Roberto Dovesi

CRYSTAL: a computational tool for the ab initio study of the electronic properties of crystals

Abstract CRYSTAL [1] computes the electronic structure and properties of periodic systems (crystals, surfaces, polymers) within Hartree-Fock [2], Density Functional and various hybrid approximations. CRYSTAL was developed during nearly 30 years (since 1976) [3] by researchers of the Theoretical Chemistry Group in Torino (Italy), and the Computational Materials Science group in CLRC (Daresbury, UK), with important contributions from visiting researchers, as documented by the main authors list and the bibliography. The basic features of the program CRYSTAL are presented, with two examples of application in the field of crystallography [4, 5].

The Calculation of the Vibrational Frequencies of Crystalline Compounds and Its Implementation in the CRYSTAL Code

The problem of numerical accuracy in the calculation of vibrational frequencies of crystalline compounds from the hessian matrix is discussed with reference to α‐quartz (SiO2) as a case study and to the specific implementation in the CRYSTAL code. The Hessian matrix is obtained by numerical differentiation of the analytical gradient of the energy with respect to the atomic positions. The process of calculating vibrational frequencies involves two steps: the determination of the equilibrium geometry, and the calculation of the frequencies themselves. The parameters controlling the truncation of the Coulomb and exchange series in Hartree–Fock, the quality of the grid used for the numerical integration of the Exchange‐correlation potential in Density Functional Theory, the SCF convergence criteria, the parameters controlling the convergence of the optimization process as well as those controlling the accuracy of the numerical calculation of the Hessian matrix can influence the obtained vibrational frequencies to some extent. The effect of all these parameters is discussed and documented. It is concluded that with relatively economical computational conditions the uncertainty related to these parameters is smaller than 2–4 cm−1. In the case of the Local Density Approximation scheme, comparison is possible with recent calculations performed with a Density Functional Perturbation Theory method and a plane‐wave basis set.

Calculation of the vibration frequencies of α-quartz: The effect of Hamiltonian and basis set

The central‐zone vibrational spectrum of α‐quartz (SiO2) is calculated by building the Hessian matrix numerically from the analytical gradients of the energy with respect to the atomic coordinates. The nonanalytical part is obtained with a finite field supercell approach for the high‐frequency dielectric constant and a Wannier function scheme for the evaluation of Born charges. The results obtained with four different Hamiltonians, namely Hartree–Fock, DFT in its local (LDA) and nonlocal gradient corrected (PBE) approximation, and hybrid B3LYP, are discussed, showing that B3LYP performs far better than LDA and PBE, which in turn provide better results than HF, as the mean absolute difference from experimental frequencies is 6, 18, 21, and 44 cm−1, respectively, when a split valence basis set containing two sets of polarization functions is used. For the LDA results, comparison is possible with previous calculations based on the Density Functional Perturbation Theory and usage of a plane‐wave basis set. The effects associated with the use of basis sets of increasing size are also investigated. It turns out that a split valence plus a single set of d polarization functions provides frequencies that differ from the ones obtained with a double set of d functions and a set of f functions on all atoms by on average less than 5 cm−1.

Hartree-Fock geometry optimisation of periodic systems with the CRYSTAL code

Abstract Results are reported on the geometry optimisation of periodic systems with the Hartree–Fock analytical gradients recently implemented in the C rystal code. Application to the structure optimisation of molecules, polymers, slabs and crystals is presented.

On the elastic properties of lithium, sodium and potassium oxide. An ab initio study

Abstract The binding energy, equilibrium lattice parameter, elastic constants and central zone phonon frequencies of Li2O, Na2O and K2O have been calculated at an ab initio level with CRYSTAL, a Hartree-Fock linear combination of atomic orbitals (LCAO) program for periodic compounds. The variational basis set has been optimized for each compound, and is reported for future reference and work. A quite satisfactory agreement with very recent experimental data is obtained for Li2O. With regard to Na2O and K2O no experimental data exist, to our knowledge, for the elastic constants and the central zone phonon frequencies, so that the present data represent the first determination of these quantities.

