Subject: Conservation of Difficulty

From: <[email-masked]>
Date: Tue, 5 Mar 1996 12:48:29 -0700 (MST)

Hi everybody,

a couple of years ago, one of my theoretical physics
professors cited the "Law of Conservation of Difficulty".
I thought I should share this law with the computational
chemistry community on the net.

As the name suggests, the law states that the difficulty 
of a problem is conserved, no matter how you reformulate it. 
I will demonstrate this by a few examples.

Take Density functional theory (DFT). We start off with the 
terribly complicated Schroedinger equation -- for the electrons
in a molecule, say. We reformulate it in a very clever
way to obtain DFT. Now what have we got? Indeed, we have a
beautiful formulation of the same problem: the basic variable
is the density, an observable that depends on three coordinates,
rather than 3*N as is the case with the wave function.
Thus, the problem has been simplified considerably.
However, all the difficulty comes back in the exchange-correlation
functional. (Remember that its functional form is unknown).

Another example is given by Molecular Mechanics. Again, the
very difficult problem of the time-dependent Schroedinger equation
is reformulated as simple classical equations of motion.
However, the difficulty is conserved. In this case, it pops up
in the necessity to obtain reliable force fields.

I suppose I have to modify the law somewhat since it is certainly
possible to make life MORE complicated (by doing lots of stupid
things). Maybe the "difficulty" is an entropy-like property?

Does anybody want to comment on the above?
If so, then I shall summarize to the net.  In particular, I
would like to get a reference ...

Yours, Georg

P.S.   Don't take me too serious on this one ...

==============================================================================
Georg Schreckenbach                      Tel: (Canada)-403-220 8204
Department of Chemistry                  FAX: (Canada)-403-289 9488
University of Calgary                    Email: [email-masked]
2500 University Drive N.W.,  Calgary,  Alberta,  Canada,  T2N 1N4
==============================================================================

---

From: <[email-masked]>
Date: Tue, 5 Mar 1996 21:04:24 -0500

> I will demonstrate this by a few examples.

	I will agree that there are very often trade offs in the
difficulty of methods.  However, I will not admit to any sort of 
law of conservation.  Let me give a counter example.

	Consider the solution of the Schrodinger equation for
the hydrogen atom.  If we did not know the exact solution we could
put incredible amounts of work into extremely accurate calculations
using DFT, GTO expansions, cubic splines, polynomials, etc.

	Now consider how difficult hydrogen atom calculations could
be even knowing the exact solution.  The logarithm and exponential
functions have proven so useful that they have been built into 
calculators and programming languages and reliably return results
to the precision of the machine.  If this were not the case, we would
have to spend quite a bit of time making sure that we are accurately
computing exponentials and logarithms, or we could have an enormous
database holding a table of logarithms ( 60 years ago every scientist and
engineer had a table of logarithms handy at all times ).

	Now extrapolate from our current state of theory.  If we knew
the exact analytic solution to the Schrodinger equation for molecules
most of what we do now would be trivial on the smallest PC.
This would still not prevent computational chemists from eating every
bit of computer power in sight as they tried to either deal with
relativistic effects or the transport of drugs through cell wall 
membranes.

	Although computers are incredibly powerful tools that I would 
not want to live without, they have also made us lazy.  In the past
such scientific problems would have either been shelved until the
mathematicians had come up with a technique for solving it or scientists
would have worked on it until a solution was found.  Although numerical
techniques allow us to jump past the development of mathematics,
sooner or later we will have to go back and get the original problem
right.  I firmly believe that the Schrodinger equation will one day
be as easily dealt with as trignometric functions on a calculator.

	Now ask your self two questions.

1.  How important would it be to work on an analytic solution to
the Schrodinger equation?

2.  What would be the chances that you could get funding for this 
project?

	I will let you draw your own conclusions.

                                Dave Young
                                [email-masked]

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     No assumption or approximation is reasonable for all cases.

 Corollary:

     Assumptions must be both rationalized and checked.
          But it is more important to check them.

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