Successive variational approach with the tensor-optimized antisymmetrized molecular dynamics for the ^5He nucleus
aa r X i v : . [ nu c l - t h ] F e b Prog. Theor. Exp. Phys. , 00000 (14 pages)DOI: 10.1093 / ptep/0000000000 Successive variational approach with thetensor-optimized antisymmetrized moleculardynamics for the He nucleus
Takayuki Myo , Mengjiao Lyu , Hiroshi Toki , and Hisashi Horiuchi General Education, Faculty of Engineering, Osaka Institute of Technology, Osaka535-8585, Japan ∗ E-mail: [email protected] Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka567-0047, Japan College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing210016, China Key Laboratory of Aerospace Information Materials and Physics, Ministry ofIndustry and Information Technology, Nanjing 210016, China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
We study He variationally as the first p -shell nucleus in the tensor-optimized antisym-metrized molecular dynamics (TOAMD) using the bare nucleon–nucleon interactionwithout any renormalization. In TOAMD, the central and tensor correlation opera-tors promote the AMD’s Gaussian wave function to a sophisticated many-body stateincluding the short-range and tensor correlations with high-momentum nucleon pairs.We develop a successive approach by applying these operators successively with up todouble correlation operators to get converging results. We obtain satisfactory results for He, not only for the ground state but also for the excited state, and discuss explicitlythe correlated Hamiltonian components in each state. We also show the importance ofthe independent optimization of the correlation functions in the variation of the totalenergy beyond the condition assuming common correlation forms used in the Jastrowapproach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index D10, D11, D13
1. Introduction
The bare nucleon–nucleon (
N N ) interaction has a strong short-range repulsion and a strongtensor force [1, 2]. In finite nuclei, the short-range repulsion produces a short-range corre-lation reducing the short-range amplitudes of nucleon pairs. The tensor force produces atensor correlation with a strong D -wave transition of nucleon pairs. The two correlationshave different characters in physics, but commonly induce the high-momentum componentsof nucleon motion in nuclei [3].The Green’s function Monte Carlo (GFMC) simulation by the Argonne group has demon-strated that they can reproduce the binding energies and low-lying energy spectra for lightnuclei up to C with the help of three-nucleon forces [4]. In GFMC, no renormalization © The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited. echnique is applied to the nuclear wave function and various nuclear properties are cal-culated directly using the resulting wave function. One of the important results in GFMCis that the one-pion exchange contribution on the nuclear binding energy is about 80% ofthe entire contribution of the two-body interaction [1]. This pion-exchange component is adominant source of the tensor force. At present, this numerical method requires extremecomputational time to be applied to heavier nuclei. It is by now highly desirable to developa new method to calculate nuclear structure with large nucleon numbers by taking care ofthe characteristics of the
N N interaction.To this end, we have introduced the tensor-optimized antisymmetrized molecular dynamics(TOAMD) [5–15], which is an analytical variational approach directly treating the correla-tions induced by nuclear force. In TOAMD, the reference wave function is taken from AMD[16], which nicely describes light nuclei with effective interactions, in particular for cluster-ing states such as the Hoyle state in C. In TOAMD, we introduce two kinds of variationalcorrelation functions of tensor- and central-operator types to treat the correlations inducedby the bare
