Electronic Structures of Anti-Ferromagnetic Tetraradicals: Ab Initio and

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The electronic structure of anti-ferromagnetic tetraradicals: an ab initio and semi-empirical study Dawei Zhang, and Chungen Liu J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00103 • Publication Date (Web): 10 Mar 2016 Downloaded from http://pubs.acs.org on March 10, 2016

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Journal of Chemical Theory and Computation

The electronic structure of anti-ferromagnetic

tetraradicals: an ab initio and semi-empirical study Dawei Zhang∗ and Chungen Liu Institute of Theoretical and Computational Chemistry, Key Laboratory of Mesoscopic Chemistry of the Ministry of Education (MOE), School of Chemistry and Chemical Engineering, Nanjing University, Nanjing, 210093, China

E-mail: [email protected]

Abstract

The energy relationships and electronic structures of the lowest-lying spin states in several anti-ferromagnetic tetraradical model systems are demonstratively studied with high-level ab initio and semi-empirical methods. The Full-CI method (FCI), the complete active space second order perturbation theory (CASPT2), and the n-electron valence state perturbation theory (NEVPT2) are employed to obtain reference results. From comparing the energy relationships predicted from the Heisenberg and Hubbard models with ab initio benchmarks, the accuracy of the widely-used Heisenberg model for anti-ferromagnetic spin-coupling in low-spin polyradicals is cautiously questioned in this work. It is found that the strength of electron correlation (|U/t|) concerning anti-ferromagnetically coupled radical centers could range widely from strong to moderate correlation regimes and could become another degree of freedom besides the spin multiplicity. Accordingly, the Heisenberg-type model works well in the regime of strong correlation, which reproduces well the energy relationships as well as the wavefunctions of all the spin states. In moderately spin-correlated tetraradicals, the results of the prototype

∗ To

whom correspondence should be addressed

1 ACS Paragon Plus Environment


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Heisenberg model deviate severely from those of multireference electron correlation ab initio methods, while the extended Heisenberg model containing four-body terms can introduce reasonable corrections and maintains its accuracy in this condition. In the weak correlation regime, both the prototype Heisenberg model and its extended forms containing higher-order correction terms will encounter difficulties. Meanwhile, the Hubbard model shows balanced accuracy from strong to weak correlation cases and can reproduce qualitatively correct electronic structures, which makes it more suitable for the study of anti-ferromagnetic coupling in polyradical systems.

1

Introduction

The electronic structure of an organic polyradical is usually characterized as the existence of a number of weakly interacting “unpaired” electrons in the molecule, which is due to an open-shell arrangement of electrons on a number of frontier molecular orbitals. For a long time, the importance of such chemical species has been well recognized in designing high-spin molecular magnets. 1–8 On the other hand, low-spin molecular polyradicals, which were considered as inherently metallic materials, received much less attention from both experimental and theoretical chemistry communities. 9–12 This is at least partially due to serious chemical instability of the polyradical structure originating from the spin-Peierls transition in quasi one-dimensional anti-ferromagnetic spin network, 13–15 which may lead to the dimerization of neighboring anti-parallel spin centers through intra- or inter-molecular chemical bond formation, resulting in a diamagnetic electronic structure. 10,16 Within the very limited examples showing satisfactory thermo-stability, a carefully designed derivative of 1,3-diradical located on a PBPB planar four-member ring has received considerable attention. Its dimerization to construct tetraradical has also been studied both experimentally and theoretically. 16–18 Reported progresses in this field are of fundamental importance toward the design of anti-ferromagnetic low-spin oligomers, polymers, as well as clusters. 19,20

On the other hand, organic molecules, such as polycyclic aromatic hydrocarbons (PAH’s) and their derivative series, may provide new resources of anti-ferromagnetic polyradicals. It has been


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noticed by the theoreticians that the polyacene systems may have open-shell singlet ground states with the conjugated chain elongating. 21–23 Besides, some well-designed conjugated alternant hydrocarbons with singlet ground state according to Ovchinnikov’s rule may exhibit strong polyradical character, due to the extra stability of benzene rings to prevent the radical electrons of methylene groups from being paired by quinonization. 24 As is well known, the typical magnitude of spin-couplings between the solitons of organic polyradicals is several Kcal/mol, 22–24 which is

significantly larger than the typical intensity of the spin-couplings in the anti-ferromagnetic metal oxides, implying different correlation intensities of radical electrons between them. For a classical chemical bond, the relative size of the hopping integral t and the on-site repul-

sion U varies with the geometry changing. 25 For example, a hydrogen molecule shows a value of | Ut | → 125 when the nuclei tend to coincide (RH

H

→ 0). This condition should be ascribed to

weak-correlation regime. With the inter-atom distance elongating, the ratio of | Ut | increases and the weight of covalent configurations increases, too. Until the molecule tend to dissociate, where RH

H

→ ∞, the ratio of | Ut | approaches infinity and the weight of covalent configurations is asymp-

totic to 100%. This condition should be ascribed to strong-correlation regime. A Mott transition

occurs between these two regimes. 26–28 For multiradicals, the electronic structures can also range from strong to moderate correlation regimes. Since the electronic structures (especially the ionicities) change significantly during this process, the electron interactions (especially the dynamic correlation) could be quite different between these regimes. 29 Besides, some recent progress in the synthesis of highly stable low-spin diradicals and multiradicals gave rise to some new radical species with surprisingly large intramolecular spin-exchange interactions up to ca. 1eV. 19 It is reasonable to suppose that the electron correlation between neighboring radical centers might be much weaker than most previously investigated radical systems with very small energy gaps between different spin states. Accordingly, it should be important to validate the popular theories whether they could afford a balanced description for these multiradical systems in a relatively weaker correlation regime. Theoretical investigation to the electronic spin interaction in an organic polyradical is com-



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monly carried out through the solution of the empirical Heisenberg model (also called Heisenberg

Dirac-Van Vleck model), 30–32

Hˆ =

∑

Ji j (2Sˆ i Sˆ j

i< j, i∼ j

1 ). 2

(1)

With the nearest-neighboring approximation, the above Heisenberg Hamiltonian is expressed as the summation of the isotropic bilinear spin-exchange interactions between adjacent radical centers. For a ferromagnetic (anti-ferromagnetic) spin-coupling between a pair of radical centers, the spinexchange constant Ji j is a positive (negative) value. 33 Computationally, J12 in a diradical system is usually determined as a half of the energy gap between the lowest singlet and triplet states. While for multiradicals, one needs to solve the Heisenberg model analytically for explicit energy expressions of several lower-lying electron states in order to obtain Ji j ’s. Due to the practical difficulties in analytical solution, such kinds of theoretical investigations could hardly be realized for systems containing more than three radical centers. 4,34 Heisenberg Hamiltonian also finds its application in constructing model Hamiltonians for the various semi-emprical quantum chemical methods based on the valence bond (VB) theory, especially for π-conjugated organic molecules and radicals. 23,34–48 In such cases, spin coupling is always anti-ferromagnetic for the neighboring pair of π-electrons. Furthermore, the strength of coupling could be up to several electron volts, significantly different from typical values of ca. 0.01 to 0.1 eV between closely contacting antiferromagnetic radical centers.

On the other hand, along with its extensive applications in condensed-matter physics as well as in quantum chemistry, the limitation of the concise Heisenberg model was noticed ever since the early years after the invention of the theory, which are mainly due to the truncated configuration space spanned by only covalent configurations. 49–58 As is well known, the Heisenberg model can not be applied to study electron and charge transportations in anti-ferromagnetic metallic systems because of the configuration space truncation. 52 Moreover, including additional higher-order terms of spin-exchange operators is recommended to improve the accuracy of the prototype Heisenberg model when studying the spin-coupling effects between magnetic metal ions in metal oxides even


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if the coupling constants J’s are very small, 54,55 or computing the energy differences between spin states of π-conjugated organic molecules containing small rings. 39,43,59,60 In computing energy differences of relevant electronic states for π-conjugated organic radicals, it was suggested that the bilinear spin-exchange Heisenberg model should only be used to describe the family of states with closely related electronic wavefunctions, which requires that the singly and doubly occupied orbitals are nearly the same for all these states. 8 For increasing the accuracy of the Heisenberg-type spin exchange models, theoretical scien

tists have suggested several kinds of higher-order corrections. Anderson has suggested a spin super-exchange theory to refine the prototype Heisenberg model, which introduces fourth-order terms ( j(Sˆ i Sˆ j )2 ) into the Hamiltonian. 61 This super-exchange theory has been widely used in discussing the spin exchange effects in metal oxide materials, 54,62–64 and can narrow the deviations of the Heisenberg results from experimental data. Besides this, Malrieu and his co-workers have systematically discussed the influence of the four-body operators to the energy relationships of metal oxide multiradicals. They have shown that the four-spin operators can provide discrepant corrections for different spin states, which can make the energy relationships more reasonable for comparison with the experimental results. 65–69 They have developed the effective Hamiltonian perturbation theory (EHPT), which suggested that the extended Heisenberg model including biquadratic (or even higher) terms could efficiently improve the accuracy of the spin-only effective Hamiltonian for a valuable comparison with other theories. 35,36,48 Based on these previous works, higher order forms of the Heisenberg-type model can be derived from the perturbation expansion of Hubbard model. 35,39 As is reported, six-order correction of Heisenberg model has already been reached, which has been proved to be important in describing benzenoid hydrocarbon systems. 35,60 In contrast with the Heisenberg-type model, the Hubbard model is set up on the full space comprised of both ionic and covalent configurations. 70 With the nearest-neighboring approximation, the Hubbard Hamiltonian can be written as,

