Symbolic Algorithm for Generating Irreducible Bases

Symbolic Algorithm for Generating Irreducible Bases of Point Groups in the Space of SO(3) Group A.A. Gusev1 , V.P. Gerdt1 , S.I. Vinitsky1 , V.L. Derbov2 , A. Góźdź3 , and A. P¸edrak3 1

Joint Institute for Nuclear Research, Dubna, Russia 2 Saratov State University, Saratov, Russia 3 Institute of Physics, Maria Curie-Skłodowska University, Lublin, Poland [email protected], [email protected], [email protected]

Abstract. A symbolic algorithm which can be implemented in computer algebra systems for generating bases for irreducible representations of the laboratory and intrinsic point symmetry groups acting in the rotor space is presented. The method of generalized projection operators is used. First the generalized projection operators for the intrinsic group acting in the space L2 (SO(3)) are constructed. The efficiency of the algorithm is investigated by calculating the bases for both laboratory and intrinsic octahedral groups irreducible representations for the set of angular momenta up to J = 10. Keywords: intrinsic point symmetry groups, method of generalized projection operators, octahedral group, irreducible representations in the space L2 (SO(3)).

1

Introduction

The rotational motion is one of the most important motions in Nature. To treat the rotational degrees of freedom in quantum objects like molecules and nuclei an intrinsic frame is commonly introduced [7,8,10,11]. In this case, the rotational motion is described on the manifold of the rotation group SO(3) which is usually parameterized by Euler angles. Molecules and nuclei can possess point symmetries [3,6]. This fact requires the construction of irreducible representations and their bases in terms of Wigner functions for arbitrary angular momentum. Particular examples of such states were already constructed “by hand” for study of the nuclear collective models [9]. However, a universal exact method is required to allow the analysis of rotational states in arbitrary molecules and nuclei [4,5]. The method of generalized projection operators (GPOs), proposed here for solving this problem, is, in principle, well known [1]. The novel idea is to use it for constructing the bases of irreducible representations of the intrinsic point groups. It means that we construct these bases in the intrinsic frame instead of the laboratory one, which is, c Springer International Publishing Switzerland 2015  V.P. Gerdt et al. (eds.): CASC 2015, LNCS 9301, pp. 166–181, 2015. DOI: 10.1007/978-3-319-24021-3_13

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in fact, required for the description of these objects. For this purpose we use the notion of the so-called intrinsic group [2]. In this paper, the symbolic algorithm implemented in computer algebra systems (CAS) for generating irreducible bases of intrinsic point groups in the group space of the group SO(3) is formulated. To perform noncommutative operator multiplication the appropriate tools of the Reduce are applied though the Maple or Mathematica can be used also. The efficiency of the algorithm is investigated by calculating the bases of the irreducible representations of the intrinsic octahedral group O for the values of angular momentum up to J = 10. The paper is organized as follows. In Section 2, the action of the rotation group in the geometrical space and the functional spaces is defined. In Section 3, the definition of intrinsic groups is given. In Section 4, the algorithm for calculating the GPOs and the bases of irreducible representations of the intrinsic finite groups are presented. In Sections 5 and 6, the algorithm is applied to the octahedral group. In the Conclusion, the results and perspectives of applications of this algorithm are summarized.

2

Rotations

We start from the precise definition of the rotation group action in both the geometrical space and the functional spaces spanned by the quantum states of a certain quantum system. Because of some discrepancies in the literature related to the definition of rotation in 3D space, we first specify the precise description of rotational angles, rotational matrix elements, and the action of rotation operators in the space L2 (SO(3)) used in the present paper. 2.1

Geometric Rotation

According to Refs. [8,10,11], the rotation of the right-handed Cartesian frame S(x, y, z) in three-dimensional geometric space R3 through the Euler angles (α, β, γ) can be defined as follows: 1◦ the initial frame S(x, y, z) is rotated about the axis Oz through the angle α; we get the new frame S(x , y  , z  = z); ◦ 2 the next step is the rotation of S(x , y  , z  = z) through the angle β about the axis Oy  ; we get the new frame S(x , y  = y  , z  ); ◦ 3 finally we perform the last rotation of S(x , y  , z  ) through the angle γ about the axis Oz  and get the frame S(x , y  , z  = z  ). The appropriate rotation matrix has the form (see [10,11]): ⎤ ⎡ cosα cosβ cosγ− sinα sinγ − cosα cosβ sinγ− sinα cosγ cosα sinβ R=⎣ sinα cosβ cosγ+ cosα sinγ − sinα cosβ sinγ+ cosα cosγ sinα sinβ ⎦ . (1) − sinβ cosγ sinβ sinγ cosβ

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The basic vectors of the rotated frame ei are related to those of the initial frame ei as  ei = Rki (α, β, γ)ek , i, k = x, y, z (2) k

and the resulting transformations of the vector coordinates that follow from Eqs. (1), (2) can be written as:  Ai = Rki (α, β, γ)Ak , (3) k

where Ai are the components of a vector in the rotated frame, and Ai are the components of the same vector in the laboratory frame. Note that there is an inconsistency between the English edition [11] of the book by Varshalovich et al. and the Russian edition [10]. As the reference book we use the English edition. 2.2

Rotation in Functional Spaces

Another problem is to define the action of the rotation group on the Cartesian basis to be consistent with the corresponding action on the space of functions depending on Euler angles, L2 (SO(3)). The Cartesian coordinates in the threedimensional Euclidean space are known to be related to the Wigner functions of the rotation group SO(3). First, the Cartesian components (Ax , Ay , Az ) of the vector A are expressed in terms of the spherical functions (see [8,11]):  Ax = 2π/3 |A|{Y1−1 (θ, ϕ) − Y11 (θ, ϕ)},   Ay = i 2π/3 |A|{Y1−1 (θ, ϕ) + Y11 (θ, ϕ)}, Az = 4π/3 |A|Y10 (θ, ϕ). (4) In turn, the spherical functions can be expressed in terms of Wigner functions  J∗ YJM (θ, ϕ) = (2J + 1)/(4π) DM0 (ϕ, θ, χ). (5) J (ϕ, θ, χ) the eigenvectors of the angular momentum operLet us denote by rMK ators in the space L2 (SO(3)) with orthonormalization conditions: √ J J∗ J J (ϕ, θ, χ) = 2J + 1 DMK (ϕ, θ, χ), rMK |rM (6) rMK  K   = δMM  δKK  .

