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nuclear cluster structure in the hoyle state , the second @xmath0 state at @xmath9 mev in @xmath2 , has been discussed for a long time by many authors  @xcite . cluster model approaches play an important role in understanding the structure and clarified that it has well developed @xmath8 cluster structure with more dilute density than that of saturation in the ground state , where @xmath3 clusters weakly interact with each other in a relative @xmath10-wave  @xcite . in the recent fifteen years , however , the understanding of the hoyle state structure has been much deepened , by an advent of a new type of microscopic cluster model wave function , which is referred to as the tohsaki - horiuchi - schuck - rpke ( thsr ) wave function  @xcite . this wave function retains a structure that constituent @xmath3 clusters are loosely bound like a gas and occupy an identical orbit , and this structured phenomenon is now called the @xmath3 condensation . one of the most important properties for the thsr wave function is to give a single and optimal configuration that is equivalent to a solution of full microscopic three - body problem  @xcite . since it is well known that the solutions of the full microscopic three - body problem via @xmath8 resonating group method ( rgm )  @xcite and via generator coordinate method ( gcm )  @xcite nicely reproduce many experimental data for the hoyle state , like the energy , width , electro - magnetic properties , etc , the equivalence leads to the conclusion that the hoyle state exists as the @xmath3 condensate composed of a weakly interacting and gaslike @xmath8 clusters .    not only the hoyle state but also some other excited states in @xmath2 trigger a special interest in recent years . for example , the second @xmath11 state was theoretically predicted by the use of the cluster models almost 40 years ago  @xcite , but it was very recently when the existence was confirmed in several experiments  @xcite , with a pioneering work by itoh _ _ as a beginning  @xcite . besides the second @xmath12 state , a new @xmath13 state is also observed at @xmath14 mev recently  @xcite . the new @xmath12 and @xmath13 states are now considered to form a rotational family with the hoyle state , though the detailed rotational structure is under question  @xcite . while in ref . @xcite a simple rotational structure based on a triangular shape of the @xmath8 clusters is assumed , it is pointed out in ref . @xcite that this is not simply considered to be an ordinary rotational band that lies on @xmath15 line , due to the @xmath3-condensate nature of the hoyle state , where the third @xmath1 state @xmath16 above the hoyle state also plays an important role .    a new experimental information is also given for the famous broad @xmath1 state observed at @xmath17 mev with a width of @xmath18 mev  @xcite . _ pointed out that the broad @xmath1 peak is decomposed into two peaks , giving the @xmath6 and @xmath7 states at @xmath19 mev and @xmath20 mev above the @xmath8 threshold , with the widths of @xmath21 mev and @xmath22 mev , respectively  @xcite . they also found a decay property that the @xmath7 state dominantly decays into @xmath23 while the @xmath6 state decays into @xmath24 . some theoretical studies consistently reproduce the resonance parameters , where semi - microscopic  @xcite or non - microscopic  @xcite @xmath8 model is adopted . in particular , in ref . @xcite , the authors applied the complex scaling method ( csm ) and analytic continuation of coupling constant ( accc ) method to the @xmath8 orthogonality condition model ( ocm ) . they suggested by extrapolation that the @xmath6 state has an @xmath10-wave dominant structure with more dilute density than that of the hoyle state . the observed decay property and resonance parameters of the @xmath6 and @xmath7 states are also reproduced by the recent calculation by the present author using an extended version of the thsr wave function , where @xmath25 correlation can be taken into account  @xcite . he further showed more directly by using the thsr wave function that the @xmath6 state is given rise to as a result of the monopole excitation from the hoyle state to have dominantly a higher nodal structure , where the @xmath3 cluster orbits around the @xmath26 in an @xmath10-wave with four nodes . he also showed