Hˆ =

∑ i∼ j

∑ ti j σ

aˆ † ˆ jσ + U iσa

∑

nˆ iα nˆ iβ ,

i


(2)


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where U denotes the on-site double-electron repulsion, and ti j is the single-electron hopping integral between nearest-neighboring atoms i and j. In comparison with the 2N configurations of the Heisenberg model, the Hubbard Hamiltonian is built up in a Hilbert space with 4N dimension, containing an enormous number of ionic configurations besides covalent configurations. As is well known, the prototype Heisenberg model can be regarded as an asymptotic case of the Hubbard 2t2

model at strong coupling limit (U ≫ |ti j |), with Ji j = Ui j . 35,39,71 The Hubbard model is capable of describing charge transportation in metal oxides as well as in organic molecules. 72,73

In this study, we present a comparative study on the accuracies of the Heisenberg-type models and Hubbard model in computing three carefully selected low-spin organic tetraradicals. The deviations of the semi-empirical models from high-level ab initio methods are presented, along with detailed analyses on the electronic wavefunctions of the six spin states (the covalent states that related to different spin arrangements of four unpaired electrons). We will demonstrate that the prototype Heisenberg model may fail in reproducing the energy gaps between these spin states when the electronic correlation between radical centers is not strong enough. The extended Heisenberg model containing four-body operators, also called the effective Hamiltonian perturbation theory containing fourth-order corrections (EHPT4), could be better in strong to moderate correlation regimes. While, the Hubbard model usually performs a balanced accuracy in both strong and weak correlation cases, and can also provide qualitatively correct information of many electron wavefunctions.

2

Computational Methodology

Ab initio calculation for open-shell systems is still a challenge in quantum chemistry. Multireference electron correlation theories, such as the complete active space self-consistent field (CASSCF) with second order perturbative corrections (CASPT2) method, are considered as a balanced choice of the computational cost and accuracy for calculating the excitation energies as well as the electronic structures of the lowest-lying states. 74,75 While Angeli, de Graaf and their co-workers


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suggest that the n-electron valence state second order perturbation theory (NEVPT2) which uses a more elaborated zeroth-order Hamiltonian, 76–81 and the difference dedicated configuration interaction method (DDCI), 63,64,69,80 are more suitable for describing the small magnetic couplings. So in this work, both NEVPT2 and CASPT2 calculations are implemented for the energy level and electronic structure calculations. The widely-used 6-311G(d) basis set with TZP quality is selected in this work, while it is considered that the diffuse functions might be significant for describing radical systems. So we have calculated the tetraradical systems in this work with both 6-311G(d) and 6-31+G(d) bases sets, and have found that the results are in quite good agreements with each other, with an RMS deviation of only 0.2Kcal/mol and a maximum deviation less than 0.4Kcal/mol for relative energy differences. The negligible deviation implies that these two bases sets are reliable for this research work. Geometry optimization and adiabatic energy differences of the lowest-lying singlet, triplet, and quintet are performed with the CASPT2 method with a minimum active space containing four radical orbitals and four electrons, using the 6-311G(d) basis set (CASPT2(4,4)/6-311G(d)). The vertical energy differences between the six spin states are then computed with partially contracted NEVPT2(12,12)/6-311G(d) method following a state-averaged CASSCF(12,12)/6-311G(d) computation of the multi-configuration SCF wavefunction. For each species, another set of vertical energy differences is also obtained from CASPT2/6-311G(d) computation for reference. The analysis of electronic wavefunctions are based on the state-averaged CASSCF(4,4)/6-311G(d) computations on relevant states. All of the calculations are implemented with MOLPRO 2010 software package. 82 In this work, both the traditional Heisenberg model and the Hubbard model are implemented to discuss the open-shell electronic structures of the tetraradicals. Besides, we also introduce the EHPT4 theory for further comparison and discussion. As is well known, the EHPT4 Hamiltonian is equivalent to the fourth-order Rayleigh-Schrödinger perturbation term of the Hubbard Hamiltonian. 35,39,59,71 If we use I, J, K to denote configurations of the model space (covalent configurations), ∑ ∑ α, β, γ to denote configurations of the outer space (ionic configurations), and Vˆ = ti j aˆ † iσaˆ jσ to i∼ j


σ


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denote the perturbation term (Hˆ 0 = U

∑ i

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aˆ † iαaˆ † iβaˆ iβ aˆ iα , Hˆ = Hˆ 0 + Vˆ ), considering that I|V |J = 0 for

any covalent configurations I and J, we can write the fourth-order term of EHPT4 Hamiltonian in

the covalent model space as, 35,39,68,71

Hˆ eff(4) =

∑ α,β,γ

I|V |α α|V |β β|V |γ γ|V |J ∑ I|V |α α|V |K K|V |β β|V |J . 2 ∆E ∆Eα ∆Eβ ∆Eγ ∆E β α α,β,K

(3)

This expression is universal for any topologies of radical clusters. If a radical system is formed by identical radical centers (that is, each Uii is equal), and the hopping integrals between any two neighboring sites i and j (ti j ) are all equal, the fourth-order operator of the EHPT4 Hamiltonian can be simplified to two well-known spin interaction terms for next-neighboring pairs (Hˆ eff(4a) )

and four-member rings (Hˆ eff(4b) ) respectively, as follows, 35,39,67,83

4

t Hˆ eff(4a) = 3 U t4

∑ [(4bi j ci j )(4Sˆ i Sˆ j 1)]

(4)

i> j

∑

Hˆ eff(4b) = 3 d i jkl U i, j,k,l

10[(Sî Sˆ j )(Sˆk Sˆ l ) + (Sˆ i Sˆ l)(Sˆ j Sˆ k) (Sˆ i Sˆ k)(Sˆ j Sˆ l )]

1 1 , + [Sˆ i Sˆ j + Sˆ j Sˆ k + Sˆ k Sˆ l + Sˆ i Sˆ l + Sˆ i Sˆ k + Sˆ j Sˆ l ] 8 2

where

bi j =

1 if sites i and j are adjacent 0 otherwise

ci j = the number of the common neighbors shared by sites i and j di jkl =

1 if bi j b jk bkl bil = 1

.

0 otherwise


(5)

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4

In these two expressions, the coefficient Ut 3 can be defined as the four-spin-exchange con

stant j. If a multi-radical system contains non-equal hopping integrals and/or non-equal on-site repulsions, its four-spin-exchange constants will be non-unique, 68 and the summation of these four-spin interaction expressions will have a more complicated form. Considering that most of the systems in this work have non-equal hopping integrals and tetrahedral topologies (which is more topologically-complicated than four-member-rings), we use the fundamental expression of Hˆ eff(4) for discussion. Its second quantization form is also derived in the Supporting Information,

which seems somewhat complicated but still convenient for computational programming. Each independent parameter (Uii and ti j ) is modulated freely in the fitting parametrization. There are several parametrization strategies for the extended Heisenberg model. One can fit the coupling constants (Ji j ) from a preliminary prototype Heisenberg model firstly, and regarded the biquadratic terms as the fourth order perturbative correction terms. The on-site repulsions (Uii ) can be decided by fitting the energy gaps of high-level ab initio calculations further. Thence √ U U the hopping integrals (ti j ) can be obtained through the equation ti j = Ji j Uii ii+U jjjj , and all of

the four-body correction terms can be derived from the perturbation expansion with the acquired coupling constants and hopping integrals. 35,39,67,83 This treatment has a clear physical meaning in accordance with the perturbation expansion conception and is adopted in this work. Alternatively, the parameters can also be decided through fitting all of the energy gaps between six spin states simultaneously, which reproduces the minimum RMS deviation from the ab initio calculations. We have noticed that these two parametrization strategies are usually qualitatively identical.

