground-stateenergy. B. In-medium similarity renormalization group
The similarity renormalization group (SRG) [3737–4040] seeksto construct a continuous unitary transformation of the Hamil-tonian in the flow parameter s , H ( s ) = U ( s ) HU † ( s ) , (9)which can be obtained by solving the flow equation dH ( s ) ds = (cid:2) η ( s ) , H ( s ) (cid:3) , (10)where the initial condition is H ( s = = H and the choice ofthe anti-Hermitian generator η ( s ) fixes the unitary transforma-tion generated over the course of the SRG evolution.When H ( s ) and η ( s ) are vacuum normal ordered, the so-called “free-space” SRG evolution of potentials can be used toreduce couplings between low and high momenta for two- andthree-nucleon potentials. These “softened” potentials exhibitimproved many-body convergence. At the same time, the evo-lution induces many-body forces, a fact one can quickly ver-ify by considering the commutator in second-quantized form.The treatment of many-body interactions in the free-spaceSRG approach is limited by the exponential cost of represent-ing the A -body Hamiltonian in a Jacobi or single-particle ba-sis, restricting this approach to the consistent evolution of two-and three-body forces [1010, 4141, 4242].In the IMSRG [1414], H ( s ) and η ( s ) are normal orderedwith respect to | Φ (cid:105) , and the expression for the commuta-tor is brought into normal order using Wick’s theorem [4343].The in-medium normal ordering captures many of the ef-fects of induced many-body interactions, which are alwayspresent in SRG evolutions, through lower-body interactionsof the normal-ordered Hamiltonian. This is the feature thatallows the IMSRG to succeed for the solution of the many-body Schrödinger equation for large systems where the SRGquickly becomes computationally intractable.The generator η ( s ) in SRG applications is typically chosento decouple certain parts of the Hamiltonian over the courseof the evolution. In the single-reference IMSRG, η ( s ) is cho-sen to suppress couplings between the reference state and itselementary excitations [1515], such that (cid:104) Φ | H ( s → ∞ ) | Φ a ··· i ··· (cid:105) = . (11)When this decoupling is achieved, the unitary transformationgenerated by the IMSRG is such that (cid:104) Φ | H | Φ (cid:105) is completelydecoupled from the rest of the Hamiltonian and, as a result, E ( s → ∞ ) is the correlated energy of the state targeted by thereference state. C. Truncation schemes
The IMSRG formalism is exact if one is able to keep trackof all induced normal-ordered many-body contributions, as itis simply a unitary transformation on the many-body Hamil-tonian that decouples the matrix element (cid:104) Φ | H | Φ (cid:105) from theremaining matrix elements. For practical calculations, the IM-SRG solution must be restricted to include only the operatorsup to some fixed particle rank. The current standard truncationfor nuclear structure applications is the IMSRG(2), where alloperators are truncated at the normal-ordered two-body level: H ( s ) = E ( s ) + f ( s ) + Γ ( s ) , (12) η ( s ) = η (1) ( s ) + η (2) ( s ) . (13)At this truncation, there are two approximations present.First, for Hamiltonians with three-body interactions, the resid-ual normal-ordered three-body part of the Hamiltonian W ( s =
0) is discarded, which is the so-called normal-ordered two-body (NO2B) approximation [4444, 4545]. Second, the commuta-tor (cid:2) η (2) ( s ) , Γ ( s ) (cid:3) has a normal-ordered three-body part, whichis discarded in the IMSRG(2). Some attempts to approx-imately capture the e ff ects of neglected induced three-bodycontributions in the IMSRG(2) have been explored [4646], but asystematic understanding has not been formed.The IMSRG(2) approximation has several desirable fea-tures as a many-body method. It scales polynomially [specifi-cally like O ( N )] in the size of the single-particle basis N . It iscomplete up to third order in MBPT, but it is also nonperturba-tive in that it resums pp / hh -ladder and ph -ring diagrams [1515].Additionally, it is size extensive, meaning that its error scaleslinearly in the size of the system. This puts the IMSRG(2) inthe same category as many-body methods like CCSD [1212] andADC(3) [4747], which are also nonperturbative and third-ordercomplete but di ff er from the IMSRG(2) in what higher-orderMBPT contributions the methods include. III. IMSRG(3)
Extending the IMSRG to the normal-ordered three-bodylevel yields the IMSRG(3) approximation. The Hamiltonian and the generator now each have a normal-ordered three-bodypart, H ( s ) = E ( s ) + f ( s ) + Γ ( s ) + W ( s ) , (14) η ( s ) = η (1) ( s ) + η (2) ( s ) + η (3) ( s ) , (15)and this makes it possible to include the initial residual three-body interactions exactly in the IMSRG(3) calculation. A. Fundamental commutators
The IMSRG truncations are typically derived and imple-mented in terms of the fundamental commutators of twomany-body operators. These fundamental commutators arethe basic operations that need to be evaluated in any IMSRGcalculation. For the commutator of a normal-ordered K -bodyoperator A ( K ) and a normal-ordered L -body operator B ( L ) , theresulting operator has di ff erent normal-ordered M -body parts C ( M ) : (cid:104) A ( K ) , B ( L ) (cid:105) = K + L − (cid:88) M = | K − L | C ( M ) . (16)The fact that M ≤ K + L − M ≤ K + L for a simple product of normal-ordered operators)ensures that the many-body expansion is “connected,” whichmeans that the IMSRG at any truncation level is size exten-sive. We isolate the di ff erent M -body parts that arise fromthe commutator of a K -body operator and an L -body opera-tor, using the following schematic notation in terms of theirmany-body ranks: [ K , L ] → M . (17)In the following, we provide the nonantisymmetrized ex-pressions for the matrix elements of the fundamental commu-tators required by the IMSRG(3). For the two- and three-bodyparts, the matrix elements must be antisymmetrized by ap-plying the appropriate two- and three-body antisymmetrizer.The expressions were derived using the automated normal-ordering tool drudge [4848], and, in cases where our expressionsdid not match those provided in Ref. [1515], the results were ver-ified by hand (see also the Appendices). [1 , → ◦ C = (cid:88) p (cid:16) A p B p − B p A p (cid:17) , (18) C (0) = (cid:88) pq ( n p ¯ n q − ¯ n p n q ) A pq B qp , (19)with ¯ n p ≡ − n p and the one-body matrix elements of theresult C . [1 , → ◦ C = (cid:88) p (cid:16) A p B p − A p B p (cid:17) , (20) C = (cid:88) pq ( n p ¯ n q − ¯ n p n q ) A pq B q p . (21) [2 , → ◦ C = (cid:88) p (cid:16) A p B p − B p A p (cid:17) , (22) C = (cid:88) pq (¯ n p ¯ n q − n p n q ) × (cid:16) A pq B pq − B pq A pq (cid:17) − (cid:88) pq ( n p ¯ n q − ¯ n p n q ) A p q B qp , (23) C = (cid:88) pqr (¯ n p ¯ n q n r + n p n q ¯ n r ) × (cid:16) A rpq B pq r − B rpq A pq r (cid:17) , (24) C (0) = (cid:88) pqrs ( n p n q ¯ n r ¯ n s − ¯ n p ¯ n q n r n s ) A pqrs B rspq . (25) [1 , → ◦ C = (cid:88) p (cid:16) A p B p − A p B p (cid:17) , (26) C = (cid:88) pq ( n p ¯ n q − ¯ n p n q ) A pq B q p . (27) [2 , → ◦ C = (cid:88) pq (¯ n p ¯ n q − n p n q ) × (cid:16) A pq B pq − A pq B pq (cid:17) + (cid:88) pq (¯ n p n q − n p ¯ n q ) A pq B q p , (28) C = (cid:88) pqr (¯ n p ¯ n q n r + n p n q ¯ n r ) × (cid:16) A r pq B pq r − A pqr B rpq (cid:17) , (29) C = (cid:88) pqrs ( n p n q ¯ n r ¯ n s − ¯ n p ¯ n q n r n s ) A pqrs B rs pq . (30) [3 , → ◦ C = (cid:88) pqr ( n p n q n r + ¯ n p ¯ n q ¯ n r ) × (cid:16) A pqr B pqr − B pqr A pqr (cid:17) + (cid:88) pqr (¯ n p ¯ n q n r + n p n q ¯ n r ) × (cid:16) A pq r B rpq − B pq r A rpq (cid:17) , (31) C = (cid:88) pqrs (¯ n p ¯ n q ¯ n r n s − n p n q n r ¯ n s ) × (cid:16) A spqr B pqr s − B spqr A pqr s (cid:17) + (cid:88) pqrs ( n p n q ¯ n r ¯ n s − ¯ n p ¯ n q n r n s ) A pq rs B rs pq , (32) C = (cid:88) pqrst ( n p n q n r ¯ n s ¯ n t + ¯ n p ¯ n q ¯ n r n s n t ) × (cid:16) A st pqr B pqrst − B st pqr A pqrst (cid:17) , (33) C (0) = (cid:88) pqrstu ( n p n q n r ¯ n s ¯ n t ¯ n u − ¯ n p ¯ n q ¯ n r n s n t n u ) × A pqrstu B stupqr . (34)The computational cost of each commutator scales naivelylike O ( N K + L + M ) in the size of the single-particle basis N . Asa result, the cost of the full IMSRG(3) solution scales like thecost of the [3, 3] → O ( N ). The full IMSRG(3)flow equations for the matrix elements of the Hamiltonian areprovided in Appendix AA along with a list of the di ff erences be-tween the expressions we provide and those given in Ref. [1515]. B. Generators
In the IMSRG(3), the extended many-body truncation in-troduces new matrix elements of the Hamiltonian that couplethe reference state and its excitations, specifically W i jkabc and W abci jk , where i , j , and k are hole-state indices and a , b , and c are particle-state indices. Below we extend the standard gen-erator definitions used in the single-reference IMSRG(2) [1515]to the three-body case, seeking to suppress these matrix ele-ments over the course of the evolution.For the imaginary-time generator, we choose the matrix el-ements of the three-body part of the generator to be η i jkabc = sgn( ∆ i jkabc ) W i jkabc , (35a) η abci jk = sgn( ∆ abci jk ) W abci jk , (35b)where we use the Møller-Plesset energy denominators ∆ i jkabc = f ii + f j j + f kk − ( f aa + f bb + f cc ) = − ∆ abci jk . (36)Similarly, the matrix elements of the three-body White gener-ator are chosen to be η i jkabc = W i jkabc ∆ i jkabc , (37a) η abci jk = W abci jk ∆ abci jk , (37b)and the matrix elements of the three-body arctan generator arechosen to be η i jkabc =
12 arctan (cid:32) W i jkabc ∆ i jkabc (cid:33) , (38a) η abci jk =
12 arctan (cid:32) W abci jk ∆ abci jk (cid:33) . (38b) C. Perturbative analysis