that the @xmath7 state has the largest @xmath27-factor in the channel of the @xmath3 cluster coupling with @xmath28 in a @xmath29-wave . this is consistent with the previous result of the @xmath8 gcm calculation  @xcite , where a large reduced width amplitude ( rwa ) from the @xmath23 channel is obtained , although their calculation fails in reproducing the observed @xmath6 state . i should also mention that the antisymmetrized molecular dynamics ( amd ) and fermionic molecular dynamics ( fmd ) calculations reproduce the @xmath7 state and predict that the state dominantly has an intrinsic configuration of bent - armed shape of the @xmath8 clusters , like a linear - chain structure that was originally proposed by morinaga  @xcite , although the observed @xmath6 state is also missing in their calculations  @xcite . in ref . @xcite , the linear - chain component of the @xmath7 state obtained by the @xmath8 ocm is identified to be @xmath30 configuration in the elliott @xmath31 model , which is calculated to be about @xmath32 % .    in this paper , i investigate the excited @xmath0 states obtained in the previous work  @xcite by using the extended thsr wave function and the so - called @xmath5-constraint method  @xcite . i focus on how much components concentrate on a single configuration of the extended thsr wave function with deformation parameters , and discuss physical natures of the states . in sec . [ sec:2 ] , the original thsr wave function is explained , and then as its natural extension the extended thsr wave function is introduced . in sec . [ sec:3 ] , the structures of the @xmath33 , @xmath6 , and @xmath7 states are discussed . squared - overlap surfaces between the states and single configurations of the thsr wave function in their deformation parameter space are calculated . [ sec:4 ] is devoted to conclusion . the original thsr wave function with deformation  @xcite is described below , @xmath34 , \nonumber \\   & & = { \cal a}\big [ \exp \big\ { - \sum_{i=1}^2 \mu_i \hspace{-0.2 cm } \sum_{k = x , y , z}\hspace{-0.1 cm } \frac{\xi_{ik}^2}{b^2 + 2\beta_{k}^2}\big\ } \phi(\alpha_1)\phi(\alpha_2)\phi(\alpha_3 ) \big],\nonumber \\ \label{eq : thsr } \end{aligned}\ ] ] with @xmath35 being the antisymmetrization operator acting on the 12 nucleons , @xmath36 the internal wave function of the @xmath37-th @xmath3 particle assuming a @xmath38 configuration , like , @xmath39,\ ] ] @xmath40 , @xmath41 , the position vectors of the @xmath37-th @xmath3 particle and of total center - of - mass , respectively , @xmath42 and @xmath43 the jacobi coordinates between the @xmath3 particles , and @xmath44 . the parameters @xmath45 and @xmath46 characterize the size of the constituent @xmath3 particle , and the size and shape of the total nucleus , respectively , though the axial symmetry @xmath47 is assumed throughout this study . the extended version of the thsr wave function which i utilize in this work is a natural extension of the original form , eq . ( [ eq : thsr ] ) , as follows : @xmath48 , \label{eq:2 } \nonumber \\\end{aligned}\ ] ] where the single parameter @xmath46 is decomposed into @xmath49 and @xmath50 corresponding to the two jacobi coordinates , @xmath51 and @xmath52 , and @xmath53 is a normalization constant . this allows us to include @xmath54-type configuration beyond the original thsr wave function , where all @xmath3 clusters are restricted to move in an identical orbit . this model wave function provides a picture that constituent clusters of a nucleus are trapped into a potential without any geometrical rigid configuration of the clusters under the constraint of antisymmetrization . the center - of - mass wave functions of the constituent clusters are assumed to have deformable gaussian shapes with widths that are variational parameters and characterize the spatial size and shape of the nucleus . this is mentioned as a `` container '' picture or non - localized concept of cluster structures in some recent publications ( see ref . @xcite and references therein ) . not only the gaslike cluster states like the hoyle state @xmath55 and @xmath56 @xmath57 but also ordinary cluster states , which had been believed to have non - gaslike localized cluster structures , such as the inversion doublet band with @xmath58 structure  @xcite , @xmath8 and @xmath59 linear - chain structure  @xcite , @xmath60 structure in @xmath61  @xcite , @xmath62 and @xmath63 structures in @xmath64  @xcite and @xmath65  @xcite , respectively , etc , can all be described by this thsr ansatz with almost @xmath66 % accuracy .    