Periodic Hartree-Fock Research Articles

Exact exchange with range-separated algorithm for thermodynamic limit of periodic Hartree-Fock theory.

The expensive cost of computing exact exchange in periodic systems limits the application range of density functional theory with hybrid functionals. To reduce the computational cost of exact change, we present a range-separated algorithm to compute electron repulsion integrals for Gaussian-type crystal basis. The algorithm splits the full-range Coulomb interactions into short-range and long-range parts, which are, respectively, computed in real and reciprocal space. This approach significantly reduces the overall computational cost, as integrals can be efficiently computed in both regions. The algorithm can efficiently handle large numbers of k points with limited central processing unit (CPU) and memory resources. As a demonstration, we performed an all-electron k-point Hartree-Fock calculation for LiH crystal with one million Gaussian basis functions, which was completed on a desktop computer in 1400 CPU hours.

Sparse tensor based nuclear gradients for periodic Hartree-Fock and low-scaling correlated wave function methods in the CP2K software package: A massively parallel and GPU accelerated implementation.

The development of novel double-hybrid density functionals offers new levels of accuracy and is leading to fresh insights into the fundamental properties of matter. Hartree-Fock exact exchange and correlated wave function methods, such as second-order Møller-Plesset (MP2) and direct random phase approximation (dRPA), are usually required to build such functionals. Their high computational cost is a concern, and their application to large and periodic systems is, therefore, limited. In this work, low-scaling methods for Hartree-Fock exchange (HFX), SOS-MP2, and direct RPA energy gradients are developed and implemented in the CP2K software package. The use of the resolution-of-the-identity approximation with a short range metric and atom-centered basis functions leads to sparsity, allowing for sparse tensor contractions to take place. These operations are efficiently performed with the newly developed Distributed Block-sparse Tensors (DBT) and Distributed Block-sparse Matrices (DBM) libraries, which scale to hundreds of graphics processing unit (GPU) nodes. The resulting methods, resolution-of-the-identity (RI)-HFX, SOS-MP2, and dRPA, were benchmarked on large supercomputers. They exhibit favorable sub-cubic scaling with system size, good strong scaling performance, and GPU acceleration up to a factor of 3. These developments will allow for double-hybrid level calculations of large and periodic condensed phase systems to take place on a more regular basis.

Full configuration interaction quantum Monte Carlo treatment of fragments embedded in a periodic mean field.

We present an embedded fragment approach for high-level quantum chemical calculations on local features in periodic systems. The fragment is defined as a set of localized orbitals (occupied and virtual) corresponding to a converged periodic Hartree-Fock solution. These orbitals serve as the basis for the in-fragment post-Hartree-Fock treatment. The embedding field for the fragment, consisting of the Coulomb and exchange potential from the rest of the crystal, is included in the fragment's one-electron Hamiltonian. As an application of the embedded fragment approach, we investigate the performance of full configuration interaction quantum Monte Carlo (FCIQMC) with the adaptive shift. As the orbital choice, we use the natural orbitals from the distinguishable cluster method with singles and doubles. FCIQMC is a stochastic approximation to the full CI method and can be routinely applied to much larger active spaces than the latter. This makes this method especially attractive in the context of open shell defects in crystals, where fragments of adequate size can be rather large. As a test case, we consider dissociation of a fluorine atom from a fluorographane surface. This process poses a challenge for high-level electronic structure models as both the static and dynamic correlations are essential here. Furthermore, the active space for an adequate fragment (32 electrons in 173 orbitals) is already quite large even for FCIQMC. Despite this, FCIQMC delivers accurate dissociation and total energies.

Reaction barriers on non-conducting surfaces beyond periodic local MP2: Diffusion of hydrogen on α-Al2O3(0001) as a test case.

The quest for "chemical accuracy" is becoming more and more demanded in the field of structure and kinetics of molecules at solid surfaces. In this paper, as an example, we focus on the barrier for hydrogen diffusion on a α-Al2O3(0001) surface, aiming for a couple cluster singles, doubles, and perturbative triples [CCSD(T)]-level benchmark. We employ the density functional theory (DFT) optimized minimum and transition state structures reported by Heiden, Usvyat, and Saalfrank [J. Phys. Chem. C 123, 6675 (2019)]. The barrier is first evaluated at the periodic Hartree-Fock and local Møller-Plesset second-order perturbation (MP2) level of theory. The possible sources of errors are then analyzed, which includes basis set incompleteness error, frozen core, density fitting, local approximation errors, as well as the MP2 method error. Using periodic and embedded fragment models, corrections to these errors are evaluated. In particular, two corrections are found to be non-negligible (both from the chemical accuracy perspective and at the scale of the barrier value of 0.72eV): the correction to the frozen core-approximation of 0.06eV and the CCSD(T) correction of 0.07eV. Our correlated wave function results are compared to barriers obtained from DFT. Among the tested DFT functionals, the best performing for this barrier is B3LYP-D3.

