Symbolic-numerical algorithm for generating cluster eigenfunctions: identical particles with pair oscillator interactions
Alexander Gusev, Sergue Vinitsky, Ochbadrakh Chuluunbaatar, Vitaly Rostovtsev, Luong Le Hai, Vladimir Derbov, Andrzej Gozdz, Evgenii Klimov
SSymbolic-numerical Algorithm for GeneratingCluster Eigenfunctions: Identical Particleswith Pair Oscillator Interactions
Alexander Gusev , Sergue Vinitsky , Ochbadrakh Chuluunbaatar ,Vitaly Rostovtsev , Luong Le Hai , , Vladimir Derbov ,Andrzej G´o´zd´z , Evgenii Klimov Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia, e-mail: [email protected] Belgorod State University, Belgorod, Russia Saratov State University, Saratov, Russia Department of Mathematical Physics, Institute of Physics,University of Maria Curie–Sk(cid:32)lodowska, Lublin, Poland Tver State University, Tver, Russia
Abstract.
The quantum model of a cluster, consisting of A identicalparticles, coupled by the internal pair interactions and affected by theexternal field of a target, is considered. A symbolic-numerical algorithmfor generating A − A identical particles inthe new symmetrized coordinates, is formulated and implemented usingthe MAPLE computer algebra system. Examples of generating the sym-metrized coordinate representation for A − . Quantum harmonic oscillator wave functions have a lot of applications in modernphysics, particularly, as a basis for constructing the wave functions of a quantumsystem, consisting of A identical particles, totally symmetric or antisymmetricwith respect to permutations of coordinates of the particles [1]. Various specialmethods, algorithms, and programs (see, e.g., [1,2,3,4,5,6,7,8,9]) were used toconstruct the desired solutions in the form of linear combinations of the eigen-functions of an A − The talk presented at the 15th International Workshop ”Computer Algebra in Sci-entific Computing 2013”, Berlin, Germany, September 9-13, 2013 a r X i v : . [ phy s i c s . c o m p - ph ] J un A promising approach to the construction of oscillator basis functions forfour identical particles was proposed in [2,3,4]. It was demonstrated that a clearalgorithm for generating symmetric (S) and antisymmetric (A) states can beobtained using the symmetrized coordinates instead of the conventional Jacobicoordinates. However, until now this approach was not generalized for a quantumsystem comprising an arbitrary number A of identical particles.We intend to develop this approach in order to describe the tunnelling ofclusters, consisting of several coupled identical particles, through repulsive po-tential barriers of a target. Previously this problem was solved only for a pairof coupled particles [11,12]. The developed approach will be also applicable tothe microscopic study of tetrahedral- and octahedral-symmetric nuclei [13] thatcan be considered in the basis of seven-dimensional harmonic oscillator eigen-functions [14]. The aim of this paper is to present a convenient formulation ofthe problem stated above and the calculation methods, algorithms, and programsfor solving it.
In this paper, we consider the quantum model of a cluster, consisting of A identical particles with the internal pair interactions, under the influence of theexternal field of a target. We assume that the spin part of the wave function isknown, so that only the spatial part of the wave function is to be considered,which can be either symmetric or antisymmetric with respect to a permutationof A identical particles [15,16,17]. The initial problem is reduced to the problemfor a composite system whose internal degrees of freedom describe an ( A − × d -dimensional oscillator, and the external degrees of freedom describe thecenter-of-mass motion of