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.     

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

A full implementation of the analytical stress tensor for 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. It is the extension of the implementation of analytical energy gradients (Lazarski et al., Journal of Computational Chemistry 2016, 37, 2518–2526) to the stress tensor for the purpose of optimization of lattice vectors. Its key component is the efficient calculation of the Coulomb contribution by combining density fitting approximation and continuous fast multipole method. For the exchange-correlation (XC) part the hierarchical numerical integration scheme (Burow and Sierka, Journal of Chemical Theory and Computation 2011, 7, 3097–3104) is extended to XC weight derivatives and stress tensor. The computational efficiency and favorable scaling behavior of the stress tensor implementation are demonstrated for various model systems. The overall computational effort for energy gradient and stress tensor for the largest systems investigated is shown to be at most two and a half times the computational effort for the Kohn–Sham matrix formation. © 2019 Wiley Periodicals, Inc.     

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.     

Accelerating Kohn‐Sham response theory using density fitting and the auxiliary‐density‐matrix method

An extension of the formulation of the atomic‐orbital‐based response theory of Larsen et al., JCP 113, 8909 (2000) is presented. This new framework has been implemented in LSDalton and allows for the use of Kohn‐Sham density‐functional theory with approximate treatment of the Coulomb and Exchange contributions to the response equations via the popular resolution‐of‐the‐identity approximation as well as the auxiliary‐density matrix method (ADMM). We present benchmark calculations of ground‐state energies as well as the linear and quadratic response properties: vertical excitation energies, polarizabilities, and hyperpolarizabilities. The quality of these approximations in a range of basis sets is assessed against reference calculations in a large aug‐pcseg‐4 basis. Our results confirm that density fitting of the Coulomb contribution can be used without hesitation for all the studied properties. The ADMM treatment of exchange is shown to yield high accuracy for ground‐state and excitation energies, whereas for polarizabilities and hyperpolarizabilities the performance gain comes at a cost of accuracy. Excitation energies of a tetrameric model consisting of units of the P700 special pigment of photosystem I have been studied to demonstrate the applicability of the code for a large system.     

Analytical nuclear excited‐state gradients for the Tamm–Dancoff approximation using uncoupled frozen‐density embedding

We report the derivation and implementation of analytical nuclear gradients for excited states using time‐dependent density functional theory using the Tamm–Dancoff approximation combined with uncoupled frozen‐density embedding using density fitting. Explicit equations are presented and discussed. The implementation is able to treat singlet as well as triplet states and functionals using the local density approximation, the generalized gradient approximation, combinations with Hartree–Fock exchange (hybrids), and range‐separated functionals such as CAM‐B3LYP. The new method is benchmarked against supermolecule calculations in two case studies: The solvatochromic shift of the (vertical) fluorescence energy of 4‐aminophthalimide on solvation, and the first local excitation of the benzonitrile dimer. Whereas for the 4‐aminophthalimide–water complex deviations of about 0.2 eV are obtained to supermolecular calculations, for the benzonitrile dimer the maximum error for adiabatic excitation energies is below 0.01 eV due to a weak coupling of the subsystems. © 2017 Wiley Periodicals, Inc.     

Improvement of initial guess via grid‐cutting for efficient grid‐based density functional calculations

We introduced an efficient initial guess method, namely the grid‐cutting, which is specialized for grid‐based density functional theory (DFT) calculations. It produces initial density and orbitals through pre‐DFT calculations in an inner simulation box made by cutting out the outer region of a full‐size one. To assess its performance, we carried out DFT calculations for small molecules included in the G2‐1 set and two large molecules with various combinations of mixing and diagonalization conditions, relative size of the inner box, and grid spacing. For all cases, the grid‐cutting method was more efficient than conventional ones such as extended Hückel, superposition of atomic densities, and linear combination of atomic orbitals. For instance, it was about 20% faster in computational time and about 45% smaller in the number of self‐consistent‐field cycles than the superposition of atomic densities because it provided high‐quality initial density and orbitals closer to the corresponding fully converged values. In addition, it showed good performance for non‐Coulombic model systems such as harmonic oscillator.     

A self-consistent coulomb bath model using density fitting

A self-consistent Coulomb bath model is presented to provide an accurate and efficient way of performing calculations for interfragment electrostatic and polarization interactions. In this method, a condensed-phase system is partitioned into molecular fragment blocks. Each fragment is embedded in the Coulomb bath due to other fragments. Importantly, the present Coulomb bath is represented using a density fitting method in which the electron densities of molecular fragments are fitted using an atom-centered auxiliary basis set of Gaussian type. The Coulomb bath is incorporated into an effective Hamiltonian for each fragment, with which the electron density is optimized through an iterative double self-consistent field (DSCF) procedure to realize the mutual many-body polarization effects. In this work, the accuracy of interfragment interaction energies enumerated using the Coulomb bath is tested, showing a good agreement with the exact results from an energy decomposition analysis. The qualitative features of many-body polarization effects are visualized by electron density difference plots. It is also shown that the present DSCF method can yield fast and robust convergence with near-linear scaling in performance with increase in system size.     