Then YJM (θ, ϕ) =

 J 1/4π rM0 (ϕ, θ, χ).

(7)

To simplify the notation, let us denote the set of Euler angles by a single letter Ω = (ϕ, θ, χ). The action of the laboratory group SO(3) in the space L2 (SO(3)) is defined by the left shift operation: gˆψ(Ω) = ψ(g −1 Ω).

(8)

Applying this operation to the components (4) treated as functions of Euler angles we get (3), i.e., the left shift operation yields the same rotation of the

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vector as produced by the geometric rotation of Cartesian coordinates. Thus, acting on functions of Euler angles we arrive at the transformation of coordinates. The same is valid for an arbitrary spherical tensor, it transforms as coordinates rather than as a function, where the standard action on a function of coordinates is assumed to be: (9) gˆf (ξ) = f (g −1 ξ). Note that the operation (8) applied to the functions (4) of Euler angles is consistent with the definition of spherical tensors, but not with the operation (9) applied to (4) as functions of Cartesian coordinates. The SO(3) action (9) on a function of Cartesian coordinates yields (x1 = x, x2 = y, x3 = z):  gˆf (x, y, z) = f gˆ−1 x, gˆ−1 y, gˆ−1 z 

   =f Rk1 (g −1 )xk , Rk2 (g −1 )xk , Rk3 (g −1 )xk k k

k    =f (10) R1k (g)xk , R2k (g)xk , R3k (g)xk . k

k

k

Obviously, the matrices of the rotation group representations calculated in the Cartesian basis, consisting of some functions fk (x, y, z), are different from the matrices calculated in the space L2 (SO(3)). It can be shown that these matrices are transposed with respect to each other. For example, in the special case of 3D representations with the basis functions fk (x1 , x2 , x3 ) = xk , where k = 1, 2, 3, Eq.(10) yields  RkT k (g)fk (x1 , x2 , x3 ), (11) gˆfk (x1 , x2 , x3 ) =  k

T

where R (g) is the transposed three-dimensional rotation matrix (1) and g = g(α, β, γ), while the geometric rotation of coordinates is  xk = gˆxk = Rk k (g)xk . (12)  k

3

Intrinsic Group

The definition of intrinsic group G is related to the so called right shift operation acting on the group manifold of a given group G. The right shift operator is chosen in the form that allows one to obtain, for convenience, the intrinsic group anti-isomorphic to the group G. This property of intrinsic groups is good for physical applications because it allows one to use most of the standard results obtained for standard groups. The idea of intrinsic groups is described in detail in Ref. [2]. Some applications of these groups to the description of quantum systems in the intrinsic frame are presented in Ref. [4]. For each element g of the group G, one can define the corresponding operator g acting in the group linear space LG as: gS = Sg,

for all S ∈ LG .

(13)

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x y Cx 1 0 Cy 0 1 Cz 0 0 Cα −1 −1 Cβ −1 Cγ

z sin θ cos θ ϕ 0 1 0 0 0 1 0 π/2 1 √0 1 0 1 √23 √13 (5π)/4

1 −1

1 −1 −1

Cδ 1 1 1 Ca 1 1 0 Cb 1 −1 0 Cc 1 0 1 Cd −1 0 1 Ce 0 1 1 Cf 0 1 −1

√ √2 √3 √2 √3 √2 3

− √13 (3π)/4 − √13 (7π)/4

1 1

√1 2 √1 2 √1 2 √1 2

√1 3

0 0

√1 2 √1 2 √1 2 −1 √ 2

π/4 π/4 (7π)/4 0 π π/2 π/2

Fig. 1. The rotational axes of the octahedral group

The group G consisting of the operators g is called the intrinsic group related to the group G. It is easy to see some implications of definition (13), for example, gh = hg

(14)

for any g ∈ G and h ∈ G. This brings us to the important property [G, G] = 0,

(15)

i.e., the actions of the laboratory and intrinsic group commute. This property fulfils the physical requirement of independence between the laboratory and intrinsic frames. Another noteworthy fact is the anti-isomorphism between the groups G and G, i.e., (16) ψG : G → G, where ψG (g) = g, ψG (g 1 g2 ) = g2 g1 . This property makes it possible to apply all already known properties of the laboratory group, such as the representations, Clebsch–Gordan coupling coefficients, etc., to the intrinsic groups. The action of the intrinsic group G on complex functions Ψ : H → C, where G is a subgroup of the group H, e.g., H = SO(3), is defined as [2]: ˆ gψ(h) = ψ(hg −1 ).

(17)

¯Γ of the intrinsic group G is equal to According to [2], the representation Δ Γ the transposed representation Δ of the laboratory group G: Δ¯Γ = (ΔΓ )T .