3

Results and Discussion

In constructing a multi-radical system, two or more radical fragments can be connected through not only conjugated bridges, but also saturated carbon atoms, 84,85 or a spiro hetero-atom. 86–89 The coupling between the radical centers could originate from both through-space and through-bond interactions. 16,90



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In this work, four model systems of tetraradical species, as shown in Figure 1, are discussed. 1 is constructed by four hydrogen atoms placed in each vertex of a regular tetrahedral. 2 is constructed by removing one hydrogen atom from each of the four vertex carbons of adamantane. 3 is formed from removing one hydrogen atom from every other carbon in cyclooctane. 4 could be taken as two combined phenalene moieties connected through a saturated spiro-carbon bridge atom. We obtain the potential energy curve for the simplest species 1 through an ab initio calculation at the level of Full-CI/Aug-cc-pVTZ. This calculation suggests a singlet ground state from the equilibrium spacing to the dissociation limit. In order to confirm the spin multiplicities of the ground states in the other three molecules, the energy minimizations of different spin states are performed at the level of CASPT2(4,4)/6-311G(d). These ab initio calculations suggest that all of the species discussed in this work have singlet ground states, with the adiabatic excitation energies presented in Table 1.

1

H

2

H

H

H

4

2

3

4 3

1

1

2

3

H H

4

4m

Figure 1: The tetraradical molecules mentioned in this work.

3.1

A Hypothetical H4 Cluster in Td Symmetry

For a concise analysis on the Heisenberg and the Hubbard models, it is convenient to take the regular tetrahedral H4 cluster as the model system, which is even simpler than the widely-used


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Table 1: The adiabatic excitation energies from ab initio optimizations.a

S0 T1 Q1

1b 0.0 +3.7 +15.5

2c 0.0 +4.3 +39.9

3c 0.0 +3.4 +12.3

4c 0.0 +1.2 +7.0

a Energies

are in Kcal/mol. Geometries are obtained from the lowest points of the potential energy curves from Full-CI/Aug-cc-pVTZ calculation. c Optimized at CASPT2(4,4)/6-311G(d) level. b

theoretical model of tetraradicals, the cubic H4 He4 cluster. 91 When the four hydrogen atoms are sufficiently separated, chemical bonding effect between them is weakened considerably, hence each of the four atoms can be regarded as a radical center. In the Td point group symmetry, each electronic interaction parameter, such as the on-site repulsion U , the hopping integral t and the spin-coupling interaction J, has only one value for setting up the semi-empirical models, so that the effective Hamiltonians are greatly simplified to allow for analytical explorations. In the prototype Heisenberg model, the relative energies of the six electronic states could be expressed in terms of J,

ES 0 = ES ′

= 6J

0

ET1 = E

T1′

= E

T1′′

= 4J

(6)

= 0.

E Q1

Here, two singlet states, S 0 and S ′ 0, forms a two-fold degenerated energy level, three triplet states, T 1 , T 1′ and T 1′′ , are three-fold degenerated, while the quintet state, Q1 , is the non-degenerating

highest energy level for any negative J. We denote the energy difference between S 0 and Q1 as ∆E S Q , and the one between T 1 and Q1 as ∆ET Q . Obviously, the prototype Heisenberg model predicts exactly a value of 1.5 for the ratio of ∆E S Q to ∆ET Q , i.e., ∆ET Q is twice of ∆E S T . Generally, the weakening of electronic correlation will increase the weight of ionic configurations in the ground state wavefunction, which are excluded from the configuration space of the Heisenberg model. For a better understanding of the dependence between the intensity of electron-



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ic correlation and the accuracy of the Heisenberg model, the full configuration Hubbard model of the tetrahedral H4 cluster is also solved analytically to express the energy differences between the six low-lying spin states in terms of U and t. In solving this 256-dimensional Hamiltonian, the Td geometrical symmetry as well as the spin symmetry must be fully employed to obtain the analytical solution. With the symmetry-adapted and spin-adapted configuration state functions (CSF’s), we can finally reduce the Hubbard Hamiltonian matrix to be block-diagonal, with the order of each sub-block being 3. Details of the solution procedure is available in the Supporting Information. Accordingly, it is convenient to express the energies of the covalent singlet, triplet and quintet

states as follows.

√ 2 t ; ES 0,E = U 32 3U 2 + 48t2 cosθS , where θS = 13 arccos √ 108U (3U 2 +48t2 )3 √ 3 2 ET1,T2 = 32 U 23 U2 + 48t2 cosθT , where θT = 13 arccos √U 236U t2 3 ; (U +48t )

(7)

E Q1,A1 = 0.

From these expressions we can easily validate the asymptotic behaviors of the Hubbard model. When t → 0 (where |U/t| → ∞, i.e., the strong correlation limit), all of the six spin states converge to zero in energy. And we find that the ratio of ∆E S Q : ∆ET Q converge asymptotically to 1.5 by using l’Hôpital’s rule, which is in consistent with the result of the Heisenberg model. When U → 0, i.e. |U/t| → 0 (the weak correlation limit), both the singlets and the triplets converge to 4t (t < 0) in energy, while the quintet remains at the zero point of the energy. So the ratio of ∆E S

Q : ∆E T Q

converges asymptotically to 1.0, which is in consistent with the result of the Hückel model. Details

of the derivation is available in the Supporting Information. Besides the semi-empirical model studies, the regular tetrahedral H4 cluster is also computed with the ab initio method at the level of Full-CI/Aug-cc-pVTZ. 92,93 After computing a series of H4 clusters with different inter-atom separations ranging from 0.7 to 6.0 Å, the plot of ∆E S

Q : ∆E T Q

ratio versus |U/t| is presented in Figure 2, where the parameter |U/t| is fitted to the Full-CI/Aug3 + cc-pVTZ potential energy curves of three low-lying valent states (11 Σ+g, 11 Σ+ u and 1 Σ u) of H2

molecule, together with the plots computed with the Hubbard model, the prototype Heisenberg


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model and the EHPT4 theory for comparison.

Hubbard Heisenberg EHPT4 Full-CI

1.5

1.4

∆ES-Q : ∆ET-Q

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.3 1.2 1.1

1.0

0.0

1.0 1.4

2.0

2.8

4.0

5.7

8.0

11.3 16.0

∞

|U/t|

Figure 2: Dependence of the ∆E S Q : ∆ET Q ratios of regular-tetrahedral H4 , from the prototype Heisenberg model, the extended Heisenberg model with 4-body operators (EHPT4), the Hubbard model as well as the ab initio calculation at Full-CI/Aug-cc-pVTZ level, to the correlation intensity |U/t|.

with all these methods found to converge asymptotThe of ∆E|U/t| → S Q : ∆E T Q computed ically toratios 1.5 when ∞ (the strong correlation limit). In suchare circumstances, the hopping of electrons between different radical centers is greatly suppressed by the on-site Coulomb repulsion U , and spin-exchange appears as the the predominant interaction between radical centers. On the other hand, when the interatomic distance shortens, the hopping integral t increases and eventually exceeds U . In this process, the ratio of ∆E S

Q : ∆E T Q gradually falls down toward 1.0

according

to the Hubbard model. Essentially, the Full-CI computation supports the statement obtained with the Hubbard model, although with minor disagreements in weak-correlation regime (|U/t| < 4.0). Besides, it is worth of noticing that in the weak correlation limit, the Hubbard model converges to

the Hückel model, and its energy gap between S 0 and T 1 states disappears. This finding can not be reproduced by any real chemical system. The ab initio computation on H4 cluster reveals that the two states reverse their order other than becoming degenerated when H H separation is shorter than 1.354Å. We incline to ascribe these discrepancies between model studies and ab initio com-

puting to the failure of the zero-differential-overlap (ZDO) approximation brought by shortening of interatomic separation. Fortunately, since the radical centers are almost always well separated in real molecular multiradicals, it is expected that the Hubbard model should be generally effec-



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tive for these systems. In comparison, the prototype Heisenberg model gives an irrelevant ratio of ∆E S

Q:

∆ET

Q

to the change of |U/t|, which might be a physically inaccurate scenario even

in the regime of moderately strong correlation, namely, for 4 < |U/t| < 8. Its error could be up to 30% compared to the values computed with the Full-CI method and the Hubbard model. Also

presented in Figure 2 is the result of the Heisenberg-type EHPT4 model including the fourth-order correction terms. Improvement over the prototype Heisenberg model is witnessed from the strong correlation regime down to the moderate correlation regime (|U/t| > 8.0). However, its accuracy declines rapidly when |U/t| becomes even smaller, which manifests the failure of the perturbative treatment of the many-body Hubbard Hamiltonian in approaching the weak correlation regime. Let us further look into the feature of the wavefunctions with varying |U/t|, which are calculated with the Hubbard model. The covalencies of the lowest spin states as a function of varying |U/t| are shown in Figure 3. It is shown that in the weak-correlation limit, the covalencies of the singlet the triplet states are 38 and 14, respectively. And the covalency of T 1 is lower than that of S 0 in the weak-correlation regime. With increasing |U/t|, both the singlet and the triplet states present stronger covalencies in their wavefunctions. In the meantime, the difference between their covalencies is decreasing. Until the |U/t| reaches 6.6, the two lines are almost touching each other, with the covalencies of both states exceeding 83.5%. Finally, when approaching the strong-correlation

limit, the covalencies of all these six spin states converge asymptotically to 100%. Accordingly, we could roughly set |U/t| = 6.6 as the lower bound for the effectiveness of the prototype Heisenberg model, which corresponds to less than 17% ionicity in S 0 and T 1 wavefunctions and ∆E S

