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A computational procedure is developed for the efficient calculation of derivatives of integrals over non-separable Gaussian-type basis functions, used for the evaluation of gradients of the total energy in quantum-mechanical simulations. The approach, based on symbolic computation with computer algebra systems and automated generation of optimized subroutines, takes full advantage of sparsity and is here applied to first energy derivatives with respect to nuclear displacements and lattice parameters of molecules and materials. The implementation in the Crystal code is presented, and the considerably improved computational efficiency over the previous implementation is illustrated. For this purpose, three different tasks involving the use of analytical forces are considered: (i) geometry optimization; (ii) harmonic frequency calculation; and (iii) elastic tensor calculation. Three test case materials are selected as representatives of different classes: (i) a metallic 2D model of the Cu(111) surface; (ii) a wide-gap semiconductor ZnO crystal, with a wurtzite-type structure; and (iii) a porous metal-organic crystal, namely the ZIF-8 zinc-imidazolate framework. Finally, it is argued that the present symbolic approach is particularly amenable to generalizations, and its potential application to other derivatives is sketched.

Cubic cuprous oxide, Cu

A general methodology has been devised and implemented into the solid-state ab initio quantum-mechanical Crystal program for studying the evolution under geophysical pressure of the elastic anisotropy of crystalline materials. This scheme, which fully exploits both translational and point symmetry of the crystal, is developed within the formal frame of one-electron Hamiltonians and atom-centered basis functions. Six silicate garnet end-members, among the most important rock-forming minerals of the Earth's mantle, are considered, whose elastic anisotropy is fully characterized under high hydrostatic compressions, up to 60 GPa. The pressure dependence of azimuthal anisotropy and shear-wave birefringence of seismic wave velocities for these minerals are accurately simulated and compared with available single-crystal measurements.

A general-purpose, fully automated, computationally efficient implementation is presented of a series of techniques for the simultaneous description of pressure and temperature effects on structural properties of materials, by means of standard ab initio simulations. Equilibrium volume, bulk modulus, thermal expansion coefficient, equation-of-state, Grüneisen parameter, constant-pressure and constant-volume specific heats are computed as a function of temperature and pressure for the simple crystal of diamond and compared with accurate experimental data. Convergence of computed properties with respect to super-cell size is critically discussed. The effect on such properties of the adopted exchange-correlation functional of the density-functional-theory is discussed by considering three different levels of approximation (including hybrids): it is found to be rather small for the temperature dependence of equilibrium volume and bulk modulus, whereas it is quite large as regards their absolute values.

Two periodic codes, CRYSTAL and CRYSCOR, are here used to simulate and characterize the librational modes of the nu(R) band of Ice XI: this band has been found experimentally to be the region of the vibrational spectrum of ordinary ice most affected by the transition from the proton-disordered (Ice Ih) to the proton-ordered (Ice XI) phase. With CRYSTAL, the problem is solved using Hartree-Fock (HF), pure Kohn-Sham (PW91) or hybrid (B3LYP) one-electron Hamiltonians: the harmonic approximation is employed to obtain the vibrational spectrum after optimizing the geometry. The B3LYP results are those in best agreement with the experiment. For a given crystalline geometry, CRYSCOR computes the energy per cell in an ab initio HF+MP2 approximation using a local-correlation approach; this technique is employed for recalculating the frequencies of the different modes identified by the B3LYP approach, by fully accounting for long range dispersive interactions. The effect of anharmonicity is evaluated separately for each mode both in the B3LYP and HF+MP2 case. The two approaches accurately reproduce the four-peak structure of the librational band. The harmonic B3LYP nu(R) bandwidth of 70 meV is lowered to 60 meV by anharmonic corrections, and becomes 57 meV in the HF+MP2 anharmonic calculation, in excellent agreement with the experimental IINS data (56-59 meV). The assignment of the librational modes is discussed.