On Entangled Singlet Pure Diradicals.

This work addresses a class of conjugated hydrocarbons that are expected to be singlet diradicals according to the topological Hückel Hamiltonian while possibly satisfying full on-bond electron pairing. These systems possess two degenerate singly occupied molecular orbitals (SOMOs), but aromaticity brought by properly positioned six-membered rings does prevent Jahn-Teller distortions. Density functional theory (DFT) calculations performed on two emblematic examples confirm the strong bond-length alternation in the closed-shell solutions and the clear spatial symmetry in the open-shell spin-unrestricted determinants, the latter solution always being found to have significantly lower energy. Since the SOMOs are here of different symmetry, the wave function is free from ionic valence-bond component, and spin decontamination of the unrestricted DFT solutions and wave function calculations at the CASSCF-plus-second-order-perturbation level confirm the expected pure diradical character of such molecules. In contrast to disjoint diradicals, the SOMOs of present systems have large amplitudes on neighbor atoms, and we propose to name them entangled pure diradicals, further providing some prescription rules for their design. Additional calculations point out the qualitative contrast between these molecules and the related diradicaloids.

Difficulty of the evaluation of the barrier height of an open-shell transition state between closed shell minima: The case of small C4n rings.

C4n cyclacenes exhibit strong bond-alternation in their equilibrium geometry. In the two equivalent geometries, the system keeps an essentially closed-shell character. The two energy minima are separated by a transition state suppressing the bond-alternation, where the wave function is strongly diradical. This paper discusses the physical factors involved in this energy difference and possible evaluations of the barrier height. The barrier given as the energy difference between the restricted density functional theory (DFT)/B3LYP for the equilibrium and the broken symmetry DFT/B3LYP of the transition state is either negative or small, in contradiction with the most reliable Wave Function Theory calculations. The minimal (two electrons in two molecular orbitals) Complete Active Space self-consistent field (CASSCF) overestimates the barrier, and the subsequent second-order perturbation cancels it. Due to the collective character of the spin-polarization effect, it is necessary to perform a full π CASSCF + second-order perturbation to reach a reasonable value of the barrier, but this type of treatment cannot be applied to large molecules. DFT procedures treating on an equal foot the closed-shell and open-shell geometries have been explored, such as Mixed-Reference Spin-Flip Time-dependent-DFT and a new spin-decontamination proposal, namely, DFT-dressed configuration interaction, but the results still depend on the density functional. M06-2X without or with spin-decontamination gives the best agreement with the accurate wave function results.

Consistent spin decontamination of broken-symmetry calculations of diradicals.

Broken-symmetry calculations of diradicals exploit the mean-field energies of determinants that are not eigenfunctions of the S2 operator, the mean value of which is close to 1 for the ms = 0 solution. This spin contamination must be corrected. Two different contributions affect ⟨S2⟩, namely, the mixing between neutral and ionic valence bond components, the so-called kinetic exchange, which decreases ⟨S2⟩, and the spin polarization of the supposedly closed shell orbitals, which increases ⟨S2⟩. The popular Yamaguchi formula is valid for the first effect but irrelevant for the second one. From a few constrained broken-symmetry calculations, one may treat separately the two contributions and apply their specific spin decontamination correction. This work proposes a consistent spin-decontaminated procedure for the evaluation of singlet-triplet gaps in diradicals.

Improved evaluation of spin-polarization energy contributions using broken-symmetry calculations.

Spin-polarization effects may play an important role in free radicals and in the magnetic coupling between radical centers. Starting from restricted open-shell calculations, i.e., a closed-shell description of the non-magnetic core electrons, a low-order perturbation expansion identifies the spin-polarization contribution to the energy of mono-radicals and to singlet-triplet energy differences in diradicals. Broken-symmetry (BS) single-determinant calculations introduce only a fraction of spin-polarization effects, and in a biased manner, since BS determinants are not spin eigenfunctions. We propose a simple technique to correctly evaluate spin-polarization energies by taking into account the effect of spin-flip components on one-hole one-particle excited configurations. Spin-decontamination corrections are shown to play a non-negligible role in the BS evaluation of bond energies. The importance of spin decontamination is illustrated in cases for which spin polarization is the leading contribution to the singlet-triplet gap, which characterizes twisted conjugated double bonds and disjoint diradicals.

Spin polarization as an electronic cooperative effect.

