Unpaired Density Research Articles

Multireference calculations on bond dissociation and biradical polycyclic aromatic hydrocarbons as guidance for fractional occupation number weighted density analysis in DFT calculations

This study explores open-shell biradical and polyradical molecular compounds based on extended multireference (MR) methods (MR-configuration interaction with singles and doubles (CISD) and MR-averaged quadratic coupled cluster (AQCC) approach) using the numbers of unpaired densities NU. These results were used to guide the analysis of the fractional occupation number weighted density (FOD) calculated within the finite temperature (FT) density functional theory (DFT) approach. As critical test examples, the dissociation of carbon–carbon (CC) single, double and triple bonds and a benchmark set of polycyclic aromatic hydrocarbons (PAHs) have been chosen. By examining single, double, and triple bond dissociations, we demonstrate the utility and accuracy but also limitations of the FOD analysis for describing these dissociation processes. In significant extension of previous work (Phys Chem Chem Phys 25: 27380–27393), the assessment of FOD applications for different classes of DFT functionals was performed examining the range-separated functionals ωB97XD, ωB97M-V, CAM-B3LYP, LC-ωPBE, and MN12-SX, the hybrid (M06-2X) functional and the double hybrid (B2P-LYP) functional. In all cases, strong correlations between NFOD and NU values are found. The major task was to develop a new linear regression formula for range-separated functionals allowing a convenient determination of the optimal electronic temperature Tel for the FT-DFT calculation. We also established an optimal temperature for the semiempirical extended tight-binding GFN2-xTB method. These findings significantly broaden the applicability of FOD analysis across various DFT functionals and semiempirical methods.

Polyradical character assessment using multireference calculations and comparison with density-functional derived fractional occupation number weighted density analysis.

The biradicaloid character of different types of polycyclic aromatic hydrocarbons (PAHs) based on small band gaps is an important descriptor to assess their opto-electronic properties. In this work, the unpaired electron densities and numbers of unpaired electrons (NU values) calculated at the high-level multireference averaged quadratic coupled-cluster (MR-AQCC) method are used to develop a test set to assess the capabilities of different biradical descriptors based on density functional theory. A benchmark collection of 29 different compounds has been selected. The DFT descriptors contain primarily the fractional occupation number weighted electron density (FOD) based on simplified thermally-assisted-occupation density functional theory (TAO-DFT) calculations, but the singlet-triplet energy difference and other descriptors denoted as y0 and nLUNO have been considered as well. After adjustment of the literature-recommended finite temperatures, a very good, detailed agreement between unpaired density and FOD analysis is observed which is also manifested in excellent statistical correlations. The other two descriptors also show good correlations even though the absolute scaling is not satisfactory. A new linear fit of FOD data to the MR-AQCC reference values leads to an improved regression relation for determining the recommended finite temperature value in dependence of the Hartree-Fock exchange. This provides the basis for fast and reliable assessment of the biradical character of many classes of PAHs without the need for performing computationally extended MR calculations.

λ-DFVB(U): A hybrid density functional valence bond method based on unpaired electron density.

In this paper, a hybrid density functional valence bond method based on unpaired electron density, called λ-DFVB(U), is presented, which is a combination of the valence bond self-consistent field (VBSCF) method and Kohn-Sham density functional theory. In λ-DFVB(U), the double-counting error of electron correlation is mitigated by a linear decomposition of the electron-electron interaction using a parameter λ, which is a function of an index based on the number of effectively unpaired electrons. In addition, λ-DFVB(U) is based on the approximation that correlation functionals in KS-DFT only cover dynamic correlation and exchange functionals mimic some amount of static correlation. Furthermore, effective spin densities constructed from unpaired density are used to address the symmetry dilemma problem in λ-DFVB(U). The method is applied to test calculations of atomization energies, atomic excitation energies, and reaction barriers. It is shown that the accuracy of λ-DFVB(U) is comparable to that of CASPT2, while its computational cost is approximately the same as VBSCF.

