Generation of Helical States - Breaking of Symmetries, Curie's Principle, and Excited States.

Herein, we deeply detail for the very first time mathematical concepts behind the generation of helical molecular orbitals (MOs) for linear chains of atoms. We first give a definition of helical MOs and we provide an index measuring how far a given helical states is from a perfect helical distribution. Structural properties of helical distribution for twisted N ${\left[N\right]}$ -cumulene and cumulene version of Möbius systems are given. We then give some simple structural assumptions as well as symmetry requirements ensuring the existence of helical MOs. Considering molecules which do not admit helical MOs, we provide a first way to induce helical states by the breaking of symmetries. We also explore an alternative way using excited conformations of given molecules as well as different electronic multiplicities.

The effect of composition on phonon softening in ABO3-type perovskites: DFT modelling.

The evolution of ferroelectric instability in ABO3 perovskites is systematically investigated for tantalates, niobates and titanates at the hybrid density-functional theory level. The influence of the A cation is analysed in terms of the frequency of the lowest F1u IR-active phonon mode at different volumes for (Cs, Rb, K, Na)TaO3, (Ba, Pb, Sn, Ge)TiO3 and (Rb, K, Na, Li)NbO3 and correlated with the ionic radius as well as the degree of hybridization in the bonds. The atomic displacement corresponding to each mode is described as a function of volume, and the static permittivity is calculated for the stable Pm3̄m phases. It is shown that the amplitude of the atomic displacements associated with the soft mode linked to the ferroelectric instability increases at a given volume when the ionic radius of the cation A decreases and when the hybridization of the B-O bond increases. This provides criteria for optimizing the dielectric properties of materials and for suggesting effective solid solutions. Tantalum perovskites presenting para-ferroelectric phase transitions, some of which are close to ambient conditions, are interesting materials for high-permittivity dielectrics in view of lead-free compounds with a high static dielectric response.

The role of spin density for understanding the superexchange mechanism in transition metal ionic compounds. The case of KMF3 (M = Mn, Fe, Co, Ni, Cu) perovskites.

In many recent papers devoted to first row transition metal fluorides and oxides, not much attention is devoted to the spin density, a crucial quantity for the determination of the superexchange mechanism, and then for the ferro-antiferromagnetic energy difference. Usually, only the eigenvalues of the system are represented, in the form of band structures or, more frequently, of density of states (DOS). When discussing the orbital ordering and the Jahn-Teller effect, simple schemes with cubes and lobes are used to illustrate the shape of the d occupancy. But the eigenvectors, and the resulting spin density function, as obtained from the calculations, are rarely shown. When represented, only a fuzzy shape that recalls the d orbital shape can be observed. On the basis of these considerations, spin density maps for 5 compounds of the KMF3 (M from Mn to Cu) family have been produced, which clearly illustrate which d orbital is singly or doubly occupied. At variance with respect to the near totality of the papers devoted to these systems, we use an all electron scheme, a Gaussian type basis set, and the Hartree-Fock Hamiltonian or the B3LYP hybrid functional (the resulting maps turn out to be very similar, in the scale used for our figures). The spin density in the five cases can easily be interpreted in terms of the shape of the d orbitals as appearing in textbooks.

Substitutional carbon defects in silicon: A quantum mechanical characterization through the infrared and Raman spectra.

The infrared (IR) and Raman spectra of eight substitutional carbon defects in silicon are computed at the quantum mechanical level by using a periodic supercell approach based on hybrid functionals, an all electron Gaussian type basis set and the CRYSTAL code. The single substitutional C s case and its combination with a vacancy (C s V and C s SiV) are considered first. The progressive saturation of the four bonds of a Si atom with C is then examined. The last set of defects consists of a chain of adjacent carbon atoms C s i , with i = 1-3. The simple substitutional case, C s , is the common first member of the three sets. All these defects show important, very characteristic features in their IR spectrum. One or two C related peaks dominate the spectra: at 596 cm-1 for C s (and C s SiV, the second neighbor vacancy is not shifting the C s peak), at 705 and 716 cm-1 for C s V, at 537 cm-1 for C s 2 and C s 3 (with additional peaks at 522, 655 and 689 for the latter only), at 607 and 624 cm-1 , 601 and 643 cm-1 , and 629 cm-1 for SiC s 2 , SiC s 3 , and SiC s 4 , respectively. Comparison with experiment allows to attribute many observed peaks to one of the C substitutional defects. Observed peaks above 720 cm-1 must be attributed to interstitial C or more complicated defects.

