V. R. Saunders

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].

Crystal field effects on the topological properties of the electron density in molecular crystals: The case of urea

The Quantum Theory of Atoms in Molecules, due to Bader, is applied to periodic systems. Results for molecular and crystalline urea are presented. Changes in both bond critical points and atomic properties due to changes of chemical environment are described. A rationale for the different lengths of the in‐plane and out‐of‐plane hydrogen bonds and for the lengthening of the CO bond in bulk urea is provided in terms of the properties of the Laplacian of the oxygen atom electron density distribution. An evaluation of molecular and atomic volume changes indicates that the decrease of molecular volume upon change of phase from gas to solid originates primarily from a contraction of the atomic basins directly involved in hydrogen bonds. Other atoms show a small expansion. The considerable decrease of oxygen and hydrogen atomic volumes is related to the mutual penetration of their van der Waals envelopes following hydrogen bond formation. The results confirm that urea is more polar in the solid phase.

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.

The direct CI method

A thorough analysis of the direct CI method as applied to the case of a general set of reference configurations coupled to all single and double substitutions is presented. It is pointed out that there is no single strategy which proves optimal under all circumstances. A variety of procedures are therefore presented together with rules to enable the selection of the most favourable under a given circumstance. Much emphasis has been placed on organizing the calculations via a series of matrix multiplications, which enables a vector or array processing computer to be used to best effect. Some consideration is given to using an atomic integral (rather than molecular integral) driven scheme for some interactions, thus removing the necessity for a complete transformation of the molecular integrals to a molecular orbital basis, and the advantages and disadvantages of so doing are discussed. Improved procedures for carrying out both full and partial transformations of the molecular integrals are described. A num...

Analytical Hartree–Fock gradients for periodic systems

We present the theory of analytical Hartree–Fock gradients for periodic systems as implemented in the code CRYSTAL. We demonstrate how derivatives of the integrals can be computed with the McMurchie–Davidson algorithm. Highly accurate gradients with respect to nuclear coordinates are obtained for systems periodic in 0, 1, 2, or 3 dimensions.

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.

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.

An investigation of self-consistent field perturbation theory applied to the calculation of nuclear spin-spin coupling constants

An initio SCF perturbation theory in the formulation of Ditchfield and Snyder has been applied to the calculation of the N.M.R. spin-spin coupling constants in CH4, NH3, H2O and HF using large gaussian basis sets, and including all second-order contributions (Fermi contact, orbital and dipolar). In accordance with most other work, the first-order term in the orbital contribution has been neglected, however. The sensitivity of the coupling constants to variations in the basis set has been studied, and the reliability of the method is discussed in the light of these results. It is shown that the Fermi contact contribution provides the largest component of the calculated coupling constants, with the dipolar term being generally negligible. However, the orbital contribution is found to be non-negligible, particularly in the case of the directly bonded couplings in HF and H2O, and is shown to be the primary reason, at the level of calculation adopted, for variations in the HH geminal couplings in the series CH...