From shepard@dirac.tcg.anl.gov Thu Jan 14 09:25:59 1993
Date: Thu, 14 Jan 93 15:25:59 CST
From: shepard@dirac.tcg.anl.gov (Ron Shepard)
Message-Id: <9301142125.AA02330@dirac.tcg.anl.gov>
To: chemistry@ccl.net
Subject: Re: Cartesian vs Internal coordinates
John Upham writes:
>>Dear All,
>> I have always assumed that given the same starting geometry any
>>program that has a choice as to how one enters it should (in the end)
>>produce the same optimised final geometry. Is this true or false ?
>>Without wishing to embarasses anyone I have at least one program
>>genearally available where different final geometries are obtained.
>>
>>Any comments (apart form use internal coordinates) ?
>>
>>john upham
>>
>>John Upham, Dept. of Chemistry, University of Reading, Berks., RG6 2AD, UK.
>>Email: scsupham%susssys1.rdg.ac.uk@uk.ac (BITnet), scsupham@rdg.susssys1 (Janet)
>>Voice: +44 734 875123 x7441 (day), Fax: +44 734 311610
Mike Frisch replies:
>If all calculations were done to infinite precision and all geometry
>optimizations were continued until the forces were exactly zero, and all
>optimizations used analytic first AND SECOND derivatives then optimizations
>starting from the same structure but using different coordinate systems
>would go to exactly the same place.
[text deleted]
>In fact, because of the differences in the Hessians, two optimizations with
>different coordinates started at the same point far from any minimum might
>fall down into different wells and to different minima. (This will not
>be the case if the two optimizations use analytic rather than approximated
>second derivatives at every point -- then every step should be the same
>regardless of coordinate system.)
>
>All of these considerations apply to comparing cartesian with internal
>coordinates and also to comparing two different sets of internal coordinates.
>
>Mike Frisch
I agree with Mike Frisch's comments regarding multiple minima.
Specifically, different optimization algorithms can lead to different
minima on a surface. This includes different hessian update
procedures in quasi-Newton approaches, different approximations to
derivatives (analytic vs. forward difference vs. central difference
vs. ...etc). However, I believe that this also applies to different
definitions of a coordinate system, even for analytic (i.e. exact)
derivatives.
Consider minimixation of the 1-dimensional function f(x)=half*x**2.
Newton-Raphson with analytic derivatives gives the update procedure
Delta_x = -x0
where x0 is the initial guess. This gives
xnew = x0 + Delta_x = 0
So that the minimum at x=0 is always reached in one iteration
regardless of the initial guess.
Now consider the "coordinate transformation"
y = exp(x)
This is a one-to-one onto mapping, so there are no problems with
multiply-defined angles or such. Now, define an F(y) such that
F(y)=f(x). That is, F(y) is the original function expressed in terms
of the new coordinate, y. The answser using the above transformation
is
F(y) = half * (ln(y))**2
The original function was minimized at x=0, whereas the new function
is minized at y=1. That is, f(0)=F(1)=0.
Again using analytic derivatives, the NR method in this new coordinate
system results in the update procedure
Delta_y = ln(y) / ( ln(y) - 1 )
= -x0 / ( 1 - x0 )
The last line gives the y update in terms of the original x coordinate
for comparison. Clearly, this does not converge in 1 iteration,
although it does converge quadratically in the neighborhood of the
minimum as all good NR methods do.
Note however that in the neighborhood of x=1 (or y=e) there will be
problems because of the denominator in the NR update expression in the
"y" coordinate system. Specifically, the NR method diverges near x=1
using the procedure in the y "coordinate", whereas it is globally
convergent, and converges in 1 iteration, when everything is done in
the "x" coordinate system.
If all of this is true for a simple 1-dimensional function, it is also
true for more complicated (3N-6)-dimensional energy surfaces. In
summary, convergence rates and convergence behavior depend on the
coordinate system.
-Ron Shepard
shepard@tcg.anl.gov