Molien generating functions and integrity bases for the action of the ${{\mathrm {SO(3)}}}$ and ${{\mathrm {O(3)}}}$ groups on a set of vectors

Abstract

The construction of integrity bases for invariant and covariant polynomials built from a set of three dimensional vectors under the ${{\mathrm {SO(3)}}}$ and ${{\mathrm {O(3)}}}$ symmetries is presented. This paper is a follow-up to our previous work that dealt with a set of two dimensional vectors under the action of the ${{\mathrm {SO(2)}}}$ and ${{\mathrm {O(2)}}}$ groups (Dhont and Zhilinskií in J Phys A Math Theor 46:455202, 2013). The expressions of the Molien generating functions as one rational function are a useful guide to build integrity bases for the rings of invariants and the free modules of covariants. The structure of the non-free modules of covariants is more complex. In this case, we write the Molien generating function as a sum of rational functions and show that its symbolic interpretation leads to the concept of generalized integrity basis. The integrity bases and generalized integrity bases for ${\mathrm {O(3)}}$ are deduced from the ${\mathrm {SO(3)}}$ ones. The results are useful in quantum chemistry to describe the potential energy or multipole moment hypersurfaces of molecules. In particular, the generalized integrity bases that are required for the description of the electric and magnetic quadrupole moment hypersurfaces of tetratomic molecules are given for the first time.

Appendices

Appendix 1: Determination of the Molien generating function

1.1 Expression via an integral

In an active transformation, a rotation of angle $\omega$ around a rotation axis whose direction is given by the unit vector ${\hat{n}}=n_x \, {\hat{e}}_x + n_y \, {\hat{e}}_y + n_z \, {\hat{e}}_z$ transforms the vector ${\mathbf {r}} = x \, {\hat{e}}_x + y \, {\hat{e}}_y + z \, {\hat{e}}_z$ into the vector ${\mathbf {r}}^\prime = x^\prime \, {\hat{e}}_x + y^\prime \, {\hat{e}}_y + z^\prime \, {\hat{e}}_z$. The initial and final coordinates are related by [48]:

$$\begin{aligned} \left( \begin{array}{c} x^\prime \\ y^\prime \\ z^\prime \end{array} \right) = R\left( \omega ,{\hat{n}}\right) \left( \begin{array}{c} x \\ y \\ z \end{array} \right) , \quad R\left( \omega ,{\hat{n}}\right) = 1_3 + N \sin \omega + N^2 \left( 1-\cos \omega \right) , \end{aligned}$$

where $1_3$ is the $3\times 3$ identity matrix and N is the $3\times 3$ matrix defined as:

$$\begin{aligned} N = \left( \begin{array}{ccc} 0 &{} -n_z &{} n_y \\ n_z &{} 0 &{} -n_x \\ -n_y &{} n_x &{} 0 \end{array} \right) . \end{aligned}$$

Using spherical coordinates to give the orientation of the rotation axis ${\hat{n}}$ gives $n_x = \sin \theta \cos \varphi$, $n_y = \sin \theta \sin \varphi$ and $n_z = \cos \theta$. The $3N\times 3N$ block matrix representation $D\left( \omega ,{\hat{n}}\right)$ of the rotation operation is:

$$\begin{aligned} D\left( \omega ,{\hat{n}}\right) =\left( \begin{array}{cccc} R\left( \omega ,{\hat{n}}\right) &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad R\left( \omega ,{\hat{n}}\right) &{}\quad \cdots &{}\quad \quad 0 \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad R\left( \omega ,{\hat{n}}\right) \end{array} \right) , \end{aligned}$$

and one easily finds that

$$\begin{aligned} \det \left( 1_{3N}-\lambda D\left( \omega ,{\hat{n}}\right) \right) = \left[ (1-\lambda )(1-2\lambda \cos \omega +\lambda ^2)\right] ^N. \end{aligned}$$

The Molien function (1) then reduces to

$$\begin{aligned} g_{\mathrm {SO(3)}}\left( N,L;\lambda \right) = \frac{2}{\pi } \frac{1}{(1-\lambda )^N} \int _{0}^{\pi } \frac{\sin \left[ \left( 2L+1\right) \frac{\omega }{2}\right] \sin \frac{\omega }{2}}{(1-2\lambda \cos \omega +\lambda ^2)^N} d\omega . \end{aligned}$$