In Ref. [1515], a perturbative analysis of the IMSRG is pre-sented for the case where the NO2B approximation and an HFreference state are used. This analysis reveals the MBPT dia-grammatic content of the many-body method, and we use it asa tool to understand the contributions of di ff erent commutatorsin the IMSRG(3). In the following we present the key ideasof the perturbative analysis and refer the reader interested in amore formal treatment to Ref. [1515].The connection from the IMSRG to MBPT is cleanly madewhen using the White generator, with the matrix elements η ia = f ia ∆ ia , (39a) η i jab = Γ i jab ∆ i jab , (39b) η i jkabc = W i jkaba ∆ i jkabc , (39c)where ∆ ia and ∆ i jab are defined analogously to Eq. (3636). Hereand in the following i , j , and k are hole single-particle indices,and a , b , and c are particle single-particle indices. Using thisgenerator, the zero-body part of the IMSRG flow equations(up to the three-body level) has three contributions from the[1, 1] →
0, [2, 2] →
0, and [3, 3] → (cid:32) dEds (cid:33) = (cid:88) ia ( η ia ( s ) f ai ( s ) − η ai ( s ) f ia ( s )) = (cid:88) ia η ia ( s ) f ai ( s ) = (cid:88) ia f ia ( s ) f ai ( s ) ∆ ia ( s ) , (40) (cid:32) dEds (cid:33) = (cid:88) i jab η i jab ( s ) Γ abi j ( s ) = (cid:88) i jab Γ i jab ( s ) Γ abi j ( s ) ∆ i jab ( s ) , (41) (cid:32) dEds (cid:33) = (cid:88) i jkabc η i jkabc ( s ) W abci jk ( s ) = (cid:88) i jkabc W i jkabc ( s ) W abci jk ( s ) ∆ i jkabc ( s ) , (42)which look remarkably similar to the second-order MBPTcorrections to the energy. Indeed, if one approximates the hole-particle block matrix elements f ia ( s ), Γ i jab ( s ), and W i jkabc ( s ) by their basic suppression behavior due to the Whitegenerator [1515], f ia ( s ) ≈ f ia ( s =
0) exp( − s ) , (43a) Γ i jab ( s ) ≈ Γ i jab ( s =
0) exp( − s ) , (43b) W i jkabc ( s ) ≈ W i jkabc ( s =
0) exp( − s ) , (43c)and one approximates the energy denominators by their initialvalues, Eqs. (4040)–(4242) can be analytically integrated to get theresults E ( s → ∞ ) ≈ (cid:88) ia f ia ( s = f ai ( s = ∆ ia ( s = , (44) E ( s → ∞ ) ≈ (cid:88) i jab Γ i jab ( s = Γ abi j ( s = ∆ i jab ( s = , (45) E ( s → ∞ ) ≈ (cid:88) i jkabc W i jkabc ( s = W abci jk ( s = ∆ i jkabc ( s = . (46)These are exactly the second-order MBPT corrections to theenergy, and this shows that these corrections are absorbed intothe IMSRG correlation energy, making the IMSRG at anymany-body truncation second-order complete in MBPT (aslong as the matrix elements are able to be captured initiallyin the many-body truncation).Extending this analysis to higher orders in MBPT requiresconsidering how the hole-particle matrix elements of f , Γ , and W change over the course of the IMSRG evolution beyondthe basic suppression of their initial values. On a high level,this corresponds to the IMSRG evolution “dressing” the one-,two-, and three-body vertices with e ff ective interaction contri-butions that generate higher-order MBPT diagrams.To make this analysis systematic, we focus on the casewhere we use an HF reference state and work in the NO2Bapproximation, where the initial o ff -diagonal matrix elementsof f and all the initial matrix elements of W are 0. Workingwith a Møller-Plesset MBPT partitioning of the initial Hamil-tonian, H = f + g Γ , (47)we have the following power-counting scheme: f pp = O ( g ) , (48) Γ pqrs = O ( g ) , (49)that is, the diagonal one-body matrix elements are O ( g ) andthe two-body matrix elements are O ( g ). In this case, thehole-particle block of f is induced by the [2, 2] → → f ia and thus η (1) are 0), and thematrix elements of W are induced by the [2, 2] → f ia = O ( g ) , (50) W pqrstu = O ( g ) , (51) Commutator Perturbative order[1 , → g [1 , → g [1 , → g [1 , → g [2 , → g [2 , → g [2 , → g [2 , → g [1 , → g [1 , → g [2 , → g [2 , → g [2 , → g [3 , → g [3 , → g [3 , → g [3 , → g TABLE I. The lowest-order perturbative contribution to the energyprovided by each of fundamental commutators. and, as a result, their contributions to the energy are both O ( g ) .Thus, the contribution of any induced two-body parts to E is suppressed by O ( g ), and the contributions of induced one-and three-body parts to E are suppressed by O ( g ). This al-lows one to quickly perturbatively estimate the importance ofdi ff erent fundamental commutators, provided in Table II.It is worth noting that the [1, 1] →
1, [1, 2] →
2, and[1, 3] → →
1, [2, 2] →
2, and [2, 3] → f , which is O ( g ). The former [1, B ] → B commutatorsare responsible for the suppression of the B -body hole-particleblocks of the Hamiltonian and play a central role in the behav-ior of the IMSRG evolution. This is intuitively similar to thecentral role the kinetic energy plays in the free-space SRG.A key result of the analysis in Ref. [1515] is that the IM-SRG(2) is complete up to third order in MBPT and containsmany fourth-order diagrams as well. At the NO2B level, theIMSRG(3) accounts for the induced three-body e ff ects, whichare what is missing for the complete inclusion of fourth-orderdiagrams in the IMSRG(2), making the IMSRG(3) fourth-order complete (at the NO2B level) [1515]. This is true both for the direct flow into the energy via, for example, the[3, 3] → g × g ) and for the indirect case via an inducedtwo-body part from, for example, the [2, 3] → → g × g × g ). D. Approximation schemes
Due to the high computational cost of full IMSRG(3) cal-culations, finding a way to approximate the IMSRG(3) trun-cation would pave the way to large model-space IMSRGcalculations that approximately include the e ff ects of three-body operators. In the following, we present approximationschemes by including in each scheme selected IMSRG(3) fun-damental commutators on top of the IMSRG(2).The first major truncation beyond IMSRG(2) we use in-cludes the minimum commutators necessary to make thetruncation fourth-order complete in MBPT. These are the[2, 2] →
3, [2, 3] →
2, [1, 3] →
3, and [3, 3] → O ( N ), putting it in the same ballpark as coupled-cluster methods with iterated triples in terms of computationalcost and diagrammatic content [1212, 2424, 4949].We note that in our studies we found that the [2, 2] → →
2, and [1, 3] → ff ects and also be numerically sta-ble. Without the [2, 2] → → → ff ects. One wouldhope to see that these higher-order e ff ects generate only smallchanges in energies and in practical calculations some “com-plete” lower-order approximation could be used.Following the first approach, including the [2, 3] → →
2, and [3, 3] → O ( N ) or less. Werefer to this truncation as the IMSRG(3)- N truncation. Theinclusion of the [2, 3] → → N truncation, whichincludes all commutators that cost O ( N ) or less. This trun-cation di ff ers from the full IMSRG(3) only by the missing[3, 3] → Commutator Cost Included in . . .IMSRG(3)-MP4 IMSRG(3)- N IMSRG(3)- N IMSRG(3)- g IMSRG(3)[2 , → O ( N ) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) [2 , → O ( N ) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) [1 , → O ( N ) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) [3 , → O ( N ) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) [2 , → O ( N ) (cid:88) (cid:88) (cid:88) (cid:88) [1 , → O ( N ) (cid:88) (cid:88) (cid:88) (cid:88) [3 , → O ( N ) (cid:88) (cid:88) (cid:88) [2 , → O ( N ) (cid:88) (cid:88) (cid:88) [3 , → O ( N ) (cid:88) (cid:88) (cid:88) [3 , → O ( N ) (cid:88) TABLE II. The computational cost of the IMSRG(3) fundamental commutators and whether they are included in various approximate and fullIMSRG(3) truncation schemes. cluding all of the IMSRG(3) commutators that are O ( g ) orless, with the exception of the [1, 2] → O ( g ) and is included in the IMSRG(2) truncation. The nexttruncation we present includes the remaining O ( g ) commuta-tors, the [2, 3] →
1, [1, 3] →
2, [2, 3] →
3, and [3, 3] → g truncation. This trunca-tion includes two commutators that cost O ( N ), making thatthe cost of the truncation. The two remaining commutatorsare O ( g ), so this is the only complete perturbatively guidedtruncation between the IMSRG(3)-MP4 and full IMSRG(3)truncations.The inclusion of specific commutators in each of the ap-proximate IMSRG(3) truncation schemes discussed above ispresented in Table IIII. IV. APPLICATIONS
In this section, we investigate the IMSRG(3) truncation andthe approximate truncations discussed in Sec. III DIII D when ap-plied to the closed-shell He and O using di ff erent nuclearHamiltonians. A. Hamiltonians and basis sets
For most of our calculations, we focus on two sets of chi-ral Hamiltonians, one using the N LO nucleon-nucleon ( NN )potential from Ref. [5151] SRG-evolved to a resolution scale λ = . − , which we refer to as the “EM 1.8” Hamiltonian,and one using the “EM 1.8 / NN and three-nucleon (3 N ) interactions. For the treat-ment of the three-body part of the NN + N Hamiltonian whenusing the EM 1.8 / / ff ected by the choice ofharder Hamiltonians. We use three sets of NN -only Hamilto- nians. One uses the N LO NN potential from Ref. [5151] (withno SRG evolution applied), which we refer to as the “EM 500”Hamiltonian based on its regulator cuto ff at Λ =
500 MeV.The other two use the N LO NN potential from Ref. [88] with Λ =
450 MeV (referred to as the “EMN 450” Hamiltonian)and
Λ =
500 MeV (referred to as “EMN 500”).In addition, we use reference states constructed from dif-ferent single-particle basis sets. Our single-particle basis ischaracterized by the maximum principal quantum number e max = (2 n + l ) max , with the radial quantum number n and theorbital angular momentum l . In the simplest case, we solvethe spherically restricted HF equations to obtain a variation-ally optimized HF solution. Where an HF reference state isused, the solution of the HF equations and the solution of theIMSRG both take place in an e max = ff ects in the construction of the single-particle ba-sis, leading to improved convergence properties and reducedsensitivity to the underlying basis frequency [5454, 5555]. We fol-low the strategy of Ref. [5555], where the one-body density ma-trix is constructed in a large model space with e NATmax . Follow-ing the construction of the basis and the transformation of theHamiltonian matrix elements, the basis and operators are trun-cated to a model space with a smaller e max , which is used forthe IMSRG solution.When using NN -only Hamiltonians, the construction of theNAT basis takes place in an e NATmax =