Fragment-Based Restricted Active Space Configuration Interaction with Second-Order Corrections Embedded in Periodic Hartree-Fock Wave Function.

We present a computational scheme for restricted-active-space configuration interaction (RASCI) calculations combined with second-order perturbation theory (RASCI-PT2) on a fragment of a periodic system embedded in the periodic Hartree-Fock (HF) wave function. This method allows one to calculate the electronic structure of localized strongly correlated features in crystals and surfaces. The scheme was implemented via an interface between the Cryscor and Q-Chem codes. To evaluate the performance of the embedding method, we explored dissociation of a fluorine atom from a lithium fluoride surface and partially fluorinated graphane layer. The results show that RASCI and RASCI-PT2 embedded in periodic HF are able to produce well-behaved potential energy surfaces and accurate dissociation energies.

Cluster-in-Molecule Local Correlation Approach for Periodic Systems.

In this article, the cluster-in-molecule (CIM) local correlation approach for periodic systems with periodic boundary condition has been developed, which allows electron-correlation calculations of various crystals computationally tractable. In this approach, the correlation energy per unit cell of a periodic system can be evaluated as the summation of the correlation contributions from electron-correlation calculations on a series of finite-sized clusters. Each cluster is defined to contain a subset of localized Wannier functions (WFs) (for the occupied space) and projected atomic orbitals (for the virtual space), which can be derived from a periodic Hartree-Fock calculation. Electron-correlation calculations on clusters at second-order Møller-Plesset perturbation theory (MP2) or coupled cluster singles and doubles (CCSD) can be performed with well-established molecular quantum chemistry packages. We perform illustrative calculations at the MP2 and CCSD levels on several types of crystals (neon lattice, carbon monoxide and ammonia crystals, two ionic liquid crystals, and diamond). The results show that CIM is a powerful framework for accurate electron-correlation calculations of crystals.

Direct inversion of the iterative subspace with contracted planewave basis functions.

Ways to reduce the computational cost of periodic electronic structure calculations by using basis functions corresponding to linear combinations of planewaves have been examined recently. These contracted planewave (CPW) basis functions correspond to Fourier series representations of atom-centered basis functions, and thus provide access to some beneficial properties of planewave (PW) and localized basis functions. This study reports the development and assessment of a direct inversion of the iterative subspace (DIIS) method that employs unique properties of CPW basis functions to efficiently converge electronic wavefunctions. This method relies on access to a PW-based representation of the electronic structure to provide a means of efficiently evaluating matrix-vector products involving the application of the Fock matrix to the occupied molecular orbitals. These matrix-vector products are transformed into a form permitting the use of direct diagonalization techniques and DIIS methods typically employed with atom-centered basis sets. The abilities of this method are assessed through periodic Hartree-Fock calculations of a range of molecules and solid-state systems. The results show that the method reported in this study is approximately five times faster than CPW-based calculations in which the entire Fock matrix is calculated. This method is also found to be weakly dependent upon the size of the basis set, thus permitting the use of larger CPW basis sets to increase variational flexibility with a minor impact on computational performance. © 2018 Wiley Periodicals, Inc.

Stability of Hydrogen Hydrates from Second-Order Møller-Plesset Perturbation Theory.

The formation of gas hydrates and clathrates critically depends on the interaction between the host water network and the guest gas species. Density functional calculations can struggle to quantitatively capture these dispersion-type interactions. Here, we report wave function-based calculations on hydrogen hydrates that combine periodic Hartree-Fock with a localized treatment of electronic correlation. We show that local second-order Møller-Plesset perturbation theory (LMP2) reproduces the stability of the different filled-ice-like hydrates in excellent agreement with experimental data. In contrast to various dispersion-corrected density functional theory implementations, LMP2 correctly identifies the pressures needed to stabilize the C0, C1, and C2 hydrates and does not find a spurious region of stability for an ice-Ih-based dihydrate. Our results suggest that LMP2 or similar approaches can provide quantitative insights into the mechanisms of formation and eventual decomposition of molecular host-guest compounds.

