Dr. Eduard Matito

Quantum Chemistry


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3-RDM approximations under the microscope

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.

Dynamic and Nondynamic Correlation Indicators

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.

All-Metal Antiaromaticity!

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).

An Aromaticity Index For Large Rings

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.

Exhaustive Benchmark Test For Natural Orbital Functional Theory

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.

Bonding Description of the Harpoon Mechanism

Posted on November 11, 2015 at 6:25 PM

The formation of molecules through the harpoon mechanism occurs from the interaction of two fragments, one with a low ionization potential (IP) and another that has a large electron affinity (EA). The reactants approach each other and, at a certain distance, an electron from the fragment with low IP harpoons the fragment with large EA, giving rise to a rapid electron-transfer process that is triggered by the Coulomb attraction exerted by the two fragments. Michael Polanyi, the father of the Nobel laureate John Polanyi, suggested the mechanism in 1932 to explain the large reaction cross sections observed in alkali atoms with halogen molecules. The reaction has been widely studied but, despite the intriguing reaction mechanism, its chemical bonding has not been so carefully analyzed.

In this work we perform a more exhaustive analysis of the bonding in the formation of molecules through harpoon mechanism using the electron sharing indices (ESI), the electron localization function (ELF) and the Laplacian of the electron density. Several diatomics are analyzed but the focus is put into the two lowest-lying singlet sigma states of LiH. The LiH in its ground state is formed by means of a harpoon mechanism, going through two avoided crossings, that change twice its bonding character. The single sigma excited state, shows a very peculiar behavior and likewise changes its bonding nature twice, as it also passes through two avoided crossings.

The work is part of Mauricio's thesis and it will be published in a special issue of Molecular Physics dedicated to the 65th birthday of Andreas Savin, who (aside from its important contributions to the density functional theory [DFT]) also contributed to the popularization, the understanding and interpretation of the ELF (e.g. the DFT version of the ELF) and introduce significant ideas in the field of the chemical bonding analysis. Happy birthday Andreas! 


Review on Aromaticity and Electron Delocalization

Posted on September 6, 2015 at 12:00 PM

We have recently published together with Ferran Feixas, Jordi Poater and Miquel Solà a review on Chemical Society Reviews (from RSC) covering the last 12 years studying the connections between aromaticity and electron delocalization. During this time we have developed a series of aromaticity indices (PDI, FLU, ING and INB) and constructed aromaticity tests (1 and 2) that allow a more reliable description of aromaticity not only in organic molecules but also inorganic molecules such as the all-metal clusters. These tools have been implemented in our own computational code (ESI-3D) and have been applied to a plethora of systems.

Miquel Solà, who has supervised the theses of the three of us (Jordi, Ferran and me) received the invitation from the editorial board of the Chemical Society Reviews who has published our work this week, including the inside cover of the journal that has been designed by Ferran.

In this link you can find the article or ask for a reprint if you don't have access.

The Electronic Structure of the Al3 Anion: Is it Aromatic?

Posted on June 19, 2015 at 9:25 PM

Our first article since I joined the Basque Country has just appeared online. The work has been carried out in close collaboration with Txema Mercero, Jesus Ugalde, Fernando Ruiperez, Xabi Lopez and Ivan Infante. The paper deals with the aromatic character of Al3- in various electronic states. Highly-accurate ab initio calculations (CASSCF) were performed in the lowest-lying singlet, triplet and quintet states. These states were found to be of significant multideterminant character, thus preventing the application of Hückel's rule or a simple molecular orbital analysis. The study of multicenter delocalization indices shed some light on the aromaticity of these species, which are found to be more aromatic than Al42- in its ground-state singlet configuration. 

This communication was published in Chem. Eur. J., whose editor selected it as a very important and invited us to publish a frontispieces, including the image you find below. Should you be interested in this work do not hesitate to request a reprint.

PhD Student position in Computational Chemistry

Posted on June 17, 2015 at 9:50 AM

PhD-student position available to perform the doctoral thesis in Donostia (Spain) under the supervision of Dr. Eduard Matito to work on some of the these subjects: (i) development of new DFT functionals; (ii) chemical bonding and aromaticity from electron delocalization measures. Dr. Matito works in the Donostia International Physics Center (DIPC) leading a group of four people that is part of a bigger laboratory working in diverse research subjects including method development, aluminum (III) interaction with bioligands, noncovalent and hydrogen-bond interactions, spin-flip methods for excited states, nanoclusters, chemical bonding and aromaticity.


