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