Toward Molecular Magnets of Organic Origin via AnionâÏ Interaction

Article pubs.acs.org/JPCA

Toward Molecular Magnets of Organic Origin via Anion−π Interaction Involving m‑Aminyl Diradical: A Theoretical Study Published as part of The Journal of Physical Chemistry virtual special issue “Mark S. Gordon Festschrift”. Debojit Bhattacharya,† Suranjan Shil,‡ Anirban Misra,§ Laimutis Bytautas,*,∥ and Douglas J. Klein† †

Department of Marine Sciences, Texas A&M University at Galveston, Galveston, Texas 77553, United States Center for Atomic Scale Materials Design, Department of Physics, Technical University of Denmark, Kgs. Lyngby 2800, Denmark § Department of Chemistry, University of North Bengal, Darjeeling, PIN. 734013, West Bengal, India ∥ Department of Chemistry, Galveston College, 4015 Avenue Q, Galveston, Texas 77550, United States ‡

S Supporting Information *

ABSTRACT: Here we study a set of novel magnetic organic molecular species with different halide ions (fluoride, chloride, bromide) absorbed ∼2 Å above or below the center of an aromatic π-ring in an m-aminyl diradical. Focus is on the nature of anion−π interaction and its impact on magnetic properties, specifically on magnetic anisotropy and on intramolecular magnetic exchange coupling. In the development of single molecule magnets, magnetic anisotropy is considered to be the most influential factor. A new insight regarding the magnetic anisotropy that determines the barrier height for relaxation of magnetization of m-aminyl diradical-derived anionic complexes is obtained from calculations of the axial zero-field-splitting (ZFS) parameter D. The noncovalent anion−π interaction strongly influences magnetic anisotropy in m-aminyl−halide diradical complexes. In particular, the change of D values from positive (for the m-aminyl diradical, m-aminyl diradical/fluoride, and maminyl diradical/chloride complexes) to negative D-values in m-aminyl diradical complexes containing bromide signals a change from oblate to prolate type of spin-density distribution. Furthermore, the noncovalent halide−π interactions lead to large values of intramolecular magnetic exchange coupling coefficients J exhibiting a ferromagnetic sign. The magnitude of J steadily increases going from anionic complexes containing fluoride to chloride and then to bromide. Relations are sought between the magnetic exchange coupling coefficients J and aromaticity, namely structural HOMA (harmonic oscillator model of aromaticity) and magnetic NICS (nucleus independent chemical shift) aromaticity indices, in particular, the NICSzz(+1) component. Finally, possible numerical checks on the conditions relating to validity of the well-known Yamaguchi’s formula for calculating the exchange coupling coefficient J in diradical systems are discussed.

1. INTRODUCTION Investigations of novel organic molecules having high-spin ground states have intensified in the last few decades yielding a significant number of exciting candidates for magnetic materials.1 The ability of such materials to display tunable macroscopic magnetic properties makes them of high interest for practical applications in spintronics and as memory or sensing materials,2−4 superconductivity,5,6 photomagnetic devices,7−9 and in other areas. Recent literature10−24 contains examples of diradical systems containing π-conjugated fragments as couplers between unpaired spin centers (SC). It is quite common to use (five- or six-membered) aromatic rings as couplers due to their ability to maintain near-planar geometries. Although the use of homoatomic (i.e., carbon-based) aromatic rings is well documented,10−15 the use of heteroatomic aromatic rings as couplers, in addition to carbon atoms also containing nitrogen, boron, sulfur, phosphorus, and other atoms, has been explored as well.16−24 It is of interest to note that, in general, in isomer classes of organic compounds the fraction of diradicals relative to nonradicals (closed-shell © XXXX American Chemical Society

systems) increases dramatically as the number of atoms increases (see, e.g., ref 25 for a case of acyclic conjugated polyenoids). Theoretical models26−28 that use low-level approximations, like Hartree−Fock (HF) theory, in general, are quite useful in describing large molecules at a low cost. The best performance of HF approximation is expected for systems of closed-shell type using restricted HF theory (RHF). For open-shell systems, a better description is often obtained from unrestricted or spinpolarized UHF theory where α and β electrons are allowed to occupy different orbitals in a single UHF determinant. More accurate energies (and densities) can be obtained using unrestricted (spin-polarized) density functional theory (DFT). Although the Kohn−Sham (KS) formulation29 of the density functional theory is conceptually and computationally very similar to the Hartree−Fock Theory, unrestricted KS-DFT Received: September 23, 2016 Revised: October 20, 2016 Published: October 20, 2016 A

DOI: 10.1021/acs.jpca.6b09666 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

