Ab Initio Modeling of MultiWall: A General Algorithm First Applied to Carbon Nanotubes.

A general, versatile and automated computational algorithm to design any type of multiwall nanotubes of any chiralities is presented for the first time. It can be applied to rolling up surfaces obtained from cubic, hexagonal, and orthorhombic lattices. Full exploitation of the helical symmetry permits a drastic reduction of the computational cost and therefore opens to the study of realistic systems. As a test case, the structural, electronic, mechanical, and transport properties of multiwall carbon nanotubes (MWCNT) are calculated using a density functional theory approach, and results are compared with those of the corresponding layered (graphene-like) precursors. The interaction between layers has a general minimum for the inter-wall distance of ≈3.4 Å, in good agreement with experimental and computed optimal distances in graphene sheets. The metallic armchair and semiconductor zigzag MWCNT are almost isoenergetic and their stability increases as the number of walls increases. The vibrational fingerprint provides a reliable tool to identify the chirality and the thickness of the nanostructures. Finally, some promising thermoelectric features of the semiconductor MWCNT are reproduced and discussed.

The CRYSTAL code, 1976-2020 and beyond, a long story.

CRYSTAL is a periodic ab initio code that uses a Gaussian-type basis set to express crystalline orbitals (i.e., Bloch functions). The use of atom-centered basis functions allows treating 3D (crystals), 2D (slabs), 1D (polymers), and 0D (molecules) systems on the same grounds. In turn, all-electron calculations are inherently permitted along with pseudopotential strategies. A variety of density functionals are implemented, including global and range-separated hybrids of various natures and, as an extreme case, Hartree-Fock (HF). The cost for HF or hybrids is only about 3-5 times higher than when using the local density approximation or the generalized gradient approximation. Symmetry is fully exploited at all steps of the calculation. Many tools are available to modify the structure as given in input and simplify the construction of complicated objects, such as slabs, nanotubes, molecules, and clusters. Many tensorial properties can be evaluated by using a single input keyword: elastic, piezoelectric, photoelastic, dielectric, first and second hyperpolarizabilities, etc. The calculation of infrared and Raman spectra is available, and the intensities are computed analytically. Automated tools are available for the generation of the relevant configurations of solid solutions and/or disordered systems. Three versions of the code exist: serial, parallel, and massive-parallel. In the second one, the most relevant matrices are duplicated on each core, whereas in the third one, the Fock matrix is distributed for diagonalization. All the relevant vectors are dynamically allocated and deallocated after use, making the code very agile. CRYSTAL can be used efficiently on high performance computing machines up to thousands of cores.

The A-center defect in diamond: quantum mechanical characterization through the infrared spectrum.

The A-center in diamond, which consists of two nitrogen atoms substituting two neighboring carbon atoms, has been investigated at the quantum mechanical level using an all-electron Gaussian type basis set, hybrid functionals and the periodic supercell approach. In order to simulate different defect concentrations, four supercells have been considered containing 32, 64, 128 and 216 atoms, respectively. The ground state is a closed shell system where the two neighboring nitrogen atoms are separated, as a consequence of the strong repulsive interaction between the lone pairs, by 2.22 Å. The calculated band gap of a perfect diamond is 5.75 eV, which is in very good agreement with the experimental value of 5.80 eV (at 0 °K); the vertical electronic transition energy from the defective band to the conduction band is 4.75 and 4.46 eV for the cells containing 128 and 216 atoms, respectively. The presence of the A-center does not affect the Raman spectrum of diamond. Several intense peaks appear on the contrary in the IR spectrum, which permit (or should permit) the identification of this defect. The four peaks proposed by Sutherland et al. (Nature, 1954, 174, 901-904) and widely accepted as fingerprints of the A-center (at 480, 1093, 1203, 1282 cm-1), and the most important features of the spectrum published by Davies 22 years later (J. Phys. C: Solid State Phys., 1976, 9, 537-542) are very well reproduced by our simulated spectrum with the largest supercell. The modes in which the nitrogen atoms are more involved are identified by the frequency shift due to the 14N → 15N isotopic substitution; the two modes corresponding to the experimental ones at 480 and 1282 cm-1 show the largest isotopic shift. The graphical animation of the modes (available at ) not only confirms this attribution, but permits also the investigation of the nature of the full set of modes.

