Posted on March 12, 2018 at 8:45 AM |

The role of aromaticity in porphyrinoids is a current subject of debate due to the intricate structure of these macrocycles, which can adopt Hückel, Möbius and even figure-eight conformers. One of the main challenges in these large π-conjugated structures is identifying the most conjugated pathway because, among aromaticity descriptors, there are very few that can be applied coherently to this variety of conformers.

As a result of a joint colaboration with Miquel Torrent-Sucarrat, Mercedes Alonso and Julia Contreras-Garcia, we have recently published a paper on Phys. Chem. Chem. Phys. studying the most aromatic circuits in porphyrinoids. The main authors of the paper are Irene Casademont (PhD student at the UPV/EHU and the DIPC) and Tatiana Woller (PhD student at the Vrije Universiteit Brussel). In this work, we have found that two new electronic aromaticity indices AV1245 and AVmin developed in our group, provide a reliable description of the aromatic pathways in a series of nine porphyrinoids. Not many indices can be used to identify the most aromatic pathway in a macrocyclic (for instance, NICS which is probably the most popular index cannot be calculated for particular circuits) and in our study we have also used BLA, BOA, FLU and HOMA. All these indices agree on the general features of these compounds, such as the fulfillment of Hückel's rule or which compounds should be more or less aromatic from the series. However, only AVmin can identify the most aromatic circuit in all the molecules. Our results evince the difficulty of finding the most aromatic pathway in the macrocycle for large porphyrinoids.

We study the effect of the exchange in DFT functionals on the description of the aromaticity of the porphyrinoids. The amount of exact exchange quantitatively changes the picture for most aromaticity descriptors, AVmin being the only exception that shows the same qualitative results in all cases.

The paper has been published in Phys. Chem. Chem. Phys.

Posted on February 20, 2018 at 7:10 AM |

Tetrahalodiboranes are enigmatic compounds, which are used for the doping of silicon with B+ ions for semiconductor device fabrication. Despite their lability and immense reactivity potential, the relative absence of tetrahalodiboranes is due to their difficult preparation, involving gas-phase synthetic steps. Recently, the first transition-metal complex of a diboranyl dianion has been prepared and characterized by the group of Prof. Braunschweig.

Our joint computational analysis with Prof. Óscar Jimenez-Halla suggests the presence of two multicenter PtB2 3c-2e bonds, similar to the two B3 3c- 2e bonds seen in Himmel’s rhomboidal B4 compound and the B2I4 unit can be seen as an olefin analogue. The compound represents the first example of intact coordination of B2X4 (X = halide) unit of any type to a metal center. These results provide a glimpse of the potentially exciting coordination chemistry of tetrahalodiboranes, about which very little is currently known.

The full work has been published in Angew. Chemie Int. Ed.

Posted on November 11, 2017 at 10:05 AM |

The so-called van der Waals (vdW) interactions are one of the weakest forces in nature and yet they govern the stability of molecules and materials, having an essential role in molecular recognition, the stability of the double-helical structure of DNA and molecular adsorption processes on surfaces, amongst others. So far, only a universal relationship between the van der Waals energy and the distance between two atoms or molecules was known, being widely used to model these interactions in physics, chemistry, and biology.

In a recent work published in Phys. Rev. A., another **universal signature of van der Waals interactions** has been unveiled. This new fingerprint of weak molecular interactions is numerically more robust than the earlier condition on the energy and could thus provide a handy tool for the development of new methods to analyze the electronic structure of molecular systems. By means of perturbation theory, we have shown that the interelectronic part of the pair density, which is the workhorse of electronic structure methods and provides a distribution of the electron pairs in the space, decays as 1/R^3 for two molecular fragments separated by a distance R. The main author of this work is Mireia Via-Nadal (PhD student at the UPV/EHU and the DIPC), to which Mauricio Rodriguez-Mayorga has also contributed.

This result opens the door to **produce density-dependent non-covalent interaction corrections in density and density matrix functional theories** (DFT and DMFT). This possibility will be explored in our laboratory.

This work has been published in Phys. Rev. A.

Posted on October 3, 2017 at 12:30 PM |

The energy usually serves as a yardstick in assessing the performance of approximate methods in computational chemistry. After all, these methods are mostly used for the calculation of the electronic energy of chemical systems. However, computational methods should be also aimed at reproducing other properties, such strategy leading to more robust approximations with a wider range of applicability.

In this work, we have suggested a battery of ten tests with the aim to analyze density matrix functional approximations (DMFAs), including several properties that the exact functional should satisfy. The tests are performed on a two-electron model system with varying electron correlation, carrying a very small computational effort. Our results not only put forward a complete and exhaustive benchmark test for DMFAs, currently lacking, but also reveal serious deficiencies of existing approximations that lead to important clues in the construction of more robust DMFAs. The main author of this work is Mauricio Rodríguez-Mayorga who received the help of several co-authors: Eloy Ramos-Cordoba, Mireia Via-Nadal and Mario Piris.

