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

Formulation of transition dipole gradients for non-adiabatic dynamics with polaritonic states

A general formulation of the strong coupling between photons confined in a cavity and molecular electronic states is developed for the state-interaction state-average spin-restricted ensemble-referenced Kohn-Sham method. The light-matter interaction is included in the Jaynes-Cummings model, which requires the derivation and implementation of the analytical derivatives of the transition dipole moments between the molecular electronic states. The developed formalism is tested in the simulations of the nonadiabatic dynamics in the polaritonic states resulting from the strong coupling between the cavity photon mode and the ground and excited states of the penta-2,4-dieniminium cation, also known as PSB3. Comparison with the field-free simulations of the excited-state decay dynamics in PSB3 reveals that the light-matter coupling can considerably alter the decay dynamics by increasing the excited state lifetime and hindering photochemically induced torsion about the C=C double bonds of PSB3. The necessity of obtaining analytical transition dipole gradients for the accurate propagation of the dynamics is underlined.

Antiaromatic Molecules as Magnetic Couplers: A Computational Quest

In this study, we investigate a set of organic diradical structures in which two oxo-verdazyl radicals are selected as radical spin centers that are connected (coupled) via six coupler molecules (CM), resulting in various magnetic (ferromagnetic (FM) or antiferromagnetic (AFM)) characteristics, as reflected by their exchange coupling constants (J). We have designed 12 diradicals with 6-antiaromatic couplers coupled with bis-oxo-verdazyl diradicals with meta-meta (m-m) and para-meta (p-m) positional connectivities. The nature of the magnetic coupling (ferromagnetic, nonmagnetic, or antiferromagnetic) and the magnitude of the exchange constant J depend on the type of coupler, the connecting point between each radical center and CM, the degree of aromaticity of the coupler, and the length of the through-bond distance between radical centers. The computed magnetic exchange coupling constants J for these diradicals at the B3LYP/6-311++G(d,p) and MN12SX/6-311++G(d,p) levels of theory are large for many of these structures, indicating strong ferromagnetic coupling (with positive J values). In some cases, magnetic couplings are observed with J > 1000 cm-1 (B3LYP/6-311++G(d,p)) and strong antiferromagnetic coupling (with negative J values) with J < -1000 cm-1 (B3LYP/6-311++G(d,p)). Similarly, in some cases, magnetic couplings are observed with J > 289 cm-1 (MN12SX/6-311++G(d,p)) and strong antiferromagnetic coupling (with negative J values) with J < -568 cm-1 (MN12SX/6-311++G(d,p)). Furthermore, while numerous studies have reported that the degree of aromaticity of molecular couplers often favors strong ferromagnetic coupling, displaying the high-spin character of diradicals in their ground states, the couplers chosen in this study are characterized as antiaromatic or nonaromatic. The current investigation provides evidence that, remarkably, antiaromatic couplers are able to enhance stability by favoring electronic diradical structures with very strong ferromagnetic coupling when the length of the through-bond distance and connectivity pattern between radical centers are selected in such a way that the FM coupling is optimized. The findings in this study offer new strategies in the design of novel organic materials with interesting magnetic properties for practical applications.

Predicting Magnetic Coupling and Spin-Polarization Energy in Triangulene Analogues

Triangulene and its analogue metal-free magnetic systems have garnered increasing attention since their discovery. Predicting the magnetic coupling and spin-polarization energy with quantitative accuracy is beyond the predictive power of today's density functional theory (DFT) due to their intrinsic multireference character. Herein, we create a benchmark dataset of 25 magnetic systems with nonlocal spin densities, including the triangulene monomer, dimer, and their analogues. We calculate the magnetic coupling (J) and spin-polarization energy (ΔE_spin) of these systems using complete active space self-consistent field (CASSCF) and coupled-cluster methods as high-quality reference values. This reference data is then used to benchmark 22 DFT functionals commonly used in material science. Our results show that, while some functionals consistently correctly predict the qualitative character of the ground state, achieving quantitative accuracy with small relative errors is currently not feasible. PBE0, M06-2X, and MN15 are predicting the correct electronic ground state for all systems investigated here and also have the lowest mean absolute error for predicting both ΔE_spin (0.34, 0.32, and 0.31 eV) and J (11.74, 12.66, and 10.64 meV). They may therefore also serve as starting points for higher-level methods such as the GW or the random phase approximation. As other functionals fail for the prediction of the ground state, they cannot be recommended for metal-free magnetic systems.