Infrared and Raman spectroscopic features of the self-interstitial defect in diamond from exact-exchange hybrid DFT calculations.

Quantum-mechanical calculations are performed to investigate the structural, electronic, and infrared (IR) and Raman spectroscopic features of one of the most common radiation-induced defects in diamond: the "dumb-bell" 〈100〉 split self-interstitial. A periodic super-cell approach is used in combination with all-electron basis sets and hybrid functionals of density-functional-theory (DFT), which include a fraction of exact non-local exchange and are known to provide a correct description of the electronic spin localization at the defect, at variance with simpler formulations of the DFT. The effects of both defect concentration and spin state are explicitly addressed. Geometrical constraints are found to prevent the formation of a double bond between the two three-fold coordinated carbon atoms. In contrast, two unpaired electrons are fully localized on each of the carbon atoms involved in the defect. The open-shell singlet state is slightly more stable than the triplet (the energy difference being just 30 meV, as the unpaired electrons occupy orthogonal orbitals) while the closed-shell solution is less stable by about 1.55 eV. The formation energy of the defect from pristine diamond is about 12 eV. The Raman spectrum presents only two peaks of low intensity at wave-numbers higher than the pristine diamond peak (characterized by normal modes extremely localized on the defect), whose positions strongly depend on defect concentration as they blue shift up to 1550 and 1927 cm(-1) at infinite defect dilution. The first of these peaks, also IR active, is characterized by a very high IR intensity, and might then be related to the strong experimental feature of the IR spectrum occurring at 1570 cm(-1). A second very intense IR peak appears at about 500 cm(-1), which, despite being originated from a "wagging" motion of the self-interstitial defect, exhibits a more collective, less localized character.

Structural, electronic and energetic properties of giant icosahedral fullerenes up to C6000: insights from an ab initio hybrid DFT study.

The properties of the (n,n) icosahedral family of carbon fullerenes up to n = 10 (6000 atoms) have been investigated through ab initio quantum-mechanical simulation by using a Gaussian type basis set of double zeta quality with polarization functions (84,000 atomic orbitals for the largest case), the hybrid B3LYP functional and the CRYSTAL14 code featuring generalization of symmetry treatment. The geometry of giant fullerenes shows hybrid features, between a polyhedron and a sphere; as n increases, it approaches the former. Hexagon rings at face centres take a planar, graphene-like configuration; the 12 pentagon rings at vertices impose, however, a severe structural constraint to which hexagon rings at the edges must adapt smoothly, adopting a bent (rather than sharp) transversal profile and an inward longitudinal curvature. The HOMO and LUMO electronic levels, as well as the band gap, are well described using power laws. The gap is predicted to become zero for n ≥ 34 (69,360 atoms). The atomic excess energy with respect to the ideal graphene sheet goes to zero following the log(Nat)/Nat law, which is well described through the continuum elastic theory applied to graphene; the limits for the adopted model are briefly outlined. Compared to larger fullerenes of the series, C60 shows unique features with respect to all the considered properties; C240 presents minor structural and energetic peculiarities, too.

Symmetry and random sampling of symmetry independent configurations for the simulation of disordered solids.

A symmetry-adapted algorithm producing uniformly at random the set of symmetry independent configurations (SICs) in disordered crystalline systems or solid solutions is presented here. Starting from Pólya's formula, the role of the conjugacy classes of the symmetry group in uniform random sampling is shown. SICs can be obtained for all the possible compositions or for a chosen one, and symmetry constraints can be applied. The approach yields the multiplicity of the SICs and allows us to operate configurational statistics in the reduced space of the SICs. The present low-memory demanding implementation is briefly sketched. The probability of finding a given SIC or a subset of SICs is discussed as a function of the number of draws and their precise estimate is given. The method is illustrated by application to a binary series of carbonates and to the binary spinel solid solution Mg(Al,Fe)2O4.

On the use of symmetry in configurational analysis for the simulation of disordered solids.

