Chemistry Reference
In-Depth Information
with wave functions with exactly this kind of oscillation. From a physical point
of view, however, core electrons are not especially important in defining
chemical bonding and other physical characteristics of materials; these prop-
erties are dominated by the less tightly bound valence electrons. From the
earliest developments of plane-wave methods, it was clear that there could
be great advantages in calculations that approximated the properties of core
electrons in a way that could reduce the number of plane waves necessary in
a calculation.
The most important approach to reducing the computational burden due to
core electrons is to use pseudopotentials. Conceptually, a pseudopotential
replaces the electron density from a chosen set of core electrons with a
smoothed density chosen to match various important physical and mathemat-
ical properties of the true ion core. The properties of the core electrons are then
fixed in this approximate fashion in all subsequent calculations; this is the
frozen core approximation. Calculations that do not include a frozen core
are called all-electron calculations, and they are used much less widely than
frozen core methods. Ideally, a pseudopotential is developed by considering
an isolated atom of one element, but the resulting pseudopotential can then
be used reliably for calculations that place this atom in any chemical environ-
ment without further adjustment of the pseudopotential. This desirable prop-
erty is referred to as the transferability of the pseudopotential. Current DFT
codes typically provide a library of pseudopotentials that includes an entry
for each (or at least most) elements in the periodic table.
The details of a particular pseudopotential define a minimum energy cutoff
that should be used in calculations including atoms associated with that pseu-
dopotential. Pseudopotentials requiring high cutoff energies are said to be
hard, while more computationally efficient pseudopotentials with low cutoff
energies are soft. The most widely used method of defining pseudopotentials
is based on work by Vanderbilt; these are the ultrasoft pseudopotentials
(USPPs). As their name suggests, these pseudopotentials require substantially
lower cutoff energies than alternative approaches.
One disadvantage of using USPPs is that the construction of the pseudopo-
tential for each atom requires a number of empirical parameters to be specified.
Current DFT codes typically only include USPPs that have been carefully
developed and tested, but they do in some cases include multiple USPPs
with varying degrees of softness for some elements. Another frozen core
approach that avoids some of the disadvantages of USPPs is the projector
augmented-wave (PAW) method originally introduced by Bl¨chl and later
adapted for plane-wave calculations by Kresse and Joubert. Kresse and
Joubert performed an extensive comparison of USPP, PAW, and all-
electron calculations for small molecules and extended solids.
1
Their work
shows that well-constructed USPPs and the PAW method give results that