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.

Characterization of the B-Center Defect in Diamond through the Vibrational Spectrum: A Quantum-Mechanical Approach.

The B-center in diamond, which consists of a vacancy whose four first nearest-neighbors are nitrogen atoms, has been investigated at the quantum-mechanical level with an all-electron Gaussian-type basis set, hybrid functionals, and the periodic supercell approach. To simulate various defect concentrations, four cubic supercells have been considered, containing (before the creation of the vacancy) 64, 216, 512, and 1000 atoms, respectively. Whereas the B-center does not affect the Raman spectrum of diamond, several intense peaks appear in the IR spectrum, which should permit us to identify this defect. It turns out that of the seven peaks proposed by Sutherland in 1954, located at 328, 780, 1003, 1171, 1332, 1372, and 1426 cm-1, and frequently mentioned as fingerprints of the B center, the first one and the last three do not appear in the simulated spectrum at any concentration. The graphical animation of the modes confirms the attribution of the remaining three and also permits investigation of the nature of the full set of modes.

Establishing the pivotal role of local aromaticity in the electronic properties of boron-nitride graphene lateral hybrids.

Within an attempt to unravel the conundrum of irregular bandgap variations in hybrids of white-graphene (hBN) and graphene (G) observed in both experiment and theory, strong proofs about the decisive role of aromaticity in their electronic properties are brought to light. Sound numerical experiments conducted on zero-, one- and two-dimensional hBNG hybrids demonstrate that upon structural and/or electronic perturbation caused by foreign doping agents, the uniformity in local cyclic electron delocalization of ideal graphene restructures locally creating carbon hexagons of contrasting cyclic electron delocalization (c.c. local aromatic patterns) which may dominate the bandgap size of the resulting systems. In addition, relying on the quantum chemical aspect of aromaticity in terms of quantitative computations of cyclic electron delocalization together with pictorial intrinsic polarizability density representations, this work provides a solid and handy rule-of-thumb to be used in qualitative and intuitive predictions. According to this empirical rule, the origin of any nonmonotonic bandgap variation observed in stoichiometric 0D (BN)n/graphene hybrids with increasing hBN segment lies in instabilities caused by partially substituted benzenoid rings formed locally at the hBNG interfaces. This relationship, established in 0D graphene flakes and extended to 1D periodic ribbons, can be used to understand and qualitatively predict conflicting bandgap variations of vacancy-free 2D periodic lattices, pointing at the property of aromaticity as the missing link needed to solve the puzzle of conflicting bandgap variations in hBNG hybrids observed in experiment.

Calculation of the dynamic first electronic hyperpolarizability ß(-ω(σ); ω(1), ω(2)) of periodic systems. Theory, validation, and application to multi-layer MoS2.

We describe our implementation of a fully analytical scheme, based on the 2n + 1 rule, for computing the coupled perturbed Hartree Fock and Kohn-Sham dynamic first hyperpolarizability tensor ß(-ωσ; ω1, ω2) of periodic 1D (polymer), 2D (slab), and 3D (crystal) systems in the CRYSTAL code [R. Dovesi et al., Int. J. Quantum Chem. 114, 1287 (2014)], which utilizes local Gaussian type basis sets. The dc-Pockels (dc-P) and second harmonic generation (SHG) tensors are included as special cases. It is verified that (i) symmetry requirements are satisfied; (ii) using LiF as an example, the infinite periodic polymer result agrees with extrapolated finite oligomer calculations and, likewise, for the build-up to a 2D slab and a 3D crystal; (iii) the values converge to the static case for low frequencies; and (iv) the Bishop-deKee dispersion formulas relating dc-P, SHG, and general processes are reproduced through quartic terms. Preliminary SHG calculations on multi-layer MoS2 satisfactorily reproduce experimental data.

The electronic structure of MgO nanotubes. An ab initio quantum mechanical investigation.