A particles in the d -dimensional Euclidean space. Forsimplicity, we restrict our consideration to the so-called s -wave approximation[11] corresponding to one-dimensional Euclidean space ( d = 1). It is shown thatthe reduction is provided by using appropriately chosen symmetrized coordinatesrather than the conventional Jacoby coordinates.The main goal of introducing the symmetrized coordinates is to provide theinvariance of the Hamiltonian with respect to permutations of A identical par-ticles. This allows construction not only of basis functions, symmetric or anti-symmetric under permutations of A − A Carte-sian coordinates of the initial particles. We refer the expansion of the solution inthe basis of such type as the Symmetrized Coordinate Representation (SCR).The paper is organized as follows. In Section 2, we present the statementof the problem in the conventional Jacobi and the symmetrized coordinates. InSection 3, we introduce the SCR of the solution of the considered problem anddescribe the appropriate algorithm implemented using the MAPLE computeralgebra system. In Section 4, we analyze some examples of generating the sym-metrized coordinate representation for A − Consider the system of A identical quantum particles with the mass m andthe set of Cartesian coordinates x i ∈ R d in the d -dimensional Euclidean space,considered as the vector ˜ x = (˜ x , ..., ˜ x A ) ∈ R A × d in the A × d -dimensionalconfiguration space. The particles are coupled by the pair potential ˜ V pair (˜ x ij )depending on the relative positions, ˜ x ij = ˜ x i − ˜ x j , similar to that of a harmonicoscillator ˜ V hosc (˜ x ij ) = mω (˜ x ij ) with the frequency ω . The whole system issubject to the influence of the potentials ˜ V (˜ x i ) describing the external field of atarget. The system is described by the Schr¨odinger equation − ¯ h m A (cid:88) i =1 ∂ ∂ ˜ x i + A (cid:88) i,j =1; i P x osc , V pair ( x ij ) = ˜ V pair ( x ij x osc ) /E osc , V hosc ( x ij ) = ˜ V hosc ( x ij x osc ) /E osc = A ( x ij ) and V ( x i ) = ˜ V ( x i x osc ) /E osc , onecan rewrite the above equation in the form − A (cid:88) i =1 ∂ ∂x i + A (cid:88) i,j =1; i 12 1 / √ 12 1 / √ − / √ · · · √ A − A √ A − A √ A − A √ A − A · · · − A − √ A − A , The inverse coordinate transformation x = J − y is implemented using the trans-posed matrix J − = J T , i.e., J is an orthogonal matrix with pairs of complexconjugate eigenvalues, the absolute values of which are equal to one. The Jacobicoordinates have the property (cid:80) A − i =0 ( y i · y i ) = (cid:80) Ai =1 ( x i · x i ) = r . Therefore, A (cid:88) i,j =1 ( x ij ) = 2 A A − (cid:88) i =0 ( y i ) − A (cid:88) i =1 x i ) = 2 A A − (cid:88) i =1 ( y i ) , so that Eq. (1) takes the form (cid:34) − ∂ ∂y + A − (cid:88) i =1 (cid:18) − ∂ ∂y i + ( y i ) (cid:19) + U ( y , ..., y A − ) − E (cid:35) Ψ ( y , ..., y A − ) = 0 ,U ( y , ..., y A − ) = A (cid:88) i,j =1; i The transformation from the Cartesian coordinates to one of the possible choicesof the symmetrized ones ξ i has the form, ξ = Cx and x = Cξ : ξ = 1 √ A (cid:32) A (cid:88) t =1 x t (cid:33) , ξ s = 1 √ A (cid:32) x + A (cid:88) t =2 a x t + √ Ax s +1 (cid:33) , s = 1 , ..., A − ,x = 1 √ A (cid:32) A − (cid:88) t =0 ξ t (cid:33) , x s = 1 √ A (cid:32) ξ + A − (cid:88) t =1 a ξ t + √ Aξ s − (cid:33) , s = 2 , ..., A, or, in the matrix form, ξ ξ ξ ... ξ A − ξ A − = C x x x ... x A − x A , C = 1 √ A · · · a a a · · · a a a a a · · · a a a a a · · · a a ... ... ... ... . . . ... ...1 a a a · · · a a a a a · · · a a , (3)where a = 1 / (1 − √ A ) < a = a + √ A . The inverse coordinate transforma-tion is performed using the same matrix C − = C , C = I , i. e., C = C T is asymmetric orthogonal matrix