From m10!frisch@uunet.UU.NET  Wed Jan 13 11:52:12 1993
Message-Id: <9301132205.AA14390@relay1.UU.NET>
Date: Wed, 13 Jan 93 16:52:12 EST
From: m10!frisch@uunet.UU.NET (Michael Frisch)
Subject: Re: Cartesian vs Internal coordinates
To: chemistry@ccl.net


    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

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.

In a practical calculation, there are two major sources of differing
answers:

1.  Unless you do analytic second derivatives at every point in your
    optimization (e.g., OPT=CALCALL in Gaussian) then the optimization
    algorithm will start with a guess for the Hessian and update it based
    on the forces at each point during the optimization.  Whether the
    initial guess for the Hessian is different for different coordinate
    systems but the same initial structure is program-dependant.  (It
    differed in Gaussian 88 and earlier, but not in Gaussian 90 and 92.
    I can't comment on other packages.)  Even if the starting Hessians
    are effectively the same for the two coordinate systems (e.g. are
    produced from cartesian force constants determined from the initial
    structure and converted to each of the two coordinate systems) the
    Hessians will be different later in the optimization, after being
    updated using any of the popular update methods.

2.  The optimization is stopped when some convergence criteria are reached.
    These typically require that both the forces and estimated next step
    size are sufficiently small.  Since the next step depends on the current
    approximate Hessian, a point might be considered close enough with the
    approximate Hessian produced by one set of coordinates and update method
    and not with either different coordinates or update method.  (This again
    is not a problem if analytic second derivatives are used at every point).
    Also, if the test on whether the forces are small enough involves
    quantities such as the maximum and/or RMS values, these depend on the
    coordinate system, and so one point might have small enough forces
    to be considered optimized in one coordinate system but not another.
    (The resulting structures from the two optimization should be pretty
    close if the convergence criteria are reasonable, but coordinates
    might differ by a couple of times the convergence criterion if the
    two approximate minima located are on opposite sides of the exact
    minimum.  Gaussian has an option Opt=Tight to make sure things are
    adequately converged for very flat surfaces, on which this problem
    is more likely to occur.)

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
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