The structural, vibrational and response properties of the (n,0) and (m,m) MgO nanotubes are computed by using a Gaussian type basis set, a hybrid functional (B3LYP) and the CRYSTAL09 code. Tubes in the range 6 ≤ n ≤ 140 and 3 ≤ m ≤ 70 were considered, being n = 2 × m the number of MgO units in the unit cell (so, the maximum number of atoms is 280). Tubes are built by rolling up the fully relaxed 2-D conventional cell (2 MgO units, with oxygen atoms protruding from the Mg plane alternately up and down by 0.38 Å). The relative stability of the (n,0) with respect to the (m,m) family, the relaxation energy and equilibrium geometry, the band gap, the IR vibrational frequencies and intensities, and the electronic and ionic contributions to the polarizability are reported. All these properties are shown to converge smoothly to the monolayer values. Absence of negative vibrational frequencies confirms that the tubes have a stable structure. The parallel component of the polarizability α(∥) converges very rapidly to the monolayer value, whereas α(⊥) is still changing at n = 140; however, when extrapolated to very large n values, it coincides with the monolayer value to within 1%. The electronic contribution to α is in all cases (α(∥) and α(⊥); 6 ≤ n ≤ 140) smaller than the vibrational contribution by about a factor of three, at variance with respect to more covalent tubes such as the BN ones, for which the ratio between the two contributions is reversed.

Ab initio analytical Raman intensities for periodic systems through a coupled perturbed Hartree-Fock/Kohn-Sham method in an atomic orbital basis. I. Theory.

We present a fully analytical formulation for calculating Raman intensities of crystalline periodic systems using a local basis set. Numerical differentiation with respect to atomic coordinates and with respect to wavevectors is entirely avoided as is the determination of crystal orbital coefficient derivatives with respect to nuclear displacements. Instead, our method utilizes the orbital energy-weighted density matrix and is based on the self-consistent solution of first- and second-order Coupled Perturbed Hartree-Fock/Kohn-Sham equations for the electronic response to external electric fields at the equilibrium geometry. This method has also been implemented in the Crystal program, which uses a Gaussian type basis set.

Ab initio analytical Raman intensities for periodic systems through a coupled perturbed Hartree-Fock/Kohn-Sham method in an atomic orbital basis. II. Validation and comparison with experiments.

In this work, we validate a new, fully analytical method for calculating Raman intensities of periodic systems, developed and presented in Paper I [L. Maschio, B. Kirtman, M. Rérat, R. Orlando, and R. Dovesi, J. Chem. Phys. 139, 164101 (2013)]. Our validation of this method and its implementation in the CRYSTAL code is done through several internal checks as well as comparison with experiment. The internal checks include consistency of results when increasing the number of periodic directions (from 0D to 1D, 2D, 3D), comparison with numerical differentiation, and a test of the sum rule for derivatives of the polarizability tensor. The choice of basis set as well as the Hamiltonian is also studied. Simulated Raman spectra of α-quartz and of the UiO-66 Metal-Organic Framework are compared with the experimental data.

Ab initio analytical infrared intensities for periodic systems through a coupled perturbed Hartree-Fock/Kohn-Sham method.

A fully analytical method for calculating Born charges and, hence, infrared intensities of periodic systems, is formulated and implemented in the CRYSTAL program, which uses a local gaussian type basis set. Our efficient formalism combines integral gradients with first-order coupled perturbed Hartree-Fock/Kohn Sham electronic response to an electric field. It avoids numerical differentiation with respect to wave vectors, as in some Berry phase approaches, and with respect to atomic coordinates. No perturbation equations for the atomic displacements need to be solved. Several tests are carried out to verify numerical stability, consistency in one, two, and three dimensions, and applicability to large unit cells. Future extensions to piezoelectricity and Raman intensities are noted.

Calculation of longitudinal polarizability and second hyperpolarizability of polyacetylene with the coupled perturbed Hartree-Fock/Kohn-Sham scheme: where it is shown how finite oligomer chains tend to the infinite periodic polymer.

The longitudinal polarizability, α(xx), and second hyperpolarizability, γ(xxxx), of polyacetylene are evaluated by using the coupled perturbed Hartree-Fock/Kohn-Sham (HF/KS) scheme as implemented in the periodic CRYSTAL code and a split valence type basis set. Four different density functionals, namely local density approximation (LDA) (pure local), Perdew-Becke-Ernzerhof (PBE) (gradient corrected), PBE0, and B3LYP (hybrid), and the Hartree-Fock Hamiltonian are compared. It is shown that very tight computational conditions must be used to obtain well converged results, especially for γ(xxxx), that is, very sensitive to the number of k(->) points in reciprocal space when the band gap is small (as for LDA and PBE), and to the extension of summations of the exact exchange series (HF and hybrids). The band gap in LDA is only 0.01 eV: at least 300 k(->) points are required to obtain well converged total energy and equilibrium geometry, and 1200 for well converged optical properties. Also, the exchange series convergence is related to the band gap. The PBE0 band gap is as small as 1.4 eV and the exchange summation must extend to about 130 Å from the origin cell. Total energy, band gap, equilibrium geometry, polarizability, and second hyperpolarizability of oligomers -(C(2)H(2))(m)-, with m up to 50 (202 atoms), and of the polymer have been compared. It turns out that oligomers of that length provide an extremely poor representation of the infinite chain polarizability and hyperpolarizability when the gap is smaller than 0.2 eV (that is, for LDA and PBE). Huge differences are observed on α(xx) and γ(xxxx) of the polymer when different functionals are used, that is in connection to the well-known density functional theory (DFT) overshoot, reported in the literature about short oligomers: for the infinite model the ratio between LDA (or PBE) and HF becomes even more dramatic (about 500 for α(xx) and 10(10) for γ(xxxx)). On the basis of previous systematic comparisons of results obtained with various approaches including DFT, HF, Moller-Plesset (MP2) and coupled cluster for finite chains, we can argue that, for the infinite chain, the present HF results are the most reliable.