with the eigenvalues λ = − λ = 1, ..., λ A = 1and det C = − 1. For A = 2, the symmetrized variables (3) are within normaliza-tion factors similar to the symmetrized Jacobi coordinates (2) considered in [9], while at A = 4 they correspond to another choice of symmetrized coordinates(¨ x , ¨ x , ¨ x , ¨ x ) T = C ( x , x , x , x ) T considered in [2,3,4] and mentioned earlierin [18,5]. We could not find a general definition of symmetrized coordinates forA-identical particles like (3) in the available literature, so we believe that in thepresent paper it is introduced for the first time. With the relations a − a = √ A , a − a √ A taken into into account, the relative coordinates x ij ≡ x i − x j of apair of particles i and j are expressed in terms of the internal A − x ij ≡ x i − x j = ξ i − − ξ j − ≡ ξ i − ,j − ,x i ≡ x i − x = ξ i − + a A − (cid:88) i (cid:48) =1 ξ i (cid:48) , i, j = 2 , ..., A. (4)So, if only the absolute values of x ij are to be considered, then there are ( A − A − / A − A − V pair ( x ij ). Thesymmetrized coordinates are related to the Jacobi ones as y = Bξ , B = JC : y y y ... y A − y A − = B ξ ξ ξ ... ξ A − ξ A − , B = · · · b b − b − b − · · · b − b − b +2 b b − b − · · · b − b − b +3 b +3 b b − · · · b − b − b +4 b +4 b +4 b · · · b − b − ... ... ... ... ... . . . ... ...0 b + A − b + A − b + A − b + A − · · · b + A − b A − , (5)where b + s = 1 / (( √ A − (cid:112) s ( s + 1)), b − s = √ A/ (( √ A − (cid:112) s ( s + 1)), and b s =(1 + s − s √ A ) / (( √ A − (cid:112) s ( s + 1)). One can see that for the center of massthe symmetrized and Jacobi coordinates are equal, y = ξ , while the relativecoordinates are related via the ( A − × ( A − 1) matrix M with the elements M ij = B i +1 ,j +1 and det M = ( − A × d , i.e., the matrix, obtained by cancellingthe first row and the first column. The inverse transformation ξ = B − y is givenby the matrix B − = ( JC ) − = CJ T = B T , i.e., B is also an orthogonal matrix.In the symmetrized coordinates Eq. (1) takes the form (cid:34) − ∂ ∂ξ + A − (cid:88) i =1 (cid:18) − ∂ ∂ξ i + ( ξ i ) (cid:19) + U ( ξ , ..., ξ A − ) − E (cid:35) Ψ ( ξ , ..., ξ A − ) = 0 , (6) U ( ξ , ..., ξ A − ) = A (cid:88) i,j =1; i 1, as followsfrom Eq. (3), i.e., the invariance of Eq. (1) under permutations x i ↔ x j at i, j = 1 , ..., A survives. Table 1. The first few eigenvalues E Sj and the oscillator S-eigenfunctions (13)at E Sj − E S ≤ E S = A − 1. We use the notations | [ i , i , ..., i A − ] (cid:105) ≡ Φ s [ i ,i ,...,i A − ] ( ξ , ..., ξ A − ) from Eqs. (8) and (10), i.e., [ i , i , ..., i A − ] assumesthe summation over permutations of [ i , i , ..., i A − ] in the layer 2 (cid:80) A − k =1 i k + A − E s ( a ) i . A=2 A=3 A=4 E Sj − E S j Φ Sj ( ξ ) j Φ Sj ( ξ , ξ ) j Φ Sj ( ξ , ξ , ξ )1 | [0] (cid:105) | [0 , (cid:105) | [0 , , (cid:105) | [2] (cid:105) | [0 , (cid:105) | [0 , , (cid:105) | [0 , (cid:105) − √ | [1 , (cid:105) | [1 , , (cid:105) | [4] (cid:105) √ | [0 , (cid:105) + | [2 , (cid:105) | [0 , , (cid:105) | [0 , , (cid:105) √ | [0 , (cid:105) − | [1 , (cid:105) − √ | [2 , (cid:105) | [1 , , (cid:105) For simplicity, consider the solutions of Eq. (6) in the internal symmetrizedcoordinates { ξ , ..., ξ A − } ∈ R A − , x i ∈ R , in the case of 1D Euclidean space( d = 1). The relevant equation describes an ( A − Φ j ( ξ , ..., ξ A − ) and the energy eigenvalues E j : (cid:34) A − (cid:88) i =1 (cid:18) − ∂ ∂ξ i +( ξ i ) (cid:19) − E j (cid:35) Φ j ( ξ , ..., ξ A − ) = 0 , E j = 2 A − (cid:88) k =1 i k + A − , (7)where the numbers i k , k = 1 , ..., A − i k = 0 , , , , ... . The eigen-functions Φ j ( ξ , ..., ξ A − ) can be expressed in terms of the conventional eigen-functions of individual 1D oscillators