Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole method: Analytical gradients

A full implementation of analytical energy gradients for molecular and periodic systems is reported in the TURBOMOLE program package within the framework of Kohn–Sham density functional theory using Gaussian‐type orbitals as basis functions. Its key component is a combination of density fitting (DF) approximation and continuous fast multipole method (CFMM) that allows for an efficient calculation of the Coulomb energy gradient. For exchange‐correlation part the hierarchical numerical integration scheme (Burow and Sierka, Journal of Chemical Theory and Computation 2011, 7, 3097) is extended to energy gradients. Computational efficiency and asymptotic O(N) scaling behavior of the implementation is demonstrated for various molecular and periodic model systems, with the largest unit cell of hematite containing 640 atoms and 19,072 basis functions. The overall computational effort of energy gradient is comparable to that of the Kohn–Sham matrix formation. © 2016 Wiley Periodicals, Inc.     

Real-time time-dependent density functional theory using density fitting and the continuous fast multipole method

An implementation of real-time time-dependent density functional theory (RT-TDDFT) within the TURBOMOLE program package is reported using Gaussian-type orbitals as basis functions, second and fourth order Magnus propagator, and the self-consistent field as well as the predictor–corrector time integration schemes. The Coulomb contribution to the Kohn–Sham matrix is calculated combining density fitting approximation and the continuous fast multipole method. Performance of the implementation is benchmarked for molecular systems with different sizes and dimensionalities. For linear alkane chains, the wall time for density matrix time propagation step is comparable to the Kohn-Sham (KS) matrix construction. However, for larger two- and three-dimensional molecules, with up to about 5,000 basis functions, the computational effort of RT-TDDFT calculations is dominated by the KS matrix evaluation. In addition, the maximum time step is evaluated using a set of small molecules of different polarities. The photoabsorption spectra of several molecular systems calculated using RT-TDDFT are compared to those obtained using linear response time-dependent density functional theory and coupled cluster methods.     

Exact relations between the electron density and external potential for systems of interacting and noninteracting electrons

The differential virial theorem (DVT) is an explicit relation between the electron density ρ( r ), the external potential, kinetic energy density tensor, and (for interacting electrons) the pair function. The time‐dependent generalization of this relation also involves the paramagnetic current density. We present a detailed unified derivation of all known variants of the DVT starting from a modified equation of motion for the current density. To emphasize the practical significance of the theorem for noninteracting electrons, we cast it in a form best suited for recovering the Kohn–Sham effective potential v_s( r ) from a given electron density. The resulting expression contains only ρ( r ), v_s( r ), kinetic energy density, and a new orbital‐dependent ingredient containing only occupied Kohn–Sham orbitals. Other possible applications of the theorem are also briefly discussed. © 2012 Wiley Periodicals, Inc.     

Energy‐surfaces from the upper bound of the Pauli kinetic energy

Based on the Kohn–Sham Pauli potential and the Kohn–Sham electron density, the upper bound of the Pauli kinetic energy is tested as a suitable replacement for the exact Pauli kinetic energy for application in orbital‐free density functional calculations. It is found that bond lengths for strong and moderately bound systems can be qualitatively predicted, but with a systematic shift toward larger bond distances with a relative error of 6% up to 30%. Angular dependence of the energy‐surface cannot be modeled with the proposed functional. Therefore, the upper bound model is the first parameter‐free functional expression for the kinetic energy that is able to qualitatively reproduce binding curves with respect to bond distortions. © 2016 Wiley Periodicals, Inc.     

About the compatibility between ansatzes and constraints for a local formulation of orbital‐free density functional theory

Functional properties that are exact for the Hohenberg–Kohn functional may turn into mutually exclusive constraints at a given level of ansatz. This is exemplarily shown for the local density approximation. Nevertheless, it is possible to reach exactly the Kohn–Sham data from an orbital‐free density functional framework based on simple one‐point functionals by starting from the Levy–Perdew–Sahni formulation. The energy value is obtained from the density‐potential pair, and therefore does not refer to the functional dependence of the potential expression. Consequently, the potential expression can be obtained from any suitable model and is not required to follow proper scaling behavior.     

Analytic derivatives for the XYG3 type of doubly hybrid density functionals: Theory,implementation, and assessment

We present a theoretical development of the equations required to perform an analytic geometry optimization of a molecular system using the XYG3 type of doubly hybrid (xDH) functionals. In contrast to the well‐established B2PLYP type of DH functionals, the energy expressions in the xDH functionals are constructed by using density and orbital information from another standard Kohn–Sham (KS) functional (e.g., B3LYP) for doing the self‐consistent field calculations. Thus, the xDH functionals are nonvariational in both the hybrid density functional part and the second‐order perturbation part, each of which requires formally to solve a coupled‐perturbed KS equation. An implementation is reported here which combines the two parts by defining a total Lagrangian such that only a single set of the Z‐vector equations need to be solved. The computational cost with our implementation is of the same order as those for the conventional Møller–Plesset theory to the second order (MP2) and B2PLYP. Systematic test calculations are provided for covalently bonded molecules as well as compounds involving the intramolecular nonbonded interactions for the main group elements. Satisfactory performance of the xDH functionals demonstrates that the extra computer time on top of the conventional KS procedure is well‐invested, in particular, when the standard KS functionals and MP2 as well, are problematic. © 2013 Wiley Periodicals, Inc.     

