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microscopic _ ab - initio _ many - body approaches have significantly progressed in recent years @xcite . nowdays , due to increased computing power and novel techniques , _ ab - initio _ calculations are able to reproduce a large number of observables for atomic nuclei with mass up to a=14 . the light nuclei have also served as a crucial site to recognize the important role of three - body forces and three - body correlations . approaches like the no - core shell model ( ncsm ) @xcite , the green s function monte carlo ( gfmc ) @xcite and the coupled - cluster theory with single and double excitations ( ccsd ) @xcite can be formally extended for heavier nuclei . however , the explosive growth in computational power , required to achieve convergent results , severely hinders the detailed _ ab - initio _ studies of heavier , a@xmath3 , nuclei . in the case of the ncsm , the slow convergence of the calculated energies is caused by the adoption of a two - body cluster approximation , which does not take many - body correlations into account . straightforward employment of the three - body and higher - body interactions dramatically complicates the problem , even for light nuclei . an alternative approach is to construct a small - space effective two - body interaction , which would account for the many - body correlations for the a - body system in a large space . attempts to include many - body correlations approximately modifying the one - body part of the effective two - body hamiltonian and employing a unitary transformation have been reported recently @xcite .    in this paper we derive a valence space ( @xmath1 ) effective two - body interaction that accounts for all the core - polarization effects available in the _ ab - initio _ ncsm wavefunctions . first , in the framework of the ncsm , we construct the effective hamiltonians on the two - body cluster level for a=6 systems in the @xmath4 space . @xmath5 represents the limit on the total oscillator quanta ( n ) above the minimum configuration . we take @xmath5 values from 2 to 12 . second , following the original idea of ref . @xcite , we employ an unitary many - body transformation and obtain the effective two - body hamiltonian in the @xmath6 space ( p - shell space ) , which exactly reproduces the lowest , @xmath6 space dominated , eigenstates of the 6-body hamiltonian in the large @xmath4 space . third , we perform ncsm calculations for a=4 and a=5 systems with the effective hamiltonian constructed on the two - body cluster level for the a=6 system and determine the core and one - body parts of the effective two - body hamiltonian for a=6 in the p - shell space . finally , the procedure is generalized for arbitrary mass number a. we analyze the properties of the constructed two - body hamiltonians , investigate their efficiency to reproduce the observables of different a - body systems calculated in large @xmath4 spaces and study the role of the effective p - shell space three - body interaction . the starting point of the no core shell model ( ncsm ) approach is the bare , exact a - body hamiltonian constrained by the harmonic oscillator ( ho ) potential @xcite :    @xmath7    where @xmath8 is the one - body ho hamiltonian @xmath9 and @xmath10 is a bare nn interaction @xmath11 modified by the term introducing a- and @xmath12-dependent corrections to offset the ho potential present in @xmath8 : @xmath13 the eigenvalue problem for the exact a - body hamiltonian ( [ homa ] ) for @xmath14 is very complicated technically , since an extremely large a - body ho basis is required to obtain converged results . however , the @xmath15 problem is considerably simpler . for many realistic nn interactions its solution in the relative ho basis with @xmath16 accounts well for the short range correlations and is a precise approximation for the infinite space ( @xmath17 ) result . this allows one to adopt the two - body cluster approximation to construct the ncsm effective two - body hamiltonian @xmath18 for an a - body system in an @xmath19 space of tractable dimension , where the lower index @xmath20 stands for the number of particles in the cluster . this approximation consists of solving eq.