Electronic Structure Methods
Electronic Structure Methods are the key tool in Theoretical Chemistry. They comprise approximations to the Schrödinger equation. The first approximation introduced is called the Born-Oppenheimer approximation. This neglects quantum effects of the nuclei of the system and treats them as fine charges. The resulting equation is called the electronic Schrödinger equation. Both a static and a time-dependent version are available. In the following approximations to the static electronic Schrödinger equation are presented.
Density Functional Theory
Density Functional Theory (DFT) was founded by Pierre Hohenberg and Walter Kohn[]. They were able to show, that there is an energy functional of the electron density and that it’s minimizer is the ground state density. The proof is a mere existence proof furnished with a variational principle.
Walter Kohn and Jiu Le Sham introduced a non-interacting particle system that shares the same density as the physical system. The advantage is an explicit expression for (parts of) the kinetic energy as well as a self-consistent scheme to find the minimum. At the heart of Kohn-Sham (KS) DFT is the so called exchange correlation (xc) functional.
It is widely believed [] that the next generation of xc functionals must be orbital dependent. A number of orbital dependent functionals have already been introduced. However, none of them have really been able to improve upon existing xc functionals.
Currently, I’m thinking about new orbital dependent functionals that will cure the deficiencies of present day xc functionals for dissociation curves as well as for weak interactions.
Optimized Effective Potential
The Optimized Effective Potential (OEP) method was introduced by Talman and Shadwick[]. In it’s original formulation it was used to obtain the best (minimizing) local potential for the Hartree-Fock (HF) energy expression.
In the context of orbital dependent functionals in DFT the OEP method has gained new interest. OEP is used to obtain the functional derivative of the orbital dependent xc functional with respect to the density.
It soon became clear that difficulties arise when OEP is applied with finite orbital basis sets[]. Different schemes to overcome the difficulties have been proposed[]. In my opinion the problems have not ultimately been resolved.
Recently, I was able to show extreme consequences for finite basis OEP. Together with Prof. Andreas Savin we were able to show that the full configuration interaction wave function is a formal solution to the OEP and the KS model.
I’m still interested in finding out to which extend it is possible to obtain the same solution for a local potential as with a non-local potential.
Density Matrix Functional Theory
Density Matrix Functional Theory (DMFT) is a fairly new theory. It is related to DFT but takes the density matrix as main argument. It was founded by Gilbert [], who proved that there is a functional of the one-particle reduced density matrix (1RDM) even for non-local external potentials. Like in DFT the proof is a mere existence proof furnished with a variational principle.
The main advantage of DMFT compared to DFT is the fractional nature of the occupation numbers. This allows a natural inclusion of correlation effects. Unfortunately, like in DFT, the form of the exact 1RDM functional remains unknown.
The most successful 1RDM functionals are called BBC1, BBC2 and BBC3. It was shown that these functionals excellently describe the dissociation curve of a few small test molecules. The most successful functional (BBC3), however, depends on user chosen parameters.
In collaboration with Prof. Baerends, Prof. Pernal and Dr. Gritsenko we developed an automated version of BBC3 called AC3. Just like its predecessors AC3 shows excellent performance for a few small test molecules.
Very recently, a new promising functional, called the power functional, was introduced. Fine tuning only one parameter, excellent results were obtained for the fundamental gap different kinds of semi-conductors.
Currently I’m working on extending the power functional to accurately describe bond breaking and bond forming processes. The new functional will be more efficient than BBC3 and AC3. I believe that the same accuracy can be achieved.
Range Separation of Electronic Interaction
Separation of the electronic interaction into a short-range interaction and a long-range interaction has been proposed by Stoll and Savin. The idea is to describe the short-range effects with a density functional and the long-range effects with wave function methods. The main advantage is that smaller basis sets are needed to obtain a certain accuracy.
Together with Prof. Katarzyna Pernal we developed two new theories. The theories allow us to use a density matrix functional for the long-range interaction. In the first theory we can use available short-range functionals. Together with a particularly simple density matrix functional we obtain excellent results for the dissociation curve of small molecules. These results are obtained at costs that are similar to state-of-the art density functionals.
The second theory to combine a short-range density functional with a long-range density matrix functional has not been tested, yet. At the moment we develop new functionals to test this theory.