Symmetry of Identical Particles, Modern Achievements in the Pauli Exclusion Principle, in Superconductivity and in Some Other Phenomena

In this review, the modern achievements in studies of the Pauli exclusion principle (PEP) and the properties of the identical particle systems when PEP is not fulfilled are discussed. The validity of conception of the spin in the framework of density functional theory (DFT) approaches is analyzed. The modern state of the recently discovered Fe-based superconductors is discussed in detail. These materials belong to the paramagnetic semimetal family and become superconductors upon doping. Recently, in 2020, room-temperature superconductivity was realized. However, from the following discussion in the SC community, it was not evident that the results of room-temperature superconductivity have been repeated by other laboratories. Thus, the question “is room temperature really achieved?” is still open. In the concluding remarks, we present the explanation of why the PEP limitations on the symmetry of identical particles system exist in nature, and following from it, some important consequences.

Electronic Structure and Magneto-Structural Correlations Study of Cu2UL Trinuclear Schiff Base Complexes: A 3d-5f-3d Case

The magnetic properties of trinuclear Schiff base complexes M₂AnLⁱ (M^II = Zn, Cu; An^IV = Th, U; Lⁱ = Schiff base; i = 1-4, 6, 7, 9), exhibiting the [M(μ-O)₂]₂U core structure with adjacent M1···U and M2···U and next-adjacent M1···M2 interactions, featuring 3d-5f-3d subsystems, have been investigated theoretically using relativistic ZORA/B3LYP computations combined with the broken symmetry (BS) approach. Bond order and natural population analyses reveal that the covalent contribution to the bonding within the Cu-O-U coordination is important thus favoring superexchange coupling between the transition metal and the uranium magnetic centers. The calculated coupling constants J_CuU between the Cu and U atoms, agree with the observed shift from the antiferromagnetic (AF) character of the L^1,2,3,4 complexes to the ferromagnetic (ferro) of the L^6,7,9 ones. The structural parameters, i.e., the Cu···U distances and the Cu-O-U angles, as well as the electronic factors driving the magnetic couplings are discussed. The analyses are supported by the study of the mixed ZnCuULⁱ and Cu₂ThLⁱ systems, where in the first complex the Cu^II (3d⁹) ion is replaced by the diamagnetic Zn^II (3d¹⁰) one, whereas in the second complex the U^IV (5f²) paramagnetic center is replaced by the diamagnetic Th^IV (5f⁰) one.

Magnetic exchange interactions in binuclear and tetranuclear iron(III) complexes described by spin-flip DFT and Heisenberg effective Hamiltonians

Low-energy spectra of single-molecule magnets (SMMs) are often described by Heisenberg Hamiltonians. Within this formalism, exchange interactions between magnetic centers determine the ground-state multiplicity and energy separation between the ground and excited states. In this contribution, we extract exchange coupling constants (J) for a set of iron (III) binuclear and tetranuclear complexes from all-electron calculations using non-collinear spin-flip time-dependent density functional theory (NC-SF-TDDFT). For 12 binuclear complexes with J-values ranging from -6 to -132 cm^-1 , our benchmark calculations using the short-range hybrid ωPBEh functional and 6-31G(d,p) basis set agree well with the experimentally derived values (mean absolute error of 4.7 cm^-1 ). For the tetranuclear SMMs, the computed J constants are within 6 cm^-1 from the experimentally derived values. We explore the range of applicability of the Heisenberg model by analyzing bonding patterns in these Fe(III) complexes using natural orbitals (NO), their occupations, and the number of effectively unpaired electrons. The results illustrate the efficiency of the spin-flip protocol for computing the exchange couplings and the utility of the NO analysis in assessing the validity of effective spin Hamiltonians.