(29)

The integral in (29) can be evaluated by the use of the product-to-sum identity $\sin \left[ \left( 2L+1\right) \omega /2\right] \sin \left( \omega /2\right) = \left\{ \cos \left( L \omega \right) -\cos \left[ \left( L+1\right) \omega \right] \right\} /2$ and the tabulated formula of Ref. [49]:

$$\begin{aligned}&\int _0^\pi \frac{\cos nx \, dx}{\left( 1-2a\cos x+a^2\right) ^m} \nonumber \\&\quad = \frac{a^{2m+n-2}\pi }{\left( 1-a^2\right) ^{2m-1}} \sum _{k=0}^{m-1} \left( \begin{array}{c} m+n-1 \\ k \end{array} \right) \left( \begin{array}{c} 2m-k-2 \\ m-1 \end{array} \right) \left( \frac{1-a^2}{a^2}\right) ^k, \quad a^2<1. \end{aligned}$$

(30)

1.2 Recursive formula for two or more vectors

The Molien generating function for $N \ge 2$ can be determined from formula (29). There is however an other approach, which gives more insight in the forthcoming construction of the integrity bases. According to the triangular conditions of the theory of angular momentum, the coupling of a set of $\left( L_1\right)$-covariants with a set of $\left( L_2\right)$-covariants generates one set of $\left( L\right)$-covariants, with $\left| L_1-L_2\right| \le L \le L_1+L_2$. Correspondingly the Molien generating function for $N_1+N_2$ vectors and final representation (L) can be computed from the Molien generating functions for $N_1$ and $N_2$ vectors, see equation (43) of Ref. [7]:

$$\begin{aligned}&g_{\mathrm {SO(3)}}\left( N_1+N_2,L;\lambda _1,\lambda _2\right) \nonumber \\&\quad = \sum _{L_1=0}^\infty \sum _{L_2=0}^\infty \varDelta \left( L_1,L_2,L\right) g_{\mathrm {SO(3)}}\left( N_1,L_1;\lambda _1\right) g_{\mathrm {SO(3)}}\left( N_2,L_2;\lambda _2\right) , \end{aligned}$$

(31)

where $\varDelta \left( L_1,L_2,L\right) =1$ if $\left| L_1-L_2\right| \le L \le L_1+L_2$ and 0 otherwise.

Appendix 2: Complex and real solid harmonics

The complex solid harmonics are homogeneous polynomials of degree l in x, y, and z. Their expression is given by [50]:

$$\begin{aligned} {{{\mathscr {Y}}}}_{l,m}\left( {\mathbf {x}}\right) = \sqrt{\frac{2l+1}{4\pi } \left( l+m\right) ! \left( l-m\right) !} \sum _k \frac{\left( -x-iy\right) ^{k+m} \left( x-iy\right) ^k z^{l-2k-m}}{2^{2k+m} \left( k+m\right) ! k! \left( l-m-2k\right) !}, \end{aligned}$$

where ${\mathbf {x}}$ is the triplet of coordinates $\left( x,y,z\right)$. The complex solid harmonics are complex-valued functions for $m \ne 0$ and satisfy the property

$$\begin{aligned} {{{\mathscr {Y}}}}_{l,m}\left( {\mathbf {x}}\right) ^* = \left( -1\right) ^m {{{\mathscr {Y}}}}_{l,-m}\left( {\mathbf {x}}\right) . \end{aligned}$$

Biedenharn and Louck define in Ref. [48] the real solid harmonics $\bar{{{\mathscr {Y}}}}_{l,m}$ through the linear combinations of complex solid harmonics [51]:

$$\begin{aligned} \bar{{{\mathscr {Y}}}}_{l,m}\left( {\mathbf {x}}\right)= & {} -\frac{1}{\sqrt{2}} \left[ {{{\mathscr {Y}}}}_{l,m}\left( {\mathbf {x}}\right) + \left( -1\right) ^m {{\mathscr {Y}}}_{l,-m}\left( {\mathbf {x}}\right) \right] , \\ \bar{{{\mathscr {Y}}}}_{l,0}\left( {\mathbf {x}}\right)= & {} {{\mathscr {Y}}}_{l,0}\left( {\mathbf {x}}\right) , \\ \bar{{{\mathscr {Y}}}}_{l,-m}\left( {\mathbf {x}}\right)= & {} \frac{i}{\sqrt{2}} \left[ {{\mathscr {Y}}}_{l,m}\left( {\mathbf {x}}\right) - \left( -1\right) ^m {{\mathscr {Y}}}_{l,-m}\left( {\mathbf {x}}\right) \right] , \end{aligned}$$

with $m \in \left\{ l,l-1,\cdots ,1\right\}$.