Tuning the Biradicaloid Nature of Polycyclic Aromatic Hydrocarbons: The Effect of Graphitic Nitrogen Doping in Zethrenes.

Zethrenes are interesting polycyclic aromatic hydrocarbons (PAHs), which possess unique optoelectronic and magnetic properties because of their singlet open-shell biradicaloid character, making them promising candidates for application in organic electronics. Tuning their properties is a key task in order to develop efficient compounds for practical use by balancing the desired biradicaloid character against its chemical instability. In this work, high-level theoretical multireference methods appropriate for the correct description of polyradicaloid systems are used to develop rules for doping of zethrenes by means of nitrogen taking heptazethrene (HZ) as a benchmark example. The results of the quantum chemical calculations have been concentrated on a series of quantitative descriptors such as unpaired densities and singlet-triplet (S-T) splittings. They clearly indicate different regions in the HZ where N-doping can either lead to strong enhancement of the biradicaloid character or to strong quenching towards a closed shell state. A wide scale of varying open-shell character is accessible from the different doping positions. It is shown that the S-T splittings correlate well with the total number of unpaired electrons in the medium range of biradicaloid character. For pronounced biradical character the S-T splitting decays to about zero with a margin of ±0.15 eV. In the opposite closed-shell limit, much larger S-T splittings of up to 3 eV are computed.

Polyradical Character of Triangular Non-Kekulé Structures, Zethrenes, p-Quinodimethane-Linked Bisphenalenyl, and the Clar Goblet in Comparison: An Extended Multireference Study.

In this work, two different classes of polyaromatic hydrocarbon (PAH) systems have been investigated in order to characterize the amount of polyradical character and to localize the specific regions of chemical reactivity: (a) the non-Kekulé triangular structures phenalenyl, triangulene and a π-extended triangulene system with high-spin ground state and (b) PAHs based on zethrenes, p-quinodimethane-linked bisphenalenyl, and the Clar goblet containing varying polyradical character in their singlet ground state. The first class of structures already have open-shell character because of their high-spin ground state, which follows from the bonding pattern, whereas for the second class the open-shell character is generated either because of the competition between the closed-shell quinoid Kekulé and the open-shell singlet biradical resonance structures or the topology of the π-electron arrangement of the non-Kekulé form. High-level ab initio calculations based on multireference theory have been carried out to compute singlet–triplet splitting for the above-listed compounds and to provide insight into their chemical reactivity based on the polyradical character by means of unpaired densities. Unrestricted density functional theory and Hartree–Fock calculations have been performed for comparison also in order to obtain better insight into their applicability to these types of complicated radical systems.

The electronic states of a double carbon vacancy defect in pyrene: a model study for graphene.

The electronic states occurring in a double vacancy defect for graphene nanoribbons have been calculated in detail based on a pyrene model. Extended ab initio calculations using the MR configuration interaction (MRCI) method have been performed to describe in a balanced way the manifold of electronic states derived from the dangling bonds created by initial removal of two neighboring carbon atoms from the graphene network. In total, this study took into account the characterization of 16 electronic states (eight singlets and eight triplets) considering unrelaxed and relaxed defect structures. The ground state was found to be of (1)Ag character with around 50% closed shell character. The geometry optimization process leads to the formation of two five-membered rings in a pentagon-octagon-pentagon (5-8-5) structure. The closed shell character increases thereby to ∼70%; the analysis of unpaired density shows only small contributions confirming the chemical stability of that entity. For the unrelaxed structure the first five excited states ((3)B3g, (3)B2u, (3)B1u, (3)Au and (1)Au) are separated from the ground state by less than 2.5 eV. For comparison, unrestricted density functional theory (DFT) calculations using several types of functionals have been performed within different symmetry subspaces defined by the open shell orbitals. Comparison with the MRCI results gave good agreement in terms of finding the (1)Ag state as a ground state and in assigning the lowest excited states. Linear interpolation curves between the unrelaxed and relaxed defect structures also showed good agreement between the two classes of methods opening up the possibilities of using extended nanoflakes for multistate investigations at the DFT level.

