Kohn-Sham Theory Research Articles

On the Simulation of Photoreactions Using Restricted Open-Shell Kohn-Sham Theory.

It is a well-established standard to describe ground-state chemical reactions at an ab initio level of multi-electron theory. Fast reactions can be directly simulated. The most widely used approach is density functional theory for the electronic structure in combination with molecular dynamics for the nuclear motion. This approach is known as ab initio molecular dynamics. In contrast, the simulation of excited-state reactions at this level of theory is significantly more difficult. It turns out that the self-consistent solution of the Kohn-Sham equations is not easily reached in excited-state simulations. The first program that solved this problem was the Car-Parrinello molecular dynamics code, using restricted open-shell Kohn-Sham theory. Meanwhile, there are alternatives, most prominently the Q-Chem code, which widens the range of applications. The present study investigates the suitability of both codes for the molecular dynamics simulation of excited-state motion and presents applications to photoreactions.

Boosting the Modeling of Infrared and Raman Spectra of Bulk Phase Chromophores with Machine Learning.

In the field of vibrational spectroscopy simulation, hybrid approximations to Kohn-Sham density-functional theory (KS-DFT) are often considered computationally prohibitive due to the significant effort required to evaluate the exchange-correlation potential in planewave codes. In this Letter, we show that by leveraging the porting of KS-DFT on GPU and incorporating machine-learning techniques, simulating IR and Raman spectra of real-life chromophores in bulk aqueous solution becomes a routine application at this level of theory.

Predicting fundamental gaps accurately from density functional theory with non-empirical local range separation.

We present an exchange-correlation approximation in which the Coulomb interaction is split into long- and short-range components and the range separation is determined by a non-empirical density functional. The functional respects important constraints, such as the homogeneous and slowly varying density limits, leads to the correct long-range potential, and eliminates one-electron self-interaction. Our approach is designed for spectroscopic purposes and closely approximates the piecewise linearity of the energy as a function of the particle number. The functional's accuracy for predicting the fundamental gap in generalized Kohn-Sham theory is demonstrated for a large number of systems, including organic semiconductors with a notoriously difficult electronic structure.

Widening of the fundamental gap in cluster GW for metal-molecular interfaces.

The GW approximation is very promising for an accurate first-principles description of charged excitations in single-molecule-metal interfaces. In the cluster approach for electronic transport across molecules, the infinite metal (with an adsorbed molecule) is replaced by a finite cluster whose volume should be incrementally increased to test the approach to the thermodynamic limit. Here we show that in GW, the approach to the thermodynamic limit will be much slower than in Kohn-Sham density-functional theory (DFT) because of the Coulomb interaction. To demonstrate this statement, we investigate spectral gaps in an ensemble of disordered sodium clusters in Kohn-Sham DFT, quasiparticle eigenvalue-self-consistent GW and Hartree-Fock. The fundamental gaps (i.e. difference between the lowest unoccupied and highest occupied level) in GW scale as N-1/3 on average, where N is the number of atoms. We demonstrate that this slow decrease artificially depletes the density of states at the Fermi level when the cluster is used to simulate a semi-infinite electrode. Therefore, the GW method cannot be taken as an out-of-the-box improvement of the DFT in cluster geometries, unless careful convergence checks are performed.

The electron-centric approach to the exchange-correlation energy.

The Kohn-Sham theory addresses the challenge of representing the kinetic energy by re-quantizing density functional theory at a level of non-interacting electrons. It transforms the many-electron problem into a fictitious non-interacting electron problem, with the many-electron effects concealed within the exchange-correlation (XC) energy, which is expressed in terms of the electron density ρ(r). Unlike the wave function, ρ(r) can be viewed as a classical quantity, and expressing the XC energy in terms of it circumvents the need for correlated wave functions. In this work, we once again employ the re-quantization strategy and determine the XC energy using a local one-particle Schrödinger equation. The ground-state eigenfunction of the corresponding Hamiltonian is a reference point (r) dependent orbital φr,σ(u, σ') which is subsequently used to generate the XC hole and the XC energy. The spin coordinate is denoted by σ and u is the electron-electron separation. The one-particle equation for φr,σ(u, σ') includes a local potential vr,σ(u, σ') that we approximate using two simple physical constraints. We assess the approximation by applying it to the helium iso-electronic series, the homogeneous electron gas, and the dissociation of the hydrogen molecule.

