On generation of the Bargmann-Moshinsky basis of SU(3) group

TL;DR

This paper presents an efficient analytical method for orthonormalising the Bargmann-Moshinsky basis of the SU(3) group, facilitating nuclear model calculations with a new Fortran implementation.

Contribution

It introduces a novel analytical orthonormalisation procedure for the BM basis and provides a practical Fortran program for nuclear physics applications.

Findings

Successful orthonormalisation of the BM basis using analytical formulas

Validated calculations of quadrupole operator components between bases

Implementation of the procedure as a Fortran program

Abstract

An efficient procedure of orthonormalisation of the Bargmann-Moshinsky (BM) basis is examined using analytical formulas of the overlap integrals of the BM basis. Calculations of components of the quadrupole operator between the both BM and the orthonormalised bases needed for construction of the nuclear models are tested. The proposed procedure is also implemented as the Fortran program.

Equations

$$H=\gamma C_2(\mathrm{SU}(3))-\kappa Q\cdot Q+\beta L\cdot L+H^{\prime \prime }(\bar{Q},\bar{L})$$

$$=(\gamma -\kappa )C_2(\text{SU(3)})+(3\kappa +\beta )L^2+H^{\prime \prime }(\bar{Q},\bar{L}),$$

$$H_{3Q}=h_{3Q}\left((\bar{Q}\otimes \bar{Q}{)}_2^3-(\bar{Q}\otimes \bar{Q}{)}_{-2}^3\right),$$

$$H_{3LQ}=h_{3LQ}\left((\bar{L}\otimes \bar{Q}{)}_2^3-(\bar{L}\otimes \bar{Q}{)}_{-2}^3\right),$$

$$\displaystyle H_{4Q}=h_{4Q}\Big{(}\sqrt{\frac{14}{5}}(\bar{Q}\otimes\bar{Q})^{4}_{0}+(\bar{Q}\otimes\bar{Q})^{4}_{-4}+(\bar{Q}\otimes\bar{Q})^{4}_{4}\Big{)},$$

$$[L_{\nu },L_{{\nu }^{\prime }}]=-\sqrt{2}{C}_{1\nu 1{\nu }^{\prime }}^{1\nu +{\nu }^{\prime }}{L}_{\nu +{\nu }^{\prime }},$$

$$[L_{\nu },Q_{{\nu }^{\prime }}]=-\sqrt{6}{C}_{1\nu 2{\nu }^{\prime }}^{2\nu +{\nu }^{\prime }}{Q}_{\nu +{\nu }^{\prime }},$$

$$[Q_{\nu },Q_{{\nu }^{\prime }}]=-3\sqrt{10}{C}_{2\nu 2{\nu }^{\prime }}^{1\nu +{\nu }^{\prime }}{L}_{\nu +{\nu }^{\prime }},$$

$$C_2(SU(3))=Q\cdot Q+3L\cdot L=4({\lambda }^2+{\mu }^2+\lambda \mu +3\lambda +3\mu )$$

$$\biggl{\langle}u_{\alpha}\biggl{|}u_{\alpha^{\prime}}\biggr{\rangle}=\left\langle\begin{array}[]{l}(\lambda,\mu)_{B}\\ \alpha,L,M\end{array}\right.\left|\begin{array}[]{l}(\lambda,\mu)_{B}\\ \alpha^{\prime},L,M\end{array}\right\rangle$$

$$D_{\lambda \mu }=\frac{1}{2}(\lambda +1)(\mu +1)(\lambda +\mu +2).$$

$$K=\mu ,\mu -2,\mu -4,...,1\text{mbox}or0,\text{mbox}for\lambda >\mu .$$

$$\alpha =\frac{1}{2}(\mu -K).$$

$$L=\lambda ,\lambda -2,\lambda -4,...,1\text{mbox}or0.$$

$$\beta=\left\{\begin{array}[]{l}0,\quad\lambda+\mu-L\mbox{ even},\\ 1,\quad\lambda+\mu-L\mbox{ odd}.\end{array}\right.$$

$$\displaystyle\left|\begin{array}[]{c}(\lambda,\mu)_{B}\\ \alpha~{}L~{}L\end{array}\right\rangle\equiv|\beta n_{0}n_{1}n_{2}\alpha\rangle=w^{\beta}(\xi_{1})^{n_{0}}(x_{10})^{n_{1}}(\mbox{\boldmath$\xi$}^{2})^{n_{2}}A^{\alpha}|0\rangle,$$

