Johnson Renormalization

[B. R. Johnson JCP 69, 4678 (1978)]

\[\mathbf{T}_n = -\frac{2(\Delta R)^2\mu}{12\hbar^2} (E\mathbf{I}-\mathbf{V}_n)\]

\[\begin{split}\begin{array}{l} \mathbf{F}_n = \mathbf{W}_n \boldsymbol{\chi}_n = [\mathbf{I} - \mathbf{T}_n] \chi \\ \left. \begin{array}{l} \mathbf{F}_{n+1} - \mathbf{U}_n\mathbf{F}_n + \mathbf{F}_{n-1} = \mathbf{0} \\ \mathbf{R}_n = \mathbf{F}_{n+1}\mathbf{F}^{-1}_n \\ \end{array} \right\} \Rightarrow \begin{array}{l} \mathbf{R}_n = \mathbf{U}_n - \mathbf{R}^{-1}_{n-1} \\ \mathbf{U}_n = 12\mathbf{W}^{-1}_n - 10\mathbf{I} \\ \end{array} \end{array}\end{split}\]

\[\begin{split}\begin{aligned} \mathbf{V} &= \left( \begin{array}{lllll} V_{00} & V_{01} & V_{02} & \ldots & V_{0\text{n}} \\ & V_{11} & V_{12} & \ldots & V_{1\text{n}} \\ & & \ddots & & \vdots \\ & & & \ddots & V_{\text{n}\text{n}}\\ \end{array} \right)\end{aligned}\end{split}\]

\(V_{ii}\) diabatic potential energy curves, \(V_{i j\neq i}\)

off-diagonal coupling terms [H. Lefebvre Brion and R. W. Field table 2.2 page 39.]

\(\Delta \Omega = 0\) homogeneous

\(\Delta \Omega = \pm 1\) heterogenous - \(J\) dependent.

Outward Solution

\[\begin{split}\begin{array}{ll} \mathbf{R}_n = \mathbf{U}_n - \mathbf{R}^{-1}_{n-1} & n = 1 \rightarrow m \text{ with}\ \mathbf{R}^{-1}_0 = 0 \\ \mathbf{F}_n = \mathbf{R}^{-1}_n\mathbf{F}_{n+1} & n = m \rightarrow 0 \text{ with}\ \mathbf{F}_\infty = \mathbf{W}_\infty \boldsymbol{\chi}_\infty \end{array}\end{split}\]

Except when \(\left| \mathbf{R}_n \right| \sim 0\) then

\(\mathbf{R}^{-1}_n\) is not well defined.

Use \(\mathbf{F}_n = \mathbf{U}_{n+1}\mathbf{F}_{n+1} - \mathbf{F}_{n+2}\)

Inward Solution (\(\hat{\ }\) - matrices)

\[\begin{split}\begin{array}{ll} \hat{\mathbf{R}}_n = \mathbf{U}_n - \hat{\mathbf{R}}^{-1}_{n+1} & n = \infty \rightarrow m \text{ with}\ \hat{\mathbf{R}}^{-1}_\infty = 0 \\ \mathbf{F}_n = \hat{\mathbf{R}}^{-1}_n \mathbf{F}_{n-1} & n = m \rightarrow \infty \ \text{ with}\ \mathbf{F}_0 = \mathbf{W}_0\boldsymbol{\chi}_0\\ \end{array}\end{split}\]

Except when \(\left| \mathbf{R}_n \right| \sim 0\) then

\(\mathbf{R}^{-1}_n\) is not well defined.

Use \(\mathbf{F}_n = \mathbf{U}_{n-1}\mathbf{F}_{n-1} - \mathbf{F}_{n-2}\)

Multiple Open Channels

\(n_{\rm open}\) linearly independent solutions:

\[\begin{split}\mathbf{R}(R=\infty) = \begin{pmatrix} 1 & 0 & \ldots & 0 \\ 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \ddots & \ldots & \vdots\\ 0 & 0 & \ldots & 1\\ \end{pmatrix} \rightarrow \text{CSE} \rightarrow \boldsymbol{\chi}(R) = \begin{pmatrix} \chi_{00} & \chi_{01} & \chi_{02} & \ldots & \chi_{0N_{\text{open}}}\\ \chi_{10} & \chi_{11} & \chi_{12} & \ldots \\ \vdots & \vdots & \vdots & & \vdots \\ \chi_{N_{\text{total}}0} & & & \ldots & \chi_{N_{\text{total}}N_{\text{open} }} \\ \end{pmatrix}\end{split}\]

Normalization

[Mies - Molecular Physics 14, 953 (1980).]

\(\boldsymbol{\chi} = \mathbf{JA} + \mathbf{NB}\)

\(\mathbf{F}^K = \boldsymbol{\chi} \mathbf{A}^{-1} = \mathbf{J} + \mathbf{NK}\)

where

\(\mathbf{K} = \mathbf{BA}^{-1} = \mathbf{U}\tan \boldsymbol{\xi} \hat{\mathbf{U}}\).

Physical solution

\(\mathbf{F}^S = \mathbf{F}^k\mathbf{U}\cos\boldsymbol{\xi} e^{\text{i} \boldsymbol{\xi}} \hat{\mathbf{U}} = \text{i}e^{-\text{i}\mathbf{k}R} - \text{i}e^{\text{i}\mathbf{k}R}\mathbf{S}\)

Determine matrices , by energy normalization of each wavefunction.

\(\chi_{ij} = \left( \frac{2\mu}{\hbar^2\pi} \right) ^{\frac{1}{2}} \frac{1}{\sqrt{k}} \left[ J_i a_{ij} + N_i b_{ij} \right]\) for potential \(i\).