Static and dynamic coupled perturbed Hartree-Fock vibrational (hyper)polarizabilities of polyacetylene calculated by the finite field nuclear relaxation method.

The vibrational contribution to static and dynamic (hyper)polarizability tensors of polyacetylene are theoretically investigated. Calculations were carried out by the finite field nuclear relaxation (FF-NR) method for periodic systems, newly implemented in the CRYSTAL code, using the coupled perturbed Hartree-Fock scheme for the required electronic properties. The effect of the basis set is also explored, being particularly important for the non-periodic direction perpendicular to the polymer plane. Components requiring a finite (static) field in the longitudinal direction for evaluation by the FF-NR method were not evaluated. The extension to that case is currently being pursued. Whereas the effect on polarizabilities is relatively small, in most cases the vibrational hyperpolarizability tensor component is comparable to, or larger than the corresponding static electronic contribution.

First-Principles Calculation of the Optical Rotatory Power of Periodic Systems: Application on α-Quartz, Tartaric Acid Crystal, and Chiral (n,m)-Carbon Nanotubes.

The self-consistent coupled-perturbed (SC-CP) method in the CRYSTAL program has been adapted to obtain electromagnetic optical rotation properties of chiral periodic systems based on the calculation of the magnetic moment induced by the electric field. Toward that end, an expression for the magnetic transition moment is developed, which involves an appropriate electronic angular momentum operator. This operator is forced to be hermitian so that the chiroptical properties are real. In our formulation, the trace of the optical rotatory power matrix is gauge-origin-invariant as long as the electric dipole transition matrix elements are obtained using the velocity (rather than position) operator. On the other hand, the component along the optic axis is invariant in general for uniaxial and biaxial crystals. Under the same conditions, these properties also do not depend on the so-called missing integers that occur in the treatment of the electric dipole moment of quasi-one-dimensional periodic systems or the analogue of missing integers for the case of higher dimensionality. Tests on a model H2O2 polymer confirm the formalism and, as desired, show that the calculated properties are independent of the size and definition of the unit cell. In addition, an empirical relation to a finite oligomer gauge-including atomic orbital (GIAO) calculation is found. Applications, with comparison to experiment, are carried for α-quartz, tartaric acid crystal, and carbon nanotubes. Future developments of this initial approach to chiroptical properties in the solid state are noted.

Brillouin spectroscopy, calculated elastic and bond properties of GaAsO4.

Experimental and theoretical studies have been performed to demonstrate the high performance of the novel piezoelectric material GaAsO(4). Hydrothermally grown single crystals of α-quartz phase GaAsO(4) were studied by Brillouin spectroscopy to determine elastic constants. Experimentally obtained values of C(11), C(66), C(33), C(44), C(14) and C(12) are 59.32, 19.12, 103.54, 30.70, 1.7, and 21.1 GPa, respectively. Elastic and piezoelectric tensors were also calculated by a first principles method in this work, leading to a very good agreement with experimental results and confirming the values of elastic components obtained indirectly such as C(14) and the negligible piezoelectric correction for C(11). The thermal behavior of the elastic constant corresponding to the [100] longitudinal L mode (C(11)) was studied up to 1137 K to estimate potential piezoelectric performance. It was found that the thermal behavior is linear up to 1273 K which is just below the thermal decomposition temperature of 1303 K. High thermal stability can be linked to the higher polarizability of large cations Ga and As because of neighboring oxygen atoms. On the basis of thermal behavior, GaAsO(4) is a promising material for high temperature piezoelectric applications.