$$n_0=L-\mu +2\alpha ,n_1=\mu -2\alpha -\beta ,n_2=(\lambda +\mu -L-2\alpha -\beta )/2.$$

$$x_{10}={\xi }_1{\eta }_0-{\xi }_0{\eta }_1,x_{1-1}={\xi }_1{\eta }_{-1}-{\xi }_{-1}{\eta }_1,x_{0-1}={\xi }_0{\eta }_{-1}-{\xi }_{-1}{\eta }_0,$$

$$\displaystyle w=\xi_{1}x_{1-1}-\xi_{0}x_{10}=\xi_{1}^{2}\eta_{-1}-\xi_{-1}\xi_{1}\eta_{1}-\xi_{0}\xi_{1}\eta_{0}+\xi_{0}^{2}\eta_{1},\quad\mbox{\boldmath$\xi$}^{2}=\xi_{0}^{2}-2\xi_{-1}\xi_{1},$$

$$A=(2x_{10}{x}_{0-1}-x_{1-1}^2)$$

$$=2{\xi }_0{\xi }_1{\eta }_0{\eta }_{-1}-2{\xi }_{-1}{\xi }_1{\eta }_0^2-2{\xi }_0^2{\eta }_{-1}{\eta }_1+2{\xi }_0{\xi }_{-1}{\eta }_0{\eta }_1-{\xi }_1^2{\eta }_{-1}^2+2{\xi }_{-1}{\xi }_1{\eta }_{-1}{\eta }_1-{\xi }_{-1}^2{\eta }_1^2,$$

$${\xi }_{\pm }=\mp \frac{1}{\sqrt{2}}({\xi }_x\pm ı{\xi }_y),{\xi }_0={\xi }_z,$$

$${\xi }_m^{+}|0⟩={\eta }_m^{+}|0⟩=0,[{\xi }_m^{+},{\xi }_n]=[{\eta }_m^{+},{\eta }_n]=(-1{)}^m{\delta }_{-m,n}.$$

$$F_M^L=\sum _{\mu \nu }{C}_{1\mu 1\nu }^{LM}({\xi }_{\mu }{\xi }_{\nu }^{+}+{\eta }_{\mu }{\eta }_{\nu }^{+}),$$

$${\xi }_{\nu }^{+}=(-1{)}^{\nu }\partial /\partial {\xi }_{-\nu },{\eta }_{\nu }^{+}=(-1{)}^{\nu }\partial /\partial {\eta }_{-\nu },$$

$$[F_{M_{1}}^{L_{1}},F_{M_{2}}^{L_{2}}]=\sqrt{(2L_{1}+1)(2L_{2}+1)}\sum_{L}((-1)^{L_{1}+L_{2}}-(-1)^{L})C_{L_{1}M_{1}L_{2}M_{2}}^{LM_{1}+M_{2}}\left\{\begin{array}[]{ccc}L_{1}&L_{2}&1\\ 1&1&1\end{array}\right\}F_{M_{1}+M_{2}}^{L},$$

$$|k_1,k_2,k_3,k_4,k_5,k_6⟩=(k_1!k_2!k_3!k_4!k_5!k_6!{)}^{-1/2}({\xi }_{-1}{)}^{k_1}({\xi }_0{)}^{k_2}({\xi }_1{)}^{k_3}({\eta }_{-1}{)}^{k_4}({\eta }_0{)}^{k_5}({\eta }_1{)}^{k_6}|0⟩,$$

$$⟨k_1^{\prime },k_2^{\prime },k_3^{\prime },k_4^{\prime },k_5^{\prime },k_6^{\prime }|k_1,k_2,k_3,k_4,k_5,k_6⟩={\delta }_{k_1^{\prime }{k}_1}{\delta }_{k_2^{\prime }{k}_2}{\delta }_{k_3^{\prime }{k}_3}{\delta }_{k_4^{\prime }{k}_4}{\delta }_{k_5^{\prime }{k}_5}{\delta }_{k_6^{\prime }{k}_6}.$$

$$w^{\beta }=\sum _{\nu }{b}_{\nu }^{\beta }({\xi }_{-1}{)}^{{\nu }_1}({\xi }_0{)}^{{\nu }_2}({\xi }_1{)}^{{\nu }_3}({\eta }_{-1}{)}^{{\nu }_4}({\eta }_0{)}^{{\nu }_5}({\eta }_1{)}^{{\nu }_6},\nu \equiv ({\nu }_1,{\nu }_2,{\nu }_3,{\nu }_4,{\nu }_5,{\nu }_6).$$

