Solution of periodic Poisson's equation and the Hartree–Fock approach for solids with extended electron states: Application to linear augmented plane wave method

We formulate a Hartree–Fock‐LAPW method for electronic band structure calculations. The method is based on the Hartree–Fock–Roothaan approach for solids with extended electron states and closed core shells where the basis functions of itinerant electrons are linear augmented plane waves. All interactions within the restricted Hartree–Fock approach are analyzed and in principle can be taken into account. In particular, we obtained the matrix elements for the exchange interactions of extended states and the crystal electric field effects. To calculate the matrix elements of exchange for extended states, we first introduce an auxiliary potential and then integrate it with an effective charge density corresponding to the electron exchange transition under consideration. The problem of finding the auxiliary potential is solved by using the strategy of the full potential LAPW approach, which is based on the general solution of periodic Poisson's equation. Here, we use an original technique for the general solution of periodic Poisson's equation and multipole expansions of electron densities. We apply the technique to obtain periodic potentials of the face‐centered cubic lattice and discuss its accuracy and convergence in comparison with other methods. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Introduction

Silicon is taken as a test system for assessing present-day feasibility of calculations for crystalline solids of near-Hartree–Fock quality. The calculations have been performed using CRYSTAL, an ab initio Hartree–Fock crystalline-orbital LCAO program for periodic systems. The influence of the computational parameters that control the truncation of infinite sums on the results has been investigated; it is shown that a reasonable accuracy (numerical errors on total energy per atom below 10^?3 a.u.) can be obtained while keeping the computational burden within manageable limits. The effect on the results of basis-set size and quality is discussed. A number of basis sets have been tested, from minimal to relatively extended sets (28 atomic orbitals per atom). The quality of the wave function has been checked using variational criteria and also through a comparison with experimental data, such as equilibrium geometry, bulk modulus, electron charge density, and electron momentum distribution. For the latter quantities, which are a measure of the accuracy of the one-electron density matrix, the best basis sets provide agreement with experiment that is almost within the experimental error. The correlation energy has been estimated using nonlocal density functionals, based on the one-electron density matrix: After this correction, the atomization energy agrees with experiment to within 2%. The generalization of the above analysis to other crystals is briefly discussed.     

Consistent Gaussian basis sets of triple‐zeta valence with polarization quality for solid‐state calculations

Consistent basis sets of triple‐zeta valence with polarization quality for main group elements and transition metals from row one to three have been derived for periodic quantum‐chemical solid‐state calculations with the crystalline‐orbital program CRYSTAL. They are based on the def2‐TZVP basis sets developed for molecules by the Ahlrichs group. Orbital exponents and contraction coefficients have been modified and reoptimized, to provide robust and stable self‐consistant field (SCF) convergence for a wide range of different compounds. We compare results on crystal structures, cohesive energies, and solid‐state reaction enthalpies with the modified basis sets, denoted as pob‐TZVP, with selected standard basis sets available from the CRYSTAL basis set database. The average deviation of calculated lattice parameters obtained with a selected density functional, the hybrid method PW1PW, from experimental reference is smaller with pob‐TZVP than with standard basis sets, in particular for metallic systems. The effects of basis set expansion by diffuse and polarization functions were investigated for selected systems. © 2012 Wiley Periodicals, Inc.     

All‐electron LCAO calculations of the LiF crystal phonon spectrum: Influence of the basis set,the exchange‐correlation functional,and the supercell size

For the first time the convergence of the phonon frequencies and dispersion curves in terms of the supercell size is studied in ab initio frozen phonon calculations on LiF crystal. Helmann–Feynman forces over atomic displacements are found in all‐electron calculations with the localized atomic functions (LCAO) basis using CRYSTAL06 program. The Parlinski–Li–Kawazoe method and FROPHO program are used to calculate the dynamical matrix and phonon frequencies of the supercells. For fcc lattice, it is demonstrated that use of the full supercell space group (including the supercell inner translations) enables to reduce essentially the number of the displacements under consideration. For Hartree–Fock (HF), PBE and hybrid PBE0, B3LYP, and B3PW exchange‐correlation functionals the atomic basis set optimization is performed. The supercells up to 216 atoms (3 × 3 × 3 conventional unit cells) are considered. The phonon frequencies using the supercells of different size and shape are compared. For the commensurate with supercell k ‐points the best agreement of the theoretical results with the experimental data is found for B3PW exchange‐correlation functional calculations with the optimized basis set. The phonon frequencies at the most non‐commensurate k ‐points converged for the supercell consisting of 4 × 4 × 4 primitive cells and ensures the accuracy 1–2% in the thermodynamic properties calculated (the Helmholtz free energy, entropy, and heat capacity at the room temperature). © 2009 Wiley Periodicals, Inc. J Comput Chem 2009     