This work has been published in Phys. Chem. Chem. Phys.

Posted on March 5, 2017 at 11:15 AM |

Last year we published the first of series a three papers dealing with the separation of dynamic (weak) and nondynamic (strong) correlation. In that work, we put forward a global correlation indicator based on natural orbital occupancies. Now, with Eloy Ramos-Cordoba we have extended this approach to account for **real-space weak and strong correlation**. By multiplying the orbital contributions to these correlation indicators by the corresponding natural orbitals, we produce a three-dimensional pictures of dynamic and nondynamic correlation.

In the picture above we produce CASSCF pictures of dynamic and nondynamic correlation for the ortho-, meta-, and para-benzine singlet diradicals, which have two unpaired electrons. These unpaired electrons are easily located in the space using the real-space nondynamic correlation indicator suggested in this work.

This work has been published in J. Chem. Theory Comput.

Posted on February 18, 2017 at 1:10 PM |

For a system of fermions subject to one and two-particle forces the exact energy can be completely expressed in terms of the second-order reduced density matrix (2-RDM). Many authors have attempted to calculate the ground-state energy from the 2-RDM because it is a much simpler object than the electronic wavefunction. The use of the variational method to calculate the energy of a system involves the modification of the 2-RDM subject to the N-representability conditions. Among them, the contracted Schrödinger equation (CSE) and the antiHermitian counterpart (ACSE) have rekindled the interest in methods without wavefunctions. Both CSE and ACSE energy expressions depend on the third-order reduced density (3-RDM), which is usually approximated from lower-order densities. The accuracy of these methods depends critically on the set of N-representability conditions enforced in the calculation and the quality of the approximate 3-RDM. There are no benchmark studies including most 3-RDM approximations and, thus far, no assessment of the deterioration of the approximations with correlation effects has been performed.

In two recent works (1 and 2) we had put forward two new approximations to the diagonal of the 3-RDM that were used to calculate 3c-indices in a series of molecules. Our approximations were compared against the Valdemoro, Nakatsuji and Mazziotti approximations, showing that one of our proposals was clearly superior to the others for the calculation of 3c-indices.

Now, in a paper recently published in *Phys. Chem. Chem. Phys.,* we introduce a series of tests (see the graphic below) to assess the performance of 3-RDM approximations in a model system with varying electron correlation effects, the three-electron harmonium atom. The results of our work put forward several limitations of the currently most used 3-RDM approximations for systems with important electron correlation effects.

Our results show that the errors of the 3-RDM approximations increase as the inverse of the confinement strength (the parameter that regulates the electron correlation effects in harmonium). All approximations fail to satisfy several N-representability conditions and show significant deviations from the trace numbers upon inclusion of electron correlation. Surprisingly, Mazziotti's 3-RDM performs remarkably bad for the doublet state and very well for the quartet state. Valdemoro's approximation shows the most promising results but provides the largest termwise errors. In the paper we give a hint to improve the performance of 3-RDM approximations.

Posted on November 11, 2016 at 11:10 AM |

The concept of electron correlation goes back as far as 1934, to the early stages of quantum-mechanics methods development, before the advent of coupled-cluster, CASSCF or density functional methods. Initially it was defined as the energy difference between the exact result and the Hartree-Fock energy but, soon enough, many different nuances were suggested. The computational lexicon now includes terms such as dynamic, static, angular, radial, short-range or long-range correlation. The most popular separation of electron correlation is done in terms of dynamic and nondynamic correlation types. The former and the latter are also known as **weak and strong correlation**, respectively. This nomenclature is often used to decide the most convenient computational tool to perform molecular simulations.

The account of electron correlation and its efficient separation into dynamic and nondynamic parts plays a key role in the development of computational methods. We have suggested a physically-sound matrix formulation to split electron correlation into dynamic and nondynamic parts using the two-particle cumulant matrix and a measure of the deviation from idempotency of the first-order density matrix. These matrices are applied to a two-electron model, giving rise to a simplified electron correlation index that (i) depends only on natural orbitals and their occupancies, (ii) can be straightforwardly decomposed into orbital contributions and (iii) splits into dynamic and nondynamic correlation parts that (iv) admit a local version. To the best of y knowledge, these expressions provide **the first separation of dynamic and nondynamic correlation based on natural orbital occupancies**. These expressions can be used in fractional-occupancy density functional theory (DFT) and density matrix functional theory (DMFT) to construct expressions that control the introduction of dynamic and nondynamic correlation.

This is the **first of a series of three works** measuring weak and strong correlation effects. This work has been published in Phys. Chem. Chem. Phys. with Eloy Ramos-Cordoba and Pedro Salvador.