since the excited states above the hoyle state were observed as resonances with non - negligible widths , it is more likely that the bound state approximation does not work well for those states . i therefore use a technique to effectively eliminate continuum components that get mixed with the resonances , so - called @xmath5-constraint method , which is also used in refs . @xcite and the effectiveness is already guaranteed . in this technique , by considering the fact that in calculations of bound states , pseudo continuum states are shown to have large root mean square ( r.m.s . ) radii , compared to those of resonances and bound states , one can remove effectively the spurious continuum components in the following way : first i solve the following equation , @xmath67 with @xmath68 , and @xmath69 , where @xmath70 is the angular - momentum projection operator onto @xmath71 . the eigenstates in eq . ( [ eq : cutoff1 ] ) can be written as , @xmath72 next i adopt , as bases to diagonalize hamiltonian , the eigenstates belonging to eigenvalues satisfying @xmath73 in eq . ( [ eq : cutoff2 ] ) , as follows : @xmath74 and obtain the eigenfunction , @xmath75 for hamiltonian , the same effective nucleon - nucleon interaction as used in ref . @xcite , volkov no .  2 force  @xcite , is adopted , where the strength parameters are slightly modified  @xcite . the cutoff radius is now taken to be @xmath76 fm as in ref .  @xcite . .energies , widths , r.m.s . radii , and monopole transition strengths . units of energies and widths are mev , of radii @xmath77 , @xmath78 @xmath79 . experimental data ( exp . ) , thsr , ocm1 , ocm2 , fmc , and amd are taken from refs . @xcite ,  @xcite ,  @xcite ,  @xcite ,  @xcite ,  @xcite , respectively . [ cols="^,<,^,^,^,^,^,^",options="header " , ]     [ tab:1 ]    in table  [ tab:1 ] , the calculated observables , energy , @xmath3-decay width , monopole transition strengths are compared with the experimental data and those of the other calculations . the present calculation consistently reproduces the corresponding experimental data . the r.m.s . radius of the @xmath6 state is the largest of the @xmath1 states and the much stronger monopole transition strength between the @xmath33 and @xmath6 states , than that between the @xmath33 and @xmath80 state ( see ref . @xcite ) , can be seen . these indicate , as discussed in ref . @xcite , that the @xmath6 state is excited by monopole transition from the hoyle state to have a higher nodal structure between @xmath56 and @xmath3 cluster .    as i mentioned in sec . [ sec:2 ] , i adopted volkov no . 2 force as an effective nucleon - nucleon interaction in this work . i also checked if the results depend on the choice of the force . for example , when i adopt volkov no . 1 force , the energies of the @xmath33 , @xmath6 , and @xmath7 states become slightly higher by about @xmath81 mev , i.e. @xmath82 mev , @xmath83 mev , and @xmath84 mev , respectively . however , i confirmed that qualitative features of these states that will be discussed in this section , such as the behaviours of squared overlaps , do not change at all . i also mention that only the thsr ansatz lists rather complete physical quantities , including the r.m.s . radii and monopole transition strengths , since the wave functions of both @xmath6 and @xmath7 states are definitely obtained . as mentioned in introduction , the amd and fmd calculations can not reproduce the @xmath6 state , and in ocm@xmath85csm@xmath85accc calculation in ref . @xcite , denoted as ocm1 in this table , the wave function of the @xmath6 state can not be obtained .     