coupling coefficient J. For a diradical, the high-spin state is triplet and the low-spin state is singlet, and J may be obtained as E(S = 1) − E(S = 0) = −2J. Thus, J > 0 indicates ferromagnetic character whereas J < 0 reflects antiferromagnetic character of the ground state. For π-conjugated planar bipartite systems the sign of J can be predicted using starred/unstarred neighboringsite alternation rule as pioneered by Ovchinnikov48 (see also refs 10, 11, and 33−41). On the one hand, in Kekulé-type molecules antiferromagnetic coupling with a large HOMO− LUMO gap is quite common with the HOMO doubly occupied (antiparallel-spins) and a singlet ground state (see, e.g., ref 49). On the other hand, non-Kekulé molecules, i.e., fully conjugated systems that cannot be represented by neighbor-paired Kekulé structures, can contain several π-nonbonded atoms.50−53 NonKekulé diradicals have two nearly degenerate singly occupied frontier MOs (SOMOs).44−46 According to Borden and Davidson,54 the two SOMOs can be either nondisjoint (spatially coincident on some of the atoms) or disjoint (with all amplitudes on different atoms). Borden and Davidson argue that for nondisjoint nonbonding SOMOs, exchange coupling is of ferromagnetic (Hund-rule) type, whereas for disjoint SOMOs antiferromagnetic exchange coupling often occurs (particularly when the two SOMO amplitudes are on adjacent atoms). Though such singlet ground states with disjoint SOMOs is observed in many systems,44,54 there are exceptions (see, e.g., refs 19 and 22), where a refined rule10,11 predicts antiferromagnetic or ferromagnetic character depending on the (shortest) lengths of coupling pathways between disjoint orbital amplitudes (thereby bringing the predictions for alternant systems into coincidence with Ovchinnikov’s rule48). Thus, the disjoint/nondisjoint SOMO’s rules should be used with care, especially, in nonalternant or heteroatomic systems.55 A remarkable feature observed in π-conjugated diradicals is correlation between the aromaticity of a central fragment (the coupler) and the ground-state multiplicity of the overall structure where two radical centers are separated by this coupler. Indeed, in many studies56−67 it has been observed that the increase in a degree of aromaticity for an aromatic coupler linking two radical centers is associated with the enhancement of high-spin (ferromagnetic) character in diradicals.18,21−23,68−72 In 2015, Ito and Nakano23 on the basis of the heteroacene models involving nitrogen atoms demonstrated that the increase in the aromaticity of the central ring(s) favors the diradical form over the zwitterionic form of resonance structures in their heteroacene systems. This evidence23 suggests that the aromaticity of central rings enhances the diradical character of nitrogen-containing heteroacenes where the unpaired spin densities are localized at nitrogen atoms. In the present study we explore the link between the exchange coupling constant J and aromaticity in terms of several aromaticity indices. We emphasize that aromaticity is, in general, a multidimensional or partially ordered characteristic73−78 (see also a comprehensive discussion by Schleyer and co-workers on this subject in ref 64 and also in recent reviews79−81). In general, aromaticity criteria can be structural, energetic, magnetic, electric, or reactivity-oriented. It is natural to expect that the exchange coupling coefficient J would correlate closely with magneto-electric descriptors, such as nuclear independent chemical shifts (NICS)64,82 or ring currents.83,84 Of the various NICS indices, the NICSzz,66,67,82 a zz-component of the NICS tensor has been found to correlate quite well with the ring currents.83,84 Also, our recent study18 found an excellent correlation between the NICSzz values and

is often more successful in describing open-shell systems, like diradicals. Individual KS-DFT implementations (or methods) vary from each other on the basis of specif ic choices for a functional form corresponding to the exchange−correlation energy.26,27 The frequent success of such DFT functionals notwithstanding, more sophisticated electronic structure theories, like multiconfiguration pair-density functional theory (MC-PDFT), sometimes are required to describe large molecules at a satisfactory level, still seeking to keep the computational cost at a minimum to make computations tractable.30 A rigorous description of molecular electronic structure within a molecular orbital (MO) framework can be obtained using the full configuration interaction (FCI) approach where a wave function of the system is represented by a superposition of all possible Slater determinants (configurations). For example, at the zeroth order the electronic structure of closedshell molecules at their equilibrium geometries is ordinarily well described by a wave function that is represented by a superposition of Slater determinants where all MOs are doubly occupied.31,32 If an ab initio wave function is dominated by a single closed-shell determinant, then the natural orbital occupation numbers are close to either 2 (doubly occupied) or 0 (unoccupied), indicating a singlet ground state. However, in more complex situations, e.g., in molecular dissociations, when a wave function has many important configurations, the NO occupation numbers can deviate drastically from either 2 or 0. A typical illustration is dissociation of the ground-state H2 molecule using a minimal basis set28 when the wave function can be well represented by two doubly occupied configurations. At the equilibrium distance, the natural orbital occupation numbers are 2 (bonding-MO) and 0 (antibonding-MO), whereas at large internuclear distances, the NO occupation numbers are 1 (bonding-MO) and 1 (antibonding-MO) representing two singly occupied hydrogen atomic orbitals as expected. Here, taking plus and minus combinations of bonding and antibonding molecular orbitals yields localized, atomic-like orbitals on each atom. Radical character can also arise on graphene nanoribbons with unpaired electron density localized near edges.33−41 In wave function-based approaches, the degree of polyradical character of such systems can be well represented by the deviation of the natural orbital occupation numbers from 2 (doubly occupied) and 0 (unoccupied), as shown by Lischka and co-workers.39−41 In contrast to closed-shell singlet-state (S = 0) molecules, the unpaired spin alignment42−44 in high-spin molecules can lead to a net magnetic moment (e.g., for a triplet state with S = 1). Here, S represents the total spin quantum number, and the spin multiplicity can be calculated as 2S + 1, yielding, for example, 3 for triplet (S = 1) and 1 for singlet (S = 0) electronic states. Diradicals45,46 having triplet ground states are popular potential candidates for the theoretical design of molecular magnets. Recent experimental research efforts by Rajca and coworkers44,47 have focused on organic molecules possessing high-spin ground states that are significantly lower (at least 2−3 kcal/mol) in energy compared to their excited low-spin states. In the design of high-spin organic molecules, it may be desirable44 to connect radical fragments or radical centers in such a way that the unpaired spins would couple ferromagnetically producing high-spin ground states (S ≥ 1). If the ferromagnetic spin coupling is strong, then the high-spin ground state is well separated from the low-spin excited states as reflected by a large (positive) magnitude of the exchangeB