14 model space. The ba-sis and Hamiltonian are truncated to an e max = / NN + N Hamiltonian, the construction of the NAT ba-sis takes place in an e NATmax =
14 model space with an additional E , max = ≥ e + e + e truncation placed on the three-bodymatrix elements. Again, the basis and Hamiltonian are trun-cated to an e max = ff ect on the result of the IMSRG solution [1515].We also experimented with generator choice in the IMSRG(3)case and found that choosing a di ff erent generator changed theresults obtained for each truncation scheme by less than 1 keV,an e ff ect much smaller than the e ff ects we discuss in the fol-lowing sections. It seems that the insensitivity to generatorchoice in the IMSRG(2) extends also to the IMSRG(3). B. Helium-4
In this section, we consider how the IMSRG solution for theground-state energy of He changes for di ff erent truncationschemes ranging from the IMSRG(2) to the full IMSRG(3)approximation. We focus our discussion on the major trunca-tions discussed in Sec. III DIII D and presented succinctly in Ta-ble IIII. In the figures like Fig. 11, these truncations are visu-ally indicated by the thicker bars. We also introduce minortruncations, which are defined as having one additional com-mutator included relative to some previous truncation scheme.For example, one minor truncation scheme we consider is theIMSRG(3)- N + [2 , → O ( N )commutators and the [2, 3] → O ( N ).The inclusion of the [3, 3] → N truncation. These minor truncations are visually indicated bythinner bars.We first focus on the case where we use the EM 1.8 NN -only Hamiltonian. For the NN -only case, we use an under-lying oscillator frequency of (cid:126) Ω =
28 MeV, which was de-termined by choosing the frequency at which the ground-stateenergy that resulted from IMSRG(2) calculations using an HFreference state was minimal. For comparison, we provide ex-act results from the full configuration interaction (FCI) diag-onalization of the e max = ff erent approximations allows us to gain insight intothe e ff ect of the many-body truncations at play.In Fig. 11, we show the ground-state energies for He ob-tained using di ff erent IMSRG truncation schemes using theEM 1.8 NN -only Hamiltonian and an HF reference state. Inboth panels, we start from the IMSRG(2) truncation and addcommutators until we reach the IMSRG(3) truncation on theright.In the left panel of Fig. 11, we follow the computationalapproach to organizing the IMSRG(3) fundamental commu-tators. At the IMSRG(2)-truncation level, the ground-stateenergy only di ff ers from the FCI result by 9 keV. The firsttruncation we consider beyond the IMSRG(2) is always theIMSRG(3)-MP4 truncation, which in all systems we investi- gated delivered a sizable repulsive correction to the energy.This is consistent with our understanding of the diagrammaticcontent of the IMSRG(2) and the nature of the missing fourth-order MBPT energy corrections. The inclusion of fundamen-tal commutators up to the IMSRG(3)- N truncation brings thecorrelated energy back down towards the FCI result. The nexttwo commutators that are included in the IMSRG(3)- N trun-cation provide significant contributions that partially cancel.The size of their individual contributions can be understoodby the fact that they are both fifth-order [ O ( g )] in our pertur-bative counting [to be compared with the O ( g ) contributionof [3, 3] →
1, which is the final commutator that contributes tothe IMSRG(3)- N ]. The contribution of the [3, 3] → ff ers from the FCIresult by 8 keV.In the right panel, we show the same information for thecase where the perturbative ordering of fundamental commu-tators is used. We see that the O ( g ) commutators addedfrom the IMSRG(3)-MP4 truncation to the IMSRG(3)- g truncation deliver contributions to the energy that are gener-ally smaller than the fourth-order shift between IMSRG(2)and IMSRG(3)-MP4 truncations and generally larger thanthe sixth-order shifts between the IMSRG(3)- g and the IM-SRG(3) truncations, which is consistent with the perturbativecounting.When discussing the contributions of commutators, it isworth noting that the contribution of an added commutatorto the energy also depends on which other commutators arealso included in that truncation. In this context, the one-by-one inclusion of fundamental commutators formally does notcommute. In practice, however, we see that the size of thecontribution of a specific commutator is not strongly sensitiveto the order in which it is included relative to other commu-tators. One can see this behavior when comparing the twopanels of Fig. 11. Of course, substantial rearrangement of thecommutators (in particular, changing the order of two commu-tators that give large contributions to the energy) can changethis picture. Our discussion, however, is built around the ma-jor truncation schemes discussed in Sec. III DIII D, restricting thefreedom we have to move commutators around in between.As far as we have seen in our explorations, the quasi-additivenature of the inclusion of commutators and their energy con-tributions seems to qualitatively hold within these restrictions.In Fig. 22, we present results for He when using the EM1.8 NN -only Hamiltonian and a NAT reference state. Thesame oscillator frequency is used as for the NN -only HF case( (cid:126) Ω =
28 MeV). The IMSRG(2) error to the FCI result isin this case 27 keV. In the left panel, following the repul-sive IMSRG(3)-MP4 corrections to the energy, we see that thecommutators added to give the IMSRG(3)- N give additionalsmall repulsive shifts to the energy. The O ( N ) commutatorsgive slightly larger attractive contributions, and the [3, 3] → ff ers from the FCI result by 9 keV. Thisis a considerable improvement over the IMSRG(2) result, al-though all of the results discussed here are quite good (sub-1%error) when compared to the total ground-state energy or the I M S R G ( ) I M S R G ( ) - M P + [ , ] → + [ , ] → I M S R G ( ) - N + [ , ] → I M S R G ( ) - N I M S R G ( ) − . − . − . − . − . − . E ( M e V ) Truncation scheme (computational) He, HF¯ h Ω = 28 MeVEM 1.8 (
N N -only) e max = 2 I M S R G ( ) I M S R G ( ) - M P + [ , ] → + [ , ] → + [ , ] → I M S R G ( ) - g + [ , ] → I M S R G ( ) Truncation scheme (perturbative) E ( s = 0) = − .
731 MeVFCI
FIG. 1. Ground-state energies of He obtained in various truncation schemes using the EM 1.8 NN -only Hamiltonian and an HF referencestate following the computational (left panel) and perturbative (right panel) truncation ordering for the fundamental commutators. Thicker,darker bars correspond to the major truncations summarized in Table IIII. Thinner, lighter bars correspond to intermediate truncations where asingle fundamental commutator has been added relative to the truncation scheme to the left. The dashed line indicates the e max = N and IMSRG(3)- g truncations. The starting HF energy is provided in the bottom right corner. I M S R G ( ) I M S R G ( ) - M P + [ , ] → + [ , ] → I M S R G ( ) - N + [ , ] → I M S R G ( ) - N I M S R G ( ) − . − . − . − . − . − . E ( M e V ) Truncation scheme (computational) He, NAT¯ h Ω = 28 MeVEM 1.8 (
N N -only) e max = 2 I M S R G ( ) I M S R G ( ) - M P + [ , ] → + [ , ] → + [ , ] → I M S R G ( ) - g + [ , ] → I M S R G ( ) Truncation scheme (perturbative) E ( s = 0) = − .
946 MeVFCI
FIG. 2. Same as Fig. 11 but using a NAT reference state. correlation energy. In the right panel, we see that the general size of energy0
20 24 28 32 36 40¯ h Ω (MeV) − . − . − . − . − . − . − . − . E ( M e V ) He, HFEM 1.8/2.0 (
N N +3 N ) e max = 2 IMSRG(2)IMSRG(3)-MP4IMSRG(3)- N IMSRG(3)- g IMSRG(3)
FIG. 3. Ground-state energies of He using the EM 1.8 / contributions follows the perturbative counting. The size of allcontributions beyond the IMSRG(3)-MP4 truncation is sub-stantially smaller than in the HF case discussed previously(note that the relative scale on the energy in the graph is iden-tical in Figs. 11 and 22). In particular, because the sixth-ordercommutator contributions are so small, the IMSRG(3)- g ap-proximates the full IMSRG(3) extremely well.Now we switch our focus to the case where we use the EM1.8 / NN + N Hamiltonian. We investigated the oscillatorfrequency sensitivity of the IMSRG(3) truncations in He us-ing an HF reference state. This system exhibits substantialfrequency dependence because NN + N Hamiltonians tend togive greater frequency dependence than their NN -only coun-terparts and the HF basis depends more strongly on the fre-quency than the NAT basis. This is because the NAT basisseeks to reduce frequency dependence by construction.In Fig. 33, we show the ground-state energy obtained us-ing several IMSRG truncations ranging from the IMSRG(2)to the IMSRG(3) for a broad range of oscillator frequencies.Generally, we find that the results for the di ff erent truncationsremain quite close together (within a spread of 300 keV) evenas the energy varies over a range of 1.5 MeV. This suggeststhat the variance in the energy is entirely due to harsh infraredand ultraviolet cuto ff s imposed by the e max = ff erences in the frequency dependence of the energy re-sulting from di ff erent IMSRG truncations.A couple systematic trends can be identified in Fig. 33. First,the IMSRG(3)-MP4 provides a repulsive contribution on topof the IMSRG(2) at all frequencies. Second, the IMSRG(3)- g and IMSRG(3) lines lie basically on top of each other, in-dicating that the IMSRG(3)- g reliably approximates the IM- I M S R G ( ) I M S R G ( ) - M P I M S R G ( ) - N I M S R G ( ) - g I M S R G ( ) - N I M S R G ( ) − . − . − . − . − . − . − . − . − . − . E ( M e V ) E ( s = 0) = − .
