AmericanChemicalSociety.com

The PRF grant has made possible a solid and ground breaking advance in electronic structure theory.

Together with co-authors Brian Landry and Adam Wasserman, a new and promising way to estimate the energies of systems of many electrons, including in molecules and quantum dots, has been constructed. The method builds on the venerable Thomas Fermi approach, which long preceded the currently wildly popular density functional theory.  These methods, including our ''Correlated Thomas Fermi'' approach to be described below, attempt to treat the electrons in an approximate and manageable way. It is acknowledged that anything close to an exact treatment is out of the question.

Density functional theory uses a fundamental theorem due to Hohenberg and Kohn,  relating the energy of a system as an exact functional of the density. The trouble is, the exact functional or even a close approximation is not known, and it is extremely probable that if it were known, it would be nearly infinitely difficult to apply. The original density functional theory, namely Thomas Fermi theory, was much less rigorous. It is a mean field theory, which means that it averages over the electron electron repulsion, such that each electron sees only and averaged field of the other electrons. As modern density functional theory is currently used, the mean field is effectively the case also.

In contrast, our method uses the exact electron electron repulsion, not mean field, together with semiclassical techniques. It has been shown  long ago that the Thomas Fermi theory is essentially semiclassical as well, but it makes the above-mentioned compromise of treating the electrons in a mean field way. By attempting to treat the electron electron interactions specifically, we hoped to arrive at a much more accurate approach. The idea is to associate the classical density of states with the quantum density, including specific interactions. This has indeed proven to be possible and accurate, with one significant caveat: the classical density of states knows nothing of particle symmetry or the fact that electrons are fermions. Therefore our raw results give us the classical density of states, not the fermion density of states. We need somehow to correct for this. In our published work based on this grant, methods were suggested for doing this, but much more progress has been made in the last few months, as reported in Brian Landry's thesis, and soon to be submitted for publication. 

The Correlated Thomas Fermi approach leads to a new question:  What state is it that is the first totally antisymmetric Fermionic state? There are states of all permutation symmeties, i.e. bosonic, ''anyonic,'' and fermionic, but the first will be bosonic.  But which one (e.g. # 27?) will be fermionic?  This is a new question in quantum chemistry, but if we have the answer we are rewarded with an accurate electronic energy. The published work consists mainly of a proof of principle, wherein this key question, namely, which is the first totally antisymmetric fermion state among the list of states starting with the bosonic, was answered separately. As a check on the method, once this information is put in, we found that the energies  obtained from our ``correlated`` Thomas Fermi theory are indeed quite accurate, especially in the strongly interacting regime, exactly where mean field theories fail miserably.  

Recent progress by Brian Landry, supported by the ACS-PRF grant, has shown that some approximations to the Fermion ordering problem are possible, by extrapolation starting from the weak coupling and strong coupling limits, which already works quite well in certain circumstances.  But further progress along these lines is needed although also quite likely, possibly putting the correlated Thomas Fermi theory in a position to compete with the popular density functional techniques.