Meta-generalized gradient approximations in time dependent generalized Kohn-Sham theory: Importance of the current density correction.

We revisit the use of Meta-Generalized Gradient Approximations (mGGAs) in time-dependent density functional theory, reviewing conceptual questions and solving the generalized Kohn-Sham equations by real-time propagation. After discussing the technical aspects of using mGGAs in combination with pseudopotentials and comparing real-space and basis set results, we focus on investigating the importance of the current-density based gauge invariance correction. For the two modern mGGAs that we investigate in this work, TASK and r2SCAN, we observe that for some systems, the current density correction leads to negligible changes, but for others, it changes excitation energies by up to 40% and more than 0.8eV. In the cases that we study, the agreement with the reference data is improved by the current density correction.

Exposing minimal composition of Kohn–Sham theory and its extendability

Reducing the many-fermion problem to a set of single-particle (s.p.) equations, the Kohn–Sham (KS) theory has provided a practical tool to implement ab initio calculations of ground-state energies and densities in many-electron systems. There have been attempts to extend the KS theory so that it could describe other physical quantities, or it could be applied to other many-fermion systems. By generalizing and reformulating the KS theory in terms of the 1-body density matrix, we expose the minimal composition of the theory that enables the reduction of the many-fermion problem to the s.p. equations. Based on the reformulation, several basic issues are reconsidered. The v- and N-representabilities for the KS theory are distinguished from those for the Hohenberg-Kohn theorem. Criteria for the extendability of the KS theory are addressed.

Chemical bond analysis for the entire periodic table: energy decomposition and natural orbitals for chemical valence in the four-component relativistic framework

The Energy Decomposition Analysis in combination with Natural Orbitals for Chemical Valence (EDA-NOCV) is a very powerful tool for the analysis of the chemical bonds. While the approach has been applied in a variety of chemical contexts, the current implementations of the EDA-NOCV scheme include relativistic effects only at scalar level, so simply neglecting the spin-orbit coupling effects and de facto limiting its applicability. In this work, we extend the EDA-NOCV method to the relativistic four-component Dirac–Kohn–Sham theory that variationally accounts for spin-orbit coupling. Its correctness and numerical stability have been demonstrated in the case of simple molecular systems, where the relativistic effects play a negligible role, by comparison with the implementation available in the ADF modelling suite (using the non-relativistic Hamiltonian and the scalar ZORA approximation). As an illustrative example we analyse the metal-ethylene coordination bond in the group 6-element series (CO)TM-CH, with TM=Cr, Mo, W, Sg, where relativistic effects are likely to play an increasingly important role as one moves down the group. The method provides a clear measure of the donation and back-donation components in coordination bonds, even when relativistic effects, including spin-orbit coupling, are crucial for understanding the chemical bond involving heavy and superheavy atoms.

Effects of dispersion corrections on the theoretical description of bulk metals

The addition of a London dispersion correction to standard Kohn-Sham density-functional theory is essential for an accurate description of noncovalent interactions. While several dispersion-corrected density functionals (DC-DFs) have shown excellent performance for hard solids at ambient conditions, their transferability to metallic systems at ambient conditions or under isotropic compression has not been systematically examined. In this study, we assess the ability of selected DC-DFs to describe the equations of state (EOSs) of selected elemental metals and intermetallic compounds up to several gigapascals of pressure. EOS-derived properties, such as the unit-cell volume, the bulk modulus, and its pressure derivative, were then evaluated with and without thermal effects and the results compared with experimental reference data. We also assess the ability of the DC-DFs to predict the phase-transition pressures for a set of intermetallic compounds. The results of this study establish that London dispersion physics, and even dispersion contributions from the core electrons, is important in the description of bulk metals.