$$b_{(0,0,0,0,0,0)}^0=1,b_{(0,0,2,1,0,0)}^1=1,b_{(1,0,1,0,0,1)}^1=-1,b_{(0,1,1,0,1,0)}^1=-1,b_{(0,2,0,0,0,1)}^1=1.$$

$$\displaystyle(x_{10})^{n_{1}}=\sum_{k_{1}=0}^{n_{1}}\left(\begin{array}[]{c}n_{1}\\ k_{1}\end{array}\right)(-1)^{k_{1}}\xi_{0}^{k_{1}}\xi_{1}^{n_{1}-k_{1}}\eta_{0}^{n_{1}-k_{1}}\eta_{1}^{k_{1}},$$

$$\displaystyle(\mbox{\boldmath$\xi$}^{2})^{n_{2}}=\sum_{k_{2}=0}^{n_{2}}\left(\begin{array}[]{c}n_{2}\\ k_{2}\end{array}\right)(-1)^{k_{2}}2^{k_{2}}\xi_{-1}^{k_{2}}\xi_{0}^{2(n_{2}-k_{2})}\xi_{1}^{k_{2}}\eta_{0}^{n_{1}-k_{1}},$$

$$\displaystyle A^{\alpha}=\sum_{s\in\Omega_{s,\alpha}}\left(\begin{array}[]{c}\alpha\\ s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}\end{array}\right)(-1)^{s_{1}+s_{3}+s_{5}+s_{7}}\xi_{-1}^{s_{2}+s_{4}+s_{6}+2s_{7}}\xi_{0}^{s_{1}+2s_{3}+s_{4}}$$

$$\times {\xi }_1^{s_1+s_2+2s_5+s_6}{\eta }_{-1}^{s_1+s_3+2s_5+s_6}{\eta }_0^{s_1+2s_2+s_4}{\eta }_1^{s_3+s_4+s_6+2s_7},$$

$$|\beta n_0{n}_1{n}_2\alpha ⟩=\sum _{\nu }\sum _{k_1=0}^{n_1}\sum _{k_2=0}^{n_2}\sum _{s\in {\Omega }_{s,\alpha }}{b}_{\nu }^{\beta }(-1{)}^{k_1+k_2+s_1+s_3+s_5+s_7}{2}^{k_2+s_1+s_2+s_3+s_4+s_6}$$

$$\displaystyle\times\left(\begin{array}[]{c}n_{1}\\ k_{1}\end{array}\right)\left(\begin{array}[]{c}n_{2}\\ k_{2}\end{array}\right)\left(\begin{array}[]{c}\alpha\\ s_{1}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}\end{array}\right)(\xi_{-1})^{\gamma_{1}}(\xi_{0})^{\gamma_{2}}(\xi_{1})^{\gamma_{3}}(\eta_{-1})^{\gamma_{4}}(\eta_{0})^{\gamma_{5}}(\eta_{1})^{\gamma_{6}}|0\rangle,$$

Full text

1 Introduction

The formalism of $\mathrm{SU(3)}$ group provides a comprehensive theoretical foundation for understanding this symmetry in nuclear structure [1, 2, 3, 4, 5]. However, the construction of the $\mathrm{SU(3)}$ bases usually can be performed analytically only for some special cases. In this respect, because of mathematical simplicity of its definition, the Bargmann–Moshinsky (BM) basis [6, 7] is especially convenient for calculation. However, the necessity to introduce the physically relevant angular momentum observable gives rise to the non-canonical group reduction $\mathrm{SU(3)}\supset\mathrm{SO(3)}\supset\mathrm{SO(2)}$. The BM vectors may be calculated from the simplest vectors which correspond to the highest angular momentum projection $M=L$, i.e. the highest weight basis vectors with respect to the SO(3) group that was proved in [7]. It should be stressed, that the analytical and what is very important an effective algorithm for construction of this basis is required for analysis of some quantum systems.

