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.

From complete to selected model spaces in determinant-based multi-reference second-order perturbation treatments.

The present paper reformulates and improves a previously proposed determinant-based second-order multi-reference perturbative formalism. Through a rather simple modification of the energy denominators, this formalism takes into account the interactions between the model space determinants, which are repeated in outer space. The method has been shown to be size-consistent when the model space is a complete active space, which is a severe limit. It is shown here that the completeness of the model space is not necessary to keep this property, provided that the zero-order function satisfies some conditions. For instance, size consistency may be obtained from truncated complete active spaces. It may even be satisfied from Singles and Doubles Configuration Interactions, provided that a coupled electron pair approximation is used in the definition of the model space wave function. The physical content of the method is illustrated by a series of model problems, showing its robustness. A major benefit of the fact that the perturbers are single determinants is the possibility to revise with full flexibility the model-space component of the wave function, i.e., to treat the feedback effect of the dynamic correlation on the valence component of the wave function.

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.

Simulation vs. Understanding: A Tension, in Quantum Chemistry and Beyond. Part C. Toward Consilience.

In the last part of our Essay, we outline a future of consilience, with a role both for fact-seekers, and for searchers for understanding. We begin by looking at theory and simulation, surrounded as they are by and interacting with experiment, especially in Chemistry. Experimenters ask questions both conceptual and numerical, and so draw the communities together. Two case studies show what brings the theoretician authors joy in this playground, and two more detailed ones make it clear that computation/simulation is anyway deeply intertwined with theory-building in what we do, or for that matter, anywhere in the profession. From a definition of science we try to foresee how simulation and theory will interact in the AI-dominated future. We posit that Chemistry's streak of creation provides in that conjoined future a link to Art, and a passage to a renewed vision of the sacred in science.

Simulation vs. Understanding: A Tension, in Quantum Chemistry and Beyond. Part B. The March of Simulation, for Better or Worse.

In the second part of this Essay, we leave philosophy, and begin by describing Roald's being trashed by simulation. This leads us to a general sketch of artificial intelligence (AI), Searle's Chinese room, and Strevens' account of what a go-playing program knows. Back to our terrain-we ask "Quantum Chemistry, † ca. 2020?" Then we move to examples of Big Data, machine learning and neural networks in action, first in chemistry and then affecting social matters, trivial to scary. We argue that moral decisions are hardly to be left to a computer. And that posited causes, even if recognized as provisional, represent a much deeper level of understanding than correlations. At this point, we try to pull the reader up, giving voice to the opposing view of an optimistic, limitless future. But we don't do justice to that view-how could we, older mammals on the way to extinction that we are? We try. But then we return to fuss, questioning the ascetic dimension of scientists, their romance with black boxes. And argue for a science of many tongues.

Simulation vs. Understanding: A Tension, in Quantum Chemistry and Beyond. Part A. Stage Setting.

We begin our tripartite Essay with a triangle of understanding, theory and simulation. Sketching the intimate tie between explanation and teaching, we also point to the emotional impact of understanding. As we trace the development of theory in chemistry, Dirac's characterization of what is known and what is needed for theoretical chemistry comes up, as does the role of prediction, and Thom's phrase "To predict is not to explain." We give a typology of models, and then describe, no doubt inadequately, machine learning and neural networks. In the second part, we leave philosophy, beginning by describing Roald's being beaten by simulation. This leads us to artificial intelligence (AI), Searle's Chinese room, and Strevens' account of what a go-playing program knows. Back to our terrain-we ask "Quantum Chemistry, † ca. 2020?" Then move to examples of AI affecting social matters, ranging from trivial to scary. We argue that moral decisions are hardly to be left to a computer. At this point, we try to pull the reader up, giving the opposing view of an optimistic, limitless future a voice. But we don't do justice to that view-how could we? We return to questioning the ascetic dimension of scientists, their romance with black boxes. Onward: In the 3rd part of this Essay, we work our way up from pessimism. We trace (another triangle!) the special interests of experimentalists, who want the theory we love, and reliable numbers as well. We detail in our own science instances where theory gave us real joy. Two more examples-on magnetic coupling in inorganic diradicals, and the way to think about alkali metal halides, show us the way to integrate simulation with theory. Back and forth is how it should be-between painfully-obtained, intriguing numbers, begging for interpretation, in turn requiring new concepts, new models, new theoretically grounded tools of computation. Through such iterations understanding is formed. As our tripartite Essay ends, we outline a future of consilience, with a role both for fact-seekers, and searchers for understanding. Chemistry's streak of creation provides in that conjoined future a passage to art and to perceiving, as we argue we must, the sacred in science.

Magnetic Coupling in the Ce(III) Dimer Ce2(COT)3.