Calculations of solid‐state 43Ca NMR parameters: A comparison of periodic and cluster approaches and an evaluation of DFT functionals

We present a computational study of magnetic‐shielding and quadrupolar‐coupling tensors of ⁴³Ca sites in crystalline solids. A comparison between periodic and cluster‐based approaches for modeling solid‐state interactions demonstrates that cluster‐based approaches are suitable for predicting ⁴³Ca NMR parameters. Several model chemistries, including Hartree–Fock theory and 17 DFT approximations (SVWN, CA‐PZ, PBE, PBE0, PW91, B3PW91, rPBE, PBEsol, WC, PKZB, BMK, M06‐L, M06, M06‐2X, M06‐HF, TPSS, and TPSSh), are evaluated for the prediction of ⁴³Ca NMR parameters. Convergence of NMR parameters with respect to basis sets of the form cc‐pVXZ (X = D, T, Q) is also evaluated. All DFT methods lead to substantial, and frequently systematic, overestimations of experimental chemical shifts. Hartree–Fock calculations outperform all DFT methods for the prediction of ⁴³Ca chemical‐shift tensors. © 2017 Wiley Periodicals, Inc.     

Ab initio periodic study of the conformational behavior of glycine helical homopeptides

Representative helicoidal conformations of polyglycine infinite chains have been investigated by using periodic boundary conditions, the B3LYP hybrid functional, and large basis sets, by means of the CRYSTAL code. The exploitation of the helix roto‐translational symmetry permits to optimize at a relatively low cost the structure of systems whose unit cell contains more than 300 atoms, much larger than the one investigated till now. In the present calculations, the helix symmetry is exploited at three levels. First, for the automatic generation of the structure. Second, for the calculation of the one‐ and two‐electron integrals that enter into the Fock matrix definition. Only the irreducible wedge of the Fock matrix is computed. Finally, for the diagonalization of the Fock matrix, where each irreducible representation is separately treated. The efficiency and accuracy of the computational scheme are documented, by considering cells containing up to 47 glycine residues. Results are compared with previous calculations and available experimental data. The role of hydrogen bonding in stabilizing polyglycine conformers is also addressed. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Attractive electron–electron interactions within robust local fitting approximations

An analysis of Dunlap's robust fitting approach reveals that the resulting two‐electron integral matrix is not manifestly positive semidefinite when local fitting domains or non‐Coulomb fitting metrics are used. We present a highly local approximate method for evaluating four‐center two‐electron integrals based on the resolution‐of‐the‐identity (RI) approximation and apply it to the construction of the Coulomb and exchange contributions to the Fock matrix. In this pair‐atomic resolution‐of‐the‐identity (PARI) approach, atomic‐orbital (AO) products are expanded in auxiliary functions centered on the two atoms associated with each product. Numerical tests indicate that in 1% or less of all Hartree–Fock and Kohn–Sham calculations, the indefinite integral matrix causes nonconvergence in the self‐consistent‐field iterations. In these cases, the two‐electron contribution to the total energy becomes negative, meaning that the electronic interaction is effectively attractive, and the total energy is dramatically lower than that obtained with exact integrals. In the vast majority of our test cases, however, the indefiniteness does not interfere with convergence. The total energy accuracy is comparable to that of the standard Coulomb‐metric RI method. The speed‐up compared with conventional algorithms is similar to the RI method for Coulomb contributions; exchange contributions are accelerated by a factor of up to eight with a triple‐zeta quality basis set. A positive semidefinite integral matrix is recovered within PARI by introducing local auxiliary basis functions spanning the full AO product space, as may be achieved by using Cholesky‐decomposition techniques. Local completion, however, slows down the algorithm to a level comparable with or below conventional calculations. © 2013 Wiley Periodicals, Inc.     