([homa ] ) for the @xmath21 body subsystem of a leading to @xmath22 the information about the total number of interacting particles a enters the bare @xmath23 hamiltonian through the second term in the right hand side of ( [ voma ] ) . next , we find the unitary transformation @xmath24 which reduces the bare @xmath23 hamiltonian in the `` infinite space '' ( @xmath25 ) to the diagonal form : @xmath26 where , for the sake of simplicity , we omit the index @xmath27 for @xmath24 and keep only the index @xmath20 indicating the order of cluster approximation . the matrix @xmath24 can be split into 4 blocks : @xmath28 where the square @xmath29 @xmath30 matrix corresponds to the p - space ( or model space ) of dimension @xmath31 , characterized by the chosen @xmath5 value . taking into account that the @xmath32 matrix has a diagonal form @xmath33 one can calculate the effective @xmath34 hamiltonian using the following formula :    @xmath35    it is easy to show by inserting eq.([homa2e ] ) into the eq.([heff1 ] ) , and taking into account eq.([udec ] ) that the unitary transformation ( [ heff1 ] ) is equivalent to the commonly used unitary transformation @xcite and that eq.([heff1 ] ) is identical to the eqs.(15,16 ) from @xcite . we note , that , by using eq.([heff1 ] ) one does not need to calculate and store a large number of matrix elements of the @xmath36-operator ( i.e. , @xmath37 ) . furthermore , the decoupling condition @xmath38 is automatically satisfied , which is obvious from the diagonal form of the @xmath32 matrix . we note that our treatment of center - of - mass motion remains the same as in the ncsm ( ref . we initiate all effective interaction developments at the a - body level , and , through a series of steps , arrive at a smaller space effective interaction appropriate for the a - body system . for this reason , our derived effective hamiltonians have their first subscript as `` a '' . the next step of the traditional ncsm prescription is to construct the full a - body hamiltonian using the effective two - body hamiltonian ( [ heff1 ] ) and to diagonalize it in the a - body @xmath5 model space . as we increase the number of nucleons , the dimension of the corresponding @xmath5 model space increases very rapidly . for instance , up - to - date computing resources allow us to go as high as @xmath39 for the lower part of the p - shell ( a=5,6 ) @xcite , while already for the upper part of the p - shell ( a@xmath4015 ) , we are limited to @xmath41 . the computational eigenvalue problem for many - body systems is complicated because of the very large matrix dimensions involved . the largest dimension of the model space that we encountered in this study for @xmath42li with @xmath43 exceeds @xmath44 . to solve this problem we have used the specialized version of the shell - model code antoine @xcite , recently adapted for the ncsm @xcite . in fact , the ncsm calculation for the a=6 system in the @xmath43 space yields nearly converged energies for the lowest states dominated by the @xmath45 components , while there is incomplete convergence for @xmath46 in @xmath41 space . therefore , considering the @xmath43 ncsm results as exact solutions for the lowest @xmath45 dominated 6-body states , we may construct the @xmath47 space hamiltonian for the a=6 system , which exactly reproduces those @xmath43 eigenvalues @xcite . moreover , if it is possible to solve the 6-body problem for a=6 , then it is possible to solve the 6-body problem for arbitrary a , using the corresponding effective hamiltonian @xmath34 obtained in the two - body cluster approximation . this means that we can determine for any a - body system the effective hamiltonian in the @xmath47 space , which accounts for 6-body cluster dynamics in the large @xmath43 space . to generalize , we start by defining the procedure for determining the effective hamiltonian matrix elements for the @xmath48-body cluster in the a - nucleon system . we do this by constructing the full @xmath48-body hamiltonian using the effective 2-body hamiltonian ( [ heff1 ] ) and diagonalizing it in the @xmath5 model space . in the spirit of eq.