The calculation of the static first and second susceptibilities of crystalline urea: A comparison of Hartree-Fock and density functional theory results obtained with the periodic coupled perturbed Hartree-Fock/Kohn-Sham scheme.

The static polarizability alpha and first hyperpolarizability beta tensors of crystalline urea and the corresponding first-(chi((1))) and second-(chi((2))) susceptibilities are calculated and compared to the same quantities obtained for the molecule by using the same code (a development version of CRYSTAL), basis set, and level of theory. In order to separate geometrical and solid state effects, two geometries are considered for the molecule in its planar conformation: (i) as cut out from the bulk structure and (ii) fully optimized. First, the effect of basis sets on computed properties is explored at the B3LYP level by employing basis sets of increasing complexity, from 6-31G(d,p) to 6-311G(2df,2pd) (Pople's family) and from DZP to QZVPPP (Thakkar/Ahlrichs/Dunning's family) on alpha and beta for both the molecule and the bulk. Then, five different levels of theory, namely, SVWN (local density approximation), PBE (generalized gradient approximation), PBE0 and B3LYP (hybrid), and Hartree-Fock are compared in combination with a TZPP basis set. Present results show that hybrid methods, in particular, B3LYP, are remarkably successful in predicting correctly both the first and second susceptibilities of urea bulk when combined at least with a triple-zeta quality basis set containing a double set of polarization functions. It is also shown that diffuse functions that are needed for molecular calculations are less crucial for the crystalline structure, as expected. Indeed, B3LYP/TZPP computed chi((1)) and chi((2)) tensor components (chi(aa) ((1))=1.107, chi(cc) ((1))=1.459, and chi((2))=-0.93 a.u.) are in very good agreement with experimental values. At variance with respect to previous periodic ab initio calculations, but in agreement with recent supermolecular results, the negative sign of chi((2)) is confirmed. Overall, static linear and nonlinear optical properties such as dielectric constants, refractive, and birefringence indices and second-harmonic generation coefficient of crystalline urea are very well reproduced by present calculations.

Calculation of the static electronic second hyperpolarizability or chi(3) tensor of three-dimensional periodic compounds with a local basis set.

The coupled perturbed Hartree-Fock (CPHF) method for evaluating static first (beta) and second (gamma) hyperpolarizability tensors of periodic systems has recently been implemented in the CRYSTAL code [Bishop et al., J. Chem. Phys. 114, 7633 (2001)]. We develop here an efficient and accurate computational protocol, along with the local basis sets needed for first and second row atoms. Application is made to several high symmetry three-dimensional systems including one (pyrope) with an 80 atom unit cell. CPHF second-order hyperpolarizabilities substantially undershoot experimental values, due to an overestimate of the band gap, but trends are satisfactorily reproduced for beta as well as gamma.

Polarization of one-dimensional periodic systems in a static electric field: sawtooth potential treatment revisited.

Various periodic piecewise linear potentials for extracting the electronic response of an infinite periodic system to a uniform electrostatic field are examined. It is shown that discontinuous potentials, such as the sawtooth, cannot be used for this purpose. Continuous triangular potentials can be successfully employed to determine both even- and odd-order (hyper)polarizabilities, as demonstrated here for the first time, although the permanent dipole moment of the corresponding long finite chain remains out of reach. Moreover, for typical highly polarizable organic systems, the size of the repeated unit has to be much larger than that of the finite system in order to obtain convergence with respect to system size. All results are illustrated both through extensive model calculations and through ab initio calculations on poly- and oligoacetylenes.

Calculation of the dielectric constant epsilon and first nonlinear susceptibility chi((2)) of crystalline potassium dihydrogen phosphate by the coupled perturbed Hartree-Fock and coupled perturbed Kohn-Sham schemes as implemented in the CRYSTAL code.

The high-frequency dielectric varepsilon and the first nonlinear electric susceptibility chi((2)) tensors of crystalline potassium dihydrogen phosphate (KH(2)PO(4)) are calculated by using the coupled perturbed Hartree-Fock and Kohn-Sham methods as implemented in the CRYSTAL code. The effect of basis sets of increasing size on varepsilon and chi((2)) is explored. Five different levels of theory, namely, local-density approximation, generalized gradient approximation (PBE), hybrids (B3LYP and PBE0), and HF are compared using the experimental and theoretical structures corresponding not only to the tetragonal geometry I4d2 at room temperature but also to the orthorhombic phase Fdd2 at low temperature. Comparison between the two phases and their optical behavior is made. The calculated results for the tetragonal phase are in good agreement with the experimental data.