Q /∆E T Q

> 1.29. Interestingly, we noticed a recent theoretical study on oligoacene using

the Hubbard model, presented by Korytár et. al. They manifested the oscillating optical gap with

increasing length of oligomer in condition of weak-correlation, which can not be reproduced by Heisenberg model. 94 However, they believed that the Heisenberg model could still be qualitatively correct when |U/t| is no less 2. This be thehydrocarbons, reason why thewhere Heisenberg tially qualitatively correct inthan applying to could conjugated typicalmodel valuesisofessen|U/t|

are around 4 to 5. 94


Page 14 of 41

Page 15 of 41

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.0 0.0

1.0

Covalency

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.0 1.4

2.0

2.8

4.0

5.7

8.0 11.3 16.0

∞

|U/t|

Figure 3: The covalencies as functions of |U/t| for S 0 and T 1 wavefunctions of H4 cluster, calculated with the Hubbard model.

3.2

Tetraradicals on a distorted tetrahedral framework

Tetraradical 2 is created from removing the hydrogen atoms connected to the four vertex carbon atoms of adamantane, which leaves the four vertex carbon atoms unsaturated and could behave as four radical centers if the dehydrogenated species basically maintained the adamantane framework. 85 Based on theoretical analyses as well as the CASPT2 calculations presented in the Supporting Information, we find that while adamantane has a Td point group symmetry, 2 undergoes the Jahn-Teller distortion to D2d symmetry, which leads to a difference of ca. 0.004Å between two groups of C C bond lengths, and a slightly larger difference of less than 0.08Å for inter-radical separations. Accordingly, the tetraradical character should be well preserved in the dehydrogenat-

ed adamantane. Since the architecture of tetraradical 2 has a D2d symmetry, the four radical orbitals are still equivalent to each other. We present in Figure 4a one radical orbital, which is localized mainly on one of the four dehydrogenated carbon atoms to form an outward dangling orbital, with smaller but non-negligible contribution from the three neighboring C C σ orbitals. Such kind of radical orbital composition implies that the differential overlaps between the radical orbitals should be



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Page 16 of 41

large enough to open wide energy separations between different spin states. We list in Table 2 the vertical relative energies of all relevant spin states computed with the equilibrium geometry of the ground state.

(a)

(b)

Figure 4: Localized radical orbitals computed with state-averaged CASSCF(4,4) method for tetraradicals 2(a) and 3(b).

Since the reliability of the CASPT2 method was seriously questioned in previous studies of electronic spin coupling in magnetic systems, 48,64,76,78–80 it is significative to compare the results of the CASPT2 method against those of the partially contracted NEVPT2 method with an enlarged active spaces spanned by 12 electrons distributed in 12 molecular orbitals. According to the results listed in Table 2, we find that the excitation energies computed with CASPT2 are in fairly good agreement with those from NEVPT2 calculation, with the largest deviation of ca. 3.3 Kcal/mol, underestimation of the excitation energies by the CASPT2 method is not observed in tetraradical 2. Moreover, the deviation in the ratio ∆E S The ratio of ∆ET

Q

to ∆E S

T

Q /∆E T Q between these

two theories is even smaller.

is 6.99 and 7.46, given respectively by NEVPT2 and CASPT2

methods, deviating far from the expected value of 2 given by the prototype Heisenberg model.

Such kind of large deviations might come from both the absence of ionic configurations in the basis space of the bilinear spin-exchange Hamiltonian of the Heisenberg model, and unbalanced dynamical correlation effects in different spin states under investigation. In some extreme cases,


Page 17 of 41

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Table 2: Vertical excitation energies of six covalent states for dehydrogenated adamantane tetraradical 2 in units of Kcal/mol.

D2d

CASPT2(4,4)

NEVPT2(12,12)

Heisenberga

EHPT4b

Hubbardc

S0 S1

-43.1 -35.6

-40.3 -32.3

-45.1 (-4.8) -36.2 (-3.9)

-39.9 (+0.4) -33.1 (-0.8)

-40.2 (+0.2) -32.6 (-0.2)

T1 T 2 (T2′ )

-38.0 -32.0

-35.3 -29.5

-30.1 (+5.2) -25.6 (+3.9)

-35.7 (-0.4) -29.1 (+0.4)

-35.4 (-0.2) -29.3 (+0.3)

0.0

0.0

0.0

0.0

0.0

1.13

1.14

1.50

1.12

1.13

Q1 ∆ES

d

Q /∆E T Q

a The relative energies computed with prototype Heisenberg model, with the

spin-exchange constants J1 = 7.5135 Kcal/mol and J2 = 5.2960 Kcal/mol, which are fitted to the NEVPT2 excitation energies at ground state geometry constrained at D2d symmetry. Discrepancy in excitation energies are in the parentheses. b The relative energies computed with the Effective Hamiltonian Perturbation Theory containing four-body operators, with the on-site repulsion of U = 174.6150Kcal/mol, the hopping integrals t12 = 27.9144Kcal/mol and t13 = 22.3665Kcal/mol, which are fitted to the NEVPT2 excitation energies. Discrepancy in excitation energies are in the parentheses. c The relative energies computed with the Hubbard model, with on-site coulomb repulsion U = 61.3163Kcal/mol, and electron hopping integrals t12 = 17.7123Kcal/mol, t13 = 13.8712Kcal/mol, which are fitted to the NEVPT2 excitation energies. d The quintet state is taken as the zero point of the energy here, since it is the common zero point of both the Heisenberg model and the Hubbard model.



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Page 18 of 41

neglect of these two effects could bring about computational errors up to several electron volts. 29,95 In contrast, the ratio of ∆ET Q to ∆E S T given by the Hubbard model is much close to the results of ab initio calculations when the three model parameters U , t1 and t2 are determined by fitting

to the four energy differences between the six spin states. Since the fitting errors of the excitation energies (shown in the parentheses after the excitation energies) are less than 0.3 Kcal/mol, we could expect that the dynamical correlation effects in the six spin states are basically balanced, and the large error of the Heisenberg model might mainly originate from its truncation of the configuration space. As an improvement over the prototype Heisenberg model, the effect of ionic configurations is partly accounted by the fourth-order correction terms in the effective Hamiltonian of EHPT4 theory, which is reflected in the significantly improved accuracy of the excitation energies in comparison with the prototype Heisenberg model. However, the fitted value of U is greatly overestimated relative to the Hubbard model, which implies that the fourth-order correction is yet insufficient in constructing an effective Hamiltonian for tetraradical 2 in the covalent configuration space.

Table 3: The coefficients of covalent configurations in the wavefunctions of spin states for dehydrogenated adamantane tetraradical 2 in D2d symmetry.a

(S z = 0) |123¯4¯ |1234 |1234 |1234 |12¯3¯4 |1¯234¯ Covalency%

S0

S1

0.5383 0.0000 0.5383 0.0000 -0.2692 0.4708 -0.2692 0.4708 -0.2692 -0.4708 -0.2692 -0.4708

(S z = 1) |1234¯ |1234 |1234 |1234

86.93% 88.66%

T1

T2

T 2′

0.4685 0.4729 0.4685 -0.4729 -0.4685 0.4729 -0.4685 -0.4729

0.4729 -0.4729 -0.4729 0.4729

87.79% 89.47%

89.47%

a Computations

are performed with the state-averaged CASSCF method, including 6 covalent spin states in the active space spanned with four electrons in the four radical orbitals. The listed covalent configurations have been transformed into the representation of localized radical orbital bases. Covalency% is computed as the summation of squared coefficients of the covalent configurations, similarly hereinafter.