The fourth-order expansion of the exchange hole and neural networks to construct exchange-correlation functionals.

The curvature Qσ of spherically averaged exchange (X) holes ρX,σ(r, u) is one of the crucial variables for the construction of approximations to the exchange-correlation energy of Kohn-Sham theory, the most prominent example being the Becke-Roussel model [A. D. Becke and M. R. Roussel, Phys. Rev. A 39, 3761 (1989)]. Here, we consider the next higher nonzero derivative of the spherically averaged X hole, the fourth-order term Tσ. This variable contains information about the nonlocality of the X hole and we employ it to approximate hybrid functionals, eliminating the sometimes demanding calculation of the exact X energy. The new functional is constructed using machine learning; having identified a physical correlation between Tσ and the nonlocality of the X hole, we employ a neural network to express this relation. While we only modify the X functional of the Perdew-Burke-Ernzerhof functional [Perdew et al., Phys. Rev. Lett. 77, 3865 (1996)], a significant improvement over this method is achieved.

Reliable Isotropic Electron-Paramagnetic-Resonance Hyperfine Coupling Constants from the Frozen-Density Embedding Quasi-Diabatization Approach.

We present a systematic benchmark of isotropic electron-paramagnetic-resonance hyperfine coupling constants calculated for radical cation and anion complexes of molecules contained in the S22 test set using the frozen-density embedding quasi-diabatization (FDE-diab) approach. The results are compared to those from Kohn-Sham density-functional theory and frozen-density embedding, employing the domain-based local pair natural orbital coupled cluster singles and doubles method as a reference. We demonstrate that our new approach outperforms frozen-density embedding in all cases and provides reliable hyperfine couplings for radical cations using rather simple generalized-gradient approximation-type functionals. By contrast, more sophisticated and computationally less efficient exchange-correlation approximations are required for Kohn-Sham density-functional theory. For the radical anions, FDE-diab can at least provide an accuracy similar to that of Kohn-Sham density-functional theory. Finally, we demonstrate the computational advantages of FDE-diab for a π-stacked benzene octamer radical cation.

Comparing correlation components and approximations in Hartree-Fock and Kohn-Sham theories via an analytical test case study.

The asymmetric Hubbard dimer is a model that allows for explicit expressions of the Hartree-Fock (HF) and Kohn-Sham (KS) states as analytical functions of the external potential, Δv, and of the interaction strength, U. We use this unique circumstance to establish a rigorous comparison between the individual contributions to the correlation energies stemming from the two theories in the {U, Δv} parameter space. Within this analysis of the Hubbard dimer, we observe a change in the signof the HF kinetic correlation energy, compare the indirect repulsion energies, and derive an expression for the "traditional" correlation energy, i.e., the one that corrects the HF estimate, in a pure site-occupation function theory spirit [Eq. (45)]. Next, we test the performances of the Liu-Burke and the Seidl-Perdew-Levy functionals, which model the correlation energy based on its weak- and strong-interaction limit expansions and can be used for both the traditional and the KS correlation energies. Our results show that, in the Hubbard dimer setting, they typically work better for the HF reference, despite having been originally devised for KS. These conclusions are somewhat in line with prior assessments of these functionals on various chemical datasets. However, the Hubbard dimer model allows us to show the extent of the error that may occur in using the strong-interaction ingredient for the KS reference in place of the one for the HF reference, as has been carried out in most of the prior assessments.