([homa2 ] ) , this yields the eigenenergies @xmath49 of the @xmath48-body system and their corresponding @xmath48 eigenvectors which make up the unitary transformation matrix @xmath50 . these @xmath48-body results can then be projected into a smaller , secondary @xmath51-space , given by @xmath52 with @xmath53 , where , similar to eqs.([udec ] ) and ( [ edec ] ) , @xmath49 and @xmath50 can be split into parts related to the two spaces , @xmath51 and @xmath54 , where @xmath55 . the new secondary effective hamiltonian then takes the following general form :    @xmath56    where the @xmath12 superscript on the left - hand side is omitted for the sake of simplicity . as stated earlier , the new index @xmath48 determines the order of the cluster approximation in the smaller @xmath51 space , _ i.e. _ , @xmath53 . because the @xmath51 space has @xmath53 , the projection into this space `` freezes '' some number of the @xmath48 nucleons into fixed single particle configurations , which can be thought of as the `` inert core '' states in the standard shell model ( ssm ) approach . consequently , it is possible to write @xmath48 as @xmath57 , where @xmath58 is the number of nucleons making up the core configuration , while @xmath59 refers to the size of valence cluster . for instance , in the case of p - shell nuclei , @xmath60 , so , if @xmath61 ( _ i.e. _ the 5-body cluster approximation ) , then the effective hamiltonian @xmath62 is simply a one - body hamiltonian ( @xmath63 ) appropriate for the a - nucleon system . similarly , for the 6-body cluster approximation , _ i.e. _ , @xmath64 , we obtain the effective hamiltonian @xmath65 , which is a two - body hamiltonian ( @xmath66 ) for the a - body system , and , so on for larger values of @xmath48 . whatever the value of @xmath59 is , the effective hamiltonian @xmath67 contains the information about the @xmath48-body dynamics in the original large @xmath19 space , since it reproduces exactly the lowest @xmath68 eigenvalues @xmath69 of the @xmath48-body hamiltonian in the @xmath19 space , where @xmath68 is a dimension of the @xmath51 space .    in the case of a doubly magic closed shell with two extra nucleons i.e. , @xmath70 , _ etc . _ , the dimension of the effective hamiltonian @xmath71 is a 2-body ( @xmath66 ) hamiltonian in the p- , sd- , pf - spaces , _ etc . _ , respectively . this means that the secondary effective hamiltonian ( [ heff2 ] ) contains only 1-body and 2-body terms , even after the _ exact _ a - body cluster transformation . this effective hamiltonian ( [ heff2 ] ) , which now contains the correlation energy of a nucleons , is the correct one - body plus two - body hamiltonian to use in a ssm calculation with inert core . the @xmath72 nucleon - spectators fully occupy the shells below the valence shell and the total a - body wave - function can be exactly factorized as the @xmath58-body `` core '' and the valence 2-body wave functions . this considerably simplifies the calculation of the effective hamiltonian , because only the @xmath6 part ( p@xmath73-space part ) of the complete @xmath4 wave function needs to be specified . utilizing the approach outlined above , we have calculated effective p - shell hamiltonians for @xmath74li , using the 6-body hamiltonians with @xmath75 and @xmath76 mev , constructed from the inoy ( inside nonlocal outside yukawa ) interaction @xcite . this is a new type of interaction , which has local behavior appropriate for traditional nn interactions at longer ranges , but exhibits a nonlocality at shorter distances . the nonlocality of the nn interaction has been introduced in order to account effectively for three - nucleon ( nnn ) interactions which correctly describe the nnn bound states @xmath77h and @xmath77he , whereas still reproducing nn scattering data with high precision . the corresponding excitation energies of p - shell dominated states and the binding energy of @xmath74li are shown in fig.[spectra6li14hw ] as a function of @xmath5 . the dimension of the configurational space for the @xmath43 case considered is 48 million ( m - scheme ) . a two orders of magnitude increase in the size of the model space , as compared to the previous @xmath78 study @xcite , allows us to determine a converged value of 31.681 mev for the @xmath42li binding energy . furthermore , the excitation energy of the highest lying p - space state , @xmath79 , is lowered by an amount of 2.1 mev in comparison to the @xmath78 case , indicating improved convergence for both the excited states and ground state for @xmath43 .    