As an example one can consider the quadrupole vibrational and rotational motions which are the most important low energy nuclear motions. The simplest $\mathrm{SU}(3)$ model Hamiltonian consists of the quadrupole-quadrupole interaction, the rotational term and potentially the other terms constructed from generators of the partner groups $G=\mathrm{SU(3)}\times\overline{\mathrm{SU(3)}}$, see [4] and references therein. A possible Hamiltonian $H$ used in this schematic nuclear model can be written as:

[TABLE]

where the second order Casimir operator $C_{2}(\mathrm{SU(3)})=Q\cdot Q+3L\cdot L$, $Q$ and $L$ are generators of $\mathrm{SU(3)}$, i.e. quadrupole and angular momentum, respectively. $\bar{Q}$ and $\bar{L}$ are generators of the intrinsic group $\overline{\mathrm{SU(3)}}$. Some examples of physically interesting forms of the interaction $H^{\prime\prime}$ can be written as

[TABLE]

where $(T_{\lambda^{\prime}}\otimes T_{\lambda})^{L}_{M}$ denotes the tensor product of two spherical tensors [8]. For, example, these interaction terms can simulate either the tetrahedral or octahedral nuclear symmetry now widely considered in nuclear physics [9]. To find the corresponding energies and quantum nuclear states one needs to solve the eigenvalue problem of the Hamiltonian (1).

To solve the eigenvalue problem for $H$ the appropriate basis constructed according to the group chain $\textrm{SU(3)}\supset\textrm{SO(3)}\supset\textrm{SO(2)}$ is required. There were several attempts to construct such bases. They were based on different group theoretical technics, for a short review see the introduction in the paper [10, 11]. In all those cases one obtains the non-orthogonal basis. This increases a complexity of calculations of the reduced matrix elements of different operators, Clebsch-Gordan coefficients, etc. It requires an adaptation of the Gram-Schmidt like orthogonalization procedure to be more effective in symbolic calculations.

We start from the BM states which are linearly independent but as in other approaches not orthonormal. We developed an effective symbolic algorithm suitable for implementation in computer algebra systems [12]. It is based on the adapted Gram-Schmidt orthonormalization procedure but using the overlap integrals calculated in an analytical form [11]. It provides the analytic construction of the desirable orthonormalized basis. Our adaptation of Gram-Schmidt orthonormalization procedure consists in construction of recursive calculation of the required quantities and the normalization integrals. These calculations do not involve any square root operation. This distinct features of the proposed orthonormalization algorithm allows for a large scale symbolic calculations [12].

Then one can calculate in this orthonormalized basis the zero component of the quadrupole operator $Q_{0}$ in the analytical form using its simpler form given in the non-canonical BM basis [13, 14]. The other components of the quadrupole operator $Q_{k}$ written in the analytical form can be obtained by making use of the Wigner-Eckart theorem with conventional $\mathrm{SO(3)}$ Clebsch-Gordan coefficients [8]. This is required theorem for a final construction of the above Hamiltonian (1) also in an analytical form.

Meanwhile, to organize a real large scale calculations of the required matrix elements of the quadrupole operator with $\lambda,\mu>5$, where $(\lambda,\mu)$ are Elliot labels denoting the irreducible representations (irrep.) of $\mathrm{SU(3)}$, one needs to have a quick algorithm implemented in Fortran that reduces computer resources. The construction and testing of such algorithm is given in the present paper.

The paper is organized as follows. In Section 2, the procedure 1 for calculation of overlap integrals of BM vectors is shown. In Section 3, the procedure 2 for orthonormalization of BM basis is given. In Section 4, the procedure 3 of an action of the quadrupole operator $Q_{0}$ onto the constructed basis is presented. In Conclusions further applications of the elaborated * procedures* are outlined.

2 Calculations of the overlap integrals of the BM basis

The effective method for constructing a non-canonical BM basis with the highest weight vectors of $\mathrm{SO(3)}$ irreducible representations corresponding to the group chain $SU(3)\supset O(3)\supset O(2)$ with commutation relations of the spherical tensors $L_{\nu}(\nu=\pm 1,0)$, $Q_{\nu}(\nu=\pm 2,\pm 1,0)$:

[TABLE]

and the Casimir operator

[TABLE]

was described in [11] and implemented as a symbolic algorithm in [12]. Let us introduce the notation for overlaps of the vectors of this basis:

[TABLE]

Here the quantum numbers $\lambda,\,\mu$ which label the irreducible representations (irreps), $\lambda,\mu=0,1,2,\dots$ and $\lambda>\mu$; $L,M$ are the quantum numbers of angular momentum ${L}\cdot{L}$ and its projection $L_{0}$ (in our case, $M=L$); $\alpha$ is the additional index that is used for unambiguously distinguishing the equivalent $\mathrm{SO(3)}$ irreps $(L)$ in a given $\mathrm{SU(3)}$ irrep $(\lambda,\,\mu)$. The dimension of an irrep of SU(3) for a given $\lambda,\mu$ can be calculated by using the following formula:

[TABLE]

In order to perform classification of the BM states (6) one should determine a set of allowed values of $\alpha$ and $L$. It is well known that the ranges of quantum numbers $\alpha$ and $L$ are determined by the values of quantum numbers $\lambda$ and $\mu$. However, the determination of the former quantities is rather cumbersome.