On modeling the induced charge in density-functional calculations for field emitters

The default assumption of many density-functional theory codes that the simulation cell is spatially periodic implies that any unbalanced charge in the cell will cause the solution to diverge, unless the imbalance is removed in some unphysical way. Periodic solution thus makes it difficult to model accurately the charge and field that are induced at the apex of a single carbon nanotube (CNT) when a background electric field is applied. We describe how the charge induced in a single cell containing 1.8 nm of the capped end of a (5,5) CNT can be calculated from a macroscopic model of the CNT with an external field acting on the whole CNT. With this method, a cell containing the CNT tip has been analyzed using the program ONETEP, a linear-scaling code that iterates the density kernel and the localized orbitals self-consistently to minimize the Helmholtz free energy. The results shown include (1) the sheath of mobile charge outside the framework of nuclei; (2) Kohn–Sham (KS) orbitals including the localized end states that are occupied when the field is applied; (3) total effective potential distribution as a function of the applied field; and (4) an induced field-enhancement factor of 50 deduced from the change of potential with the applied field. The computation also shows that (5) the charge density in zero field extends into the potential barrier over a distance of at least 0.12 nm beyond the Fermi equipotential, consistent with KS theory for the boundary between emitter and barrier.

Metal-organic framework supported single-site nickel catalysts for butene dimerization

Homotopic sites in a well-controlled environment are not only ideal systems for mechanistic studies, but also allow optimal control of catalytic transformations. Sites having only a single metal cation and sites consisting of metal oxo complexes with few nickel (Ni) cations supported on the nodes of UiO-66 metal–organic framework (Ni-UiO-66) are studied for 1-butene dimerization. Monomeric Ni sites, which bind to the Zr6 node via two Zr-OH(μ3) linkages, are active and selective for the dimerization of 1-butene to linear and mono-branched C8 isomers. Ni oxo complexes with few Ni cations show lower activity and promote the oligomerization of transiently formed C8 isomers. Kohn-Sham density function theory calculations combined with spectroscopic measurements and kinetic analyses indicate that dimerization follows a Cossee-Arlman reaction mechanism.

High harmonic spectra computed using time-dependent Kohn–Sham theory with Gaussian orbitals and a complex absorbing potential

High harmonic spectra for H2 and H2 + are simulated by solving the time-dependent Kohn-Sham equation in the presence of a strong laser field using an atom-centered Gaussian representation of the density and a complex absorbing potential. The latter serves to mitigate artifacts associated with the finite extent of the basis functions, including spurious reflection of the outgoing electronic wave packet. Interference between the outgoing and reflected waves manifests as peak broadening in the spectrum as well as the appearance of spurious high-energy peaks after the harmonic progression has terminated. We demonstrate that well-resolved spectra can be obtained through the use of an atom-centered absorbing potential. As compared to grid-based algorithms, the present approach is more readily extensible to larger molecules.

A robust and efficient line search for self-consistent field iterations

We propose a novel adaptive damping algorithm for the self-consistent field (SCF) iterations of Kohn-Sham density-functional theory, using a backtracking line search to automatically adjust the damping in each SCF step. This line search is based on a theoretically sound, accurate and inexpensive model for the energy as a function of the damping parameter. In contrast to usual SCF schemes, the resulting algorithm is fully automatic and does not require the user to select a damping. We successfully apply it to a wide range of challenging systems, including elongated supercells, surfaces and transition-metal alloys.

Vertical ionization potential benchmarks from Koopmans prediction of Kohn–Sham theory with long-range corrected (LC) functional* *In memory of Enrico Clementi.