The starting point for a quantum mechanical investigation of disordered systems usually implies calculations on a limited subset of configurations, generated by defining either the composition of interest or a set of compositions ranging from one end member to another, within an appropriate supercell of the primitive cell of the pure compound. The way in which symmetry can be used in the identification of symmetry independent configurations (SICs) is discussed here. First, Pólya's enumeration theory is adopted to determine the number of SICs, in the case of both varying and fixed composition, for colors numbering two or higher. Then, De Bruijn's generalization is presented, which allows analysis of the case where the colors are symmetry related, e.g. spin up and down in magnetic systems. In spite of their efficiency in counting SICs, neither Pólya's nor De Bruijn's theory helps in solving the difficult problem of identifying the complete list of SICs. Representative SICs are obtained by adopting an orderly generation approach, based on lexicographic ordering, which offers the advantage of avoiding the (computationally expensive) analysis and storage of all the possible configurations. When the number of colors increases, this strategy can be combined with the surjective resolution principle, which permits the efficient generation of SICs of a problem in |R| colors starting from the ones obtained for the (|R| - 1)-colors case. The whole scheme is documented by means of three examples: the abstract case of a square with C(4v) symmetry and the real cases of the garnet and olivine mineral families.

Ab initio modeling of TiO2 nanotubes.

TiO(2) nanotubes constructed from a lepidocrocite-like TiO(2) layer were investigated with ab initio methods employing the periodic CRYSTAL code. The dependence of strain energies, structural and electronic properties on the tube diameter was investigated in the 18-57 A range. Nanotubes constructed by a (0,n) rollup proved to be the most stable at all diameters. All three types of rollup undergo significant reconstruction at diameters <25 A. All investigated structures possess a high ( approximately 5.4 eV) band gap compared to bulk TiO(2) phases (3.96 and 4.63 eV for rutile and anatase calculated with the same functional and basis set).

On the use of symmetry in the ab initio quantum mechanical simulation of nanotubes and related materials.

Nanotubes can be characterized by a very high point symmetry, comparable or even larger than the one of the most symmetric crystalline systems (cubic, 48 point symmetry operators). For example, N = 2n rototranslation symmetry operators connect the atoms of the (n,0) nanotubes. This symmetry is fully exploited in the CRYSTAL code. As a result, ab initio quantum mechanical large basis set calculations of carbon nanotubes containing more than 150 atoms in the unit cell become very cheap, because the irreducible part of the unit cell reduces to two atoms only. The nanotube symmetry is exploited at three levels in the present implementation. First, for the automatic generation of the nanotube structure (and then of the input file for the SCF calculation) starting from a two-dimensional structure (in the specific case, graphene). Second, the nanotube symmetry is used for the calculation of the mono- and bi-electronic integrals that enter into the Fock (Kohn-Sham) matrix definition. Only the irreducible wedge of the Fock matrix is computed, with a saving factor close to N. Finally, the symmetry is exploited for the diagonalization, where each irreducible representation is separately treated. When M atomic orbitals per carbon atom are used, the diagonalization computing time is close to Nt, where t is the time required for the diagonalization of each 2M x 2M matrix. The efficiency and accuracy of the computational scheme is documented.

Single-layered chrysotile nanotubes: A quantum mechanical ab initio simulation.

Chrysotile single-layered nanotubes, obtained by wrapping the Mg(3)Si(2)O(5)(OH)(4) lizardite monolayer along the (n,-n) hexagonal lattice vector, are simulated at the ab initio level by using an all electron 6-31G( *) basis set and the B3LYP functional for n varying from 14 to 24 (the nanotube radius R referred to the oxygen connecting the Mg and Si layers increases from 20 to 35 A). Because of the full exploitation of the helical symmetry, recently implemented in the CRYSTAL code, the computational cost for the full self-consistent field (SCF) and gradient calculation increases only by a factor of 2 and 1.2, respectively, when passing from the lizardite monolayer [18 atoms and 236 AOs (atomic orbitals) in the unit cell] to the (24, -24) tube (864 atoms and 11,328 AOs). The total energy of the tubes is always larger than that of the lizardite monolayer; the difference DeltaE decreases very rapidly with n; for the largest tube here considered (n=24) DeltaE is as small as 2.7 kJ/mol per formula unit (f.u.); extrapolating to larger n values, at about R=50 A, DeltaE becomes smaller than 1 kJ mol f.u. Very large energy gains are observed for small n values during optimization after rolling, mainly due to the rotation of the SiO(4) tetrahedra that are in the inner part of the cylinder ("normal rolling"); such a rigid rotation accounts for about 85% of the overall relaxation energy. "Inverse rolling" tubes (SiO(4) on the external wall of the tube) are shown to be less stable than the corresponding "normal" tubes.