Posted on May 6, 2016 at 3:25 PM |

*Aromaticity* is an ubiquitous term in chemistry referring to the cyclic electron delocalization that leads to energy stabilization, among other particular properties. Its antonym is *antiaromaticity*, which was coined by Breslow to refer to situations where "electronic delocalization is destabilizing". In the past years, the synthesis of new aromatic compounds of inorganic nature has shaken the traditional concept of aromaticity that has been extended to include new species such all-metal aromatic clusters. Nowadays, there is a number of new inorganic species commonly referred as aromatic, whereas there is very few antiaromatic molecules that are not organic.

The group of Prof. Wang and Prof. Boldyrev have worked in the synthesis and characterization of some well-known new aromatic species such as (Al4)2-, and they reported the existence of an all-metal antiaromatic molecule containing an antiaromatic (Al4)4- unit that could not be realized in the lattice. Now, together with Prof. Sun's group from the Changchun Institute of Applied Chemistry, they have synthesized an all-metal cluster, [Ln(Sb4)3]3– (Ln=La, Y, Ho, Er, or Lu), that contains three Sb4 antiaromatic units.

They invite me to collaborate in their research and help characterizing the aromaticity of the synthesized species using the indices developed in the group. The calculations were conclusive for [La(Sb4)3]3–: the compound can be formally drawn as La3+ and three (Sb4)2- units, in accord with simple electron count rules. However, the electronic arrangement in the (Sb4)2- unit within the [La(η4-Sb4)3]3- cluster is very different from the isolated (Sb4)2- molecule, which is aromatic. The delocalization index between the (Sb4)2- unit and La in the cluster is 1.2, suggesting a strong η4-interaction between both that probably hinders the internal ring delocalization that existed in the aromatic free (Sb4)2- moiety and prompted its aromatic character. Conversely, the multicenter indices values suggest that the electronic structure of the Sb4 unit within [Ln(Sb4)3]3– is similar to cyclobutadiene, the prototypical antiaromatic molecule.

Our work has been published this week in Angewandte Chemie and has been commented in Chemical & Engineering News (here in Spanish).

Posted on February 12, 2016 at 11:25 AM |

Porphyrins are ubiquitous pigments in nature, playing prominent roles in several biological processes. Expanded-porphyrins are also versatile compounds giving rise to a plethora of twisted Möbius aromatic structures that give rise to Hückel-to-Möbius topological switches. These molecules can be utilized in a number of applications, including molecular electronics, such as the preparation of photovoltaic cells (dye-sensitised solar cells), nonlinear optical materials or electroluminescene displays, biomimetic catalysis and medicine.

The salient features of porphyrins and expanded porphyrins are related to its long conjugated π-electron closed circuits that furnish these molecules with a certain aromatic character. The presence of conjugated circuits in these large molecular rings calls for aromaticity measures that can be unambiguously used in rings of arbitrary size and do not suffer from severe limitations.

The aromaticity is a key property of these species that can be used to guide the synthesis and the design of new molecules. There are very few aromaticity indices that can be applied to large macrocycles. Thus far, the studies dealing with these species analyzed the aromaticity using the harmonic-oscillator model of aromaticity (HOMA), the in-house aromatic fluctuation index (FLU) or the nuclear-independent chemical shifts (NICS). These indices suffer from serious drawbacks that could lead to erroneous conclusions: HOMA or FLU are reference-based indices and therefore lead to spurious results when applied to reactivity studies; whereas NICS is known to be size-dependent and it is highly affected by the currents of metals present in metalated porphyrins. A perfect candidate to characterize porphyrins are multicenter indices (MCI), which were shown as the most reliable ones according to a series of aromaticity tests we designed. Unfortunately, MCI suffers from a series of problems (mostly numerical precision and large computational cost) that prevents its application in large rings. The smallest porphyrin already requires 16-center MCI calculations, which are both computationally expensive and very inaccurate.

In a paper recently accepted in the special issue of Phys. Chem. Chem. Phys. devoted to Electron delocalization and aromaticity: "Celebrating the 150th Anniversary of the Kekulé Benzene Structure" I introduce a new electronic aromaticity index, AV1245. AV1245 consists of the average of the 4-center MCI values along the ring that keep a positional relationship of 1,2,4,5.

AV1245 measures the extent of transferability of the delocalized electrons between bonds 1-2 and 4-5, which is expected to be large in conjugated circuits and, therefore, in aromatic molecules. A new algorithm for the calculation of MCI for large rings is also introduced and used to produce the data for the calibration of the new aromaticity index. **AV1245 does not rely on reference values**, **does not suffer from large numerical precision** errors, and it does not present any limitation on the nature of atoms, the molecular geometry or the level of calculation. It is a **size-extensive measure with a small computational** cost that grows linearly with the number of ring members. Therefore, it is **specially suited** to study the aromaticity of large molecular rings as those occurring in **porphyrins** or belt-shaped Möbius structures (**expanded porphyrins**). The work includes the analysis of AV1245 in free-base and several bis-metalated Pd [32]octaphyrins (1,0,1,0,1,0,1,0) shown in the picture above.

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.