DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science.

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Secondary Kinetic Peak in the Kohn-Sham Potential and Its Connection to the Response Step.

We consider a prototypical 1D model Hamiltonian for a stretched heteronuclear molecule and construct individual components of the corresponding KS potential, namely, the kinetic, the N - 1, and the conditional potentials. These components show very special features, such as peaks and steps, in regions where the density is drastically low. Some of these features are quite well-known, whereas others, such as a secondary peak in the kinetic potential or a second bump in the conditional potential, are less or not known at all. We discuss these features building on the analytical model treated in Giarrusso et al. J. Chem. Theory Comput. 2018, 14, 4151. In particular, we provide an explanation for the underlying mechanism which determines the appearance of both peaks in the kinetic potential and elucidate why these peaks delineate the region over which the plateau structure, due to the N - 1 potential, stretches. We assess the validity of the Heitler-London Ansatz at large but finite internuclear distance, showing that, if optimal orbitals are used, this model is an excellent approximation to the exact wave function. Notably, we find that the second natural orbital presents an extra node very far out on the side of the more electronegative atom.

Self-Consistent-Field Method for Correlated Many-Electron Systems with an Entropic Cumulant Energy.

A self-consistent field method is presented within density matrix functional theory. The computational cost for a correlated many-electron calculation is reduced to that of the self-consistent-field Hartree-Fock method, while the accuracy still reaches that of sophisticated configuration interaction based methods. In this method, the two-electron cumulant energy is measured with an information entropy associated with the Fermi-Dirac distribution of the occupation numbers. An eigenvalue equation for the orbitals is obtained, with the eigenvalues (orbital energies) connected to the occupation numbers through the Fermi-Dirac distribution. The occupation numbers for the strongly occupied orbitals are very close to the natural orbital occupation numbers from wave function methods. It covers in a single scheme the nondynamical correlation in weak or breaking bonds as well as the dynamical correlation at all distances. The method is well suited to large-scale potential energy surface calculation and molecular dynamics simulation.

Origin of the Enhanced Binding Capability toward Axial Nitrogen Bases of Ni(II) Porphyrins Bearing Electron-Withdrawing Substituents: An Electronic Structure and Bond Energy Analysis.

Axial coordination to metalloporphyrins is important in many biological and catalytic processes. Experiments found the axial coordination of nitrogenous bases to nickel(II) porphyrins to be strongly favored by electron-withdrawing substituents such as perfluorophenyls at the meso carbon positions. Careful analysis of the electronic structure reveals that the natural explanation in terms of density change of the nickel(II) porphyrin system (in particular the metal), does not apply. Electron density changes, by the assumed inductive or polarizing effects on the metal or on the porphyrin ring system, are slight. The effect is caused by a remarkable through-space electric field effect on the metalloporphyrin system, originating from the charge distribution inside the perfluorphenyl groups (mostly the C-F dipoles).

Density functional approximations for orbital energies and total energies of molecules and solids.

The relation of Kohn-Sham (KS) orbital energies to ionization energies and electron affinities is different in molecules and solids. In molecules, the local density approximation (LDA) and generalized gradient approximations (GGA) approximate the exact ionization energy (I) and affinity (A) rather well with self-consistently calculated (total energy based) ILDFA and ALDFA, respectively. The highest occupied molecular orbital (HOMO) energy and lowest unoccupied molecular orbital (LUMO) energy, however, differ significantly (by typically 4-6 eV) from these quantities, ϵHLDFA(mol)>-I(mol)≈-ILDFA(mol), ϵLLDFA(mol)<-A(mol)≈-ALDFA(mol). In solids, these relations are very different, due to two effects. The (almost) infinite extent of a solid makes the difference of orbital energies and (L)DFA calculated ionization energy and affinity disappear: in the solid state limit, ϵH(L)DFA(solid)=-I(L)DFA(solid) and ϵL(L)DFA(solid)=-A(L)DFA(solid). Slater's relation ∂E/∂ni = ϵi for local density functional approximations (LDFAs) [and Hartree-Fock (HF) and hybrids] is useful to prove these relations. The equality of LDFA orbital energies and LDFA calculated -ILDFA and -ALDFA in solids does not mean that they are good approximations to the exact quantities. The LDFA total energies of the ions with a delocalized charge are too low, hence ILDFA(solid) < I and ALDFA(solid) > A, due to the local-approximation error, also denoted delocalization error, of LDFAs in extended systems. These errors combine to make the LDFA orbital energy band gap considerably smaller than the exact fundamental gap, ϵLLDFA(solid)-ϵHLDFA(solid)=ILDFA(solid)-ALDFA(solid)

Light-induced water splitting by titanium-tetrahydroxide: a computational study.