Calculation of exciton couplings based on density functional tight-binding coupled to state-interaction state-averaged ensemble-referenced Kohn-Sham approach

We introduce the combination of the density functional tight binding (DFTB) approach, including onsite correction (OC) and long-range corrected (LC) functional and the state-interaction state-averaged spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS or SSR) method with extended active space involving four electrons and four orbitals [LC-OC-DFTB/SSR(4,4)], to investigate exciton couplings in multichromophoric systems, such as organic crystals and molecular aggregates. We employ the LC-OC-DFTB/SSR(4,4) method to calculate the excitonic coupling in anthracene and tetracene. As a result, the LC-OC-DFTB/SSR(4,4) method provides a reliable description of the locally excited (LE) state in a single chromophore and the excitonic couplings between chromophores with reasonable accuracy compared to the experiment and the conventional SSR(4,4) method. In addition, the thermal fluctuation of excitonic couplings from dynamic nuclear motion in an anthracene crystal with LC-OC-DFTB/SSR(4,4) shows a similar fluctuation of excitonic coupling and spectral density with those of first-principle calculations. We conclude that LC-OC-DFTB/SSR(4,4) is capable of providing reasonable features related to LE states, such as Frenkel exciton with efficient computational cost.

Towards the engineering of a photon-only two-stroke rotary molecular motor

The rational engineering of photoresponsive materials, e.g., light-driven molecular motors, is a challenging task. Here, we use structure-related design rules to prepare a prototype molecular rotary motor capable of completing an entire revolution using, exclusively, the sequential absorption of two photons; i.e., a photon-only two-stroke motor. The mechanism of rotation is then characterised using a combination of non-adiabatic dynamics simulations and transient absorption spectroscopy measurements. The results show that the rotor moiety rotates axially relative to the stator and produces, within a few picoseconds at ambient T, an intermediate with the same helicity as the starting structure. We discuss how such properties, that include a 0.25 quantum efficiency, can help overcome the operational limitations of the classical overcrowded alkene designs.

Generalized Formulation of the Density Functional Tight Binding-Based Restricted Ensemble Kohn-Sham Method with Onsite Correction to Long-Range Correction

We present a generalized formulation for the combination of the density functional tight binding (DFTB) approach and the state-interaction state-average spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS or SSR) method by considering onsite correction (OC) as well as the long-range corrected (LC) functional. The OC contribution provides more accurate energies and analytic gradients for individual microstates, while the multireference character of the SSR provides the correct description for conical intersections. We benchmark the LC-OC-DFTB/SSR method against various DFTB calculation methods for excitation energies and conical intersection structures with π/π* or n/π* characters. Furthermore, we perform excited-state molecular dynamics simulations with a molecular rotary motor with variations of LC-OC-DFTB/SSR approaches. We show that the OC contribution to the LC functional is crucial to obtain the correct geometry of conical intersections.

Study of Self-Interaction Errors in Density Functional Calculations of Magnetic Exchange Coupling Constants Using Three Self-Interaction Correction Methods

