Kohn-Sham theory for ground-state ensembles

Local Hybrid Functional Applicable to Weakly and Strongly Correlated Systems

The recent idea (Wodyński, A.; Arbuznikov, A. V.; Kaupp M. J. Chem. Phys. 2021, 155, 144101) to augment local hybrid functionals by a strong-correlation (sc) factor obtained from the adiabatic connection in the spirit of the KP16 model has been extended and applied to generate the accurate sc-corrected local hybrid functional scLH22t. By damping small values of the ratio between nondynamical and dynamical correlation entering the correction factor, it has become possible to avoid double counting of nondynamical correlation for weakly correlated situations and thereby preserve the excellent accuracy of the underlying LH20t local hybrid for such cases almost perfectly. On the other hand, scLH22t improves substantially over LH20t in reducing fractional-spin errors (FSEs), in providing improved spin-restricted bond dissociation curves, and in treating some typical systems with multireference character. The obtained FSEs are similar to those of the KP16/B13 model and slightly larger than for B13, but performance for weakly correlated systems is better than for these two related methods, which are also difficult to use self-consistently. The recent DM21 functional based on the training of a deep neural network still performs somewhat better than scLH22t but allows no physical insights into the origins of reduced FSEs. Examination of local mixing functions (LMFs) for the corrected scLH22t and uncorrected LH20t functionals provides further insights: in weakly correlated situations, the LMF remains essentially unchanged. Strong-correlation effects manifest in a reduction of the LMF values in certain regions of space, even to the extent of producing negative LMF values. It is suggested that this is the mechanism by which also DM21, which may be viewed as a range-separated local hybrid, is able to reduce FSEs.

Chemical potential, derivative discontinuity, fractional electrons, jump of the Kohn-Sham potential, atoms as thermodynamic open systems, and other (mis)conceptions of the density functional theory of electrons in molecules

Many references exist in the density functional theory (DFT) literature to the chemical potential of the electrons in an atom or a molecule. The origin of this notion has been the identification of the Lagrange multiplier μ = ∂E/∂N in the Euler-Lagrange variational equation for the ground state density as the chemical potential of the electrons. We first discuss why the Lagrange multiplier in this case is an arbitrary constant and therefore cannot be a physical characteristic of an atom or molecule. The switching of the energy derivative ("chemical potential") from -I to -A when the electron number crosses the integer, called integer discontinuity or derivative discontinuity, is not physical but only occurs when the nonphysical noninteger electron systems and the corresponding energy and derivative ∂E/∂N are chosen in a specific discontinuous way. The question is discussed whether in fact the thermodynamical concept of a chemical potential can be defined for the electrons in such few-electron systems as atoms and molecules. The conclusion is that such systems lack important characteristics of thermodynamic systems and do not afford the definition of a chemical potential. They also cannot be considered as analogues of the open systems of thermodynamics that can exchange particles with an environment (a particle bath or other members of a Gibbsian ensemble). Thermodynamical (statistical mechanical) concepts like chemical potential, open systems, grand canonical ensemble etc. are not applicable to a few electron system like an atom or molecule. A number of topics in DFT are critically reviewed in light of these findings: jumps in the Kohn-Sham potential when crossing an integer number of electrons, the band gap problem, the deviation-from-straight-lines error, and the role of ensembles in DFT.

Ensemble Density Functional Theory of Neutral and Charged Excitations : Exact Formulations, Standard Approximations, and Open Questions

Recent progress in the field of (time-independent) ensemble density-functional theory (DFT) for excited states are reviewed. Both Gross-Oliveira-Kohn (GOK) and N-centered ensemble formalisms, which are mathematically very similar and allow for an in-principle-exact description of neutral and charged electronic excitations, respectively, are discussed. Key exact results, for example, the equivalence between the infamous derivative discontinuity problem and the description of weight dependencies in the ensemble exchange-correlation density functional, are highlighted. The variational evaluation of orbital-dependent ensemble Hartree-exchange (Hx) energies is discussed in detail. We show in passing that state-averaging individual exact Hx energies can lead to severe (although solvable) v-representability issues. Finally, we explore the possibility of using the concept of density-driven correlation, which has been introduced recently and does not exist in regular ground-state DFT, for improving state-of-the-art correlation density-functional approximations for ensembles. The present review reflects the efforts of a growing community to turn ensemble DFT into a rigorous and reliable low-cost computational method for excited states. We hope that, in the near future, this contribution will stimulate new formal and practical developments in the field.