Other linear combinations of complex solid harmonics exist in the litterature. Steinborn [52] or Blanco et al. [53] use:

$$\begin{aligned} \tilde{{\mathscr {Y}}}_{l,m}\left( {\mathbf {x}}\right)= & {} \frac{\left( -1\right) ^m}{\sqrt{2}} \left[ {\mathscr {Y}}_{l,m}\left( {\mathbf {x}}\right) + \left( -1\right) ^m {\mathscr {Y}}_{l,-m}\left( {\mathbf {x}}\right) \right] , \\ \tilde{{\mathscr {Y}}}_{l,0}\left( {\mathbf {x}}\right)= & {} {\mathscr {Y}}_{l,0}\left( {\mathbf {x}}\right) , \\ \tilde{{\mathscr {Y}}}_{l,-m}\left( {\mathbf {x}}\right)= & {} \frac{\left( -1\right) ^m}{i\sqrt{2}} \left[ {\mathscr {Y}}_{l,m}\left( {\mathbf {x}}\right) - \left( -1\right) ^m {\mathscr {Y}}_{l,-m}\left( {\mathbf {x}}\right) \right] , \end{aligned}$$

with $m \in \left\{ l,l-1,\ldots ,1\right\}$. The two definitions of real solid harmonics are identical except for m even where the two definitions give expressions with opposite sign. In the main text, we use $\tilde{{\mathscr {Y}}}_{l,m}$ as real solid harmonics. Their expression for up to $L=3$ are given below:

$$\begin{aligned}&\sqrt{4\pi } \tilde{{\mathscr {Y}}}_{0,0}\left( {\mathbf {x}}\right) = 1\\&\sqrt{\frac{4\pi }{3}} \left( \begin{array}{c} \tilde{{\mathscr {Y}}}_{1,1}\left( {\mathbf {x}}\right) \\ \tilde{{\mathscr {Y}}}_{1,0}\left( {\mathbf {x}}\right) \\ \tilde{{\mathscr {Y}}}_{1,-1}\left( {\mathbf {x}}\right) \end{array} \right) = \left( \begin{array}{c} x \\ z \\ y \end{array} \right) \\&\sqrt{\frac{4\pi }{5}} \left( \begin{array}{c} \tilde{{\mathscr {Y}}}_{2,2}\left( {\mathbf {x}}\right) \\ \tilde{{\mathscr {Y}}}_{2,1}\left( {\mathbf {x}}\right) \\ \tilde{{\mathscr {Y}}}_{2,0}\left( {\mathbf {x}}\right) \\ \tilde{{\mathscr {Y}}}_{2,-1}\left( {\mathbf {x}}\right) \\ \tilde{{\mathscr {Y}}}_{2,-2}\left( {\mathbf {x}}\right) \end{array} \right) = \left( \begin{array}{c} \frac{\sqrt{3}}{2}\left( x^2 - y^2\right) \\ \sqrt{3}xz \\ \frac{1}{2}\left( 2z^2-x^2-y^2\right) \\ \sqrt{3}yz \\ \sqrt{3}xy \end{array} \right) \\&\sqrt{\frac{4\pi }{7}} \left( \begin{array}{c} \tilde{{\mathscr {Y}}}_{3,3}\left( {\mathbf {x}}\right) \\ \tilde{{\mathscr {Y}}}_{3,2}\left( {\mathbf {x}}\right) \\ \tilde{{\mathscr {Y}}}_{3,1}\left( {\mathbf {x}}\right) \\ \tilde{{\mathscr {Y}}}_{3,0}\left( {\mathbf {x}}\right) \\ \tilde{{\mathscr {Y}}}_{3,-1}\left( {\mathbf {x}}\right) \\ \tilde{{\mathscr {Y}}}_{3,-2}\left( {\mathbf {x}}\right) \\ \tilde{{\mathscr {Y}}}_{3,-3}\left( {\mathbf {x}}\right) \end{array} \right) = \left( \begin{array}{c} \frac{\sqrt{10}}{4} x\left( x^2-3y^2\right) \\ \frac{\sqrt{15}}{2}z\left( x^2-y^2\right) \\ \frac{\sqrt{6}}{4} \left( 4z^2-x^2-y^2\right) x \\ \frac{1}{2}z\left( 2z^2-3x^2-3y^2\right) \\ \frac{\sqrt{6}}{4}\left( 4z^2-x^2-y^2\right) y \\ \sqrt{15}xyz \\ \frac{\sqrt{10}}{4} y\left( 3x^2-y^2\right) \end{array} \right) \end{aligned}$$