in eq . ( [ eq : ovlp1 ] ) with @xmath86 , i.e. for the @xmath33 state , in two parameter space , @xmath87 . black solid curves are drawn in a step of @xmath88 and red dotted curves , which cover the region of @xmath89 , are in a step of @xmath90 . ]     in eq . ( [ eq : ovlp2 ] ) with @xmath86 , i.e. for the @xmath33 state , in two parameter space , @xmath87 . black solid curves are drawn in a step of @xmath88 and red dotted curves , which cover the region of @xmath91 , are in a step of @xmath90 . ]    in ref . @xcite , the author investigated the hoyle band @xmath92 and the @xmath6 and @xmath7 states by using the extended thsr wave function . all the states obtained in his calculations are derived as solutions of the hill - wheeler equation eq . ( [ eq : wf ] ) , where many bases of thsr - type configurations are superposed . this will , however , make it unclear how the thsr picture mentioned above is realized for those states , since the superposition of the thsr configurations might break its original picture . thus , in order to investigate how the thsr picture holds for the @xmath1 states , i calculate the following two quantities : @xmath93 and @xmath94 the former is the squared overlap of the @xmath33 @xmath95 , @xmath6 @xmath96 and @xmath7 @xmath97 states in eq . ( [ eq : wf ] ) with the single configuration of the angular - momentum - projected thsr wave function , @xmath98 . the latter is also the squared overlap of the @xmath1 states in eq . ( [ eq : wf ] ) with the single configuration of the angular - momentum - projected thsr wave function but the lower @xmath1 states components are subtracted by the projection operator defined as , @xmath99 , from the single thsr wave function . @xmath100 is a normalization constant of the wave function @xmath101 .    in fig . [ fig:1 ] , i show the contour map of the squared overlap @xmath102 in two parameter space @xmath47 and @xmath103 with @xmath104 . one can see that maximum value of the squared overlap amounts to @xmath105 at @xmath106 fm and @xmath107 fm . while this large value indicates that the hoyle state is expressed by a single configuration of the thsr wave function with @xmath108 , the hoyle state is orthogonal to the ground state , and therefore the orthogonality condition should be imposed on the thsr wave function to describe the hoyle state . i then calculate the second quantity of the squared overlap discussed above , @xmath109 with @xmath104 . figure  [ fig:2 ] shows the contour map in two parameter space @xmath87 . the maximal value is found to be @xmath110 at @xmath111 , which is surprisingly large value , almost @xmath66 % . this large value was already found before in refs . @xcite with the thsr ansatz of @xmath108 , but i should note that in the present case this large value is also obtained in the calculation with larger model space @xmath112 , where @xmath113 and @xmath3 asymmetric configuration is allowed for . the fact that nevertheless this large value is again obtained strongly suggests that in the hoyle state the @xmath8 particles , democratically , without a strong @xmath114 correlation , take an identical motion , so that the @xmath8 condensate state is realized . one can also find that the region denoted by dotted curve in red , where the squared overlap is more than @xmath115 , is widely ranged from prolately @xmath116 to oblately @xmath117 deformed shapes , passing through the spherical one @xmath118 , with large @xmath119 values . this supports the idea that the hoyle state does not have any definite intrinsic shape but have a gaslike configuration of the @xmath8 particles . i should also note that the projection operator @xmath120 , which removes the ground state component with compact structure , plays a role as a repulsive force to prevent the @xmath8 clusters from being resolved and to form well developed @xmath8 cluster structure for the hoyle state , due to its orthogonality condition . this is essentially the same as the situation of @xmath56 , where the antisymmetrization operator @xmath35 removes the pauli forbidden states , to construct a structural repulsive core between the two @xmath3 clusters .     in eq . ( [ eq : ovlp1 ] ) with @xmath121 , i.e. for the @xmath6 state , in two parameter space , @xmath87 . black solid curves are drawn in a step of @xmath88 and red dotted curves , which cover the region of @xmath122 , are in a step of @xmath90 . ]     in eq . ( [ eq : ovlp2 ] ) with @xmath121 , i.e. for the @xmath6 state , in two parameter space , @xmath87 . black solid curves are drawn in a step of @xmath88 and red dotted curves , which cover the region of @xmath123 , are in a step of @xmath90 . ]     in eq . ( [ eq : ovlp2 ] ) with @xmath121 , i.e. for the @xmath6 state , in two parameter space , @xmath124 , where the maximal values of @xmath125 obtained by varying @xmath50 are shown . the region of @xmath126 is only shown in a step of @xmath127 of the contour lines . ] next i show