N N interaction. The correlation functions are multiplied by the AMD wavefunction and the resulting many-body wave functions are used to calculate the Hamiltonianmatrix elements completely. This prescription ensures the energy-variational principle fornuclei in TOAMD.So far, TOAMD has been applied to s -shell nuclei within the double products of thecorrelation functions, and we have successfully reproduced the properties of s -shell nuclei [6–9]. It is noted that in TOAMD, multiple products of the correlation functions are introducedand each correlation function in each term is independently optimized to minimize the totalenergy of the nucleus, which indicates the flexibility of the correlation functions. Owing tothis property, we have shown that the binding energies in TOAMD are better than the valuesobtained in the Jastrow correlation method using common forms of the correlation functionsfor every pair [7, 8]. We have recently applied the concept of TOAMD to the variationaldescription of nuclear matter starting from the N N interaction [17, 18].With the success in s -shell nuclei and nuclear matter, it is important to apply TOAMDto p -shell nuclei, where the Argonne group is only the one to perform complete variationalcalculations without any renormalization [1]. There are several theoretical studies for p -shellnuclei using no-core shell model and lattice simulations with a renormalization method. Inthese studies, the nuclear force is constructed first in the chiral perturbation theory forvarious cut-off parameters for the pion-exchange interaction, with which high-momentumcomponents are controlled [19]. The similarity renormalization group is used further to softenthe interaction [20, 21], and then no-core shell model is used to perform diagonalization ofa large Hamiltonian matrix in a limited space [22]. Lattice simulations are performed withthe effective field theory using the chiral nuclear force [23]. In these methods, it is essentialto avoid the high-momentum components in nuclei caused by the interaction to make theconvergence of solutions faster with respect to the model space. The present TOAMD methodtakes quite a different approach, where we treat the bare correlations explicitly as much aspossible without any truncation and discuss the properties of the obtained nuclear wavefunction directly. There are other methods where the N N correlations are directly treated[24–27], and in some cases they attempt calculations of the p -shell nucleus, He, with thebare
N N interaction [26]. he purpose of this paper is to show the ability of TOAMD by applying central and tensorcorrelation operators successively to obtain a satisfactory result for p -shell nuclei. As a firsttrial of TOAMD to p -shell nuclei, we calculate He as a five-body problem. We investigatethe structure of this nucleus within the double products of the correlation functions. The roleof each correlation function is analysed focusing on their independent optimization. We alsoextend TOAMD by superposing the different reference AMD wave functions in a generatorcoordinate method, which enables us to obtain the excited states as well as the ground statesimultaneously. We discuss the structures of ground and excited states of He in a unifiedTOAMD framework.In TOAMD, the increase of the variational accuracy is straightforward by successivelyadding the higher orders of the correlation functions in the wave function. The necessarymatrix elements at any order of TOAMD are calculated in the analytical form. In general,an increase of the order of TOAMD requires more computer resources.In Sect. 2, we explain the nuclear models of AMD and TOAMD. In Sect. 3, we present theresults of the s -shell nuclei H and He, and the p -shell nucleus He in TOAMD. A summaryis given in Sect. 4.
2. Tensor-optimized antisymmetrized molecular dynamics (TOAMD)
We explain the formulation of TOAMD [5, 9]. The reference AMD wave function Φ
AMD isthe Slater determinant of nucleons with mass number A asΦ AMD = 1 √ A ! det ( A Y i =1 ψ σ i τ i ( r i ) ) , (1) ψ στ ( r ) = (cid:18) νπ (cid:19) / e − ν ( r − D ) χ σ χ τ . (2)The nucleon wave function ψ στ ( r ) has a Gaussian wave packet with a common range param-eter ν and a centroid position D , a spin part χ σ , and an isospin part χ τ . In this study, χ σ isthe up or down component and χ τ is a proton or neutron. The AMD wave function Φ AMD has a set of D = { D i } with i = 1 , . . . , A with the condition of P Ai =1 D i = .We introduce two kinds of the two-body correlation functions, F D for the tensor force and F S for the short-range repulsion, to make the correlated wave function. These functions aredefined as F D = X t =0 A X i AMD and superpose these components. Wedefine the TOAMD wave function with the single correlation functions asΦ singleTOAMD = (1 + F S + F D ) × Φ AMD . (5)We can increase the order of the correlation function in TOAMD by further adding thedouble products of F D and F S asΦ TOAMD = (1 + F S + F D + F S F S + F D F S + F D F D ) × Φ AMD . (6)In the present study, we use this form of TOAMD for calculations of nuclei up to the secondorders, which are based on the power series expansion in terms of correlations F D and F S independently. It is noted that all of F D and F S in each term in Eq. (6) are independentand variationally determined, which means that there are four kinds of F D and four kindsof F S , while we use the common notations of F D and F S in the equations for simplicity.This property of TOAMD brings a flexibility of the correlation functions in comparisonwith the so-called Jastrow method, in which common correlation functions are assumed inall nucleon pairs [28]. As an extension of Eq. (6), we can successively increase the orderof power expansion to triple products such as F D F D F S when we increase the variationalaccuracy of the solutions. This extension is feasible and would require more computationalresources for numerical calculation in future. The form of wave function in TOAMD givenin Eq. (6) is general and commonly used for all nuclei.We use the Hamiltonian with a two-body bare N N interaction V for mass number A as H = A X i t i − T c . m . + A X i AMD . The operators e H and e N are the correlated Hamiltonian and norm operator, respectively. They involve themultiple products of operators such as F † HF and F † F , where F stands for F D and F S .We evaluate the matrix elements of the correlated operators with the AMD wave function.The operators e H and e N consist of various products of correlation functions and they areindividually expanded into many-body operators using the cluster expansion technique [5, 9]. ig. 1 Diagrams of the cluster expansion of F † F from two-body to four-body terms,where the vertical and horizontal lines represent the particles and the correlation function F , respectively. The numbers in the square brackets describe the particle index specifyingthe configuration of each diagram [8]. Fig. 2 Diagrams of the cluster expansion of F † V F from two-body to six-body terms,where V represents a two-body operator with dotted lines.In the case of F † F , this product is expanded into two-body, three-body and four-bodyoperators as shown in the diagrams of Fig. 1. For the two-body interaction V , the correlatedinteraction F † V F gives up to six-body operators and their diagrams are shown in Fig. 2. Forone-body operators such as kinetic energy and square radius, the correlated operators give upto five-body ones and their diagrams are shown in Fig. 3. Similarly, the correlated operators F † F † T F F and F † F † V F F give up to nine-body and ten-body operators, respectively.In TOAMD, we adopt all of the resulting many-body operators in the cluster expansionof the correlated operators and calculate their many-body matrix elements using the AMDwave function. This treatment is important to keep TOAMD as a variational framework.The calculation of the matrix elements of many-body operators is performed analytically forany order of the multiple products of the correlation functions [5, 9]. In general, higher-body ig. 3 Diagrams of the cluster expansion of F † T F from two-body to five-body terms,where T represents a one-body operator with solid circles.operators in the cluster expansion tend to require a computational cost to calculate theirmatrix elements.The TOAMD wave function has two kinds of variational functions, the AMD wave func-tion in Eq. (1) and the correlation functions F in Eqs. (3) and (4). We determine thesefunctions under the Ritz variational principle for the total energy as δE = 0 in Eq. (8). Inthe determination of the radial distribution of F D and F S , we use the Gaussian expansionto express the pair functions f tD ( r ) and f t,sS ( r ): f tD ( r ) = N G X n =1 C tn e − a tn r , (9) f t,sS ( r ) = N G X n =1 C t,sn e − a t,sn r , (10)where a tn , a t,sn , C tn , and C t,sn are variational parameters with the index n . We take thenumber of Gaussian functions N G = 6 to get converging solutions. For the Gaussian ranges a tn , a t,sn , we search for their optimized values in a wide range. The coefficients C tn and C t,sn are linear parameters and are determined by diagonalizing the Hamiltonian matrix. Typicaldistributions of f tD ( r ) and f t,sS ( r ) are shown for s -shell nuclei in Ref. [8].For the double products of correlation functions such as F D F D , the products of two Gaus-sian