The monomer [Ce(COT)2]- and the dimer [Ce2(COT)3], with Ce(III) and COT = 1,3,5,7-cyclooctatetraenide, are studied by quantum chemistry calculations. Due to the large spin-orbit coupling, the ground state of the monomer is a strong mixing of σ and π states. The experimental isotropic coupling in the dimer was evaluated by Walter et al. to be J = -7 cm-1 (with a Heisenberg Hamiltonian [Formula: see text]) with a small anisotropic coupling of 0.02 cm-1. The coupling between the two Ce(III) in the dimer is calculated using CI methods. The low energy part of the spectra are modeled by spin Hamiltonians. All spin Hamiltonians parameters are deduced from ab initio calculations. g factors are calculated for both the pseudodoublet of the monomer and the pseudotriplet of the dimer and their sign have been determined. The magnetic coupling in the dimer is rationalized by a model based on crystal field theory. The kinetic and exchange contributions arising from the different configurations to the isotropic and anisotropic couplings are evaluated. It is shown that the main contribution to isotropic coupling is kinetic and originates from the fσ-fσ interaction due to the large transfer integral between those orbitals. However, the fπ-fπ interaction plays a non-negligible role. The anisotropic coupling originates from the difference of exchange energy of states arising from the fσfπ configuration and is, in no matter, related to the anisotropy of the local magnetic moments as already pointed by van Vleck for a fictitious s-p system. The analysis of the natural orbitals evidences a superexchange mechanism through a σCH* orbital of the bridging cycle favored by a local 4fσ/5dσ hybridization and that the δ type orbitals, both the HOMOs of the ligands and the virtual fδ orbitals of the cerium atoms play an important polarization role, and to a less extend the π type orbitals, the HOMOs-1 of the ligands, and the metal fπ orbitals.

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.

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.

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.

Alternative definition of excitation amplitudes in multi-reference state-specific coupled cluster.

A central difficulty of state-specific Multi-Reference Coupled Cluster (MR-CC) in the multi-exponential Jeziorski-Monkhorst formalism concerns the definition of the amplitudes of the single and double excitation operators appearing in the exponential wave operators. If the reference space is a complete active space (CAS), the number of these amplitudes is larger than the number of singly and doubly excited determinants on which one may project the eigenequation, and one must impose additional conditions. The present work first defines a state-specific reference-independent operator T∼^m which acting on the CAS component of the wave function |Ψ0m⟩ maximizes the overlap between (1+T∼^m)|Ψ0m⟩ and the eigenvector of the CAS-SD (Singles and Doubles) Configuration Interaction (CI) matrix |ΨCAS-SDm⟩. This operator may be used to generate approximate coefficients of the triples and quadruples, and a dressing of the CAS-SD CI matrix, according to the intermediate Hamiltonian formalism. The process may be iterated to convergence. As a refinement towards a strict coupled cluster formalism, one may exploit reference-independent amplitudes provided by (1+T∼^m)|Ψ0m⟩ to define a reference-dependent operator T^m by fitting the eigenvector of the (dressed) CAS-SD CI matrix. The two variants, which are internally uncontracted, give rather similar results. The new MR-CC version has been tested on the ground state potential energy curves of 6 molecules (up to triple-bond breaking) and two excited states. The non-parallelism error with respect to the full-CI curves is of the order of 1 mEh.

A Jeziorski-Monkhorst fully uncontracted multi-reference perturbative treatment. I. Principles, second-order versions, and tests on ground state potential energy curves.

The present paper introduces a new multi-reference perturbation approach developed at second order, based on a Jeziorski-Mokhorst expansion using individual Slater determinants as perturbers. Thanks to this choice of perturbers, an effective Hamiltonian may be built, allowing for the dressing of the Hamiltonian matrix within the reference space, assumed here to be a CAS-CI. Such a formulation accounts then for the coupling between the static and dynamic correlation effects. With our new definition of zeroth-order energies, these two approaches are strictly size-extensive provided that local orbitals are used, as numerically illustrated here and formally demonstrated in the Appendix. Also, the present formalism allows for the factorization of all double excitation operators, just as in internally contracted approaches, strongly reducing the computational cost of these two approaches with respect to other determinant-based perturbation theories. The accuracy of these methods has been investigated on ground-state potential curves up to full dissociation limits for a set of six molecules involving single, double, and triple bond breaking together with an excited state calculation. The spectroscopic constants obtained with the present methods are found to be in very good agreement with the full configuration interaction results. As the present formalism does not use any parameter or numerically unstable operation, the curves obtained with the two methods are smooth all along the dissociation path.

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.

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.

The "Fermi hole" and the correlation introduced by the symmetrization or the anti-symmetrization of the wave function.

The impact of the antisymmetrization is often addressed as a local property of the many-electron wave function, namely that the wave function should vanish when two electrons with parallel spins are in the same position in space. In this paper, we emphasize that this presentation is unduly restrictive: we illustrate the strong non-local character of the antisymmetrization principle, together with the fact that it is a matter of spin symmetry rather than spin parallelism. To this aim, we focus our attention on the simplest representation of various states of two-electron systems, both in atomic (helium atom) and molecular (H2 and the π system of the ethylene molecule) cases. We discuss the non-local property of the nodal structure of some two-electron wave functions, both using analytical derivations and graphical representations of cuttings of the nodal hypersurfaces. The attention is then focussed on the impact of the antisymmetrization on the maxima of the two-body density, and we show that it introduces strong correlation effects (radial and/or angular) with a non-local character. These correlation effects are analyzed in terms of inflation and depletion zones, which are easily identifiable, thanks to the nodes of the orbitals composing the wave function. Also, we show that the correlation effects induced by the antisymmetrization occur also for anti-parallel spins since all Ms components of a given spin state have the same N-body densities. Finally, we illustrate that these correlation effects occur also for the singlet states, but they have strictly opposite impacts: the inflation zones in the triplet become depletion zones in the singlet and vice versa.

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.