The easiest way to get the allowed values of $\alpha$ and $L$ is by using the following procedures:

Step 1. Firstly we start with choosing some particular value of the quantum number $\mu$. For the following consideration, it is convenient to introduce auxiliary label $K$ [3] which varies within the range

[TABLE]

The label $K$ is related to $\alpha$ by

[TABLE]

So, for every fixed $\mu$, the set of possible values of $K$ can be obtained directly from Ref. (8). Now, the set of allowed values of $\alpha$ may be determined from these $K$ values using relation (9).

Step 2. In the case $K=0$, that may occur only for even values of $\mu$, the allowed values of $L$ are determined by the label $\lambda$:

[TABLE]

Step 3. In the case $K\neq 0$ the $L_{\mathrm{min}}=K$. Since for every particular $\mu$, there is a number of possible $K$ numbers, according to (8) there exists a number of the corresponding numbers $\alpha$. It means that for every particular $\mu$, there will be a number of pairs $(\alpha,L_{\mathrm{min}})$. The maximal value of $L$ is defined by the expression $L_{\mathrm{max}}=\mu-2\alpha+\lambda-\beta$, where

[TABLE]

To determine $L_{\mathrm{max}}$ it is convenient to consider two alternatives: $\lambda-L\mbox{ is even}$ and $\lambda-L\mbox{ is odd}$. In both cases, the label $\beta$ is defined by the given value $\mu$ value. The number $L_{\mathrm{max}}$ is determined in a similar manner. An illustrative example for calculation of allowed values of $\alpha$ and $L$ is presented in Table 1 of ref [12]. It should be noted that the set of allowed values of $L$ for overlap integrals is given by intersection of these sets for the corresponding $<bra|$ and $|ket>$ vectors.

The highest-weight vector of the BM basis of the SO(3) multiplets for all kinds of irreducible representation of SU(3) can be written in the form [11]

[TABLE]

that differ from the states Eq. (3.8) given in [7] in the definition of the number $\alpha$ and coincide up to a phase factor $(-1)^{\alpha}$. Here the set of numbers $n_{0}$, $n_{1}$, $n_{2}$ is given in terms of the above defined $\lambda$, $\mu$, $\alpha$, $L$ and $\beta$

[TABLE]

The corresponding operators are determined by the following relations:

[TABLE]

via two SO(3) spherical vectors belong to two independent SU(3) representations [8]

[TABLE]

which we consider as the vector-boson creation operators $\xi_{m}$ and $\eta_{m}$ with $(L,M)=(1,M=1,0,-1)$. Then the states (14) are polynomials constructed from these operators which act on the vacuum state denoted by $|0\rangle$. The pairs of creation $\xi_{m}$ and $\eta_{m}$, and annihilation $\xi_{m}^{+}$ and $\eta_{m}^{+}$ vector-boson operators are defined by relations

[TABLE]

With the help of $\xi$ and $\eta$ one can construct the irreducible tensor operators

[TABLE]

where $C_{1\mu 1\nu}^{LM}$ are Clebsch–Gordan coefficients [8]. The vectors \mbox{\boldmath\xi}^{+} and \mbox{\boldmath\eta}^{+} can be chosen in the form

[TABLE]

i.e. the vectors $\xi$, $\eta$ and \mbox{\boldmath\xi}^{+}, \mbox{\boldmath\eta}^{+} can be considered as creation and annihilation operators of two distinct kinds of vector-boson in Fock representation.

The tensor operators satisfy the following commutation relations

[TABLE]

where $\{...\}$ is the 6j–Wigner symbol [8]. If we introduce $L_{m}=-\sqrt{2}F_{m}^{1}$ and $Q_{k}=-\sqrt{6}F_{k}^{2}$, we can see that operators $L_{m}$ $(m=0,\pm 1)$ and $Q_{k}$ $(k=0,\pm 1,\pm 2)$ satisfy the standard commutation relations of SU(3)group (5). It is evident that the operators $L_{m}$ $(m=0,\pm 1)$ define the algebra of angular momentum SO(3) and the operators $Q_{k}$ $(k=0,\pm 1,\pm 2)$ extend this algebra to SU(3) algebra.