Ab initio simulation of structure and surface energy of low-index surfaces of stoichiometric α-Fe2O3

The structure and surface energy of a series of low-index surfaces of stoichiometric α-Fe2O3 (hematite) are investigated using the periodic Hartree–Fock approach with an a posteriori correction of the correlation energy. The simulations show that, amongst the modeled facets, (011¯2) and (0001) are the most stable surfaces of hematite, which is consistent with the fact that the latter are the dominant growth faces exposed on natural α-Fe2O3. The Fe-terminated (0001) surface is shown to exhibit a large relaxation of the surface atoms. It is argued that this arises mainly due to the fact that the surface cations are located opposite empty cation sites in the filled-filled-unfilled cation sequence along the c-axis. In contrast, the (011¯2) plane cuts the crystal through a plane of empty cation sites, thus giving rise to relatively small relaxations and surface energies. The small relaxations and concomitant exposure of five-coordinate cation sites may be important for the catalytic activity of hematite. The simulations also show that the relative stability of the investigated surfaces changes after a full lattice relaxation with the (0001) and (112¯6) facets relaxing disproportionately large. Wherever possible, the simulations are compared with previous simulation data and experimental results. A Wulff–Gibbs construction is also presented.

Fragment-Based Direct-Local-Ring-Coupled-Cluster Doubles Treatment Embedded in the Periodic Hartree-Fock Solution.

We present a periodic/finite-cluster interface for fragment-based direct local ring-coupled-cluster doubles (d-LrCCD) calculations embedded in the periodic mean field. The fragment is defined by a set of Wannier functions (WFs), obtained from a periodic Hartree-Fock calculation. The pair-specific virtual space is spanned by projected atomic orbitals (PAOs) truncated to pair domains. The computational procedure is initiated by a periodic local Møller-Plesset (LMP2) calculation. A subset of the WF pairs is then subsequently subjected to a finite-cluster d-LrCCD treatment using the local coupled cluster program of Molpro; this subset is specified by an interorbital cutoff distance. The orbital, pair, and domain lists, as well as other essential quantities needed for d-LrCCD such as the Fock and overlap matrices, and the electron repulsion integrals (ERIs) in the basis of WFs and PAOs are evaluated in the periodic framework and passed to Molpro via an interface. These periodic quantities provide the correct periodic mean-field embedding for the fragment d-LrCCD. Moreover, no expensive orbital transformations involving orbital coefficients related to large supporting clusters are necessary. ERIs appearing in the d-LrCCD diagrams are factorized via density fitting, which enables an efficient processing of the corresponding terms via three-index intermediates. The corresponding 3-index and the metric 2-index ERIs involving auxiliary functions are also computed and transformed to the WF-PAO basis (the 3-index ERI) on the periodic side. Although the direct ring-CCD method itself is not generally more accurate than MP2, it is more stable in the case of small band gap systems, as it sums up the ring diagrams to infinite order. Furthermore, this interface is a first step toward a high-level fragment-based quantum chemical treatment such as local CCSD(T) within a periodic embedding that is treated at a lower level. As two test examples we study the physisorption of H2 and argon on graphane.

Weak interactions in Graphane/BN systems under static electric fields—A periodic ab-initio study.

Ab-initio calculations via periodic Hartree-Fock (HF) and local second-order Møller-Plesset perturbation theory (LMP2) are used to investigate the adsorption properties of combined Graphane/boron nitride systems and their response to static electric fields. It is shown how the latter can be used to alter both structural as well as electronic properties of these systems.

Understanding the ε and ζ High-Pressure Solid Phases of Oxygen. Systematic Periodic Density Functional Theory Studies Using Localized Atomic Basis.