Taking as an example the simple CH3 radical, this work demonstrates the cooperative character of the spin-polarization phenomenon of the closed-shell core in free radicals. Spin polarization of CH σ bonds is not additive here, as spin polarization of one bond enhances that of the next bond. This cooperativity is demonstrated by a series of configuration interaction calculations converging to the full valence limit and is rationalized by analytic developments. The same phenomenon is shown to take place in those diradicals where spin polarization plays a major role, as illustrated in square planar carbo-cyclobutadiene C12H4. The treatment of cooperativity represents a challenge for usual post-Hatree-Fock methods.

Dodecacene Generated on Surface: Reopening of the Energy Gap.

The acene series represents a model system to investigate the intriguing electronic properties of extended π-electron structures in the one-dimensional limit, which are important for applications in electronics and spintronics and for the fundamental understanding of electronic transport. Here, we present the on-surface generation of the longest acene obtained so far: dodecacene. Scanning tunneling spectroscopy gives access to the energy position and spatial distribution of its electronic states on the Au(111) surface. We observe that, after a progressive closing of the gap and a stabilization to about 1 eV at the length of decacene and undecacene, the energy gap of dodecacene unexpectedly increases to 1.4 eV. Considering the acene series as an exemplary general case, we discuss the evolution with length of the single tunneling resonances in comparison with ionization energy, electronic affinity, and optical gap.

Oxidative addition of carbon dioxide into mesoionics.

This work examines the prospect of making stable mesoionic compounds of the type mesomeric betaine R+-CO2- from direct oxidative additions of carbon dioxide to suitably-delocalized singlet carbene moieties, with bold objectives of carbon sequestration and overall energy storage. A set of possible candidates for such mesoionic compounds is theoretically explored through DFT calculations, inspecting coupling paths, thermodynamic and kinetic stabilities, and geometric and electronic structural features. Among others, the addressed cationic parts include aromatic rings in their broader sense, phenalene systems, and odd linear polyenic chains. Various structurally-close neutral alternatives such as oxiranones or carbene-acid forms are also considered. In the linear polyenic chain family, there is stark contrast between 4N + 1 and 4N - 1 lengths, with ensuing substantial consequences for stabilities and structures. Amino substitutions can favor mesoionic arrangements through their cation-stabilizing π-donor properties, further supported by possible strong intramolecular hydrogen bonds, but they can also contribute to weaken their kinetic stability through the existence of stable neutral imino alternatives. All in all, mesoionics including tropylium, phenalene, or 4N + 1 odd polyene frames as cationic parts could be reasonable targets.

Qualitative Views on the Polyradical Character of Long Acenes.

Spin-symmetry breaking appears in the DFT treatment of polyacenes, beyond a certain length, the critical length depending on the exchange-correlation potential. This phenomenon may be attributed to an instability with respect to HOMO-LUMO mixing, which suggests a diradical character of long acenes. However, the increase of the S2 operator with acene length questions this simple view. It is shown that this increase cannot be attributed to spin polarization of the inner MOs, and that a second symmetry breaking takes place for the pentadecacene, with four unpaired electrons centered at the first and third quarters of the chain. The spin density distributions of broken-symmetry solutions support a qualitative picture in terms of tetra-methylene hexacenes separated by Clar sextets. A strategy is proposed to identify local symmetry breaking in polyradical systems.

Electronic Resonances and Gap Stabilization of Higher Acenes on a Gold Surface.

On-surface synthesis provides a powerful method for the generation of long acene molecules, making possible the detailed investigation of the electronic properties of single higher acenes on a surface. By means of scanning tunneling microscopy and spectroscopy combined with theoretical considerations, we discuss the polyradical character of the ground state of higher acenes as a function of the number of linearly fused benzene rings. We present energy and spatial mapping of the tunneling resonances of hexacene, heptacene, and decacene, and discuss the role of molecular orbitals in the observed tunneling conductance maps. We show that the energy gap between the first electronic tunneling resonances below and above the Fermi energy stabilizes to a finite value, determined by a first diradical electronic perturbative contribution to the polyacene electronic ground state. Up to decacene, the main contributor to the ground state of acenes remains the lowest-energy closed-shell electronic configuration.

Predicting the Open-Shell Character of Polycyclic Hydrocarbons in Terms of Clar Sextets.

