The Emergence of the Hexagonal Lattice in Two-Dimensional Wigner Fragments.

At very low density, the electrons in a uniform electron gas spontaneously break symmetry and form a crystalline lattice called a Wigner crystal. But which type of crystal will the electrons form? We report a numerical study of the density profiles of fragments of Wigner crystals from first principles. To simulate Wigner fragments, we use Clifford periodic boundary conditions and a renormalized distance in the Coulomb potential. Moreover, we show that high-spin restricted open-shell Hartree-Fock theory becomes exact in the low-density limit. We are thus able to accurately capture the localization in two-dimensional Wigner fragments with many electrons. No assumptions about the positions where the electrons will localize are made. The density profiles we obtain emerge naturally when we minimize the total energy of the system. We clearly observe the emergence of the hexagonal crystal structure, which has been predicted to be the ground-state structure of the two-dimensional Wigner crystal.

Electronic Structure of Pentagonal Carbon Nanocones: An ab Initio Study.

In this work, we investigate the electronic structure of a particular class of carbon nanocones having a pentagonal tip and C5v symmetry. The ground-state nature of the wave function for these structures can be predicted by the recently proposed generalized Hückel rule that extends the original Hückel rule for annulenes to this class of carbon nanocones. In particular, the structures here considered can be classified as closed-shell or anionic/cationic closed-shells, depending on the geometric characteristics of the cone. The goal of this work is to assess the relationship between the electronic configuration of these carbon nanocones and their ability to gain or lose an electron as well as their adsorption capability. For this, the geometry of these structures in the neutral or ionic forms, as well as systems containing either one lithium or fluorine atom, was optimized at the DFT/B3LYP level. It was found that the electron affinity, ionization potential, and the Li or F adsorption energy present an intimate connection to the ground-state wave function character predicted by the generalized Hückel rule. In fact, a peculiar oscillatory energy behavior was discovered, in which the electron affinity, ionization energy, and adsorption energies oscillate with an increase in the nanocone size. The reasoning behind this is that if the anion is closed-shell, then the neutral nanocone will turn out to be a good electron acceptor, increasing the electron affinity and lithium adsorption energy. On the other hand, in the case of a closed-shell cation, this means that the neutral nanocone will easily lose an electron, leading to a smaller ionization potential and higher fluorine adsorption energy.

Solution to the Thomson Problem for Clifford Tori with an Application to Wigner Crystals.

In its original version, the Thomson problem consists of the search for the minimum-energy configuration of a set of point-like electrons that are confined to the surface of a two-dimensional sphere (S2) that repel each other according to Coulomb's law, in which the distance is the Euclidean distance in the embedding space of the sphere, i.e., R3. In this work, we consider the analogous problem where the electrons are confined to an n-dimensional flat Clifford torus Tn with n = 1, 2, 3. Since the torus Tn can be embedded in the complex manifold Cn, we define the distance in the Coulomb law as the Euclidean distance in Cn, in analogy to what is done for the Thomson problem on the sphere. The Thomson problem on a Clifford torus is of interest because supercells with the topology of a Clifford torus can be used to describe periodic systems such as Wigner crystals. In this work, we numerically solve the Thomson problem on a square Clifford torus. To illustrate the usefulness of our approach, we apply it to Wigner crystals. We demonstrate that the equilibrium configurations we obtain for large numbers of electrons are consistent with the predicted structures of Wigner crystals. Finally, in the one-dimensional case, we analytically obtain the energy spectrum and the phonon dispersion law.

Mapping of Hückel zigzag carbon nanotubes onto independent polyene chains: Application to periodic nanotubes.

The electric polarizability and the spread of the total position tensors are used to characterize the metallic vs insulator nature of large (finite) systems. Finite clusters are usually treated within the open boundary condition formalism. This introduces border effects, which prevent a fast convergence to the thermodynamic limit and can be eliminated within the formalism of periodic boundary conditions. Recently, we introduced an original approach to periodic boundary conditions, named Clifford boundary conditions. It considers a finite fragment extracted from a periodic system and the modification of its topology into that of a Clifford torus. The quantity representing the position is modified in order to fulfill the system periodicity. In this work, we apply the formalism of Clifford boundary conditions to the case of carbon nanotubes, whose treatment results in a particularly simple zigzag geometry. Indeed, we demonstrate that at the Hückel level, these nanotubes, either finite or periodic, are formally equivalent to a collection of non-interacting dimerized linear chains, thus simplifying their treatment. This equivalence is used to describe some nanotube properties as the sum of the contributions of the independent chains and to identify the origin of peculiar behaviors (such as conductivity). Indeed, if the number of hexagons along the circumference is a multiple of three, a metallic behavior is found, namely a divergence of both the (per electron) polarizability and total position spread of at least one linear chain. These results are in agreement with those in the literature from tight-binding calculations.

