Chemistry Reference
In-Depth Information
available electronic states along a series of lines in reciprocal space that
typically form a closed loop beginning and ending at the
G
point. Band struc-
ture diagrams can be routinely calculated from plane-wave DFT calculations.
Attention must be paid in calculations of this kind to the placement of
k
points,
as the electronic states must be evaluated as a series of
k
points spaced close
together along the special directions in reciprocal space relevant for the
band structure diagram.
8.2 LOCAL DENSITY OF STATES AND ATOMIC CHARGES
To interpret the electronic structure of a material, it is often useful to under-
stand what states are important in the vicinity of specific atoms. One standard
way to do this is to use the local density of states (LDOS), defined as the
number of electronic states at a specified energy weighted by the fraction of
the total electron density for those states that appears in a specified volume
around a nuclei. Typically, this volume is simply taken to be spherical; so to
calculate the LDOS we must specify the effective radii of each atom of interest.
This definition cannot be made unambiguously. If a radius that is too small is
used, information on electronic states that are genuinely associated with the
nuclei will be missed. If the radius is too large, on the other hand, the
LDOS will include contributions from other atoms.
Figures 8.7 and 8.8 show the local DOS for O and Si atoms in bulk quartz.
In these calculations, the radii of O and Si were set to 1.09 and 1.0
˚
, respect-
ively. The total DOS for this material was shown in Fig. 8.5. Each LDOS is
split into contributions from
s
and
p
bands in the electronic structure. It can
be seen that the band at the top of the valence band is dominated by
p
states
from O atoms, with a small contribution from Si
p
states. The band located
5-10 eV below the valence band edge is a mixture of O atom
p
states and
Si atom
s
and
p
states. The band furthest below the valence band edge is a
mixture of Si and O
s
states.
It is often convenient to think of atoms within bulk materials or molecules
as having net charges. In an ionic material such as NaCl, for example, it is
conventional to associate charges of
þ
1 and
2
1 (in units of electron
charge) to Na and Cl atoms. The ambiguity in defining the volumes used
for LDOS calculations illustrates why making this assignment from a calcu-
lated electron density is not necessarily a simple task. This is a longstanding
challenge in computational chemistry in general, not just for plane-wave
DFT calculations, and many different methods have been explored as possible
solutions. Many of the most widely used methods, such as Mulliken analysis
and the ChelpG method, rely on having a well-defined basis set of local func-
tions, so they are not typically available within plane-wave DFT calculations.
