Lorenzo Maschio

Periodic local MP2 method for the study of electronic correlation in crystals: Theory and preliminary applications

A computational technique for solving the MP2 equations for periodic systems using a local‐correlation approach and implemented in the CRYSCOR code is presented. The Hartree‐Fock solution provided by the CRYSTAL program is used as a reference. The motivations for the implementation of the new code are discussed, and the techniques adopted are briefly recalled. With respect to the original formulation (Pisani et al., J Chem Phys 2005, 122, 094113), many new features have been introduced in CRYSCOR to improve its efficiency and robustness. In particular, an adaptation of the density fitting scheme to translationally periodic systems is described, based on Fourier transformation techniques. Three examples of application are provided, concerning the CO2 crystal, proton transfer in ice XI, and the adsorption of methane on MgO (001). The results obtained with the periodic LMP2 method for these systems appear more reliable than the ones obtained using density functional theory.

Intermolecular interaction energies in molecular crystals: comparison and agreement of localized Møller-Plesset 2, dispersion-corrected density functional, and classical empirical two-body calculations.

A comparative analysis of the intermolecular energy for a data set including 60 molecular crystals with a large variety of functional groups has been carried out using three different computational approaches: (i) a method based on a physically meaningful empirical partition of the interaction energy (PIXEL), (ii) density functional methods with a posteriori empirical correction for the dispersion interactions (DFT-D), and (iii) a full periodic ab initio quantum mechanical method based on Møller-Plesset perturbation theory for the electron correlation using localized crystal orbitals (LMP2). Due to the large computational cost, LMP2 calculations have been restricted to a subset of seven molecular crystal comprising benzene, formic acid, formamide, succinic anhydride, urea, oxalic acid, and nitroguanidine, and the results compared with PIXEL and DFT-D data as well as with the experimental data show excellent agreement among all adopted methods. This shows that both DFT-D and PIXEL approaches are robust predictive tools for studying molecular crystals. A detailed analysis shows a very similar dispersion contribution of the two methods across the 60 considered molecular crystals. The study also confirms that pure DFT shows serious deficiencies in properly handling molecular crystals in which the dispersive contribution is large. Due to the negligible requested computational resources, PIXEL is the method of choice in screening of a large number of molecular crystals, an essential step to predict crystal polymorphism or to study crystal growth processes. DFT-D can then be used to refine the ranking emerged from PIXEL calculations due to its general applicability and robustness in properly handling short-range interactions.

Local-MP2 electron correlation method for nonconducting crystals.

Rigorous methods for the post-HF (HF-Hartree-Fock) determination of correlation corrections for crystalline solids are currently being developed following different strategies. The CRYSTAL program developed in Torino and Daresbury provides accurate HF solutions for periodic systems in a basis set of Gaussian type functions; for insulators, the occupied HF manifold can be represented as an antisymmetrized product of well localized Wannier functions. This makes possible the extension to nonconducting crystals of local correlation linear scaling On techniques as successfully and efficiently implemented in Stuttgarts MOLPRO program. These methods exploit the fact that dynamic electron correlation effects between remote parts of a molecule (manifesting as dispersive interactions in intermolecular perturbation theory) decay as an inverse sixth power of the distance R between these fragments, that is, much more quickly than the Coulomb interactions that are treated already at the HF level. Translational symmetry then permits the crystalline problem to be reduced to one concerning a cluster around the reference zero cell. A periodic local correlation program (CRYSCOR) has been prepared along these lines, limited for the moment to the solution of second-order Moller-Plesset equations. Exploitation of point group symmetry is shown to be more important and useful than in the molecular case. The computational strategy adopted and preliminary results concerning five semiconductors with tetrahedral structure (C, Si, SiC, BN, and BeS) are presented and discussed.

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.

Performance of six functionals (LDA, PBE, PBESOL, B3LYP, PBE0, and WC1LYP) in the simulation of vibrational and dielectric properties of crystalline compounds. The case of forsterite Mg2SiO4

The performance of six different density functionals (LDA, PBE, PBESOL, B3LYP, PBE0, and WC1LYP) in describing the infrared spectrum of forsterite, a crystalline periodic system with orthorhombic unit cell (28 atoms in the primitive cell, Pbmn space group), is investigated by using the periodic ab initio CRYSTAL09 code and an all‐electron Gaussian‐type basis set. The transverse optical (TO) branches of the 35 IR active modes are evaluated at the equilibrium geometry together with the oscillator strengths and the high‐frequency dielectric tensor ϵ∞. These quantities are essential to compute the dielectric function ϵ(ν), and then the reflectance spectrum R(ν), which is compared with experiment. It turns out that hybrid functionals perform better than LDA and GGA, in general; that B3LYP overperforms WC1LYP and, in turn, PBE0; that PBESOL is better than PBE; that LDA is the worst performing functional among the six under study.

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.

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.

Periodic local Møller–Plesset second order perturbation theory method applied to molecular crystals: Study of solid NH3 and CO2 using extended basis sets

We have calculated the equilibrium geometry, formation energy, and bulk modulus of two molecular bulk crystals, NH(3) and CO(2), at the periodic post-Hartree-Fock correlated level. The dependence of the results on the basis set has been analyzed, by employing basis sets up to aug-cc-pVQZ quality. In the calculations, we used the periodic local Møller-Plesset second order perturbation theory (LMP2), implemented in the CRYSCOR program. Multipolar expansion techniques, as well as density fitting, are employed in this code to reduce the number of and to factorize the required electron repulsion integrals; as a consequence of that, the computational cost for the correlation part of the calculations is comparable to that of the Hartree-Fock. Auxiliary calculations performed on molecular dimers are also reported to verify the accuracy of the LMP2 approach and of the basis sets used. Furthermore, the effect of spin-component scaling has been investigated for the two crystals. One intention of the present paper is also to lay out and specify the computational setup, which is generally applicable for accurate CRYSCOR calculations on molecular crystals.

Local MP2 with Density Fitting for Periodic Systems: A Parallel Implementation.

A parallel implementation is presented for the evaluation of local second-order Møller-Plesset perturbation theory (LMP2) energies in periodic, nonconducting crystalline systems with a density-fitting approximation of two-electron repulsion integrals. Peculiarities of the periodic case with respect to parallel LMP2 implementations in molecular codes, such as the use of translational and point symmetry, impose different strategies in order to achieve good parallel performance. The implementation is benchmarked on a few systems, representing a choice of the most interesting solid state quantum-chemistry problems where the MP2 approach can be decisive. Good parallel efficiency of the algorithms is demonstrated for up to 54 processors. Test systems include a metal organic framework (MOF-5) 3D crystalline structure with a triple-ζ-quality basis set: this is the largest calculation performed so far with 106 atoms, 532 correlated electrons, and 2884 atomic orbitals per unit cell.

Second Order Local Møller-Plesset Perturbation Theory for Periodic Systems: the CRYSCOR Code

Abstract This article reviews the periodic LMP2 method and its implementation in the CRYSCOR code. The main steps of the LMP2 calculations and the techniques employed are briefly described. Illustrative single-point calculations for three TiO2 polymorphs: rutile, anatase and brookite in their experimental geometry are performed. It is shown that the method scales linearly with respect to the number of atoms per unit cell, and can be applied to relatively complex periodic systems. The LMP2 method in contrast to DFT positions rutile slightly lower in the energy than anatase. Brookite is found to be the most stable. However, since the energy differences are relatively small, a further investigation of the influence of the domain-sizes, geometry, zero-point vibrations, etc. on the relative stability of these systems is required.