The Wigner localization of interacting electrons in a one-dimensional harmonic potential.

In this work, we study the Wigner localization of interacting electrons that are confined to a quasi-one-dimensional harmonic potential using accurate quantum chemistry approaches. We demonstrate that the Wigner regime can be reached using small values of the confinement parameter. To obtain physical insight in our results, we analyze them with a semi-analytical model for two electrons. Thanks to electronic-structure properties such as the one-body density and the particle-hole entropy, we are able to define a path that connects the Wigner regime to the Fermi-gas regime by varying the confinement parameter. In particular, we show that the particle-hole entropy, as a function of the confinement parameter, smoothly connects the two regimes. Moreover, it exhibits a maximum that could be interpreted as the transition point between the localized and delocalized regimes.

Multilayer Graphtriyne Membranes for Separation and Storage of CO2: Molecular Dynamics Simulations of Post-Combustion Model Mixtures.

The ability to remove carbon dioxide from gaseous mixtures is a necessary step toward the reduction of greenhouse gas emissions. As a contribution to this field of research, we performed a molecular dynamics study assessing the separation and adsorption properties of multi-layered graphtriyne membranes on gaseous mixtures of CO2, N2, and H2O. These mixtures closely resemble post-combustion gaseous products and are, therefore, suitable prototypes with which to model possible technological applications in the field of CO2 removal methodologies. The molecular dynamics simulations rely on a fairly accurate description of involved force fields, providing reliable predictions of selectivity and adsorption coefficients. The characterization of the interplay between molecules and membrane structure also permitted us to elucidate the adsorption and crossing processes at an atomistic level of detail. The work is intended as a continuation and a strong enhancement of the modeling research and characterization of such materials as molecular sieves for CO2 storage and removal.

Toward a Generalized Hückel Rule: The Electronic Structure of Carbon Nanocones.

In this work, we investigate a particular class of carbon nanocones, which we name graphannulenes, and present a generalized Hückel rule (GHR) that predicts the character of their ground state based on simply the three topological indices that uniquely define them. Importantly, this rule applies to both flat and curved systems, encompassing a wide variety of known structures that do not satisfy the "classic" 4n + 2 rule such as coronene, corannulene, and Kekulene. We test this rule at the Hückel level of theory for a large number of systems, including structures that are convex and flat, with a saddle-like geometry, and at the CASSCF level of theory for a selected representative subset. All the performed calculations support the GHR that we propose in this work.

Wigner localization in two and three dimensions: An ab initio approach.

In this work, we investigate the Wigner localization of two interacting electrons at very low density in two and three dimensions using the exact diagonalization of the many-body Hamiltonian. We use our recently developed method based on Clifford periodic boundary conditions with a renormalized distance in the Coulomb potential. To accurately represent the electronic wave function, we use a regular distribution in space of Gaussian-type orbitals and we take advantage of the translational symmetry of the system to efficiently calculate the electronic wave function. We are thus able to accurately describe the wave function up to very low density. We validate our approach by comparing our results to a semi-classical model that becomes exact in the low-density limit. With our approach, we are able to observe the Wigner localization without ambiguity.

The localization spread and polarizability of rings and periodic chains.

The localization spread gives a criterion to decide between metallic and insulating behavior of a material. It is defined as the second moment cumulant of the many-body position operator, divided by the number of electrons. Different operators are used for systems treated with open or periodic boundary conditions. In particular, in the case of periodic systems, we use the complex position definition, which was already used in similar contexts for the treatment of both classical and quantum situations. In this study, we show that the localization spread evaluated on a finite ring system of radius R with open boundary conditions leads, in the large R limit, to the same formula derived by Resta and co-workers [C. Sgiarovello, M. Peressi, and R. Resta, Phys. Rev. B 64, 115202 (2001)] for 1D systems with periodic Born-von Kármán boundary conditions. A second formula, alternative to Resta's, is also given based on the sum-over-state formalism, allowing for an interesting generalization to polarizability and other similar quantities.

