Prerequisite CG coefficient
With the phase convention of CG coefficients and the phase convention of spherical harmonics, you can define Generalized Spherical Harmonics
\begin{equation}
\mathcal{Y}_{l_1,l_2}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2) = \sum_{m_1, m_2} \begin{bmatrix}l_1 & l_2 & L\\ m_1 & m_2 & M\end{bmatrix} Y_{l_1 m_1}( \hat{\boldsymbol{\mathbf{r}}} _1) Y_{l_2 m_2} ( \hat{\boldsymbol{\mathbf{r}}} _2)
\end{equation}
Since the CG coefficient is not zero only when $m_1 + m_2 = M$ is $0022$, the summation changes from a double summation to a single summation. Upper and lower limits
parity
The function of the parity operator $\Pi$ is to multiply all the arguments by $-1$ to get the function. The eigenfunction is a function of all central symmetry or anti-cars, and the eigenvalues are respectively $\pm 1$.
The spherical harmonic function is the eigenvector of the parity operator, and the eigenvalue is $(-1)^l$ (eq. 8 ). It is easy to find that the generalized spherical harmonic function is also the eigenvector of the parity operator, the eigenvalue For $(-1)^{l_1+l_2}$
\begin{equation}
\Pi \mathcal{Y}_{l_1,l_2}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2) = \mathcal{Y}_{l_1,l_2}^{L,M}(- \hat{\boldsymbol{\mathbf{r}}} _1, - \hat{\boldsymbol{\mathbf{r}}} _2) = (-1)^{l_1+l_2} \mathcal{Y}_{l_1,l_2}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2)
\end{equation}
Exchange symmetry
The generalized spherical harmonic function does not have commutative symmetry1 (Unless $l_1 = l_2$)
\begin{equation}
P_{12} \mathcal{Y}_{l_1,l_2}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2) = \mathcal{Y}_{l_1,l_2}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _2, \hat{\boldsymbol{\mathbf{r}}} _1) =
(-1)^{l_1+l_2-L} \mathcal{Y}_{l_2,l_1}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2)
\end{equation}
Can also be written as
\begin{equation}
\mathcal{Y}_{l_1,l_2}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2) =
(-1)^{l_1+l_2-L} \mathcal{Y}_{l_2,l_1}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _2, \hat{\boldsymbol{\mathbf{r}}} _1)
\end{equation}
But we can easily construct symmetric (+) or antisymmetric (-) functions
\begin{equation}
\mathcal{Y}_{l_1,l_2}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2) \pm (-1)^{l_1+l_2-L}\mathcal{Y}_{l_2,l_1}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2)
\end{equation}
\begin{equation}
\Psi( \boldsymbol{\mathbf{r}} _1, \boldsymbol{\mathbf{r}} _2) =
R(r_1, r_2)\mathcal{Y}_{l_1,l_2}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2) - R(r_1, r_2)\mathcal{Y}_{l_2,l_1}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2) \qquad (l_1+l_2-L = odd)
\end{equation}
Prove that the symmetry of the CG coefficient can be used (exchange the two columns on the left, eq. 7 )
\begin{equation}
\begin{aligned}
\mathcal{Y}_{l_1,l_2}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _2, \hat{\boldsymbol{\mathbf{r}}} _1)
&= \sum_{m_1, m_2} \begin{bmatrix}l_1 & l_2 & L\\ m_1 & m_2 & M\end{bmatrix} Y_{l_1, m_1}( \hat{\boldsymbol{\mathbf{r}}} _2)Y_{l_2, m_2}( \hat{\boldsymbol{\mathbf{r}}} _1) \\
&= (-1)^{l_1+l_2-L} \sum_{m_1, m_2} \begin{bmatrix}l_2 & l_1 & L\\ m_2 & m_1 & M\end{bmatrix} Y_{l_2, m_2}( \hat{\boldsymbol{\mathbf{r}}} _1) Y_{l_1, m_1}( \hat{\boldsymbol{\mathbf{r}}} _2)\\
& = (-1)^{l_1+l_2-L} \mathcal{Y}_{l_2,l_1}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2)
\end{aligned} \end{equation}
From
eq. 3 can also get the exchange relationship
\begin{equation}
\left[L^2, P_{12}\right] = \left[L_z, P_{12}\right] = 0
\end{equation}
That is, exchanging two particles does not change the total angular momentum.
If the operation type operator (parity, translation, exchange) is exchanged with a certain physical quantity operator, it means that the wave function changes the conservation of physical quantity through this operation. If a certain operator in Hamilton (such as the dipole of the electric field in the direction of $z$) is traded with a certain physical quantity operator (such as $L_z$), it means that the wave function propagates through the propagator and the physical quantity is conserved.
Orthogonality
Express eq. 10 with the symbols here, which is
\begin{equation}
\int \mathcal{Y}_{l_1', l_2'}^{L', M'}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2) \mathcal{Y}_{l_1, l_2}^{L, M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2) \,\mathrm{d}{\Omega_1} \,\mathrm{d}{\Omega_2} = \delta_{l_1, l_1'} \delta_{l_2, l_2'} \delta_{L, L'} \delta_{M, M'}
\end{equation}
Other properties
\begin{equation}
\mathcal{Y}_{l_1,l_2}^{L,-M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2) = (-1)^{l_1 + l_2 + L + M} \mathcal{Y}_{l_1,l_2}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2)^*
\end{equation}
Among them, $*$ represents complex conjugate. The derivation is as follows,
\begin{equation}
\begin{aligned}
\mathcal{Y}_{l_1,l_2}^{L,-M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2) &= \sum_{m_1+m_2 = M} \begin{bmatrix}l_1 & l_2 & L \\ -m_1 & -m_2 & -M\end{bmatrix} Y_{l_1,-m_1}( \hat{\boldsymbol{\mathbf{r}}} _1) Y_{l_2, -m_2} ( \hat{\boldsymbol{\mathbf{r}}} _2)\\
&= (-1)^{l_1+l_2+L} \sum_{m_1+m_2 = M} \begin{bmatrix}l_1 & l_2 & L \\ m_1 & m_2 & M\end{bmatrix} Y_{l_1,-m_1}( \hat{\boldsymbol{\mathbf{r}}} _1) Y_{l_2, -m_2} ( \hat{\boldsymbol{\mathbf{r}}} _2)\\
&= (-1)^{l_1+l_2+L+M} \sum_{m_1+m_2 = M} \begin{bmatrix}l_1 & l_2 & L \\ m_1 & m_2 & M\end{bmatrix} Y_{l_1,m_1}^*( \hat{\boldsymbol{\mathbf{r}}} _1) Y_{l_2, m_2}^* ( \hat{\boldsymbol{\mathbf{r}}} _2)\\
&= (-1)^{l_1+l_2+L+M} \mathcal{Y}_{l_1,l_2}^{L,M}( \hat{\boldsymbol{\mathbf{r}}} _1, \hat{\boldsymbol{\mathbf{r}}} _2)^*
\end{aligned} \end{equation}
1. ^ where the negative sign in front of $l_1+l_2-L$ can be turned into a plus sign, but it is still written as a negative sign, because although the spherical harmonic function requires $L$ to be an integer, when it involves electrons $L$ may be a half-integer when spinning, but it cannot become a plus sign at this time.