The experimentally characterized ε and ζ phases of solid oxygen are studied by periodic Hartree-Fock (HF) and Density Functional Theory calculations at pressures from 10 to 160 GPa using different types of exchange-correlation functionals with Gaussian atomic basis sets. Full geometry optimizations of the monoclinic C2/m (O2)4 unit cell were done to study the evolution of the structural and electronic properties with pressure. Vibrational calculations were performed at each pressure. While periodic HF does not predict the ε-ζ phase transition in the considered range, Local Density approximation and Generalized Gradient approximation methods predict too low transition pressures. The performance of hybrid functional methods is dependent on the amount of non-local HF exchange. PBE0, M06, B3PW91, and B3LYP approaches correctly predict the structural and electronic changes associated with the phase transition. GGA and hybrid functionals predict a pressure range where both phases coexist, but only the latter type of methods yield results in agreement with experiment. Using the optimized (O2)4 unit cell at each pressure we show, through CASSCF(8,8) calculations, that the greater accuracy of the optimized geometrical parameters with increasing pressure is due to a decreasing multireference character of the unit cell wave function. The mechanism of the transition from the non-conducting to the conducting ζ phase is explained through the Electron Pair Localization Function, which clearly reveals chemical bonding between O2 molecules in the ab crystal planes belonging to different unit cells due to much shorter intercell O2-O2 distances.

On the DFT ground state of crystalline bromine and iodine.

We report on an erroneous ground state within common density functional theory (DFT) methods for the solid elements bromine and iodine. Phonon computations at the GGA level for both molecular crystals yield imaginary vibrational modes, erroneously indicating dynamic instability-that fact alone could easily pass as a computational artefact, but these imaginary modes lead to energetically more favorable and dynamically stable structures, made up of infinite monoatomic chains. In contrast, meta-GGA and hybrid functionals yield the correct energetic order for bromine, while for iodine, most global hybrids do not improve the GGA result significantly. The qualitatively correct answer, in both cases, is given by the long-range corrected hybrid LC-ωPBE, the Minnesota functionals M06L and M06, and by periodic Hartree-Fock and MP2 theory. This poor performance of economic DFT functionals should be kept in mind, for example, during global structure optimizations of systems with significant contributions from halogen bonds.

Periodic Hartree-Fock and hybrid density functional calculations on the metallic and the insulating phase of (EDO-TTF)2PF6.

The insulating and conducting phases of (EDO-TTF)2PF6 were studied by all electron, periodic Hartree-Fock and hybrid density functional calculations. Electronic properties, such as the electronic band structure, the density of states and the Fermi surface are discussed in relation to the metal-insulator transition in this material. The nature of conduction is confirmed in both phases from their band structures and density of states. The hybrid DFT band gaps are in good agreement with experiment. Interactions are discussed on the basis of band dispersion in the inter-stack, intra-stack and inter-sheet directions. We discuss the phase transition in terms of the Peierls mechanism and our results fully support this view.

Incrementally Corrected Periodic Local MP2 Calculations: I. The Cohesive Energy of Molecular Crystals.

A method for accurate calculations of the cohesive energy of molecular crystals is presented. The cohesive energy is evaluated as a sum of several components. The major contribution is captured by periodic Hartree-Fock (HF) coupled with the local Møller-Plesset perturbation theory of second order (LMP2) with a triple-ζ basis set. Post-MP2 corrections and corrections for the basis set incompleteness are calculated from inexpensive incremental calculations with finite clusters. This is an essential improvement with respect to the periodic LMP2 method and allows for results of benchmark quality for crystalline systems. The proposed technique is superior to the standard incremental scheme as concerns the cluster size and basis set convergence of the results. In contrast to the total energy or electron correlation energy, which are evaluated in standard incremental calculations, post-MP2 and basis set corrections are rather insensitive to approximations and converge quickly both in terms of the order of the increments and the number of terms at a given order. Evaluation of the incremental corrections within the sub-kJ/mol precision requires computing very few of the most compact two-center and three-center non-embedded clusters, making the whole correction scheme computationally inexpensive. This method as well as alternative routes to compute the cohesive energy via the incremental scheme are tested on two molecular crystals: carbon dioxide (CO2) and hydrogen cyanide (HCN).

The Loewenstein rule: the increase in electron kinetic energy as the reason for instability of Al–O–Al linkage in aluminosilicate zeolites

Problem of Al–O–Al linkage in aluminosilicate materials or Al-avoidance is discussed for two zeolite structures (phillipsite and brewsterite) exchanged with Mg2+ cations. All models are fully optimized at periodic Hartree–Fock and hybrid density functional theory levels (the CRYSTAL code). Their properties are then calculated at the periodic level with the same basis sets. The reasons for the instability of the zeolite structures including the Al–O–Al moieties are interpreted on the basis of cell energy decomposition. This destabilization comes from an increase in kinetic energy if the Al–O(Mg)–Al moieties are present in zeolites. This effect is discussed in parallel with already known variational evidences in favor of dominating role of kinetic energy for stability of molecular systems.