(18)

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Table 1. Values of n1 , n2 and n3 of the rotational angles of R(C∗ ) from Eq. (29) N 1 5 9 13 17 21

C∗ E C3α −1 C3α C4x −1 C4y C2c

n1 0 2 3 3 2 0

n2 0 1 1 1 1 1

n3 0 3 0 1 2 2

N 2 6 10 14 18 22

C∗ C2x C3β −1 C3β C4y −1 C4z C2d

n1 0 0 3 0 0 2

n2 2 1 1 1 0 1

n3 2 3 2 0 3 0

N 3 7 11 15 19 23

C∗ C2y C3γ −1 C3γ C4z C2a C2e

n1 0 2 1 0 0 1

n2 2 1 1 0 2 1

n3 0 1 0 1 1 1

N 4 8 12 16 20 24

C∗ C2z C3δ −1 C3δ −1 C4x C2b C2f

n1 0 0 1 1 0 3

n2 0 1 1 1 2 1

n3 2 1 2 3 3 3

It means that to construct representations of intrinsic groups in the space L2 (H) one can use the representation matrices ΔΓ obtained for the group G ⊂ H, but in the other space. For example, for H = SO(3), one can use the appropriate Cartesian basis fk (x, y, z), where k = 1, 2, . . . , dim(Γ ).

4

Generalized Projection Operators

The technique of projecting on the spaces of irreducible representation of a given group G is described, e.g., in [1]. The idea is based on using the generalized projection operators (GPOs) for the irreducible representations of locally compact groups. In this paper, we restrict ourselves to finite groups. The projection method is very useful for obtaining orthogonal bases of irreducible representations in the required spaces of functions. It can be applied to both laboratory groups and intrinsic groups. The main idea is to decompose the Hilbert space K = L2 (G) of square integrable functions, defined in the group manifold of the group G, into a set of orthogonal irreducible subspaces that are invariant with respect to the action of the group G, irreducible subspaces Kκ : K= Kκ . (19) κ

This technique requires using the GPOs defined as:  dim(Γ )  Γ Γ Pˆab Δab (g) gˆ, = card(G)

(20)

g∈G

where dim(Γ ) denotes the dimension of the irreducible representation Γ of the group G, card(G) is the order of the group G (i.e., the number of its elements), ΔΓab (g) denotes the matrix elements of the irreducible representation Γ for the element g. The symbol gˆ denotes a unitary operator of the group G, acting in the space K. Γ have the following properties (see [1], Chap. 7, §3): The operators Pˆab

† Γ Γ Pˆab = Pˆba ,

Γ ˆΓ  Γ Pa b = δΓ Γ  δba Pˆab Pˆab .

(21)

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A.A. Gusev et al. Table 2. The multiplication table of operators C1 ,...,C24 numbered in Table 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

2 1 4 3 8 7 6 5 11 12 9 10 16 22 19 13 21 20 15 18 17 14 24 23

3 4 1 2 6 5 8 7 12 11 10 9 23 17 20 24 14 19 18 15 22 21 13 16

4 3 2 1 7 8 5 6 10 9 12 11 24 21 18 23 22 15 20 19 14 17 16 13

5 7 8 6 9 11 12 10 1 3 4 2 17 15 24 22 19 16 13 23 20 18 21 14

6 8 7 5 12 10 9 11 3 1 2 4 14 20 16 21 18 24 23 13 15 19 22 17

7 5 6 8 10 12 11 9 4 2 1 3 22 18 13 17 20 23 24 16 19 15 14 21

8 6 5 7 11 9 10 12 2 4 3 1 21 19 23 14 15 13 16 24 18 20 17 22

9 12 10 11 1 4 2 3 5 8 6 7 19 24 14 18 13 22 17 21 23 16 20 15

10 11 9 12 4 1 3 2 7 6 8 5 20 13 21 15 24 17 22 14 16 23 19 18

11 10 12 9 2 3 1 4 8 5 7 6 15 23 22 20 16 14 21 17 24 13 18 19

12 13 9 16 11 24 10 23 3 15 2 18 4 20 1 19 6 22 7 21 5 14 8 17 18 2 16 9 17 8 19 1 23 10 21 7 14 5 22 6 13 12 24 11 15 3 20 4

14 21 17 22 16 24 13 23 19 15 20 18 8 3 11 6 1 9 12 10 4 2 7 5

15 20 19 18 22 14 17 21 13 24 23 16 10 8 4 11 5 1 2 3 6 7 12 9

16 13 23 24 19 20 18 15 14 17 22 21 1 11 5 2 12 6 8 7 10 9 4 3

17 22 14 21 24 16 23 13 18 20 15 19 7 1 10 5 3 12 9 11 2 4 8 6

18 19 20 15 17 21 22 14 24 13 16 23 9 6 1 12 7 4 3 2 8 5 11 10

19 18 15 20 14 22 21 17 16 23 24 13 12 5 3 9 8 2 1 4 7 6 10 11

20 15 18 19 21 17 14 22 23 16 13 24 11 7 2 10 6 3 4 1 5 8 9 12

21 14 22 17 23 13 24 16 20 18 19 15 6 2 12 8 4 10 11 9 1 3 5 7

22 17 21 14 13 23 16 24 15 19 18 20 5 4 9 7 2 11 10 12 3 1 6 8

23 24 16 13 20 19 15 18 21 22 17 14 4 12 7 3 11 8 6 5 9 10 1 2

24 23 13 16 18 15 19 20 17 14 21 22 3 10 6 4 9 5 7 8 11 12 2 1

Γ These properties imply that the operator Pˆbb is a true projection operator: Γ † Γ Γ Γ Γ (Pˆbb ) = Pˆbb , Pˆbb Pˆbb = Pˆbb . Let us describe step by step the procedure of constructing the orthogonal basis of irreducible representation using the concept of GPOs (see [1], Chap. 7, §4A). This approach is rather general, but for the purpose of this paper we restrict our consideration to the subgroup G of the intrinsic rotation group G ⊂ SO(3). Let us introduce the orthogonal basis in the space K = L2 (SO(3)) as the set J , where K = of Wigner functions (6). For fixed J and M , the vectors |rMK −J, −J + 1, . . . , J, span the orthonormal basis in the subspace KJ . Note that for any subgroup of SO(3), the subspaces related to different angular momentum quantum numbers J and J  are orthogonal reducible subspaces. For this reason, our procedure can be applied to each subspace KJ separately. The GPOs in this example are constructed in accordance with Eq. (20). The algorithm [1] consists of the following steps:

ˆ¯ Γ K, where P ˆ¯ Γ is the projection Step 1. Choose one of the subspaces KbΓ = P bb bb operator. The label Γ denotes the required irreducible representation, b is an arbitrary but fixed index.