Water oxidation by Ti(OH)4 in the ground and excited states was investigated using density functional (ΔSCF, TDDFT) methods gauged by the coupled cluster (CCSD, CCSD(T)) calculations. O2 and H2 are generated in a reaction sequence that starts with Ti(OH)4 reacting with H2O. This reaction can proceed by either nucleophilic attack by H2O or by H-atom abstraction from H2O. The nucleophilic attack has high energy barriers (40-120 kcal mol(-1)) in both the ground and excited states. On the other hand, H abstraction is effected by Ti(OH)4 in the excited state with a low energy barrier (4-8 kcal mol(-1)), generating OH˙. This is the rate-limiting barrier in the chain of O2 formation reactions proposed in this work. The production of free OH˙ radicals is not energetically feasible in the ground state. By absorbing two photons, two hydroxyl radicals are produced, which then form H2O2. By a stepwise H-abstraction from H2O2 and OOH˙, O2 is generated by absorbing two more photons. In each H-abstraction reaction a Ti(OH)4 is consumed and a Ti(OH)3H2O is produced. H2 production can proceed thermally from the latter in a very exothermic (68-105 kcal mol(-1)) bimolecular reaction. The solvent effects, modelled by explicit water molecules, have a limited influence on the reactivity.

Real-space representation of electron correlation in π-conjugated systems.

π-electron conjugation and aromaticity are commonly associated with delocalization and especially high mobility of the π electrons. We investigate if also the electron correlation (pair density) exhibits signatures of the special electronic structure of conjugated systems. To that end the shape and extent of the pair density and derived quantities (exchange-correlation hole, Coulomb hole, and conditional density) are investigated for the prototype systems ethylene, hexatriene, and benzene. The answer is that the effects of π electron conjugation are hardly discernible in the real space representations of the electron correlation. We find the xc hole to be as localized (confined to atomic or diatomic regions) in conjugated systems as in small molecules. This result is relevant for density functional theory (DFT). The potential of the electron exchange-correlation hole is the largest part of vxc, the exchange-correlation Kohn-Sham potential. So the extent of the hole directly affects the orbital energies of both occupied and unoccupied Kohn-Sham orbitals and therefore has direct relevance for the excitation spectrum as calculated with time-dependent DFT calculations. The potential of the localized xc hole is comparatively more attractive than the actual hole left behind by an electron excited from a delocalized molecular orbital of a conjugated system.

The importance of large-amplitude motions for the interpretation of mid-infrared vibrational absorption and circular dichroism spectra: 6,6'-dibromo-[1,1'-binaphthalene]-2,2'-diol in dimethyl sulfoxide.

Using the 6,6'-dibromo-[1,1'-binaphthalene]-2,2'-diol molecule and its vibrational absorption (VA) and vibrational circular dichroism (VCD) spectra measured in deuterated dimethyl sulfoxide as example, we present a first detailed study of the effects induced in VCD spectra by the large-amplitude motions of solvent molecules loosely bound to a solute molecule. We show that this type of perturbation can induce significant effects in the VA and VCD spectra. We also outline a computational procedure that can effectively model the effects induced in the spectra and at the same time provide detailed structural information regarding the relative orientations of moieties involved in a solute-solvent molecular complex.

Solvent induced enhancement of enantiomeric excess: a case study of the Henry reaction with cinchona thiourea as the catalyst.

Enantiomeric excess (ee) in asymmetric catalysis may be strongly dependent on the solvent. The reaction product may range from an almost racemic mixture to an ee of over 90% for different solvents. We study this phenomenon for the C-C coupling reaction between nitromethane and benzaldehyde (the Henry reaction) with cinchona thiourea as the catalyst, where solvents that are strong Lewis bases induce a high ee. We show that the effect of the solvent does not consist of a change in the reaction mechanism. Instead, the solvation "prepares" the molecule, which is very flexible, in a specific conformation. The reaction barriers in this conformer are not lower than for other conformers, but are sufficiently differentiated between the enantiomers to give rise to a large ee. It is the strong Lewis basicity of the solvent that leads to the clear preference in solution for the "asymmetric" conformer. Although general rules or predictions for how solvent effects could be harnessed to produce a desired ee in general would be hard to formulate, this study does show that it is in this case (and presumably in many other cases as well) specific solute-solvent interactions rather than effects of the dielectric continuum of the solvent that are the root cause of the solvent effect. This is in agreement with experiment for the Henry reaction.