Appendix 3: Molien generating function for four vectors

The Molien generating function for four vectors can be written as a single rational function:

$$\begin{aligned} g_{\mathrm {SO(3)}}^{\left( a\right) }\left( 4,L;\lambda \right) = \frac{{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( a\right) }\left( 4,L;\lambda \right) }{( 1-\lambda ^2 )^9}, \end{aligned}$$

(32)

with its numerator equal to:

$$\begin{aligned} {\mathscr {N}}_{\mathrm {SO(3)}}^{\left( a\right) }\left( 4,L;\lambda \right)= & {} \frac{(L+3)(L+2)(L+1)}{6} \lambda ^L + \frac{(L+3)(L+2)L}{2} \lambda ^{L+1} \nonumber \\&+\,\frac{(L+3)(L+2)(L+1)}{6} \lambda ^{L+2} - \frac{(L+3)(L-2)(5L+4)}{6} \lambda ^{L+3} \nonumber \\&-\, \frac{(L+3)(L-2)(5L+1)}{6} \lambda ^{L+4} + \frac{L(L-1)(L-2)}{6} \lambda ^{L+5} \nonumber \\&+\, \frac{(L+1)(L-1)(L-2)}{2} \lambda ^{L+6} + \frac{L(L-1)(L-2)}{6} \lambda ^{L+7}. \end{aligned}$$

(33)

The Molien generating function (32) has non-negative coefficients in its numerator for $L\in \left\{ 0,1,2\right\}$. Negative coefficients appear for $L\ge 3$. However, the Molien function can be rewritten as Eq. (34), which has only non-negative coefficients in the numerator for $L\in \left\{ 2,3,4\right\}$,

$$\begin{aligned}&g_{\mathrm {SO(3)}}^{\left( b\right) }\left( 4,L;\lambda \right) \nonumber \\&\quad = \frac{\frac{(L+3)(L+2)(L+1)}{6} \lambda ^L + 4(2L+1) \lambda ^{L+1} + (-\frac{1}{6}L^3-L^2+\frac{37}{6}L+3) \lambda ^{L+2}}{(1-\lambda ^2)^9} \nonumber \\&\qquad +\,\frac{2(L-2)(2L+1) \lambda ^{L+1} - \frac{(L-2)(L^2-16L-9)}{6} \lambda ^{L+2}}{(1-\lambda ^2)^8} \nonumber \\&\qquad +\,\frac{\frac{L(L-1)(L-2)}{2} \lambda ^{L+1} + \frac{(L+1)(L-1)(L-2)}{2} \lambda ^{L+2} + \frac{L(L-1)(L-2)}{6} \lambda ^{L+3}}{(1-\lambda ^2)^7}, \end{aligned}$$

(34)

as (35), which has only non-negative coefficients in the numerator for L between 5 and 16,

$$\begin{aligned}&g_{\mathrm {SO(3)}}^{\left( c\right) }\left( 4,L;\lambda \right) \nonumber \\&\quad = \frac{4(2L+1) \lambda ^L + 4(2L+1) \lambda ^{L+1}}{(1-\lambda ^2)^9} \nonumber \\&\qquad +\,\frac{(\frac{1}{6}L^3+L^2-\frac{37}{6}L-3) \lambda ^L + 2(L-2)(2L+1) \lambda ^{L+1} - \frac{(L-2)(L^2-16L-9)}{6} \lambda ^{L+2}}{(1-\lambda ^2)^8} \nonumber \\&\qquad +\,\frac{\frac{L(L-1)(L-2)}{2} \lambda ^{L+1} + \frac{(L+1)(L-1)(L-2)}{2} \lambda ^{L+2} + \frac{L(L-1)(L-2)}{6} \lambda ^{L+3}}{(1-\lambda ^2)^7}, \end{aligned}$$

