—Engineers tend to prefer a “by the numbers”
methodology that is founded on “linear logic”—i.e., one consisting of welldefined
steps that can be spelled out in advance. (This approach also appeals to
program managers, who want to design their programs in terms of milestones.)
—Scientists
are better able to cope with a “research” approach, which involves unknown
elements, where each step of the process depends upon the results of
previous steps, and where the methods often have to be invented as one
proceeds. (Unfortunately, program managers generally don’t like this
approach.)
The
engineering approach is appropriate and useful when a reliable methodology
actually exists, especially when one is doing more or less the same
thing over and over—like building a bridge, for example. That is not the
case in EOS modeling. Yes, there are some routine tasks that can be
handled in this way. But research is essential for a really good EOS (assuming
one actually cares enough to put in the required effort).
A
person attempting to approach EOS modeling from an engineering viewpoint
might envision the following steps: 1—identify what input parameters and
other data are needed to “calibrate” the model; 2—obtain data from existing
databases; 3—carry out new experiments, where necessary; 4—construct an
input file for the modeling code; 5—let the code compute and output the
EOS. This simplistic scenario could be made more sophisticated by allowing
for decision trees, iterations, etc., but I think the basic idea is clear.
This
scenario may sound reasonable, but it has many limitations and weaknesses.
EOS modeling is not just a matter of fitting data. Even if it were, it
would be impossible to obtain all the data that would be required; many
regions are inaccessible to experiment, and many quantities cannot be
measured experimentally. A good model explicitly treats the chemical and
physical phenomena that govern the material behavior, and an understanding
of those phenomena cannot be directly obtained from experiments. Existing
models also have limitations; they sometimes need to be modified or even
replaced.
In
short, a good EOS can only be created when the modeler devotes time and
energy into research. He/she must study and learn something about
the material to be modeled, explore different ideas about phase transitions
and changes in chemical structure, try various options to see what gives
the best results, improve on and even invent theories. This process cannot
be carried out “by the numbers.”
DFT—The Gold
Standard?
Numerical
calculations are all the rage in EOS modeling at the present time. Many
people view them as the cutting edge of EOS work, a kind of “gold
standard,” against which all other EOS models are to be measured. Some even
appear to believe that my approach to EOS modeling will ultimately become
obsolete.
I will
conclude this essay by explaining how I disagree with others on this issue,
why I think these numerical calculations are not as rigorous and
trustworthy as they are perceived to be.
I will
confine my remarks to DFT/MD, the most popular numerical method at the
present time. It uses density functional theory (DFT) to calculate the
electronic structure and free energy of a system (nuclei + electrons) as a
function of the nuclear coordinates. This free energy function is then used
as the potential energy surface for the motions of the nuclei, which are
calculated using molecular dynamics (MD). This method is sometimes called
“quantum molecular dynamics” (QMD). That term is more compact (and sounds more
classy) than DFT/MD, but it is misleading, because MD has no QM
corrections. (However, some DFT calculations use lattice dynamics instead
of MD.)
A
misconception, commonly found in the literature, is that the density
functional is the main source of error in these calculations. Efforts to
improve the calculations typically focus on the density functional.
Numerical issues and corrections to classical MD are also acknowledged as
sources of error. However, there are other approximations that should also
be considered and are virtually always ignored.
Single
Potential Approximation
As
noted above, the DFT/MD method assumes a single potential surface for the
nuclear motion. This approximation is absolutely essential to the viability
of the method. An exact treatment of the problem, even when the
BornOppenheimer approximation is applied, would employ a separate
potential surface and manifold of nuclear wave functions for every
electronic configuration of the system—a completely intractable
calculation, even with the fastest modern computers.
Where
does this approximation come from, and how is it justified? It is usually attributed to a 1965 paper by David Mermin; but Mermin was
not concerned with justifying this approximation, only with applying the
DFT theorem to the electronic free energy. The
justification can be traced to a 1957 paper by Robert Zwanzig,
who showed it to be the leading term in a high temperature expansion;
he took the classical limit of the nuclear motions while retaining the QM
treatment of the electrons. Zwanzig also derived
a firstorder correction term; but this correction is never even mentioned
in DFT papers, let alone tried out.
It
should be noted that “high temperature”—a qualitative term at best—does not
guarantee the accuracy of the single potential approximation. Quantum
corrections are likely to be important for the rotational and vibrational
degrees of freedom, when molecules are present. Even more important: the
nature of the electronic potential surface and the nuclear motions will
depend markedly upon the state of dissociation in a hot fluid. Therefore,
corrections to the single potential surface should be most important for
systems undergoing changes in chemical structure with temperature and/or
pressure.
Treatment
of Localized States
Another
problem with DFT is that it uses wave functions and statistical formulas
that do not give correct results for localized states. I have discussed
this issue elsewhere, and it is too complicated to discuss in detail in
this essay. So I will only make some brief remarks.
The
problem with the wave functions can be illustrated by a simple case: the
molecular orbital treatment of the H_{2} molecule does not give
correct results for the separated atoms. The problem arises from the single
determinant wave function. This problem cannot be corrected by modifying
the density functional or by using the unrestricted KohnSham
approximation. (See the book by Koch and Holthausen for further
discussion.)
The
problem with the statistical formulas can be illustrated by a collection of
isolated H atoms. It is easy to show that there are 2^{N} possible
configurations for the ground state of this system. The standard treatment,
in which band orbitals are constructed as linear combinations of the atomic
orbitals, gives 4^{N} configurations, making the entropy off by a
factor of 2. (And the error becomes even larger for many other elements.)
That’s
all I wanted to say. I’m going to stop here and say goodbye and best wishes.