Kohn–Sham potentials by an inverse Kohn–Sham equation and accuracy assessment by virial theorem

In the recent study, the authors have proposed an integral equation for solving the inverse Kohn–Sham problem. In the present paper, the integral equation is numerically solved for one-dimensional model of a He atom and an H₂ molecule in the electronic ground states. For this purpose, we propose an iterative solution algorithm avoiding the inversion of the kernel of the integral equation. To quantify the numerical accuracy of the calculated exchange-correlation potentials, we evaluate the exchange and correlation energies based on the virial theorem as well as the reproduction of the exact ground-state electronic energy. The results demonstrate that the numerical solutions of our integral equation for the inverse Kohn–Sham problem are accurate enough in reproducing the Kohn–Sham potential and in satisfying the virial theorem.     

Calculation of core-level electron spectra of ionic liquids

On the example of 40 ion pairs (5 cations times 8 anions), this study demonstrates how the core-level binding energy values can be calculated and used to plot theoretical spectra at low computational cost using density functional theory methods. Three approaches for obtaining the binding energy values are based on delta Kohn–Sham (ΔKS) calculations, 1s KS orbital energies, and atomic charges. The ΔKS results show reasonable agreement with the available experimental X-ray photoelectron data. The 1s KS orbital energies correlate well with the ΔKS results. Atomic charge correlation with ΔKS is improved by accounting for the charges of neighboring atoms. Assignment of binding energies to atoms and the applicability of the mentioned methods to model systems of ionic liquids are discussed.     

Density functional theory: Approximating quasiparticle density functional

A constructive approach for deriving the approximating quasiparticle energy density functional is proposed. As a matter of fact, the proposed approach is the direct development of the Kohn–Sham quasiparticle concept and the Levy–Valone approach. The approach presented takes into account a pseudopotential character of the exchange-correlation part of the density functional and results in a system of functional equations to obtain ground-state energies of many-electron systems.     

Density functional calculations of potential energy surface and charge transfer integrals in molecular triphenylene derivative HAT6

We investigate the effect of structural fluctuations on charge transfer integrals, overlap integrals, and site energies in a system of two stacked molecular 2,3,6,7,10,11-hexakishexyloxytriphenylene (HAT₆), which is a model system for conducting devices in organic photocell applications. A density functional based computational study is reported. Accurate potential energy surface calculations are carried out using an improved meta-hybrid density functional to determine the most stable configuration of the two weakly bound HAT₆ molecules. The equilibrium parameters in terms of the twist angle and co-facial separation are calculated. Adopting the fragment approach within the Kohn–Sham density functional framework, these parameters are combined to a lateral slide, to mimic structural/conformational fluctuations and variations in the columnar phase. The charge transfer and spatial overlap integrals, and site energies, which form the matrix element of the Kohn–Sham Hamiltonian are derived. It is found that these quantities are strongly affected by the conformational variations. The spatial overlap between stacked molecules is found to be of considerable importance since charge transfer integrals obtained using the fragment approach differ significantly from those using the dimer approach.     

Neural network and ReaxFF comparison for Au properties

We have studied how ReaxFF and Behler–Parrinello neural network (BPNN) atomistic potentials should be trained to be accurate and tractable across multiple structural regimes of Au as a representative example of a single‐component material. We trained these potentials using subsets of 9,972 Kohn‐Sham density functional theory calculations and then validated their predictions against the untrained data. Our best ReaxFF potential was trained from 848 data points and could reliably predict surface and bulk data; however, it was substantially less accurate for molecular clusters of 126 atoms or fewer. Training the ReaxFF potential to more data also resulted in overfitting and lower accuracy. In contrast, BPNN could be fit to 9,734 calculations, and this potential performed comparably or better than ReaxFF across all regimes. However, the BPNN potential in this implementation brings significantly higher computational cost. © 2016 Wiley Periodicals, Inc.     

An analysis of nonlocal density functionals in chemical bonding

In this work, we carry out an analysis of the gradient-corrected density functionals in molecules that are used in the Kohn–Sham density functional approach. We concentrate on the special features of the exchange and correlation energy densities and exchange and correlation potentials in the bond region. By comparing to the exact Kohn–Sham potential, it is shown that the gradient-corrected potentials build in the required peak in the bond midplane, but not completely correctly. The gradient-corrected potentials also exhibit wrong asymptotic behavior. Contributions from different regions of space (notably bond and outer regions) to nonlocal bonding energy contributions are investigated by integrating the exchange and correlation energy densities in various spatial regions. This provides an explanation of why the gradient corrections reduce the local density approximation (LDA ) overbinding of molecules. It explains the success of the presently used nonlocal corrections, although it is possible that there is a cancellation of errors, too much repulsion being derived from the bond region and too little from the outer region. © John Wiley & Sons, Inc.     