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The Journal of Physical Chemistry A the exchange coupling constant J for a number of diradicals with heteroatomic couplers (containing boron and nitrogen atoms). In general, negative values of NICS indicate aromaticity whereas positive values of NICS reflect antiaromatic character. One should keep in mind, however, that in some special cases, for example, involving the d-orbitals of transition metal atoms, the correlation between structural aromaticity index like HOMA and its magnetic counterpart NICS might exhibit opposing trends.85,86 Also, some findings87,88 seem to suggest that negative values of NICS may not reflect aromatic character in some specialized cases, most notably involving transition metals. Nevertheless, with a possible exception of such special cases, the overall usefulness of the NICS index in describing aromaticity and its correlation with the J coefficient for organic systems has been well documented.18,64,82 Furthermore, as has been shown recently in ab initio studies by Trinquier and Malrieu,89 aromaticity can prevent electron pairing and impose polyradical character in certain conjugated alternant hydrocarbons. Indeed, such is the (common) case of the higher polyacenes, which, though having a singlet ground state (according to Ovchinnikov’s rule48), demonstrate90 a strong radicaloid character in central rings with a countervailing aromatic character in noncentral rings. A similar effect accounts36 for unpaired π-electron density near suitable (e.g., “zig-zag”) edges of graphene. A promising avenue to enhance ferromagnetic coupling between unpaired spin centers connected by a π-conjugated coupler is to moderate it via noncovalent ion−π interaction. Ali and Oppeneer reported68 that at equilibrium geometries the interactions between an anion (F−, Cl−, and Br−) and coupler fragment in 1,3-phenylene-based bis(aminoxyl) diradical yield a significant increase in exchange coupling constant J, whereas the interaction between a cation (Li+, Na+, K+) and coupler fragments in 1,3-phenylene-based bis(aminoxyl) diradical yields a slight decrease in J values, making these systems less ferromagnetic. Thus, this study68 suggests that the use of interaction between anion and π-conjugated coupler can be an effective way to enhance ferromagnetic interaction between radical centers. Inspired by the result of Ali and Oppeneer,68 in this work we pursue the anion/coupler noncovalent interaction strategy in exploring the effect of anions not only in enhancing ferromagnetism but also, most importantly, in influencing magnetic anisotropy of organic diradical complexes. Although Ali and Oppeneer68 discussed the influence of noncovalent cation/anion−π interactions on the magnetic exchange phenomenon in the case of 1,3-phenylene-based bis (aminoxyl) diradical, these authors have not explored magnetic anisotropy. Magnetic anisotropy is considered to be the most influential factor in the design of single molecule magnets because it controls the barrier height for relaxation of magnetization.91−106 Due to the critical importance of magnetic anisotropy in development of molecular magnets, in this study we focus on the magnetic anisotropy in terms of the axial zero-field-splitting (ZFS) parameter D. To this end, we choose m-aminyl diradical as our parent system (Figure 1), which has been recently isolated by Rajca and co-workers.12 Starting with the parent structure of the m-aminyl diradical, we explore magnetic properties in its original form as well as under the influence of noncovalent interactions due to the presence of halides F−, Cl−, and Br−. In the m-aminyl diradical complexes the halide ions F−, Cl−, and Br− are located directly above the center of the six-membered ring (A) (Figure 2) of the parent

Figure 1. m-Aminyl diradical synthesized by Rajca and co-workers12 taken as a parent diradical in this study. Ring “A” is the aromatic sextet above (and below) which the halides (F−, Cl−, and Br−) are placed and n-Pr represents the n-propyl group.

Figure 2. Optimized distances (d and d*) of the anions from the center of the π-ring surface. The arrow shows the direction of charge transfer from halide ions to the sextet ring (A). Here, X is the representative of halide ions. (a) depicts anion−π interaction when a halide is placed above the aromatic π-ring. (b) represents a situation when two halides of the same type are placed on both sides of the aromatic π-ring.

diradical (Figure 1) at a distance of about 2 Å. In addition, we also consider a Br−/m-aminyl diradical/Br− complex with the aminyl diradical being sandwiched by two Br− ions. It should be mentioned that, in general, noncovalent interaction between an electron deficient π-system and an anion is of importance in supramolecular chemistry,107 playing a major role not only in solid state but also in solution.108 The common classical charge transfer of anion−π systems offers an exciting avenue to investigate the magnetic properties of various molecular complexes, and X-ray crystallography may be used to demonstrate anion−π interactions.109,110 The judicial design of efficient anion receptors is an important task in investigations of environmental, biological and medical significance.111 Indeed, various ion interactions with neutral closed-shell systems as well as diradicals have been a subject of intense research for many years. For example, pH levels are highly relevant to the stability of certain organomercury compounds112 and can also influence high-to-low-spin ground-state switchover in some organic compounds.113,114 Thus, the current investigation of the noncovalent interactions between various halide anions and the central πring in the m-aminyl diradical12 should be of interest, keeping in mind its potential use as strong organic magnets. C