functions in Eq. (9) are treated as single basis functions with an amplitude of C tn C t ′ n ′ . Inthe same manner, the products of C tn and C t,sn , C tn C t ′ n ′ , C tn C t ′ ,sn ′ , and C t,sn C t ′ ,s ′ n ′ become linearparameters.Finally, we rewrite the TOAMD wave function in Eq. (6) in a linear combination formusing the coefficients e C α of the Gaussian expansion of the correlation functions:Φ TOAMD = X α =0 e C α Φ TOAMD ,α , (11) H αβ = h Φ TOAMD ,α | H | Φ TOAMD ,β i ,N αβ = h Φ TOAMD ,α | Φ TOAMD ,β i , here the labels α and β are the set of the Gaussian index n and the labels s and t in thecorrelation functions, and the summation includes all the single and double correlated states.The case with labels of α = β = 0 indicates the AMD wave function. The Hamiltonian andnorm matrix are H αβ and N αβ , respectively. We solve the generalized eigenvalue problem todetermine the total energy E and the coefficients e C α in Eq. (11): X β =0 ( H αβ − E N αβ ) e C β = 0 . (12)We explain the procedure to evaluate the matrix elements of the correlated Hamiltonianand norm using the AMD wave function in Eq. (8). We express the N N interaction V as a sum of Gaussians, similar to the correlation functions. In the cluster expansion of e H and e N , many-body operators have various combinations of the square of the interparticlecoordinates r ij in the Gaussians. We perform a Fourier transformation of each Gaussian withthe individual momentum k , which results in the product of the plane waves, e i k · r i e − i k · r j .We calculate the single-particle matrix elements of the plane waves with various momentain AMD using Eq. (2). Using these matrix elements, we perform multiple integration overall momenta and obtain the correlated matrix elements. Typical analytical expressions ofthe many-body matrix elements are given in Refs. [5, 9].In the present study, we adopt the intrinsic AMD wave function Φ AMD in Eq. (1), which isa mixed state of the J π components. In TOAMD, the fraction of the J π components in AMDis changeable in the total wave function given in Eq. (6) due to the degrees of freedom ofthe correlation functions F . The function F is expanded in a linear combination form usingthe Gaussian functions in Eqs. (9) and (10), and the expansion coefficients are optimizedby solving the energy eigenvalue problem in Eq. (12) and depend on the eigenstates ofTOAMD. In general, the function F works to ensure that the TOAMD wave function hasgood quantum numbers. We have checked this property in H and He using the non s -waveconfigurations of Φ AMD , and the results of TOAMD within F show their ground states withpositive parity.We can extend TOAMD superposing the various AMD wave functions in the generatorcoordinate method (GCM). We express the different AMD wave functions Φ AMD ,k with theindex k , in which the set of Gaussian centroid positions D is different in Eq. (1). The numberof the AMD wave functions is N GCM . In TOAMD, the correlation functions also depend onthe index k . We call this method TOAMD+GCM and the corresponding total wave functionΦ GCM is given as Φ GCM = N GCM X k =1 Φ TOAMD ,k , (13)Φ TOAMD ,k = (1 + F S,k + F D,k + F S,k F S,k + F D,k F S,k + F D,k F D,k ) × Φ AMD ,k = X α e C α,k Φ TOAMD ,α,k . (14)The amplitudes { e C α,k } are determined in the minimization of the total energy E GCM as aneigenvalue problem in the same form as Eq. (12). able 1 Total energies E of H ( 12 + ) and He (0 + ) in TOAMD with the AV6 ′ potentialin units of MeV. Two kinds of ν values in the Gaussian are used. One is ν = 0 . 15 fm − optimized for He and the other is the optimized ν for each nucleus, denoted as “Optimized ν ”. ν = 0 . 15 Optimized ν GFMC [27, 30, 31] H − . − . 93 ( ν = 0 . − . He − . − . 01 ( ν = 0 . − . Table 2 Total energies E of the ground state of He in TOAMD with AV6 ′ potential inunits of MeV. We add the correlation terms successively.AMD + S + D + SS + DS + DD . 72 10 . − . − . − . − . 3. Results s -shell nuclei We start from the single configuration of AMD in TOAMD in Eq. (6). We discuss theapplicability of TOAMD showing the results of s -shell nuclei, H and He, in which s -waveconfigurations of the AMD wave function are used with the centroid parameters D = forall nucleons. This condition has been shown to be variationally favored [6]. In Table. 1,we summarize the energies of two nuclei with two kinds of the range parameter ν in theGaussian wave packet in Eq. (2); One is ν = 0 . 