Wednesday, September 21, 2011

The case for effective Hamiltonians

When trying to understand complex molecular materials, the dominant approach in chemistry to is to do DFT-based calculations for the system of interest. However, a case needs to be made for an alternative "physics" approach. Recent Anthony Jacko, Ben Powell and I wrote an article Models of organometallic complexes which makes the case below for effective Hamiltonians.

Another approach to modeling the optoelectronic properties of organometallic complexes is to construct a model with fewer states but an accurate treatment of the electronic correlations. This contrasts with first principles calculations, which include several basis states for each atom but neglect some electronic correlations. The small number of degrees of freedom in such semi-empirical models allows one to make fewer approximations on the interactions and correlations in the model. It also allows one to identify key trends that describe broad classes of materials. This approach has proven itself incredibly powerful in wide areas of materials science. For example, the Anderson single impurity model can describe a wide range of systems including magnetic impurities in metals, quantum dots in semiconductor heterostructures, carbon nanotubes , and single molecule transistors.47,48
In principal an effective model Hamiltonian is found by starting with the exact Hamiltonian and ‘integrating out’ high energy states.49 This procedure is computationally expensive,50 so often one simply chooses a reduced basis set, motivated by the physical processes one wishes to capture.49 DFT can be used to estimate the values (or trends in values) of some of the parameters of these effective models. The model Hamiltonian can then be solved, retaining correlations that the approximate DFT functional does not include.51–55
Identifying the frontier orbitals which dominate the photophysics is one of the most significant steps of the effective model approach. In this reduced basis set one can define an effective Hamiltonian with just a few parameters. Conjugated polyenes have been investigated in this way via the Hückel, Hubbard, Heisenberg and Pariser-Parr-Pople models.49 This approach has been applied to organometallic complexes, for example mixed valence binuclear systems including magnetic atoms in proteins (Hubbard and double exchange models; Ref. 56), molecular magnets (Ref. 57), Anderson impurity models for cobalt based valence tautomers (Ref. 58), and a series of metal-coredbipyridine complexes (Ref. 22). It has also been shown recently that this approach naturally explains the sensitivity of the photophysical properties of organometallic complexes to small chemical changes.18
To correctly describe the character of the excited states the model must capture the key interactions. There are many important features of the system that might be included in such a model, for example electronic ‘hopping’ terms between the frontier molecular orbitals, direct Coloumb interactions between electrons in those orbitals, spin interactions, and relativistic effects such as spin–orbit coupling. The relative energy scales of these various interactions will define the composition of the excited states and therefore their properties. 

I welcome comments and suggestions on any review articles that persuasively make the case for effective Hamiltonians as an important tool in materials modelling.

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