(35)

and as Eq. (36), which has only non-negative coefficients in the numerator for $L\ge 17$,

$$\begin{aligned}&g_{\mathrm {SO(3)}}^{\left( d\right) }\left( 4,L;\lambda \right) \nonumber \\&\quad = \frac{4(2L+1) \lambda ^L + 4(2L+1) \lambda ^{L+1}}{(1-\lambda ^2)^9} \nonumber \\&\qquad +\,\frac{2(L-3)(2L+1) \lambda ^L + 2(L-2)(2L+1) \lambda ^{L+1}}{(1-\lambda ^2)^8} \nonumber \\&\qquad +\,\frac{\frac{(L-2)(L^2-16L-9)}{6} \lambda ^L + \frac{L(L-1)(L-2)}{2} \lambda ^{L+1} + \frac{(L+1)(L-1)(L-2)}{2} \lambda ^{L+2} + \frac{L(L-1)(L-2)}{6} \lambda ^{L+3}}{(1-\lambda ^2)^7}. \end{aligned}$$

(36)

Appendix 4: Molien generating function for five vectors

The Molien generating function for five vectors can be written as a single rational function:

$$\begin{aligned} g_{\mathrm {SO(3)}}^{\left( a\right) }\left( 5,L;\lambda \right) = \frac{{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( a\right) }\left( 5,L;\lambda \right) }{\left( 1-\lambda ^2\right) ^{12}}, \end{aligned}$$

(37)

with the numerator equal to:

$$\begin{aligned}&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( a\right) }\left( 5, L ; \lambda \right) = \frac{(L+4)(L+3)(L+2)(L+1)}{24} \lambda ^L + \frac{(L+4)(L+3)(L+2)L}{6} \lambda ^{L+1} \nonumber \\&\qquad +\, \frac{(L+4)(L+3)(L+2)(L+1)}{8} \lambda ^{L+2} - \frac{(L+4)(L+3)(L^2-3L-\frac{5}{2})}{3} \lambda ^{L+3} \nonumber \\&\qquad -\, \frac{(L+4)(L+3)(L-3)(7L+2)}{12} \lambda ^{L+4} - \frac{(L+4)(L-3)(2L+1)}{2} \lambda ^{L+5} \nonumber \\&\qquad +\, \frac{(L+4)(L-2)(L-3)(7L+5)}{12} \lambda ^{L+6} + \frac{(L-2)(L-3)(L^2+5L+\frac{3}{2})}{3} \lambda ^{L+7} \nonumber \\&\qquad - \,\frac{L(L-1)(L-2)(L-3)}{8} \lambda ^{L+8} - \frac{(L+1)(L-1)(L-2)(L-3)}{6} \lambda ^{L+9} \nonumber \\&\qquad -\, \frac{L(L-1)(L-2)(L-3)}{24} \lambda ^{L+10}. \end{aligned}$$

(38)

The six next alternative expressions of the Molien generating function are written as a sum over four rational functions:

$$\begin{aligned}&g_{\mathrm {SO(3)}}^{\left( x\right) }\left( 5,L;\lambda \right) \\&\quad = \frac{{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( x\right) ,1}\left( 5,L;\lambda \right) }{\left( 1-\lambda ^2\right) ^{12}} + \frac{{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( x\right) ,2}\left( 5,L;\lambda \right) }{\left( 1-\lambda ^2\right) ^{11}}\\&\qquad +\, \frac{{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( x\right) ,3}\left( 5,L;\lambda \right) }{\left( 1-\lambda ^2\right) ^{10}}\\&\qquad +\, \frac{{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( x\right) ,4}\left( 5,L;\lambda \right) }{\left( 1-\lambda ^2\right) ^{9}}, \, x \in \left\{ b,c,d,e,f,g\right\} , \end{aligned}$$

and provide for any value of L at least one expression with only non-negative coefficients in the numerators. An heuristic to derive these numerators is presented in Sect. 4.2.2.

