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The Schrödinger equation, the fundamental basis of wavefunction-based methods, cannot be solved in closed form per se. One can, however, solve it by using a finite number of closed-form basis sets (and the corresponding closed-form Slater determinants) and letting the basis set tend toward the complete basis set; this is similar to how most mathematical functions, that cannot be written in closed form per se, can be written as limits of sequences of closed-form functions.

It is currently unknown whether the exact exchange-correlation functional in DFT can be written in closed form or not. However, it should be able to be written as some limit of some sequence closed-form formulae, since the end result, i.e. the solution, is (vide supra) and the exchange-correlation functional can be written in a closed form expression when the end result is explicitly included as a term. (I'm a mathematics student, so I know what I'm talking about)

Now comes my main question- is my reasoning correct; i.e. is there any explicit series of closed-form "approximate DFT functionals' that mathematically converges to the exact potential?

UPDATE: After reading the answer that a sequence of neural networks can converge to the exact potential, which was not the type of answer that I wanted, I'm editing this post to re-ask a more intended question.

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There is one, although I'm afraid it is not what you want. The exchange energy is a continuous and non-singular functional of the density, so thanks to the universal approximation theorem, it can be approximated by a one-layer neural network that takes the density as input, to arbitrary precision. One therefore has a sequence of increasingly large neural networks that converge to the exact exchange functional. Taking the functional derivative of this sequence gives a series of better and better approximations to the exact exchange potential, and they converge pointwise to the exact exchange potential.

Likewise, one can prove that there is an explicit closed-form sequence that converges to the exact exchange-correlation potential, or the exact kinetic potential, etc.

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  • $\begingroup$ +1 So does that mean one day a neural network could potentially replace DFT codes, and provide close to exact wavefunctions? That sounds exciting! $\endgroup$
    – S R Maiti
    Jun 3, 2022 at 11:12
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    $\begingroup$ @SRMaiti Yes that's possible. But don't be too optimistic, as the convergence of this sequence of neural networks may be extremely slow, so that e.g. you have to use trillions of neurons to get the desired accuracy. This is related to the fact that calculating the wavefunction up to a given accuracy is NP-hard. $\endgroup$
    – wzkchem5
    Jun 3, 2022 at 16:04
  • $\begingroup$ If we assume the exact functional is smooth, it could be approximated with neural networks. Unfortunately, we still don't know the exact functional and we also have no way to find neural networks that give out systematic approximations to the exact solution. $\endgroup$
    – Susi Lehtola
    Jun 10, 2022 at 17:24
  • $\begingroup$ The universal approximation theorem only assumes continuity, not smoothness (although smoothness undoubtedly makes approximation easier). That's why I've taken the detour of approximating the energy functional and then taking the functional derivative, since the exact potential is not necessarily continuous w.r.t. changes of the density. As for your second sentence, we actually can systematically approximate the exact solution, by calculating the exact energy of more and more systems and then fit larger and larger neural networks against these data. $\endgroup$
    – wzkchem5
    Jun 10, 2022 at 20:02
  • $\begingroup$ Of course, the obvious problem is that one needs exponentially large training sets (and therefore neuron counts) to arrive at a certain accuracy. Besides, while the approach is systematic, it is not ab initio. $\endgroup$
    – wzkchem5
    Jun 10, 2022 at 20:10
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The exact exchange potential is not a problem. We know the exact exchange functional, and we also know how to build an optimized effective potential from it.

The problem is many-particle quantum mechanics, that is, correlation. The whole division into exchange vs correlation arises from the Hartree-Fock model: the model by definition captures all exchange, and the remaining bit is called correlation.

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A quick Google search led me to a very recent(2019) open-access article, readable here, that such a series/sequence does exist. Again, my being a mathematics student, I could understand the basic idea of the article- one can pick a set of some small molecules with some desired property, calculate the pair ((ab initio density, exact functional)) for each molecule, and use machine learning techniques to extrapolate the pairs into an approximate functional that is intended to be used in any molecule, of any size, posessing the aforementioned property.

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