We know that there is convergence issue of DFT method on study real molecular system. It would be conducive if people could pre-determine if the SCF procedure is converge analytically before starting the calculation.
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2$\begingroup$ +1. But maybe a duplicate of this: materials.stackexchange.com/a/837/5 ? $\endgroup$– Nike DattaniMay 28, 2020 at 20:16
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1$\begingroup$ I don’t think the question is the same. The criteria is an artificial threshold but here my question is that if SCF procedure could be converged in principle. $\endgroup$– Paulie BaoMay 28, 2020 at 20:19
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2$\begingroup$ The other question is asking about what to check for to determine convergence (energy change, or density change), whereas you're asking if there's something that can be looked at before doing the SCF, that will tell you whether or not the SCF will converge. Again I worry that there's no answer, but we can see what people say. This question is very related: materials.stackexchange.com/questions/495/… and it's by the same person that answered the question linked in my first comment. $\endgroup$– Nike DattaniMay 28, 2020 at 20:23
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1$\begingroup$ Well the AO initial guess does not tend to work for big basis sets. For big basis sets, you are often better off to converge with a smaller basis set, then use the SCF density from the small basis set as an initial guess for the bigger basis set. $\endgroup$– Nike DattaniMay 28, 2020 at 20:45
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1$\begingroup$ I don't have a good answer myself but maybe this would be worth a look if you haven't read it already: arxiv.org/abs/1302.6022 $\endgroup$– Kevin J. MayMay 29, 2020 at 1:43
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2 Answers
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This question is a bit ill-defined: what do you mean by "the self-consistent field procedure"? If you mean the original Roothaan procedure, then the question makes sense, but it is uninteresting: nobody uses the Roothaan procedure, since it usually doesn't converge, and you need to do something smarter like use damping or other convergence acceleration schemes.
But, these are different methods, and now you would have to study each of them separately.
Still, it is possible to make any self-consistent field calculation converge simply by switching from iterative diagonalization to direct energy minimization. Here, you rewrite the problem in terms of iterative orbital rotations, and what you get is the minimization of a scalar function f(theta) in Cartesian space, which is a well-understood problem in numerical analysis. There are methods for minimization without gradients (e.g. the Nelder-Mead "amoeba" method), with gradients (e.g. steepest descent and conjugate gradients, and preconditioned versions thereof), and with Hessians (e.g. Newton-Raphson and trust region methods). These methods are proven to always converge to an extremum, and you'll just need to check whether you're at a local minimum or not just as if you use some kind of iterative diagonalization.
For details, you can refer to our recent open access overview paper: Molecules 2020, 25 (5), 1218.
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$\begingroup$ Thanks for providing me the review literature. It enriches my life especially during the quarantine. It is good to clarify different types of SCF procedures, but I think the "acceleration schemes" might not affect the condition of convergence. It can be analogous to the difference of thermodynamics and kinetics. e.g. spontaneous chemical reactions could have extreme slow rate. The "acceleration schemes" might affect the path (number of iterations to reach convergence) but I think there might be a fixed condition (like Gibbs free energy) to determine if SCF could converge. $\endgroup$ May 29, 2020 at 10:52
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1$\begingroup$ No, acceleration schemes definitely affect the condition of convergence, since they change the way the SCF is undertaken. $\endgroup$ May 30, 2020 at 6:53
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If by "the SCF method" you mean the simple SCF, the answer is: no, usually it does not converge (unless the problem is very simple; basically the gap is huge, so that the system does not respond very much to potentials). The damped SCF problem on the other hand does converge, for damping parameters small enough (unless you run into fractional occupations). The trouble is that it's not trivial to find out a priori what is a good damping parameter, so you have to couple that with a globalization strategy (like line searches). A good review paper is https://arxiv.org/abs/1905.02332, and for technical aspects about SCF/DM convergence I will shamelessly cite a recent preprint of mine: https://arxiv.org/abs/2004.09088
