Tuesday, June 23, 2015

A Practicable Real-Space Measure and Visualization of Static Electron-Correlation Effects

Grimme, S. and Hansen, A., Angew Chem Int Edit, (2015)
Contributed by Tobias Schwabe

The question of how to deal with multireference (MR) cases in DFT has a longstanding history. Of course, the exact functional would also include multireference effects (or non-dynamical/non-local/static electron correlaton, as these effects are also called) and no special care is needed. But when it comes to today's density functional approximations (DFAs) within the Kohn-Sham framework, everything is a little bit more complicated. For example, Baerends and co-workers have shown that is the exchange part in GGA-DFAs that actually accounts for static electron correlation.[1] These studies, among others, led to the conclusion that the (erroneous) electron self-interaction in DFAs accounts for some of the MR character in a system. A good review about how these things are interconnected can be found in Ref. [2].

Instead of searching for better and better DFAs, another approach to the problem is to apply ensemble DFT which introduces the free electron energy and also the concept of entropy into DFT.[3] The key concept here is to allow for fractional occupation numbers in Kohn-Sham orbitals and to look at the system at T > 0 K. In case of systems with MR character which cannot be described with a single Slater determinant fractional occupation will result (for e.g. when computing natural occupation numbers). The interesting thing about ensemble DFT is that it allows to find these numbers directly via a variational approach without computing an MR wavefunction first.

Grimme and Hansen now turned this approach into a tool for a qualitative analysis of molecular systems. They do so by plotting what they call the fractional orbital density (FOD). That is, only those molecular orbitals with non-integer occupation numbers contribute to the density – and only at a finite temperature. This density vanishes completely at T = 0 K. So, the analysis literally shows MR hot spots. Integrating the FOD yields also an absolute scalar which allows to quantify the MR character and to compare different molecules. Due to the authors, this value correlates well with other values which attempts to provide such information.

A great advantage of the approach is that now the MR character can be located (geometrically) within the molecule. The findings presented in the application part of the paper go along well with chemical intuition. The analysis might help to visualize and to interpret MR phenomenon. The tool can provide insight when the nature of the electronic structure is not obvious – for example, when dealing with biradicals in a singlet spin state. It might also be a good starting point to identify relevant regions/orbitals which should be included when one wants to treat a system on a higher level than DFT, for example with WFT-in-DFT based on projector techniques. Last but not least, it can help to identify chemical systems to which standard DFAs should not (or only with great care) be applied.

References:

[1] a) O. V. Grittsenko, P. R. T. Schipper, and E. J. Baerends, J. Chem. Phys. (1997), 107, 5007 b) P. R. T. Schipper, O. V. Grittsenko, and E. J. Baerends, Phys. Rev. A (1998), 57, 1729 c) P. R. T. Schipper, O. V. Grittsenko, and E. J. Baerends, J. Chem. Phys. (1999), 111, 4056

[2] A. J. Cohen, P. Mori-Sánchez, and W. Yang., Chem. Rev. (2012), 112, 289

[3] R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford University Press (1989)