Cytoclonal Pharmaceutics Inc.
What are Nonlinear Systems
The term "nonlinear" is probably one of the most over used terms inmathematics. In the case of chaos theory, a nonlinear equation is one of the formx = f(x)Computational chemists should recognize this as exactly what a self consistent field (SCF) calculation is. In the SCF method, an initial set of orbitals is used to generate a new set of orbitals and the procedure is repeated until some convergence criteria is met.
The branch of mathematics called "chaos theory" which has been created and explored over the last decade is the study of equations andsystems of equations of this type. To some extent the mathematicians aretacking new names onto behavior that computational chemists have been seeing for years, but they are also learning more about why these systemsbehave the way they do. It is hoped that, in time, the work being doneby mathematicians will provide the insight necessary to create betterSCF convergence methods.
The purpose of this document is two fold. First it attempts tomake the connections between the field of chaos theory and SCF calculations.Second it gives suggestions on how to fix calculations with convergenceproblems.
What Nonlinear Systems Do
If one chooses a nonlinear equation, picks a starting value forthe nonlinear variable then iterates there are several possible outcomes.- After a number of iterations, the value returned by the equation may be the value that was put in on that iteration. This is what chemists desire - a converged solution.
- The values could be almost repeating but not quite. These are calledLorenz attractor systems in chaos theory.
- The values produced from one iteration to the next may oscillate between2 values, 4 values or any other power of 2. (I am not aware of any examplesother than powers of 2.)
- The values produced may be random within some fixed range. Random numbergenerators use this property intentionally, but it is rather annoying whenan SCF calculation does so.
- The values produced may be random and not bounded within any upper or lowerlimits.
It stands to reason that the total energy may follow any of these patterns during SCF iterations. We have encountered oscillatingand random behavior in the convergence of open shell transition metalcompounds but have never tried to determine if the random values werebounded or not.
How to Control Nonlinear Behavior
Changing the constants in a non-linear equation would be roughly equivalent to switching the basis set used for an SCF calculation. Since a particular basis set is often chosen for a desired accuracy and speed, this is not generally the most practical solution to a convergence problem. Plots of behavior vs constant values are the bifurcation diagrams which are found in many explanations of chaos theory.Another way of changing the constants in an SCF calculation isto change the geometry a bit. Often pulling a bond length a bit shorterthan expected is effective (say making the length 90% of the expectedvalue). lengthening bond lengths a bit and avoiding eclipsed or gaucheconformations are second and third best. Once you have a converged wave function move the geometry back where you want it and use theconverged function as the initial guess.
Changing the initial value of variables is equivalent to usinga different initial guess in an SCF calculation. The best initial guess is usually a converged SCF calculation for a different state of thesame molecule or a slightly different geometry of the same molecule. This can be a very effective way to circumvent convergence problems. In the worst case it may be necessary to construct an initial guess by hand in order to ensure that the nodal properties of all of the orbitals are correct for the desired electronic state of the molecule. The construction of the virtual orbitals as well as the occupied orbitals can have a significant effect on convergence. Chaos theorists will try many starting points and color code a plot by which solution is obtained for each starting point, this is called fractal geometry (i.e. the famous Mandelbrot diagram).
There are quite a number of ways to effectively change the equation in an SCF calculation. These include switching computationmethods, using level shifting and using forced convergence methods.
Switching between Hartree-Fock (HF), semiempirical, generalized valence bond (GVB), multi-configuration self consistent field (MCSCF), complete active space self consistent field (CASSCF) and Moller-Plesset calculations (MPn) will change the convergence properties.Configuration interaction (CI) and coupled cluster (CC) calculations usually start with an SCF calculation, thus will not circumvent problemswith an SCF. In general higher levels of theory tend to be harder to converge. Ease of convergence as well as calculation speed are why lower level calculations are usually used to generate the initial guess for higher level calculations.