Clifford Boundary Conditions: A Simple Direct-Sum Evaluation of Madelung Constants.

We propose a simple direct-sum method for the efficient evaluation of lattice sums in periodic solids. It consists of two main principles: (i) the creation of a supercell that has the topology of a Clifford torus, which is a flat, finite, and borderless manifold; (ii) the renormalization of the distance between two points on the Clifford torus by defining it as the Euclidean distance in the embedding space of the Clifford torus. Our approach does not require any integral transformations nor any renormalization of the charges. We illustrate our approach by applying it to the calculation of the Madelung constants of ionic crystals. We show that the convergence toward the system of infinite size is monotonic, which allows for a straightforward extrapolation of the Madelung constant. We are able to recover the Madelung constants with a remarkable accuracy, and at an almost negligible computational cost, i.e., a few seconds on a laptop computer.

Tuning the magnetic properties of beryllium chains.

In this work we explore the effect of confining beryllium chains inside carbon nanotubes. Linear Ben systems are characterized by two states originating from the presence of edge orbitals localized at the chain extremities. The two spins occupying these orbitals are, in the gas phase, antiferromagnetically coupled, with the magnetic coupling J decaying exponentially as a function of increasing length of the chain. When inserted into narrow carbon nanotubes, the linear geometry is found to be more stable than the more compact cluster conformation favored for the isolated case: the lack of space inside the cavity prevents the chain from folding. Most importantly, the presence of the surrounding nanotube not only preserves the linear structure of Ben, but affects its magnetic properties too. In particular it was found that the magnetic coupling between the ground and the first excited state can be modulated according to the nanotube diameter as well as the chain length, and our calculations suggest a possible direct relationship between these parameters and J. This behavior can be exploited to engineer a composite Ben@CNT system with the magnetic coupling tuned by construction, with interesting potential applications.

Distributed Gaussian orbitals for the description of electrons in an external potential.

In this work, we demonstrate the viability of using distributed Gaussian orbitals as a basis set for the calculation of the properties of electrons subjected to an external potential. We validate our method by studying one-electron systems for which we can compare to exact analytical results. We highlight numerical aspects that require particular care when using a distributedGaussian basis set. In particular, we discuss the optimal choice for the distance between two neighboring Gaussian orbitals. Finally, we show how our approach can be applied to many-electron problems.

The Electronic Structure of Beryllium Chains.

We present an ab initio theoretical study of quasi one-dimensional beryllium chains, Be N, from an electronic structure perspective for N = 3, 4,···, 12. In particular, linear and cyclic systems were compared by using high-quality coupled-cluster formalism. Both linear and cyclic species were found to be local minima on the corresponding potential energy surface, for all the considered values of N. The linear geometry is the most stable one only in the case of Be4. Several indicators (energy gap, position spread tensor, locality of the molecular orbitals) clearly show that both linear and cyclic one-dimensional structures, unlike three-dimensional bulk beryllium, have a covalent insulating nature.

Signatures of Wigner localization in one-dimensional systems.

We propose a simple and efficient approach to study Wigner localization in one-dimensional systems using ab initio theory. In particular, we propose a suitable basis for the study of localization which consists of equally spaced overlapping gaussians. We illustrate our approach with full-configuration interaction which yields exact results for a given basis set. With our approach, we were able to study up to 8 electrons with full-configuration interaction. Finally, we propose the total-position spread tensor and the total electron entropy as convenient quantities to obtain signatures of Wigner localization.

N⁻3 azide anion confined inside finite-size carbon nanotubes.

In this work, the confinement of an N[Formula: see text] azide anion inside finite-size single-wall zigzag and armchair carbon nanotubes of different diameters has been studied by wave function and density functional theory. Unrelaxed and relaxed interaction energies have been computed, resulting in a favorable interaction between the guest and host system. In particular, the largest interaction has been observed for the confinement in an armchair (5,5) carbon nanotube, for which a natural population analysis as well as an investigation based on the molecular electrostatic potential has been carried out. The nature of the interaction between the two fragments appears to be mainly electrostatic, favored by the enhanced polarizability of the nanotube wall treated as a finite system and passivated by hydrogen atoms. The results obtained are promising for possible applications of this complex as a starting point for the stabilization of larger polynitrogen compounds, suitable as a high-energy density material.

