Chemistry Reference
In-Depth Information
We cannot continue this process indefinitely because in an
N
-dimensional space
we can onlymake a vector orthogonal to at most (
N
2
1) other vectors. So tomake
this a well-defined algorithm, we have to restart the process of defining search
directions after some number of iterations less than
N
.
The conjugate-gradient method is a powerful and robust algorithm. Using it
in a practical problem involves making choices such as how to select step
lengths during each iteration and how often to restart the process of orthogo-
nalizing search directions. Like the quasi-Newton method, it can be used to
minimize functions by evaluating only the function and its first derivatives.
3.3.3 What Do I Really Need to Know about Optimization?
We have covered a lot of ground about numerical optimization methods, and, if
you are unfamiliar with these methods, then it is important to distill all these
details into a few key observations. This is not just a mathematical tangent
because performing numerical optimization efficiently is central to getting
DFT calculations to run effectively. We highly recommend that you go back
and reread the seven-point summary of properties of numerical optimization
methods at the end of Section 3.3.1. You might think of these (with apologies
to Stephen Covey
2
) as the “seven habits of effective optimizers.” These points
apply to both one-dimensional and multidimensional optimization. We further
recommend that as you start to actually perform DFT calculations that you
actively look for the hallmarks of each of these seven ideas.
3.4 DFT TOTAL ENERGIES—AN ITERATIVE
OPTIMIZATION PROBLEM
The most basic type of DFT calculation is to compute the total energy of a
set of atoms at prescribed positions in space. We showed results from many
calculations of this type in Chapter 2 but have not said anything about how
they actually are performed. The aim of this section is to show that this kind
of calculation is in many respects just like the optimization problems we
discussed above.
As we discussed in Chapter 1, the main aim of a DFT calculation is to find
the electron density that corresponds to the ground-state configuration of the
system,
r
(
r
). The electron density is defined in terms of the solutions to the
Kohn-Sham equations,
c
j
(
r
), by
r
(
r
)
¼
X
j
c
j
(
r
)
c
j
(
r
)
:
(3
:
21)
