Effective local potentials for density and density-matrix functional approximations with non-negative screening density.

A way to improve the accuracy of the spectral properties in density functional theory (DFT) is to impose constraints on the effective, Kohn-Sham (KS), local potential [J. Chem. Phys. 136, 224109 (2012)]. As illustrated, a convenient variational quantity in that approach is the "screening" or "electron repulsion" density, ρrep, corresponding to the local, KS Hartree, exchange and correlation potential through Poisson's equation. Two constraints, applied to this minimization, largely remove self-interaction errors from the effective potential: (i) ρrep integrates to N - 1, where N is the number of electrons, and (ii) ρrep ≥ 0 everywhere. In this work, we introduce an effective "screening" amplitude, f, as the variational quantity, with the screening density being ρrep = f2. In this way, the positivity condition for ρrep is automatically satisfied, and the minimization problem becomes more efficient and robust. We apply this technique to molecular calculations, employing several approximations in DFT and in reduced density matrix functional theory. We find that the proposed development is an accurate, yet robust, variant of the constrained effective potential method.

Evaluating the performance of ReaxFF potentials for sp2 carbon systems (graphene, carbon nanotubes, fullerenes) and a new ReaxFF potential.

We study the performance of eleven reactive force fields (ReaxFF), which can be used to study sp2 carbon systems. Among them a new hybrid ReaxFF is proposed combining two others and introducing two different types of C atoms. The advantages of that potential are discussed. We analyze the behavior of ReaxFFs with respect to 1) the structural and mechanical properties of graphene, its response to strain and phonon dispersion relation; 2) the energetics of (n, 0) and (n, n) carbon nanotubes (CNTs), their mechanical properties and response to strain up to fracture; 3) the energetics of the icosahedral C60 fullerene and the 40 C40 fullerene isomers. Seven of them provide not very realistic predictions for graphene, which made us focusing on the remaining, which provide reasonable results for 1) the structure, energy and phonon band structure of graphene, 2) the energetics of CNTs versus their diameter and 3) the energy of C60 and the trend of the energy of the C40 fullerene isomers versus their pentagon adjacencies, in accordance with density functional theory (DFT) calculations and/or experimental data. Moreover, the predicted fracture strain, ultimate tensile strength and strain values of CNTs are inside the range of experimental values, although overestimated with respect to DFT. However, they underestimate the Young's modulus, overestimate the Poisson's ratio of both graphene and CNTs and they display anomalous behavior of the stress - strain and Poisson's ratio - strain curves, whose origin needs further investigation.

Density inversion method for local basis sets without potential auxiliary functions: inverting densities from RDMFT.

A density inversion method is presented, to obtain the constrained, optimal, local potential that has a prescribed asymptotic behaviour and reproduces optimally any given ground-state electronic density. This work builds upon the method of [Callow et al., J. Chem. Phys., 2020, 152, 164114.] and differs in the expansion of the screening density in orbital basis element products instead of basis functions of an additional auxiliary set. We demonstrated the method by applying it to densities from DFT, Hartree-Fock, CAS-SCF and RDMFT calculations. For RDMFT, we demonstrate that density inversion offers a viable single-particle description by comparing the ionization potentials for atomic and molecular systems to the corresponding experimental values. Finally, we show that with the present method, accurate correlation potentials can be obtained from the inversion of accurate densities.

Generalized Kohn-Sham equations with accurate total energy and single-particle eigenvalue spectrum.

We propose a new generalized Kohn-Sham or constrained hybrid method, where the exchange potential is the (equally weighted) average of the nonlocal Fock-exchange term and the self-interaction-corrected exchange potential, as obtained from our constrained minimization method of semi-local approximations. The new method gives an accurate single-particle eigenvalue spectrum with an average deviation between (the negative of) the valence orbital eigenvalues and the experimental ionization potentials of about 0.5 eV, while the deviation of core orbitals is within 2 eV. The improvement in the eigenvalue spectrum is achieved with a minimal increase in the total energy.

Width Dependent Elastic Properties of Graphene Nanoribbons.

The mechanical response of graphene nanoribbons under uniaxial tension, as well as its dependence on the nanoribbon width, is presented by means of numerical simulations. Both armchair and zigzag edged graphene nanoribbons are considered. We discuss results obtained through two different theoretical approaches, viz. density functional methods and molecular dynamics atomistic simulations using empirical force fields especially designed to describe interactions within graphene sheets. Apart from the stress-strain curves, we calculate several elastic parameters, such as the Young's modulus, the third-order elastic modulus, the intrinsic strength, the fracture strain, and the Poisson's ratio versus strain, presenting their variation with the width of the nanoribbon.

