We thank Dr. Andreas Tillack (Oak Ridge National Laboratory), Dr. Lewis Johnson (University of Washington), and Dr. Bruce Robinson (University of Washington) for developing the program for coarse-grained Monte Carlo simulations that was used to generate the structure of the molecular system presented in the Representative Results section. A.A.K. and P.F.G. are supported by a Collaborative research award from the College of Science, CSU East Bay. M.H. is supported by a Forever Pioneer fellowship from the Center for Student Research, CSU East Bay. C.M.I. and S.S. are supported by the U.S. Department of Defense (Proposal 67310-CH-REP) under the Air Force Office of Scientific Research Organic Materials Division.

1. Splitting the multi-molecular system into individual molecules
<ol>
	<li>Generate the structure of the system for which the excitonic Hamiltonian needs to be constructed in Tripos MOL2 molecular file format. This structure can be a snapshot from a molecular dynamics or Monte Carlo simulation of the system.</li>
	<li>If all molecules in the system consist of the same number of atoms, use the python 2.7 script getMonomers.py to generate files that contain the Cartesian coordinates for atoms in individual molecules tha...

The method presented here allows for multiple customizations. For instance, it is possible to modify the parameters of the DFT and TD-DFT calculations, including the density functional, basis set, and specific definition of the atomic point charges.
Using long-range corrected functionals, such as &#969;B97X, &#969;B97XD, or &#969;PBE, is recommended in order to obtain reasonable transition densities for transitions with charge-transfer character. It may be interesting to study to what extent t...

In the past two decades, solids and films that are made from aggregated organic molecules have found multiple applications in optoelectronic devices. Devices based on such materials have many attractive properties, including small weight, flexibility, low power consumption, and potential for cheap production using inkjet printing. Displays based on organic light emitting diodes (OLEDs) are replacing liquid crystalline displays as state of the art for mobile phones, laptops, television sets, and other electronic devices1,2,3,4. The importance of OLEDs for lighting applications is expected to increase in the coming years4. The performance of organic photovoltaic devices is steadily improving, with power conversion efficiencies above 16% recently reported for single-junction organic solar cells5. Organic materials also have the potential to disrupt other technologies, such as fiber-optic communications, where their use enables the development of electro-optic modulators with extremely high bandwidths of 15 THz and above6,7.
A major challenge in optimizing solid-state molecular materials for applications in optoelectronics is that typically their properties strongly depend on the nanoscale structure of the material. The production process allows defining the nanostructure of a material to some extent by using controlled growth techniques, such as chemical vapor deposition,8 templating of optically active molecules onto another material (i.e., a polymer matrix9,10), thermal annealing11,12, etc. However, nanoscale disorder is intrinsic to most molecular materials and usually cannot be eliminated entirely. Therefore, understanding how disorder affects the properties of a material and finding ways to engineer it for optimal performance is essential for the rational design of organic optoelectronic materials.
The degree of disorder in molecular materials is usually too great to treat it as a perturbation of a periodic crystalline structure with an electronic structure that can be described by band theory. On the other hand, the number of molecules that must be included in a simulation to reproduce the properties of a bulk material or a film is too great to use first principles quantum chemical methods like density functional theory (DFT)13,14 and time-dependent density functional theory (TD-DFT)15,16. Organic molecules with applications in optoelectronics typically have relatively large &#960;-conjugated systems; many also have donor and acceptor groups. Capturing the correct charge-transfer behavior in such molecules is essential to calculating their optoelectronic properties, but it can only be accomplished using long-range corrected hybrid functionals in TD-DFT17,18,19,20. Calculations that use such functionals scale super linearly with the size of the system and, at present, they are only practical for modeling the optoelectronic properties of individual organic molecules or small molecular aggregates that can be described using no more than ~104 atomic basis functions. A simulation method that could describe disordered materials that consist of large numbers of chromophores would be very useful for modeling these systems.
The magnitude of intermolecular interactions in molecular materials is often comparable to or smaller than the order of variation in the energetic parameters (such as the eigenstate energies or excitation energies) between individual molecules that make up the material. In such cases, multiscale modeling is the most promising approach to understanding and optimizing the optoelectronic properties of large disordered molecular systems21,22,23. This approach uses first-principles quantum chemical methods (usually DFT and TD-DFT) to accurately calculate the properties of individual molecules that compose the material. The Hamiltonian of a material sample that is large enough to represent the bulk molecular material (perhaps, by employing periodic boundary conditions) is then constructed using the parameters that were calculated for individual molecules. This Hamiltonian can then be used to calculate the optoelectronic parameters of a large molecular aggregate, a thin film, or a bulk molecular material.
Exciton models are a subclass of multiscale models in which excited states of a molecular material are represented in a basis of excitons: electron-hole pairs that are bound by Coulomb attraction24,25. For modeling many excited state processes, it is sufficient to only include Frenkel excitons26, where the electron and the hole are localized on the same molecule. Charge transfer excitons, where the electron and the hole are localized on different molecules, may need to be included in some cases (e.g., when modeling charge separation in donor-acceptor systems)27,28. Although exciton models are multiscale models that can be parametrized using only first-principle calculations on individual molecules, they still account for intermolecular interactions. The two primary interaction types that they can account for are (a) excitonic couplings between molecules that characterize the ability of excitons to delocalize across or transfer between molecules and (b) electrostatic polarization of the electron density on a molecule by the charge distribution on surrounding molecules. We have previously shown that both of these factors are important for modeling the optical and electro-optic properties of molecular aggregates, such as the optical absorption spectra29 and first hyperpolarizabilities30.
In this paper, we present a protocol for parametrizing exciton models that can be used to calculate the optical spectra and other optoelectronic properties of large molecular aggregates and bulk molecular materials. The excitonic Hamiltonian is assumed to be a tight-binding Hamiltonian24,25,
<img alt="Equation 1" src="/files/ftp_upload/60598/60598equ01.jpg" />
where &#949;i is the excitation energy of the ith molecule in the material, bij is the excitonic coupling between the ith and the jth molecules,&#160;&#226;i&#8224; and&#160;&#226;i are the creation and annihilation operators, respectively, for an excited state on the ith molecule in the material. The excitonic Hamiltonian parameters are found using TD-DFT calculations that are performed on individual molecules that make up the material. In these TD-DFT calculations, the charge distribution on all other molecules in the material is represented by electrostatic embedding of atomic point charges to account for electrostatic polarization of a molecule&#8217;s electronic density. The excitation energies, &#949;i, for individual molecules are taken directly from the TD-DFT calculation output. The excitonic couplings, bij, between molecules are calculated using the transition density cube method31, with the ground-to-excited state transition densities for the interacting molecules taken from the output of a TD-DFT calculation in Gaussian32 and post-processed using the Multiwfn multifunctional wavefunction analyzer33. For simulating the properties of bulk molecular solids, periodic boundary conditions may be applied to the Hamiltonian.
The current protocol requires that the user have access to the Gaussian32 and Multiwfn33 programs. The protocol has been tested using Gaussian 16, revision B1 and Multiwfn version 3.3.8, but should also work for other recent versions of these programs. In addition, the protocol uses a custom C++ utility and a number of custom python 2.7 and Bash scripts, the source code for which is provided under the GNU General Public License (Version 3) at https://github.com/kocherzhenko/ExcitonicHamiltonian. The calculations are intended to be performed on a machine running an operating system from the Unix/Linux family.