the contour map of the squared overlap for the @xmath6 state , @xmath128 in fig . [ fig:3 ] and @xmath129 in fig . [ fig:4 ] in two parameter space @xmath47 and @xmath103 . in fig . [ fig:3 ] , the contour lines more than @xmath88 are denoted by dotted ones in red , in a step of @xmath90 . one can see that the @xmath6 state does not have any squared overlap that is more than 0.1 in internal region of @xmath47 and @xmath103 . it only has at most @xmath130 in an outside region around @xmath131 fm , @xmath132 fm . i also check the squared overlap in the model space of @xmath133 , i.e. @xmath134 in four parameter space , but no large squared overlap more than 0.19 can be obtained .    however , this situation changes drastically when i consider the thsr model space of its single configuration that is orthogonal to the hoyle state as well as to the ground state , i.e. @xmath135 , where the projection operator is defined as @xmath136 . then in fig . [ fig:4 ] , i show the squared overlap @xmath129 in two parameter space , @xmath87 . the contour lines more than @xmath137 are denoted by dotted ones in red , in a step of @xmath90 . one can see that contrary to fig . [ fig:3 ] , very large squared overlap appears in the internal region . the largest one amounts to @xmath138 at @xmath139 . one can also see that this large value quickly decreases toward around @xmath111 , which corresponds to the point giving the maximum squared overlap for the hoyle state in fig . [ fig:2 ] . this is of course due to the effect of the orthogonalization operator @xmath140 . these results clearly indicate that in order to describe the @xmath6 state , the orthogonalization to the hoyle state as well as to the ground state is the most essential . this implies that the @xmath6 state is intimately related to the hoyle state , like its family . due to this orthogonalization , the @xmath6 state is considered to exist as an excitation mode from the hoyle state with respect to the internal motions of the @xmath8 clusters .    in the previous paper ref . @xcite , for the @xmath6 state , the large rwa of @xmath141 component with a higher node than in the hoyle state is calculated . also as shown in table  [ tab:1 ] , the much larger monopole strength from the hoyle state than a single particle strength , @xmath142 , is obtained . from these results , it was concluded that the @xmath6 state is given rise to as a result of the strong monopole transition from the hoyle state , to have a higher nodal structure . these results are consistent with the present results , giving the interpretation that the monopole transition or vibration mode is brought about by the orthogonalization to the hoyle state , to necessarily provide the higher nodes in the internal motions of the @xmath8 clusters .    in order to further investigate how @xmath6 state contains the @xmath25 correlation , i then calculate the squared overlap in four parameter space , @xmath143 . for a given @xmath49 value , the maximal squared overlap obtained by varying @xmath50 value is shown in the contour map of fig . [ fig:5 ] , as a function of two parameters @xmath144 . the region giving more than the squared overlap of @xmath145 is only shown in fig . [ fig:5 ] with contour lines in a step of @xmath127 . while the largest value appears at oblately deformed region , like fig . [ fig:4 ] , it is also found that there appear two maxima in the prolately deformed region , which are about @xmath146 , at @xmath147 and @xmath148 . the corresponding @xmath50 values are @xmath149 and @xmath150 , respectively . since @xmath49 and @xmath50 correspond to the deformation parameters of @xmath56 and the remaining @xmath3-cluster motion , respectively , and @xmath56 is expressed to have prolately deformed value for @xmath49 , this large squared overlap values indicate that the @xmath6 state has sizable @xmath25 correlation , which is also consistent with the previous calculation of the @xmath25 rwa .     