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WIP Theory

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Joshua S Brown edited this page Sep 26, 2020 · 2 revisions

WARNING

This is a work in progress

DISCLAIMER

1. In this section I hope to clarify what and how the transfer integral is calculated. Be aware that I am presenting my understanding, I will try my best to be accurate, but the reader is always advised to refer to peer-reviewed literature. 2. If you notice inconsistencies in your understanding and the theory presented here please contact me so that it can be corrected or the documentation can be made clearer.

Introduction

The charge transfer integral is written mathematically as:

J= ❬ Ψ A | Ĥ | Ψ B ❭

Here, ΨA and ΨB represent the wavefunctions of two different quantum states which are orthonormal and Ĥ represents the Hamiltonian operator. J represents the electronic coupling between states A and B. The larger the coupling the larger the probability that a charge in state A will move to state B and vice versa.

There are different methods for calculating the charge transfer integrals that make use of various approximations, these include the energy splitting in dimer method among others [1].

The calculation for the DIPRO method, from my understanding was first clearly articulated by E. Valeev in 2006 [2]. It was developed to help address the failure of the energy splitting in dimer method to account for polarization effects.

Energy Splitting in Dimer Method

It would be helpful at this point to explain what the energy splitting dimer method does. It makes a few important assumptions:

  1. the two molecules are chemically symmetric

  2. the two molecules are spatially symmetric

Now if we remember that when we bring two molecules together the Pauli exclusion principle states that no more than two electrons can occupy the same orbital and hence the orbitals will split. Take the simple case of two Helium atoms that approach one another, at a distance they will each have a single occupied orbital. As the atoms approach the orbital will split into two orbitals, the highest occupied molecular orbital (HOMO) and HOMO-1 of the dimer.

The magnitude of the split in energy is a result of the energy level of the HOMO of each of the Helium atoms which we will call the site energy. The second contribution arises from the interaction of the two HOMO levels. It is this interaction that we call the transfer integral. In this simple case because both atoms are identical, they will have the same site energies. This means that the split between the HOMO and HOMO-1 is purely a result of the transfer integral. As powerful as this technique is for calculating the transfer integral its limitations are very apparent.

Say you have two molecules that are not chemically and spatially symmetric, the dimer splitting method in this case cannot resolve the correct contributions from the site energies and transfer integrals as you cannot assume the site energies and or the transfer integrals will be equal. Furthermore, note that we have assumed that the HOMO and HOMO-1 energy split is purely a result of the interaction of the HOMO orbitals of the two isolated Helium atoms. This is also a limitation. For instance, if you start with an isolated molecule which has a HOMO and HOMO-1 that are close in energy, when the molecule forms a dimer both orbitals of the HOMO and HOMO-1 may interact it will then be difficult to correctly resolve which contributions to the Pauli splitting arise from which transfer integrals.

Accounting for Polarization

References

  1. J. Kirkpatrick, “An approximate method for calculating transfer integrals based on the ZINDO Hamiltonian,” Int. J. Quantum Chem., vol. 108, no. 1, pp. 51–56, 2008.
  2. E. F. Valeev, V. Coropceanu, D. a da Silva Filho, S. Salman, and J.-L. Brédas, “Effect of electronic polarization on charge-transport parameters in molecular organic semiconductors.,” J. Am. Chem. Soc., vol. 128, no. 30, pp. 9882–6, Aug. 2006.
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