Perturbation Theory Treatment of Spin-Orbit Coupling II: A Coupled Perturbed Kohn-Sham Method

A noncanonical coupled perturbed Kohn-Sham density functional theory (KS-DFT)/Hartree-Fock (HF) treatment of spin-orbit coupling (SOC) is provided. We take the scalar-relativistic KS-DFT/HF solution, obtained with a relativistic effective core potential, as the zeroth-order approximation. Explicit expressions are given for the total energy through the 4th order, which satisfy the 2n + 1 rule. Second-order expressions are provided for orbital energies and density variables of spin-current DFT. Test calculations are carried out on the halogen homonuclear diatomic and hydride molecules, including 6p and 7p elements, as well as open-shell negative ions. The computed properties through second or third order match well with those from reference two-component self-consistent field calculations for total and orbital energies as well as spin-current densities. In only one case (At₂^-) did a significant deviation occur for the remaining density variables. Our coupled perturbation theory approach provides an efficient way of adding the effect of SOC to a scalar-relativistic single-reference KS-DFT/HF treatment, in particular because it does not require diagonalization in the two-component spinor basis, leading to saving factors on the number of required floating-point operations that may exceed one order of magnitude.

Ultrafast Spectroscopy of Photoactive Molecular Systems from First Principles: Where We Stand Today and Where We Are Going

Computational spectroscopy is becoming a mandatory tool for the interpretation of the complex, and often congested, spectral maps delivered by modern non-linear multi-pulse techniques. The fields of Electronic Structure Methods, Non-Adiabatic Molecular Dynamics, and Theoretical Spectroscopy represent the three pillars of the virtual ultrafast optical spectrometer, able to deliver transient spectra in silico from first principles. A successful simulation strategy requires a synergistic approach that balances between the three fields, each one having its very own challenges and bottlenecks. The aim of this Perspective is to demonstrate that, despite these challenges, an impressive agreement between theory and experiment is achievable now regarding the modeling of ultrafast photoinduced processes in complex molecular architectures. Beyond that, some key recent developments in the three fields are presented that we believe will have major impacts on spectroscopic simulations in the very near future. Potential directions of development, pending challenges, and rising opportunities are illustrated.

On The Density Functional Theory Treatment of Lanthanide Coordination Compounds: A Comparative Study in a Series of Cu-Ln (Ln = Gd, Tb, Lu) Binuclear Complexes

The nontrivial aspects of electron structure in lanthanide complexes, considering ligand field (LF) and exchange coupling effects, have been investigated by means of density functional theory (DFT) calculations, taking as a prototypic case study a series of binuclear complexes [LCu(O₂COMe)Ln(thd)₂], where L^2- = N,N'-2,2-dimethyl-propylene-di(3-methoxy-salicylidene-iminato) and Ln = Tb, Lu, and Gd. Particular attention has been devoted to the Cu-Tb complex, which shows a quasi-degenerate nonrelativistic ground state. Challenging the limits of density functional theory (DFT), we devised a practical route to obtain different convergent solutions, permuting the starting guess orbitals in a manner resembling the run of the β electron formally originating from the f⁸ configuration of the Tb(III) over seven molecular orbitals (MOs) with predominant f-type character. Although the obtained states cannot be claimed as the DFT computed split of the ⁷F multiplet, the results are yet interesting numeric experiments, relevant for the ligand field effects. We also performed broken symmetry (BS) DFT estimation of exchange coupling in the Cu-Gd system, using different settings, with Gaussian-type and plane-wave bases, finding a good match with the coupling parameter from experimental data. We also caught BS-type states for each of the mentioned series of different states emulated for the Cu-Tb complex, finding almost equal exchange coupling parameters throughout the seven LF-like configurations, the magnitude of the J parameter being comparable with those of the Cu-Gd system.