The Kohn–Sham density functional theory (KS-DFT) with the long-range corrected (LC) functional is applied to the benchmark dataset of 401 valence ionization potentials (IPs) of 63 small molecules of Chong, Gritsenko and Baerends (the CGB set). The vertical IP of the CGB set are estimated as negative orbital energies within the context of the Koopmans’ prediction using the LCgau-core range-separation scheme in combination with PW86–PW91 exchange–correlation functional. The range separation parameter μ of the functional is tuned to minimize the error of the negative HOMO orbital energy from experimental IP. The results are compared with literature data, including ab initio IP variant of the equation-of-motion coupled cluster theory with singles and doubles (IP-EOM-CCSD), the negative orbital energies calculated by KS-DFT with the statistical averaging of orbital potential, and those with the QTP family of functionals. The optimally tuned LC functional performs better than other functionals for the estimation of valence level IP. The mean absolute deviations (MAD) from experiment and from IP-EOM-CCSD are 0.31 eV (1.77%) and 0.25 eV (1.46%), respectively. LCgau-core performs quite well even with fixed μ (not system-dependent). A μ value around 0.36 bohr−1 gives MAD of 0.40 eV (2.42%) and 0.33 eV (1.96%) relative to experiment and IP-EOM-CCSD, respectively. The LCgau-core-PW86–PW91 functional is an efficient alternative to IP-EOM-CCSD and it is reasonably accurate for outer valence orbitals. We have also examined its application to core ionization energies of C(1s), N(1s), O(1s) and F(1s). The C(1s) core ionization energies are reproduced reasonably [MAD of 46 cases is 0.76 eV (0.26%)] but N(1s), O(1s) and F(1s) core ionization energies are predicted less accurately.

A general justification for hybrid functionals in DFT by means of linear response theory* *To the memory of Enrico Clementi, an outstanding human being; in recognition of his pioneering work in quantum chemistry and in tribute to his leadership and long-lasting accomplishments.

In the present work, resorting to linear response theory, we examine the plausibility of postulating Kohn–Sham (KS)-type equations which contain, by definition, an effective hybrid potential made up by some arbitrary mixture of local and non-local terms. In this way a general justification for the construction of hybrid functionals is provided without resorting to arguments based on the adiabatic connection, the generalized KS theory or the Levy’s constrained search (or its variations). In particular, we examine the cases of single-hybrid functionals, derived from non-local exchange and of double-hybrid functionals, emerging from non-local second-order expressions obtained from the KS perturbation theory. A further generalization for higher-order hybrid functionals is also included.

Approximate functionals in hypercomplex Kohn–Sham theory

The recently developed hypercomplex Kohn–Sham (HCKS) theory shows great potential to overcome the static/strong correlation issue in density functional theory (DFT), which highlights the necessity of further exploration of the HCKS theory toward better handling many-electron problem. This work mainly focuses on approximate functionals in HCKS, seeking to gain more insights into functional development from the comparison between Kohn–Sham (KS) DFT and HCKS. Unlike KS-DFT, HCKS can handle different correlation effects by resorting to a set of auxiliary orbitals with dynamically varying fractional occupations. These orbitals of hierarchical correlation (HCOs) thus contain distinct electronic information for better considering the exchange–correlation effect in HCKS. The test on the triplet–singlet gaps shows that HCKS has much better performance as compared to KS-DFT in use of the same functionals, and the systematic errors of semi-local functionals can be effectively reduced by including appropriate amount of the HCO-dependent Hartree–Fock exchange. In contrast, KS-DFT shows large systematic errors, which are hardly reduced by the functionals tested in this work. Therefore, HCKS creates new channels to address to the strong correlation issue, and further development of functionals that depend on HCOs and their occupations is necessary for the treatment of strongly correlated systems.

Incorporation of the Fermi-Amaldi Term into Direct Energy Kohn-Sham Calculations.

In direct energy Kohn-Sham (DEKS) theory, the density functional theory electronic energy equals the sum of occupied orbital energies, obtained from Kohn-Sham-like orbital equations involving a shifted Hartree exchange-correlation potential, which must be approximated. In the present study, the Fermi-Amaldi term is incorporated into approximate DEKS calculations, introducing the required -1/r contribution to the exchange-correlation component of the shifted potential in asymptotic regions. It also provides a mechanism for eliminating one-electron self-interaction error, and it introduces a nonzero exchange-correlation component of the shift in the potential that is of appropriate magnitude. The resulting electronic energies are very sensitive to the methodologies considered, whereas the highest occupied molecular orbital energies and exchange-correlation potentials are much less sensitive and are similar to those obtained from DEKS calculations using a conventional exchange-correlation functional.