If we ascribe the inaccuracy of the Heisenberg model mainly to missing the effect of ionic configurations, an analysis on the electronic wavefuntions of relevant spin states could supply a di-


Page 19 of 41

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rect viewpoint on the origin of the imbalanced description by Heisenberg model. For the purpose of qualitative understanding, the wavefunctions are computed with the state-averaged CASSCF method using the minimal active configuration space. Since the Q1 state is purely covalent within the scheme of the minimal active configuration space (ΨQ1 = |1234 ), Table 3 lists the wavefunctions of the rest five spin states. Instead of the complete wavefunctions available in the Supporting

Information, we show in the table the coefficients of every covalent configuration. The covalency of each state is computed as the summation of squared coefficients of the covalent configurations. Obviously, the listed five states are mainly composed of covalent configurations (ca. 88%). In the Hubbard theory of Td model cluster shown in Figure 3, similar covalencies should fall into the regime of moderate to strong correlation with |U/t| 8. Since the CASSCF wavefunctions are calculated with the minimal active space, the ionicity of these states might be considerably

underestimated, and this system should be ascribed to moderate correlation regime. According to our calculation, |U/t|’s are determined as 3.5 and 4.4 corresponding to t1 and t2 respectively by Hubbard theory, which confirms this conjecture. It is surprising that the strength of electron

correlation related to the radical centers is comparable to those in normal carbon-carbon π-bonds, where the typical values of |U/t| are usually in the range from 2.0 to 5.0, and the prerequisite of the prototype Heisenberg model may cease to exist. We notice that the highest quintet is purely

covalent, but all the singlets and triplets have similar ionicities. So the Heisenberg-type effective Hamiltonian is accurate for the quintet, but the full-configuration theories like Hubbard model contain more interactions between covalent and ionic configurations for the lower-spin states than Heisenberg-type theories do, which lower their energy levels synchronously and enlarge the ratio of ∆ET

Q : ∆E S T ,

as is demonstrated in Figure 5.

3.3

Tetraradical on a square framework

Rectangularly positioned four-spin structure has been rigorously studied previously in Malrieu’s group. 25,67 Earlier studies revealed that the ionicities and the energy order of the low-lying states are related to the inter-radical distances in the square cluster of Li4 . 25 More recent investigation on



Q1

Q1

T1 S0

≈7:1

Energy

2:1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 41

Containing covalent-ionic configuration interactions Heisenberg scheme

T1 S0

Hubbard scheme

Figure 5: The graphic representation for the influence of covalent-ionic configuration interactions on the energy ratio of ∆ET Q : ∆E S T . spin-ladder cuprates indicated that the fourth-order correction (four-spin cyclic exchange) to the prototype Heisenberg Hamiltonian is necessary, with the interaction constant being up to 20% of the bilinear spin exchange constant. 67 For the sake of simplicity, a squared tetraradical model will be studied in this section. The

prototype Heisenberg model for this model will include both adjacent (J12 ) and diagonal (J13 ) spin interactions. The analytical solution of the six spin states is derived as follows, and since the adjacent constant J12 should be larger than the diagonal J13 , the energy level order should be,

= 0

E Q1

ET2 = ET2′ = 2J12 + 2J13

ES 1

= 2J12 + 4J13

ET1

= 4J12

ES 0

= 6J12

(8)

Evidently, the ratio of ∆E S 0

Q1

to ∆ET 1

Q1

from prototype Heisenberg theory is also exactly

equal to 1.5, which is just the same as in the cases of tetraradical 1 and 2. We notice that if the

diagonal spin exchange is neglected, the six spin states will be split into four energy levels with an equal interval of 2J12 , in sequence of S 0 , T 1 , S 1 , T 2 , T ′ , and the highest Q1 state. If the diagonal any longer, and the six spin states spin exchange is considered, S 1 , T 2 and T ′ will not degenerate 2 2


Page 21 of 41

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are split to five energy levels, with T 2 and T ′ 2 still degenerated due to symmetry constrain. Table 4: The vertical energy differences between six covalent states of tetraradical 3 in units of Kcal/mol, computed at the ground state equilibrium geometry.

S0

CASPT2(4,4) -12.3

NEVPT2(12,12) -11.9

Heisenberg a -12.3 (-0.4)

EHPT4b -11.9 (+0.0)

Hubbardc -11.9 (-0.0)

T1

-9.0

-8.7

-8.2 (+0.6)

-8.8 (-0.0)

-8.7 (+0.0)

S1 T 2 (T 2′ )

-4.7 -4.4

-4.5 -4.2

-4.4 (+0.0) -4.3 (-0.0)

-4.5 (+0.0) -4.2 (-0.0)

-4.5 (-0.0) -4.2 (+0.0)

0.0

0.0

0.0

0.0

0.0

1.36

1.36

1.50

1.36

1.37

Q1 ∆ES

d

Q /∆E T Q

a The relative energies computed with prototype Heisenberg model, with the spin-exchange constants J 12

= 2.0464 Kcal/mol and J13 = 0.0831 Kcal/mol, obtained from fitting to the NEVPT2 excitation energies. Discrepancy in excitation energies are in the parentheses. b The relative energies computed with the Effective Hamiltonian Perturbative Theory containing four-body operators, with the on-site repulsion U = 118.1566 Kcal/mol and the electron hopping integrals t12 = 11.3714 Kcal/mol, t13 = 0, which are fitted to the NEVPT2 excitation energies. Discrepancy in excitation energies are in the parentheses. c The relative energies computed with the Hubbard model, with on-site coulomb repulsion U = 92.0861 Kcal/mol, and electron hopping integrals t12 = 10.0639 Kcal/mol, t13 0, which are fitted to the NEVPT2 excitation energies. d The quintet state is taken as the zero point of the energy, as the previous example.

The equilibrium geometry of the ground singlet is optimized with the CASPT2(4,4) method, following a preliminary state-specific CASSCF(4,4) calculation in the active space spanned with four electrons in the four radical orbitals (available in the Supporting information). Then the excitation energies are computed on this geometry with NEVPT2(12,12) and CASPT2(4,4) methods, both of which are performed following preliminary state-averaged CASSCF calculations including six covalent spin states in the corresponding active space. One of the four equivalent radical orbitals, obtained with Boys localization of the pseudo-canonical CASSCF orbitals, is displayed in Figure 4b. We can find that each radical orbital of 3 is essentially a well-localized p-type orbital on one radical carbon atom, with a minor attendance of C H σ-bonds on the neighboring methylene groups. It is expected that, in comparison with the situation of tetraradical 2, the more-weakly-

overlapped radical orbitals should lead to much smaller energy splitting here. It could be found in Table 4 that the CASPT2 method within minimal active space agrees well with the NEVPT2 method within enlarged active space in computing the vertical energy differences



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

between the six spin states, which is similar to the case of tetraradical 2. The ratio of ∆E S ∆ET

Q

Page 22 of 41

Q

to

is reported as 1.36 from both NEVPT2 and CASPT2 calculations, which is much less

deviated from the “ideal” Heisenberg value of 1.5. The Heisenberg model presents much better

fitting accuracy here. Therefore, it is expected that the electron correlation related to the four radical centers in tetraradical 3 should be ascribed to the strong correlation condition. In addition, the Hubbard model can also perfectly reproduce the energy gaps with negligible errors. | tU12 | related to the correlation intensities between the nearest-neighboring radical centers is determined as 9.21 by fitting to the energy differences computed with the NEVPT2 method. Interestingly, the diagonal hopping t13 obtained by numerical fitting is almost zero. We notice that the nearest-neighboring radical separation in the ground state (S 0 ) equilibrium geometry of 3 is 2.75Å, √ which is approximately 0.4Å longer than in tetraradical 2. The diagonal radical separation is 2

times of the nearest-neighboring one. This large separation implies that the nearest-neighboring spin interaction through saturated CH2 groups should be the predominant effect, while the diagonal spin exchanges should be greatly suppressed. For this model example, accuracy improvement over the Heisenberg-type model is also achieved by the Effective Hamiltonian Perturbation Theory. We notice that this improvement is significant in comparison with the case of tetraradical 2, and the EHPT4 theory can fit the results of high-level ab initio theories perfectly with negligible deviation. These results imply that the perturbation expansion of Heisenberg-type effective Hamiltonian converges more rapidly in strong correlation regime than in weak to moderate correlation regimes. Accordingly, including the higher-order spin-exchange terms will be more effective in improving the accuracy of the Heisenberg-type model in this condition. Anyway, the prototype Heisenberg model is already good enough in qualitatively understanding the electronic structures of the six spin states in tetraradical 3. The validity of the Heisenberg model concerning tetraradical 3 could also be confirmed by inspecting the CASSCF wavefunctions of the six spin states, which are presented in Table 5. One can find that even in the S 0 state, which shows the highest ionicity, the covalent configurations still contribute more than 92% weight in the electronic wavefunction. T 1 is the second lowest state and


Page 23 of 41

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shows ca. 6% ionicity. Above them, the three near-degenerate states S 1 and T 2 , T ′ 2 show even lower ionicities of ca. 3%. Due to the negligible attendance of ionic configurations in these spin

states, it is not difficult to understand that the Heisenberg-type effective Hamiltonian is reliable for a balanced description of all these six spin states.