The diverse manifold of electronic states generated by a single carbon defect in a graphene sheet: multireference calculations using a pyrene defect model.

Detailed calculations have been performed on the electronic states occurring in a single vacancy defect model based on pyrene from which one of the central carbon atoms has been removed. Complete active space self-consistent field and multireference configuration interaction with singles and doubles calculations have been performed using the 6-31G and 6-31G* basis sets. Two types of defect geometries have been defined: 1) The unrelaxed defect structure based on pyrene and 2) a relaxed structure. In total 12 electronic states have been computed for the unrelaxed structure at C2v symmetry, comprising four singlets, triplets and quintets each. The lowest six states are formed from singlet and triplet states and appear in a rather narrow gap of ∼0.6 eV. The lowest quintet state is found 1.43 eV above the (3) B1 ground state. As predicted from Jahn-Teller distortions, a CC bond is formed between dangling carbon bonds in the (1, 3) B1 states, leading to the formation of a five-membered ring. The (1, 3) A2 states show initial repulsive behavior along the bond formation coordinate until an avoided crossing is reached by which these states are furnished with CC bonding character so that finally also in these cases a CC bond is established. Linear interpolation curves between the initial unrelaxed defect structure and the final optimized structure are used to give an overview of the evolution of electronic states and the occurrence of avoided crossings. Out-of-plane structures are investigated with special emphasis on the carbon atom containing a dangling bond in the relaxed structure. Unpaired electron densities are used to characterize the electronic structure of the different states.

Electronic Structure and Effectively Unpaired Electron Density Topology in closo-Boranes: Nonclassical Three-Center Two-Electron Bonding.

This article provides a detailed study of the structure and bonding in closo-borane cluster compounds X2B3H3 (X = BH(-), P, SiH, CH, N), with particular emphasis on the description of the electron distribution using the topology of the quantum many-body effectively unpaired density. The close relationship observed between the critical points of this quantity and the localization of the electron cloud allows us to characterize the nonclassical bonding patterns of these systems. The obtained results confirm the suitability of the local rule to detect three-center two-electron bonds, which was conjectured in our previous study on boron hydrides.

Topology of the Effectively Paired and Unpaired Electron Densities for Complex Bonding Patterns: The Three-Center Two-Electron Bonding Case.

Our previously reported local formalism of the electron density decomposition into effectively paired and unpaired densities is applied to electron deficient molecular systems possessing complex bonding patterns. It is shown that the unpaired density is not only near the nuclear positions, like in classical bonds, but also spills out over the bonding regions, to compensate the electron deficiency. Topological information obtained from the effectively unpaired density, which may not be directly observed from the total density, allows us to establish a procedure to detect complex interactions. This study is complemented with results arising from nonlocal formalism of topological population analyses. The conclusions from both formalisms are in complete agreement and permit to interpret the well-known structural information from Lipscomb styx numbers going beyond it in cases where the electronic description becomes ambiguous, pointing out the subtle information contained in the unpaired density. Numerical results for three-center two-electron bondings in the boranes B2H6, B4H10, B5H9, and B5H11 are reported.

Electron-density topology in molecular systems: Paired and unpaired densities

This work studies the partitioning of the electron density into two contributions which are interpreted as the paired and the effectively unpaired electron densities. The topological features of each density field as well as of the total density are described localizing the corresponding critical points in simple selected molecules (local formalism). The results show that unpaired electron-density concentrations occur out of the topological bonding regions whereas the paired electron densities present accumulations inside those regions. A comparison of these results with those arising from population analysis techniques (nonlocal or integrated formalisms) is reported.

ChemInform Abstract: Characterization of Singly Reduced Iron(II) Porphyrins.