Candidates should have completed a BSc. and a MSc., preferably in chemistry or physics, and have a good background in computational and theoretical chemistry. The candidate will enjoy a three-year contract of 16,422€/year (minimal salary in Spain ca. 9,034.20€) at the University of the Basque Country in Donostia. Donostia, the 2016 European Capital of Culture, is one of Spain’s most attractive, charming and popular cities, a sophisticated coastal gem situated in the north of the peninsula surrounded by hills and offering a lively beach front, a range of natural beauty and a unique gastronomical experience. The fellowship includes the payment of the doctoral studies, the possibility to perform short stays in other research centers and a fourth year of contract (19,000€/year) if the candidate obtains her/his PhD degree before the end of the third year. In addition, the group normally covers the expenses to attend a conference or a summer school every year.


The deadline to submit paperwork is the 29th of June at 15:00. Applicants should contact Dr. Matito ([email protected]) for futher details as soon as possible (the application procedure requires some paperwork that usually takes some days to obtain). Please include a CV with your e-mail.

More details in this website.

NLOP-DFT awarded with the MINECO project

Posted on May 21, 2015 at 9:50 AM

In the last years we have been working together with Josep Maria Luis on the ability of density functional theory (DFT) to reproduce nonlinear optical properties (NLOP). We started by studing the so-called catastrophe of DFT to reproduce polarizabilities and second-order polarizability on long conjugated chains as the size of chain increases. We supervised together the Bachelor Thesis of Natalia Abulí and recently the Erasmus stay of Sebastian Sitkiewicz, both devoted to this research topic. 

DFT has found applications on a wide variety of scientific areas due to its remarkable combination of efficiency and accuracy. It is being applied with success to the fields of bioinorganic chemistry, material science, drug design, biochemistry and nano-technology, among many others. The exact density functional expressions of many energy components have not been found, and as a result, the construction of new functionals in DFT has become a complicated task, often untangled by the recourse of fitting parameters with the aid of experimental results. In this regard, the errors in DFT calculations are hardly predictable and for each new scientific challenge functionals must be calibrated against the expensive standard ab initio methods to assess their performance. DFT has reached a state of saturation, and the design of new strategies for constructing DFT functionals is now of utmost importance.

Last year we put together an ambitious research project to construct new DFT functionals that do not suffer from the above-mentioned drawback. We based the strategy on the same idea I suggested for DFTCorr, which combined with Josep Maria outstanding expertise on NLOP, resulted in the research projecte entitled: "DFT functionals for the calculation of nonlinear optical properties" (acronym: NLOPDFT). This project was submitted to 2014's call "Generación de Conocimiento" of the Spanish Ministry (MINECO) with the following team: Sebastian Sitkiewicz, Mauricio Rodríguez-Mayorga, Eloy Ramos-Córdoba, Marc Garcia-Borràs, Josep Maria Luis and myself. 

The goal of NLOPDFT is to use a genuinely new strategy to design density functionals for the calculation of NLOPs, which will lead ultimately to an all-purpose functional that yields reasonably accurate results in most applications. This strategy is physically motivated and consists in using variable amounts of the components of the exchange-correlation functionals regulated by electron correlation measures that enter into the functional expressions. Overcoming one of the most important DFT pitfalls —the description of NLOPs—, promises to furnish new functionals with the flexibility to accurately describe a wider range of properties, paving the way towards the development of all-purpose functionals. This project gives the recipe to construct an unprecedented class of local hybrid functionals and range-separate hybrids with more flexibility than its predecessors. Our preliminary results following this strategy show a drastic improvement of hybrid functionals. This project shall leap forward DFT development and make an impact in a number of fields where DFT has found applications. 


A few weeks ago, the MINECO decided to grant our project with 60,000€ and a PhD fellowship. Therefore, there is an opening for a PhD position to work in this project. The PhD fellowship is a four-year grant including a free one-year master in the MaCMOM master organized by the IQCC at the University of Girona. Interested candidates should have an excellent academic track record, preferably with a good background on computational chemistry. Prospective candidates should send their CV by e-mail to either JosepM ([email protected]) or myself ([email protected]).


[Polarizability and second-order polarizabilities in polyacetylene chains. From: Limacher et al. JCP 130, 194114 (2009)]