Using the GVB Ansatz to develop ensemble DFT method for describing multiple strongly correlated electron pairs

Ensemble density functional theory (DFT) furnishes a rigorous theoretical framework for describing the non-dynamic electron correlation arising from (near) degeneracy of several electronic configurations. Ensemble DFT naturally leads to fractional occupation numbers (FONs) for several Kohn-Sham (KS) orbitals, which thereby become variational parameters of the methodology. The currently available implementation of ensemble DFT in the form of the spin-restricted ensemble-referenced KS (REKS) method was originally designed for systems with only two fractionally occupied KS orbitals, which was sufficient to accurately describe dissociation of a single chemical bond or the singlet ground state of biradicaloid species. To extend applicability of the method to systems with several dissociating bonds or to polyradical species, more fractionally occupied orbitals must be included in the ensemble description. Here we investigate a possibility of developing the extended REKS methodology with the help of the generalized valence bond (GVB) wavefunction theory. The use of GVB enables one to derive a simple and physically transparent energy expression depending explicitly on the FONs of several KS orbitals. In this way, a version of the REKS method with four electrons in four fractionally occupied orbitals is derived and its accuracy in the calculation of various types of strongly correlated molecules is investigated. We propose a possible scheme to ameliorate the partial size-inconsistency that results from perfect spin-pairing. We conjecture that perfect pairing natural orbital (NO) functionals of reduced density matrix functional theory (RDMFT) should also display partial size-inconsistency.

Restricted Ensemble-Referenced Kohn-Sham versus Broken Symmetry Approaches in Density Functional Theory: Magnetic Coupling in Cu Binuclear Complexes

The performance of density functional theory in estimating the magnetic coupling constant in a series of Cu(II) binuclear complexes is investigated by making use of two open shell formalisms:  the broken symmetry and the spin-restricted ensemble-referenced Kohn-Sham methods. The strong dependence of the calculated magnetic coupling constants with respect to the exchange-correlation functional is confirmed and found to be independent of whether spin symmetry is imposed or not. The use of a method which guarantees the spin state does not improve the correlation with the experiment and indeed shows some worsening due to an overestimation of the ferromagnetic interactions. However, with the present exchange-correlation functionals, a rather systematic deviation is found. Therefore, it would be possible to develop improved density functionals which will allow for a rigorous treatment of open shell systems in density functional theory.

Assessment of approximate computational methods for conical intersections and branching plane vectors in organic molecules

Quantum-chemical computational methods are benchmarked for their ability to describe conical intersections in a series of organic molecules and models of biological chromophores. Reference results for the geometries, relative energies, and branching planes of conical intersections are obtained using ab initio multireference configuration interaction with single and double excitations (MRCISD). They are compared with the results from more approximate methods, namely, the state-interaction state-averaged restricted ensemble-referenced Kohn-Sham method, spin-flip time-dependent density functional theory, and a semiempirical MRCISD approach using an orthogonalization-corrected model. It is demonstrated that these approximate methods reproduce the ab initio reference data very well, with root-mean-square deviations in the optimized geometries of the order of 0.1 Å or less and with reasonable agreement in the computed relative energies. A detailed analysis of the branching plane vectors shows that all currently applied methods yield similar nuclear displacements for escaping the strong non-adiabatic coupling region near the conical intersections. Our comparisons support the use of the tested quantum-chemical methods for modeling the photochemistry of large organic and biological systems.

Ensemble DFT Approach to Excited States of Strongly Correlated Molecular Systems

Ensemble density functional theory (DFT) is a novel time-independent formalism for obtaining excitation energies of many-body fermionic systems. A considerable advantage of ensemble DFT over the more common Kohn-Sham (KS) DFT and time-dependent DFT formalisms is that it enables one to account for strong non-dynamic electron correlation in the ground and excited states of molecular systems in a transparent and accurate fashion. Despite its positive aspects, ensemble DFT has not so far found its way into the repertoire of methods of modern computational chemistry, probably because of the perceived lack of practically affordable implementations of the theory. The spin-restricted ensemble-referenced KS (REKS) method is perhaps the first computationally feasible implementation of the ideas behind ensemble DFT which enables one to describe accurately electronic transitions in a wide class of molecular systems, including strongly correlated molecules (biradicals, molecules undergoing bond breaking/formation), extended π-conjugated systems, donor-acceptor charge transfer adducts, etc.

Frozen density embedding with non-integer subsystems' particle numbers

We extend the frozen density embedding theory to non-integer subsystems' particles numbers. Different features of this formulation are discussed, with special concern for approximate embedding calculations. In particular, we highlight the relation between the non-integer particle-number partition scheme and the resulting embedding errors. Finally, we provide a discussion of the implications of the present theory for the derivative discontinuity issue and the calculation of chemical reactivity descriptors.