Comment on “density and physical current density functional theory” by Xiao‐Yin Pan and Viraht Sahni

A recent paper by Xiao‐Yin Pan and Viraht Sahni [Int. J. Quant. Chem. 110, 2833 (2010)] claims that current density functional theory should be based on the physical current density rather than the paramagnetic current density, as in the standard Vignale‐Rasolt formulation. In this comment we show that the claims in the paper by Pan and Sahni are erroneous. © 2012 Wiley Periodicals, Inc.     

Long‐range analysis of density fitting in extended systems

Density fitting scheme is analyzed for the Coulomb problem in extended systems from the correctness of long‐range behavior point of view. We show that for the correct cancellation of divergent long‐range Coulomb terms it is crucial for the density fitting scheme to reproduce the overlap matrix exactly. It is demonstrated that from all possible fitting metric choices the Coulomb metric is the only one which inherently preserves the overlap matrix for infinite systems with translational periodicity. Moreover, we show that by a small additional effort any non‐Coulomb metric fit can be made overlap‐preserving as well. The problem is analyzed for both ordinary and Poisson basis set choices. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Analytical gradients for subsystem density functional theory within the slater‐function‐based amsterdam density functional program

We present a new implementation of analytical gradients for subsystem density‐functional theory (sDFT) and frozen‐density embedding (FDE) into the Amsterdam Density Functional program (ADF). The underlying theory and necessary expressions for the implementation are derived and discussed in detail for various FDE and sDFT setups. The parallel implementation is numerically verified and geometry optimizations with different functional combinations (LDA/TF and PW91/PW91K) are conducted and compared to reference data. Our results confirm that sDFT‐LDA/TF yields good equilibrium distances for the systems studied here (mean absolute deviation: 0.09 Å) compared to reference wave‐function theory results. However, sDFT‐PW91/PW91k quite consistently yields smaller equilibrium distances (mean absolute deviation: 0.23 Å). The flexibility of our new implementation is demonstrated for an HCN‐trimer test system, for which several different setups are applied. © 2016 Wiley Periodicals, Inc.     

First‐order correlation‐kinetic contribution to Kohn–Sham exchange charge density function in atoms,using quantal density functional theory approach

Using the static exchange‐correlation charge density concept, the total integrated exchange‐charge density function is calculated within the nonrelativistic spin‐restricted exchange‐only (i) optimized effective potential model, and (ii) nonvariational local potential derived from the exchange‐only work potential within the quantal density functional theory, for the ground‐state isoelectronic series: Ga⁺, Zn, Cu^?; In⁺, Cd, Ag^?; and Tl⁺, Hg, Au^?. The difference between the exchange charge density function derived from these potentials is employed to evaluate the first‐order correlation‐kinetic contribution to the integrated exchange charge density. This contribution is found to be important for both the intra‐ and inter‐shell regions. Screening effects on the contribution due to the nd¹⁰ (n = 3–5) subshells are discussed through comparisons with similar calculations on Ca, Sr, and Ba, wherein nd¹⁰ electrons are absent. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

New theoretically predicted RDX‐ and β‐HMX‐based high‐energy‐density molecules

Theoretically new high‐energy‐density materials (HEDM) in which the hydrogens on RDX and β‐HMX (hexahydro‐1,3,5‐trinitro‐1,3,5‐triazine and octahydro‐1,3,5,7‐tetranitro‐1,3,5,7‐tetrazocine, respectively) were sequentially replaced by (N NO₂)x functional groups were designed and evaluated using density functional theory calculations in combination with the Kamlet–Jacobs equations and an atoms‐in‐molecules (AIM) analysis. Improved detonation properties and reduced sensitivity compared to RDX and β‐HMX were predicted. Interestingly, the RDX and β‐HMX derivatives having one attached N NO₂ group [RDX‐(NNO₂)1 and HMX‐(NNO₂)1] showed excellent detonation properties (detonation velocities: 9.529 and 9.575 km·s⁻¹, and detonation pressures: 40.818 and 41.570 GPa, respectively), which were superior to the parent compounds. Sensitivity estimations obtained by calculating impact sensitivities and HOMO‐LUMO gaps indicated that RDX‐(NNO₂)1 and HMX‐(NNO₂)1 were less stable than RDX and HMX but more stable than any of the other derivatives. This method of sequential NNO₂ group attachment on conventional HEDMs offers a firm basis for further studies on the design of new explosives. Furthermore, the newly found structures may be promising candidates for better HEDMs.     