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Table 3. Elements of octahedral group C∗ ⎛

R(C∗ ) ⎞ 1 0 0 E ⎝ 0 1 0⎠ ⎛ 0 0 1⎞ 0 0 −1 C3α ⎝ 1 0 0 ⎠ ⎛ 0 −1 0 ⎞ 0 1 0 −1 ⎝ C3α 0 0 −1 ⎠ −1 0 0⎞ ⎛ 1 0 0 C4x ⎝ 0 0 −1 ⎠ ⎛ 0 1 0⎞ 0 0 −1 −1 ⎝ 0 1 0⎠ C4y ⎛ 1 0 0⎞ 0 0 1 C2c ⎝ 0 −1 0 ⎠ 1 0 0

C∗ ⎛

C∗ ⎛

C∗ ⎛

C2x

C2y

C2z

C3β −1 C3β

C4z −1 C4z

C2d

R(C∗ ) ⎞ 1 0 0 ⎝ 0 −1 0 ⎠ ⎛ 0 0 −1 ⎞ 0 0 1 ⎝ −1 0 0 ⎠ ⎛ 0 −1 0 ⎞ 0 −1 0 ⎝ 0 0 −1 ⎠ ⎛ 1 0 0⎞ 0 −1 0 ⎝ 1 0 0⎠ ⎛ 0 0 1⎞ 0 1 0 ⎝ −1 0 0 ⎠ ⎛ 0 0 1⎞ 0 0 −1 ⎝ 0 −1 0 ⎠ −1 0 0

C3γ −1 C3γ

C4y C2a C2e

R(C∗ ) ⎞ −1 0 0 ⎝ 0 1 0⎠ ⎛ 0 0 −1 ⎞ 0 0 −1 ⎝ −1 0 0 ⎠ ⎛ 0 1 0⎞ 0 −1 0 ⎝ 0 0 1⎠ ⎛ −1 0 0 ⎞ 0 0 1 ⎝ 0 1 0⎠ ⎛ −1 0 0 ⎞ 0 1 0 ⎝ 1 0 0⎠ ⎛ 0 0 −1 ⎞ −1 0 0 ⎝ 0 0 1⎠ 0 1 0

C3δ −1 C3δ

−1 C4x

C2b C2f

R(C∗ ) ⎞ −1 0 0 ⎝ 0 −1 0 ⎠ ⎛ 0 0 1⎞ 0 0 1 ⎝ 1 0 0⎠ ⎛ 0 1 0⎞ 0 1 0 ⎝ 0 0 1⎠ ⎛ 1 0 0⎞ 1 0 0 ⎝ 0 0 1⎠ ⎛ 0 −1 0 ⎞ 0 −1 0 ⎝ −1 0 0 ⎠ ⎛ 0 0 −1 ⎞ −1 0 0 ⎝ 0 0 −1 ⎠ 0 −1 0

One can directly check the orthogonality of the spaces KbΓ , for different b. ˆ¯ Γ u ∈ KΓ , P ˆ¯ Γ u|P ˆ¯ Γ u = u|P ˆ¯ Γ P ˆ¯ Γ ˆ¯ Γ u ∈ KΓ and |P ˆ¯ Γ Let |P cc c cc bb b bb bb cc u = δbc u|Pbc u. Step 2. Construct in an arbitrary manner the basis in the subspace KbΓ , we denote it by u1 , u2 , . . . , udim(KΓb ) . J Since the vectors |rMK  are basic vectors in KJ , the projected vectors |K = ˆ ¯ Γ |rJ  for K = −J, −J + 1, . . . , J span the the projected subspace KΓ . The P bb MK Jb Γ because they can vectors |K do not form an orthonormal basis in the space KJb be neither orthogonal, nor even linearly independent. To solve the problem we use the symmetric orthonormalization procedure. The overlaps J ˆ¯ Γ |rJ   |P (22) K|K   = rMK bb MK form a finite (2J + 1) × (2J + 1) dimensional Hermitian matrix referred to as Gramm matrix, which we denote here by N . Solving the eigenvalue problem for the matrix N allows one to find the orthonormal basis, in which this matrix is diagonal: N wt (K) = λt wt (K). (23) The coefficients wt (K) allow for the construction of the states, which in the Generator Coordinate Method [7] are named the “natural states”, and which furnish the required basis in our space. For λt = 0 one gets the following basis of orthonormal states:  ˆ¯ Γ |rJ , |ut  = At wt (K)P (24) bb MK K

where J, M, Γ , and b are fixed, and At is the normalization coefficient.

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A.A. Gusev et al. Table 4. Classes, Cartesian bases, and character table for octahedral group O

Irr\Classes C1 C2 C3 C4 C5 Cartesian bases Γ1 = A1 1 1 1 1 1 v1 = R2 = x2 + y 2 + z 2 Γ2 = A2 1 1 1 −1 −1 v1 = xyz √ Γ3 = E 2 −1 2 0 0 v1 = 3(x2 − y 2 ), v2 = (2z 2 − x2 − y 2 ) Γ4 = T2 3 0 −1 −1 1 v1 = xy, v2 = yz, v3 = xz Γ5 = T1 3 0 −1 1 −1 v1 = x, v2 = y, v3 = z −1 −1 −1 −1 C1 = {E}; C2 = {C3α , C3β , C3γ , C3δ , C3α , C3β , C3γ , C3δ }; C3 = {C2x , C2y , C2z }; −1 −1 −1 C4 = {C4x , C4y , C4z , C4x , C4y , C4z }; C5 = {C2a , C2b , C2c , C2d , C2e , C2f }

Let us calculate the scalar products:  J ˆ¯ Γ J wt (K  ) wt (K)rMK ut |ut  = At At  |Pbb |rMK  = K  ,K

= At At



wt (K  ) λt wt (K) = δt t At At λt

K



|wt (K)|2 .