A frontier orbital study with ab initio molecular dynamics of the effects of solvation on chemical reactivity: solvent-induced orbital control in FeO-activated hydroxylation reactions.

Solvation effects on chemical reactivity are often rationalized using electrostatic considerations: the reduced stabilization of the transition state results in higher reaction barriers and lower reactivity in solution. We demonstrate that the effect of solvation on the relative energies of the frontier orbitals is equally important and may even reverse the trend expected from purely electrostatic arguments. We consider the H abstraction reaction from methane by quintet [EDTAH(n)·FeO]((n-2)+), (n = 0-4) complexes in the gas phase and in aqueous solution, which we examine using ab initio thermodynamic integration. The variation of the charge of the complex with the protonation of the EDTA ligand reveals that the free energy barrier in gas phase increases with the negative charge, varying from 16 kJ mol(-1) for [EDTAH4·FeO](2+) to 57 kJ mol(-1) for [EDTAHn·FeO](2-). In aqueous solution, the barrier for the +2 complex (38 kJ mol(-1)) is higher than in gas phase, as predicted by purely electrostatic arguments. For the negative complexes, however, the barrier is lower than in gas phase (e.g., 45 kJ mol(-1) for the -2 complex). We explain this increase in reactivity in terms of a stabilization of the virtual 3σ* orbital of FeO(2+), which acts as the dominant electron acceptor in the H-atom transfer from CH4. This stabilization originates from the dielectric screening caused by the reorientation of the water dipoles in the first solvation shell of the charged solute, which stabilizes the acceptor orbital energy for the -2 complex sufficiently to outweigh the unfavorable electrostatic destabilization of the transition-state relative to the reactants in solution.

Assessment of density functional methods for reaction energetics: iridium-catalyzed water oxidation as case study.

We investigate basis set convergence for a series of density functional theory (DFT) functionals (both hybrid and nonhybrid) and compare to coupled-cluster with single and double excitations and perturbative triples [CCSD(T)] benchmark calculations. The case studied is the energetics of the water oxidation reaction by an iridium-oxo complex. Complexation energies for the reactants and products complexes as well as the transition state (TS) energy are considered. Contrary to the expectation of relatively weak basis set dependence for DFT, the basis set effects are large, for example, more than 10 kcal mol(-1) difference from converged basis for the activation energy with "small" basis sets (DZ/6-31G** for Ir/other atoms, or SVP) and still more than 6 kcal mol(-1) for def2-TZVPP/6-31G**. Inclusion of the dispersion correction in DFT-D3 schemes affects the energies of reactant complex (RC), TS, and product complex (PC) by almost the same amount; it significantly improves the complexation energy (the formation of RC), but has little effect on the activation energy with respect to RC. With converged basis, some pure GGAs (PBE-D3, BP86-D3) as well as the hybrid functional B3LYP-D3 are very accurate compared to benchmark CCSD(T) calculations.

Understanding solvent effects in vibrational circular dichroism spectra: [1,1'-binaphthalene]-2,2'-diol in dichloromethane, acetonitrile, and dimethyl sulfoxide solvents.

We present a combined experimental and computational investigation of the vibrational absorption (VA) and vibrational circular dichroism (VCD) spectra of [1,1'-binaphthalene]-2,2'-diol. First, the sensitive dependence of the experimental VA and VCD spectra on the solvent is demonstrated by comparing the experimental spectra measured in CH(2)Cl(2), CD(3)CN, and DMSO-d(6) solvents. Then, by comparing calculations performed for the isolated solute molecule to calculations performed for molecular complexes formed between solute and solvent molecules, we identify three main types of perturbations that affect the shape of the VA and VCD spectra when going from one solvent to another. These sources of perturbations are (1) perturbation of the Boltzmann populations, (2) perturbation of the electronic structure, and (3) perturbation of the normal modes.

On the equivalence of conformational and enantiomeric changes of atomic configuration for vibrational circular dichroism signs.