Pseudodiagonalization-based wavefunction optimization with contracted planewave basis functions

Electronic structure calculations representing the molecular orbitals (MOs) with contracted planewave basis functions (CPWBFs) have been reported recently. CPWBFs are Fourier-series representations of atom-centered basis functions. The mathematical features of CPWBFs permit the construction of matrix–vector products, FC _o , involving the application of the Fock matrix, F , to the set of occupied MOs, C _o , without the explicit evaluation of F . This approach offers a theoretical speed-up of M/n over F -based methods, where M and n are the number of basis functions and occupied MOs, respectively. The present study reports methodological advances that permit FC _o -based optimization of wavefunction formed from CPWBFs. In particular, a technique is reported for optimizing wavefunctions by combining pseudodiagonalization techniques based on an exact representation of FC _o , approximate information regarding the virtual orbital energies, and direct inversion of the iterative subspace optimization schemes to guide the wavefunction to a converged solution. This method is found to speed-up wavefunction optimizations by factors of up to ~6 − 8 over F -based optimization methods while providing identical results. Further, the computational cost of this technique is relatively insensitive to basis set size, thus providing further benefits in calculations using large CPWBF basis sets. The results of density functional theory calculations show that this method permits the use of hybrid exchange-correlation (XC) functionals with a small increase in effort over analogous calculations using generalized gradient approximation XC functionals. © 2019 Wiley Periodicals, Inc.     

Performance of numerical atom‐centered basis sets in the ground‐state correlated calculations of noncovalent interactions: Water and methane dimer cases

Numerical atom‐centered basis sets (orbitals) (NAO) are known for their compactness and rapid convergence in the Hartree–Fock and density‐functional theory (DFT) molecular electronic‐structure calculations. To date, not much is known about the performance of the numerical sets against the well‐studied Gaussian‐type bases in correlated calculations. In this study, one instance of NAO [Blum et al., The Fritz Haber Institute ab initio Molecular Simulations Package (FHI‐aims), 2009] was thoroughly examined in comparison to the correlation‐consistent basis sets in the ground‐state correlated calculations on the hydrogen‐bonded water and dispersion‐dominated methane dimers. It was shown that these NAO demonstrate improved, comparing to the unaugmented correlation‐consistent based, convergence of interaction energies in correlated calculations. However, the present version of NAO constructed in the DFT calculations on covalently‐bound diatomics exhibits enormous basis‐set superposition error (BSSE)—even with the largest bases. Moreover, these basis sets are essentially unable to capture diffuse character of the wave function, necessary for example, for the complete convergence of correlated interaction energies of the weakly‐bound complexes. The problem is usually treated by addition of the external Gaussian diffuse functions to the NAO part, what indeed allows to obtain accurate results. However, the operation increases BSSE with the resulting hybrid basis sets even further and breaks down the initial concept of NAO (i.e., improved compactness) due to the significant increase in their size. These findings clearly point at the need in the alternative strategies for the construction of sufficiently‐delocalized and BSSE‐balanced purely‐numerical bases adapted for correlated calculations, possible ones were outlined here. For comparison with the considered NAOs, a complementary study on the convergence properties of the correlation‐consistent basis sets, with a special emphasis on BSSE, was also performed. Some of its conclusions may represent independent interest. © 2013 Wiley Periodicals, Inc.     

Bond centred functions in relativistic and non-relativistic calculations for diatomics

In this paper, we discuss the performance of molecular basis sets consisting of atomic centred (AC) functions augmented with bond centred (BC) functions in relativistic and non-relativistic calculations carried out at the Hartree–Fock and several correlated levels of approximation. While usually non-correlated calculations employing BC functions can be performed at a lower computational cost as compared with those making use of energy optimized AC basis sets, the correlated calculations are always more accurate and less expensive with the latter. It is demonstrated that both correlated or non-correlated calculations always benefit from the addition of a few BC functions with a moderate increase of computational effort. The performance of basis sets containing even-tempered BC functions is also studied and their usage is advocated in case of relativistic calculations.     

Consistent gaussian basis sets of double‐ and triple‐zeta valence with polarization quality of the fifth period for solid‐state calculations

Consistent basis sets of double‐ and triple‐zeta valence with polarization quality for the fifth period have been derived for periodic quantum‐chemical solid‐state calculations with the crystalline‐orbital program CRYSTAL. They are an extension of the pob‐TZVP basis sets, and are based on the full‐relativistic effective core potentials (ECPs) of the Stuttgart/Cologne group and on the def2‐SVP and def2‐TZVP valence basis of the Ahlrichs group. We optimized orbital exponents and contraction coefficients to supply robust and stable self‐consistent field (SCF) convergence for a wide range of different compounds. The computed crystal structures are compared to those obtained with standard basis sets available from the CRYSTAL basis set database. For the applied hybrid density functional PW1PW, the average deviations of calculated lattice constants from experimental references are smaller with pob‐DZVP and pob‐TZVP than with standard basis sets. © 2018 Wiley Periodicals, Inc.     