(25)

K

This result means that the states |ut  for which the eigenvalue λt = 0 are the zero vectors, and for the cases when λt = 0 the normalization coefficient is equal to At = (λt K |wt (K)|2 )−1/2 . Note that since the eigenequation (23) is, in fact, ˆ¯ Γ , the eigenequation for the matrix representation of the projection operator P bb the eigenvalues are equal to λt = 0, 1. Thus, we obtain the required basis in the Γ projected subspace KJb . ˆ¯ Γ to each of the vectors u , i.e., P ˆ¯ Γ u , Step 3. Apply successively the operators P t ab ab t where Γ and b are fixed and a = 1, 2, . . . , dim(Γ ). As a result, we get the required basis states of the irreducible representation Γ of the group G in the rotor space:  ˆ¯ Γ |rJ . |Γ a; tJ = At wt (K)P (26) ab MK K

The projection of the angular momentum M and the quantum number b can be chosen in an arbitrary manner. These states satisfy the usual conditions Γ  a ; t J  |Γ a; tJ = δΓ  Γ δa a δt t δJ  J .

(27)

The last but not the least issue is how to construct the representations ΔΓ to be used in the GPOs. These representations have to be consistent with the action of group operators, the elements of GPOs. This problem was solved in Ref. [9]. In table (28), the relations between the action of the rotation group and the appropriate matrix representations are shown. ΔΓ denotes the irreducible representation of the laboratory group, obtained using the Cartesian basis fk (x, y, z), where k = 1, 2, . . . , dim(Γ ). The right column of table (28) shows which representation should be used in the GPOs while acting in the space spanned by the basis shown in the left column [9]:

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Table 5. The operators in the octahedral group representations Γ1 , Γ2 and Γ3 C∗ E , C2x , C2y , C2z C3α , C3β , C3γ , C3δ −1 −1 −1 −1 C3α , C3β , C3γ , C3δ −1 C4x , C4x , C2e , C2f −1 C4y , C4y , C2c , C2d −1 C4z , C4z , C2a , C2b

Γ1 (C∗ ) Γ2 (C∗ )     1 1     

1 1 1 1 1

         

1 1 −1 −1 −1

    

Γ3 (C ∗ ) 1 0 0 √1  −1/2 3/2 √ −1/2 − 3/2 √  −1/2 − 3/2 √ 3/2 −1/2 √  √1/2 − 3/2 − 3/2 √−1/2  √1/2 3/2 3/2 −1/2 −1 0 0 1 

rotation group action matrices required in projection operators gˆfk (x, y, z) = fk (g −1 (x, y, z)) ΔΓ (g) J J −1 gˆrMK (Ω) = rMK (g Ω) (ΔΓ )T (g) J J −1 ΔΓ (g) gˆ¯rMK (Ω) = rMK (Ωg )

(28)

Here g¯ denotes an element of the intrinsic group SO(3).

5

Example of Using the Algorithm for the Octahedral Group

Below we consider only one example, the octahedral point group. However, the formalism and the algorithm is quite general and can be applied to any point group. In principle, all considerations of the previous sections can be applied to a wider class of groups, namely, to any finite group. 5.1

Construction of Elements of the Octahedral Group from Its Generators

The octahedral group O consists of 24 rotations grouped into five classes Cn . The element Cnζ denotes the right-handed rotation through the angle 2π/n around the axis ζ. The direction of the required rotation axes and the directional angles of their unit vectors are shown in Fig. 1. The notation corresponds to Appendix D in [3]. Step O1. All three-dimensional matrices representing the elements R(C∗ ) of the group O were calculated using the matrix R(α, β, γ), Eq. (1) R(C∗ ) = R(α = πn1 /2, β = πn2 /2, γ = πn3 /2),

(29)

with the values n1 , n2 and n3 specified in Table 1. The angles α = πn1 /2, β = πn2 /2 and γ = πn3 /2 were calculated using the formulas [11] sin(β/2)= sin θ sin(ω/2), tan((α+γ)/2)= cos θ tan(ω/2), (α−γ)/2=ϕ−π/2,

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A.A. Gusev et al. Table 6. Operators of the octahedral group representation Γ4

C∗ ⎛ Γ4 (C∗ ) ⎞ 1 0 0 E ⎝ 0 1 0⎠ ⎛ 0 0 1⎞ 0 0 −1 C3α ⎝ −1 0 0 ⎠ ⎛ 0 1 0⎞ 0 −1 0 −1 ⎝ C3α 0 0 1⎠ ⎛ −1 0 0 ⎞ 0 0 −1 C4x ⎝ 0 −1 0 ⎠ ⎛ 1 0 0⎞ 0 −1 0 −1 ⎝ 1 0 0⎠ C4y ⎛ 0 0 −1 ⎞ 0 −1 0 C2c ⎝ −1 0 0 ⎠ 0 0 1

C∗ ⎛ Γ4 (C∗ ) ⎞ −1 0 0 C2x ⎝ 0 1 0 ⎠ ⎛ 0 0 −1 ⎞ 0 0 −1 C3β ⎝ 1 0 0 ⎠ ⎛ 0 −1 0 ⎞ 0 1 0 −1 ⎝ C3β 0 0 −1 ⎠ −1 0 0⎞ ⎛ 0 1 0 C4y ⎝ −1 0 0 ⎠ ⎛ 0 0 −1 ⎞ −1 0 0 −1 ⎝ 0 0 −1 ⎠ C4z ⎛ 0 1 0⎞ 0 1 0 C2d ⎝ 1 0 0 ⎠ 0 0 1