We examine the role of self-interaction error (SIE) removal on the evaluation of magnetic exchange coupling constants. In particular, we analyze the effect of scaling down the self-interaction correction (SIC) for three nonempirical density functional approximations (DFAs) namely, the local spin density approximation, the Perdew-Burke-Ernzerhof generalized gradient approximation, and the recent SCAN family of meta-GGA functionals. To this end, we employ three one-electron SIC methods: Perdew-Zunger SIC [Perdew, J. P.; Zunger, A. Phys. Rev. B, 1981, 23, 5048.], the orbitalwise scaled SIC method [Vydrov, O. A. et al. J. Chem. Phys. 2006, 124, 094108.], and the recent local scaling method [Zope, R. R. et al. J. Chem. Phys. 2019, 151, 214108.]. We compute the magnetic exchange coupling constants using the spin projection and nonprojection approaches for sets of molecules composed of dinuclear and polynuclear H···He models, organic radical molecules, and chlorocuprate and compare these results against accurate theories and experiment. Our results show that for the systems that mainly consist of single-electron regions, PZSIC performs well, but for more complex organic systems and the chlorocuprates, an overcorrecting tendency of PZSIC combined with the DFAs utilized in this work is more pronounced, and in such cases, LSIC with kinetic energy density ratio performs better than PZSIC. Analysis of the results in terms of SIC corrections to the density and to the total energy shows that both density and energy correction are required to obtain an improved prediction of magnetic exchange couplings.

Description of Sudden Polarization in the Excited Electronic States with an Ensemble Density Functional Theory Method

Sudden polarization (SP) is one of the manifestations of electron transfer in the electronically excited states of molecules. Proposed initially to explain the unusual reactivity of photoexcited olefins, SP often occurs in the excited states of molecules possessing strongly correlated diradical ground state. Theoretical description of SP involves mixing between the singly excited and the doubly excited zwitterionic states, which makes it inaccessible with the use of the popular linear-response time-dependent density functional theory methods. In this work, an extended variant of the state-interaction state-averaged spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS, or SSR) method is applied to study SP in a number of organic diradical systems. To this end, the analytical derivative formalism is derived and implemented for the SSR(3,2) method (see the main text for explanation of the acronym), which enables the automatic geometry optimization and obtains the relaxed density matrices as well as the electron binding energies and respective Dyson's orbitals. Application of the new method to SP in the lowest singlet excited state of ethylene agrees with the results obtained previously with the use of multireference methods of wavefunction theory. A number of interesting manifestations of SP are observed, such as the charge transfer in photoexcited tetramethyleneethene (TME) diradical mediated by the vibrational motion and conductivity switching in the excited state of a donor-acceptor dyad placed in an external electric field.

Signatures of Conical Intersection Dynamics in the Time-Resolved Photoelectron Spectrum of Furan: Theoretical Modeling with an Ensemble Density Functional Theory Method

The non-adiabatic dynamics of furan excited in the ππ* state (S2 in the Franck-Condon geometry) was studied using non-adiabatic molecular dynamics simulations in connection with an ensemble density functional method. The time-resolved photoelectron spectra were theoretically simulated in a wide range of electron binding energies that covered the valence as well as the core electrons. The dynamics of the decay (rise) of the photoelectron signal were compared with the excited-state population dynamics. It was observed that the photoelectron signal decay parameters at certain electron binding energies displayed a good correlation with the events occurring during the excited-state dynamics. Thus, the time profile of the photoelectron intensity of the K-shell electrons of oxygen (decay constant of 34 ± 3 fs) showed a reasonable correlation with the time of passage through conical intersections with the ground state (47 ± 2 fs). The ground-state recovery constant of the photoelectron signal (121 ± 30 fs) was in good agreement with the theoretically obtained excited-state lifetime (93 ± 9 fs), as well as with the experimentally estimated recovery time constant (ca. 110 fs). Hence, it is proposed to complement the traditional TRPES observations with the trXPS (or trNEXAFS) measurements to obtain more reliable estimates of the most mechanistically important events during the excited-state dynamics.

