mod_rclm Module

Transform matrix to spherical harmonics representation.

• (KEY=1) From real to complex spherical harmonics basis
• (KEY=1) From complex to real spherical harmonics basis

Transformation matrix is $A$: $Y_{real} = A Y_{cmplx}$ $$\begin{equation} A= \begin{bmatrix} I\frac{i}{\sqrt{2}} & 0 & -J\frac{i}{\sqrt{2}} \\ 0 & 1 & 0 \\ J\frac{1}{\sqrt{2}} & 0 & I\frac{1}{\sqrt{2}}\end{bmatrix} \end{equation}$$ (of dimension $2l+1$), where $I$ is the $l\times l$ unit matrix, $J$ is the $l\times l$ antidiagonal unit matrix $$\begin{equation} J = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix} \( A_{mm'} = \int Y_{lm} Y^{c *}_{lm'} d\Omega\) \end{equation}$$ Transformation rule: Complex --> Real (VC --> VR) $$\begin{equation} VR_{mm'} = \sum_{m1,m2} A_{m,m1} VC_{m1,m2} A^*_{m'm2} \end{equation}$$ with $$\begin{equation} VC_{m1,m2} = \int Y^{c *}_{m2} V Y^{c}_{m1} \end{equation}$$ LDIM corresponds to the dimension of the array VMAT as VMAT(2*LDIM+1,2*LDIM+1). LL is the angular momentum l, corresponding to the part of VMAT used -- VMAT(2*LL+1,2*LL+1) Result returns in again in VMAT (original VMAT is destroyed) @note Warning Original VMAT is destroyed.