<table>
	<tbody>
		<tr>
			<td>Gaussian 16, revision B1</td>
		</tr>
		<tr>
			<td>Multiwfn version 3.3.8</td>
		</tr>
		<tr>
			<td>GNU compiler collection version 9.2</td>
		</tr>
		<tr>
			<td>python 2.7.0</td>
		</tr>
	</tbody>
</table>

excitonic hamiltonians for calculating optical absorption spectra and optoelectronic properties of molecular aggregates and solids

Rational design of disordered molecular aggregates and solids for optoelectronic applications relies on our ability to predict the properties of such materials using theoretical and computational methods. However, large molecular systems where disorder is too significant to be considered in the perturbative limit cannot be described using either first principles quantum chemistry or band theory. Multiscale modeling is a promising approach to understanding and optimizing the optoelectronic properties of such systems. It uses first-principles quantum chemical methods to calculate the properties of individual molecules, then constructs model Hamiltonians of molecular aggregates or bulk materials based on these calculations. In this paper, we present a protocol for constructing a tight-binding Hamiltonian that represents the excited states of a molecular material in the basis of Frenckel excitons: electron-hole pairs that are localized on individual molecules that make up the material. The Hamiltonian parametrization proposed here accounts for excitonic couplings between molecules, as well as for electrostatic polarization of the electron density on a molecule by the charge distribution on surrounding molecules. Such model Hamiltonians can be used to calculate optical absorption spectra and other optoelectronic properties of molecular aggregates and solids.

In this section we present representative results for computing the optical absorption spectrum of an aggregate of six YLD 124 molecules, shown in Figure 3a, where the structure of the aggregate was obtained from a coarse-grained Monte Carlo simulation. YLD 124 is a prototypical charge-transfer chromophore that consists of an electron-donating group of diethyl amine with tert-butyldimethylsilyl protecting groups that is connected via a &#960; -conjugated bridge to the electron accep...

Watch this Scientific Journal Video about Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids at JoVE.com

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids

Here, we present a protocol for parametrizing a tight-binding excitonic Hamiltonian for calculating optical absorption spectra and optoelectronic properties of molecular materials from first-principles quantum chemical calculations.

Rational design of disordered molecular aggregates and solids for optoelectronic applications relies on our ability to predict the properties of such ...