019 MeV He, NAT¯ h Ω = 32 MeVEM 1.8/2.0 (
N N +3 N ) e max = 2 FIG. 4. Ground-state energies of He obtained in various trunca-tion schemes using the EM 1.8 / N and IMSRG(3)- g truncations. Thestarting energy of the NAT reference state is provided in the bottomright corner. SRG(3). The same cannot be said for the IMSRG(3)- N . Fi-nally, the IMSRG(3) results always lie below the IMSRG(3)-MP4 results.In Fig. 44, we present the He ground-state energies obtainedin various IMSRG truncation schemes using the EM 1.8 / (cid:126) Ω =
32 MeV was determined by choosing thefrequency at which the HF IMSRG(2) energy result was mini-mal for this Hamiltonian (see Fig. 33). Overall, the correctionso ff ered by approximate IMSRG(3) truncations are larger inmagnitude than in the NN -only case, with the IMSRG(2) andIMSRG(3) results di ff ering by 112 keV (compare with the dif-ference of 36 keV in the NN -only case). We see similar trendsas in the NN -only case, with a large repulsive correction fromthe IMSRG(3)-MP4 truncation and a smaller repulsive correc-tion from the IMSRG(3)- N . The O ( N ) fifth-order commu-tators provide attractive corrections, and the final IMSRG(3)result lands between the IMSRG(3)- N and IMSRG(3)- g re-sults, as indicated by the blue band. C. Oxygen-16
In this section, we consider the IMSRG solution for theground-state energy of O. We first focus on the case wherewe use the EM 1.8 NN -only Hamiltonian. In this case, we1 I M S R G ( ) I M S R G ( ) - M P + [ , ] → + [ , ] → I M S R G ( ) - N + [ , ] → I M S R G ( ) - N I M S R G ( ) − . − . − . − . − . − . − . E ( M e V ) Truncation scheme (computational) O, HF¯ h Ω = 24 MeVEM 1.8 (
N N -only) e max = 2 I M S R G ( ) I M S R G ( ) - M P + [ , ] → + [ , ] → + [ , ] → I M S R G ( ) - g + [ , ] → I M S R G ( ) Truncation scheme (perturbative) E ( s = 0) = − .
625 MeVextrap. FCI
FIG. 5. Ground-state energies of O obtained in various truncation schemes using the EM 1.8 NN -only Hamiltonian and an HF referencestate following the computational (left panel) and perturbative (right panel) truncation ordering for the fundamental commutators. Thicker,darker bars correspond to the major truncations summarized in Table IIII. Thinner, lighter bars correspond to intermediate truncations where asingle fundamental commutator has been added relative to the truncation scheme to the left. The dashed line indicates the e max = N and IMSRG(3)- g truncations. The starting HF energy is provided in the bottom right corner. use an oscillator frequency of (cid:126) Ω =
24 MeV. For NN -onlyresults, we provide for comparison extrapolated FCI results.These results were obtained from a series of CI calculationswith increasing N max (the model space truncation for the ap-proach) from 0 to 8 using the kshell code [5757]. The resultsfrom N max = N max → ∞ extrapolated value [5858]. The uncer-tainty in the extrapolation was assessed by leaving out one ofthe N max =
2, 4, 6 points and fitting the exponential to the re-maining three points (the highest-quality N max = O as ob-tained from di ff erent truncation schemes when using an HFreference state. The IMSRG(2) result di ff ers from the exactresult by about 180 keV, which corresponds to an error of1.8% in the correlation energy. The IMSRG(3)-MP4 approxi-mation provides a large, repulsive correction to the IMSRG(2)result. In the left panel, we see that the [2, 3] → N truncation provides a small, butsignificant attractive correction and the [2, 3] → N delivers most of the remainingattraction needed to produce the IMSRG(3) result. The finalIMSRG(3) result di ff ers from the extrapolated FCI result byonly 32 keV, which corresponds to an error of about 0.3% inthe correlation energy. In the right panel, we see that the per- turbative counting of commutators continues to be predictive,with the smallest contributions belonging to the sixth-ordercommutators. As a result, the IMSRG(3)- g result lies quiteclose to the IMSRG(3) result.In Fig. 66, we switch to a NAT reference state, still consider-ing O using the EM 1.8 NN -only Hamiltonian. The di ff er-ence between the IMSRG(2) result and the exact result is only16 keV, making the IMSRG(2) result in this case remarkablygood. The correction provided by the IMSRG(3)-MP4 trun-cation is still repulsive, but considerably smaller than in theHF case. In the left panel, we see that again the [2, 3] → → N and IMSRG(3)- N truncations, respectively. The final IMSRG(3) result di ff ersfrom the extrapolated FCI result by 28 keV, quite similar tothe di ff erence in the HF case. The right panel shows that con-vergence to the IMSRG(3) result in the perturbative countingapproach is systematic in this case as well.Switching to the EM 1.8 / Ousing a NAT reference state, where the underlying oscillatorfrequency is (cid:126)
Ω =
20 MeV. In this case, the IMSRG(3)-MP4 truncation result is about 270 keV more repulsive thanthe IMSRG(2) result, and the IMSRG(3)- N provides onlysmall corrections to the IMSRG(3)-MP4 result. These resultsdi ff er substantially from those obtained from the remaining2 I M S R G ( ) I M S R G ( ) - M P + [ , ] → + [ , ] → I M S R G ( ) - N + [ , ] → I M S R G ( ) - N I M S R G ( ) − . − . − . E ( M e V ) Truncation scheme (computational) O, NAT¯ h Ω = 24 MeVEM 1.8 (
N N -only) e max = 2 I M S R G ( ) I M S R G ( ) - M P + [ , ] → + [ , ] → + [ , ] → I M S R G ( ) - g + [ , ] → I M S R G ( ) Truncation scheme (perturbative) E ( s = 0) = − .
441 MeVextrap. FCI
FIG. 6. Same as Fig. 55 but using a NAT reference state. truncation schemes, which contain all the O ( N ) fifth-ordercommutators. Of the systems we studied, this is the sys-tem with the largest contribution by these commutators, mak-ing the IMSRG(3)- g , for example, a substantial improve-ment over the IMSRG(3)- N due to its inclusion of thesehigher-cost fifth-order commutators that are neglected in theIMSRG(3)- N . We see that again the large band resultingfrom the IMSRG(3)- N and IMSRG(3)- g results includes theIMSRG(3) result. D. Analysis of truncation performance
Next, we consider the relative performance of the di ff erentIMSRG truncations over all systems considered. These trendsare summarized in Fig. 88. In this figure, we compare the cor-relation energy, defined as E corr = E ( s → ∞ ) − E ( s = , (52)for the IMSRG(2) and approximate IMSRG(3) truncationsrelative to the IMSRG(3) correlation energy. The vertical lineat x = . Ocase with the EM 1.8 NN -only Hamiltonian and the NAT ref-erence state). This intuitively matches the expected behaviorof the many-body expansion, where including higher many-body ranks in the many-body expansion allows the truncatedmethods to systematically approach the exact result. In this figure and the following discussion, we frame things relativeto the IMSRG(3) results, as the IMSRG(3) truncation is the“most complete” IMSRG result we have available.Considering the performance of the IMSRG(2) relative tothe IMSRG(3), we see that the di ff erence in the correlationenergy is about 1–2% for most systems. This also makes itclear how unusually good the IMSRG(2) results are in the ex-ceptional O NN -only NAT case, where the di ff erence in theIMSRG(2) and IMSRG(3) results is closer to 0.1%. We alsosee that the IMSRG(2) results are systematically overboundrelative to the IMSRG(3) results.Turning our attention to the IMSRG(3)-MP4 truncation, wefind that these results di ff er from the IMSRG(3) results byup to 1%. The results are also all less bound than the IM-SRG(3) results, making the IMSRG(2) and IMSRG(3)-MP4results lower and upper bounds on the IMSRG(3) result. Con-sidering that the IMSRG(3)-MP4 is the least computationallyexpensive approximate IMSRG(3) truncation we considered,this provides a relatively cheap way to set a weak bound onwhere the IMSRG(3) result lands. In the case where the many-body expansion converges systematically, this bound shouldalso encompass the e ff ects of higher orders in the many-bodyexpansion.Turning our attention to the next two truncations, theIMSRG(3)- N and IMSRG(3)- g truncations, we find thatthe IMSRG(3)- N results are generally less bound than theIMSRG(3) results by about 0.5% (1% in one case) and theIMSRG(3)- g results are generally more bound by about0.1%. The gray bands in Fig. 88 show the range of energiesbounded by the results from these two truncations, where we3 I M S R G ( ) I M S R G ( ) - M P I M S R G ( ) - N I M S R G ( ) - g I M S R G ( ) - N I M S R G ( ) − . − . − . − . − . − . − . E ( M e V ) E ( s = 0) = − .
240 MeV O, NAT¯ h Ω = 20 MeVEM 1.8/2.0 (
N N +3 N ) e max = 2 FIG. 7. Ground-state energies of O obtained in various trunca-tion schemes using the EM 1.8 / N and IMSRG(3)- g truncations. Thestarting energy of the NAT reference state is provided in the bottomright corner. see that these bands always contain the IMSRG(3) results.The IMSRG(3)- N is of comparable expense and quality tothe IMSRG(3)-MP4 truncation. However, the IMSRG(3)- g is considerably more expensive and nearly as expensive asthe full IMSRG(3). This means that even once large-scaleIMSRG(3)-MP4 and IMSRG(3)- N are possible IMSRG(3)- g calculations may still be out of reach. Still, if bothIMSRG(3)- N and IMSRG(3)- g calculations are possible,these can be used to provide a robust bound on what the IM-SRG(3) results could be. E. Performance for harder Hamiltonians
In Fig. 99, we show the error to the exact FCI ground-stateenergy of He for the harder NN -only Hamiltonians for cal-culations using major and minor truncations schemes goingfrom the IMSRG(2) approximation to the IMSRG(3) approx-imation. The correlation energies for these Hamiltonians areabout 8 to 10 MeV, approximately double that of the EM 1.8and EM 1.8 / He. We also note that theEM 500 Hamiltonian gives an unbound HF solution with apositive HF energy.We see that for all three Hamiltonians, the IMSRG(2)overbinds the system substantially relative to the exact result.These errors of about 350 to 500 keV correspond to errors of .
99 1 .
00 1 .