Oscillating convergence in an SCF calculation is usually an oscillation between wave functions that are close to different statesor a mixing of states. Thus oscillating convergence can often be helpedby level shifting, which artificially raises the energies of thevirtual orbitals. Level shifting may or may not help in cases ofrandom convergence.
Most programs will stop trying to converge a problem after a certain number of iterations. In a few rare cases the wave function will converge if given more than the default number of iterations.
Most SCF programs do not actually compute orbitals from the previous iteration orbitals in the way that is described in introductorydescriptions of the SCF method. Most programs use a convergence excelleration method which is designed to reduce the number of iterationsnecessary to converge to a solution. The method of choice is usuallyPulay's DIIS method (Direct Inversion of the Iterative Subspace).Some programs also give the user the capability to modify the DIIS method,such as putting in a dampening factor. These modifications can beuseful for fixing convergence problems but they usually require asignificant amount of experience to know how best to modify the procedure.Turning off the DIIS extrapolation can help a calculation converge, butusually requires more iterations.
Some programs contain alternative convergence methods that aredesigned to force even the most difficult problems to converge. Thesemethods are often called direct covergence or quadratic convergencemethods. While these methods almost always work, they often requirea very large number of iterations and thus a very large amount of CPU time.
What to Try First
If you have an SCF calculation that failed to converge, which of these tricks should you try first? Here are my suggestions with number one being the first thing I try and etc.- Try a different initial guess (using the "guess" keyword in Gaussian).For an open shell system, try converging the closed shell ion of the same molecule then using that as an initial guess for the open shellcalculation. Adding electrons may give more reasonable virtual orbitalsbut as a general rule of thumb cations are easier to converge than anions.
- Another initial guess method is to first run the calculation witha small basis set then get the initial guess for a larger basis setcalculation from the previous wave function.
- Try level shifting ( "SCF=Vshift" in Gaussian).
- If the SCF is approaching, but not reaching, the convergence criteria relax or ignore the convergence criteria. This is usually done forgeometry optimizations which do not converge at the initial geometry thenoften converge as they approach the equilibrium geometry.
- Some programs use reduced accuracy integrals to speed the SCF. Usingfull integral accuracy may be necessary for systems with diffuse functions,long range interactions or low energy excited states.
- Try changing the geometry. First slightly shortening a bond lengththen slightly lengthening a bond length then shifting the conformationa bit.
- Consider trying a different basis set.
- Consider doing the calculation at a different level of theory. Thisisn't always practical, but beyond this point the increased number ofiterations may make the computation time as much as using a higher levelof theory anyway.
- Turn off the DIIS extrapolation ( "SCF=NoDIIS" in Gaussian ). You should probably give the calculation more iterations along with this.
- Give the calculation more SCF iterations ( "SCF(MaxCyc=N)" where N isthe number of iterations in Gaussian ). This seldom helps but the nextoption often uses so many iterations that it's worth a try.
- Use a forced convergence method. (In Gaussian "SCF=QC" is usuallythe best but on rare occasions "SCF=DM" will be faster). Don't forgetto give the calculation an extra thousand iterations or so. The wavefunction obtained by these methods should be tested to make sureit is a minimum and not just a stable point ( see the "stable" keywordin Gaussian ).
- See if the software documentation suggests any other ways to changethe DIIS method. You may well have to run hundreds of calculations toget enough experience with this to know what works when and how muchto change it by.
Further Information
The manuals accompanying many software packages contain discussions ofhow to handle convergence difficulties.There is a very small discussion of handling convergence problems in
T. Clark "A Handbook of Computational Chemistry" Wiley-Interscience (1985)
A good introduction to chaos theory is
J. Gleick "Chaos: Making a New Science" Viking (1987)
A more mathematical treatment of chaos theory is
S. H. Strogatz "Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry and Engineering" Addison Wesley (1994)
An expanded version of this article will be published in"Computational Chemistry: A Practical Guide for Applying Techniquesto Real World Problems" by David Young, which will be available fromJohn Wiley & Sons in the spring of 2001.
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