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2. COMPUTATIONAL DETAILS The halide anions (F−, Cl−, and Br−) are placed at about 2 Å above (and below in the case of two Br− ions) the ring center of the parent diradical to produce four halide/m-aminyl diradical complexes (Figures 1 and 2 and Table 1). The geometries of

core do not span the same space. Moreover, the a and b orbitals do not span the same space (for the nT and BS cases), which is weaker than requiring that the canonical SCF-like orbtials be the same in these two cases, it being understood that the canonicalness condition of most computations enable mixings of these two near-degenerate SOMOs. There are different possible checks that the Yamaguchi conditions are close to being satisfied. For instance, the expectation ⟨nT|(S⃗)·(S⃗)|nT⟩ = (S2̂ )nT should be close to that for a (pure) triplet, namely, S(S + 1) = 2. And further, ⟨BS|(S⃗)· (S⃗)|BS⟩ = (Ŝ2)BS should be close to that for a state that is half-and-half triplet and singlet, namely 1 1 1(1 + 1) + 2 0(0 + 1) = 1. Another check involves consid2 eration of the fractions wS of spin-S contributions to expectation values that are made by the different wave functions (to be used in the Yamaguchi formula). At least for the nominal triplet |nT⟩, one can bound its excess-spin weight (i.e., its net weights exceeding S = 1) w>1 = w2 + w3 + ..., if one has available the expectation ⟨Ŝ2⟩nT. Indeed,

Table 1. Optimized Distances (Å) from the Center of the Ring “A” (Figure 1)a diradical 2 3 4 5

(parent−F−) (parent−Cl−) (parent−Br−) (Br−−parent−Br−)

UB3LYP

UM06-2X

2.47625 3.38430 3.34714 3.66740/3.93132

2.58004 3.04066 3.17501 3.34411/3.43508

Basis sets are 6-31G(d,p). Here, “parent” structure implies m-aminyl diradical.

a

these five diradicals (parent, parent−F−, parent−Cl−, parent− Br−, Br−−parent−Br−; here “parent” structure implies m-aminyl diradical) are fully optimized at the UB3LYP/6-31G(d,p)115−118 level of theory. Further, the geometries of these five diradicals have been fully optimized using the M06-2X functional.119,120 The magnetic exchange interaction between two spin centers is expressed by the Heisenberg effective Hamiltonian, Ĥ = −2JS1̂ ·S2̂

2

⟨S ̂ ⟩nT =

S≥1

1 2̂ 1 ⟨S − 2⟩nT = 4 4

(1)

(E BS − ET ) 2 2 (⟨S ̂ ⟩T − ⟨S ̂ ⟩BS )

=

(3)

Note that when the ⟨Ŝ2⟩T value is very close to 2, eq 1 leads to ΔES−T ≈ 2J. The Yamaguchi formula of eq 2 is based on assumptions, that there are nominal triplet (nT) and broken-symmetry (BS) states of a form and

∑ wS(nT)· S(S + 1) − 2 4

(6)

1 2̂ 1 ⟨S ⟩nT − 4 2

(7)

Many other (partial) checks are also conceivable, but the S2̂ expectations and w>1 values are ordinarily available in most of quantum chemical packages in use today. The degeneracy of the triplet (S = 1) microstates MS = +1, 0, −1 at the nonrelativistic level of theory can be lifted upon the inclusion of dipolar spin−spin couplings.129 This effect is known as the zero-field-splitting (ZFS). In the DFT framework, the ZFS tensor may be obtained using the Pederson and Khanna method.130 The majority of calculations performed in this study are done by the Gaussian’09W131 quantum chemical package, whereas the axial ZFS parameter D is calculated by the ORCA132 package of programs using UB3LYP functional and 6-31G(d,p) basis set (unrestricted Kohn−Sham DFT framework). The ZFS value arising from the spin−spin (SS) interactions can be estimated through an effective spin Hamiltonian Ĥ ZFS = Σ μ,ν D μν Ŝ μ Ŝ ν , where D μν are the components of the ZFS tensor, Ŝμ is the μth Cartesian component of the total electron spin operator.133 The ZFS129,133−139 axial parameter D is determined from the tensor 1 components as D = Dzz − 2 (Dxx + Dyy) The method of Pederson and Khanna is known to yield a correct sign of the ZFS parameter D.133−137 The zero-field splitting arises from anisotropic electron spin distribution, with the axial parameter D reflecting the energetic separation between MS levels 0 and ±1. If two unpaired spins are localized in parts of a molecule that are separated by a large distance, the magnitude of D is expected to be small (because of its dipolar distance

where EBS and ET are energies of the broken-symmetry singlet (BS) and the unrestricted (nominally) triplet states. Here, ⟨S2̂ ⟩T and ⟨Ŝ2⟩BS are the corresponding average spin-square values. One can estimate the energy separation between the spinprojected singlet and the unrestricted DFT triplet (pure singlet−triplet splitting ΔES−T = ES − ET), by applying Ginsberg’s128 formula:

|nT⟩ = |core × aα × bα⟩

S≥1

But as all the coefficients in this last sum are ≥1, it is seen that this last sum is ≥w>1, for our nT state. That is,

(2)

2

(5)

∑ {S(S + 1) − 2}·wS(nT)

S≥2

w> 1(nT) ≤

ΔES − T = ⟨S ̂ ⟩T × J

S≥2

and

where Ŝ1 and S2̂ are local spin operators on the two component radicals and the singlet−triplet energy gap is given by 2J. The positive and negative sign of J reflects ferromagnetic and antiferromagnetic interactions, respectively.121−125 The J values were calculated using the well-known Yamaguchi formula126,127 J=