Rather unexpected spin-symmetry breakings of mean-field single determinants occur in singlet ground states of many families of alternating conjugated hydrocarbons which accept a full on-bond electron pairing. These symmetry breakings may be seen as an indication of the existence of unpaired electrons. Although qualitative, the concept of disjoint electronic sextets proposed by Clar (hereafter called CS) is at least a heuristic tool for predicting various features of fused polybenzenic hydrocarbons. The present work shows that identifying the preferred CS distribution enables one to rationalize the existence of one or several spin-symmetry breakings, i.e., the existence and the number of unpaired electrons in alternant fused polycyclic hydrocarbons via a simple recipe for the prediction of these features. This recipe is based on comparison between various distributions of CS on the molecular frame, subject to a restriction concerning the fragments of the graph that do not belong to the CS. This rule is successfully confronted to UDFT calculations and to a recently proposed criterion predicting the possible occurrence of spin-symmetry breaking from the topological Hückel Hamiltonian. The confrontation runs on series of rhombuses, periacenes, anthenes, and other graphene flakes or nanoribbons. The CS distribution definitely offers a qualitative guide to look at the possible occurrence of (multiple) symmetry breakings in polycyclic architectures which are commonly seen as closed-shell singlets.

Spreading out spin density in polyphenalenyl radicals.

As suggested by simple topological arguments, monoradical arrangements of properly-oriented polycondensed phenalenyl units can produce highly-delocalized spin distributions. This work examines under which geometrical conditions and to which extent these flat distributions take place. UDFT calculations performed on various instances gathering up to 19 such fused phenalene units confirm the spin-density spreading over entire conjugated skeletons. This occurs, however, with more or less uniformity depending on the compacity of the arrangement, and simple linear stretches do not actually support the delocalized picture. Thermodynamic stability of the radicals is assessed, which follows their delocalization degree. Monocations and monoanions derived from these delocalized radicals are expected to support likewise delocalized charges, as checked by various calculations on corresponding closed-shell ions. Altogether, these attributes might lead to possible applications in organic design, electronic devices, and spintronics.

Can a Topological Approach Predict Spin-Symmetry Breaking in Conjugated Hydrocarbons?

The closed-shell mean-field single determinants of large alternant hydrocarbons are frequently unstable with respect to a possible spin-symmetry breaking which produces different orbitals for the α and ß electrons, either in Hartree-Fock or in Kohn-Sham DFT calculations. The present work shows that one may easily predict whether such a symmetry breaking will take place from the elementary topological Hückel Hamiltonian which introduces a simple hopping integral t. The demonstration makes use of the simplest representation of the bielectronic repulsion, namely, the Hubbard bielectronic operator, reduced to an on-site repulsion U, and takes benefit of the mirror theorem. A recipe is proposed to determine the relevant t/U ratio for a given exchange-correlation potential. The symmetry-breaking phenomenon first concerns the mixing between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), but it may eventually run on other pairs of mirror orbitals. These symmetry breakings may take place while the other molecular orbitals keep a closed-shell character. The spin polarization of these MOs, appearing in typical unrestricted mean-field calculations, is an induced and amplifying effect, which has to be distinguished from the symmetry breaking itself. Special attention is paid to the possible appearance of multiple symmetry breakings, leading to a polyradical character. The model is tested on six series of polycyclic hydrocarbons. This elementary approach sheds new arguments on the debate concerning the di- or polyradical character of polyacenes.

Communication: Proper use of broken-symmetry calculations in antiferromagnetic polyradicals.

The present comment formulates some recommendations regarding the use of broken-symmetry Unrestricted Density Functional Theory (UDFT) solutions in those polyradical architectures predicted to be of ground-state singlet character according to Ovchinnikov's rule. It proposes a procedure to identify the number of open shells, to reach the relevant Ms = 0 solution, and to estimate the low-energy spectrum of the states which keeps this number of open shells.

Towards Magnetic Carbo-meric Molecular Materials.

Numerous studies have underlined the putative diradical character of π-conjugated molecules that can be described by closed-shell Lewis structures, for instance, p-dimethylene p-n phenylenes, or long polyacenes. In the latter compounds, the only way to save the aromaticity of the six-membered rings is to give up the Lewis electron pairing in the singlet biradical ground state. The present work considers the possibility of doing the same by using the basic C2 units of carbo-meric architectures. A series of acyclic and cyclic carbo-meric architectures is studied by using UB3LYP DFT broken-symmetry calculations, including spin decontaminations and subsequent geometry optimization of the singlet diradical. The C2 units are shown to stabilize the singlet biradical by spin delocalization, two of them playing approximately the same role as one radical-insulating 1,4 phenylene moiety. The results are generalized to the investigation of open-shell polyradical singlet states of rigid hydrocarbon structures, the symmetry and rigidity of which can assist cooperativity and self spin polarization effect. Several synthesis targets with challenging magnetic/spin properties are suggested in the carbo-mer series.