Optical absorption, electron paramagnetic resonance (EPR), and resonance Raman (RR) spectra are reported for the one-electron reduction products of a series of Fe/sup II/ porphyrins. The porphyrins include 5,10,15,20-tetraphenylporphyrin (TPP), 2,7,12,17-tetrabromo-5,10,15,20-tetraphenylporphyrin (TPPBr/sub 4/), 2,7,12-tricyano-5,10,15,20-tetraphenylporphyrin (TPP(CN)/sub 3/), and 2,7,12,17-tetracyano-5,10,15,20-tetraphenylporphyrin (TPP(CN)/sub 4/). The iron complexes of these porphyrins represent a series in which the reduction potential is successively shifted to more positive values. The RR and EPR data demonstrate that for all of the complexes reduction results in a low-spin configuration for the metal ion. For (FeTPP)/sup -/ and (FeTPPBr/sub 4/)/sup -/, the unpaired electron resides in the metal d/sub z/sup 2// orbital; however, reduction results in the transfer of a significant amount of paired electron density to the macrocyle through ..pi.. back-bonding. Destabilization the metal d/sub z/sup 2// orbital via ligation of a single pyridine or CO molecule is insufficient to push the unpaired electron from the metal ion to the macrocycle. For (FeTPP(CN)/sub 3/)/sup -/ and (FeTPP(CN)/sub 4/)/sup -/, the unpaired electron resides primarily on the porphyrin ring although a small amount of unpaired density is shared with the metal ion through ..pi..-orbital interactions. The extensive interaction between the metal and porphyrin ..pi.. orbitals which is present in all ofmore » the complexes provides a mechanism for enhancing the oscillator strength of formally forbidden charge-transfer transitions. It is suggested that these charge-transfer absorptions are primarily responsible for the complicated optical spectra of the reduced complexes.« less

Characterization of singly reduced iron(II) porphyrins

Optical absorption, electron paramagnetic resonance (EPR), and resonance Raman (RR) spectra are reported for the one-electron reduction products of a series of Fe/sup II/ porphyrins. The porphyrins include 5,10,15,20-tetraphenylporphyrin (TPP), 2,7,12,17-tetrabromo-5,10,15,20-tetraphenylporphyrin (TPPBr/sub 4/), 2,7,12-tricyano-5,10,15,20-tetraphenylporphyrin (TPP(CN)/sub 3/), and 2,7,12,17-tetracyano-5,10,15,20-tetraphenylporphyrin (TPP(CN)/sub 4/). The iron complexes of these porphyrins represent a series in which the reduction potential is successively shifted to more positive values. The RR and EPR data demonstrate that for all of the complexes reduction results in a low-spin configuration for the metal ion. For (FeTPP)/sup -/ and (FeTPPBr/sub 4/)/sup -/, the unpaired electron resides in the metal d/sub z/sup 2// orbital; however, reduction results in the transfer of a significant amount of paired electron density to the macrocyle through ..pi.. back-bonding. Destabilization the metal d/sub z/sup 2// orbital via ligation of a single pyridine or CO molecule is insufficient to push the unpaired electron from the metal ion to the macrocycle. For (FeTPP(CN)/sub 3/)/sup -/ and (FeTPP(CN)/sub 4/)/sup -/, the unpaired electron resides primarily on the porphyrin ring although a small amount of unpaired density is shared with the metal ion through ..pi..-orbital interactions. The extensive interaction between the metal and porphyrin ..pi.. orbitals which is present in all ofmore » the complexes provides a mechanism for enhancing the oscillator strength of formally forbidden charge-transfer transitions. It is suggested that these charge-transfer absorptions are primarily responsible for the complicated optical spectra of the reduced complexes.« less

Impurities and Charge Compensation in Some Molybdenium Oxides

AbstractThe role of impurities in the defect formation in MoO3 and Na2Mo2O7 crystals is discussed. The EPR spectra of the X‐ray and uv‐irradiation centres are investigated. The constants of the appropriate spin‐Hamiltonians are obtained. It is shown that the localization radius of an electron (or a hole) exceeds the interatomic distances, therefore the EPR spectra have a low symmetry with noncoincident axes of the A‐ and g‐tensors. The unpaired density distribution depends on the characteristics of the lattice and impurity cations such as the electronegativity, the coordination number, and the degree of the local environment symmetry distortion.