Using the above definitions (17) and (18), we determine the standard boson basis

[TABLE]

which is orthonormal

[TABLE]

Then we expand the vectors (14) in the terms of basis (19). As the first step we apply the multipliers of (14) in the variables $\xi_{-1}$, $\xi_{0}$, $\xi_{1}$, $\eta_{-1}$, $\eta_{0}$, $\eta_{1}$. In fact we use only operator expansion. Because $\beta=0$ or $\beta=1$, we write

[TABLE]

By comparing (16) and (21), we obtain that the sum in (21) contains one term for $\beta=0$ and four terms for $\beta=1$:

[TABLE]

From (16) using the multinomial theorem, we also calculate the needed powers of the operators

[TABLE]

where a set of indices runs within the range $\Omega_{s,\alpha}=\{s_{1},...,s_{7}|s_{1}\geq 0,...,s_{7}\geq 0,s_{1}+...+s_{7}=\alpha\}$.

So, we have the highest-weight vector of the BM basis

[TABLE]

where the set multi-indices $\gamma_{1}$,…,$\gamma_{6}$ is determined by the relations

[TABLE]

From (31) in the boson representation (19) we obtain the required BM states

[TABLE]

where the coefficients $B(\beta,n_{0},n_{1},n_{2},\alpha,\nu,k_{1},k_{2},s)$ have the following form

[TABLE]

Step 4. The overlap integrals are determined by the relation

[TABLE]

where domains of summation are determined by the definitions (21) and (24).

Output of Step 4. The overlap integral (48) has been calculated with algorithm implemented in Fortran. The obtained results for $\lambda=0,...,10$ and $\mu=1,...,10$ for $\lambda\geq\mu$ and corresponding sets of values $\alpha$ and $L$ (for example, see Table 1 of ref. [12]) coincide up to 10 digits with results of calculations obtained with help of symbolic algorithm [12] implemented in Wolfram Mathematica.

3 Orthogonalisation of the BM basis

Let us construct the orthonormal basis in the space spanned by the non-canonical BM vectors (6), $(M=L)$. For this purpose, we propose a bit more efficient form of the Gram–Schmidt orthonormalisation procedure

[TABLE]

Here multiplicity index $i$ is introduced to distinguish the orthonormalized states. The symbols $A_{i,\alpha}^{(\lambda,\mu)}(L)$ denotes the matrix elements of the upper triangular matrix of the BM basis orthonormalization coefficients. These coefficients fulfill the following condition

[TABLE]

Because the BM vectors (6) are linearly independent, one can require the orthonormalization properties for the vectors (49)

[TABLE]

Step 5. Gramian of a set of BM eigenvectors $u_{\alpha_{\max}},...,u_{0}$ from r.h.s. of (49) in notations (6)

[TABLE]

Set of orthogonal (not orthonormal) vectors are calculated in the following way [15]:

[TABLE]

where $t=0,1,...,\alpha_{\max}$. In result we have a set of the orthonormal vectors and coefficients:

[TABLE]

Output of Step 5. The required set of the coefficients $A_{i,\alpha}^{(\lambda,\mu)}(L)$ are the components of the orthonormal vector $|z_{i}\rangle$, at $i=\alpha_{\max},\alpha_{\max}-1,...,0$ has been calculated with the above algorithm implemented in Fortran. Note, the orthonormalization procedure is performed in the reverse order with respect to the one adopted in [15], that allows us to obtain the same orthonormal basis as in the papers [11, 12]. The obtained results for $\lambda,\mu=0,...,10$, where $\lambda\geq\mu$ coincide up to 10 digits with results of calculations obtained by the symbolic algorithm [12] implemented in Wolfram Mathematica.

4 Action of the the quadrupole operator onto the orthonormal basis

Step 6. Following the paper [14], we determine the action of the zero component $Q_{0}$ of the second order generator of $\mathrm{SU(3)}$ group onto the BM basis vectors

[TABLE]

The coefficients $a_{s}^{(k)}$ can be calculated as in [13] and in [14]. To calculate action of $Q_{0}$ onto the orthogonal BM basis vectors (49), we determine the inverse transformation $\tilde{A}_{i,\alpha}^{(\lambda,\mu)}(L)$ taken from the formula (49)

[TABLE]

where the following relations take place

[TABLE]

Using (56), (57), and (58), we obtain the action of the zero component $Q_{0}$ of the quadrupole operator onto the orthogonal BM basis vectors as

[TABLE]

where the coefficients $q_{i,j,k}^{(\lambda,\mu)}(L)$ are calculated by the formula

