# Accuracy trade-off between energy and density in DFT calculations

Consider the following situation that the exchange-correlation term (e.g., B3LYP) overestimates the energy in a system (i.e., a simpler one (e.g., LDA) is sufficient in the system). In this case, my question is that, will the obtained density be incorrect, that is, will the parameters in LCAO (i.e., orbital superposition) be incorrect even if the energy accuracy calculated by B3LYP is very good? In other words, does good accurate energy calculated using a complex energy functional always guarantee the correct density? I am wondering that an overestimated energy functional can output the accurate energy even if the LCAO parameters (i.e., coefficient in orbital superposition) are incorrect and this will lead to the incorrect density.

I also found the discussion and in the Science paper ("Density functional theory is straying from the path toward the exact functional" and its comment). Especially in recent years in machine learning research, very complicated deep learning models with over ten million parameters have been developed and successfully predicted the energy of various molecules (e.g., molecular graph message passing model and neural network potential) However, such deep models seem to completely ignore the density. I am wondering that such overparameterized models are good or not for both energy and density.

If a functional always gives the exact energy for arbitrary external potentials, then it is easy to see that the functional must always give the exact density. Consider two DFT calculations, one with an external potential $$V(\mathbf{r})$$, and another with the potential $$V(\mathbf{r}) + \epsilon\delta(\mathbf{r}_0)$$, where $$\epsilon$$ is infinitesimal. By perturbation theory, the energy difference between the two systems is $$-\epsilon\rho(\mathbf{r}_0) + O(\epsilon^2)$$. So if the functional gives exact energies, it must give exact $$\rho(\mathbf{r}_0)$$ at an arbitrary point $$\mathbf{r}_0$$.