Machine Learning-Assisted Excited State Molecular Dynamics with the State-Interaction State-Averaged Spin-Restricted Ensemble-Referenced Kohn-Sham Approach

We present a machine learning-assisted excited state molecular dynamics (ML-ESMD) based on the ensemble density functional theory framework. Since we represent a diabatic Hamiltonian in terms of generalized valence bond ansatz within the state-interaction state-averaged spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS) method, we can avoid singularities near conical intersections, which are crucial in excited state molecular dynamics simulations. We train the diabatic Hamiltonian elements and their analytical gradients with the SchNet architecture to construct machine learning models, while the phase freedom of off-diagonal elements of the Hamiltonian is cured by introducing the phase-less loss function. Our machine learning models show reasonable accuracy with mean absolute errors of ∼0.1 kcal/mol and ∼0.5 kcal/mol/Å for the diabatic Hamiltonian elements and their gradients, respectively, for penta-2,4-dieniminium cation. Moreover, by exploiting the diabatic representation, our models can predict correct conical intersection structures and their topologies. In addition, our ML-ESMD simulations give almost identical result with a direct dynamics at the same level of theory.

Computation of Molecular Electron Affinities Using an Ensemble Density Functional Theory Method

The computation of electron attachment energies (electron affinities) was implemented in connection with an ensemble density functional theory method, the state-interaction state-averaged spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS or SSR) method. With the use of the extended Koopmans' theorem, the electron affinities and the respective Dyson orbitals are obtained directly for the neutral molecule, thus avoiding the necessity to compute the ionized system. Together with the EKT-SSR (extended Koopmans' theorem-SSR) method for ionization potentials, which was developed earlier, EKT-SSR for electron affinities completes the implementation of the EKT-SSR formalism, which can now be used for obtaining electron detachment as well as the electron attachment energies of molecules in the ground and excited electronic states. The extended EKT-SSR method was tested in the calculation of several closed-shell molecules. For the molecules in the ground states, the EKT-SSR energies of Dyson's orbitals are virtually identical to the energies of the unoccupied orbitals in the usual single-reference spin-restricted Kohn-Sham calculations. For the molecules in the excited states, EKT-SSR predicts an increase of the most positive electron affinity by approximately the amount of the vertical excitation energy. The electron affinities of a number of diradicals were calculated with EKT-SSR and compared with the available experimental data. With the use of a standard density functional (BH&HLYP), the EKT-SSR electron affinities deviate on average by ca. 0.2 eV from the experimental data. It is expected that the agreement with the experiment can be improved by designing density functionals parametrized for ionization energies.

Computation of Molecular Ionization Energies Using an Ensemble Density Functional Theory Method

Computation of the ionization energies and of the respective Dyson orbitals based on the use of the extended Koopmans theorem (EKT) is implemented in connection with an ensemble density functional theory (eDFT) method, the state-interaction state-averaged spin-restricted ensemble-referenced Kohn-Sham (SI-SA-REKS or SSR) method. The new methodology enables fast computation of the ionization energies and evaluation of the respective Dyson orbitals, the square norms of which are related with the ionization probabilities, in the ground and excited electronic states of molecules. As the application of EKT recycles the intermediate quantities from the SSR analytical energy gradient, evaluation of the ionization energies and probabilities can be carried out on-the-fly during the nonadiabatic molecular dynamics simulations. This opens up a perspective for fast theoretical simulation of the time-resolved photoelectron spectroscopy observations. In the present work, the new methodology is tested in the computation of the ionization energies and Dyson orbitals of several molecules in the ground and excited electronic states, including strongly correlated species, such as the ozone molecule, dissociating chemical bonds, and conical intersections.

DFTB+, a software package for efficient approximate density functional theory based atomistic simulations

DFTB+ is a versatile community developed open source software package offering fast and efficient methods for carrying out atomistic quantum mechanical simulations. By implementing various methods approximating density functional theory (DFT), such as the density functional based tight binding (DFTB) and the extended tight binding method, it enables simulations of large systems and long timescales with reasonable accuracy while being considerably faster for typical simulations than the respective ab initio methods. Based on the DFTB framework, it additionally offers approximated versions of various DFT extensions including hybrid functionals, time dependent formalism for treating excited systems, electron transport using non-equilibrium Green's functions, and many more. DFTB+ can be used as a user-friendly standalone application in addition to being embedded into other software packages as a library or acting as a calculation-server accessed by socket communication. We give an overview of the recently developed capabilities of the DFTB+ code, demonstrating with a few use case examples, discuss the strengths and weaknesses of the various features, and also discuss on-going developments and possible future perspectives.