[TABLE]

and ${\tilde{A}}_{i,\alpha}^{(\lambda,\mu)}(L)$ are elements of the inverse and the transpose of the matrix $A^{(\lambda,\mu)}$

[TABLE]

4.1 Calculations of the action of the quadrupole operator zero

component onto the BM basis

Let us calculate action of the operator $Q_{0}$ according Eq. (56)

[TABLE]

where the coefficients $\tilde{a}_{s}^{(k)}\equiv\tilde{a}_{s}^{(k)}(L)$ are related to $a_{s}^{(k)}\equiv a_{s}^{(k)}(L)$ by the factors:

[TABLE]

We rewrite the BM basis vectors (14) in the form

[TABLE]

where $c_{1}=w$, $c_{2}=\xi_{1}$, $c_{3}=x_{10}$, c_{4}=\mbox{\boldmath\xi}^{2}, $c_{5}=A$, and the powers $n_{0}=L-\mu+2\alpha$, $n_{1}=\mu-2\alpha-\beta$, $n_{2}=(\lambda+\mu-L-2\alpha-\beta)/2$, and $\beta$ are defined by relations (16) and (11). The expression $\xi_{-1},\xi_{1},\eta_{-1},\eta_{0}$ in terms of $c_{1},...,c_{5}$ give us the formal expressions: $\xi_{1}=c_{2}$, $\eta_{0}=c_{3}/c_{2}+\xi_{0}\eta_{1}/c_{2}$, $\xi_{-1}=-c_{4}/2/c_{2}+\xi_{0}^{2}/2/c_{2}$, $\eta_{-1}=(c_{1}-(-\xi_{0}c_{3}-(1/2)\xi_{0}^{2}\eta_{1}+(1/2)\eta_{1}c_{4}))/c_{2}^{2}$, and the following relation among the components $c_{1},...,c_{5}$:

[TABLE]

Firstly, we calculate action of operators $Q_{0}$, $L_{-1}$ and $L_{-1}^{2}$ over eigenvectors (81) and express them in terms of components $c_{1},...,c_{5}$. Using the above relations, implemented in REDUCE we obtain action of operators $Q_{0}$, $L_{-1}$ and $L_{-1}^{2}$ onto eigenvectors (81). For even $\lambda{+}\mu{-}L$ we get:

[TABLE]

and for odd $\lambda{+}\mu{-}L$ we get:

[TABLE]

Using the above actions of the operators $Q_{0}$, $L_{-1}$ and $L_{-1}^{2}$ on eigenvectors (81) and extracting the coefficients at $c_{1},...,c_{5}$, we arrive to a set of equations with respect to unknown coefficients: $\tilde{a}_{1}^{(0)}$, $\tilde{a}_{0}^{(0)}$, $\tilde{a}_{{-}1}^{(0)}$, $\tilde{a}_{0}^{(1)}$, $\tilde{a}_{1}^{(1)}$, $\tilde{a}_{{-}1}^{(1)}$, $\tilde{a}_{0}^{(2)}$, $\tilde{a}_{1}^{(2)}$, $\tilde{a}_{1}^{(2)}$  . From formula (70) and from action of the operators $Q_{0}$, $L_{-1}$ and $L_{-1}^{2}$ on functions (81) we obtain:

[TABLE]

The solution of the set of equations obtained by extraction of coefficients at the same powers of the operators $c_{1}$,…,$c_{5}$ gives values of unknown coefficients $\tilde{a}_{1}^{(0)}$, $\tilde{a}_{0}^{(0)}$, $\tilde{a}_{{-}1}^{(0)}$, $\tilde{a}_{0}^{(1)}$, $\tilde{a}_{1}^{(1)}$, $\tilde{a}_{{-}1}^{(1)}$, $\tilde{a}_{0}^{(2)}$, $\tilde{a}_{1}^{(2)}$, $\tilde{a}_{1}^{(2)}$. Finally, from (78) we arrive to needed values of $a_{1}^{(0)}$, $a_{0}^{(0)}$, $a_{{-}1}^{(0)}$, $a_{0}^{(1)}$, $a_{1}^{(1)}$, $a_{{-}1}^{(1)}$, $a_{0}^{(2)}$, $a_{1}^{(2)}$, $a_{1}^{(2)}$. In result, the action of zero component $Q_{0}$ of the quadrupole operator onto the nonorthogonal BM basis (14), i.e. the coefficients $a_{s}^{(k)}$ of expansion (56), reads as:

[TABLE]

Note, there are five misprints in formulas (2.4) of Ref. [14], that are corrected in the above expressions (98). They are marked by the square brackets.