We study systematically the vibrational circular dichroism (VCD) spectra of the conformers of a simple chiral molecule, with one chiral carbon and an "achiral" alkyl substituent of varying length. The vibrational modes can be divided into a group involving the chiral center and its direct neighbors and the modes of the achiral substituent. Conformational changes that consist of rotations around the bond from the next-nearest neighbor to the following carbon, and bond rotations further in the chain, do not affect the modes around the chiral center. However, conformational changes within the chiral fragment have dramatic effects, often reversing the sign of the rotational strength. The equivalence of the effect of enantiomeric change of the atomic configuration and conformational change on the VCD sign (rotational strength) is studied. It is explained as an effect of atomic characteristics, such as the nuclear amplitudes in some vibrational modes as well as the atomic polar and axial tensors, being to a high degree determined by the local topology of the atomic configuration. They reflect the local physics of the electron motions that generate the chemical bonds rather than the overall shape of the molecule.

Hydroxylation catalysis by mononuclear and dinuclear iron oxo catalysts: a methane monooxygenase model system versus the Fenton reagent Fe(IV)O(H2O)5(2+).

Hydroxylation of aliphatic C-H bonds is a chemically and biologically important reaction, which is catalyzed by the oxidoiron group FeO(2+) in both mononuclear (heme and nonheme) and dinuclear complexes. We investigate the similarities and dissimilarities of the action of the FeO(2+) group in these two configurations, using the Fenton-type reagent [FeO(2+) in a water solution, FeO(H(2)O)(5)(2+)] and a model system for the methane monooxygenase (MMO) enzyme as representatives. The high-valent iron oxo intermediate MMOH(Q) (compound Q) is regarded as the active species in methane oxidation. We show that the electronic structure of compound Q can be understood as a dimer of two Fe(IV)O(2+) units. This implies that the insights from the past years in the oxidative action of this ubiquitous moiety in oxidation catalysis can be applied immediately to MMOH(Q). Electronically the dinuclear system is not fundamentally different from the mononuclear system. However, there is an important difference of MMOH(Q) from FeO(H(2)O)(5)(2+): the largest contribution to the transition state (TS) barrier in the case of MMOH(Q) is not the activation strain (which is in this case the energy for the C-H bond lengthening to the TS value), but it is the steric hindrance of the incoming CH(4) with the ligands representing glutamate residues. The importance of the steric factor in the dinuclear system suggests that it may be exploited, through variation in the ligand framework, to build a synthetic oxidation catalyst with the desired selectivity for the methane substrate.

Cu(bipy)2+/TEMPO-catalyzed oxidation of alcohols: radical or nonradical mechanism?

In the oxidation of alcohols with TEMPO as catalyst, the substrate has alternatively been postulated to be oxidized but uncoordinated TEMPO(+) (Semmelhack) or Cu-coordinated TEMPO(•) radical (Sheldon). The reaction with the Cu(bipy)(2+)/TEMPO cocatalyst system has recently been claimed, on the basis of DFT calculations, to not be a radical reaction but to be best viewed as electrophilic attack on the alcohol C-H(α) bond by coordinated TEMPO(+). This mechanism combines elements of the Semmelhack mechanism (oxidation of TEMPO to TEMPO(+)) and the Sheldon proposal ("in the coordination sphere of Cu"). The recent proposal has been challenged on the basis of DFT calculations with a different functional, which were reported to lead to a radical mechanism. We carefully examine the results for the two functionals and conclude from both the calculated energetics and from an electronic structure analysis that the results of the two DFT functionals are consistent and that both lead to the proposed mechanism with TEMPO not acting as radical but as (coordinated) positive ion.

On the origin dependence of the angle made by the electric and magnetic vibrational transition dipole moment vectors.

The concept of robustness of rotational strengths of vibrational modes in a VCD spectrum has been introduced as an aid in assignment of the absolute configuration with the help of the VCD spectrum. The criteria for robustness have been based on the distribution around 90° of the angles ξ(i) between electric and magnetic transition dipoles of all the modes i of a molecule. The angles ξ(i) (not, of course, the rotational strengths) are, however, dependent on the choice of origin. The derived criteria are for the center of mass chosen as the origin of the coordinate system. We stress in this note that application of the derived criteria assumes that excessive translation of the coordinate origin is not applied. Although the ξ(i) angles are not very sensitive to the position of the origin, very small displacements (a few Å) are not a problem, excessive translation of the origin does have considerable effect on the ξ(i) angles. In this note we quantify this effect and demonstrate how the distribution of ξ(i) angles is affected. Although it is possible to recalibrate the robustness criteria for the angles for a specific (large) displacement, we recommend that such displacement simply be avoided. It is to be noted that some modeling software does yield output with excessively displaced coordinate origin; this should be checked and corrected.

