# Coupling models
The off-diagonal matrix elements of the one-exciton system Hamiltonian ($\hat{H}$) is determined by the selected coupling model.

Current implemented models are:
- **TDC**: Transition Dipole Coupling model
- **NN_Mix_TDC**: Nearest-Neighbor modified TDC
- **Jansen_TDC**: Jansen $\phi/\psi$ angle map modified TDC
- **Zero**: Zero coupling for testing purpose

## Transition Dipole Coupling (TDC) model
The transition dipole moment between the ground state (g) and an one-exiton state ($\nu$) is defined below, where the $\hat{\mu}$ is the dipole operator:
$$
\vec{\mu} = \langle \nu | \hat{\mu} | g \rangle
$$

The transition dipole coupling model assumes a through-space coupling between two transition dipoles. A simple physical picture of such coupling is a measure of on how the observed vibration mode affected by the electric field change due to the neighboring vibrations. Using the multiple expansion and take the lowest order, we further simplify the coupling ($\beta$) as a pair of transition dipoles:

$$
\beta = C*[\frac{\vec{\mu}_1 \cdot \vec{\mu}_2}{\vec{r}}
- \frac{3(\vec{\mu}_1 \cdot \vec{r})(\vec{\mu}_2 \cdot \vec{r})}{\vec{r}^3}]
$$

The prefactor $C$ is a scaling constant. The exact value depends on what unit of the transition dipole is used. For more information please see the foot note [^1]

As for how to calculate transition dipole and it's relationship with the dipole derivative, I would suggest further reading on a well-written note by T. Daniel Crawford, which is titled [Vibrational Transition Moments and Dipole Derivatives](http://www.ccl.net/cca/documents/ir_intensities-1.pdf).

To give you a sense of coupling size, here is an example of a general peptide system. The typical transition dipole strength for the Amide-I vibrational mode of a peptide bond is 3.144 Debye [^2]. Using this number, a di-peptide model $\beta \approx 10 cm^{-1}$ (order-of-magnitude wise), which varies according to their relative orientation.

## Nearest neighbor modified TDC model
Coming soon...

## Jensan map modified TDC model
Coming soon...

## Other possible models
There are other semi-implemented models:

- **Cho_PB**: M. H. Cho's coupling map for parallel $\beta$-sheet
- **Cho_APB**: M. H. Cho's coupling map for anti-parallel $\beta$-sheet
- **TDC+Cho_APB**: Same as Cho_APB but use TDC for non-$\beta$-sheet segment
- **TDC_PBC**: TDC with periodic boundary condition

You can find them under the folder here:

'/SpecFunc/MolecularResponse/'

[^1]: Chem. Phys. Lett. v106, #6, p613
[^2]: https://doi.org/10.1021/jp9072203
###### tags: `COSMOSS` `Theory`