15 fm − , which is adopted to minimize thetotal energy of He. Another choice is that ν is optimized for each nucleus. It is found thatthe energy differences between the two kinds of ν values are small. This means that thesolutions are less dependent on ν in TOAMD. This is because the correlation functions canoptimize the TOAMD wave function to minimize the total energy. The GFMC calculationprovides − . 95 MeV for H and − . 85 MeV for He after subtracting the Coulomb forcecontribution, which amounts to 0 . He [27]. We see that the energiesobtained in TOAMD are close to these values in two nuclei as shown in Table 1. From thesecomparisons, our TOAMD solutions provide sufficiently reliable energies for s -shell nuclei,and we are able to discuss the spectroscopic properties of each nucleus.When we want to increase the variational accuracy to be close to rigorous calculations suchas GFMC, more correlation terms such as the triple products of F , would be successivelyadded, although more computing power would be demanded. He with single configuration of AMD We calculate He in TOAMD with ν = 0 . 15 fm − and start from the single He+ n clus-ter configuration of the AMD wave function with an s -wave state of He, namely, D i =(0 , , − d/ 5) for i = 1 , . . . , D = (0 , , d/ 5) for the Gaussian centroid parameters witha cluster distance d . It is noted that the He nucleus in He is not the s -wave state inTOAMD due to the correlation functions F , which induce the excitation of nucleons fromthe s -wave state. He+n E He E n e r gy [ M e V ] Correlation functionsFree F Fig. 4 Total energy E of the ground state of He with the AV6 ′ potential by addingeach correlation term successively. The solid circles are the results with full variation of thecorrelation functions F . The open circles are the results assuming the common correlationfunctions. The dashed horizontal line indicates the He+ n threshold energy. -60-40-20020 AMD +S +D +SS +DS +DDC K/2T E n e r gy [ M e V ] Correlation functionsCommon F Free F Fig. 5 Components of the kinetic energy K , central force C , and tensor force T in theground state of He with the AV6 ′ potential. The half value of kinetic energy is shown denotedas K/ 2. The solid circles are the results with full variation of the correlation functions F .The open circles are the results assuming the common correlation functions.We take the cluster distance d as 0 . p -orbit due to theantisymmetrization and the parity is obtained as − . 00 in the AMD wave function Φ AMD .In Table 2, the total energy of the He ground state is shown by adding the correlationterms successively, which corresponds to extending the variational space. The same resultsare shown using the solid line in Fig. 4. We start from the case of the AMD wave function,where the resulting energy is very high as denoted by AMD in Fig. 4. We then add thecorrelation functions step by step and see the converging behavior in energy. The label of +S eans the addition of the F S component and the TOAMD wave function is (1 + F S ) Φ AMD .The +DD case is the full component of the TOAMD wave function defined in Eq. (6). Thefinal energy is − . He+ n threshold energy by about 5.7MeV because of the unbound nature of the He system.In this study, we do not perform parity projection on the TOAMD wave function, sincethe final state has an expectation value of parity of − . 00. Hence, the resulting state has oddparity, and we can identify it with the p -wave state of He. According to the GFMC resultswith AV6 ′ [30, 31], the energy of He is − . / − state and − . / − state. The p -orbit splitting is small at around 80 keV. From this fact and alsoconsidering the optimization of the quantum number in TOAMD by the correlation functions F explained in Sect. 2, we estimate that the effect of angular momentum projection is smallin the present calculation of TOAMD. Among the 3 / − and 1 / − components, we considerthat the resulting ground state of He can be dominated by the 3 / − state, which is favoredin energy.In Fig. 5, the Hamiltonian components of the He ground state are shown by addingcorrelation terms in a similar way to that shown in Fig. 4. It is found that as correlation termsare added successively, every component increases its magnitude and more correlations areinvolved in the wave function. In particular, the enhancement of the kinetic energy indicatesthe inclusion of