Charge-Shift Bonding: A New and Unique Form of Bonding

Charge-shift bonds (CSBs) constitute a new class of bonds different than covalent/polar-covalent and ionic bonds. Bonding in CSBs does not arise from either the covalent or the ionic structures of the bond, but rather from the resonance interaction between the structures. This Essay describes the reasons why the CSB family was overlooked by valence-bond pioneers and then demonstrates that the unique status of CSBs is not theory-dependent. Thus, valence bond (VB), molecular orbital (MO), and energy decomposition analysis (EDA), as well as a variety of electron density theories all show the distinction of CSBs vis-à-vis covalent and ionic bonds. Furthermore, the covalent-ionic resonance energy can be quantified from experiment, and hence has the same essential status as resonance energies of organic molecules, e.g., benzene. The Essay ends by arguing that CSBs are a distinct family of bonding, with a potential to bring about a Renaissance in the mental map of the chemical bond, and to contribute to productive chemical diversity.

DFT Investigations of the Magnetic Properties of Actinide Complexes

Over the past 25 years, magnetic actinide complexes have been the object of considerable attention, not only at the experimental level, but also at the theoretical one. Such systems are of great interest, owing to the well-known larger spin–orbit coupling for actinide ions, and could exhibit slow relaxation of the magnetization, arising from a large anisotropy barrier, and magnetic hysteresis of purely molecular origin below a given blocking temperature. Furthermore, more diffuse 5f orbitals than lanthanide 4f ones (more covalency) could lead to stronger magnetic super-exchange. On the other hand, the extraordinary experimental challenges of actinide complexes chemistry, because of their rarity and toxicity, afford computational chemistry a particularly valuable role. However, for such a purpose, the use of a multiconfigurational post-Hartree-Fock approach is required, but such an approach is computationally demanding for polymetallic systems—notably for actinide ones—and usually simplified models are considered instead of the actual systems. Thus, Density Functional Theory (DFT) appears as an alternative tool to compute magnetic exchange coupling and to explore the electronic structure and magnetic properties of actinide-containing molecules, especially when the considered systems are very large. In this paper, relevant achievements regarding DFT investigations of the magnetic properties of actinide complexes are surveyed, with particular emphasis on some representative examples that illustrate the subject, including actinides in Single Molecular Magnets (SMMs) and systems featuring metal-metal super-exchange coupling interactions. Examples are drawn from studies that are either entirely computational or are combined experimental/computational investigations in which the latter play a significant role.

Theoretical modelling of the dynamics of primary photoprocess of cyclopropanone

Photodecomposition of cyclopropanone is investigated by static quantum chemical calculations and non-adiabatic molecular dynamics (NAMD) simulations.

Analytical derivatives of the individual state energies in ensemble density functional theory method. I. General formalism

The state-averaged (SA) spin restricted ensemble referenced Kohn-Sham (REKS) method and its state interaction (SI) extension, SI-SA-REKS, enable one to describe correctly the shape of the ground and excited potential energy surfaces of molecules undergoing bond breaking/bond formation reactions including features such as conical intersections crucial for theoretical modeling of non-adiabatic reactions. Until recently, application of the SA-REKS and SI-SA-REKS methods to modeling the dynamics of such reactions was obstructed due to the lack of the analytical energy derivatives. In this work, the analytical derivatives of the individual SA-REKS and SI-SA-REKS energies are derived. The final analytic gradient expressions are formulated entirely in terms of traces of matrix products and are presented in the form convenient for implementation in the traditional quantum chemical codes employing basis set expansions of the molecular orbitals. The implementation and benchmarking of the derived formalism will be described in a subsequent article of this series.