01 1 . E corr /E corr,IMSRG(3)16 O, NAT
N N +3 N O, NAT
N N -only O, HF
N N -only He, NAT
N N +3 N He, NAT
N N -only He, HF
N N -only
IMSRG(2)IMSRG(3)-MP4 IMSRG(3)- N IMSRG(3)- g FIG. 8. Ratios of correlation energies obtained in IMSRG(2) and ap-proximate IMSRG(3) calculations relative to the IMSRG(3) correla-tion energies for di ff erent systems discussed in Secs. IV BIV B and IV CIV C.The gray band indicates the range spanned by the IMSRG(3)- N andIMSRG(3)- g results. N and IMSRG(3)- g truncations brings theIMSRG results within 100 keV of the exact results, a sub-1%error in the correlation energy. The higher-cost and higher-order corrections bring relatively small corrections, and thefinal IMSRG(3) results remain within 100 keV of exact ener-gies for all three Hamiltonians. In Fig. 1010, we show the resultsfor O. The approximate IMSRG(3) truncations systemati-cally improve over the IMSRG(2), and the final IMSRG(3) re-sults di ff er from the exact results by just over 100 keV, whichis an error of about 0.5% in the correlation energy for bothHamiltonians. For the EM 500 Hamiltonian in the e max = O does not con-verge. The IMSRG(3) improves upon this by delivering con-verged results that di ff er from exact results by about 3%, sta-bilizing the solution of IMSRG flow equations.We see that the IMSRG(3) o ff ers substantial, systematicimprovements over the IMSRG(2). These improvements arelargely already present in approximate IMSRG(3) truncationswith lower computational cost, such as the IMSRG(3)- N . Wenote that the IMSRG(3) is not able to achieve as small of er-rors for these harder Hamiltonians as it is able to achieve forthe EM 1.8 Hamiltonian with errors of up to 0.6% in the corre-lation energy. This suggests that the many-body expansion inthe IMSRG converges more slowly when using harder Hamil-tonians (as one would also expect from perturbative argu-4 I M S R G ( ) I M S R G ( ) - M P + [ , ] → + [ , ] → I M S R G ( ) - N + [ , ] → I M S R G ( ) - N I M S R G ( ) − . − . − . − . − . . . . . ∆ E F C I ( M e V ) Truncation scheme (computational) He, HF¯ h Ω = 28 MeV e max = 2 I M S R G ( ) I M S R G ( ) - M P + [ , ] → + [ , ] → + [ , ] → I M S R G ( ) - g + [ , ] → I M S R G ( ) Truncation scheme (perturbative)FCIEM 500EMN 450EMN 500
FIG. 9. Di ff erences of ground-state energies of He obtained in various truncation schemes to exact FCI results using several unevolved chiralHamiltonians (see text for details) and an HF reference state following the computational (left panel) and perturbative (right panel) truncationordering for the fundamental commutators. Thicker, darker bars correspond to the major truncations summarized in Table IIII. Thinner, lighterbars correspond to intermediate truncations where a single fundamental commutator has been added relative to the truncation scheme to theleft. I M S R G ( ) I M S R G ( ) - M P + [ , ] → + [ , ] → I M S R G ( ) - N + [ , ] → I M S R G ( ) - N I M S R G ( ) − . − . − . − . − . . . . . ∆ E F C I ( M e V ) Truncation scheme (computational) / IMSRG(2): − .
618 MeV O, HF ~ Ω = 24 MeV e max = 2 I M S R G ( ) I M S R G ( ) - M P + [ , ] → + [ , ] → + [ , ] → I M S R G ( ) - g + [ , ] → I M S R G ( ) Truncation scheme (perturbative)extrap. FCIEMN 450EMN 500
FIG. 10. Same as Fig. 99 but for O. ments). Still, the convergence behavior of the IMSRG many- body expansion is systematic in the cases discussed here, and5the general trends discussed in Sec. IV DIV D continue to hold. V. SUMMARY AND OUTLOOK
We performed the first systematic study of the inclusion ofthree-body operators in the IMSRG in small model spaces.To this end, we presented the fundamental commutators, thebasic computational building blocks for the IMSRG, requiredfor the IMSRG(3) approximation and introduced new trunca-tions that include subsets of these commutators to understandif one can reliably approximate the IMSRG(3). We appliedthe full and approximate IMSRG(3) truncations to the closed-shell He and O using NN -only and NN + N chiral Hamilto-nians with the Hartree-Fock and natural orbital single-particlebases.When considering NN -only systems, we compared the IM-SRG(2) and IMSRG(3) results to exact results in the samemodel space obtained from FCI calculations for He and fromextrapolated FCI for O. We found that the IMSRG(3) er-ror to the (extrapolated) FCI correlation energy was consis-tently about 0.3% for the softest Hamiltonian considered andup to 0.6% for harder Hamiltonians. Moreover, the IMSRG(3)results improved systematically over the IMSRG(2) results,where the error to the (extrapolated) FCI results varied quitesignificantly for di ff erent bases and systems. This suggeststhat the many-body expansion in the IMSRG, which we havetaken to the three-body-operator level in this work, is well-behaved.We also considered the performance of various lower-costapproximate IMSRG(3) truncations relative to the full IM-SRG(3) approximation. We used the perturbative analysis ofRef. [1515] to investigate the expected size of contributions ofterms that are included in certain truncations and neglectedin others. We found that this perturbative analysis was ableto explain the size of contributions to the ground-state en-ergy by individual terms quite well. As a result, the energies calculated using approximate IMSRG(3) truncations that in-cluded commutators based on their estimated perturbative im-portance systematically converged to the full IMSRG(3) re-sult. The major truncation we considered in this approach,the IMSRG(3)- g , reproduced the full IMSRG(3) results withvery small errors for both NN -only and NN + N Hamiltoni-ans, across all frequencies, single-particle bases, and systemsconsidered.We also considered the organization of IMSRG(3) trunca-tions based on computational cost. The key major truncationof this approach, the IMSRG(3)- N , has a lower computa-tional cost than the IMSRG(3)- g truncation. The IMSRG(3)- N truncation generally saw smaller errors relative to the fullIMSRG(3) than the IMSRG(2), but the large contributions ofmissing commutators prevented its performance from beingas good as that of the IMSRG(3)- g truncation. The energyrange given by the results from these two major IMSRG(3)truncation schemes (IMSRG(3)- N and IMSRG(3)- g ) con-tained the full IMSRG(3) result in all of the cases we studied.These IMSRG(3) approximations o ff er possibilities for per-forming approximate IMSRG(3) calculations where full IM-SRG(3) calculations are no longer feasible and for studyingthe theoretical uncertainty due to the many-body truncation inIMSRG calculations. The challenge going from here is theimplementation of full and approximate IMSRG(3) calcula-tions for model spaces where nuclear Hamiltonians are con-verged. ACKNOWLEDGMENTS
We thank S. R. Stroberg for numerical checks to validateour implementation and P. Arthuis, H. Hergert, S. R. Stroberg,and J. M. Yao for useful discussions. This work was supportedin part by the Deutsche Forschungsgemeinschaft (DFG, Ger-man Research Foundation) – Project-ID 279384907 – SFB1245 and by the Max Planck Society.
Appendix A: IMSRG(3) flow equations
The uncoupled (or m -scheme) IMSRG(3) flow equations are given by dEds = (cid:88) pq ( n p ¯ n q − ¯ n p n q ) η pq f qp + (cid:88) pqrs ( n p n q ¯ n r ¯ n s − ¯ n p ¯ n q n r n s ) η pqrs Γ rspq + (cid:88) pqrstu ( n p n q n r ¯ n s ¯ n t ¯ n u − ¯ n p ¯ n q ¯ n r n s n t n u ) η pqrstu W stupqr , (A1) d f ds = (cid:88) p (cid:16) η p f p − f p η p (cid:17) + (cid:88) pq ( n p ¯ n q − ¯ n p n q ) (cid:16) η pq Γ q p − f pq η q p (cid:17) + (cid:88) pqr (¯ n p ¯ n q n r + n p n q ¯ n r ) (cid:16) η rpq Γ pq r − Γ rpq η pq r (cid:17) + (cid:88) pqrs ( n p n q ¯ n r ¯ n s − ¯ n p ¯ n q n r n s ) (cid:16) η pqrs W rs pq − Γ pqrs η rs pq (cid:17) + (cid:88) pqrst ( n p n q n r ¯ n s ¯ n t + ¯ n p ¯ n q ¯ n r n s n t ) (cid:16) η st pqr W pqrst − W st pqr η pqrst (cid:17) , (A2)6 d Γ ds = (1 − P ) (cid:88) p (cid:16) η p Γ p − f p η p (cid:17) − (1 − P ) (cid:88) p (cid:16) η p Γ p − f p η p (cid:17) + (cid:88) pq (¯ n p ¯ n q − n p n q ) (cid:16) η pq Γ pq − Γ pq η pq (cid:17) − (1 − P )(1 − P ) (cid:88) pq ( n p ¯ n q − ¯ n p n q ) η p q Γ qp + (cid:88) pq ( n p ¯ n q − ¯ n p n q ) (cid:16) η pq W q p − f pq η q p (cid:17) +
12 (1 − P ) (cid:88) pqr (¯ n p ¯ n q n r + n p n q ¯ n r ) (cid:16) η r pq W pq r − Γ r pq η pq r (cid:17) −
12 (1 − P ) (cid:88) pqr (¯ n p ¯ n q n r + n p n q ¯ n r ) (cid:16) η pqr W rpq − Γ pqr η rpq (cid:17) + (cid:88) pqrs (¯ n p ¯ n q ¯ n r n s − n p n q n r ¯ n s ) (cid:16) η spqr W pqr s − W spqr η pqr s (cid:17) +
14 (1 − P )(1 − P ) (cid:88) pqrs ( n p n q ¯ n r ¯ n s − ¯ n p ¯ n q n r n s ) η pq rs W rs pq , (A3) d W ds = P (12 / P (45 / (cid:88) p (cid:16) η p Γ p − Γ p η p (cid:17) + P (12 / (cid:88) p (cid:16) η p W p − f p η p (cid:17) − P (45 / (cid:88) p (cid:16) η p W p − f p η p (cid:17) + P (12 / (cid:88) pq (¯ n p ¯ n q − n p n q ) (cid:16) η pq W pq − Γ pq η pq (cid:17) − P (45 / (cid:88) pq (¯ n p ¯ n q − n p n q ) (cid:16) η pq W pq − Γ pq η pq (cid:17) + P (12 / P (45 / (cid:88) pq (¯ n p n q − n p ¯ n q ) (cid:16) η pq W q p − Γ pq η q p (cid:17) + (cid:88) pqr ( n p n q n r + ¯ n p ¯ n q ¯ n r ) (cid:16) η pqr W pqr − W pqr η pqr (cid:17) + P (12 / P (45 / (cid:88) pqr (¯ n p ¯ n q n r + n p n q ¯ n r ) (cid:16) η pq r W rpq − W pq r η rpq (cid:17) , (A4)where the permutation operator P pq exchanges the indices p and q in the following expression. We further define permutationoperators P ( pq / r ) ≡ − P pr − P qr and P ( p / qr ) ≡ − P pq − P pr . The action of the permutation operators in Eqs. (A3A3) and (A4A4)ensures the antisymmetry of two- and three-body matrix elements over the course of the IMSRG evolution. We note that the m -scheme IMSRG(3) flow equations agree with those in Ref. [1515], except for the following typo:1. The occupation numbers in the term on the third row of Eq. (A2A2) are corrected.Our expressions di ff er somewhat because we do not use the Hermiticity of the Hamiltonian and the anti-Hermiticity of thegenerator to manipulate the terms. We note that there is no possible reduction in the computational cost obtainable by thesemanipulations. We also provide a list of corrections between our m -scheme IMSRG(3) fundamental commutators and those inRef. [1515]:1. Our expression for the [1, 3] → / → → A and B is Hermitian and the other is anti-Hermitian. Appendix B: Spherical fundamental commutators
In practice, the IMSRG(3) framework is applied to closed-shell systems with a spherical reference state. Given the sharedrotational symmetry of the reference state and nuclear Hamiltonians, one can choose a spherical single-particle basis and use7angular-momentum-coupling techniques to significantly reduce the storage and computational cost of the IMSRG(3) solution.
