Posted on November 17, 2015 at 10:55 AM |

The most popular method in computational chemistry is density functional theory (DFT). For the past thirty years DFT has become the preferred method over *ab initio* calculations due to its cost-efficiency favorable ratio. Despite its success, the development of new DFT functionals is an arduous task because the physical interactions entering the Hamiltonian have to be expressed in terms of the electron density and, such expressions only exist for the external potential (e.g. the attractive electron-nucleus Coulombic potential of molecular systems). On the other hand, the density matrix functional theory (DMFT), which uses the first-order reduced density matrix (1-RDM), permits an easier construction of the Hamiltonian because the only energy component that needs to be approximated is the electron-electron repulsion (Vee). Namely, since the Hartree-Fock expression of Vee in terms of the 1-RDM is well known, only the correlation part of Vee is actually needed to construct a DMFT functional. It is thus only natural that DMFT functionals provide quite accurate energy predictions compared to DFT ones. The downside is that DMFT are computationally more expensive than DFT calculations. In the past years, there has just been a resurged interest in the DMFT functionals based on natural orbitals, in a framework known as the natural orbital functional theory (NOFT). The groups of Baerends, Lathiotakis, Csányi, Goedecker, Umrigar and Piris, among others, have been actively suggesting NOFT functionals.

Particularly interesting are the series of functionals developed by Mario Piris, PNOF1-PNOF6, who in the past ten years has put forward functionals that can deal with nondynamic correlation effects as efficiently as the well-stablished complete active space self-consistent field (CASSCF) method. Lately, his efforts focus on introducing larger dynamic correlation effects in his functionals. We have colaborated with Mario in the past, trying to find stringent conditions that DMFT functionals should fulfill or testing the limits of applicability of his latest functional, PNOF6.

Despite the promising research behind NOFT functionals, thus far there has been no attempt to calibrate the functionals available in the literature. This information is not only useful to reveal the limitations of current functionals but it is also important to provide a test set that can be used in order to improve the new NOFT functionals. In a work that has just been accepted in *J. Chem. Phys.*, we have used harmonium (a well-known friend from past investigations) to calibrate a set of 14 NOFT functionals together with Prof. Jerzy Cioslowski (who published relevant papers on this topic, see e.g. this and that) and Mario.

Our results reveal that most functionals performance poorly within different electron correlation regimes. Notwithstanding, the PNOF functionals perform better than most functionals, as the following plot of the correlation Vee energy in the singlet state of four-electron harmonium shows:

PNOF6 gives very accurate results in the weak- and moderate-correlation regimes and it is reasonable accurate in the strong correlation regime. In this sense, **PNOF6 can be regarded as the best performing functional of the series**, although our results show there is still a long way to go to achieve chemical accuracy for arbitrary correlation regimes.

The present approach not only uncovers the flaws and patent failures of the functionals but, even more importantly, allows for pinpointing their root causes. Since the approximate values of U are computed at exact 1-RDM, the **test requires minimal programming**, and thus is particularly useful in quick screening of new functionals. In conjuction with the previously described DMFT stringent conditions based on the local spin, these tools will be used to construct more robust natural orbital functionals.

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