∑ S(S + 1)·wS = 2w1 + ∑ wS·S(S + 1)

|BS⟩ = |core × aα × bβ⟩ (4)

where the cores are completely singlet and are the same for each of these two states. As a result, |nT⟩ is a pure triplet and |BS⟩ is an equal mixture of singlet and triplet, and yet further the Yamaguchi formula then gives the splitting between the singlet and triplet. The UHF wave functions, |nT⟩ and |BS⟩ are not precisely of this form in eq 4, because the core is different in the two cases, the core is not a pure singlet, and the spaces spanned by the (spin-free) a and b orbitals are different in the two cases. Indeed, in computational practice the core is in each case a single Slater determinant, and the spin−σ orbitals in the D


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coefficient (spin-coupling constant) J is one of the most frequently used quantities for describing magnetic properties of molecular systems. On the one hand, when the J value is large and positive, it means that triplet ground state is well below the lowest singlet state. On the other hand, when the value of J is negative, it implies that the singlet state is the ground state. The data for exchange coupling coefficients J at the UB3LYP/631G(d,p) level of theory are displayed in Table 2. Also, in

dependence). The value of D is expected to be positive for an oblate spin distribution where a flattening in one direction is observed, whereas D should be negative for a prolate spindensity distribution. For a positive D value the components MS = ±1 are higher in energy than the MS = 0 level, whereas for negative D the components MS = ±1 are lower than the MS = 0 component. To quantify the degree of aromaticity of the central ring in our halide/m-aminyl diradical complexes, we have used several aromaticity indices, namely, the harmonic oscillator model of aromaticity (HOMA) and different nuclear independent chemical shifts (NICS). We have calculated the harmonic oscillator model of aromaticity as an effective, structure-based aromaticity index. The HOMA index140,141 is defined as α HOMA = 1 − n

Table 2. Optimized UB3LYP Level Absolute Energies (E, hartree), ⟨S2⟩, and Intramolecular Magnetic Exchange Coupling Constants (J, cm−1) Using the 6-31G(d,p) Basis Set for All the Diradicalsa UB3LYP/6-31G(d,p)

n

diradical

2

∑ (R 0 − R i ) i=1

1 (parent)

(8)

where α is an empirical constant for a particular bond in a π system, n is the number of bonds taken in the summation, Ri is the calculated length of a specific bond in a given ring, and R0 is the optimal aromatic bond length taken from literature. Here, we use α = 98.89 Å−2 and R0 = 1.397 Å to calculate the HOMA index of a central ring in m-aminyl diradical-complexes. We have also calculated the NICS index which represents a magnetic index of aromaticity, as proposed by Schleyer and coworkers.63 The aromaticity index NICS represents a negatively signed absolute shielding measured at any desired point of a given system. A more negative value of NICS implies a higher degree of aromaticity. We used gauge independent atomic orbital (GIAO) methodology142 to calculate our NICS data for the ring labeled as “A” (Figure 1). The NICS(0) symbol implies that the probe is placed at the center of the ring, the NICS(+1) symbol means that the probe is 1 Å above the ring center on the same side as the halide ion, and the NICS(−1) symbol implies that the probe is on the opposite side (at −1 Å from the ring center) of the halide ion. The reason behind the calculation of the NICS index at 1 Å above the ring is that the effect of the ring current due to π-electrons is larger and also the effect of σ aromaticity is negligible at this distance. We have also calculated the zz-component of NICS, i.e., NICSzz, namely NICSzz(+1) and NICSzz(−1). The value of NICSzz(+1) represents the zz-component of NICS tensor calculated at 1 Å above the ring center of m-aminyl diradical complex, and the value of NICSzz(−1) is that 1 Å below the ring center. In our earlier studies18 we found that the NICSzz(+1) index has exhibited a strong correlation with the exchange coupling coefficient J as well as with aromaticity index HOMA whereas no such correlations have been observed using the total NICS(+1) index. The advantage of the zz-component NICSzz compared to the total NICS index in describing aromaticity of π-delocalized systems has also been emphasized by Schleyer and co-workers.64 Thus, in this study we shall focus on the NICSzz(+1) component of the NICS index rather than total indices (NICS(0), NICS(−1), and NICS(+1)) to examine a possible correlation of with the exchange coupling constant J in our five diradical complexes.

2 (parent−F−) 3 (parent−Cl−) 4 (parent−Br−) 5 (Br−−parent−Br−) a

E ⟨S2⟩ E ⟨S2⟩ E ⟨S2⟩ E ⟨S2⟩ E ⟨S2⟩

BS

triplet

J, cm−1

−1356.64244 0.978 −1456.49149 1.024 −1816.91163 1.001 −3928.42952 1.013 −6500.14403 0.982

−1356.65351 2.068 −1456.50339 2.059 −1816.92437 2.064 −3928.44261 2.062 −6500.15755 2.066

2229 2524 2630 2739 2727

Here, “parent” structure implies the m-aminyl diradical.