1.1 Numerators of the Molien generating function (b)

$$\begin{aligned}&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( b\right) ,1}\left( 5,L;\lambda \right) \\&\quad = \frac{(L+4)(L+3)(L+2)(L+1)}{24} \lambda ^L + 20(2L+1) \lambda ^{L+1} \\&\qquad +\, \left( -\frac{1}{24}L^4-\frac{5}{12}L^3-\frac{35}{24}L^2+\frac{455}{12}L+19\right) \lambda ^{L+2} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( b\right) ,2}\left( 5,L;\lambda \right) \\&\quad = \left( \frac{1}{6}L^4+\frac{3}{2}L^3+\frac{13}{3}L^2-36L-20\right) \lambda ^{L+1} \\&\qquad -\, \frac{(L-3)(L^3+13L^2-406L-208)}{24} \lambda ^{L+2} \\&\qquad - \,\frac{(L-3)(L^3+12L^2-58L-30)}{6} \lambda ^{L+3} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( b\right) ,3}\left( 5,L;\lambda \right) \\&\quad =- \frac{(L-2)(L-3)(L^2-81L-40)}{24} \lambda ^{L+2} - \frac{(L-2)(L-3)(L^2-10L-6)}{6} \lambda ^{L+3} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( b\right) ,4}\left( 5,L;\lambda \right) \\&\quad = \frac{L(L-1)(L-2)(L-3)}{4} \lambda ^{L+2} + \frac{(L+1)(L-1)(L-2)(L-3)}{6} \lambda ^{L+3} \\&\qquad +\, \frac{L(L-1)(L-2)(L-3)}{24} \lambda ^{L+4} \end{aligned}$$

1.2 Numerators of the Molien generating function (c)

$$\begin{aligned}&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( c\right) ,1}\left( 5,L;\lambda \right) \\&\quad = \frac{(L+4)(L+3)(L+2)(L+1)}{24} \lambda ^L + 20(2L+1) \lambda ^{L+1} \\&\qquad +\, \left( -\frac{1}{24}L^4-\frac{5}{12}L^3-\frac{35}{24}L^2+\frac{455}{12}L+19\right) \lambda ^{L+2} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( c\right) ,2}\left( 5,L;\lambda \right) \\&\quad = 5(2L+1)(2L-7) \lambda ^{L+1} - \frac{(L-3)(L^3+13L^2-406L-208)}{24} \lambda ^{L+2} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( c\right) ,3}\left( 5,L;\lambda \right) \\&\quad = \frac{(L-3)(L^3+12L^2-58L-30)}{6} \lambda ^{L+1} \\&\qquad - \,\frac{(L-2)(L-3)(L^2-81L-40)}{24} \lambda ^{L+2} \\&\qquad -\, \frac{(L-2)(L-3)(L^2-10L-6)}{6} \lambda ^{L+3} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( c\right) ,4}\left( 5,L;\lambda \right) \\&\quad = \frac{L(L-1)(L-2)(L-3)}{4} \lambda ^{L+2} + \frac{(L+1)(L-1)(L-2)(L-3)}{6} \lambda ^{L+3} \\&\qquad +\, \frac{L(L-1)(L-2)(L-3)}{24} \lambda ^{L+4} \end{aligned}$$

1.3 Numerators of the Molien generating function (d)

$$\begin{aligned}&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( d\right) ,1}\left( 5,L;\lambda \right) \\&\quad = 20(2L+1) \lambda ^L + 20(2L+1) \lambda ^{L+1} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( d\right) ,2}\left( 5,L;\lambda \right) \\&\quad = \left( \frac{1}{24}L^4+\frac{5}{12}L^3+\frac{35}{24}L^2-\frac{455}{12}L-19\right) \lambda ^L + 5(2L+1)(2L-7) \lambda ^{L+1} \\&\qquad - \,\frac{(L-3)(L^3+13L^2-406L-208)}{24} \lambda ^{L+2} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( d\right) ,3}\left( 5,L;\lambda \right) \\&\quad = \frac{(L-3)(L^3+12L^2-58L-30)}{6} \lambda ^{L+1} - \frac{(L-2)(L-3)(L^2-81L-40)}{24} \lambda ^{L+2} \\&\qquad -\, \frac{(L-2)(L-3)(L^2-10L-6)}{6} \lambda ^{L+3} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( d\right) ,4}\left( 5,L;\lambda \right) \\&\quad = \frac{L(L-1)(L-2)(L-3)}{4} \lambda ^{L+2} + \frac{(L+1)(L-1)(L-2)(L-3)}{6} \lambda ^{L+3} \\&\qquad +\, \frac{L(L-1)(L-2)(L-3)}{24} \lambda ^{L+4} \end{aligned}$$

