Coupling Matrix

In order to construct the coupling matrix, we need the explicit form of matrix elements of \( {W}^{Jp}_{\gamma, \gamma'}(R) \).

Matrix elements of the interaction potential are given as $$\begin{equation} \label{eq:pes_contribution} {V}_{\gamma,\gamma'}^{Jp}(R) = \sum_{{\lambda}} g_{{\lambda},\gamma,\gamma'}^{Jp} v_{{\lambda},\gamma,\gamma'} (R) . \end{equation}$$

Here, \( v_{\boldsymbol{\lambda},\gamma,\gamma'} (R) \) denotes the radial coupling terms of the potential energy surface:

$$\begin{equation} \label{eq:radial_terms} v_{{\lambda},\eta, \eta'} (R)= \int_{0}^{\infty} \mathrm{d}r \chi_{\eta}(r) \left[ \frac{2\lambda + 1}{2} \int_{0}^{\pi}\mathrm{d}\theta V(R, r, \theta) P_{\lambda} (\cos\theta) \right] \chi_{\eta'}(r), \end{equation}$$

i.e. the 3D molecule - atom PES, \( V(R, r, \theta) \), averaged over Legendre polynomials, \( P_{\lambda} (\cos\theta)\), and rovibrational wave functions of the isolated molecule, \( \chi_{\eta}(r) \), where \( \eta = v, j\). Note that they are independent of \( J \) and \( p \).

\( g_{{\lambda},\gamma,\gamma'}^{Jp} \) are the algebraic coefficients of the expansion which stem from matrix elements of Legendre polynomials, \( P_{\lambda} (\cos\theta) \) in the \( \gamma \) basis. They are given as

$$\begin{equation} \label{eq:algebraic_coeffs} g_{{\lambda},\gamma,\gamma'}^{Jp} = \delta_{\bar{\Omega},\bar{\Omega}'} (-1)^{\bar{\Omega}} \sqrt{(2j+1)(2j'+1)} \begin{pmatrix} j & j' & \lambda \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} j & j' & \lambda \\ \bar{\Omega} & -\bar{\Omega} & 0 \end{pmatrix}, \end{equation}$$

where quantities in brackets are the 3-j symbols.

Algebraic coefficients are \(R\)-independent and are calculated only once per each block. Note that they are block-diagonal with respect to \(\bar{\Omega}\).

Procedure check_nonzero_pes_matrix_elements first checks how many matrix elements fulfill the \( \delta_{\bar{\Omega},\bar{\Omega}'} \) condition - this is saved in the number_of_nonzero_pes_matrix_elements variable. On top of that, for each non-zero matrix element, the number of non-zero terms in the sum over \(\lambda\) is checked and saved in number_of_nonzero_algebraic_coefficients variable. The coefficients are calculated with prepare_pes_matrix_elements procedure and saved in the nonzero_coupling_coefficients variable. The corresponding \( \lambda \) indices are written indirectly in nonzero_legendre_indices array which holds indices to legendre_indices.

The centrifugal term, \( \mathbf{L}^{2\,Jp}_{\gamma, \gamma'} \), is block-diagonal with respect to the rotational quantum number \( j \). Terms diagonal with respect to \( \bar{\Omega} \) are given as $$\begin{equation} \label{eq:cent-mat-diag} (\mathbf{L}^{2})^{Jp}_{v' j' \bar{\Omega}' ; v j \bar{\Omega}} = J(J+1)+j(j+1)- 2\bar{\Omega}^{2} \end{equation}$$

The centrifugal term additionally introduces couplings with \( \bar{\Omega} \) and \( \bar{\Omega}' = \bar{\Omega} \pm 1 \) blocks

$$\begin{equation} \label{eq:cent-mat-off} (\mathbf{L}^{2})^{Jp}_{v' j' \bar{\Omega}' \pm 1 ; v j \bar{\Omega}} = -\sqrt{\left(1+\delta_{\bar{\Omega}0}\right) \left(1+\delta_{\bar{\Omega}\pm1, 0}\right)} \sqrt{\left[J(J+1)- \bar{\Omega}(\bar{\Omega}+1)\right]\left[j(j+1)- \bar{\Omega}(\bar{\Omega}+1)\right]} \end{equation}$$