Table 2 are listed the total energy values (in hartree) for brokenspin (BS) state and triplet electronic states as well as their corresponding ⟨S2̂ ⟩ values. It is of interest to note that for the singlet states we have ⟨Ŝ2⟩ = 0 whereas for triplet states one has ⟨Ŝ2⟩ = 2. As one can see from Table 2, for UHF triplet states we have ⟨S2̂ ⟩ values that are very close to 2 whereas for UHF broken-symmetry states the values are very close to 1, i.e., halfway between a singlet and a triplet. This then provides a good check for the validity of the Yamaguchi formula, and in addition, from formula 7 we see that our (nominal) triplet state has a weight fraction of nontriplet contribution of no more than ≈0.8% for our various molecular species. The last column in Table 2 lists the J values (cm−1) for each diradical, as calculated using eq 2. The spin-projected singlet and triplet energy splittings for each of our five diradicals can be estimated using eq 3. Although the UB3LYP/6-31G(d,p) level of theory has been used extensively in describing open-shell systems, it is of interest to explore the sensitivity of our calculations with respect to different DFT functionals and/or basis sets. To this end, we selected UM06-2X functional119,120 that performed quite well in our recent studies of cis−trans conformers of coupled bis-imino nitroxide diradicals.9 The UM06-2X calculations have been performed using 6-31G(d,p) and 6311++G(d,p) basis sets. The relevant data are displayed in Table 3. When comparing the J data for diradicals 1−5 in Tables 2 and 3, one can see that the neutral structure of maminyl diradical has J values that are 10−20% smaller compared to those of anionic complexes. Furthermore, using the 631G(d,p) basis set, UB3LYP level of theory yields J values that exhibit a very slight but increasing trend going from system 2 to 4, whereas the UM06-2X level of theory yields J values that are practically constant and identical for these complexes. Finally, the results for the UM06-2X level of theory and larger basis set, i.e., 6-311++G(d,p), show (Table 3) that all anionic species exhibit J values that are about 700 cm−1 larger (i.e., more

3. RESULTS AND DISCUSSION 3.1. Ground-State Multiplicity and Exchange Coupling Coefficients. Electronic structure theory based on DFT functionals has been used extensively for predicting magnetic properties of organic diradicals. The exchange coupling E


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Table 3. Total Energies (E, hartree), ⟨S2⟩, and Intramolecular Magnetic Exchange Coupling Constants (J, cm−1) Using the UM06-2X Level of Theory and 6-31G(d,p) and 6-311++G(d,p) Basis Sets for the Diradical Complexesa UM06-2X/6-31G(d,p) diradical 1 (parent) 2 (parent−F−) 3 (parent−Cl−) 4 (parent−Br−) 5 (Br−−parent−Br−) a

E ⟨S2⟩ E ⟨S2⟩ E ⟨S2⟩ E ⟨S2⟩ E ⟨S2⟩

UM06-2X/6-311++G(d,p)

BS

triplet

J

BS

triplet

J

−1356.00862 0.983 −1455.82961 1.007 −1816.25014 0.996 −3927.87599 1.003 −6499.66857 1.022

−1356.02089 2.070 −1455.84420 2.072 −1816.26469 2.069 −3927.89074 2.068 −6499.68551 2.071

2477

−1356.30877 0.977 −1456.19085 0.999 −1816.59888 0.987 −3930.58122 0.993 −6504.77994 0.980

−1356.32010 2.073 −1456.20510 2.076 −1816.61322 2.072 −3930.59573 2.072 −6504.79521 2.074

2269

3007 2976 3040 3544

2904 2901 2951 3063

Here, “parent” structure implies the m-aminyl diradical.

Figure 3. Spin density plots of m-aminyl diradical systems investigated in the present work at the UB3LYP/6-31G(d,p) level of theory. The hydrogen atoms are removed due to picture clarity. The mauve and the sea green colors represent (α) and (β) spin densities, respectively.

ferromagnetic) compared to the original m-aminyl diradical. Thus, these findings suggest that in the quest for diradical systems having large singlet−triplet gap and high stability at room temperature, the anionic m-aminyl diradical complexes should be of high practical interest, especially keeping in mind that these diradicals can be excellent candidates to serve as building blocks for high-spin polyradicals12,44 in the development of robust magnetic polymers.143−146

To summarize, all the results obtained using UB3LYP and UM06-2X functionals and basis sets described above clearly demonstrate that the halides F−, Cl−, and Br− significantly enhance the ferromagnetic character of the m-aminyl diradical via noncovalent interaction. For the sake of simplicity, in the present study we primarily focus on results obtained using the UB3LYP/6-31G(d,p) level of theory to discuss magnetic properties of our diradical systems. F