Preparation of hydrogen, fluorine and chlorine doped and co-doped titanium dioxide photocatalysts: a theoretical and experimental approach.

Titanium dioxide (TiO2) has a strong photocatalytic activity in the ultra-violet part of the spectrum combined with excellent chemical stability and abundance. However, its photocatalytic efficiency is prohibited by limited absorption within the visible range derived from its wide band gap value and the presence of charge trapping states located at the band edges, which act as electron-hole recombination centers. Herein, we modify the band gap and improve the optical properties of TiO2 via co-doping with hydrogen and halogen. The present density functional theory (DFT) calculations indicate that hydrogen is incorporated in interstitial sites while fluorine and chlorine can be inserted both as interstitial and oxygen substitutional defects. To investigate the synergy of dopants in TiO2 experimental characterization techniques such as Fourier transform infrared (FTIR), X-ray diffraction (XRD), X-ray and ultra-violet photoelectron spectroscopy (XPS/UPS), UV-Vis absorption and scanning electron microscopy (SEM) measurements, have been conducted. The observations suggest that the oxide's band gap is reduced upon halogen doping, particularly for chlorine, making this material promising for energy harvesting devices. The studies on hydrogen production ability of these materials support the enhanced hydrogen production rates for chlorine doped (Cl:TiO2) and hydrogenated (H:TiO2) oxides compared to the pristine TiO2 reference.

Uniaxially Strained Graphene: Structural Characteristics and G-Mode Splitting.

The potential use of graphene in various strain engineering applications requires an accurate characterization of its properties when the material is under different mechanical loads. In this work, we present the strain dependence of the geometrical characteristics at the atomic level and the Raman active G-band evolution in a uniaxially strained graphene monolayer, using density functional theory methods as well as molecular dynamics atomistic simulations for strains that extend up to the structural failure. The bond length and bond angle variations with strain, applied either along the zigzag or along the armchair direction, are discussed and analytical relations describing this dependence are provided. The G-mode splitting with strain, as obtained by first principles' methods, is also presented. While for small strains, up to around 1%, the G-band splitting is symmetrical in the two perpendicular directions of tension considered here, this is no longer the case for larger values of strains where the splitting appears to be larger for strains along the zigzag direction. Further, a crossing is observed between the lower frequency split G-mode component and the out-of-plane optical mode at the Γ point for large uniaxial strains (>20%) along the zigzag direction.

Improving the exchange and correlation potential in density-functional approximations through constraints.

We review and expand on our work to impose constraints on the effective Kohn-Sham (KS) potential of local and semi-local density-functional approximations. Constraining the minimisation of the approximate total energy density-functional invariably leads to an optimised effective potential (OEP) equation, the solution of which yields the KS potential. We review briefly our previous work on this and demonstrate with numerous examples that despite the well-known mathematical issues of the OEP with finite basis sets, our OEP equations are numerically robust. We demonstrate that appropriately constraining the 'screening charge' which corresponds to the Hartree, exchange and correlation potential not only corrects its asymptotic behaviour but also allows the exchange and correlation potential to exhibit a non-zero derivative discontinuity, a feature of the exact KS potential that is necessary for the accurate prediction of band-gaps in solids but very hard to capture with semi-local approximations.

Density-inversion method for the Kohn-Sham potential: Role of the screening density.

We present a method to invert a given density and find the Kohn-Sham (KS) potential in Density Functional Theory (DFT) that shares the density. Our method employs the concept of screening density, which is naturally constrained by the inversion procedure and thus ensures that the density being inverted leads to a smooth KS potential with correct asymptotic behavior. We demonstrate the applicability of our method by inverting both local and non-local (Hartree-Fock and coupled cluster) densities; we also show how the method can be used to mitigate the effects of self-interactions in common DFT potentials with appropriate constraints on the screening density.

Defect processes in F and Cl doped anatase TiO2.

Titanium dioxide represents one of the most widely studied transition metal oxides due to its high chemical stability, non-toxicity, abundance, electron transport capability in many classes of optoelectronic devices and excellent photocatalytic properties. Nevertheless, the wide bang gap of pristine oxide reduces its electron transport ability and photocatalytic activity. Doping with halides and other elements has been proven an efficient defect engineering strategy in order to reduce the band gap and maximize the photocatalytic activity. In the present study, we apply Density Functional Theory to investigate the influence of fluorine and chlorine doping on the electronic properties of TiO2. Furthermore, we present a complete investigation of spin polarized density functional theory of the (001) surface doped with F and Cl in order to elaborate changes in the electronic structure and compare them with the bulk TiO2.

Interfacing tetrapyridyl-C60 with porphyrin dimers via π-conjugated bridges: artificial photosynthetic systems with ultrafast charge separation.

We report on the synthesis, characterization and photophysical properties of a donor-bridge-acceptor supramolecular hybrid system, consisting of a tetrapyridyl fullerene derivative (C60-tpyr) as electron acceptor, with the four pyridyl groups as part of oligophenyleneethynylene/phenylenevinylene bridges, and zinc porphyrin dimers (ZnP)2 as electron donor species. Based on the metal-to-ligand coordination between the zinc metal centers of (ZnP)2 and the four pyridyl entities of C60-tpyr, a strong binding constant (5 × 105 M-1) for the formation of C60-tpyr·[(ZnP)2]2 was evidenced. Insights into the electronic interactions between the photoactive (ZnP)2 units and C60-tpyr emanated from complementary physicochemical assays, which were further supported by theoretical calculations. Notably, the absorption and emission titration assays revealed strong interactions between the electron donor and acceptor species within C60-tpyr·[(ZnP)2]2, both in the ground and excited state. Moreover, femtosecond and nanosecond laser photolysis transient absorption measurements were performed and provided solid evidence for intramolecular electron transfer processes derived from the singlet excited state of (ZnP)2 to C60-tpyr. Comparison with systems in which either four monomeric zinc porphyrins (ZnP) were complexed with C60-tpyr or a (ZnP)2 was coordinated with a dipyridylfullerene revealed the beneficial role of C60-tpyr in increasing the lifetime of charge-separation.

Structure of the first order reduced density matrix in three electron systems: A generalized Pauli constraints assisted study.

We investigate the structure of the one-body reduced density matrix of three electron systems, i.e., doublet and quadruplet spin configurations, corresponding to the smallest interacting system with an open-shell ground state. To this end, we use configuration interaction (CI) expansions of the exact wave function in Slater determinants built from natural orbitals in a finite dimensional Hilbert space. With the exception of maximally polarized systems, the natural orbitals of spin eigenstates are generally spin dependent, i.e., the spatial parts of the up and down natural orbitals form two different sets. A measure to quantify this spin dependence is introduced and it is shown that it varies by several orders of magnitude depending on the system. We also study the ordering issue of the spin-dependent occupation numbers which has practical implications in reduced density matrix functional theory minimization schemes, when generalized Pauli constraints (GPCs) are imposed and in the form of the CI expansion in terms of the natural orbitals. Finally, we discuss the aforementioned CI expansion when there are GPCs that are almost "pinned."

Atomistic potential for graphene and other sp2 carbon systems.

We introduce a torsional force field for sp2 carbon to augment an in-plane atomistic potential of a previous work [G. Kalosakas et al., J. Appl. Phys., 2013, 113, 134307] so that it is applicable to out-of-plane deformations of graphene and related carbon materials. The introduced force field is fit to reproduce density-functional-theory calculation data of appropriately chosen structures. The aim is to create a force field that is as simple as possible so it can be efficient for large scale atomistic simulations of various sp2 carbon structures without significant loss of accuracy. We show that the complete proposed potential reproduces characteristic properties of fullerenes and carbon nanotubes. In addition, it reproduces very accurately the out-of-plane acoustic and optical modes of graphene's phonon dispersion as well as all phonons with frequencies up to 1000 cm-1.

Towards a formal definition of static and dynamic electronic correlations.

Some of the most spectacular failures of density-functional and Hartree-Fock theories are related to an incorrect description of the so-called static electron correlation. Motivated by recent progress in the N-representability problem of the one-body density matrix for pure states, we propose a method to quantify the static contribution to the electronic correlation. By studying several molecular systems we show that our proposal correlates well with our intuition of static and dynamic electron correlation. Our results bring out the paramount importance of the occupancy of the highest occupied natural spin-orbital in such quantification.

Structural deformations of two-dimensional planar structures under uniaxial strain: the case of graphene.

In the present work, a method for the study of the structural deformations of two dimensional planar structures under uniaxial strain is presented. The method is based on molecular mechanics using the original stick and spiral model and a modified one which includes second nearest neighbor interactions for bond stretching. As we show, the method allows an accurate prediction of the structural deformations of any two dimensional planar structure as a function of strain, along any strain direction in the elastic regime, if structural deformations are known along specific strain directions, which are used to calculate the stick and spiral model parameters. Our method can be generalized including other strain conditions and not only uniaxial strain. We apply this method to graphene and we test its validity, using results obtained from ab initio density functional theory calculations. What we find is that the original stick and spiral model is not appropriate to describe accurately the structural deformations of graphene in the elastic regime. However, the introduction of second nearest neighbor interactions provides a very accurate description.