the high-momentum component in the wave function induced by short-range repulsion and tensor force by the correlation functions. In TOAMD, we can discussexplicitly the contribution of each Hamiltonian component at each step of the variationalspace without any renormalization, which is the advantage of TOAMD.In TOAMD, the correlation functions F S and F D are optimized independently in eachterm of Eq. (6) at each variational space. It is interesting to see this effect on the solutionsof He, and for this purpose we perform the following calculation. First, F S and F D aredetermined in the single correlation function of TOAMD as Φ singleTOAMD , defined in Eq. (5).Second, keeping the radial form of F S and F D with Gaussian expansion in all correlationterms, we perform the calculation including double correlation functions, where the weightsof the double correlation functions are variational parameters. This calculation correspondsto the Jastrow correlation method in which every nucleon pair in nuclei is correlated bythe common correlation function. Under this condition, TOAMD provides the energies of He as − . He [8]. This amount indicates the importance of the independentoptimization of the correlation functions in TOAMD and this property contributes to thegood energy convergence. We show this effect step by step at each correlation level usingopen circles connected by dashed lines in Figs. 4 and 5. In the full TOAMD calculation withup to the F D F D term, each Hamiltonian component show clear differences by more than 5MeV. Hence, the variation of both correlation functions of F S and F D contributes to theoptimization of the total wave function. He with the generator coordinate method (GCM) We extend TOAMD by superposing the TOAMD basis states having different AMD wavefunctions according to Eq. (13) and investigate this effect on He. In TOAMD+GCM, wecan discuss the structures of not only the ground state but also the excited state of He. He+n E n e r gy [ M e V ] Correlation functions2 Fig. 6 Total energy E GCM of the ground (1 st ) and excited (2 nd ) states of He with the AV6 ′ potential by adding each correlation term successively. The solid (open) circles indicate theground (excited) state. The horizontal arrow on the right-hand side indicates the theoreticalthreshold energy of the H+ d state. Table 3 Total energies E GCM of the ground and excited states of He in TOAMD+GCMwith the AV6 ′ potential in units of MeV.+ S + D + SS + DS + DD ground state 7 . − . − . − . − . . 47 0 . − . − . − . d between He and n being 0.2fm, 0.5 fm, and 1.0 fm. This choice is sufficient to include the effect of the valence-neutronmotion of He without continuum states. It is noted that, when we start from the differentAMD configurations of H+ d with the s -wave configurations for both H and d , the resultingenergies are almost the same as those obtained in the He+ n case for both the ground andexcited states. This fact indicates that the correlation functions in TOAMD can optimizethe total wave function even starting from different reference AMD wave functions.We show the results of He in TOAMD+GCM in Table 3, in which the successive additionof correlation terms is performed. The results are also shown in Fig. 6. For the groundstate, its energy is obtained as − . 63 MeV, which shows an energy gain of 0.4 MeV fromthe value of TOAMD with a single AMD wave function. This result indicates that most ofthe important correlations are described by the correlation functions in TOAMD and theGCM effect on the ground state energy is not large. In Table 4, we show the Hamiltoniancomponents of He. The difference between the results for the single AMD case shown inTable 2 and GCM is less than 2 MeV for each component. Hence we can conclude that theGCM effect is not large in TOAMD because of the optimization of the correlation functionswith even a single AMD wave function.In the GCM calculation, we newly obtained the excited state of He, which has dominantlya positive parity component of 93%. The total energy, Hamiltonian components, and radius able 4 Hamiltonian components and radius of the ground and excited states of He inTOAMD+GCM in units of MeV and fm for energy and radius, respectively. The results of He are also shown for comparison.Kinetic Central Tensor Radiusground state 99 . − . − . 46 2 . . − . − . 24 2 . He 90 . − . − . 