Table 5: The coefficients of covalent configurations in the wavefunctions of low-lying singlet and triplet states for tetraradical 3 constrained in D4h symmetry.a

(S z = 0) |12¯34¯ |1234 |1234 |1234 |12¯3¯4 |1¯234¯ Covalency%

S0

S1

0.5543 0.0000 0.5543 0.0000 -0.2771 0.4924 -0.2771 0.4924 -0.2771 -0.4924 -0.2771 -0.4924

(S z = 1) |123¯4 |1234 |1234 |1234

92.17% 96.99%

T1

T2

T 2′

0.4845 0.4928 -0.4845 0.4928 -0.4845 -0.4928 0.4845 -0.4928

0.4928 -0.4928 0.4928 -0.4928

93.89% 97.15%

97.15%

a Computations

are performed with state-averaged CASSCF(4,4) including six spin states, with the active space been constructed with four electrons and four radical orbitals. The listed covalent configurations have been transformed into the representation of four localized radical orbitals enumerated in (counter-)clockwise direction, as is marked in Figure 1. Covalency% is computed as the sum of squared coefficients of the covalent configurations.

3.4

Tetraradical on the spiro-phenalene framework

High spin multiradicals could be stabilized by spin delocalization, such as in the case of fused polybenzenic hydrocarbon polyradicals. 4,96 After saturating one vertex carbon atom, phenalene becomes a diradical, which can be used as the building block to construct radical clusters or polymers if only appropriate coupling units were selected. 96 Tetraradical 4 is such an example that has a spiro-phenalene framework, with a spiro-carbon atom being the spin-coupling bridge. The whole π-conjugated system is well separated into two sub-systems, each of which is confined in one 1,8dimethylenenaphthene (DMN) moiety. It is not difficult to understand from the Ovchinnikov’s rule that each DMN moiety has a high-spin diradical character. In addition, since the two delocalized radical orbitals of one moiety share most of the atomic orbitals, spin interaction between them is expected to be strong enough to open a wide energy gap between the triplet ground state and



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Page 24 of 41

the singlet spin state. This is confirmed through high-level ab initio computations of the T 0 S 1 energy gap on model system 4m, which is 30.02 Kcal/mol and 27.61 Kcal/mol at the levels of

NEVPT2(6,6)/6-311G(d) and CASPT2(6,6)/6-311G(d) respectively.

0

+1

0

+2

0

+1

0

-2

0

0

0

+2

-1

-1

0

-3

0

-3

0

+2

0

-2

+2

-2

The singly-occupied HMOs of the moiety

The localized radical orbitals of spiro-Phenalene

Figure 6: The four radical orbitals in tetraradical 4, obtained through Boys localization of the pseudo-canonical orbitals computed with CASSCF(4,4). The corresponding singly-occupied nonbonding Hückel Molecular Orbitals (HMOs) of the moiety are also placed for comparison.

The singly-occupied frontier orbitals of 4 shown in Figure 6 are obtained from Boys localization of the CASSCF(4,4) pseudo-canonical orbitals. Obvious similarity could be found between these orbitals and the radical orbitals of a DMN moiety, except that the radical orbitals in 4 have small tails on the side of the opposite DMN moiety. Accordingly, the inter-moiety spin coupling is mainly through the σ π hyper-conjugation between the conjugated π systems and the C C σ bonds connected to the spiro-carbon atom, which could bring about non-negligible kinetic ex-

change that prevails over spin polarization effect. 84 In consequence, the inter-moiety spin-coupling is anti-ferromagnetic, leading to a singlet ground state of the whole molecule. There are totally four kinds of spin-exchange constants need to be determined in order to set up the Heisenberg Hamiltonian, i.e., the ferromagnetic intra-moiety coupling constant J13 = J24 and three anti-ferromagnetic inter-moiety coupling constants J12 , J34 and J14 = J23 , the subscripts of which refer to the corresponding radical orbitals shown in Figure 6. Since the intra-moiety spin

couplings should be significantly larger and ferromagnetic, J13 and J24 are equal and positive, and the absolute value of them is much greater than the other three coupling constants. Analytical


Page 25 of 41

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solution of this model is available in the Supporting Information, and the first-order Maclaurin

expansions leading to simplified forms of energy expressions for the six spin states are listed as,

ES 0

23 (J12 + J 34) + 3J14

ES 1

4J 13 + 21 (J 12 + J 34) + J14

ET1

(J12 + J34 ) + 2J14

ET2

2J13 + (J12 + J34 )

(9)

ET3 = 2J13 + 2J14 .

With the prototype Heisenberg theory, we can formulate the correlation diagram of all these six states in the whole 4 molecule with the spin states in each DMN moiety. As is displayed in Figure 7, the six spin states of tetraradical 4 can be divided into 3 groups. Among them, S 0 , T 1 , and Q1 , which belong to the bottom group, originate from the spin coupling of the two triplet ground states of DMN moieties. In the middle group, T 2 and T 3 are related to the coupling of the S 1 state in one moiety with the T 0 state in the opposite. The S 1 state is on the top of these spin states, which corresponds to simultaneous excitation of both DMN moieties. When inter-moiety coupling constants are negligibly small compared to the intra-moiety coupling constant J13 , the three groups of states are split by 2J13 approximately. Correspondingly, with first-order Maclaurin expansion, the energy splitting between the three states in the lowest group, S 0 , T 1 , and Q1 , is irrelevant to the ferromagnetic coupling constant J13 . Moreover, we get the same relationship of energy gaps between the three lowest states as in the case of tetraradical 2 and 3. That is, ∆E S is 1.5 times of ∆ET

Q

Q.

On the other hand, since the intra-moiety coupling is ferromagnetic and is significantly larger than the antiferromagnetic coupling between two moieties, the tetraradical 4 can also be treated as an antiferromagnetic system constituted by two coupled S = 1 spin sites. We have followed the spin Hamiltonian in the S = 1 model space in the work of Moreira and his coworkers, 62 and introduce the on-site ferromagnetic effective exchange K for describing this molecule. We have found that the prototype S = 1 spin Heisenberg theory suggests a ratio of ∆E S


Q:

∆ET

Q

being


S1

Energy

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 41

≈2J13

S1

S1

T3

T2

2J13

≈2J13

2J13

Q1 T0 S0

T0

T1

L-Moiety

spiro-Phenalene

R-Moiety

Figure 7: The electronic states correlation diagram of spiro-phenalene and its DMN moieties.

1.5, which is coincident with the simplified Heisenberg interpretation of four S = 12 spin model. We have also noticed that, the biquadratic spin-exchange correction j(Sˆ 1 Sˆ 2 )2 to the prototype Heisenberg Hamiltonian is essential for S = 1 spin systems in many cases. 54,62–64,97,98

For the commonly-used full-configuration semi-empirical theories, as the prototype Hubbard model and the Pariser-Parr-Pople (PPP) model, 99–101 the orbital exchange integral K is neglected 2

due to the zero differential overlap (ZDO) approximation, always leading to a negative J = 2tU corresponding to the antiferromagnetic coupling between neighboring sites. Malrieu and his cowork

ers have pointed out that a positive orbital exchange integral K must be introduced in the Hubbard and PPP Hamiltonian to obtain the positive J for describing the ferromagnetic coupling as follows, 62–64

Hˆ ExtHubbard =

∑

∑ ti j

i∼ j

σ

∑ ∑ 1 ∑ † aˆ † iσ aˆ jσ + Uii nˆ iα nˆ iβ Ki j aˆ † a ˆ ˆ jσ′ aˆ iσ . iσ′ jσ a 2 ′ σσ i

(10)

i∼ j,FM

With this supplement, the extended Hubbard model can be used to discuss the tetraradical 4


Page 27 of 41

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containing ferromagnetic intra-moiety coupling,

J13 = K13

t213 U11

t2 13 . U33

(11)

The Hubbard Hamiltonian is block-diagonalized in the space of symmetry- and spin- adapted CS

F’s as follows, and can be easily solved either analytically or numerically.

|ΦCovI S ,A = 1

√

1 ¯ ¯ 12 (2|1234

¯ ¯ |1234 ¯ ¯ |1234 ¯ ¯ |1234 ¯ ¯ |1234 ¯¯ ) + 2|1234

1 ¯¯ ¯ ¯ ¯¯ ¯¯ |ΦSCovII ,A1 = 2 (|1234 + |1234 |1234 |1234 ) 1 ¯ ¯ ¯¯ ¯ ¯ ¯¯ |ΦSIon ,A1 = 2 (|1134 |1134 + |2234 |2234 ) 1 ¯ ¯ ¯ ¯ |ΦCovI T ,A1 = 2 (|1234 |1234 + |1234 |1234 ) 1 ¯ ¯ ¯ ¯ |ΦCovII T ,A1 = 2 (|1234 |1234 |1234 + |1234 ) √ 1 ¯ ¯ |ΦIon T ,A1 = 2 (|1134 + |2234 ) 1 ¯ ¯ ¯ ¯ |ΦCov T ,B2 = 2 (|1234 + |1234 |1234 |1234 ) √ 1 (|1134 ¯ |2234 ¯ ) |ΦIon T ,B2 = 2

|ΦCov = |1234 . of tetraradical 2 and 3, it is quite difficult to make an accurate Q,Athe Different from conditions 1

description for tetraradical 4 with high-level multireference electron-correlation ab initio theories, due to the remarkable interaction between radical electrons and the numerous π-electrons. A larger active space is elementary for a quantitatively-correct result. So we push both the CASPT2(12,12)/6311G(d) and the NEVPT2(12,12)/6-311G(d) calculations for vertical energy differences instead of

the computations within minimal active space. From examining their results, we consider that although the deviation between these two theories is larger than in the previous examples, they are yet in fairly good qualitative agreement on proportional relationships of the energy gaps. So we still accept the results of NEVPT2(12,12)/6-311G(d) calculation as the benchmark for evaluating the effectiveness of the semi-empirical methods. All of the computed relative energies of the six spin states are presented in Table 6, with every theory mentioned above, including the prototype Heisenberg model and the Hubbard model in S = 12 model space, the prototype Heisenberg model,