Conditions for Describing Triplet States in Reduced Density Matrix Functional Theory.

We consider necessary conditions for the one-body reduced density matrix (1RDM) to correspond to a triplet wave function of a two-electron system. The conditions concern the occupation numbers and are different for the high spin projections, Sz = ±1, and the Sz = 0 projection. Hence, they can be used to test if an approximate 1RDM functional yields the same energies for both projections. We employ these conditions in reduced density matrix functional theory calculations for the triplet excitations of two-electron systems. In addition, we propose that these conditions can be used in the calculation of triplet states of systems with more than two electrons by restricting the active space. We assess this procedure in calculations for a few atomic and molecular systems. We show that the quality of the optimal 1RDMs improves by applying the conditions in all the cases we studied.

Orbitals from local RDMFT: Are they Kohn-Sham or natural orbitals?

Recently, an approximate theoretical framework was introduced, called local reduced density matrix functional theory (local-RDMFT), where functionals of the one-body reduced density matrix (1-RDM) are minimized under the additional condition that the optimal orbitals satisfy a single electron Schrödinger equation with a local potential. In the present work, we focus on the character of these optimal orbitals. In particular, we compare orbitals obtained by local-RDMFT with those obtained with the full minimization (without the extra condition) by contrasting them against the exact NOs and orbitals from a density functional calculation using the local density approximation (LDA). We find that the orbitals from local-RMDFT are very close to LDA orbitals, contrary to those of the full minimization that resemble the exact NOs. Since local RDMFT preserves the good quality of the description of strong static correlation, this finding opens the way to a mixed density/density matrix scheme, where Kohn-Sham orbitals obtain fractional occupations from a minimization of the occupation numbers using 1-RDM functionals. This will allow for a description of strong correlation at a cost only minimally higher than a density functional calculation.

Graphene allotropes under extreme uniaxial strain: an ab initio theoretical study.

Using density functional theory calculations, we study the response of three representative graphene allotropes (two pentaheptites and octagraphene) as well as graphene, to uniaxial strain up to their fracture limit. Those allotropes can be seen as distorted graphene structures formed upon periodically arranged Stone-Walles transformations. We calculate their mechanical properties (Young's modulus, Poisson's ratio, speed of sound, ultimate tensile strength and the corresponding strain), and we describe the pathways of their fracture. Finally, we study strain as a factor for the conversion of graphene into those allotropes upon Stone-Walles transformations. For specific sets of Stone-Walles transformations leading to an allotrope, we determine the strain directions and the corresponding minimum strain value, for which the allotrope is more favorable energetically than graphene. We find that the minimum strain values which favor those conversions are of the order of 9-13%. Moreover, we find that the energy barriers for the Stone-Walles transformations decrease dramatically under strain, however, they remain prohibitive for structural transitions. Thus, strain alone cannot provide a synthetic route to these allotropes, but could be a part of composite procedures for this purpose.

Generalized Pauli constraints in reduced density matrix functional theory.

Functionals of the one-body reduced density matrix (1-RDM) are routinely minimized under Coleman's ensemble N-representability conditions. Recently, the topic of pure-state N-representability conditions, also known as generalized Pauli constraints, received increased attention following the discovery of a systematic way to derive them for any number of electrons and any finite dimensionality of the Hilbert space. The target of this work is to assess the potential impact of the enforcement of the pure-state conditions on the results of reduced density-matrix functional theory calculations. In particular, we examine whether the standard minimization of typical 1-RDM functionals under the ensemble N-representability conditions violates the pure-state conditions for prototype 3-electron systems. We also enforce the pure-state conditions, in addition to the ensemble ones, for the same systems and functionals and compare the correlation energies and optimal occupation numbers with those obtained by the enforcement of the ensemble conditions alone.

Quasi-particle energy spectra in local reduced density matrix functional theory.

Recently, we introduced [N. N. Lathiotakis, N. Helbig, A. Rubio, and N. I. Gidopoulos, Phys. Rev. A 90, 032511 (2014)] local reduced density matrix functional theory (local RDMFT), a theoretical scheme capable of incorporating static correlation effects in Kohn-Sham equations. Here, we apply local RDMFT to molecular systems of relatively large size, as a demonstration of its computational efficiency and its accuracy in predicting single-electron properties from the eigenvalue spectrum of the single-particle Hamiltonian with a local effective potential. We present encouraging results on the photoelectron spectrum of molecular systems and the relative stability of C20 isotopes. In addition, we propose a modelling of the fractional occupancies as functions of the orbital energies that further improves the efficiency of the method useful in applications to large systems and solids.