Chemistry Reference
In-Depth Information
used an initial geometry that was chemically implausible. The details of how
this type of failure will unfold are dependent on the details of the optimization
algorithm, but this is something that can and will happen with almost
any optimization method if you use poor initial estimates for the geometries
of the atoms in which you are interested. The critical lesson here is that
expending effort to create good initial estimates for atomic coordinates will
greatly increase the speed of your DFT calculations and in many cases
make a decisive difference in whether your calculations can even converge
to an energy minimum.
As a second example of optimizing the positions of atoms within a super-
cell, we will optimize the geometry of a molecule of CO
2
. If we again use a
cubic supercell with side length
L
angstroms, we can create a CO
2
molecule
by placing a C atom at fractional coordinates (0,0,0) and O atoms at (
þd
/
L
,0,0) and (
2
d
L
,0,0). Optimizing the energy of the molecule from this initial
state with
d
1.3
˚
and the same stopping criterion that we used for the
N
2
calculations gives us an optimized C-O bond length of 1.17
˚
and an
O-C-O bond angle of 180
/
.
Our CO
2
results seem quite reasonable, but can we trust this result?
Remember that we defined our stopping criterion by the magnitude of the
force on the atoms. Let us examine the force on one of the O atoms for the con-
figuration we used in our calculations. If we write this force as
f ¼
(
f
x
,
f
y
,
f
z
),
then
by symmetry
,
f
y
¼ f
z
¼
0, regardless of what value of
d
we choose. This
means that as the geometry of the molecule is iteratively updated during energy
minimization, the C-O bond lengths will be varied but the O-C-O bond
angle will remain fixed at 180
o
, the value we defined in our initial configur-
ation. Saying this another way, the bond angle in our calculation is not
really a converged result, it is an inevitable result of the symmetry we imposed
in our original estimate for the molecule's structure.
A more reliable approach to optimizing the geometry of this molecule is to
choose an initial bond angle that does not make components of the forces
vanish by symmetry alone. We can easily do this by starting from a configur-
ation with a C atom at fractional coordinates (0,0,0) and O atoms at (
þa/L
,
b/L
,0) and (
2
a/L
,
b/L
,0). Minimizing the energy of the molecule starting
from this structure with
a
1.2
˚
and
b
0.1
˚
, a configuration with an
O-C-O bond angle of 107.5
8
, gives us a converged result with C-O bond
lengths of 1.17
˚
and a O-C-O bond angle of 179.82
8
. This geometry is
extremely close to the one we found starting from a linear molecule. We
should not expect the two geometries to be
exactly
the same; they were deter-
mined using iterative optimization methods that were halted once we were
“sufficiently close” to the exact solution as dictated by the stopping criterion
we used. It is quite reasonable to draw the conclusion from these calculations
that our DFT calculations have predicted that CO
2
is a linear molecule (i.e., it
8
