Calculation of electronic densities of states for conjugated polymer chains, with off-diagonal disorder parameterised by statistical mechanics.
- Given an arbitrary effective potential energy landscape U=f(\theta). [The Urge]
- Specified as a functional form (
U(theta)=( E0 * sin(theta*pi/180.0)^2 ) #P3HT like) [1] - OR read in from tabulated data (i.e. a Quantum-Chemical potential energy scan), fitting a Chebyshev Polynomial [2] with a Vandermonde matrix [3] via the excellent ApproxFun [4] package.
- Specified as a functional form (
- Integrate (monte carlo direct sampling) to get a (statistical mechanical) partition function, Z=\sum e^{U/kBT}
- Use this partition function to generate random samples of \theta
- Use a model for the transfer integral between two neighbouring units (i.e. monomers in a polymer chain, J~cos(theta)) to build a tridiagonal tight-binding Hamiltonian
- Solve this tridiagonal Hamiltonian with
Sturm sequencemethods, which are linear in time and require no memory. [The Storm]
Method developed in these codes are discussed in these talk slides, but I'm afraid it's pretty incoherent without the talk (and not much better with...).
https://speakerdeck.com/jarvist/2016-03-pvcdt-jarvistmoorefrost-from-atoms-to-solar-cells?slide=82
The only published application of this method we applied it to the P3HT system, treating the P3HT as non-interacting chains. Unfortunately, we found that it didn't agree particularly well with the detailed Molecular Dynamics of the rest of the paper. I suspect this is due to the poor model for the inter-monomer potential (we need an effective potential that includes steric hindrance + entropic effects of the sidechains).
Parameter free calculation of the subgap density of states in poly(3-hexylthiophene)
Jarvist M. Frost, James Kirkpatrick, Thomas Kirchartz and Jenny Nelson
Faraday Discuss., 2014,174, 255-266
http://dx.doi.org/10.1039/C4FD00153B
Extension of these methods to molecular crystals.
- If tri-diagonal, use linear scaling Sturm sequencies to generate a historgrammed DoS
- If not - currently just standard linear algebra methods. Though perhaps Arrowhead methods for 2D/3D systems in the future?
- [1] This should really be a 'free energy' not a potential, taking in an entropic contribution + all enthalpic contributions. However, you can approximate it by using the torsional potential as you would use in part of a molecular dynamics forcefield. For instance for P3HT, Raos FF paper (Moreno et al. J.Phys.Chem.B 2010), contains a 'full' potential energy Fig 4.a. puts a barrier at 90 degress of ~3.0 kCal / mol = 126 meV.
- [2] The Numerical Recipes description is pretty useful. https://en.wikipedia.org/wiki/Approximation_theory#Chebyshev_approximation
- [3] Mathematicians keep all the good stuff to themselves. A superior method of fitting a polynomial, with a specially constructed matrix. https://en.wikipedia.org/wiki/Vandermonde_matrix#Applications
- [4] https://github.com/ApproxFun/ApproxFun.jl