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

HˆSExtHubbard 2 =0,S =0,A z

ΦCovI S ,A1 |

1

|ΦSCovI ,A1

|ΦSCovII ,A1

2K13

0

|ΦSIon √ ,A1 3t12

0

2K13

t12

√ 3t12 |ΦTCovI ,A1

t12 |ΦTCovII ,A1

2K13 0 √ 2t12

0 0 √ 2t12

U11 K13 |ΦTIon √ ,A1 2t √ 12 2t12 U11 K13

Page 28 of 41

ΦCovII S ,A1

|

Φ Ion

S ,A1 |

HˆSExtHubbard 2 =2,S =1,A z

ΦCovI T ,A1 | CovII ΦT ,A1 | Φ Ion T ,A1 |

1

Hˆ SExtHubbard 2 =2,S =1,B z

ΦCov T ,B2 | Ion ΦT ,B2 |

2

|ΦCov T ,B2

|ΦIon T ,B2

0 0

0

Hˆ SExtHubbard 2 =6 S =2 A ΦCov Q,A1 |

1

|ΦCov Q A1

U11 K13

2K13

the EHPT4 model and the Hubbard model in S = 1 model space, and high-level ab initio methods

as CASPT2(12,12) and NEVPT2(12,12). From Table 6 we find that the ratio of ∆E S Q : ∆ET Q computed with CASPT2 and NEVPT2 methods are 1.31 and 1.27 respectively. All the semi-empirical methods are parameterized through fitting to the energy gaps from NEVPT2 calculation. The prototype Heisenberg model provides the ratio of ∆E S Q : ∆ET Q being 1.5 whether in S = 2 or in S = 1 model spaces. The biquadratic 1 because the number of the parameters to correction cannot be implemented in S = 1 model space be determined is more than the number of2 the energy relationship conditions. While in the more concise S = 1 space, the biquadratic form of Heisenberg-type effective Hamiltonian provide this ratio to be 1.51 and 1.68, lying on the fitting strategies to be selected. The biquadratic correction cannot provide balanced improvement for the fitting accuracy here if all the six spin states are fitted synchronously in the parametrization. This unexpected phenomenon implies imbalanced covalencies of the spin states in tetraradical 4, which may bring difficulties for Heisenberg-type theories. Finally, the Hubbard-type models in the S = 21 and S = 1 model spaces provide the ratios of 1.33 and 1.39 respectively, which are significantly better than the Heisenberg-type models.


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Table 6: The vertical energy differences of six spin states in units of Kcal/mol, which are computed at the ground state equilibrium geometry of tetraradical 4.

S = 12 model space

CASPT2(12,12)

NEVPT2(12,12)

S0 T1 Q1

-8.3 -6.4 0.0

-6.7 -5.3 0.0

-6.8(-0.1) -4.5(+0.8) 0.0

T2 T3

19.2 20.9

25.8 28.3

24.6(-1.2) 27.1(-1.2)

24.0(-1.8) 29.1(+0.7)

S1

41.1

52.7

54.0(+1.2)

53.2(+0.5)

∆ES Q /∆ET Q S = 1 model space

1.31 Heisenbergc

1.27 EHPT4d

1.50 EHPT4e

1.33 Hubbard f

S0 T1 Q1

-7.1(-0.4) -4.7(+0.6) 0.0

-7.1(-0.4) -4.7(+0.6) 0.0

-7.1(-0.4) -4.2(+1.1) 0.0

-6.6(+0.1) -4.8(+0.5) 0.0

T2 T3

25.8(+0.0) 25.8(-2.5)

25.8(+0.0) 25.8(-2.5)

26.0(+0.0) 26.0(-2.4)

24.4(-1.5) 28.7(+0.3)

S1

54.0(+1.3)

54.0(+1.3)

53.8(+1.1)

53.3(+0.6)

1.50

1.51

1.68

1.39

Heisenberg a

Hubbardb -6.4(+0.4) -4.8(+0.5) 0.0

∆ES

Q /∆E T Q

relative energies computed with prototype S = 12 Heisenberg model, with the spin-exchange constants J13 = 14.0614Kcal/mol, (J12 + J34 ) = 3.5397Kcal/mol and J14 = 0.5028Kcal/mol, which are fitted to the NEVPT2 excitation energies. Discrepancy in excitation energies are in the parentheses. b The relative energies computed with Malrieu-type S = 1 Hubbard model, with the on-site repulsion U 11 = 2 64.1539Kcal/mol, the exchange integral K13 = 14.5438Kcal/mol, and the hopping integral t12 = 13.2151Kcal/mol (the other hopping integrals t34 , t13 = t24 and t14 = t23 equal zero due to the mismatching symmetries of the local orbitals), which are fitted to the NEVPT2 excitation energies. Discrepancy in excitation energies are in the parentheses. c The relative energies computed with the concise Heisenberg model describing the coupling between two S = 1 spins, with the spin-exchange constant J = 2.3671Kcal/mol, and the orbital exchange integral K = 14.0928 Kcal/mol, which are fitted to the NEVPT2 excitation energies. Discrepancy in excitation energies are in the parentheses. d The relative energies computed with the Effective Hamiltonian Perturbation Theory of S = 1 spins containing biquadratic spin-spin interaction terms. The orbital exchange integral and the spin-exchange constant are taken from the previous fitting of the concise Heisenberg model, with the additionally fitted biquadratic exchange j = 0.0109Kcal/mol. Discrepancy in excitation energies are in the parentheses. e The relative energies computed with the Effective Hamiltonian Perturbation Theory of S = 1 spins containing biquadratic spin-spin interaction terms. All of the constants are fitted directly to the NEVPT2 excitation energies to reproduce a minimum RMS deviation for all the six spin states, finding the orbital exchange integral K = 13.9203Kcal/mol, the spin-exchange constant J = 2.1060Kcal/mol and the biquadratic exchange j = 0.2565Kcal/mol. Discrepancy in excitation energies are in the parentheses. f The relative energies computed with Hubbard-type model in the S = 1 model space. The Hamiltonian comes from the published work of Moreira and his coworkers, 62 with the on-site repulsion U = 202.3397Kcal/mol, the exchange integral K = 14.3259Kcal/mol, and the hopping integrals t = 20.4984Kcal/mol, t′ = 4.7737Kcal/mol, which are fitted to the NEVPT2 excitation energies. Discrepancy in excitation energies are in the parentheses. a The



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Page 30 of 41

Table 7: The coefficients of covalent configurations in the wavefunctions of low-lying singlet and triplet states for tetraradical 4 constrained in D2d symmetry.a

(S z = 0) |1¯23¯4 |1234 |1234 |1234 |1234 |1234 Covalency%

S0

-0.5510 -0.5510 -0.2932 -0.2932 -0.2577 -0.2577 91.19%

(S z = 1) T1 -0.0587 |12¯34 0.4245 -0.4245 -0.0587 |1234 0.5315 0.4769 |1234 0.4769 |1234 -0.5315 -0.4182 -0.4182 81.17% 92.54% S1

T2

T3

-0.5489 0.4994 0.5489 0.4994 0.3290 -0.4994 -0.3290 -0.4994

81.90% 99.77%

a Computations

are performed with the state-averaged CASSCF method, including completely 36 electronic states in the active space spanned with four electrons in the four radical orbitals. The listed covalent configurations have been transformed into the representation of localized radical orbital bases. Covalency% is computed as the sum of squared coefficients of the covalent configurations.