Electron pair density in the lowest 1Σ(u)(+) and 1Σ(g)(+) states of H2.

We demonstrate and advocate the use of observable quantities derived from the two-electron reduced density matrix - pair densities, conditional densities, and exchange-correlation holes--as signatures of the type of electron correlation in a chemical bond. The prototype cases of the lowest (1)Σ(u)(+) and (1)Σ(g)(+) states of H(2), which exhibit large variation in types of bonding, ranging from strongly ionic to covalent, are discussed. Both the excited (1)Σ(g)(+) and (1)Σ(u)(+) states have been interpreted as essentially consisting of (natural) orbital configurations with an inner electron in a contracted 1sσ(g) orbital and an outer electron in a diffuse (united atom type, Rydberg) orbital. We show that nevertheless totally different correlation behavior is encountered in various states when comparing them at a common internuclear distance. Also when following one state along the internuclear distance coordinate, strong variation in correlation behavior is observed, as expected. Switches between ionic to covalent character of a state occur till very large distances (40 bohrs for states approaching the 1s3[script-l] asymptotic limit, and 282 bohrs for states approaching the 1s4[script-l] limit).

Counterpoise correction is not useful for short and Van der Waals distances but may be useful at long range.

This article investigates the errors in supermolecule calculations for the helium dimer. In a full CI calculation, there are two errors. One is the basis set superposition error (BSSE), the other is the basis set convergence error (BSCE). Both of the errors arise from the incompleteness of the basis set. These two errors make opposite contributions to the interaction energies. The BSCE is by far the largest error in the short range and larger than (but much closer to) BSSE around the Van der Waals minimum. Only at the long range, the BSSE becomes the larger error. The BSCE and BSSE largely cancel each other over the Van der Waals well. Accordingly, it may be recommended to not include the BSSE for the calculation of the potential energy curve from short distance till well beyond the Van der Waals minimum, but it may be recommended to include the BSSE correction if an accurate tail behavior is required. Only if the calculation has used a very large basis set, one can refrain from including the counterpoise correction in the full potential range. These results are based on full CI calculations with the aug-cc-pVXZ (X = D, T, Q, 5) basis sets.

An abiotic analogue of the diiron(IV)oxo "diamond core" of soluble methane monooxygenase generated by direct activation of O2 in aqueous Fe(II)/EDTA solutions: thermodynamics and electronic structure.

We study the generation of a dinuclear Fe(IV)oxo species, [EDTAH·FeO·OFe·EDTAH](2-), in aqueous solution at room temperature using Density Functional Theory (DFT) and Ab Initio Molecular Dynamics (AIMD). This species has been postulated as an intermediate in the multi-step mechanism of autoxidation of Fe(II) to Fe(III) in the presence of atmospheric O(2) and EDTA ligand in water. We examine the formation of [EDTAH·FeO·OFe·EDTAH](2-) by direct cleavage of O(2), and the effects of solvation on the spin state and O-O cleavage barrier. We also study the reactivity of the resulting dinuclear Fe(IV)oxo system in CH(4) hydroxylation, and its tendency to decompose to mononuclear Fe(IV)oxo species. The presence of the solvent is shown to play a crucial role, determining important changes in all these processes compared to the gas phase. We show that, in water solution, [EDTAH·FeO·OFe·EDTAH](2-) (as well as its precursor [EDTAH·Fe·O(2)·Fe·EDTAH](2-)) exists as stable species in a S = 4 ground spin state when hydrogen-bonded to a single water molecule. Its structure comprises two facing Fe(IV)oxo groups, in an arrangement similar to the one evinced for the active centre of intermediate Q of soluble Methane Monooxygenase (sMMO). The inclusion of the water molecule in the complex decreases the overall symmetry of the system, and brings about important changes in the energy and spatial distribution of orbitals of the Fe(IV)oxo groups relative to the gas phase. In particular, the virtual 3σ* orbital of one of the Fe(IV)oxo groups experiences much reduced repulsive orbital interactions from ligand orbitals, and its consequent stabilisation dramatically enhances the electrophilic character of the complex, compared to the symmetrical non-hydrated species, and its ability to act as an acceptor of a H atom from the CH(4) substrate. The computed free energy barrier for H abstraction is 28.2 kJ mol(-1) (at the BLYP level of DFT), considerably below the gas phase value for monomeric [FeO·EDTAH](-), and much below the solution value for the prototype hydrated ferryl ion [FeO(H(2)O)(5)](2+).