New parallel optimal‐parameter fast multipole method (OPFMM)

The three key translation equations of the fast multipole method (FMM) are deduced from the general polypolar expansions given earlier. Simplifications are introduced for the rotation‐based FMM that lead to a very compact FMM formalism. The optimum‐parameter searching procedure, a stable and efficient way of obtaining the optimum set of FMM parameters, is established with complete control over the tolerable error (ε). This new procedure optimizes the linear scaling with respect to the number of particles for a given ε. In addition, a new parallel FMM algorithm, which requires virtually no internode communication, is suggested that is suitable for the parallel construction of Fock matrices in electronic structure calculations. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1484–1501, 2001     

Moment expansion of the linear density‐density response function

We present a low rank moment expansion of the linear density‐density response function. The general interacting (fully nonlocal) density‐density response function is calculated by means of its spectral decomposition via an iterative Lanczos diagonalization technique within linear density functional perturbation theory. We derive a unitary transformation in the space of the eigenfunctions yielding subspaces with well‐defined moments. This transformation generates the irreducible representations of the density‐density response function with respect to rotations within SO(3). This allows to separate the contributions to the electronic response density from different multipole moments of the perturbation. Our representation maximally condenses the physically relevant information of the density‐density response function required for intermolecular interactions, yielding a considerable reduction in dimensionality. We illustrate the performance and accuracy of our scheme by computing the electronic response density of a water molecule to a complex interaction potential. © 2015 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     

A density functional theory study on the catalytic mechanism of hydroxycinnamoyl‐CoA hydratase‐lyase

Hydroxycinnamoyl‐CoA hydratase‐lyase (HCHL), a particular member of the crotonase superfamily, catalyzes the bioconversion of feruloyl‐CoA to the important flavor and fragrance compound vanillin. In this article, the catalytic mechanism of HCHL has been studied by using hybrid density functional theory method with simplified models. The calculated results reveal that the mechanism involves the hydration of the C?C double bond of feruloyl‐CoA and thence the cleavage of C? C single bond of β‐hydroxythioester. The hydration step is a typical concerted process, whereas C? C bond cleavage follows a concerted but asynchronous mechanism. The calculated energy barrier of hydration reaction is only slightly lower than that of cleavage process, implying both of two processes are rate limiting. By using three substrate analogs, the substrate specificity of HCHL was further examined. It is found that the p‐hydroxyl group of aromatic ring is necessary for the catalytic reaction. © 2013 Wiley Periodicals, Inc.     

Kinetic energy density for orbital‐free density functional calculations by axiomatic approach

An axiomatic approach is herein used to determine the physically acceptable forms for general D‐dimensional kinetic energy density functionals (KEDF). The resulted expansion captures most of the known forms of one‐point KEDFs. By statistically training the KEDF forms on a model problem of noninteracting kinetic energy in 1D (six terms only), the mean relative accuracy for 1000 randomly generated potentials is found to be better than the standard KEDF by several orders of magnitudes. The accuracy improves with the number of occupied states and was found to be better than for a system with four occupied states. Furthermore, we show that free fitting of the coefficients associated with known KEDFs approaches the exactly analytic values. The presented approach can open a new route to search for physically acceptable kinetic energy density functionals and provide an essential step toward more accurate large‐scale orbital free density functional theory calculations.     

Electron‐pair density decomposition for core–valence separable systems

The electron pair density of a core‐valence separable system can be decomposed into three parts: core‐core, core‐valence, and valence‐valence. The core‐core part has a Hartree‐Fock like structure. The core‐valence part can be written as Γ_cv (1,2) = γ_c (1,1)γ_v (2,2) ? γ_c (1,2)γ_v (2,1) + γ_c (2,2)γ_v (1,1) ? γ_c (2,1)γ_v (1,2), where only the 1‐matrices from the core and valence orbitals contribute. The valence‐valence part is left to be determined from the reduced frozen‐core type wave function, which often contains the essential information on the electron correlation and the chemical bond. We demonstrate the analysis to the ground state of negative ion Li^? and 2¹Σ_u⁺ excited state of the Li₂ molecule. © 2012 Wiley Periodicals, Inc.     

Interplay of Electronic and Steric Effects to Yield Low‐Temperature CO Oxidation at Metal Single Sites in Defect‐Engineered HKUST‐1

In contrast to catalytically active metal single atoms deposited on oxide nanoparticles, the crystalline nature of metal‐organic frameworks (MOFs) allows for a thorough characterization of reaction mechanisms. Using defect‐free HKUST‐1 MOF thin films, we demonstrate that Cu⁺/Cu²⁺ dimer defects, created in a controlled fashion by reducing the pristine Cu²⁺/Cu²⁺ pairs of the intact framework, account for the high catalytic activity in low‐temperature CO oxidation. Combining advanced IR spectroscopy and density functional theory we propose a new reaction mechanism where the key intermediate is an uncharged O₂ species, weakly bound to Cu⁺/Cu²⁺. Our results reveal a complex interplay between electronic and steric effects at defect sites in MOFs and provide important guidelines for tailoring and exploiting the catalytic activity of single metal atom sites.