as Φ j ( ξ , ..., ξ A − ) = (cid:88) A − (cid:80) k =1 i k + A − E j β j [ i ,i ,...,i A − ] ¯ Φ [ i ,i ,...,i A − ] ( ξ , ..., ξ A − ) , (8)¯ Φ [ i ,i ,...,i A − ] ( ξ , ..., ξ A − ) = A − (cid:89) k =1 ¯ Φ i k ( ξ k ) , ¯ Φ i k ( ξ k ) = exp( − ξ k / H i k ( ξ k ) √ π √ i k √ i k ! , where H i k ( ξ k ) are Hermite polynomials [19]. Generally the energy level E f =2 f + A − f = (cid:80) A − k =1 i k , of an ( A − p = ( A + f − /f ! / ( A − Φ [ i ,i ,...,i A − ] ( ξ , ..., ξ A − ). This de-generacy allows further symmetrization by choosing the appropriate coefficients β ( j )[ i ,i ,...,i A − ] . Degeneracy multiplicity p of all states with the given energy E j Table 2. The first few eigenvalues E Aj and the oscillator A-eigenfunctions (13)at E Aj − E A ≤ E A = A − 1. We use the notations | [ i , i , ..., i A − ] (cid:105) ≡ Φ a [ i ,i ,...,i A − ] ( ξ , ..., ξ A − ) from Eq. (11), i.e., [ i , i , ..., i A − ] assumes the sum-mation over the multiset permutations of [ i , i , ..., i A − ] in the layer 2 (cid:80) A − k =1 i k + A − E s ( a ) i . A = 2, E A = 3 A = 3, E A = 8 A = 4, E A = 15 E Aj − E A j Φ Aj ( ξ ) j Φ Aj ( ξ , ξ ) j Φ Aj ( ξ , ξ , ξ )1 | [1] (cid:105) | [0 , (cid:105) + √ | [1 , (cid:105) | [0 , , (cid:105) | [3] (cid:105) √ | [0 , (cid:105) + | [1 , (cid:105)− √ | [2 , (cid:105) | [0 , , (cid:105) | [0 , (cid:105) − √ | [2 , (cid:105) | [1 , , (cid:105) | [5] (cid:105) √ | [0 , (cid:105) + √ | [1 , (cid:105) | [0 , , (cid:105) − | [2 , (cid:105) + √ | [3 , (cid:105) | [0 , , (cid:105) √ | [0 , (cid:105) − √ | [2 , (cid:105) | [1 , , (cid:105) defined by formula p = (cid:88) (cid:80) A − k =1 i k + A − E j N β , N β = ( A − / N υ (cid:89) k =1 υ k ! , (9)where N β is the number of multiset permutations (m.p.) of [ i , i , ..., i A − ], and N υ ≤ A − i k in the multiset [ i , i , ..., i A − ],and υ k is the number of repetitions of the given value i k . Step 1. Symmetrization with respect to permutation of A − particles For the states Φ sj ( ξ , ..., ξ A − ) ≡ Φ s [ i ,i ,...,i A − ] ( ξ , ..., ξ A − ), symmetric with re-spect to permutation of A − i = [ i , i , ..., i A − ], the coefficients β i [ i (cid:48) ,i (cid:48) ,...,i (cid:48) A − ] in Eq. (8) are β i [ i (cid:48) ,i (cid:48) ,...,i (cid:48) A − ] = (cid:40) √ N β , if [ i (cid:48) , i (cid:48) , ..., i (cid:48) A − ] is a m. p. of [ i , i , ..., i A − ],0 , otherwise . (10)The states Φ aj ( ξ , ..., ξ A − ) ≡ Φ a [ i ,i ,...,i A − ] ( ξ , ..., ξ A − ), antisymmetric withrespect to permutation of A − Φ aj ( ξ , ..., ξ A − ) = 1 (cid:112) ( A − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ Φ i ( ξ ) ¯ Φ i ( ξ ) · · · ¯ Φ i A − ( ξ )¯ Φ i ( ξ ) ¯ Φ i ( ξ ) · · · ¯ Φ i A − ( ξ )... ... . . . ...¯ Φ i ( ξ A − ) ¯ Φ i ( ξ A − ) · · · ¯ Φ i A − ( ξ A − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (11)i.e., the coefficients β ( i )[ i (cid:48) ,i (cid:48) ,...,i (cid:48) A − ] in (8) are expressed as β ( i )[ i (cid:48) ,i (cid:48) ,...,i (cid:48) A − ] = ε i (cid:48) ,i (cid:48) ,...,i (cid:48) A − / (cid:112) ( A − , 23 1 Fig. 1. Profiles of the first eight oscillator S-eigenfunctions Φ S [ i ,i ] ( ξ , ξ ), at A = 3 in the coordinate frame ( ξ , ξ ). The lines correspond to pair collision x = x , x = x and x = x of the projection ( x , x , x ) → ( ξ , ξ ), markedonly in the left upper panel with ‘23’, ‘12’, and ‘13’, respectively. The additionallines are nodes of the eigenfunctions Φ S [ i ,i ] ( ξ , ξ ).where ε i (cid:48) ,i (cid:48) ,...,i (cid:48) A − is a totally antisymmetric tensor. This tensor is defined asfollows: ε i (cid:48) ,i (cid:48) ,...,i (cid:48) A − = +1( − i (cid:48) , i (cid:48) , ..., i (cid:48) A − is an even (odd) permutation ofthe numbers i < i < ... < i A − , and ε i (cid:48) ,i (cid:48) ,...,i (cid:48) A − = 0 otherwise, i.e., when sometwo numbers in the set i (cid:48) , i (cid:48) , ..., i (cid:48) A − are equal. Therefore, for antisymmetricstates the numbers i k in Eq. (7) take the integer values i k = k − , k, k + 1 , ... , k = 1 , ..., A − s and a are used for the functions, symmetric (antisymmet-ric) under permutations of A − S and A are used for the functions,symmetric (asymmetric) under permutations of A initial Cartesian coordinates.This is actually the symmetry with respect to permutation of identical parti-cles themselves; in this sense, S and A states may be attributed to boson- andfermion-like particles. However, we prefer to use the S (A) notation as morerigorous. Step 2. Symmetrization with respect to permutation of A particles For A = 2, the symmetrized coordinate ξ corresponds to the difference x − x ofCartesian coordinates, so that a function even (odd) with respect to ξ appears tobe symmetric (antisymmetric) with respect to the permutation of two particles x ↔ x . Hence, even (odd) eigenfunctions with corresponding eigenvalues E sj =2(2 n ) + 1 ( E aj = 2(2 n + 1) + 1) describe S (A) solutions.For A ≥ 3, the functions, symmetric (antisymmetric) with respect to permu-tations of Cartesian coordinates x i +1 ↔ x j +1 , i, j = 0 , ..., A − Φ S ( A ) ( ..., x i +1 , ..., x j +1 , ... ) ≡ Φ S ( A ) ( ξ ( x , ..., x A ) , ..., ξ A − ( x , ..., x A )) Fig. 2. The same as in Fig. 1, but for the first eight oscillator A-eigenfunctions Φ A [ i ,i ] ( ξ , ξ ), at A = 3. = ± Φ S ( A ) ( ..., x j +1 , ..., x i +1 , ... )become symmetric (antisymmetric) with respect to permutations of symmetrizedcoordinates ξ i ↔ ξ j , i, j = 1 , ..., A − Φ S ( A ) ( ..., ξ i , ..., ξ j , ... ) = ± Φ S ( A ) ( ..., ξ j , ..., ξ i , ... ) , as follows from Eq. (4). However, the converse statement is not valid, Φ s ( a ) ( ..., ξ i , ..., ξ j , ... ) = ± Φ s ( a ) ( ..., ξ j , ..., ξ i , ... ) (cid:54)⇒ Φ s ( a ) ( x , ..., x i +1 , ... ) = ± Φ s ( a ) ( x i +1 , ..., x , ... ) , because we deal with a projection map( ξ , ..., ξ A − ) T = ˆ C ( x , ..., x A ) T , (12)which is implemented by the ( A − × ( A ) matrix ˆ C with the matrix elementsˆ C ij = C i +1 ,j , obtained from (3) by cancelling the first row. Hence, the func-tions, symmetric (antisymmetric) with respect to permutations of symmetrizedcoordinates, are divided into two types, namely, the S (A) solutions, symmetric(antisymmetric) with respect to permutations x ↔ x j +1 at j = 1 , ..., A − Φ S ( A ) ( x , ..., x j +1 , ... ) = ± Φ S ( A ) ( x j +1 , ..., x , ... )and the other s (a) solutions, Φ s ( a ) ( x , ..., x i +1 , ... ) (cid:54) = ± Φ s ( a ) ( x i +1 , ..., x , ... ),which should be eliminated. These requirements are equivalent to only one per-mutation x ↔ x , as follows from (4), which simplifies their practical imple-mentation. With these requirements taken into account in the Gram–Schmidt process, implemented in the symbolic algorithm SCR , we obtained the requiredcharacteristics of S and A eigenfunctions, Φ S ( A ) i ( ξ , ..., ξ A − ) = (cid:88) (cid:80) A − k =1 i k + A − E s ( a ) i α S ( A ) i [ i ,i ,...,i A − ] Φ s ( a )[ i ,i ,...,i A − ] ( ξ , ..., ξ A − ) . (13) The algorithm SCR :Input : A is the number of identical particles; i max is defined by the maximal value of the energy E i max ;( ξ , ..., ξ A − ) and ( x , ..., x A ) are the symmetrized and the Cartesian coordinates; Output : Φ S ( A ) i ( ξ , ..., ξ A − ) and Φ S ( A ) i ( x , ..., x A ) are the total symmetric (antisymmet-ric) functions (13) in the above coordinates connected by (12); Local : E s ( a ) i ≡ E S ( A ) i = 2 (cid:80) A − k =1 i k + A − i + 1) th eigenenergy; i