1. Primer on angular-momentum coupling
We o ff er a brief introduction to the concepts of angular-momentum coupling and the associated notation. For a more detailedtreatment of the formalism of angular-momentum coupling, we refer readers to Refs. [5959, 6060].The single-particle basis is chosen to consist of spherical states | p (cid:105) ≡ | ξ p j p m p (cid:105) ≡ | ˜ pm p (cid:105) , (B1)with the total angular momentum j p , the angular-momentum projection m p , and the remaining quantum numbers that character-ize the state ξ p . In nuclear applications, ξ = ( n , l , t ), with the radial quantum number n , the orbital angular momentum l , and theisospin projection t . The reduced single-particle index ˜ p is a collective index for all the quantum numbers of the state besides m p and always has an associated j p . These spherical states are eigenstates of the one-body total angular momentum squared J and the z -projection of the one-body total angular momentum J z .When using a spherical single-particle basis, the one-body matrix elements of operators that are scalars under rotations inspace and spin (as is the case for the Hamiltonian and the generator in the IMSRG) (cid:104) ξ p j p m p | O | ξ q j q m q (cid:105) = (cid:104) ˜ pm p | O | ˜ qm q (cid:105) (B2)are diagonal in j p = j q ≡ J O and in m p = m q ≡ M O and independent of M O . This allows for the compact representation of theone-body matrix elements as O J O ˜ p ˜ q ≡ (cid:104) ξ p , j p = J O , m p = j p | O | ξ q , j q = J O , m q = j q (cid:105) , (B3)where the single-particle indices now only run over reduced indices. We have introduced a channel notation where the superscript J O indicates that the matrix elements are partitioned into channels where matrix elements in each channel are nonzero only when j p = j q = J O . While it is conventional to use j , J , and J for one-, two-, and three-body angular momenta, respectively, we optinstead to use j only for single-particle angular momenta and J for all angular momenta that appear in one-, two-, and three-bodyangular-momentum channels.The antisymmetric two-body states | pq (cid:105) ≡ a † p a † q | (cid:105) (B4)may be coupled to two-body total angular momentum J using the Clebsch-Gordan coe ffi cients C JMj p m p j q m q = (cid:104) ( ˜ p ˜ q ) JM | pq (cid:105) , (B5)yielding the coupled two-body states | ( ˜ p ˜ q ) JM (cid:105) = (cid:88) m p m q C JMj p m p j q m q | pq (cid:105) , (B6)which are eigenstates of two-body J and J z .When using coupled two-body states, the two-body matrix elements of scalars under rotations in space and spin (cid:104) ( ˜ p ˜ q ) J pq M pq | O | (˜ r ˜ s ) J rs M rs (cid:105) (B7)are diagonal in J pq = J rs ≡ J O and in M pq = M rs ≡ M O and independent of M O . This allows for the compact representation ofthese coupled matrix elements as O J O ˜ p ˜ q ˜ r ˜ s ≡ (cid:104) ( ˜ p ˜ q ) J pq = J O , M pq = J pq | O | (˜ r ˜ s ) J rs = J O , M rs = J rs (cid:105) , (B8)where the single-particle indices again only run over reduced indices, and the matrix elements have a channel structure thatspecifies to which total angular momentum J O the bra and ket states are coupled.This approach is quickly extended to three-body states | pqr (cid:105) ≡ a † p a † q a † r | (cid:105) , (B9)8where the angular momenta j p and j q are coupled to an intermediate two-body angular momentum J pq that is then coupled with j r to the three-body angular momentum J , yielding the coupled three-body states | [( ˜ p ˜ q ) J pq ˜ r ] JM (cid:105) = (cid:88) m p m q M pq m r C J pq M pq j p m p j q m q C JMJ pq M pq j r m r | pqr (cid:105) , (B10)which are eigenstates of the three-body J and J z . Here, we made a choice to couple the p and q indices first and then the r index. One could also couple two di ff erent indices in the first coupling step and then couple the remaining index last to arriveat valid eigenstates of J and J z . One arrives at a similar representation for the coupled three-body matrix elements of a scalaroperator, O ( J O , J pq , J st )˜ p ˜ q ˜ r ˜ s ˜ t ˜ u ≡ (cid:104) [( ˜ p ˜ q ) J pq ˜ r ] J pqr = J O , M pqr = J pqr | O | [( ˜ s ˜ t ) J st ˜ u ] J stu = J O , M stu = J stu (cid:105) , (B11)with J O = J pqr = J stu and M pqr = M stu . The channel structure of three-body coupled matrix elements is complicated by theappearance of the intermediate couplings J pq and J st , which do not have to be equal.Angular-momentum coupling allows one to reduce the working equations of a theory to expressions that depend only on thecoupled matrix elements discussed above. The substantial reduction in storage requirements due to working with coupled matrixelements and in computational cost by having any purely geometric dependence on angular-momentum projection analyticallysimplified is essential to making IMSRG(3) calculations tractable.For this work, we used the automated angular-momentum-coupling tool amc [6161] to generate coupled expressions for thefundamental commutators. The generated expressions and their implementations were validated by evaluating the coupled anduncoupled implementations for the same input and observing that the same coupled matrix elements were produced.
2. Coupled expressions for fundamental commutators
In the following, we present the coupled expressions for the fundamental commutators required for the IMSRG(3). We dropthe tilde from reduced single-particle indices, as all matrix elements are coupled matrix elements, and thus all indices on thematrix elements are reduced single-particle indices.The expressions are nonantisymmetrized, so the resulting two- and three-body coupled matrix elements must be antisym-metrized by applying the appropriate antisymmetrizer to the bra and ket indices. The antisymmetrization of two-body braindices is given by ¯ O J O pqrs ≡ A O J O pqrs = (cid:16) O J O pqrs − ( − j p + j q − J O O J O qprs (cid:17) , (B12)where A is the two-body antisymmetrizer and the output matrix elements ¯ O J O pqrs are antisymmetric under exchange of p and q .If the input matrix elements O J O pqrs are already antisymmetric in p and q , the antisymmetrization does nothing and the input andoutput matrix elements are identical. Similarly, the antisymmetrization of two-body ket indices is given by¯ O J O pqrs ≡ O J O pqrs A = (cid:16) O J O pqrs − ( − j r + j s − J O O J O pqsr (cid:17) . (B13)The antisymmetrization of three-body bra indices is given by¯ O ( J O , J pq , J st ) pqrstu ≡ A O ( J O , J pq , J st ) pqrstu = (cid:32) O ( J O , J pq , J st ) pqrstu + ˆ J pq (cid:88) J ˆ J j p j q J pq j r J O J O ( J O , J , J st ) rqpstu − ( − j q + j r − J pq ˆ J pq (cid:88) J ( − J ˆ J j q j p J pq j r J O J O ( J O , J , J st ) prqstu − ( − j p + j q − J pq ˆ J pq (cid:88) J ˆ J j q j p J pq j r J O J O ( J O , J , J st ) rpqstu − ( − j q + j r ˆ J pq (cid:88) J ( − J ˆ J j p j q J pq j r J O J O ( J O , J , J st ) qrpstu − ( − j p + j q − J pq O ( J O , J pq , J st ) qprstu (cid:33) , (B14)with the three-body antisymmetrizer A , ˆ J ≡ √ J +
1, and the Wigner 6j symbols j j j j j j . O ( J O , J pq , J st ) pqrstu ≡ O ( J O , J pq , J st ) pqrstu A = (cid:32) O ( J O , J pq , J st ) pqrstu + ˆ J st (cid:88) J ˆ J j s j t J st j u J O J O ( J O , J pq , J ) pqruts − ( − j t + j u − J st ˆ J st (cid:88) J ( − J ˆ J j t j s J st j u J O J O ( J O , J pq , J ) pqrsut − ( − j s + j t − J st ˆ J st (cid:88) J ˆ J j t j s J st j u J O J O ( J O , J pq , J ) pqrust − ( − j t + j u ˆ J st (cid:88) J ( − J ˆ J j s j t J st j u J O J O ( J O , J pq , J ) pqrtus − ( − j s + j t − J st O ( J O , J pq , J st ) pqrtsu (cid:33) . (B15) a. [1 , → ◦ C J C = (cid:88) p (cid:16) A J C p B J C p − B J C p A J C p (cid:17) , (B16) C (0) = (cid:88) J p ˆ J p (cid:88) pq ( n p ¯ n q − ¯ n p n q ) A J p pq B J p qp . (B17) b. [1 , → ◦ C J C = (cid:88) J A (cid:88) p (cid:16) A J A p B J C p − A J A p B J C p (cid:17) , (B18) C J C = J C (cid:88) J B ˆ J B (cid:88) J A (cid:88) pq ( n p ¯ n q − ¯ n p n q ) A J A pq B J B q p . (B19) c. [2 , → ◦ C ( J C , J , J )123456 = − J ˆ J (cid:88) p j j p J j J C J (cid:16) A J p B J p − B J p A J p (cid:17) , (B20) C J C = D J C + E J C , (B21) D J C = (cid:88) pq (¯ n p ¯ n q − n p n q ) (cid:16) A J C pq B J C pq − B J C pq A J C pq (cid:17) , (B22) E J (cid:48) C = (cid:88) pq ( n p ¯ n q − ¯ n p n q ) A J (cid:48) C pq B J (cid:48) C pq , (B23) C J C =