Model studies of the optical rotation,and theoretical determination of its sign for β‐pinene and trans‐pinane

We have carried out extensive studies on the basis set dependence of the calculated specific optical rotation (OR) in molecules at the level of the time–dependent Hartree–Fock and density functional approximations. To reach the limits of the basis set saturation, we have devised an artificial model, the asymmetrically deformed (chiral) methane (CM) molecule. This small system permits to use basis sets which are prohibitively large for real chiral molecules and yet shows all the important features of the basis set dependence of the OR values. The convergence of the OR has been studied with n‐aug‐cc‐pVXZ basis sets of Dunning up to the 6–ζ. In a parallel series of calculations, we have used the recently developed large polarized (LPolX) basis sets. The relatively small LPolX sets have been shown to be competitive to very large n‐aug‐cc‐pVXZ basis sets. The conclusions reached in calculations of OR in CM concerning the usefulness of LPolX basis sets have been further tested on (S)‐methyloxirane and (S)‐fluoro‐oxirane. The smallest set of the LPolX family (LPol–ds) has been found to yield OR values of similar quality as those obtained with much larger Dunning's aug‐cc‐pVQZ basis set. These results have encouraged us to carry out the OR calculations with LPol–ds basis sets for systems as large as β‐pinene and trans‐pinane. In both cases, our calculations have lead to the correct sign of the OR value in these molecules. This makes the relatively small LPol–ds basis sets likely to be useful in OR calculations for large molecules. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Ab initio investigation of ground‐state properties of group‐12 fluorides

The performance of wavefunction‐based correlation methods in theoretical solid‐state chemistry depends on reliable Hartree–Fock (HF) results for infinitly extended systems. Therefore, we optimized basis sets of valence‐triple‐ζ quality based on HF calculations for the periodic system of group‐12‐metal difluorides. Scalar‐relativistic effects were included in the case of the metal‐ions by applying small‐core pseudopotentials. To assess the quality of the proposed basis sets, the structural parameters, bulk moduli as well as cohesive and lattice energies of the systems were evaluated at the HF and the density functional theory levels. In addition to these two mean‐field approaches and to assess further employment of our basis sets to wavefunction‐based correlation methods we performed periodic local MP2 computations. Finally, the possibilities of pressure induced structural phase transitions occurring in the ZnF₂, CdF₂, and HgF₂ were investigated. © 2014 Wiley Periodicals, Inc.     

Intermediate Electrostatic Field for the Generalized Elongation Method

An intermediate electrostatic field is introduced to improve the accuracy of fragment‐based quantum‐chemical computational methods by including long‐range polarizations of biomolecules. The point charge distribution of the intermediate field is generated by a charge sensitivity analysis that is parameterized for five different population analyses, namely, atoms‐in‐molecules, Hirshfeld, Mulliken, natural orbital, and Voronoi population analysis. Two model systems are chosen to demonstrate the performance of the generalized elongation method (ELG) combined with the intermediate electrostatic field. The calculations are performed for the STO‐3G, 6‐31G, and 6‐31G(d) basis sets and compared with reference Hartree–Fock calculations. It is shown that the error in the total energy is reduced by one order of magnitude, independently of the population analyses used. This demonstrates the importance of long‐range polarization in electronic‐structure calculations by fragmentation techniques.     

Theoretical study of the structure and antimicrobial activity of horminone

The structural and electronic parameters of the horminone molecule, an abietan diterpene quinone, were studied by means of all‐electron calculations using Hartree–Fock and density functional theory‐based methods, as implemented in the Gaussian98 program. The 6‐31G orbital basis sets were used for the C, H, O, and Mg atoms. The results allow the identification of the negative site of horminone (HM) most favorable for its binding to the Mg²⁺ ion. The HM–Mg²⁺ complex is assumed to play a significant role in the antibacterial activity. First, it penetrates the membrane cell. Then, through its interaction with rRNA, it inhibits the protein synthesis in several types of bacteria. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 93: 411–421, 2003     

DFT study of the gas‐phase thermal decomposition kinetics of 2‐ethoxypyridine into 2‐pyridone

The mechanism of the gas‐phase elimination kinetics of 2‐ethoxypyridine has been studied through the electronic structure calculations using density functional methods: B3LYP/6‐31G(d,p), B3LYP/6‐31++G(d,p), B3PW91/6‐31G(d,p), B3PW91/6‐31++G(d,p), MPW1PW91/6‐31G(d,p), MPW1PW91/6‐31++G(d,p), PBEPBE/6‐31G(d,p), PBEPBE/6‐31++G(d,p), PBE1PBE1/6‐31G(d,p), and PBE1PBE1/6‐31++G(d,p). The elimination reaction of 2‐ethoxypyridine occurs through a six‐centered transition state geometry involving the pyridine nitrogen, the substituted carbon of the aromatic ring, the ethoxy oxygen, two carbons of the ethoxy group, and a hydrogen atom, which migrates from the ethoxy group to the nitrogen to give 2‐pyridone and ethylene. The reaction mechanism appears to occur with the participation of π‐electrons, similar to alkyl vinyl ether elimination reaction, with simultaneous ethylene formation and hydrogen migration to the pyridine nitrogen producing 2‐pyridone. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Perturbation theoretical investigation of electron correlation effects in infinite metallic and semiconducting polymers