49 1 . − 11 MeV, which is very close tothe threshold energy consisting of H and d of about − 10 MeV, indicated by the arrow inFig. 6. Experimentally, there is an excited 3 / + state located just above the H+ d thresholdenergy by about 50 keV with a very small decay width of 75 keV [32]. The obtained radiusof the excited state is shown to be larger than that of the ground state by about 0.3 fmas shown in Table 4 under the bound state approximation. This could be an indication ofthe clustering state. We consider that the resulting excited state could be a candidate forthe H+ d clustering state. For the spin quantum number of the state, there might be amixing of the 5 / + state in the spectral function, which is experimentally located abovethe 3 / + state by about 2.3 MeV, although the 3 / + state is regarded as a dominantcomponent energetically. We will investigate the detailed structure of this interesting statein a forthcoming paper.We compare the Hamiltonian components between He and He to see the effect of a lastneutron in Table 4. The ground state of He shows larger values in magnitude for everyHamiltonian component, indicating more correlations than those of He by the interactionwith a last neutron. On the other hand, the excited state of He shows smaller interactioncomponents than those of He. This trend indicates the possibility of a different configurationfrom the He+ n one in the excited state. It is important to note that the TOAMD frameworkcan treat the ground and excited states in a unified manner with a diagonalization of theHamiltonian matrix under the variational principle. 4. Summary We have developed a new variational method of “tensor-optimized antisymmetrized molec-ular dynamics” (TOAMD) for nuclei, which is a successive approach to treat bare nucleon–nucleon interactions without any renormalizations of the wave function and the interaction.Based on the successful results for s -shell nuclei, in this paper we have reported the firstapplication of TOAMD to the p -shell nucleus, He, solving a five-body problem. We preparethe reference AMD wave function of He with a He+ n cluster configuration and multiplythe variational correlation functions of central and tensor types successively with up to thedouble products.In TOAMD, the products of the Hamiltonian and the correlation function are expanded ina series of many-body operators using the cluster expansion. We explicitly treat all of theseoperators in the calculation of the matrix elements in TOAMD without any truncation ofthe higher-body operators. This is an important point to keep the variational principle in the OAMD wave function. Owing to this advantage, we directly discuss the nuclear propertiesobtained in TOAMD without any transformation.We further performed superposition of the TOAMD basis states with different He– n distances of the AMD wave function. It was found that effect of superposition on the totalenergy is less than 0.5 MeV. This indicates that the important correlations are alreadydescribed by the correlation functions.We obtained not only the total energies but also the Hamiltonian components of the groundand excited states of He, and discussed the effects of each correlation function on thesequantities. It is found that correlation functions always increase the contributions of centraland tensor forces and the high-momentum components of He. We compared our calculationswith those assuming common forms of the correlation functions similar to the Jastrow ansatz,which provides an energy loss of 4 MeV. This means that independent optimization of thecorrelation functions is essential, which can be general in the nucleon–nucleon correlationsin nuclei.It would be interesting to investigate the role of tensor correlation on the LS splittingof He. In our previous study with the tensor-optimized shell model [33, 34], we discussedthis effect with the bare AV8 ′ potential; the 3 / − state gains a tensor force of more thanabout 4 MeV than the 1 / − state, which results in a larger kinetic energy in 3 / − thanthe 1 / − state by about 7 MeV. The obtained splitting energy including central and LS components finally becomes 3 MeV. In TOAMD, we will perform a similar analysis of thetensor correlation in relation to the clustering of He in the future. Acknowledgements We thank Professor Kiyomi Ikeda for fruitful discussions on this project. This work was sup-ported by JSPS KAKENHI Grants No. JP18K03660. 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