12 1ˆ J C (cid:88) J pq ˆ J pq (cid:88) pqr (¯ n p ¯ n q n r + n p n q ¯ n r ) (cid:16) A J pq rpq B J pq pq r − B J pq rpq A J pq pq r (cid:17) , (B24) C (0) = (cid:88) J pq ˆ J pq (cid:88) pq ( n p n q ¯ n r ¯ n s − ¯ n p ¯ n q n r n s ) A J pq pqrs B J pq rspq , (B25)where we split the [2, 2] → A (2) and0 B (2) in Eq. (B23B23) (the A and B objects) are obtained by a Pandya transformation [6262], O J (cid:48) O ≡ − (cid:88) J O ˆ J O j j J (cid:48) O j j J O O J O . (B26)The Pandya transformation is its own inverse, so the output Pandya-transformed matrix elements in Eq. (B23B23) ( E J (cid:48) C ) must bePandya transformed again to arrive at the standard coupled matrix elements ( E J C ) that contribute in Eq. (B21B21) to obtain the full[2, 2] → d. [1 , → ◦ C ( J C , J , J )123456 = (cid:88) J A (cid:88) p (cid:16) A J A p B ( J C , J , J )12 p − A J A p B ( J C , J , J )12345 p (cid:17) , (B27) C J C = J C (cid:88) J B ˆ J B (cid:88) J A (cid:88) pq ( n p ¯ n q − ¯ n p n q ) A J A pq B ( J B , J C , J C )12 q p . (B28) e. [2 , → ◦ C ( J C , J , J )123456 = D ( J C , J , J )123456 + E ( J C , J , J )123456 , (B29) D ( J C , J , J )123456 = (cid:88) pq (¯ n p ¯ n q − n p n q ) (cid:16) A J pq B ( J C , J , J ) pq − A J pq B ( J C , J , J )123 pq (cid:17) , (B30) E ( J C , J , J )123456 = (cid:88) J A , J B , J qp ( − J B + J C ˆ J A ˆ J B ˆ J qp (cid:88) pq (¯ n p n q − n p ¯ n q )( − j + j q × j j J qp j p j q J A J J J qp j p j q J B J qp J J J C j j A J A pq B ( J B , J , J )12 q p , (B31) C J C = − ( − J C ˆ J C (cid:88) J pq , J B ˆ J pq ˆ J B (cid:88) pqr (¯ n p ¯ n q n r + n p n q ¯ n r ) × ( − j + j j j J C j r J B J pq A J pq r pq B ( J B , J pq , J C ) pq r − ( − j + j j j J C j r J B J pq A J pq pqr B ( J B , J C , J pq )12 rpq , (B32) C J C =
14 1ˆ J C (cid:88) J pq , J B ˆ J B (cid:88) pqrs ( n p n q ¯ n r ¯ n s − ¯ n p ¯ n q n r n s ) A J pq pqrs B ( J B , J pq , J pq ) rs pq , (B33)where we split the [2, 3] → f. [3 , → ◦ C ( J C , J , J )123456 = D ( J C , J , J )123456 + E ( J C , J , J )123456 , (B34) D ( J C , J , J )123456 = (cid:88) J pq (cid:88) pqr ( n p n q n r + ¯ n p ¯ n q ¯ n r ) (cid:16) A ( J C , J , J pq )123 pqr B ( J C , J pq , J ) pqr − B ( J C , J , J pq )123 pqr A ( J C , J pq , J ) pqr (cid:17) , (B35) E ( J (cid:48) C , J , J )126453 = (cid:88) J pq (cid:88) pqr (¯ n p ¯ n q n r + n p n q ¯ n r ) (cid:18) A ( J (cid:48) C , J pq , J ) pqr B ( J (cid:48) C , J , J pq )126 pqr − B ( J (cid:48) C , J pq , J ) pqr A ( J (cid:48) C , J , J pq )126 pqr (cid:19) , (B36) C J C = D J C + E J C , (B37)1 D J C =
16 1ˆ J C (cid:88) J pqr ˆ J pqr (cid:88) J pq (cid:88) pqrs (¯ n p ¯ n q ¯ n r n s − n p n q n r ¯ n s ) (cid:16) A ( J pqr , J C , J pq )12 spqr B ( J pqr , J pq , J C ) pqr s − B ( J pqr , J C , J pq )12 spqr A ( J pqr , J pq , J C ) pqr s (cid:17) , (B38) E J C = − ( − j + j + J C (cid:88) J A , J B ˆ J A ˆ J B ( − J A + J B (cid:88) J pq , J rs (cid:88) J ˆ J (cid:88) pqrs ( n p n q ¯ n r ¯ n s − ¯ n p ¯ n q n r n s ) × J pq J rs J j j J A J pq J rs J j j J B j j J C j j J A ( J A , J pq , J rs ) pq rs B ( J B , J rs , J pq ) rs pq , (B39) C J C =
112 1ˆ J C (cid:88) J pqr , J pq , J st ˆ J pqr (cid:88) pqrst ( n p n q n r ¯ n s ¯ n t + ¯ n p ¯ n q ¯ n r n s n t ) (cid:16) A ( J pqr , J st , J pq ) st pqr B ( J pqr , J pq , J st ) pqrst − B ( J pqr , J st , J pq ) st pqr A ( J pqr , J pq , J st ) pqrst (cid:17) , (B40) C (0) = (cid:88) J pqr , J pq , J st ˆ J pqr (cid:88) pqrstu ( n p n q n r ¯ n s ¯ n t ¯ n u − ¯ n p ¯ n q ¯ n r n s n t n u ) A ( J pqr , J pq , J st ) pqrstu B ( J pqr , J st , J pq ) stupqr . (B41)Here we split the [3, 3] → → A (3) and B (3) in Eq. (B36B36) (the A and B objects) are obtained by the three-body analog of the Pandya transformation, O ( J (cid:48) O , J , J )126453 ≡ − (cid:88) J O ˆ J O J j J (cid:48) O J j J O O ( J O , J , J )123456 . (B42)The output Pandya-transformed matrix elements in Eq. (B36B36) must be Pandya transformed again to arrive at the standard matrixelements that contribute in Eq. (B34B34) to obtain the full [3, 3] → [1] K. Hebeler, J. D. Holt, J. Menéndez, and A. Schwenk, “Nuclearforces and their impact on neutron-rich nuclei and neutron-richmatter,” Annu. Rev. Nucl. Part. Sci. , 457 (2015)Annu. Rev. Nucl. Part. Sci. , 457 (2015).[2] T. D. Morris, J. Simonis, S. R. Stroberg, C. Stumpf, G. Ha-gen, J. D. Holt, G. R. Jansen, T. Papenbrock, R. Roth,and A. Schwenk, “Structure of the lightest tin isotopes,”Phys. Rev. Lett. , 152503 (2018)Phys. Rev. Lett. , 152503 (2018).[3] H. Hergert, “A guided tour of ab initio nuclear many-body the-ory,” Front. Phys. , 379 (2020)Front. Phys. , 379 (2020).[4] E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, “ModernTheory of Nuclear Forces,” Rev. Mod. Phys. , 1773 (2009)Rev. Mod. Phys. , 1773 (2009).[5] R. Machleidt and D. R. Entem, “Chiral e ff ective field theoryand nuclear forces,” Phys. Rep. , 1 (2011)Phys. Rep. , 1 (2011).[6] K. Hebeler, S. K. Bogner, R. J. Furnstahl, A. Nogga,and A. Schwenk, “Improved nuclear matter cal-culations from chiral low-momentum interactions,”Phys. Rev. C , 031301(R) (2011)Phys. Rev. C , 031301(R) (2011).[7] A. Ekström, G. R. Jansen, K. A. Wendt, G. Hagen,T. Papenbrock, B. D. Carlsson, C. Forssén, M. Hjorth-Jensen, P. Navrátil, and W. Nazarewicz, “Accurate nu-clear radii and binding energies from a chiral interaction,”Phys. Rev. C , 051301(R) (2015)Phys. Rev. C , 051301(R) (2015).[8] D. R. Entem, R. Machleidt, and Y. Nosyk, “High-quality two-nucleon potentials up to fifth order of the chiral expansion,”Phys. Rev. C , 024004 (2017)Phys. Rev. C , 024004 (2017).[9] E. Epelbaum, H. Krebs, and P Reinert, “High-precision nu-clear forces from chiral EFT: State-of-the-art, challenges andoutlook,” Front. Phys. , 98 (2020)Front. Phys. , 98 (2020).[10] K. Hebeler, “Three-nucleon forces: Implementationand applications to atomic nuclei and dense matter,”Phys. Rep. , 1 (2021)Phys. Rep. , 1 (2021).[11] W. G. Jiang, A. Ekström, C. Forssén, G. Hagen, G. R.Jansen, and T. Papenbrock, “Accurate bulk properties of nu-clei from A = ∞ from potentials with ∆ isobars,” (2020),arXiv:2006.16774arXiv:2006.16774. [12] G. Hagen, T. Papenbrock, M. Hjorth-Jensen, and D. J.Dean, “Coupled-cluster computations of atomic nuclei,”Rep. Prog. Phys. , 096302 (2014)Rep. Prog. Phys. , 096302 (2014).[13] S. Binder, J. Langhammer, A. Calci, and R. Roth, “Ab initiopath to heavy nuclei,” Phys. Lett. B , 119 (2014)Phys. Lett. B , 119 (2014).[14] K. Tsukiyama, S. K. Bogner, and A. Schwenk, “In-medium Similarity Renormalization Group for Nuclei,”Phys. Rev. Lett. , 222502 (2011)Phys. Rev. Lett. , 222502 (2011).[15] H. Hergert, S. K. Bogner, T. D. Morris, A. Schwenk,and K. Tsukiyama, “The In-Medium Similarity Renormal-ization Group: A Novel Ab Initio Method for Nuclei,”Phys. Rep. , 165 (2016)Phys. Rep. , 165 (2016).[16] S. R. Stroberg, H. Hergert, S. K. Bogner, and J. D. Holt,“Nonempirical interactions for the nuclear shell model: An up-date,” Annu. Rev. Nucl. Part. Sci. , 307 (2019)Annu. Rev. Nucl. Part. Sci. , 307 (2019).[17] W. H. Dickho ff and C. Barbieri, “Self-consistentGreen’s function method for nuclei and nuclear matter,”Prog. Part. Nucl. Phys. , 377 (2004)Prog. Part. Nucl. Phys. , 377 (2004).[18] V. Somà, P. Navrátil, F. Raimondi, C. Barbieri, and T. Duguet,“Novel chiral Hamiltonian and observables in light andmedium-mass nuclei,” Phys. Rev. C , 014318 (2020)Phys. Rev. C , 014318 (2020).