What happens when you want to do charge-fitting on large molecules?

Electronic energy transfer (EET) plays an important role in natural systems, most prominently as an essential step in photosynthesis(Scholes 2003). Related to EETs are excitonic interactions, i.e. the interaction between excited states on different chromophores. Of particular recent interest are excitonic resonance states that arise when an excitation is a combination of locally excited states on different chromophores giving rise to a delocalized excited state.

The quantity of interest in EETs are the electronic couplings that determines how much the different chromophores interact. The larger this coupling is, the more delocalized the excited state can be. In order to understand the photophysics in large multichromophoric systems, the electronic coupling are essential to evaluate and understand.

There are essentially two ways to calculate the couplings:

- taking into accout Coulomb, exchange and overlap terms.
- transformation to diabatic state.

In this work, we use the first approach.

For two chromophores \(A\) and \(B\), the upper limit to the electronic coupling between adiabatic states \(i\) and \(j\) is one half of the splitting energy \(\Delta E_{ij}\) \[\label{eqn:halfsplit} |J_{ij}| = \frac{|\Delta E_{ij}|}{2} = \frac{|E_i - E_j|}{2}.\] For symmetry-equivalent chromophores, the couplings in eq \ref{eqn:halfsplit} is the exact coupling.(Blancafort 2014) Computing the coupling through eq \ref{eqn:halfsplit} is impractical for large chromophores because of the computational demand and especially if surroundings needs to be taken into account also.

It is possible to partition the couplings into three contributions: Coulombic, exchange and overlap. Commonly, the chromophores are separed enough such that the latter two terms can be ignored. The resulting Coulomb-couplings are usually evaluated using transition densities of each chromophore. That is, the electronic coupling, \(J_{ij}\), between the excited state \(i\) on chromophore \(A\) and the excited state \(j\) on chromophore \(B\), \(J_{ij}\), is related to the transition densities \(\rho(\mathbf{r})_{A,i}^T\) and \(\rho(\mathbf{r}')_{B,j}^T\) of chromophore \(A\) and \(B\), respectively. The coupling is evaluated as \[\label{eqn:coupling_exact}
J_{ij} = \int \int \mathrm{d}\mathbf{r} \mathrm{d}\mathbf{r}' \frac{\rho(\mathbf{r})_{A,i}^T \rho(\mathbf{r}')_{B,j}^T}{|\mathbf{r}-\mathbf{r}'|}.\] Rather than using the transition densities directly, it is possible to represent them using a set of atomic point charges, \(\{q^T\}\), that have been fitted to reproduce the electrostatic potential of the transition densities, \(\rho(\mathbf{r})^T\), i.e. **TrESP**. The couplings are evaluated as \[J_{ij} \approx \sum_{a\in A} \sum_{b\in B} \frac{q_{a,i}^T q_{b,j}^T}{|\mathbf{R}_a - \mathbf{R}_b|}.\] The fitted charges are constrained to reproduce an overall charge of 0 (no charge is being generated or removed during excitaiton - only moved around) and the transition dipole associated with an excitation \(i\), \(\mu^T\). A final expression for the calculation of the coupling is by using the transition dipoles directly. Here, the transition dipoles are assumed to be located in the center of mass of each chromophore yielding the classical Förster coupling expressed as \[\label{eqn:coupling_charges}
J_{ij} \approx \mu_{A,i}^T \nabla^2 \frac{1}{|\mathbf{R}_A - \mathbf{R}_B|} \mu_{B,j}^T.\] For larger molecules and at shorter distances, the Förster couplings are expected to give poor results.

*direct* potential from the \(j\)’th excited state of chromophore \(B\) onto the atomic sites \(a\) in chromophore \(A\) is \[\phi_a^T (B_j) = \sum_{b\in B} \frac{q^T_{b,j}}{|\mathbf{R}_a - \mathbf{R}_b|}\].

\[|\Psi_i\rangle\]

\[\hat{V} = \hat{V}_{AB} + \sum_{N}\left[ \hat{V}_{AN} + \hat{V}_{BN} + \sum_{N'} \hat{V}_{NN'} \right]\]

We write a general excited state as \[|A\rangle \approx |A^0\rangle + |\delta A^0\rangle\] where the first order correction is \[|\delta A^0\rangle = \sum_N \sum_{k\in N} \frac{V_{A^0,N^k}}{E_{A^0} - E_{N^k}} |N^k\rangle + \sum_{j>1\in B} \frac{V_{A^0,B^j}}{E_{A^0} - E_{B^j}} |B^k\rangle\] \[\] \[\]

We have calculated the electronic coupling, \(J\), between the the excited states of an ergosterol dimer. The ergosterol molecules are separated by distances of 6Å, 7Å and 8Å. To obtain reference values for the coupling, supermolecule calculations were carried out in Gaussian at the CAM-B3LYP/6-31G level of theory. We have tested the numerical requirements of the more approximate method using transition density fitted charges.

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