min = 0 for the symmetric and i min = ( A − for the antisymmetric case; Φ j ≡ Φ s ( a )[ i ,i ,...,i A − ] ( ξ , ..., ξ A − ) and Φ j ≡ Φ s ( a )[ i ,i ,...,i A − ] ( x , ..., x A ) are the func-tions, symmetric (antisymmetric) with respect to A − p s ( a ) ≡ p i ; s ( a ) and p S ( A ) ≡ p i ; S ( A ) are the degeneracy factors of the energy levels E s ( a ) i and E S ( A ) i for s(a) and S(A) functions, respectively; p i ; min ( p i ; max ) and P i ; min ( P i ; max ) are the lowest (highest) numbers of s(a) andS(A) functions, belonging to the energy levels E s ( a ) i and E S ( A ) i , respectively; { ¯ α j } and { α S ( A ) pj } are the sets of intermediate and desired coefficients;1.1 j := 0; for i from i min to i max do ;1.2: p i ; min := j + 1;1.3: for each sorted i , i , ..., i A − , 2 (cid:80) A − k =1 i k + A − E s ( a ) i do j := j + 1;construction Φ j ( ξ , ..., ξ A − ) = Φ sj ( ξ , ..., ξ A − ) from (8), (10)or Φ j ( ξ , ..., ξ A − ) = Φ aj ( ξ , ..., ξ A − ) from (11) Φ j ( x , ..., x A ) =subs(( ξ , ..., ξ A − ) → ( x , ..., x A ) , Φ j ( ξ , ..., ξ A − )); end for p i ; max := j ; p i ; s ( a ) = p i ; max − p i ; min + 1; end for P min = 1; for i from i min to i max do P i ; min = P min ;2.3.: Φ ( ξ , ..., ξ A − ) = (cid:80) p i ; max j = p i ; min ¯ α j Φ j ( ξ , ..., ξ A − ); Φ ( x , ..., x A ) = (cid:80) p i ; max j = p i ; min ¯ α j Φ j ( x , ..., x A );2.4.: Φ ( x , x , ..., x A ) :=change( x ↔ x , Φ ( x , x , ..., x A )));2.5.: Φ ( x , x , ..., x A ) ∓ Φ ( x , x , ..., x A ) = 0 , → (¯ α pj , j = p i ; min , ..., p i ; max , p = 1 , ..., p i ; S ( A ) ) ;2.6.: P i ; max = P i ; min − p i ; S ( A ) ; Fig. 3. Upper panel: Profiles of the oscillator S-eigenfunctions Φ S [1 , , ( ξ , ξ , ξ ), Φ S [0 , , ( ξ , ξ , ξ ) and A-eigenfunction Φ A [0 , , ( ξ , ξ , ξ ), at A = 4 (left, middle,and right panels, respectively ). Some maxima and minima positions of thesefunctions are connected by black and gray lines and duplicated in lower panels:two tetrahedrons forming a stella octangula for Φ S [1 , , ( ξ , ξ , ξ ), a cube and anoctahedron for Φ S [0 , , ( ξ , ξ , ξ ), and a polyhedron with 20 triangle faces (only8 of them being equilateral triangles) for Φ A [0 , , ( ξ , ξ , ξ ).2.7.: Gram–Schmidt procedure for Φ ( ξ , ..., ξ A − ) → Φ S ( A ) p ( x , x , ..., x A ) = (cid:80) p i ; max j = p i ; min α S ( A ) pj Φ j ( x , x , ..., x A ); Φ S ( A ) p ( ξ , ..., ξ A − ) = (cid:80) p i ; max j = p i ; min α S ( A ) pj Φ j ( ξ , ..., ξ A − ),at p = P i ; min , ..., P i ; max ; end for The SCR algorithm was implemented in MAPLE 14 on Intel Core i5 CPU 6603.33GHz, 4GB 64 bit, to generate first 11 symmetric (antisymmetric) functionsup to ∆E j = 12 at A = 6 with CPU time 10 seconds (600 seconds), that togetherwith a number of functions in dependence of number of particles given in Table3 demonstrates efficiency and complexity of the algorithm.The examples of generated total symmetric and antisymmetric ( A − A = 4, the first four states from Table 1 are similar to those of the translation-invariant model without excitation of the center-of-mass variable [3]. Table 3. The degeneracy multiplicities p from (9), p s = p a and p S = p A ofs-, a-, S-, and A-eigenfunctions of the oscillator energy levels ∆E j = E • j − E • , • = ∅ , s, a, S, A . A=3 A=4 A=5 A=6 ∆E j p p s , p a p S , p A p p s , p a p S , p A p p s , p a p S , p A p p s , p a p S , p A As an example, in Figs. 1 and 2 we show isolines of the first eight S andA oscillator eigenfunctions Φ S [ i ,i ] ( ξ , ξ ) and Φ A [ i ,i ] ( ξ , ξ ) for A = 3, calcu-lated at the second step of the algorithm. One can see that the S (A) oscillatoreigenfunctions are symmetric (antisymmetric) with respect to reflections fromthree straight lines. The first line (labelled ‘23’) corresponds to the permutation( x , x ) and is rotated by π/ ξ . Thesecond and the third lines (labelled ‘12’ and ‘13’) correspond to the permuta-tions ( x , x ) and ( x , x ) and are rotated by π/ D in R and the 3-body permutation group S (A = 3).Figure 3 shows examples of profiles of S and A oscillator eigenfunctions for A = 4. Note that four maxima (black) and four minima (grey) of the S eigenfunc-tion Φ S [1 , , ( ξ , ξ , ξ ) are positioned at the vertices of two tetrahedrons forminga stella octangula , with the edges shown by black and grey lines, respectively.Eight maxima and six outer minima for S eigenfunction Φ S [0 , , ( ξ , ξ , ξ ) arepositioned at the vertices of a cube and an octahedron, the edges of which areshown by black and grey lines, respectively. The positions of twelve maxima ofthe A oscillator eigenfunction, Φ A [0 , , ( ξ , ξ , ξ ) coincide with the vertices of apolyhedron with 20 triangle faces (only 8 of them being equilateral triangles)and 30 edges, 6 of them having the length 2.25 and the other having the length2.66. The above shapes of eigenfunctions illustrate the isomorphism between thetetrahedron group T d in R and the 4-particle permutation group S (A = 4),discussed in [2] in the case of d = 3.The degeneracy multiplicity (9), i.e., number p of all states with the givenenergy E j of low part of spectra, the numbers p s ( p a ) of the states, symmetric(antisymmetric) under permutations of A − the total numbers p S ( p A ) of the states, symmetric (antisymmetric) under per-mutations of A initial Cartesian coordinates are summarized in Table 3. Notethat the S and A states with E (cid:48) = E S,A + 2 do not exist. The numbers p s ( p a )are essentially smaller than the total number p of all states, which simplifies theprocedure of constructing S (A) states with possible excitation of the center-of-mass degree of freedom and allows the use of a compact basis with the reduceddegeneracy p S ( p A ) of the S (A) states in our final calculations. For clarity, inthe case A = 3, d = 1, the S(A)-type functions generated by the SCR algorithm,in polar coordinates ξ = ρ cos ϕ , ξ = ρ sin ϕ are expressed as: Φ S ( A ) k,m ( ρ, ϕ ) = C km ( ρ ) m/ exp( − ρ / L mk ( ρ ) cossin (3 m ( ϕ + π/ , where C km is the normalization constant, L mk ( ρ ) are the Laguerre polynomials[19], k = 0 , , ... , m = 0 , , ... for S states, while m = 1 , , ... for A states, that areclassified by irr of the D m -symmetry group. The corresponding energy levels E S ( A ) k,m = 2(2 k +3 m +1) = E s ( a )[ i ,i ] = 2( i + i +1) have the degeneracy multiplicity K + 1, if the energy E S ( A ) k,m − E S ( A )1 = 12 K + K (cid:48) , where K (cid:48) = 0 , , , , , 14. Forexample, in Figs. 1 and 2 we show the wave functions Φ S , ( ρ, ϕ ) and Φ S , ( ρ, ϕ )(or Φ A , ( ρ, ϕ ) and Φ A , ( ρ, ϕ )) labelled with 6 and 7, corresponding to the energylevels E S ( A ) k,m − E S ( A )1 = 12 with the degeneracy K = 2, while the functionslabelled with 1 , , , , , K = 1). So, the eigenfunctions ofthe A-identical particle system in one dimension are degenerate in accordancewith [21], and this result disagrees with nondegenerate ansatz solutions [10]. We considered a model of A identical particles bound by the oscillator-typepotential under the influence of the external field of a target in the new sym-metrized coordinates. The constructive SCR algorithm of symmetrizing or an-tisymmetrizing the A − A − A − A − D and T d for A = 3 and A = 4 shapes is displayed. It is shownthat one can use the presented SCR algorithm, implemented using the MAPLEcomputer algebra system, to construct the basis functions in the closed