Correlation energy,correlated electron density,and exchange‐correlation potential in some spherically confined atoms

We report correlation energies, electron densities, and exchange‐correlation potentials obtained from configuration interaction and density functional calculations on spherically confined He, Be, Be²⁺, and Ne atoms. The variation of the correlation energy with the confinement radius R_c is relatively small for the He, Be²⁺, and Ne systems. Curiously, the Lee–Yang–Parr (LYP) functional works well for weak confinements but fails completely for small R_c. However, in the neutral beryllium atom the CI correlation energy increases markedly with decreasing R_c. This effect is less pronounced at the density‐functional theory level. The LYP functional performs very well for the unconfined Be atom, but fails badly for small R_c. The standard exchange‐correlation potentials exhibit significant deviation from the “exact” potential obtained by inversion of Kohn–Sham equation. The LYP correlation potential behaves erratically at strong confinements. © 2016 Wiley Periodicals, Inc.     

Quadrupole and octupole Cauchy moments of the atoms through argon

Quadrupole and octupole Cauchy moments of the atoms through argon are calculated using the hydrodynamic formulation of time-dependent Kohn–Sham theory. The exchange-correlation energy density functional is approximated by a gradient expansion for atoms that has an explicit dependence upon the number of electrons. The first-order corrections to the Kohn–Sham amplitudes and phases are found by seeking variational solutions of the derived sequential set of functionals. The trial functions employed contain both linear and nonlinear variational parameters and are thus flexible enough to provide rapid convergence to the multipole polarizabilities. The resulting Cauchy moments provide information that allows the calculation of various properties that result from the linear interaction of atoms with a time-varying electric field. © 1994 John Wiley & Sons, Inc.     

Electron transport in Nano‐Gold–Silicon interfaces

The electron transport characteristics of gold–silicon interfaces are studied using a combined ab initio approach of the Green's function for electron transfer and quantum density functional theory (DFT) for finite and extended systems. The Kohn–Sham Hamiltonian of an extended cluster or molecule and the density of states (DOS) of bulk Si and Au are used to construct the interface Hamiltonian to obtain the DOS, electron transmission, and current–voltage characteristics of the interface. Diode behavior is observed with electron conduction when the gold side is positively biased with a threshold of 0.8 V. The presence of molecules trapped at the interface and the geometry of the metal atoms strongly affect the conductance, implying difficult or even impossible theory–experiment validations. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Analytical energy gradient evaluation in relativistic and nonrelativistic density functional calculations

The expressions of analytical energy gradients in density functional theory and their implementation in programs are reported. The evaluation of analytical energy gradients can be carried out in the fully 4-component relativistic, approximate relativistic, and nonrelativistic density functional calculations under local density approximation or general gradient approximation with or without frozen core approximation using different basis sets in our programs. The translational invariance condition and the fact that the one-center terms do not contribute to the energy gradients are utilized to improve the calculation accuracy and to reduce the computational effort. The calculated results of energy gradients and optimized geometry as well as atomization energies of some molecules by the analytical gradient method are in very good agreement with results obtained by the numerical derivative method.     

Acceleration of catalyst discovery with easy,fast, and reproducible computational alchemy

The expense of quantum chemistry calculations significantly hinders the search for novel catalysts. Here, we provide a tutorial for using an easy and highly cost‐efficient calculation scheme, called alchemical perturbation density functional theory (APDFT), for rapid predictions of binding energies of reaction intermediates and reaction barrier heights based on the Kohn‐Sham density functional theory (DFT) reference data. We outline standard procedures used in computational catalysis applications, explain how computational alchemy calculations can be carried out for those applications, and then present benchmarking studies of binding energy and barrier height predictions. Using a single OH binding energy on the Pt(111) surface as a reference case, we use computational alchemy to predict binding energies of 32 variations of this system with a mean unsigned error of less than 0.05 eV relative to single‐point DFT calculations. Using a single nudged elastic band calculation for CH₄ dehydrogenation on Pt(111) as a reference case, we generate 32 new pathways with barrier heights having mean unsigned errors of less than 0.3 eV relative to single‐point DFT calculations. Notably, this easy APDFT scheme brings no appreciable computational cost once reference calculations are performed, and this shows that simple applications of computational alchemy can significantly impact DFT‐driven explorations for catalysts. To accelerate computational catalysis discovery and ensure computational reproducibility, we also include Python modules that allow users to perform their own computational alchemy calculations.