in eq . ( [ eq : ovlp1 ] ) with @xmath151 , i.e. for the @xmath7 state , in two parameter space , @xmath87 . black solid curves are drawn in a step of @xmath88 and red dotted curves , which cover the region of @xmath152 , are in a step of @xmath90 . ]    , which gives the maximal squared overlap , @xmath153 , in fig . [ fig:6 ] . ] finally i discuss the structure of the @xmath7 state . as i mentioned in introduction , in the amd , fmd , and @xmath8 gcm calculations , the @xmath6 state may be missing and only the @xmath7 state was obtained . the dominant intrinsic configuration of the @xmath7 state in the amd and fmd calculations shows a bent - armed structure of the @xmath8 clusters , resembling the linear - chain structure . on the other hand , in the ocm@xmath85csm@xmath85accc and @xmath8 gcm calculations , the @xmath7 state has a structure that the @xmath3 cluster predominantly couples in a @xmath29-wave with the @xmath28 core . the same result is also given by the rwa analysis in the present ansatz in ref . the problem is whether the interpretation of the linear - chain - like structure for the @xmath7 state is reasonable or not , since the amd and fmd calculations can not reproduce the @xmath6 and @xmath7 state simultaneously . actually it was recently reported by suhara _ , that the orthogonality condition to the lower states plays an important role for the survival of the linear - chain structure state in @xmath2  @xcite .    in fig . [ fig:6 ] , i show the contour map of the squared overlap for the @xmath7 state , @xmath154 , in two parameter space , @xmath87 . the contour lines giving more than @xmath155 are denoted by dot in red , in a step of @xmath90 . this contour map has a characteristic feature , where the strongly prolate deformation is only allowed to have a non - negligible squared overlap amplitude . except for this prolately deformed region , the squared overlap is less than @xmath88 . the largest value is @xmath153 , which is not so much large but clearly indicates the @xmath8 linear - chain structure . i can consider this situation as follows : it is shown that the extremely prolately - deformed shaped thsr wave function has very small overlap with the other shaped thsr wave function ( see fig .  2 in ref . @xcite ) . since the configuration space other than the extremely prolately - deformed region is already used by the hoyle state and the @xmath6 state ( see fig . [ fig:1 ] and fig . [ fig:3 ] ) , as well as by the ground state for the more compact region , the @xmath7 state has no choice but using the remaining configuration , to result in having the extremely prolately - deformed shape , i.e. linear - chain structure . i also mention that this feature of the @xmath7 state is quite different from the behaviour of the hoyle state , in which the @xmath46 parameter space giving the large squared overlap is widely spanned as shown in fig . [ fig:1 ] . while the feature for the hoyle state implies that any definite intrinsic wave function for this state is difficult to be uniquely determined , the feature of the @xmath7 state allows for the definite intrinsic shape . the parameter values giving the maximal squared overlap is calculated to be @xmath156 , which is close to @xmath157 that was obtained in a rather ideal one - dimension situation in ref . @xcite . in ref . @xcite , it is discussed that largely prolately - deformed thsr wave function shows one - dimensional @xmath3 condensate of @xmath8 clusters , which is fairly different from the ordinary picture of the linear - chain state with rigid - body @xmath8-cluster configuration arranged in a line in a spatially localized way . thus i can say that the present @xmath7 state has the one - dimensional @xmath3 condensate structure by around @xmath158 % , where the @xmath8 clusters are loosely trapped into a prolately deformed potential like a one - dimensional gas .     in eq . ( [ eq : ovlp2 ] ) with @xmath151 , i.e. for the @xmath7 state , in two parameter space , @xmath87 , in a step of @xmath88 . ]    figure . [ fig:7 ] shows the intrinsic density profile generated from the single thsr wave function , before angular - momentum projection , with @xmath156 , which gives the maximal squared overlap in fig . [ fig:6 ] . this density distribution shows a clear linear - chain structure of the @xmath8 clusters , i.e. localized @xmath3 clusters , with an extended long tail along the @xmath159-direction . as is discussed in ref . @xcite , this comes from the inter-@xmath3 pauli repulsion , as a kinematical effect , which makes this object look like keeping localized clustering . however , dynamics prefers a one - dimensional gas , according to the potential picture and the character of the thsr wave function mentioned above . this is particularly expressed as the long