Roberto Orlando

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.

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...

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.

The Periodic Hartree‐Fock Method and Its Implementation in the Crystal Code

The present chapter discusses the Hartree-Fock (HF) method for periodic systems with reference to its implementation in the Crystal program. The HF theory is shortly recalled in its Closed Shell (CS), Unrestricted (UHF) and Restricted open shell (RHF) variants; its extension to periodic systems is illustrated. The general features of Crystal, the periodic ab initio linear combination of atomic orbitals (LCAO) program, able to solve the CS, RHF and UHF, as well as Kohn-Sham equations, are presented. Three examples illustrate the capabilities of the Crystal code and the quality of the HF results in comparison with those obtained with the Local Density Approximation using the same code and basis set: NiO in its ferro-magnetic and anti-ferromagnetic structure, trapped electron holes in doped alkaline earth oxides, and F-centres in LiF.

Ab initio Hartree-Fock calculations for periodic compounds: application to semiconductors

The ab initio Hartree-Fock crystalline orbital program CRYSTAL is applied to diamond, silicon, BN, BP, SiC and AlP. The effects of the computational parameters controlling the accuracy of the infinite Coulomb and exchange series are analysed; the performances of five standard (but re-optimised in the valence part) molecular basis sets (STO-3G; 3-21G; 3-21G*; 6-21G; 6-21G*) are documented with reference to equilibrium binding energy, lattice parameter and bulk modulus. The analysis is then extended, with the largest basis set, to transverse optical phonon frequencies, band-structure and charge-density data. The results show trends similar to those expected from molecular calculations; typically, the mean lattice parameter and bulk modulus errors obtained at a 6-21G* level are about +1% and +10%, respectively.

Coupled perturbed Hartree-Fock for periodic systems: The role of symmetry and related computational aspects

A general and efficient implementation of the coupled perturbed Hartree-Fock (CPHF) scheme in the CRYSTAL06 code that applies to systems periodic in one dimension (polymers), two dimensions (slabs), three dimensions (crystals) and, as a limiting case, zero dimension (molecules) is presented. The dielectric tensor of large unit cell systems such as boehmite (gamma-AlOOH, 8 atoms/cell), calcite (CaCO3, 10 atoms/cell), and pyrope (Mg3Al2Si3O12, 80 atoms/cell) has been computed. Results are well converged with respect to the computational parameters, in particular, to the number of k points in the reciprocal space and tolerances used in the truncation of the Coulomb and exchange series, showing that the same standard computational conditions used for the self-consistent-field (SCF) step can also be used safely in a CPHF calculation. Point symmetry, being so important in determining crystal properties, also reduces dramatically the computational cost both of the preliminary SCF step and the CPHF calculation, so that the dielectric tensor for large unit cell systems such as pyrope can be computed within 2 CPU hours on a single processor PC.

CO adsorption on MgO crystals: Hartree-fock calculations for regular adlayers on a (001) lattice plane

The regular adsorption of CO on a MgO(001) monolayer has been investigated at an “ab initio Hartree-Fock crystalline-orbital level, with a basis set containing 9 atomic orbitals per atom. Both vertical bindings to Mg (through oxygen and carbon) have been considered and the C-O as well as the Mg-CO or Mg-OC distances have been optimized. At the 1/2 surface coverage (one CO molecule every two surface Mg ions) adsorption energies of 4.13 (carbon binding) and 3.29 (oxygen binding) kcal/mol were obtained, to be compared with an experimental adsorption heat of 3.6 kcal/mol. Similar energies were obtained for the 1/4 coverage, whereas the 1/1 adsorption is less favoured, due to the repulsion between the CO molecules. The charge redistributions in the CO molecule and in the substrate are considered, as well as the changes occurring in the density of states for the interacting systems.