C∗ ⎛ Γ4 (C∗ ) ⎞ −1 0 0 C2y ⎝ 0 −1 0 ⎠ ⎛ 0 0 1⎞ 0 0 1 C3γ ⎝ −1 0 0 ⎠ ⎛ 0 −1 0 ⎞ 0 −1 0 −1 ⎝ C3γ 0 0 −1 ⎠ 1 0 0⎞ ⎛ −1 0 0 C4z ⎝ 0 0 1 ⎠ ⎛ 0 −1 0 ⎞ 1 0 0 C2a ⎝ 0 0 −1 ⎠ ⎛ 0 −1 0 ⎞ 0 0 −1 C2e ⎝ 0 1 0 ⎠ −1 0 0

C∗ ⎛ Γ4 (C∗ ) ⎞ 1 0 0 C2z ⎝ 0 −1 0 ⎠ ⎛ 0 0 −1 ⎞ 0 0 1 C3δ ⎝ 1 0 0 ⎠ ⎛ 0 1 0⎞ 0 1 0 −1 ⎝ C3δ 0 0 1⎠ ⎛ 1 0 0⎞ 0 0 1 −1 ⎝ C4x 0 −1 0 ⎠ −1 0 0⎞ ⎛ 1 0 0 C2b ⎝ 0 0 1 ⎠ ⎛ 0 1 0⎞ 0 0 1 C2f ⎝ 0 1 0 ⎠ 1 0 0

with θ and ϕ shown in Fig. 1 for the rotation Cn∗ through the angle ω = 2π/n. As a result, we obtain the required R(Cn∗ ) presented in Table 3. The multiplication table of the elements C1 ,...,C24 of the octahedral group (Table 2) uses the direct numeration/coding 1, 2, .., 24 of Table 1. 5.2

Construction of Irreducible Representations of the Octahedral Group O in the Cartesian Bases

Step O2. Let us describe the construction of irreducible representations in the Cartesian bases. There are five classes and five irreducible representations of the octahedral group O, the notations of which, similar to those of Ref. [3], are presented in Table 4 together with the Cartesian basis vectors taken from Ref. [6], where the appropriate action is applied according to the first row of Eq. (28). The matrices Γ∗ (C∗ ) of the operators C∗ describing the action of the representation Γ∗ on the Cartesian basis vectors vi of Table 4 are sought in the form:  C∗ vj (x, y, z) = (Γ∗ (C∗ ))ij vi (x, y, z). (30) i

The unknown coefficients (Γ∗ (C∗ ))ij in Eq. (30) are calculated by direct substitution of this relation

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177

Table 7. Operators of the octahedral group representation Γ5 C∗ ⎛ Γ5 (C∗ ) ⎞ 1 0 0 E ⎝ 0 1 0⎠ ⎛ 0 0 1⎞ 0 0 −1 C3α ⎝ 1 0 0 ⎠ ⎛ 0 −1 0 ⎞ 0 1 0 −1 ⎝ C3α 0 0 −1 ⎠ −1 0 0⎞ ⎛ 1 0 0 C4x ⎝ 0 0 −1 ⎠ ⎛ 0 1 0⎞ 0 0 −1 −1 ⎝ 0 1 0⎠ C4y ⎛ 1 0 0⎞ 0 0 1 C2c ⎝ 0 −1 0 ⎠ 1 0 0

C∗ ⎛ Γ5 (C∗ ) ⎞ 1 0 0 C2x ⎝ 0 −1 0 ⎠ ⎛ 0 0 −1 ⎞ 0 0 1 C3β ⎝ −1 0 0 ⎠ ⎛ 0 −1 0 ⎞ 0 −1 0 −1 ⎝ C3β 0 0 −1 ⎠ ⎛ 1 0 0⎞ 0 0 1 C4y ⎝ 0 1 0 ⎠ ⎛ −1 0 0 ⎞ 0 1 0 −1 ⎝ −1 0 0 ⎠ C4z ⎛ 0 0 1⎞ 0 0 −1 C2d ⎝ 0 −1 0 ⎠ −1 0 0

C∗ ⎛ Γ5 (C∗ ) ⎞ −1 0 0 C2y ⎝ 0 1 0 ⎠ ⎛ 0 0 −1 ⎞ 0 0 −1 C3γ ⎝ −1 0 0 ⎠ ⎛ 0 1 0⎞ 0 −1 0 −1 ⎝ C3γ 0 0 1⎠ −1 0 0⎞ ⎛ 0 −1 0 C4z ⎝ 1 0 0 ⎠ ⎛ 0 0 1⎞ 0 1 0 C2a ⎝ 1 0 0 ⎠ ⎛ 0 0 −1 ⎞ −1 0 0 C2e ⎝ 0 0 1 ⎠ 0 1 0

C∗ ⎛ Γ5 (C∗ ) ⎞ −1 0 0 C2z ⎝ 0 −1 0 ⎠ ⎛ 0 0 1⎞ 0 0 1 C3δ ⎝ 1 0 0 ⎠ ⎛ 0 1 0⎞ 0 1 0 −1 ⎝ C3δ 0 0 1⎠ ⎛ 1 0 0⎞ 1 0 0 −1 ⎝ C4x 0 0 1⎠ ⎛ 0 −1 0 ⎞ 0 −1 0 C2b ⎝ −1 0 0 ⎠ ⎛ 0 0 −1 ⎞ −1 0 0 C2f ⎝ 0 0 −1 ⎠ 0 −1 0