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The Journal of Physical Chemistry A 3.2. Spin-Density Distribution. Tables 2 and 3 display the data for the broken-symmetry state (BS) and nominal triplet for each of the five m-aminyl diradical-derived systems. One can estimate the energy separation between the spin-projected singlet and the unrestricted DFT triplet (representing pure singlet−triplet splitting ΔES−T = ES − ET), using the formula in eq 3 after the spin-coupling coefficient J is obtained from Yamaguchi’s formula in eq 2. Because J coefficients are positive for all five complexes, the positive value of ΔES−T indicates that triplets lie below singlet states, thus yielding a triplet as the ground state for all five systems. A further indication of the dominance of a triplet ground state in these systems can be seen from spin-density plots displayed in Figure 3. The spindensity plots in Figure 3 are determined as differences between (α) and (β) atomic spin density contributions. From the graphical representation of spin-density plots shown in Figure 3 one can observe an alternating spin-density distribution pattern in all five diradical complexes, a pattern characteristic of spinpolarization and triplet ground states.10,11,33,34,147 In terms of frontier molecular orbitals, different singly occupied molecular orbitals (SOMOs) may arise if the energetic separation between two SOMOs, having orbital energies ESMO2 and ESMO1 is sufficiently small. In particular, on the basis of arguments put forward by Hund that lead to the establishment of the well-known Hund’s rule,148 when the ΔESMO‑SMO = (ESMO2 − ESMO1) value is sufficiently small (i.e., when there is near-degeneracy), the ground state is expected to be a triplet. Note that all orbitals discussed in this study represent Kohn− Sham type29 molecular orbitals. SOMO1 and SOMO2, as well as LUMOs for all five diradical systems are shown in Figure 4. In 1968, Hoffmann, Zeiss, and Van Dine149 using extended Hückel calculations to provide an empirical criterion stating that if ΔESMO‑SMO < 1.5 eV, then two nonbonding electrons will favor occupying different molecular orbitals with parallel-spin orientations resulting in a triplet ground state. More recently, Constantinides and co-workers,147 based on calculations using B3LYP level of theory for a set of about 14 polyheteroacenes concluded that systems with ΔESMO‑SMO > 1.3 eV are clear singlets. Molecules with small ΔESMO‑SMO values have a tendency to have large positive ΔES−T = ES − ET (singlet− triplet splittings, here ES = singlet energy and ET = triplet energy) and vice versa, molecules with large ΔESOMO−SOMO values have small or negative ΔES−T’s. In the present study we find that SOMO−SOMO energy gaps are very small and for any of our diradical structures do not exceed 0.8 eV (SOMO1, SOMO2, and LUMO total orbital energies are listed in Figure 4) and, according to the criteria presented above, are consistent with the fact that these diradical structures have triplet ground states. 3.3. Aromaticity and Magnetism. Both aromaticity and magnetism are of fundamental significance in discussions of physical and chemical properties of molecular systems. Unlike magnetism, however, aromaticity is best described as a partially ordered quantity having a multidimensional character.73−78 It has been observed18,21−23,68−72 that there is a close correlation between the aromaticity of coupler fragments that link the unpaired-spin centers and exchange coupling coefficients J in diradicals. The majority of the evidence suggests that an increase in aromaticity of such couplers favors a ferromagnetic trend. Although various aromaticity descriptors exist in the literature, our recent study of borazine couplers18 suggests that NICSzz and HOMA indices should be well suited to describe a correlation between magnetic properties in terms of exchange

Figure 4. Molecular orbitals (SOMO1, SOMO2, and LUMO) of the diradicals (iso-value 0.02). Below each orbital its orbital energy is listed in electronvolts. The results correspond to the UB3LYP level of theory using the 6-31G(d,p) basis set. The maroon and the green color represent different signs (phases) of the orbital coefficients.

coupling constants J and aromaticity of the π-conjugated couplers in our diradical complexes. The most relevant among the NICS indices was found to be the NICSzz(+1) component. The relevant data are listed in Tables 2, 3, and 4, and displayed graphically in Figure 5, where panel A represents the (UB3LYP/6-31G(d,p)) data and panel B represents the (UM06-2X/6-31G(d,p)) results. It is clear that the (UB3LYP/6-31G(d,p)) data reflect a higher degree of correlation between J and NICSzz(+1) than the (UM06-2X/ 6-31G(d,p)) results. However, on the average, both DFT functionals exhibit the same trend, namely, that as NICSzz(+1) values become more negative, indicating an increasing degree of aromaticity of a coupler, the J values become larger, reflecting the increase in ferromagnetic character of a diradical complex. G


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Table 4. Various Nuclear Independent Chemical Shift (NICS) Values (ppm) for the Different Prototype Systems Considered in This Study Evaluated at Optimized Geometriesb at the UB3LYP/6-31G(d,p) Level of Theorya UB3LYP diradical 1 2 3 4 5

(parent) (parent−F−) (parent−Cl−) (parent−Br−) (Br−−parent−Br−)

UM06-2X

NICS(0)

NICS(+1)/ NICS(−1)

NICSzz(+1)/ NICSzz(−1)

NICS(0)

NICS(+1)/NICS(−1)

NICSzz(+1)/ NICSzz(−1)

−0.0817 0.6721 0.1489 0.2647 0.0236

−3.4775 −2.7190/−2.4734 −2.9489/−3.2563 −2.7853/−3.1840 −2.7549/−3.1779

−2.5392 −5.3613/−1.6173 −7.6200/−2.7172 −9.5351/−2.8847 −9.7402/−7.8475

−0.7747 1.3703 1.2237 1.4416 1.3660

−3.7064 −3.2721/−2.1103 −2.9965/−2.9069 −2.7370/−3.0000 −2.4849/−2.7454

−3.0239 −8.5134/−2.6352 −11.9708/−1.6903 −12.8532/−3.2519 −12.4035/−10.5995

Here, “parent” structure implies m-aminyl diradical. bThe NICS(0), NICS(+1), NICS(−1), NICSzz(+1) and NICSzz(−1) are the representative of the NICS values at the center, 1 Å below and 1 Å above of the sextet ring ‘A’ of the parent aminyl diradical. Note that 1 Å above of the ring “A” means that the dummy atom is placed in between the sextet “A” and the halide anion. NICS(+1) and NICSzz (+1) are the values of NICS when a probe is placed 1 Å above the plane of the ring toward the anion, but NICS(−1) and NICSzz(−1) are the values of NICS when a probe is placed 1 Å below the plane of the ring in opposite to the anion.