This protocol facilitates the construction of excitonic Hamiltonians for the efficient calculation of both optical absorption spectra and the more complex optoelectronic properties of bulk molecular materials. Our technique breaks down extremely computationally intensive quantum chemical calculations on bulk molecular materials into much more manageable calculations of single molecules that are performed using common quantum chemical software. Our method can help computationally guide the design of optoelectronic devices using organic materials such as photovoltaic cells or optical switches for fiber optic communications.New users should carefully follow the procedure as described including the suggested filenaming convention and should check that each step has been completed without error before moving on. For the splitting of a multi-molecular system into individual molecules, use the Python 2.7 script getMonomers. py to generate files that contain the Cartesian coordinates for atoms in individual molecules that compose the system.Specify the name of the file that contains the geometry of the system and the number of atoms in each individual molecule that makes up the system using the command as indicated. To generate ground state point charges for atoms in individual molecules, set up a plain text file called chargeOptions. txt with the options for Gaussian density functional theory calculation of the atomic point charges in the ground state of an electrically neutral molecule.To obtain a reasonably accurate charge distribution for transitions with charge transfer character, use a long range corrected density functional, a sufficiently large basis set that includes at least depolarization functions on non-hydrogen atoms, a superfine integration grid, and a very tight self-consistent field convergence criterion. Include the Nosymm keyword in the input file to ensure that the atomic coordinates in the Gaussian output file are written in input orientation. Set up the Gaussian input files for all of the individual molecules that make up the system using the parameters in the file chargeOptions.txt using the indicated Bash script. Then run the Gaussian calculations specifying the output filename to be the same as the input. com filename but with the extension log.Use the Python 2.7 script getCHelpG. py to extract the CHelpG atomic point charges from the Gaussian output files with the extension log. To calculate the excitation energies and transition densities of individual molecules in the material in the presence of an electrostatic environment, set up a plain text file named monomerOptions.txt with a parameter set as for the calculation of the atomic point charges and with a low threshold for printing eigenvector components ideally to at least the order of one times 10 to the negative five. Set up the Gaussian input files for the calculation of the excitation energies and transition densities of all of the individual molecules in the material in the presence of an electrostatic environment represented by the point charges on all other molecules in the material and name the file monomer_n_wCh. com where n is the monomer number.Then run the Gaussian calculations specifying the output filename to be the same as the input. com filename but with the extension log. The calculation will also save a checkpoint file with the same filename but with the extension chk.For excitation energy extraction for bright states of individual molecules that make up the system from the Gaussian output files, copy the excitation energies for the bright excited states of individual monomers from the Gaussian output files with the extension log to a plain text file called all_energies.txt. In the file all_energies. txt, keep only the column that contains the numerical values of the excitation energies.To calculate the excitonic couplings for all pairs of molecules that make up the molecular system, first use the form check utility from the indicated Bash script to convert the checkpoint files to human readable format. Use the Python 2.7 script switchSign. py that takes the name of the Gaussian output file with the extension log and the number of excited states n included in the calculation as input parameters.Use the Multiwfn multi-functional wavefunction analyzer to write the transition density cube file based on the Gaussian formatted checkpoint file with the extension fchk and the processed Gaussian output file with the extension log2. To efficiently generate setup files with Multiwfn processing options for all fchk files in the current directory, use the makeOpt. sh Bash script.The files will have the same names as the fchk files with the extension opt. Then generate the transition density cube files in a single batch using the indicated Bash script and convert the cub files to files that explicitly specify the coordinates of the centers of all cubes on the grid and the values of the transition density inside the cube using the cubeFormat. py Python 2.7 script.Run the command as indicated to use the fcub files to calculate the excitonic couplings between all pairs of molecules in the system using the transition density cube method. Once the calculations are complete, create an empty file called all_couplings. txt and use the Bash script as indicated to combine all of the excitonic couplings into a single file.To set up the excitonic Hamiltonian, use the setUpHam. py Python 2.7 script and the indicated terminal command to combine the excited state energies in the all_energies. txt file and the excitonic couplings in the all_couplings.txt file into a single file that contains the complete excitonic Hamiltonian matrix. Here, the optical absorption spectrum of an aggregate of six YLD 124 molecules obtained from a cross-grained Monte Carlo simulation that was used to calculate the excitonic Hamiltonian of the molecules is shown. In this table, the Hamiltonian for this system constructed as demonstrated can be observed.Because there are six molecules with only a single bright excited state for each molecule, a six by six excitonic Hamiltonian was generated resulting in six transitions. The exciton model and TDDFT spectra calculated using the WB97X density functional with the G31G*basis set also have similar shapes as characterized by Pearson's product-moment correlation coefficient. Excitonic Hamiltonians constructed using our protocol can be parameterized with any quantum chemical method allowing study of how approximations for specific methods affect the computation accuracy for various optoelectronic parameters.We have used this method to model the optical absorption spectra and the first hyperpolarizabilities of molecular aggregates with efforts ongoing to accurately model the properties of bulk molecular solids.

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