Non-Convergence of the XC functional in the sense of a Taylor series

Question

When it comes to improving the accuracy of DFT calculations, there is a well known hierarchy:

• starting from LDA using $E_{xc}[n]$,
• proceeding with GGA and its $E_{xc}[ n, \nabla n]$ dependency
• and finally meta-GGAs with $E_{xc}[ n, \nabla n, \nabla^2n~\text{or}~\nabla^2\phi_i]$

I always wondered why higher order derivatives are not used but never really looked into it. Recently I heard from an expert in the field that it can be shown that $E_{xc}[n]$ can not converge in the sense of a Taylor series.

How? Why?

I just couldn't find more on this matter.

Answer 1

I agree with the comment by @jheindel. One issue is that the derivatives in GGA and meta-GGA functionals are not used like a Taylor series. The derivatives are used to obey more known constraints and to provide hooks to parameterize on.

I'll now try to address the spirit of your question, supposing that the derivatives were used to construct a Taylor series. The Taylor series is used to approximate a function, the energy functional, in this case. As you know, the approximation improves with higher orders of terms in the series (assuming the Taylor series converges). However, that does not mean that the function that is being approximated is actually correct.

Concretely: constructing a Taylor expansion for the exchange-correlation functional of a homogenous electron gas would simply yield better and better approximations of the energy functional for a homogeneous electron gas, which is still incorrect. That is, $E_{xc}^{\mathrm{HEG}}[n^\mathrm{exact}] \neq E_{xc}^\mathrm{exact}$ except for the case of the homogenous electron gas.

Answer 2

It's somewhat ambiguous to talk about approximating the XC energy as a Taylor expansion. A Taylor expansion is a series involving a fixed number of variables, and contains successively higher (non-negative integer) powers of the variables. But the hierarchy that you mentioned involves an increasing number of variables, where each variable is a successively higher order derivative of the density, and the density derivatives can enter the expression in arbitrary forms (fractional powers, exponentials, logarithms, trigonometric functions, ...). The latter is obviously quite different from a Taylor expansion.

In fact, if the XC energy is not restricted to a single spatial integral, but is allowed to take the following double integral form

$$ \begin{align} E_{xc} = \int f\left(\int g(\mathbf{r},\mathbf{r'},n(\mathbf{r})) d\mathbf{r}\right)d\mathbf{r'} \tag{1} \end{align} $$

then the exact XC energy is recovered already at the LDA level (!). The reason is simple: express both integrals as infinite sums, so that the formula resembles a single-layer neural network with an infinite number of neurons, where $f$ is the activation function. Then invoke the Universal Approximation Theorem. QED.

Of course, most of the time we are talking about the single integral form

$$ \begin{align} E_{xc} = \int \epsilon(n(\mathbf{r}),\nabla n(\mathbf{r}),\ldots) d\mathbf{r} \tag{2} \end{align} $$

(give or take a factor $n(\mathbf{r})$ in front of $\epsilon$, depending on the convention). While I'm not sure if the result will not converge at all when the maximum order of gradient approaches infinity, it's easy to construct examples where the result converges to a wrong value. Consider two infinite potential wells, each containing a helium atom. Now we place these two wells at a non-zero but finite distance with each other, and ask what will be the interaction energy of the two helium atoms, before the electronic densities have time to change due to the interaction. A semilocal functional, which only involves the density and its (arbitrary order) derivatives, will yield a zero interaction energy, since the density and its derivatives at an arbitrary point in one of the wells are not altered by the mere act of placing the two wells together. However, the real interaction energy is negative, due to the dispersion interaction between the helium atoms. This proves that semilocal functionals can never be exact, and when we push the accuracy sufficiently far, non-local terms like VV10 correlation must be included.