 C∗ vj (x, y, z) = vj R(C∗ )11 x+R(C∗ )12 y+R(C∗ )13 z,

(31)  R(C∗ )21 x+R(C∗ )22 y+R(C∗ )23 z, R(C∗ )31 x+R(C∗ )32 y+R(C∗ )33 z ,

with the appropriate permutations of the indexes i and j: Γij (C∗ ) = Rji (C∗ ) = Rij (C∗ )T . From Eqs. (30) and (31) we get the following expression:  vj R(C∗ )11 x+R(C∗ )12 y+R(C∗ )13 z, R(C∗ )21 x+R(C∗ )22 y+R(C∗ )23 z,   (Γ∗ (C∗ ))ij vi (x, y, z) = 0. (32) R(C∗ )31 x+R(C∗ )32 y+R(C∗ )33 z − i

After equating in Eq. (32) the coefficients of monomials xi1 y i2 z i3 to zero we arrive at the set of equations with respect to the required coefficients (Γ∗ (C∗ ))ij . The irreducible representations Γ1 ,..., Γ5 of the octahedral group found with this algorithm are presented in Tables 5–7. Step O3. The characters χΓ (C∗ ) of representation Γ∗ (C∗ ) are calculated as χΓ (C∗ ) = Tr(Γ∗ (C∗ )) =



dim(Γ∗ ) k=1

and are presented in Table 4.

(Γ∗ (C∗ ))kk

(33)

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Remark 1. The obtained octahedral group rotation matrices R(C∗ ) and representations Γ1 ,..., Γ5 differ from the analogous results by Cornwell [3]. They differ by transposition because our rotation matrix R(α, β, γ) from Eq. (1) and those of Cornwell’s paper differ by transposition, too. 5.3

GPOs Implementation for Standard and Intrinsic Point Groups

Now we proceed to the explicit form of the GPOs required for point groups. Though in the rest of this paper we consider only intrinsic groups, we start from the standard (laboratory) groups for comparison. Laboratory Groups. To construct the bases in the space of rotational functions for the laboratory point groups we start from calculating the action of the operators (20) belonging to the standard point group, using the appropriate representation from (28), on the functions (6): JK (Ω) = RnpqM

J  dim(Γn )  J (Γn (g)T )pq gˆ rMK (Ω) = card(G)  g∈G

lab JK AnpqM  M

dim(Γn )  J = (Γn (g)T )pq DM  M (g), card(G)

lab JK J AnpqM  M rM  K (Ω), M =−J

(34)

g∈G

where the appropriate matrices of the representations Γn , obtained from the action in the space of functions of the Cartesian coordinates, are substituted instead of the general symbols ΔΓ . Here M = −J, −J + 1, . . . , J that provides full decomposition of the rotor space for a given J. Intrinsic Groups. To construct the appropriate bases in the space of rotational functions for intrinsic point groups we have to use the action of the operators (20) belonging to the intrinsic group, using the appropriate representation from (28), on functions (6): JM (Ω) = RnpqK

J  dim(Γn )  int JM J J Γn (g)pq gˆ¯ rMK (Ω) = AnpqKK  rMK  (Ω), card(G)  g∈G

int JM AnpqKK 

dim(Γn )  J = Γn (g)pq DKK  (g), card(G)

K =−J

(35)

g∈G

where the representation matrices Γn , obtained from the action in the space of functions of the Cartesian coordinates, are used to replace the general symbols ΔΓ . Here K runs over the full range K = −J, −J + 1, . . . , J that provides the decomposition of the rotor space for a given J into irreducible representations. J Remark 2. From relations Γn = Γn , (Γn (g))T = Γn (g −1 ) and DKK  (g) = J  (DK  K (g)) (see (36)) for the octahedral group follows that the constituent matrices (34) and (35) of rotational functions for laboratory and intrinsic groups  lab JM  T int Anpq . are hermitian conjugate with respect to each other AJM npq =

Symbolic Algorithm for Generating Irreducible Bases of Point Groups

179

Consider step by step the algorithm from Section 4 for explicit construction of irreducible bases for the intrinsic octahedral group. Step 1. First one needs to organize the loop running over all irreducible representation Γn = Γ1 , Γ2 , ..., Γnmax of the group G. In Eq. (35) choose a single (though we are doing the loop over all q0 ) fixed q = q0 , q0 = 1, ..., dim(Γn ) such JM (Ω) is not identically equal to zero, if that there exists K, for which Rnq 0 q0 K possible. It can happen so that either for given q0 or for given representation Γn JM all Rnq (Ω) = 0. 0 q0 K JM Step 2. Given the set of vectors u ˜nK (Ω) = Rnq (Ω), where K = −J, −J + 0 q0 K 1, . . . , J one needs to choose among them a set of linearly independent vectors. Let us assume the quantum number K = K1 , K2 , . . . , Ks , where s ≤ 1, 2, . . . , 2J + 1, to number the linearly independent vectors u ˜nq0 K (Ω)

u ˜nq0 K (Ω) =

JM Rnq (Ω) 0 q0 K

=

J 

J AJM nq0 KK  rMK  (Ω),

K = K1 , ..., Ks (q0 ),

K  =−J

AJM nq0 KK  =

dim(Γn )  J Γn (g)q0 q0 DKK  (g) at q = q0 = p. card(G) g∈G

J J Here the Wigner functions Dmm  (g)=Dmm (α=πn1 /2, β=πn2 /2, γ=πn3 /2) for rotational operators R(C∗ ) of the octahedral group O are calculated using values n1 , n2 , n3 from Table 1 by the formula

⎧ (−1)−m(n1 +n3 )/2 δmm , n2 ⎪ ⎪ ⎨ (−1)J (−1)−m(n1 −n3 +2)/2 δ n2 m,−m , J  Dmm (g) = n2 (−1)−mn1 /2 dJmm (π/2)(−1)−m n3 /2 , ⎪ ⎪ ⎩   (−1)−mn1 /2 (−1)m−m dJmm (π/2)(−1)−m n3 /2 , n2

= 0, = 2, (36) = 1, = 3,

J where for calculation of dJmm (π/2)=Dmm  (α=0, β=π/2, γ=0) we use [10,11]

(−1)m−m dJmm (π/2) = 2J





(J + m)! (J − m)! (J + m )! (J − m )!

min(J+m ,J−m)

×



k=max(0,m −m)

(−1)k (J + m )!(J − m )! . k!(k + m − m )!(J + m − k)!(J − m − k)!