[19] J. D. Holt, J. Menéndez, J. Simonis, and A. Schwenk, “Three-nucleon forces and spectroscopy of neutron-rich calcium iso-topes,” Phys. Rev. C , 024312 (2014)Phys. Rev. C , 024312 (2014).[20] A. Tichai, J. Langhammer, S. Binder, and R. Roth, “Hartree-Fock many-body perturbation theory for nuclear ground-states,” Phys. Lett. B , 283 (2016)Phys. Lett. B , 283 (2016).[21] A. Tichai, P. Arthuis, T. Duguet, H. Hergert, V. Somá, andR. Roth, “Bogoliubov Many-Body Perturbation Theory forOpen-Shell Nuclei,” Phys. Lett. B , 195 (2018)Phys. Lett. B , 195 (2018).[22] A. Tichai, R Roth, and T. Duguet, “Many-body perturbationtheories for finite nuclei,” Front. Phys. , 164 (2020)Front. Phys. , 164 (2020).[23] T. A. Lähde, E. Epelbaum, H. Krebs, D. Lee, U.-G. Meißner,and G. Rupak, “Lattice E ff ective Field Theory for Medium-Mass Nuclei,” Phys. Lett. B , 110 (2014)Phys. Lett. B , 110 (2014). [24] Y. S. Lee, S. A. Kucharski, and R. J. Bartlett,“A coupled cluster approach with triple excitations,”J. Chem. Phys. , 5906 (1984)J. Chem. Phys. , 5906 (1984).[25] J. Noga and R. J. Bartlett, “The full CCSDT model for molecu-lar electronic structure,” J. Chem. Phys. , 7041 (1987)J. Chem. Phys. , 7041 (1987).[26] G. E. Scuseria and H. F. Schaefer, “A new implementationof the full CCSDT model for molecular electronic structure,”Chem. Phys. Lett. , 382 (1988)Chem. Phys. Lett. , 382 (1988).[27] P. Piecuch and M. Włoch, “Renormalized coupled-clustermethods exploiting left eigenstates of the similarity-transformed Hamiltonian,” J. Chem. Phys. , 224105 (2005)J. Chem. Phys. , 224105 (2005).[28] A. G. Taube and R. J. Bartlett, “Improving uponCCSD(T): Λ CCSD(T). I. Potential energy surfaces,”J. Chem. Phys. , 044110 (2008)J. Chem. Phys. , 044110 (2008).[29] S. Binder, P. Piecuch, A. Calci, J. Langhammer, P. Navrátil, andR. Roth, “Extension of coupled-cluster theory with a nonitera-tive treatment of connected triply excited clusters to three-bodyHamiltonians,” Phys. Rev. C , 054319 (2013)Phys. Rev. C , 054319 (2013).[30] G. Hagen, G. R. Jansen, and T. Papenbrock, “Struc-ture of Ni from first-principles computations,”Phys. Rev. Lett. , 172501 (2016)Phys. Rev. Lett. , 172501 (2016).[31] J. Simonis, S. R. Stroberg, K. Hebeler, J. D. Holt, andA. Schwenk, “Saturation with chiral interactions and conse-quences for finite nuclei,” Phys. Rev. C , 014303 (2017)Phys. Rev. C , 014303 (2017).[32] M. Miorelli, S. Bacca, G. Hagen, and T. Papenbrock, “Comput-ing the dipole polarizability of Ca with increased precision,”Phys. Rev. C , 014324 (2018)Phys. Rev. C , 014324 (2018).[33] S. Kaufmann, J. Simonis, S. Bacca, J. Billowes, M. L.Bissell, K. Blaum, et al. , “Charge Radius of the Short-Lived Ni and Correlation with the Dipole Polarizability,”Phys. Rev. Lett. , 132502 (2020)Phys. Rev. Lett. , 132502 (2020).[34] S. J. Novario, P. Gysbers, J. Engel, G. Hagen, G. R. Jansen,T. D. Morris, P. Navrátil, T. Papenbrock, and S. Quaglioni,“Coupled-cluster calculations of neutrinoless double-beta de-cay in Ca,” (2020), arXiv:2008.09696arXiv:2008.09696.[35] H. Hergert, J. M. Yao, T. D. Morris, N. M. Parzu-chowski, S. K. Bogner, and J. Engel, “Nuclear Struc-ture from the In-Medium Similarity Renormalization Group,”J. Phys. Conf. Ser. , 012007 (2018)J. Phys. Conf. Ser. , 012007 (2018).[36] C. Li and F. A. Evangelista, “Connected three-body terms insingle-reference unitary many-body theories: Iterative and per-turbative approximations,” J. Chem. Phys. , 234116 (2020)J. Chem. Phys. , 234116 (2020).[37] F. Wegner, “Flow-equations for Hamiltonians,”Ann. Phys. , 77 (1994)Ann. Phys. , 77 (1994).[38] S. D. Glazek and K. G. Wilson, “Renormalization of Hamilto-nians,” Phys. Rev. D , 5863 (1993)Phys. Rev. D , 5863 (1993).[39] S. K. Bogner, R. J. Furnstahl, and R. J. Perry, “Similar-ity Renormalization Group for Nucleon-Nucleon Interactions,”Phys. Rev. C , 061001(R) (2007)Phys. Rev. C , 061001(R) (2007).[40] S. K. Bogner, R. J. Furnstahl, and A. Schwenk,“From low-momentum interactions to nuclear structure,”Prog. Part. Nucl. Phys. , 94 (2010)Prog. Part. Nucl. Phys. , 94 (2010).[41] E. D. Jurgenson, P. Navrátil, and R. J. Furnstahl, “Evolution ofNuclear Many-Body Forces with the Similarity Renormaliza-tion Group,” Phys. Rev. Lett. , 082501 (2009)Phys. Rev. Lett. , 082501 (2009). [42] K. Hebeler, “Momentum-space evolution of chiral three-nucleon forces,” Phys. Rev. C , 021002(R) (2012)Phys. Rev. C , 021002(R) (2012).[43] G. C. Wick, “The Evaluation of the Collision Matrix,”Phys. Rev. , 268 (1950)Phys. Rev. , 268 (1950).[44] G. Hagen, T. Papenbrock, D. J. Dean, A. Schwenk, A. Nogga,M. Włoch, and P. Piecuch, “Coupled-cluster theory for three-body Hamiltonians,” Phys. Rev. C , 034302 (2007)Phys. Rev. C , 034302 (2007).[45] R. Roth, S. Binder, K. Vobig, A. Calci, J. Langhammer, andP. Navrátil, “Medium-Mass Nuclei with Normal-Ordered Chi-ral NN +
3N Interactions,” Phys. Rev. Lett. , 052501 (2012)Phys. Rev. Lett. , 052501 (2012).[46] T. D. Morris,
Systematic improvements of ab-initio in-mediumsimilarity renormalization group calculations , Ph.D. thesisPh.D. thesis,Michigan State University, East Lansing (2016).[47] A. Cipollone, C. Barbieri, and P. Navrátil, “Isotopic chainsaround oxygen from evolved chiral two- and three-nucleon in-teractions,” Phys. Rev. Lett. , 062501 (2013)Phys. Rev. Lett. , 062501 (2013).[48] J. Zhao and G. E. Scuseria,https: // github.com / tschijnmo / drudge (2021).[49] J. D. Watts and R. J. Bartlett, “Economical triple excitationequation-of-motion coupled-cluster methods for excitation en-ergies,” Chem. Phys. Lett. , 81 (1995)Chem. Phys. Lett. , 81 (1995).[50] S. R. Stroberg, private communication (2020).[51] D. R. Entem and R. Machleidt, “Accurate charge-dependentnucleon-nucleon potential at fourth order of chiral perturbationtheory,” Phys. Rev. C , 041001(R) (2003)Phys. Rev. C , 041001(R) (2003).[52] S. R. Stroberg, https: // github.com / ragnarstroberg / imsrg (2021).[53] M. R. Strayer, W. H. Bassichis, and A. K. Kerman, “CorrelationE ff ects in Nuclear Densities,” Phys. Rev. C , 1269 (1973)Phys. Rev. C , 1269 (1973).[54] A. Tichai, J. Müller, K. Vobig, and R. Roth, “Natu-ral orbitals for ab initio no-core shell model calculations,”Phys. Rev. C , 034321 (2019)Phys. Rev. C , 034321 (2019).[55] J. Hoppe, A. Tichai, M. Heinz, K. Hebeler, andA. Schwenk, “Natural orbitals for many-body expansion meth-ods,” Phys. Rev. C , 014321 (2021)Phys. Rev. C , 014321 (2021).[56] T. D. Morris, N. M. Parzuchowski, and S. K. Bogner, “Magnusexpansion and in-medium similarity renormalization group,”Phys. Rev. C , 034331 (2015)Phys. Rev. C , 034331 (2015).[57] N. Shimizu, T. Mizusaki, Y. Utsuno, and Y. Tsunoda, “Thick-Restart Block Lanczos Method for Large-Scale Shell-ModelCalculations,” Comput. Phys. Commun. , 372 (2019)Comput. Phys. Commun. , 372 (2019).[58] R. Roth, J. Langhammer, A. Calci, S. Binder, andP. Navrátil, “Similarity-Transformed Chiral NN +
3N Inter-actions for the Ab Initio Description of C and O,”Phys. Rev. Lett. , 072501 (2011)Phys. Rev. Lett. , 072501 (2011).[59] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii,
Quantum Theory of Angular MomentumQuantum Theory of Angular Momentum (World Scientific, Sin-gapore, 1988).[60] J. Suhonen,
From Nucleons to Nucleus: Concepts of Micro-scopic Nuclear Theory (Springer Berlin Heidelberg, Berlin,Heidelberg, 2007).[61] A. Tichai, R. Wirth, J. Ripoche, and T. Duguet, “Sym-metry reduction of tensor networks in many-body the-ory I. Automated symbolic evaluation of
S U (2) algebra,”Eur. Phys. J. A , 272 (2020)Eur. Phys. J. A , 272 (2020).[62] S. P. Pandya, “Nucleon-Hole Interaction in jj Coupling,”Phys. Rev.103
S U (2) algebra,”Eur. Phys. J. A , 272 (2020)Eur. Phys. J. A , 272 (2020).[62] S. P. Pandya, “Nucleon-Hole Interaction in jj Coupling,”Phys. Rev.103 , 956 (1956)Phys. Rev.103