Abinitio approach to molecular crystals: A periodic Hartree–Fock study of crystalline urea

The electronic structure of crystalline urea (two molecules, 16 atoms per unit cell) is investigated at an ab initio level with CRYSTAL, a Hartree–Fock linear combination of atomic orbitals (LCAO) program for periodic systems. The influence of the basis set and of the computational parameters which control the treatment of the Coulomb and exchange series and the reciprocal space integration is documented; results include total and interaction energy, Mulliken analysis data and interaction (solid minus molecules) density maps, band structure, and density of states. The crystal field modifies the electronic structure of the isolated molecule, the main effect being an increase in the ionicity of bonds. The interaction energy obtained with a 6‐21** basis set is 28 kcal/mol, (16 kcal/mol after a correction of the basis set superposition error by using the counterpoise method) to be compared with 21±0.5 kcal/mol from experiment. This preliminary application shows that accurate ab initio calculations of hydrogen...

AB initio hartree-fock study of the MgO(001) surface

Abstract The MgO(001) surface has been studied with an ab initio Hartree-Fock crystalline orbital LCAO program. An optimized basis set containing nine atomic orbitals (three s and six p) per atom has been used. The semi-infinite crystal has been simulated by a slab containing three planes (six atoms per cell). In agreement with the most recent LEED experiments, no relaxation is found and the “rumpling” is very small (1% of the nearest neighbour separation in the bulk). The analysis of the electron charge distribution and of the ion multipoles shows that the fully ionic character found for the bulk is maintained at the surface, and that the anion deformation is very small. A surface energy of 1.43 J/m 2 was obtained.

The calculation of static polarizabilities of 1-3D periodic compounds. the implementation in the crystal code

The Coupled Perturbed Hartree–Fock (CPHF) scheme has been implemented in the CRYSTAL06 program, that uses a gaussian type basis set, for systems periodic in 1D (polymers), 2D (slabs), 3D (crystals) and, as a limiting case, 0D (molecules), which enables comparison with molecular codes. CPHF is applied to the calculation of the polarizability α of LiF in different aggregation states: finite and infinite chains, slabs, and cubic crystal. Correctness of the computational scheme for the various dimensionalities and its numerical efficiency are confirmed by the correct trend of α: α for a finite linear chain containing N LiF units with large N tends to the value for the infinite chain, N parallel chains give the slab value when N is sufficiently large, and N superimposed slabs tend to the bulk value. CPHF results compare well with those obtained with a saw‐tooth potential approach, previously implemented in CRYSTAL. High numerical accuracy can easily be achieved at relatively low cost, with the same kind of dependence on the computational parameters as for the SCF cycle. Overall, the cost of one component of the dielectric tensor is roughly the same as for the SCF cycle, and it is dominated by the calculation of two‐electron four‐center integrals.

A general method to obtain well localized Wannier functions for composite energy bands in linear combination of atomic orbital periodic calculations

A method for obtaining spatially localized crystalline orbitals starting from delocalized Bloch functions is proposed. The method, that has been implemented in the LCAO CRYSTAL code, is intrinsic and general for nonconducting systems, and provides a set of well localized Wannier functions that can be used for applications that take advantage of their localized character. Examples are given that illustrate the performances and efficiency of the proposed scheme.

Ab-initio prediction of materials properties with CRYSTAL: MOF-5 as a case study

MOF-5 is by far the most relevant member of the new class of metal–organic framework materials and has been adopted as a case study to show that reliable ab initio prediction of materials properties of complex systems can be obtained by means of a solid state computational tool like the CRYSTAL code. Structure, electronic properties and vibrational frequencies of MOF-5 computed at the B3LYP level of theory are reported and discussed. Animations representing MOF-5 vibrations are available at the web site: http://www.crystal.unito.it/vibs/mof5