Following section 4 one needs to orthogonalize this set of vectors. The result of orthogonalization using the Gram–Schmidt procedure can be written as: unq0 t (Ω) =

J 

¯ JM u˜nq0 K (Ω). B nq0 tK

K=−J

After the orthogonalization for every n = 1, ..., nmax , where nmax is the number of representations of the point group, we obtain the set of s orthonormalized vectors

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which we denote by unq0 1 (Ω), unq0 2 (Ω), . . ., unq0 t , . . . , unq0 s (Ω), t = 1, ..., s(q0 ). J These vectors unq0 t (Ω) are decomposed using the basis rMK (Ω) uJM nq0 t (Ω)

=

J 

JM J Bnq  rMK  (Ω), 0 tK

JM Bnq  = 0 tK

K  =−J

J 

¯ JM AJM  , (37) B nq0 tK nq0 KK

K=−J

where t = 1, ..., s(q0 ), q0 = 1, ..., dim(Γn ). Step 3. Now for the irreducible representation Γn of the group G and for every vector uJM nq0 t (Ω), t = 1, ..., s(q0 ) at fixed q0 = 1, ..., dim(Γn ) we apply the projection operator (35) for every p = 1, 2, ..., dim(Γn ):  JM ¯ Γn uJM (Ω) = dim(Γn ) v¯ntpq (Ω) = P Γn (g)pq0 gˆ¯ uJM nq nq0 t (Ω) pq t 0 0 0 card(G) g∈G

JM (Ω) = v¯ntpq 0

J 

J AˇJM ntpq0 K  rMK  (Ω),

t = 1, ..., s(q0 ),

(38)

K  =−J

AˇJM ntpq0 K  =

J 

JM Bnq 0 tK

K=−J

dim(Γn )  J Γn (g)pq0 DKK  (g), card(G) g∈G

J DKK  (g)

JM where K = K1 , K2 , . . . , Ks (q0 ), and Bnq were calculated by for0 tK mulas (37). Using the orthonormalization conditions (6) for the basis functions J 2 (Ω), we calculate the normalization factor Nntpq and the required set of rMK 0 orthonormalized vectors v¯ntpq0 (Ω) JM JM (Ω)=¯ vntpq (Ω)/Nntpq0 , vntpq 0 0

2 JM JM Nntpq =¯ vntpq |¯ vntpq = 0 0 0

J 

2 (AˇJM ntpq0 K ) . (39)

K=−J

The above algorithm was realized in the form of the program implemented in the computer algebra system Reduce. The typical running time of calculating the irreducible representations Γ1 ,...,Γ5 for the octahedral group for J ≤ 10 is 980 seconds using the PC Inter Pentium CPU 1.50 GHz 4 GB 64 bit Windows 8. Results of Step 3. In the following, the output of the program for the irreducible representation Γ5 is presented. The first set of vectors (40) corresponds to the basis furnishing representations of the laboratory group the second one (41) of the intrinsic group. The index t distinguishes among equivalent representations. JM (Ω) given by Eqs. (38), (39) For example, the orthonormalized vectors vntpq 0 after the projection (37) at q0 = 3 read as: √ −5 √ −5 √ −5 5K =−( 42r1K (Ω)+9r3K (Ω)+ 5r5K (Ω))/ 128, v5113 √ −5 √ −5 √ −5 √ 5K =−( 30r1K (Ω)− 35r3K (Ω)+ 63r5K (Ω))/ 128, v5213 √ +5 √ +5 √ +5 5K =i( 42r1K (Ω)−9r3K (Ω)+ 5r5K (Ω))/ 128, (40) v5123 √ +5 √ +5 √ +5 √ 5K v5223 =i( 30r1K (Ω)+ 35r3K (Ω)+ 63r5K (Ω))/ 128, +5 5K v5133 =(r4K (Ω)),

5K 5 v5233 = (r0K (Ω));

Symbolic Algorithm for Generating Irreducible Bases of Point Groups

√ 5− √ 5− √ 5− 5M v5113 =−( 42rM1 (Ω)+9rM3 (Ω)+ 5rM5 (Ω))/ 128, √ 5− √ 5− √ 5− √ 5M =−( 30rM1 (Ω)− 35rM3 (Ω)+ 63rM5 (Ω))/ 128, v5213 √ 5+ √ 5+ √ 5+ 5M =−i( 42rM1 (Ω)−9rM3 (Ω)+ 5rM5 (Ω))/ 128, v5123 √ 5+ √ 5+ √ 5+ √ 5M =−i( 30rM1 (Ω)+ 35rM3 (Ω)+ 63rM5 (Ω))/ 128, v5223 5+ 5M v5133 =(rM4 (Ω)),

181

(41)

5M 5 v5233 =(rM0 (Ω)),

where √ √ J± ±J J J J J rMK (Ω)=(rMK (Ω) + rM−K (Ω))/ 2, rMK (Ω)=(rMK (Ω) + r−MK (Ω))/ 2.

6

Conclusion

We present the symbolic algorithm for calculating irreducible representations of intrinsic as well as for laboratory point groups in the rotor space L2 (SO(3)), which can be implemented in computer algebra systems. The bases of these representations are required for calculating spectra and electromagnetic transitions in molecular and nuclear physics. The program is now prepared for calculating both the laboratory and the intrinsic octahedral groups, which is typically considered as the highest rotation point group (without the space inversion). It can be adopted for any subgroup of the octahedral group. The work was partially supported by the Russian Foundation for Basic Research (RFBR) (grants Nos. 14-01-00420 and 13-01-00668) and the Bogoliubov– Infeld program.

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