in the ssm an effective two - body hamiltonian for a nucleus with mass number a is represented in terms of three components : @xmath80 where @xmath81 is the inert core part associated with the interaction of the nucleons occupying closed shells , @xmath82 is the one - body part corresponding to the interaction of valence nucleons with core nucleons , and @xmath83 is the two - body part referring to the interaction between valence particles . it is usually assumed that the core and one - body parts are constant for an arbitrary number of valence particles and that only the two - body part @xmath83 may contain mass dependence that includes effects of three - body and higher - body interactions .    to represent the @xmath84 hamiltonian in the ssm format , we develop a valence cluster expansion ( vce ) , @xmath85 where the lower index , k , stands for the k - body interaction in the @xmath59-body valence cluster ( @xmath86 ) ; the first upper index a for the mass dependence ; and the second upper index , @xmath87 for the number of particles contributing to the corresponding k - body part . thus , we consider the more general case of allowing the core ( k=0 ) , one - body ( k=1 ) and other k - body parts to vary with the mass number a. this appears necessary to include the a - dependence of the excitations of the core ( @xmath58 ) nucleons treated in the original n@xmath88 basis space . for the a=6 case the two - body valence cluster ( 2bvc ) approximation is exact : @xmath89 where the core part , @xmath90 , is defined as the ground state @xmath91 energy of @xmath92he calculated in the @xmath19 space with the tbmes of the primary effective hamiltonian , @xmath93 for a=6 . then the one - body part , @xmath94,is determined as @xmath95 the tbmes of the one - body part , @xmath94 , @xmath96 may be represented in terms of single particle energies ( spe ) , @xmath97 : @xmath98 where the index a ( as well as b , c , and d ) denotes the set of single particle ho quantum numbers @xmath99 , upper index stands for proton ( p ) and neutron ( n ) , and the e(@xmath100li , j ) , e(@xmath100he , j ) are ncsm energies of the lowest @xmath101 and @xmath102 states calculated in the @xmath4 space for the @xmath103-body system using the tbmes of the @xmath104 effective hamiltonian , @xmath105 , which includes coulomb energy . finally , the two - body part @xmath106 is obtained by subtracting of two hamiltonians : @xmath107 it is worth noting that since the coulomb energy is included in the original hamiltonian , the proton - proton ( pp ) , neutron - neutron ( nn ) and proton - neutron ( pn ) @xmath108 tbmes of the two - body part , @xmath109 , have different values . the pn tbmes of the core , one - body and two - body parts of the expanded hamiltonian for @xmath42li are listed in the table i.    [ cols="^,^,^,^,^,^ , > , > , > , > , > , > , > , > , > , > , > , > " , ]     obtained results indicate that accounting for the effective 3-body interactions considerably improves the agreement with the exact ncsm for the @xmath110he , does not bring much change for @xmath111he and yields worse results for @xmath112he ( see fig.[he8 - 9 ] ) .    performing a similar calculation with the effective interaction obtained in the 3bvc approximation starting from the cd - bonn interaction @xcite , we obtained results which are shown in fig.