Kekulé versus Lewis: when aromaticity prevents electron pairing and imposes polyradical character.

Some conjugated alternant hydrocarbons, of singlet ground state according to Ovchinnikov's rule, may exhibit strong polyradical character, despite admitting complete pairing of electrons in bond orbitals between adjacent atoms. Typical organizations of this kind are encountered in polycyclic frames supporting two or more extracyclic methylene groups. Lewis bond pairing would require quinonization of six-membered rings, whereas safeguarding aromaticity proves sufficient to impose ground-state open-shell character, that is, the existence of unpaired electrons, providing the number of benzene rings to be quinonized is larger than two. Several examples built as variations around para-polyphenylene frames are examined through unrestricted DFT (UDFT) calculations, using various methods for spin decontamination of wavefunctions, geometries, and singlet-triplet energy gaps. They all illustrate how it is possible to conceive architectures that can be written with a closed-shell bond pairing, although they exhibit a large number of unpaired electrons. The same analyses also apply to systems in which quinonization would not kill but only reduce the number of unpaired electrons.

When a single hole aligns several spins: double exchange in organic systems.

The double exchange is a well-known and technically important phenomenon in solid state physics. Ionizing a system composed of two antiferromagnetically coupled high-spin units, the ground state of which is a singlet state, may actually produce a high-spin ground state. This work illustrates the possible occurrence of such a phenomenon in organic chemistry. The here-considered high-spin units are triangulenes, the ground state of which is a triplet. Bridging two of them through a benzene ring produces a molecular architecture of singlet ground state. A careful exploitation of a series of unrestricted density functional calculations enables one to avoid spin contamination in the treatment of the doublet states and shows that under ionization the system becomes of quartet multiplicity in its ground state. The possibility to align more than three spins from conjugated hydrocarbon polyradicals is explored, considering partially hydrogenated triangulenes. A dramatic example shows that ionization of a singlet ground state molecule may generate a decuplet.

In Search of Organic Compounds Presenting a Double Exchange Phenomenon.

The objective of this paper is to design a consistent series of organic molecules that may present a double exchange mechanism and study their low energy spectrum using spin unrestricted Density Functional Theory. For this purpose, organic tetra-methylene methane units having an S = 1 spin ground state and diamagnetic organic bridges are taken as building blocks for constructing molecules having two or more magnetic units. When biunit systems are ionized, the ground state of the resulting molecular ions may be either a quartet, if the spectrum is ruled by a double exchange mechanism, or a doublet, if it obeys the logic of a monoelectronic picture. A strategy based on the physical analysis of the leading interactions is followed in order to energetically favor a high-spin ground state. It is shown that the most promising compounds involve bridges that have both a large gap between the highest occupied and the lowest unoccupied molecular orbitals and small coefficients on the atoms to which the magnetic units are connected. While the followed strategy enables one to conceive organic compounds exhibiting a double exchange phenomenon, it is shown that the electronic mechanism ruling the spectrum of such organic double exchange compounds is different from that of their inorganic homologues. A new method to reconstruct the spectrum of low energy from various spin unrestricted DFT solutions is proposed and applied. Finally monodimensional and bidimensional periodic lattices based on the most promising organic architecture are suggested.

A recipe for geometry optimization of diradicalar singlet states from broken-symmetry calculations.

The equilibrium geometries of the singlet and triplet states of diradicals may be somewhat different, which may have an influence on their magnetic properties. The single-determinantal methods, such as Hartree-Fock or Kohn-Sham density functional theory, in general rely on broken-symmetry solutions to approach the singlet-state energy and geometry. An approximate spin decontamination is rather easy for the energy of this state but is rarely performed for its geometry optimization. We suggest simple procedures to estimate the optimized geometry and energy of a spin-decontaminated singlet, the accuracies of which are tested on a few organic diradicals. This technique can be generalized to interactions between higher-spin units or to multispin systems.