The theoretical account of the ligand field bonding regime and magnetic anisotropy in the DySc2N@C80 single ion magnet endohedral fullerene

Methodological advances for the treatment of electron structure and properties of the f-type ions embedded in fullerenes are presented.

Assessment of Density Functional Methods for Obtaining Geometries at Conical Intersections in Organic Molecules

A number of commonly available density functionals have been tested for their ability to describe the energetics and the geometry at conical intersections in connection with the spin-restricted ensemble referenced Kohn–Sham (REKS) method. The minimum energy conical intersections have been optimized for several molecular systems, which are widely used as paradigmatic models of photochemical rearrangements and models of biological chromophores. The results of the calculations are analyzed using the sign-change theorem of Longuet-Higgins and a method of elementary reaction coordinates of Haas et al. The latter approach helps to elucidate the differences between the geometries at conical intersections as predicted by the multireference wave function ab initio methods and by the density functional methods. Overall, the BH&HLYP density functional yields the best results for the conical intersection geometries and energetics.

Guaranteed convergence of the Kohn-Sham equations

A sufficiently damped iteration of the Kohn-Sham (KS) equations with the exact functional is proven to always converge to the true ground-state density, regardless of the initial density or the strength of electron correlation, for finite Coulomb systems. We numerically implement the exact functional for one-dimensional continuum systems and demonstrate convergence of the damped KS algorithm. More strongly correlated systems converge more slowly.

Assessment of Density Functional Theory for Describing the Correlation Effects on the Ground and Excited State Potential Energy Surfaces of a Retinal Chromophore Model

In the quest for a cost-effective level of theory able to describe a large portion of the ground and excited potential energy surfaces of large chromophores, promising approaches are rooted in various approximations to the exact density functional theory (DFT). In the present work, we investigate how generalized Kohn-Sham DFT (GKS-DFT), time-dependent DFT (TDDFT), and spin-restricted ensemble-DFT (REKS) methods perform along three important paths characterizing a model retinal chromophore (the penta-2,4-dieniminium cation) in a region of near-degeneracy (close to a conical intersection) with respect to reference high-level multiconfigurational wave function methods. If GKS-DFT correctly describes the closed-shell charge transfer state, only TDDFT and REKS approaches give access to the open-shell diradical, one which sometimes corresponds to the electronic ground state. It is demonstrated that the main drawback of the usual DFT-based methods lies in the absence of interactions between the charge transfer and the diradicaloid configurations. Hence, we test a new computational scheme based on the State-averaged REKS (SA-REKS) approach, which explicitly includes these interactions into account. The State-Interaction SA-REKS (SI-SA-REKS) method significantly improves on the REKS and the SA-REKS results for the target system. The similarities and differences between DFT and wave function-based approaches are analyzed according to (1) the active space dimensions of the wave function-based methods and (2) the relative electronegativities of the allyl and protonated Schiff base moieties.

A multiconfigurational hybrid density-functional theory

We propose a multiconfigurational hybrid density-functional theory which rigorously combines a multiconfiguration self-consistent-field calculation with a density-functional approximation based on a linear decomposition of the electron-electron interaction. This gives a straightforward extension of the usual hybrid approximations by essentially adding a fraction λ of exact static correlation in addition to the fraction λ of exact exchange. Test calculations on the cycloaddition reactions of ozone with ethylene or acetylene and the dissociation of diatomic molecules with the Perdew-Burke-Ernzerhof and Becke-Lee-Yang-Parr density functionals show that a good value of λ is 0.25, as in the usual hybrid approximations. The results suggest that the proposed multiconfigurational hybrid approximations can improve over usual density-functional calculations for situations with strong static correlation effects.

Coordinate space translation technique for simulation of electronic process in the ion-atom collision

Recently we developed a theoretical model of ion-atom collisions, which was made on the basis of a time-dependent density functional theory description of the electron dynamics and a classical treatment of the heavy particle motion. Taking advantage of the real-space grid method, we introduce a "coordinate space translation" technique to allow one to focus on a certain space of interest such as the region around the projectile or the target. Benchmark calculations are given for collisions between proton and oxygen over a wide range of impact energy. To extract the probability of charge transfer, the formulation of Lüdde and Dreizler [J. Phys. B 16, 3973 (1983)] has been generalized to ensemble-averaging application in the particular case of O((3)P). Charge transfer total cross sections are calculated, showing fairly good agreements between experimental data and present theoretical results.

