Symmetrized Vibrational-Rotational Basis for Collective Nuclear Models

The generalized projection operators for the intrinsic group acting in the space L2(SO(3)) and in the space spanned by the eigenfunctions of a multidimensional harmonic oscillator are constructed. New symbolic-numerical algorithm implemented in computer algebra system for generating irreducible representations of the point symmetry groups in the rotor + shape vibrational space of a nuclear collective model in the intrinsic frame is presented. The efficiency of the algoritm is investigated by calculating the bases of irreducible representations subgroup D̅4y of octahedral group in the intrinsic frame of a quadrupole-octupole collective nuclear model. The discrete variable representation algorithm proposed for solving eigenvalue problem, describing vibrational-rotational motion of collective nuclear model in intrinsic frame.


Introduction
The most efficient way of construction of collective models in nuclear and molecular physics is using of the so called intrinsic frame [1] The most useful is the rotating intrinsic frame. However, to construct vibrational-rotational basis of the collective models in an intrinsic frame one needs to notice some important features of this notion [2] and the technique of generalized projection operators [3].
In this paper we propose a simple and effective algorithm for numerical construction of such basis for the vibrational + rotational collective motion based on algorithms elaborated in refs. [4,5]. We consider application of the algorithm on the example of construction of the rotationvibrational quadrupole-octupole basis of the irreducible representation (irr.) A1 = Γ 1 of the point group D 4y (in classification of Ref. [6]) in the intrinsic frame considered in the zero and onephonon approximation, as in ref. [7]. Structure of the paper is following. In Section 2, a setting of intrinsic frame and symmetrization groups are presented In Section 3, a setting of quadrupoleoctupole harmonic oscillator in rotating frame is formulated. In Section 4, parametric vibration basis functions of irr. A1 = Γ 1 of intrinsic dihedral D 4y point group are derived. In Section 5, an 1