analyticalform. However, for practical calculations of matrix elements between the basisstates, belonging to the lower part of the spectrum, this is not necessary. Theapplication of the developed approach and algorithm for solving the problemof tunnelling clusters through barrier potentials of a target is considered in ourforthcoming paper [22]. The proposed approach can be adapted to the analysis of tetrahedral-symmetric nuclei, quantum diffusion of molecules and micro-clustersthrough surfaces, and the fragmentation in producing neutron-rich light nuclei.The authors thank Professor V.P. Gerdt for collaboration. The work was sup-ported partially by grants 13-602-02 JINR, 11-01-00523 and 13-01-00668 RFBRand the Bogoliubov-Infeld program. References 1. Moshinsky, M., Smirnov, Y.F.: The harmonic oscillator in modern physics. InformaHealth Care, Amsterdam (1996)2. Kramer, P., Moshinsky, M.: Group theory of harmonic oscillators (III). States withpermutational symmetry. Nucl. Phys. 82, 241–274 (1966)3. Aguilera-Navarro, V.C., Moshinsky, M., Yeh, W.W.: Harmonic-oscillator statesand the α particle I. Form factor for symmetric states in configuration space. Ann.Phys. 51, 312–336 (1969)4. Aguilera-Navarro, V.C., Moshinsky, M., Kramer, P.: Harmonic-oscillator states andthe α particle II. Configuration-space states of arbitrary symmetry. Ann Phys. 54,379–393 (1969)5. L´evy-Leblond, J.-M.: Global and democratic methods for classifying N particlestates. J. Math. Phys. 7, 2217–2229 (1966).6. Neudatchin, V.G., Smirnov, Yu.F.: Nucleon clusters in the light nuclei. Nauka,Moscow (1969) (in Russian)7. Novoselsky, A., Katriel, J.: Non-spurious harmonic oscillator states with arbitrarysymmetry. Ann. Phys. 196, 135–149 (1989)8. Wildermuth, K., Tang, Y.C.: A unified theory of the nucleus. Academic Press, NewYork (1977)9. Kamuntaviˇcius, G.P., Kalinauskas, R.K., Barrett, B.R., Mickeviˇcius, S., Germanas,D.: The general harmonic-oscillator brackets: compact expression, symmetries,sums and Fortran code. Nucl. Phys. A 695, 191–201 (2001)10. Wang, Zh., Wang, A., Yang, Y., Xuechao, Li: Exact eigenfunctions of N-bodysystem with quadratic pair potential. arXiv:1108.1607v4 (2012)11. Pen’kov, F.M.: Metastable states of a coupled pair on a repulsive barrier. Phys.Rev. A 62, 044701–1–4 (2000)12. Gusev, A.A., Vinitsky, S.I., Chuluunbaatar, O., Gerdt, V.P., Rostovtsev, V.A.:Symbolic-numerical algorithms to solve the quantum tunneling problem for a cou-pled pair of ions. LNCS 6885, 175–191 (2011)13. Dobrowolski, A., G´o´zd´z, A., Mazurek, K., Dudek, J.: Tetrahedral symmetry innuclei: new predictions based on the collective model. International Journal ofModern Physics E Vol. 20, No. 02, pp. 500-50614. Dobrowolski, A., Szulerecka, A., G´o´zd´z, A. Electromagnetic transitions in hypo-thetical tetrahedral and octahedral bands; G´o´zd´z, A.: Hidden symmetries in intrin-sic frame; in: Proc 19th Nuclear Physics Workshop in Kazimierz Dolny, September,2012, http://kft.umcs.lublin.pl/wfj/archive/2012/proceedings.php15. Fock, V.A.: N¨aherungsmethode zur L¨osung des quantenmechanischenMehrk¨orperproblems, Zs. Phys. 61, 126–148 (1930).16. Hamermesh, M.: Group theory and its application to physical problems. Dover,New York (1989)17. Kanada-En’yo, Yo., Hidaka, Yo.: α -cluster structure and density waves in oblatenuclei. Phys. Rev. C , 014313–1–16 (2011)518. Jepsent, D.W., Hirschfelder, J.O.: Set of co-ordinate systems which diagonalize thekinetic energy of relative motion, Proc. Natl. Acad. Sci. U.S.A. 45, 249–256 (1959);19. Abramovits, M., Stegun, I.A.: Handbook of Mathematical Functions, Dover, NewYork (1972), p. 1037.20. Baker, G.A., Jr., Degeneracy of the n-dimensional, isotropic, harmonic oscillator.Phys. Rev.103