Anisotropic displacement parameters for molecular crystals from periodic Hartree–Fock and density functional theory calculations

Fully periodic Hartree–Fock and density functional theory calculations have been used to compute the anisotropic displacement parameters (ADPs) of molecular crystals at different temperatures by using the CRYSTAL code. Crystalline urea was adopted as a benchmark system to investigate the dependence on basis set and Hamiltonian. The results were compared with ADPs derived from neutron diffraction experiments. The approach can estimate the internal ADPs, corresponding to the contributions of high-frequency intramolecular vibrations, and for these internal contributions the results are almost independent of the basis set and Hamiltonian. Much larger variations and discrepancies from neutron diffraction experiments are seen for the external, low-frequency modes, which become dominant at higher temperatures. The approach was then tested on benzene and urotropine. Finally, ADPs of l-alanine were predicted at the B3LYP/6-31G(d,p) level of theory. The total ADPs, including low-frequency external modes, are underestimated, but can be brought into good agreement with the experimental ADPs by introducing a Grüneisen parameter, which partly accounts for anharmonicity of the potential energy surface, but likely also contains contributions from other deficiencies of the calculations.

Ab Initio Hartree-Fock Simulation of R-Plane Sapphire

Abstract R-plane sapphire is the preferred substrate material for silicon-based radiation-hard devices and RF integrated circuits. Ab initio periodic Hartree-Fock simulations of r-plane sapphire slabs are presented with a particular focus on the surface relaxation and the surface energy. The calculations show that there is a considerable relaxation of the four outermost atomic layers of r-plane sapphire. A comparison with c-, a- and m-plane sapphire shows that the surface energy of fully relaxed r-plane sapphire is considerably larger than the surface energy of c-plane sapphire, but similar to the surface energies of a- and m-plane sapphire. The implications of these findings for chemical mechanical polishing (CMP) and electrolytic in-process dressing (ELID) grinding of sapphire are discussed briefly.

Molecular dynamic study of the melting temperature in MgF 2 with the fluorite structure

The melting temperature of MgF 2 with cubic fluorite structure has been calculated by the constant temperature and pressure molecular dynamics (MD) simulation using the well-tested effective pair-wise potentials, which consist of the Coulomb, dispersion, and repulsion interaction by varying temperature from 300 to 2500 K. It is found that the potential parameters for MgF 2 derived from ab initio periodic Hartree-Fock calculations are very successful in reproducing accurately the DFT-GGA combined with quasi-harmonic Debye model calculated volumes of the cubic fluorite-type MgF 2 over a wide range of temperature at room pressure. Our simulated melting temperature of cubic fluorite-type MgF 2 is very close to the actual melting temperature 1539 K. Meanwhile, the radial distribution functions of Mg–Mg, F–F, and Mg–F ion pairs near the melting temperature are investigated from the isobaric and isothermal ensemble.

Wave-function-based approach to quasiparticle bands: Insight into the electronic structure ofc-ZnS

Ab initio wave-function-based methods are employed for the study of quasiparticle energy bands of zinc-blende ZnS, with focus on the Zn 3d "semicore" states. The relative energies of these states with respect to the top of the S 3p valence bands appear to be poorly described as compared to experimental values not only within the local density approximation (LDA), but also when many-body corrections within the GW approximation are applied to the LDA or LDA+U mean-field solutions [T. Miyake, P. Zhang, M. L. Cohen, and S. G. Louie, Phys. Rev. B 74, 245213 (2006)]. In the present study, we show that for the accurate description of the Zn 3d states a correlation treatment based on wave function methods is needed. Our study rests on a local Hamiltonian approach which rigorously describes the short-range polarization and charge redistribution effects around an extra hole or electron placed into the valence respective conduction bands of semiconductors and insulators. The method also facilitates the computation of electron correlation effects beyond relaxation and polarization. The electron correlation treatment is performed on finite clusters cut off the infinite system. The formalism makes use of localized Wannier functions and embedding potentials derived explicitly from prior periodic Hartree-Fock calculations. The on-site and nearest-neighbor charge relaxation lead to corrections of several eV to the Hartree-Fock band energies and gap. Corrections due to long-range polarization are of the order of 1.0 eV. The dispersion of the Hartree-Fock bands is only little affected by electron correlations. We find the Zn 3d "semicore" states to lie about 9.0 eV below the top of the S 3p valence bands, in very good agreement with values from valence-band x-ray photoemission.