Thursday, February 04, 2016

What is density functional theory? part 2 - approximations

So, DFT contains a deep truth:   Somehow just the electronic density as a function of position within a system in its lowest energy state contains, latent within it, basically all of the information about that ground state.  This is the case even though you usually think that you should need to know the actual complex electronic wavefunction \(\Psi(\mathbf{r})\), and the density (\(\Psi^{*}\Psi\)) seems to throw away a bunch of information.

Moreover, thanks to Kohn and Sham, there is actually a procedure that lets you calculate things using a formalism where you can ignore electron-electron interactions and, in principle, get arbitrarily close to the real (including interaction corrections) density.  In practice, life is not so easy.  We don't actually know how to write down a readily computable form of the complete Kohn-Sham functional.  Some people have very clever ideas about trying to finesse this, but it's hard, especially since the true functional is actually nonlocal - it somehow depends on correlations between the density (and its spatial derivatives) at different positions.   In our seating chart analogy, we know that there's a procedure for finding the true optimal seating even without worrying about the interactions between people, but we don't know how to write it down nicely.  The correct procedure involves looking at whether each seat is empty or full, whether its neighboring seats are occupied, and even potentially the coincident occupation of groups of seats - this is what I mean by nonlocal.

Fig. from here.
We could try a simplifying local approximation, where we only care about whether a given chair is empty or full.  (If you try to approximate using a functional that depends only on the local density, you are doing LDA (the local density approximation)).  We could try to be a bit more sophisticated, and worry about whether a chair is occupied and how much the occupancy varies in different directions.  (If you try to incorporate the local density and its gradient, you are doing GGA (the generalized gradient approximation)).   There are other, more complicated procedures that add in additional nonlocal bits - if done properly, this is rigorous.  The real art in this business is understanding which approximations are best in which regimes, and how to compute things efficiently.

So how good can this be?  An example is shown in the figure (from a summer school talk by my friend Leeor Kronik).  The yellow points indicate (on both axes) the experimental values of the ionization energies for the various organic molecules shown.  The other symbols show different calculated ionization energies plotted vs. the experimental values.  A particular mathematical procedure with a clear theoretical justification (read the talk for details) that mixes in long-range and short-range contributions gives the points labeled with asterisks, which show very good agreement with the experiments.

Next time:  The conclusion, with pitfalls, perils, and general abuses of DFT.

1 comment:

Nanoday said...

I am not that great in math but thanks to you I didnt have any problem understanding your formula and a huge thanks.