Output of Step 6. The set of matrix elements $q_{i,j,k}^{(\lambda,\mu)}(L)$ has been calculated with algorithm implemented in Fortran. The obtained results for $\lambda,\mu=0,...,10$, where $\lambda\geq\mu$ (the corresponding sets of values $\alpha$ and $L$ (for example see tables 1 of ref. [12])) coincide up to 10 digits with results of calculations obtained with help of symbolic algorithm [12] implemented in Wolfram Mathematica. Note that the coefficients $q_{i,j,k}^{(\lambda,\mu)}(L)$ for up to $\mu=3$ were calculated as well and their values are equal to those presented in Table 1 of Ref. [14] except values for the following sets of indices: $k=0$,$\mu=3$: $q_{1,1,k}^{(\lambda,\mu)}(L)$ for $\lambda-L$ even, $q_{1,0,k}^{(\lambda,\mu)}(L)$ and $q_{0,1,k}^{(\lambda,\mu)}(L)$ for $\lambda-L$ odd; $k=1$, $\mu=3$: $q_{1,1,k}^{(\lambda,\mu)}(L)$ for $\lambda-L$ odd, $q_{0,0,k}^{(\lambda,\mu)}(L)$ and $q_{1,0,k}^{(\lambda,\mu)}(L)$ for $L=\lambda+1$, $\lambda-L$ odd, $q_{0,0,k}^{(\lambda,\mu)}(L)$ for $L=\lambda+2$, $\lambda-L$ even. The corrected expressions are shown in Appendix.

4.2 Calculations of action of the quadrupole operator onto the

orthonormalized BM basis

The matrix elements of the quadrupole operators, generators of the group SU(3), can be expressed as the reduced matrix elements by means of the Wigner–Eckart theorem

[TABLE]

The corresponding reduced matrix elements are determined by the formula

[TABLE]

where the coefficients $q^{(\lambda,\mu)}_{i,j,k}(L)$ are defined by (60). In this definition, $k\geq 0$. Dimension of subspace of the ket vectors $|(\lambda,\mu)iLM\rangle$ at fixed $\lambda$ and $\mu$ are defined by formula (7). The dimension of this subspace determines the complexity of the above algorithms, i.e., requirements on the computer memory and execution time.

5 Conclusions

We present the practical algorithm implemented in Fortran for constructing the non-canonical Bargmann–Moshinsky (BM) basis with the highest weight vectors of $\mathrm{SO(3)}$ irreps., which can be used for calculating spectra and electromagnetic transitions in molecular and nuclear physics. The orthonormalisation algorithm [15] applied in Section 3, as well as recursion algorithm [12], allows one to calculate the orthonormalized BM basis in Fortran and Wolfram Mathematica, respectively. The distinct advantage of such orthonormalisation is that it does not involve any square root operation on the expressions coming from the previous recursion steps of the conventional Gram–Schmidt algorithm. This makes the proposed method very suitable for large-scale calculations of spectral characteristics (especially close to resonances) of quantum systems under consideration and to study their analytical properties for understanding the dominant symmetries. The formalism of partner groups $G=\mathrm{SU(3)}\times\overline{\mathrm{SU(3)}}$ allows for simulation of the intrinsic properties of quantum systems (also nuclei), including their intrinsic symmetries. The presented nuclear SU(3) model is extended and allows for additional intrinsic structure, especially it allows to construct terms having required point symmetries. Calculations of spectral characteristics of the above nuclei models and study of their dominant symmetries will be done in our next publications.

The work was partially supported by the Bogoliubov-Infeld program, Votruba-Blokhintsev program, the RUDN University Program 5-100 and grant of Plenipotentiary of the Republic of Kazakhstan in JINR. AD&PMK are grateful to Prof. A. Góźdź for hospitality during visits in Institute of Physics, Maria Curie-Skłodowska University.

Appendix

There are misprints in expressions of the coefficients $q_{i,j,k}^{(\lambda,3)}$ in Table 1 of Ref. [14], that are corrected in the below expressions. For $\lambda-L$ even:

[TABLE]

For $\lambda{-}L$ odd:

[TABLE]

[TABLE]

For $L=\lambda{+}1$ and $\lambda{-}L$ odd:

[TABLE]

[TABLE]

For $L=\lambda{+}2$ and $\lambda{-}L$ even:

[TABLE]

where

[TABLE]

[TABLE]

Thus, table 1 of Ref. [14] together with the above expressions have been used to test the Step 6 of the proposed procedure implemented in Fortran.