The discrepancy between the Heisenberg-type models and the multireference electron correlation methods might be ascribed to the absence of the ionic configurations in the model space of their effective Hamiltonian. According to the computed CASSCF wavefunctions presented in Table 7, we notice that tetraradical 4 shows very imbalanced ionicities of the spin states in comparison with the previously discussed 2 and 3. Among the lowest group of the states, S0 and T1 show slightly less than 9% weight of ionic configurations, while the Q1 is a pure covalent state. In the middle group, the T2 state shows about 18% ionicity, while the T3 is almost purely covalent. On top of them, S1 state shows the highest ionicity of up to ca. 19%. Considering that the imbalanced covalency of the spin states usually lowers the accuracy of

Heisenberg-type theories, here we want to make a further discussion for the electronic structure of tetraradical 4 with the full-configuration semi-empirical Hubbard model. Bastardis et. al. have detailedly studied the coupling between two S = 1 spin sites. 63,64 They have found that the deviation of the ∆E S

Q : ∆E T Q ratio from the

“ideal” Heisenberg model can be due to the imbalanced con-

tributions of the non-Hund configurations (as the singlet states of the moieties here) to the lowest

group of spin states. From the electronic structures calculated by Hubbard model we can learn further that this deviation could be mainly due to the imbalanced interactions between the lowest coCovI valent configurations (|ΦCovI S ,A1 and |Φ T ,A1 ) and the ionic configurations with the same irreducible


Page 31 of 41

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Ion representations (|ΦIon S ,A1 and |Φ T ,A1 ). The direct interactions between two pairs of covalent con-

ˆ ExtHubbard |ΦCovII = 0, ΦCovI |Hˆ ExtHubbard |ΦCovII = 0), figurations are relatively small ( ΦCovI S ,A1 |H S ,A1 T ,A1 T ,A1

while the two higher covalent configurations can influence the energy of the ground spin states weakly through the indirect interactions via the ionic configurations. Since the ionic configurations “contaminate” the covalent S0 and T1 spin states and lower their energy levels, the ratio of ∆E S

Q

: ∆ET

Q

is increased by this contamination, which cannot be reproduced by the

Heisenberg-type theories in pure covalent model space, as is encountered in tetraradical 2.

When analysing the CASSCF wavefunctions, we are surprised to find that the T2 and T3 states, similarly formed with a singlet of one moiety and a triplet of the opposite, have quite different ionicities. This obvious difference indicate a strong interaction between ionic and covalent configurations, and can be also well interpreted by the Hubbard model. It is easy to know that T1 and T2 states belong to A1 irreducible representation of D2d point group, while T3 belongs to B2 CovII symmetry. Besides the two covalent CSF’s (|ΦCovI T ,A1 and |Φ T ,A1 ), there exists another ionic CSF

(|ΦIon T ,A1 ) that also belongs to A1 symmetry. Obviously, configuration interactions between this ionic CSF and two covalent CSF’s increase considerably the ionicities of the covalent T1 (mainly CovII constituted by the lowest CSF |ΦCovI T ,A1 ) and T2 (mainly constituted by |Φ T ,A1 ) states, and create another ionic-configuration predominant triplet state T5 (mainly constituted by |ΦIon ). TheIonreais that, the B2 symmetry-adapted ionic ) son for the almost pure covalency of T3 state T ,A1 CSF (|Φ interacts hardly with the covalent CSF (|ΦCov ), due to the zero off-diagonal elements in the 2 2 ˆ ExtHubbard |ΦIon = 0). Accordingly, state T3 Hamiltonian matrix of the Hubbard model ( ΦCov T,B2 T ,B2|H T ,B2

( |ΦCov T,B2the other B2 triplet (T4 T ,B2 ) is almost purely covalent, while

|ΦIon T ,B2 ) is almost purely

ionic. The negligible “contamination” of ionic configuration in the wavefunction of T3 reminds us of the rationality of the ZDO approximation in the Hubbard model. We can find information of the configuration interaction from the off-diagonal elements of the extended Hubbard Hamiltonian. More detailed analysis is available in the Supporting Information. Since the interactions between covalent and ionic configurations lead to more diverse ionicities in the six spin states of tetraradical 4 than in the conditions of tetraradical 2 and 3 discussed pre-



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Page 32 of 41

viously, the Heisenberg-type models could hardly give a balanced description for all these six spin states here as good as before, which brings about much larger deviations from high-level ab initio methods in computing energy gaps. From these discussions we may also learn that the Hubbard model can reproduce the electronic structures as well as the energy gaps of the spin states semiquantitatively, and is helpful for understanding the electronic properties of the antiferromagnetic multiradicals.

3.5

The effectiveness of the Heisenberg-type model

At the end of this work, we want to summarize the relationship between the effectiveness of the Heisenberg-type model and the correlation intensity of the system. As is well known, the prototype Heisenberg Hamiltonian is equivalent to a second-order perturbation expansion of Hubbard ∑ ∑ small ea ˆ Hamiltonian. 35,39,71 This expansion demands the “perturbation term” i ∼ j ti j σ aˆ † iσ jσ

∑ nough comparison with the zeroth-order Hamiltonian ˆ iβ for its efficiency. is well iα n which the known,inthe effective Hamiltonian series could be regardedUas ai nˆpower series of |t/U |, in As 35,39 number of the summation terms increases sharply with the order of the expansion ascending. This indicates that the Heisenberg model containing higher-order terms as well as its prototype would be efficient only the correlation (|U/t|) is large enough. In the strong lation regime where thewhen correlation intensity intensity |U/t| is significantly large, the series can have correa fast convergence to Hubbard model. With the correlation intensity decreasing, its convergence rate decreases. Until the decreasing of the power series of |t/U | and the increasing of the summation terms reaching a balance, corresponding to somewhere in the weak to moderate regime, the series

will become divergent from then on. In this condition, both the prototype Heisenberg effective

Hamiltonian and its higher-order extended forms are invalid, as has been shown in Figure 2. From the previous calculations, we have already noticed that the characteristic correlation intensities of the anti-ferromagnetic tetraradical oligomers can range widely from strong to moderate correlation regimes, as is shown in Figure 8. So we can say that, although the widely-used empirical Heisenberg model is correct in reflecting the physics of magnetic systems where spin is


Page 33 of 41

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the predominant degree of freedom, its effectiveness and accuracy should be paid more attention

sometimes when the system is considered to be in the weak to moderate correlation regimes.

1

2

4

3

0.0

1.0 1.4

Weak correlation regime

2.0

2.8

4.0

5.7

8.0

Moderate correlation regime

11.3 16.0

∞

Strong correlation regime

Figure 8: The characteristic correlation intensities of the tetraradical examples in this work.

4

Conclusion

The anti-ferromagnetic spin-coupling exists not only in polyradical systems, but also in almost every covalent chemical bond, whether it is strong or weak. 86 The intensity of electron correlation (|U/t|) between unpaired electrons in a polyradical system may also range widely from strong to weak correlation regimes. In the strong correlation regime, the “perturbative” hopping integral t is

negligible with respect to the on-site coulomb repulsion integral U . As a result, the wavefunctions of the spin states show negligible weights of ionic configurations, hence the Heisenberg model is a good approximation to the full-configuration models. In such cases, the energy gaps between the spin states as well as the electronic wavefunctions can be well reproduced with the prototype Heisenberg model. However, in the moderate correlation regime, remarkable ionicities are usually witnessed in the electronic wavefunctions. Hence the accuracy of the Heisenberg model may break down due to the incompleteness of the configuration basis set, leading to qualitatively incorrect description of the energy gaps and the electronic structures of various spin states. In this condition, the biquadratic or even higher order terms of spin-exchange interaction could still introduce considerable corrections for the energy values through the form of perturbation, and should be introduced to increase the accuracy and to expand the applicability of the Heisenberg-type effec-

tive Hamiltonian theory. In the weak correlation regime, both the prototype Heisenberg model and



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Page 34 of 41

its extended forms might encounter difficulties, due to the failure of perturbation expansion they based on. Meanwhile, the full-configuration Hubbard model exhibits balanced accuracies from strong to weak correlation regimes, and can reproduce the electronic structures (especially for the ionicities of the states) correctly, which makes it more suitable for the study of anti-ferromagnetic coupling in polyradical systems. 102 Besides this, the PPP model including long range coulomb repulsions could also be recommended for discussing low-spin multi-radicals. 103 The only weakness

of these full-configuration models is their more-quickly-growing computational costs along with the increasing of molecular sizes.

Acknowledgement

This work is supported by China NSF (Grant Nos. 21173116, 21373109, 21473088), National Basic Research Program (Grant No. 2011CB808501). The authors are thankful to Prof. Shuhua Li, Prof. Jing Ma and Prof. Haibo Ma for their constructive suggestions on this work.

Supporting Information Available

Please see the Supporting Information for the derivation of the EHPT expansion, the comparison of the bases sets, the optimized geometries and the adiabatic energy differences of the molecules, the CASSCF wavefunctions and the ionicities of the spin states, as well as the analytical solutions of the Heisenberg model and the Hubbard model mentioned in this work. This material is available free of charge via the Internet at http://pubs.acs.org/.

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