Description of ground and excited electronic states by ensemble density functional method with extended active space

An extended variant of the spin-restricted ensemble-referenced Kohn-Sham (REKS) method, the REKS(4,4) method, designed to describe the ground electronic states of strongly multireference systems is modified to enable calculation of excited states within the time-independent variational formalism. The new method, the state-interaction state-averaged REKS(4,4), i.e., SI-SA-REKS(4,4), is capable of describing several excited states of a molecule involving double bond cleavage, polyradical character, or multiple chromophoric units. We demonstrate that the new method correctly describes the ground and the lowest singlet excited states of a molecule (ethylene) undergoing double bond cleavage. The applicability of the new method for excitonic states is illustrated with π stacked ethylene and tetracene dimers. We conclude that the new method can describe a wide range of multireference phenomena.

Singlet-triplet energy gaps and the degree of diradical character in binuclear copper molecular magnets characterized by spin-flip density functional theory

Molecular magnets, defined here as organic polyradicals, can be used as building blocks in the fabrication of novel and structurally diverse magnetic light-weight materials. We present a theoretical investigation of the lowest spin states of several binuclear copper diradicals. In contrast to previous studies, we consider not only the energetics of the low-lying states (which are related to the exchange-coupling parameter within the Heisenberg-Dirac-van-Vleck model), but also the character of the diradical states themselves. We use natural orbitals, their occupations, and the number of effectively unpaired electrons to quantify bonding patterns in these systems. We compare the performance of spin-flip time-dependent density functional theory (SF-TDDFT) using various functionals and effective core potentials against the wave function based approach, equation-of-motion spin-flip coupled-cluster method with single and double substitutions (EOM-SF-CCSD). We find that SF-TDDFT paired with the PBE50 and B5050LYP functionals performs comparably to EOM-SF-CCSD, with respect to both singlet-triplet gaps and states' characters. Visualization of frontier natural orbitals shows that the unpaired electrons are localized on copper centers, in some cases exhibiting slight through-bond interaction via copper d-orbitals and p-orbitals of neighboring ligand atoms. The analysis reveals considerable interactions between the formally unpaired electrons in the antiferromagnetic diradicaloids, meaning that they are poorly described by the Heisenberg-Dirac-van-Vleck model. Thus, for these systems the experimentally derived exchange-coupling parameters are not directly comparable with the singlet-triplet gaps. This explains systematic discrepancies between the computed singlet-triplet energy gaps and the exchange-coupling parameters extracted from experiment.

A QTAIM and stress tensor investigation of the torsion path of a light-driven fluorene molecular rotary motor

The utility of the QTAIM/stress tensor analysis method for characterizing the photoisomerization of light driven molecular rotary machines is investigated on the example of the torsion path in fluorene molecular motor. The scalar and vector descriptors of QTAIM/stress tensor reveal additional information on the bonding interactions between the rotating units of the motor, which cannot be obtained from the analysis of the ground and excited state potential energy surfaces. The topological features of the fluorene motor molecular graph display that, upon the photoexcitation a certain increase in the torsional stiffness of the rotating bond can be attributed to the increasing topological stability of the rotor carbon atom attached to the rotation axle. The established variations in the torsional stiffness of the rotating bond may cause transfer of certain fraction of the torsional energy to other internal degrees of freedom, such as the pyramidalization distortion. © 2016 Wiley Periodicals, Inc.

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.

Electronic Structure and Magnetic Properties of Dioxo-Bridged Diuranium Complexes with Diamond-Core Structural Motifs: A Relativistic DFT Study