[he8 - 9cd ] . note , that the effective cd - bonn interaction constructed in the 2bvc approximation considerably underbinds the he isotopes in comparison to the exact ncsm results . the subsequent employment of the 3bvc approximation compensates these large differences and yields much better results for @xmath112he . however , to draw more quantitative conclusion about the 3-body and higher - body effective interactions , one needs to perform exact diagonalization using the 3bmes . we will evaluate this effect in future studies . within the ncsm approach we can calculate , by exact projection , full a - nucleon dependent tbmes ( and 3bmes ) . these a - dependent tbmes ( and 3bmes ) can be separated into core , one - body and two - body ( and three - body ) parts , all of which are also a - dependent , contrary to the ssm approach . when these a - dependent effective one- and two - body ( and three - body ) interactions are employed in ssm calculations , they exactly reproduce full ncsm calculations for a=6 ( a=7 ) isobars and yield results in good agreement with full ncsm calculations for @xmath113 performed in large basis spaces . our results for @xmath113 , which include the 3-body effective interaction , indicate that 3- and higher - body effective interactions may play an important role in determining their binding energies and spectra . future investigations will be extended to include effective 3-body interactions exactly and to explore other physical operators , such as transition operators and em moments . we thank the institute for nuclear theory at the university of washington for its hospitality and the department of energy for partial support during the development of this work . b.r.b . and a.f.l . acknowledge partial support of this work from nsf grants phy0244389 and phy0555396 ; p.n . acknowledges support in part by the u.s . doe / sc / np ( work proposal n. scw0498 ) and u.s department of energy grant de - fg02 - 87er40371 ; j.p.v . acknowledges support from u.s . department of energy grants de - fg02 - 87er40371 and de - fc02 - 07er41457 ; and the work of i.s . was performed under the auspices of the u.s . doe . prepared by llnl under contract de - ac52 - 07na27344 . thanks the gesellschaft fr schwerionenforschung mbh darmstadt , germany , for its hospitality during the preparation of this manuscript and the alexander von humboldt stiftung for its support . 100 p. navratil , v. g. gueorgiev , j.p . vary , w. e. ormand , and a. nogga , phys . . lett . * 99 * , 042501 ( 2007 ) . a. nogga , p.navratil , b.r.barrett , j.p.vary , phys . rev . c. * 73 * , 064002 ( 2006 ) . i. stetcu , b.r.barrett , p.navratil , j.p.vary , phys . rev . c. * 71 * , 044325 ( 2005 ) . p. navratil and w.e.ormand , phys . lett . * 88 * , 152502 ( 2002 ) ; phys . rev . c. * 68 * , 034305 ( 2003 ) . p. navratil , j.p.vary , b.r.barrett , phys . * 84 * , 5728 ( 2000 ) ; phys . c. * 62 * , 054311 ( 2000 ) . s. c. pieper , k. varga , and r. b. wiringa , phys . c 66 , 044310 ( 2002 ) . s. pieper and r. b. wiringa , annu . nucl . part . sci . * 51 * , 53 ( 2001 ) . k.kowalski , d.j.dean , m.hjorth-jensen , t.papenbrock , p.piecuch , phys . 92 , 132501 ( 2004 ) . s. fujii , t. mizusaki , t. otsuka , t. sebe and a. arima , phys . b650 * , 9 ( 2007 ) . p. navratil , m. thoresen , and b. r. barrett , phys . rev . c. * 55 * , r573 ( 1997 ) . susumu okubo , prog . 12 , no.5 , 603 ( 1954 ) . kenji suzuki , prog . 68 , no.1 , 246 ( 1982 ) . p. navratil and e.caurier , phys . rev . c. * 69 * , 014311 ( 2004 ) . e. caurier and f. nowacki , acta . b30 , ( 1999 ) 705 . e. caurier , g. martinez - pinedo , f. nowacki , a. poves , j. retamosa , and a. p. zuker , phys . c * 59 * , 2033 ( 1999 ) . e. caurier , p. navratil , w. e. ormand , and j.p . vary , phys . c * 64 * , 051301(r ) ( 2001 ) . d. r. entem and r. machleidt , phys . rev . c. * 68 * , 041001(r ) ( 2003 ) . p. doleschall , phys . * 69 * , 054001 ( 2004 ) . p. doleschall , i. borbely , z. papp , and w. plessas , phys . rev . c. * 67 * , 0064005 ( 2003 ) . b. r. barrett and m.w . kirson , nucl . a * 418 * , 145 ( 1970 ) . j. p. vary , p. u. sauer and c. w. wong , phys . rev . c. * 7 * , 1776 ( 1973 ) . r. machleidt , f. sammarruca , and y. song , phys . rev . c. * 53 * , 1483 ( 1996 ) .
we construct effective 2- and 3-body hamiltonians for the p - shell by performing @xmath0 _ ab initio _ no - core shell model ( ncsm ) calculations for a=6 and 7 nuclei and explicitly projecting the many - body hamiltonians onto the @xmath1 space . 
 we then separate these effective hamiltonians into 0- , 1- and 2-body contributions ( also 3-body for a=7 ) and analyze the systematic behavior of these different parts as a function of the mass number a and size of the ncsm basis space . 
 the role of effective 3- and higher - body interactions for @xmath2 is investigated and discussed .