Møller–Plesset-type perturbation theory is applied to calculate electron correlation effects in infinite, periodic, metallic, and semiconducting polymers. Different possible choices of the zeroth-order Hamiltonian (symmetry adapted Hartree–Fock, spin-, and charge-density waves), the influence of the size of the atomic basis sets, lattice and conduction band sums are investigated. In the case of semiconducting systems a very efficient version of the method using optimally localized Wannier functions has been worked out. Applications include model calculations for infinite atomic hydrogen chains, the investigation of the role of correlation effects in the bond alternation and metal-insulator phase transition in polyene, different one-and many-particle effects contributing to the single-particle gap in polyene, and the calculation of correlated quasiparticle band structures using the electronic polaron model.     

A general,recursive, and open‐ended response code

We present a new implementation of a recent open‐ended response theory formulation for time‐ and perturbation‐dependent basis sets (Thorvaldsen et al., J. Chem. Phys. 2008, 129, 214108) at the Hartree–Fock and density functional levels of theory. A novel feature of the new implementation is the use of recursive programming techniques, making it possible to write highly compact code for the analytic calculation of any response property at any valid choice of rule for the order of perturbation at which to include perturbed density matrices. The formalism is expressed in terms of the density matrix in the atomic orbital basis, allowing the recursive scheme presented here to be used in linear‐scaling formulations of response theory as well as with two‐ and four‐component relativistic wave functions. To demonstrate the new code, we present calculations of the third geometrical derivatives of the frequency‐dependent second hyperpolarizability for HSOH at the Hartree–Fock level of theory, a seventh‐order energy derivative involving basis sets that are both time and perturbation dependent. © 2014 Wiley Periodicals, Inc.     

Grid‐based density functional calculations of many‐electron systems

Exploratory variational pseudopotential density functional calculations are performed for the electronic properties of many‐electron systems in the 3D cartesian coordinate grid (CCG). The atom‐centered localized gaussian basis set, electronic density, and the two‐body potentials are set up in the 3D cubic box. The classical Hartree potential is calculated accurately and efficiently through a Fourier convolution technique. As a first step, simple local density functionals of homogeneous electron gas are used for the exchange‐correlation potential, while Hay‐Wadt‐type effective core potentials are employed to eliminate the core electrons. No auxiliary basis set is invoked. Preliminary illustrative calculations on total energies, individual energy components, eigenvalues, potential energy curves, ionization energies, and atomization energies of a set of 12 molecules show excellent agreement with the corresponding reference values of atom‐centered grid as well as the grid‐free calculation. Results for three atoms are also given. Combination of CCG and the convolution procedure used for classical Coulomb potential can provide reasonably accurate and reliable results for many‐electron systems. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Relative energies of dioxo mu-oxo molybdenum complexes from various fragmentation strategies

A thiocyanatomolybdenum(VI ) dioxo μ‐oxo complex dimer bearing 4,4′‐di‐tBu‐2,2′‐bipyridine ligands has shown exceptional oxidizing ability. It exists as a meso form (symmetrical) and a d,l pair (unsymmetrical) in the crystal unit cell and also in equilibrium in solution. Which oxygen atom and which configuration are predominantly involved in the process of oxygen atom transfer is a question whose answer would certainly help experimentalists. From ab initio theoretical calculations we analyzed the electronic differences between the two configurations. The large number of atoms (101) restricts the choice of theoretical methods. We give results for second‐order Møller–Plesset (MP2) and DFT with the hybrid functional B3LYP, with and without pseudopotentials, with double‐ζ basis sets plus polarization functions. Although the structures of the two types of configurations are quite different, we show they have practically the same energy. Similarly, no significant differences were found for electronic atomic populations on oxygen and surrounding atoms. To facilitate future studies on the process of oxygen atom transfer, we compare the entire molecules to smaller entities obtained by fragmentations or by so‐called hybrid methods. We show that the tBu groups, and even sometimes the pyridyl rings, play only a minor role in determining the electronic structure. Concerning the energy difference between the two configurations, the MP2 results appear more consistent than the B3LYP results.