Intrinsic frame and symmetrization group
The classical rotation is well understood phenomenon in which the orientation of a body is changing with time. Contrary, the quantum rotation allows to determine only the probability of a given orientation and there is no time variable in the wave function. The notion of the quantum rotational motion allows to define the rotating intrinsic frame, for the collective variables {α (lab) λµ } defined by the equation of surface [1]: The corresponding collective variables in the intrinsic frame we denote by {α λµ }. They can be obtained by the quantum rotation of the laboratory collective variables {α assuming, in addition, that the rotation group SO(3) R (Ω) parameters, represented by the Euler angles Ω = (Ω 1 , Ω 2 , Ω 3 ), are considered as a part of intrinsic variables. The intrinsic variables α λ are invariant in respect to the laboratory rotationsR(Ω). It is important to notice that inclusion of the Euler angles into the set of intrinsic variables makes this set of variables redundant, 3 variable more than needed. It implies that, the definition of the intrinsic frame requires three additional conditions which leads the same number of variables in both frames F k (α, Ω) = 0, where k = 1, 2, 3. ( In this way one can obtain a new description of a physical system, e.g. a nucleus, in which the rotational motion can be directly described by the Euler angles, i.e. we start to work in the rotating frame. An interesting question is how to investigate symmetries of a nucleus in the intrinsic frame. It is clear, that the transformations furnishing an intrinsic symmetry group have to be defined in the intrinsic frame.
A convenient definition of intrinsic groups was formulated in [2] in the following form: for each element g of the group G, one can define a corresponding operatorĝ in the group linear space L G as:ĝ where all elements inside the ket vectors S = g∈G c g g, here c g are the complex numbers, form a group algebra of the group G. In this definition the notion of the group linear space L G is used. This space is defined as the linear space spanned by all possible formal linear combinations of the elements of the group G It looks like the group algebra mentioned above, but, it is important that the elements of L G have to be considered only as vectors, not as the elements of the group algebra. The group formed by the collection of the operatorsĝ is called the intrinsic group G related to the group G. One of the most important property of the intrinsic group G is that this group commutes with its partner group G (this can be called a laboratory group, because it acts in the laboratory frame): [G, G] = 0.
The groups G and G are antyisomorphic. The required anti-isomorphism between the partner groups G and G is given by This property suggests that the partner groups G and G have a lot of common properties as e.g. similar structure of representations, decompositions of the Kronecker products, the Clebsch-Gordan coefficients and many others.
As an example let us consider a relation among representations of both groups. Because the partner groups commute one can find common basis |Γmk for representations of the group G and the group G. The representations are defined as: To compare both representation one can use as the basis in the form of the generalized projection operators (elements of the group linear space where dim[Γ] denotes the dimension of the representation Γ and card(G) is the number of elements in the group G. This allows to calculate (8) where ∆ A bit different are definitions of irreducible tensors in respect to the laboratory group G and the intrinsic group G. By definition the irreducible tensors in respect to the laboratory group G transform asĝ The tensors in respect to the intrinsic group G, due to the anti-isomorphism between both groups, have to be defined in the following waŷ As an example, let us consider the action of the intrinsic group in the collective space consisted of the square integrable functions of the deformation parameters and the Euler angles. The intrinsic rotation operatorsR(ḡ 1 ,ḡ 2 ) ∈ SO(3) α × SO(3) Ω (the indices α and Ω show the variables which are affected by the corresponding group) are defined as followŝ The action of the group SO(3) α onto the deformation variables is a bit non-standard and is given by the following equation where D λ µ µ are Wigner functions [8]. The intrinsic group SO(3) corresponding to the laboratory rotation group SO(3) defined in the laboratory frame consists of all rotationsR(ḡ,ḡ) for which the deformation parameters and the Euler angles are rotated with the same angles. The required anti-isomorphism between the partner groups SO(3) and SO (3) is given by (7). It is important to notice that, in general, not all transformations (ḡ 1 , are allowed in the intrinsic frame. They are allowed if they do not break the conditions which define the intrinsic frame (3) For example, in the case of the quadrupole collective variables α 2 with the standard Bohr condition which define the intrinsic frame: α 2,±1 = 0 and α 22 = α 2−2 , the allowed intrinsic rotationsR(ḡ 1 ,ḡ 2 ) ∈ SO(3) α × SO(3) Ω have to fulfil the following conditionŝ where the second argument represents the unit element of the group SO(3) Ω . The Bohr conditions allow for the arbitrary rotationsḡ 2 ∈ SO(3) Ω . Using the conditions (18) the allowed rotations of the deformation parameters α have to satisfy the following equations In this case, the octahedral point group O α ⊂ SO(3) α acting only on the variables α provide the solution of the set of equations (19). In practice, the transformation from the laboratory frame to the intrinsic frame is not a oneto-one function. In principle, there are two possibilities to achieve uniqueness of transformation from the laboratory to the intrinsic frame: • first, one can define the appropriate region of the intrinsic collective variables in which the transformation from the laboratory to intrinsic frame is a one-to-one function, • second, one can allow for the whole range of collective variables but then one needs to fulfil some special conditions for physical states.
where α = {α λµ } and which leave invariant the laboratory variables as function of the intrinsic variables: where α (lab) (α, Ω) =R(Ω −1 )α, see (2). The group G S we call the symmetrization group. The symmetrization group decomposes the collective manifold into orbits of physically equivalent points.
Let the function Ψ (lab) denotes a state vector of a nucleus in the laboratory frame. The corresponding state vector in the intrinsic frame has to fulfil the obvious equation which represents the fact that the wave function of the physical system written in the laboratory frame has to be well and uniquely defined function. However, after transformation of Eq. (22) with the elements of the symmetrization group we potentially change Ψ and do not change the laboratory state vector: To omit the obvious contradiction, we have to define the uniqueness condition for the states in the intrinsic frame Ψ(α , Ω ) = Ψ(α, Ω).
This condition we will call the symmetrization condition. It can be expressed as invariance of the intrinsic state vectors in respect to all transformationsh ∈ G S , hΨ(α, Ω) = Ψ(α, Ω), where the group G S is the symmetrization group for a given intrinsic frame.
In quantum mechanics the collective states in the laboratory/intrinsic frame belong the spaces of square integrable functions. In the laboratory frame the quantum state space is usually represented by the whole space L 2 ({α (lab) λµ }), however, the quantum state space in the intrinsic frame has to be restricted to the subspace of functions satisfying the symmetrization condition (25): K S ⊂ L 2 ({α λµ }). This implies that in calculations we cannot use any standard basis, but we have to construct the basis dedicated to the subspace K S .

Generalized Projection Operators
To construct the appropriate bases in the space of vibrational functions (32) for the intrinsic point groups G we have to use the action of the generalized projection operatorsP Γ ab corresponding to the intrinsic group: where the matrices of the representations ∆ Γ (g) ab 0 obtained from the action of g ∈ G in the space of basis functions of the Cartesian variables are used.

Algorithm for construction of the orthonormal vibrational-rotational basis
Calculation of the parametric vibrational-rotational basis functions of irr. A1=Γ 1 of D 4y .
Step 1. To calculate overlaps, B JN n 1 ,...,n 6 ,K,n 1 ,...,n 6 ,K = Ψ JN n 1 ,...,n 6 ,K |Ψ JN n 1 ,...,n 6 ,K , Step 2. Solving the eigenvalue problem of the matrix B allows to find the orthonormal basis in which this matrix is diagonal: For λ t = 0 one gets the following basis of orthonormal states: where J, M, Γ and b fixed, and C t is the normalization coefficient.
Step 3. To calculate the scalar products: u t |u t = C t C t s ,K ,s,K X t (s , K ) X t (s, K) Ψ JN n 1 ,...,n 6 ,K |Ψ JN n 1 ,...,n 6 ,K = = C t C t s,K X t (s, K) λ t X t (s, K) = δ t t C t C t λ t s,K |X t (s, K)| 2 .
It means that the states |u t for which the eigenvalue λ t = 0 are the zero vectors, and for the cases when λ t = 0 the normalization coefficient is equal: C t = λ t s,K |X t (s, K)| 2 −1/2 . Note that because the eigenequation (38) is, in fact, the eigenequation for the matrix representation of the projection operatorP ...
During the integration we use Gauss-Hermite quadratures of ν th order at ν = 10: