WHAT IS DENSITY FUNCTIONAL THEORY? - Density Functional Theory - page 29

with errors larger than 1 eV being common when comparing with experimen-
tal data. A subtle feature of this issue is that it has been shown that even the
formally exact Kohn-Sham exchange-correlation functional would suffer
from the same underlying problem. k
Another situation where DFT calculations give inaccurate results is associ-
ated with the weak van der Waals (vdW) attractions that exist between atoms
and molecules. To see that interactions like this exist, you only have to think
about a simple molecule like CH 4 (methane). Methane becomes a liquid at suf-
ficiently low temperatures and high enough pressures. The transportation of
methane over long distances is far more economical in this liquid form than
as a gas; this is the basis of the worldwide liquefied natural gas (LNG) industry.
But to become a liquid, some attractive interactions between pairs of CH 4 mol-
ecules must exist. The attractive interactions are the van der Waals interactions,
which, at the most fundamental level, occur because of correlations that exist
between temporary fluctuations in the electron density of one molecule and the
energy of the electrons in another molecule responding to these fluctuations.
This description already hints at the reason that describing these interactions
with DFT is challenging; van der Waals interactions are a direct result of
long range electron correlation. To accurately calculate the strength of these
interactions from quantum mechanics, it is necessary to use high-level
wave-function-based methods that treat electron correlation in a systematic
way. This has been done, for example, to calculate the very weak interactions
that exist between pairs of H 2 molecules, where it is known experimentally that
energy of two H 2 molecules in their most favored geometry is 0.003 eV
lower than the energy of the same molecules separated by a long distance. 8
There is one more fundamental limitation of DFT that is crucial to appreci-
ate, and it stems from the computational expense associated with solving the
mathematical problem posed by DFT. It is reasonable to say that calculations
that involve tens of atoms are now routine, calculations involving hundreds of
atoms are feasible but are considered challenging research-level problems, and
calculations involving a thousand or more atoms are possible but restricted to
a small group of people developing state-of-the-art codes and using some
of the world's largest computers. To keep this in a physical perspective, a
droplet of water 1 m m in radius contains on the order of 10 11 atoms. No con-
ceivable increase in computing technology or code efficiency will allow DFT
k Development of methods related to DFT that can treat this situation accurately is an active area
of research where considerable progress is being made. Two representative examples of this kind
of work are P. Rinke, A. Qteish, J. Neugebauer, and M. Scheffler, Exciting Prospects for Solids:
Exact Exchange Based Functional Meet Quasiparticle Energy Calculations, Phys. Stat. Sol. 245
(2008), 929, and J. Uddin, J. E. Peralta, and G. E. Scuseria, Density Functional Theory Study of
Bulk Platinum Monoxide, Phys. Rev. B , 71 (2005), 155112.