Ground-state degeneracies leave recognizable topological scars in the electronic density

In Kohn-Sham density functional theory (KS DFT) a fictitious system of noninteracting particles is constructed having the same ground-state (GS) density as the physical system of interest. A fundamental open question in DFT concerns the ability of an exact KS calculation to spot and characterize the GS degeneracies in the physical system. In this Letter we provide theoretical evidence suggesting that the GS density, as a function of position on a 2D manifold of parameters affecting the external potential, is "topologically scarred" in a distinct way by degeneracies. These scars are sufficiently detailed to enable determination of the positions of degeneracies and even the associated Berry phases. We conclude that an exact KS calculation can spot and characterize the degeneracies of the physical system.

Excitation energies from spin-restricted ensemble-referenced Kohn-Sham method: a state-average approach

A time-independent density functional approach to the calculation of excitation energies from the ground states of molecules typified by the strong nondynamic electron correlation is suggested. The new method is based on the use of the spin-restricted ensemble-referenced Kohn-Sham formalism [Filatov, M.; Shaik, S. Chem. Phys. Lett. 1999, 304, 429] for the calculation of the ground state. In the new method, the average energy of the ground state and a state created by a single excitation thereof is minimized with respect to the Kohn-Sham orbitals and their fractional occupation numbers. The lowest singlet excitation energies obtained with the help of the new formalism for a number of model systems, such as the hydrogen molecule with stretched bond, twisted ethylene, and twisted hexa-1,3,5-triene, are compared with the results of the time-dependent density functional theory, with the results of ab initio CASSCF/CASPT2 calculations, and with the experimental data. Applicability of the new method to the description of photochemical reactions is discussed.

Orbital currents in the Colle-Salvetti correlation energy functional and the degeneracy problem

Popular density functionals for the exchange-correlation energy typically fail to reproduce the degeneracy of different ground states of open-shell atoms. As a remedy, functionals which explicitly depend on the current density have been suggested. We present an analysis of this problem by investigating functionals that explicitly depend on the Kohn-Sham orbitals. Going beyond the exact-exchange approximation by adding correlation in the form of the Colle-Salvetti functional, we show how current-dependent terms enter the Colle-Salvetti expression and their relevance is evaluated. A very good description of the degeneracy of ground states for atoms of the first and second rows of the Periodic Table is obtained.

Towards understanding performance differences between approximate density functionals for spin states of iron complexes

Density functional theory has been widely used to investigate the structural and electronic properties of heme-containing proteins such as cytochrome P450. Nevertheless, recent studies have shown that approximate exchange-correlation energy density functionals can incorrectly predict the stability order of spin states in, for instance, iron-containing pyridine and imidazole systems. This raises questions about the validity of earlier theoretical studies. In this work, we systematically investigate a few typical inorganic and organic iron-containing complexes and try to understand the performance difference of various density functionals. Two oxidation states of iron, Fe(II) and Fe(III), with different spin states and both adiabatic and vertical structures are considered. A different description of the outmost molecular orbital is found to play the crucial role. Local density and generalized gradient based functionals bias the lower spin state and produce a more localized frontier orbital that is higher in energy than the hybrid functionals. Energy component analysis has been performed, together with comparison of numerous structural and electronic properties. Implications of the present work to the theoretical study of heme-containing biological molecules and other spin-related systems are discussed.

On the degeneracy of atomic states within exact-exchange (spin-) density functional theory

The problem of degenerate ground states of open-shell atoms is investigated in spin-restricted and spin-unrestricted density functional theories using the exact-exchange energy functional. For the spin-unrestricted case, spurious energy splittings of the order of 2-3 kcal/mol are found for atoms of the second and third periods which are larger than the splittings obtained from recently proposed approximate exchange functionals depending explicitly on the current density. In remarkable contrast, for spin-restricted calculations the degeneracy of different atomic ground states is recovered to within less than 0.6 kcal/mol.

Current-dependent extension of the Perdew-Burke-Ernzerhof exchange-correlation functional

The probability current density is used in addition to the electron density and its gradient as a variable in the construction of an exchange-correlation functional. Starting from the Perdew-Burke-Ernzerhof generalized gradient approximation, we employ exact conditions to build a nonempirical exchange functional. Matching the correlation functional to that for exchange yields a current-dependent approximation for correlation. The resulting functional is given in a simple closed form. Application of this approximation to open shell atoms eliminates the artificial level splitting of formally degenerate states observed with generalized gradient approximations.