1.4 Numerators of the Molien generating function (e)

$$\begin{aligned}&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( e\right) ,1}\left( 5,L;\lambda \right) \\&\quad = 20(2L+1) \lambda ^L + 20(2L+1) \lambda ^{L+1} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( e\right) ,2}\left( 5,L;\lambda \right) \\&\quad = \left( \frac{1}{24}L^4+\frac{5}{12}L^3+\frac{35}{24}L^2-\frac{455}{12}L-19\right) \lambda ^L + 5(2L+1)(2L-7) \lambda ^{L+1} \\&\qquad -\, \frac{(L-3)(L^3+13L^2-406L-208)}{24} \lambda ^{L+2} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( e\right) ,3}\left( 5,L;\lambda \right) \\&\quad = (L-3)(2L-7)(2L+1) \lambda ^{L+1} - \frac{(L-2)(L-3)(L^2-81L-40)}{24} \lambda ^{L+2} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( e\right) ,4}\left( 5,L;\lambda \right) \\&\quad = \frac{(L-2)(L-3)(L^2-10L-6)}{6} \lambda ^{L+1} + \frac{L(L-1)(L-2)(L-3)}{4} \lambda ^{L+2} \\&\qquad +\, \frac{(L+1)(L-1)(L-2)(L-3)}{6} \lambda ^{L+3} + \frac{L(L-1)(L-2)(L-3)}{24} \lambda ^{L+4} \end{aligned}$$

1.5 Numerators of the Molien generating function (f)

$$\begin{aligned}&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( f\right) ,1}\left( 5,L;\lambda \right) \\&\quad = 20(2L+1) \lambda ^L + 20(2L+1) \lambda ^{L+1} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( f\right) ,2}\left( 5,L;\lambda \right) \\&\quad = 5(2L+1)(2L-9) \lambda ^L + 5(2L+1)(2L-7) \lambda ^{L+1} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( f\right) ,3}\left( 5,L;\lambda \right) \\&\quad = \frac{(L-3)(L^3+13L^2-406L-208)}{24} \lambda ^L + (L-3)(2L+1)(2L-7) \lambda ^{L+1} \\&\qquad -\, \frac{(L-2)(L-3)(L^2-81L-40)}{24} \lambda ^{L+2} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( f\right) ,4}\left( 5,L;\lambda \right) \\&\quad = \frac{(L-2)(L-3)(L^2-10L-6)}{6} \lambda ^{L+1} + \frac{L(L-1)(L-2)(L-3)}{4} \lambda ^{L+2} \\&\qquad +\, \frac{(L+1)(L-1)(L-2)(L-3)}{6} \lambda ^{L+3} + \frac{L(L-1)(L-2)(L-3)}{24} \lambda ^{L+4} \end{aligned}$$

1.6 Numerators of the Molien generating function (g)

$$\begin{aligned}&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( g\right) ,1}\left( 5,L;\lambda \right) \\&\quad = 20(2L+1) \lambda ^L + 20(2L+1) \lambda ^{L+1} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( g\right) ,2}\left( 5,L;\lambda \right) \\&\quad = 5(2L+1)(2L-9) \lambda ^L + 5(2L+1)(2L-7) \lambda ^{L+1} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( g\right) ,3}\left( 5,L;\lambda \right) \\&\quad = 2(L-3)(L-6)(2L+1) \lambda ^L + (L-3)(2L+1)(2L-7) \lambda ^{L+1} \\&{\mathscr {N}}_{\mathrm {SO(3)}}^{\left( g\right) ,4}\left( 5,L;\lambda \right) \\&\quad = \frac{(L-2)(L-3)(L^2-81L-40)}{24} \lambda ^L + \frac{(L-2)(L-3)(L^2-10L-6)}{6} \lambda ^{L+1} \\&\qquad +\, \frac{L(L-1)(L-2)(L-3)}{4} \lambda ^{L+2} + \frac{(L+1)(L-1)(L-2)(L-3)}{6} \lambda ^{L+3} \\&\qquad + \,\frac{L(L-1)(L-2)(L-3)}{24} \lambda ^{L+4} \end{aligned}$$