Increasing Radical Character of Large [n]cyclacenes Unveiled by Wave Function Theory.

We have investigated the radicality and the vertical singlet-triplet energy gap of [n]cyclacenes (cyclic polyacenes) as a function of the system size for n even, from 6 to 22. The calculations are performed using the complete active space self-consistent field method and second-order n-electron valence perturbation theory. We present a systematic way for the selection of the active space in order to have a balanced description of the wave function as the size of the system increases. Moreover, we provide didactic insight into the failure of an approach based on a minimal active space. We find that the ground state is an open-shell singlet and its multireference character increases progressively with n. The singlet-triplet gap decreases as a function of the system size and approaches a finite positive value for the limit n → ∞. Finally, an analysis based on the one-particle reduced density matrix suggests a polyradical character for the largest cyclacenes.

The Spin-Partitioned Total-Position Spread Tensor: An Application To Diatomic Molecules.

The spin partition (SP) of the total-position spread (TPS) tensor is applied to the case of a few light diatomic molecules at full configuration interaction (FCI) level. It appears that the SP-TPS tensor gives informations that are complementary with respect to the corresponding spin-summed (SS) quantity. The spin-summed total position-spread tensor (SS-TPS, Λ) is defined as the second moment cumulant of the total position operator, and the SP-TPS is its partition in equal (Λαα+ßß) and different spin (Λαß+ßα) contributions. Then, the SS-TPS allows description of the molecule charge mobility, while the SP-TPS allows description of the spin delocalization. The most relevant Cartesian-component for both tensors (SS-TPS and SP-TPS) is the component along the chemical bond (Λ(∥)), and it was found that its behavior was related to the type of interaction involved. For covalent bonds the SP-TPS has a squared growth when the bond is stretched, while for ionic bonds there exists a faster-than-linear growth after the avoided-crossing between the covalent and the ionic states. Other exotic bonds, like He2 and Be2, were also considered, and a particular spin delocalization was able to describe the different character of the two weakly bonded molecules, and specially the multireference character of the wave function along the dissociative potential energy curve.

The total position-spread tensor: spin partition.

The Total Position Spread (TPS) tensor, defined as the second moment cumulant of the position operator, is a key quantity to describe the mobility of electrons in a molecule or an extended system. In the present investigation, the partition of the TPS tensor according to spin variables is derived and discussed. It is shown that, while the spin-summed TPS gives information on charge mobility, the spin-partitioned TPS tensor becomes a powerful tool that provides information about spin fluctuations. The case of the hydrogen molecule is treated, both analytically, by using a 1s Slater-type orbital, and numerically, at Full Configuration Interaction (FCI) level with a V6Z basis set. It is found that, for very large inter-nuclear distances, the partitioned tensor growths quadratically with the distance in some of the low-lying electronic states. This fact is related to the presence of entanglement in the wave function. Non-dimerized open chains described by a model Hubbard Hamiltonian and linear hydrogen chains Hn (n ≥ 2), composed of equally spaced atoms, are also studied at FCI level. The hydrogen systems show the presence of marked maxima for the spin-summed TPS (corresponding to a high charge mobility) when the inter-nuclear distance is about 2 bohrs. This fact can be associated to the presence of a Mott transition occurring in this region. The spin-partitioned TPS tensor, on the other hand, has a quadratical growth at long distances, a fact that corresponds to the high spin mobility in a magnetic system.

The spin-partitioned total position-spread tensor: An application to Heisenberg spin chains.

The spin partition of the Total Position-Spread (TPS) tensor has been performed for one-dimensional Heisenberg chains with open boundary conditions. Both the cases of a ferromagnetic (high-spin) and an anti-ferromagnetic (low-spin) ground-state have been considered. In the case of a low-spin ground-state, the use of alternating magnetic couplings allowed to investigate the effect of spin-pairing. The behavior of the spin-partitioned TPS (SP-TPS) tensor as a function of the number of sites turned to be closely related to the presence of an energy gap between the ground-state and the first excited-state at the thermodynamic limit. Indeed, a gapped energy spectrum is associated to a linear growth of the SP-TPS tensor with the number of sites. On the other hand, in gapless situations, the spread presents a faster-than-linear growth, resulting in the divergence of its per-site value. Finally, for the case of a high-spin wave function, an analytical expression of the dependence of the SP-TPS on the number of sites n and the total spin-projection Sz has been derived.