tail elongated along the @xmath159-direction in this figure . finally i show in fig . [ fig:8 ] the contour map in the orthogonal space , @xmath160 as usual , in two parameter space @xmath87 . the largest squared overlap value increases up to @xmath146 and the @xmath108 value to give the maximum slightly moves toward spherical region , which is @xmath161 , but still shows very strongly prolately - deformed shape . on the one hand , this large value again supports that the @xmath7 state has a linear - chain shape . on the other hand , the second and third maxima also appear at @xmath162 and @xmath163 , where the maximal values are @xmath164 and @xmath165 , respectively . the former and the latter correspond to the oblately deformed shape and the spherical shape , respectively . this may suggest that due to the orthogonalization operator @xmath166 , some other correlations to unstabilize the linear - chain structure , like a bending mode , which allows for a spherical shape at a certain probability , take part in , as is discussed in ref .  @xcite . in conclusion , i investigated the excited @xmath0 states in @xmath2 by using the extended thsr wave function with @xmath5-constraint method . in particular , i focused on the @xmath6 and @xmath7 states , which were recently found in experiment . the physical properties of the states , relationship with the hoyle state , and similarities and differences between them were discussed by calculating the squared overlap with the single configurations of the extended thsr wave function . the @xmath6 state was found to appear as a result of the orthogonalization to the hoyle state as well as to the ground state , so that the strong monopole transition or vibrational transition is induced . the state is considered to be a family of the hoyle state with a higher nodal structure in internal motions of the @xmath8 clusters . the @xmath7 state was shown to have a linear - chain structure as a dominant configuration , where the @xmath8 clusters move rather freely in a much elongated way along the @xmath159-axis , i.e. like a one - dimensional gas , though the density distribution shows a localized @xmath8 linear - chain structure , due to the inter-@xmath3 pauli repulsion . besides the linear - chain configuration , some other correlations like a bending mode also seem to be mixed . the author wishes to thank b. zhou , h. horiuchi , t. suhara , a. tohsaki , g. rpke , p. schuck , t. yamada for many helpful discussions . this work is financially supported by hpci project and jsps kakenhi grant number 25400288 . the support of beihang university under the `` zhuoyue 100 talents '' program is gratefully acknowledged . h. horiuchi , prog . . phys . * 51 * , 1266 ( 1974 ) ; 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background : :    two new @xmath0 states are recently observed around a    few mev above the hoyle state ( the second @xmath1 state in    @xmath2 ) . the characters of them are only poorly discussed in    theory and are still mysterious . 
 purpose : :    i give for the first time a comprehensive understanding of the    structures of the @xmath1 states by analyzing their wave    functions , and discuss relationship with the hoyle state ,    similarities , and differences between the states . 
 method : :    i extend a microscopic @xmath3-cluster model called    tohsaki - horiuchi - schuck - rpke ( thsr ) wave function so as to    incorporate @xmath4 asymmetric configuration    explicitly . the so - called @xmath5-constraint method to    effectively eliminate spurious continuum components is also used . 
 results : :    the @xmath6 state is shown to have a very large squared    overlap with a single configuration of the extended thsr wave function    in an orthogonal space to the hoyle state as well as to the ground    state . 
 the @xmath7 state has a maximal squared overlap    with a single extended thsr wave function with an extremely    prolately - deformed shape . 
 conclusions : :    the @xmath6 state appears as a family of the hoyle state    to have a higher nodal structure in the internal motions of the    @xmath8 clusters , due to the orthogonalization to the    hoyle state . 
 the @xmath7 state dominantly has a    linear - chain structure , where the @xmath8 clusters move    freely in a non - localized way , like a one - dimensional gas of the    @xmath8 clusters .