Electronic structures and magnetic properties of the binuclear bis(μ-oxo) U(IV)/U(IV) K2[{(((nP,Me)ArO)3tacn)U(IV)}2(μ-O)2] and U(V)/U(V) [{(((nP,Me)ArO)3tacn)U(V)}2(μ-O)2] (tacn = triazacyclononane, nP = neopentyl) complexes, exhibiting [U(μ-O)2U] diamond-core structural motifs, have been investigated computationally using scalar relativistic Density Functional Theory (DFT) combined with the Broken Symmetry (BS) approach for their magnetic properties. Using the B3LYP hybrid functional, the BS ground state of the pentavalent [U(V)(μ-O)2U(V)] 5f(1)-5f(1) complex has been found of lower energy than the high spin (HS) triplet state, thus confirming the antiferromagnetic character in agreement with experimental magnetic susceptibility measurements. The nonmagnetic character observed for the tetravalent K2[U(IV)(μ-O)2U(IV)] 5f(2)-5f(2) species is also predicted by our DFT calculations, which led practically to the same energy for the HS and BS states. As reported for related dioxo diuranium(V) systems, superexchange is likely to be responsible for the antiferromagnetic coupling through the π-network orbital pathway within the (μ-O)2 bridge, the dissymmetrical structure of the U2O2 core playing a determining role. In the case of the U(IV) species, our computations indicate that the K(+) counterions are likely to play a role for the observed magnetic property. Finally, the MO analysis, in conjunction with NPA and QTAIM analyses, clarify the electronic structures of the studied complexes. In particular, the fact that the experimentally attempted chemical oxidation of the U(V) species does not lead straightforwardly to binuclear complexes U(VI) is clarified by the MO analysis.

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.

Quantum chemical investigation of meta-xylylene based one-dimensional polymer chain

We have investigated unsubstituted and methyl substituted polyradical chains of meta-xylylene by using density functional theory-broken symmetry methodology (DFT-BS). Optimization of geometry in the high-spin and low-spin states have been done at B3LYP/6-31G(d,p) and M06-2X/6-31G(d,p) levels in unrestricted methodology. Single-point calculations on the high-spin optimized geometries have been done by using the 6-311G(d,p) basis set. Each polyradical has been found to be nonplanar with a high-spin ground state. Each has a coupling constant larger than thermal energy. For each group of polyradicals, the coupling constant has been found to exponentially decrease with increase in the number of phenylene groups. The B3LYP infinite chain limit has been estimated for both the unsubstituted and substituted polyradicals. The individual inter-radical-site coupling constants have been estimated for the triradicals and tetraradicals using HDVV Hamiltonian in ORCA 3.0.1 code. These are also generally large and positive, revealing a strong intersite ferromagnetic interaction. The intersite coupling constant too decreases with increasing distance between the radical centers. Finally, we have used CRYSTAL09 package for calculations on the infinitely long one-dimensional and periodic polyradical chains. The coupling constants estimated from the periodic calculations are quite large at about 500 cm(-1) and somewhat greater than the limiting values calculated for the polyradicals with an increasing number of phenylene groups. This happens as the individual polyradicals of increasing size progressively deviate from periodicity, and thereby lessens the strength of through-bond spin-spin coupling. The calculated band gap of ∼4.5 eV indicates that the infinitely long one-dimensional chains must be ferromagnetic and electron insulators.

Description of Conical Intersections with Density Functional Methods

Conical intersections are perhaps the most significant mechanistic features of chemical reactions occurring through excited states. By providing funnels for efficient non-adiabatic population transfer, conical intersections govern the branching ratio of products of such reactions, similar to what the transition states do for ground-state reactivity. In this regard, intersections between the ground and the lowest excited states play a special role, and the correct description of the potential energy surfaces in their vicinity is crucial for understanding the mechanism and dynamics of excited-state reactions. The methods of density functional theory, such as time-dependent density functional theory, are widely used to describe the excited states of large molecules. However, are these methods suitable for describing the conical intersections or do they lead to artifacts and, consequently, to erroneous description of reaction dynamics? Here we address the first part of this question and analyze the ability of several density functional approaches, including the linear-response time-dependent approach as well as the spin-flip and ensemble formalisms, to provide the correct description of conical intersections and the potential energy surfaces in their vicinity. It is demonstrated that the commonly used linear-response time-dependent theory does not yield a proper description of these features and that one should instead use alternative computational approaches.

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.