a

Figure 5. (A) Plot of exchange-coupling constants J (cm−1) versus NICSzz(+1) (units of ppm) at the UB3LYP/6-31G(d,p) level of theory. (B) Plot of exchange-coupling constants J (cm−1) versus NICSzz(+1) (units of ppm) at the UM06-2X/6-31G(d,p) level of theory. Here, NICSzz(+1) represents NICSzz values when a probe is placed 1 Å above the center of the hexagon ring “A” toward the anion (see text).

correlation between HOMA and the J constant (6-31G(d,p) basis set). Namely, as HOMA values come closer to 1 (highest degree of aromaticity), the values of J become larger, implying greater ferromagnetic character. Of course, as discussed in our earlier study,18 it is somewhat expected that the correlation between the exchange coupling coefficient J and various aromaticity indices, i.e., HOMA and NICSzz(+1), may not be perfect for complex heteroatomic systems, such as considered in this study. 3.4. Zero-Field Splitting. It is well recognized91−97 that magnetic anisotropy is the most influential factor in design of single-molecule magnets because it controls the barrier height for (slow) relaxation of magnetization. The sign and magnitude of the axial zero-field splitting parameter (D) plays a significant role101 in this context. As mentioned earlier, for the m-aminyl diradical Rajca and co-workers12 reported the experimental ZFS parameter D of 1.8 × 10−2 cm−1. The calculation of the axial parameter D using B3LYP/EPR-II level of theory yields a slightly larger value (relative to the experiment) of 3.42 × 10−2 cm−1. Despite this small deviation, the positive sign for D (>0) clearly indicates an oblate-like shape for the overall spin density (i.e., f lattening in one direction). Rajca et al.12 also noted that, by contrast, a similar system, m-phenylene−nitroxide diradical, exhibits a prolate-like overall spin density (i.e., elongated shape in

This observation indicates the same trend that has been observed in our earlier study18 for borazine couplers. The correlation between another aromaticity index, HOMA, and the exchange coupling constant J also indicates a reasonable correlation between these quantities for the set of our five diradical systems (see the data in Table 5 and Figure 6). The data obtained by both DFT functionals, i.e., UB3LYP and UM06-2X, exhibit a similar behavior in describing the Table 5. Harmonic Oscillator Model of Aromaticity (HOMA) Values Using UB3LYP/6-31G(d,p) and UM062X/6-31G(d,p) Levels of Theorya HOMA diradical 1 2 3 4 5


UB3LYP/6-31G(d,p)

UM06-2X/6-31G(d,p)

0.8566 0.8750 0.8806 0.8790 0.9158

0.8767 0.9070 0.8897 0.8897 0.9145

a

Results correspond to the six membered ring above which the anions are placed. HOMA model constants are α = 98.89 Å−2 and R0 = 1.397 Å (see text). Here, “parent” structure implies m-aminyl diradical. H


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Figure 6. (A) Plot of exchange-coupling constants J (cm−1) versus HOMA aromaticity indices at UB3LYP level of theory. (B) Plot of exchangecoupling constants J (cm−1) versus HOMA aromaticity indices at UM06-2X level of theory. All calculations use 6-31G(d,p) basis sets.

energy than the MS = 0 level.100,102 In general, especially for very high-spin S states, a negative value of the ZFS-parameter D assures that the maximum spin projection (MS = ±S) along the z-axis will be the lowest in energy, whereas smaller magnitude MS projections will be higher in energy. As a consequence, a classical rotation of the projection from MS = +S to MS = −S will require an excitation over a significant energy barrier.101−103 Thus, it is generally accepted that, in the absence of quantum tunneling, molecules having D < 0 values are of critical importance in the design of single-molecule or singlechain magnets because the ground state for such systems exhibits magnetic bistability.101 In particular, an energy barrier (also known as the magnetic anisotropy energy barrier) that separates MS = +S and MS = −S sublevels is equal102 to |S2·D| (for integer S). Thus, for the triplet state (S = 1), the energy barrier between sublevels MS = +1 and MS = −1 is equal to |2D|. For the parent m-aminyl diradical our calculated ZFS axial parameter D of 3.42 × 10−2 cm−1 practically coincides with the value reported by Rajca et al.12 The calculated values for the axial ZFS parameter D in m-aminyl diradical complexes 2 and 3 containing F− and Cl− ions, respectively, exhibit positive values of a similar magnitude of about 3.2 × 10−2 cm−1. On the contrary, for m-aminyl diradical/Br− and Br−/m-aminyl diradical/Br− complexes, the calculations yield negative values of −0.60 × 10−2 and −0.93 × 10−2 cm−1, respectively (Table 6). The change of axial parameter D values going from positive (for structures 1, 2, and 3) to negative in Br−-containing maminyl diradical complexes 4 and 5, indicates a transition from oblate-like overall spin-density to prolate-like overall spin-density distribution (Figure 3). Such oblate-to-prolate spin transition (with reorientation of the ZFS tensor) has been often discussed in the context of building conjugated diradical-based systems from monomers to polymer chains.101−106 Furthermore, the magnitude of the axial parameter D for Br−-containing maminyl diradical complexes 4 and 5 is signif icantly smaller compared to the values for structures representing parent maminyl diradical, F−-containing m-aminyl diradical, and Cl−containing m-aminyl diradical complexes, respectively. One can reason that the spin−spin interaction between unpaired spins makes use of bromide’s more diffused valence atomic orbitals

one direction) with a negative value of parameter D (D < 0). In fact, there are many examples of molecular systems exhibiting negative D values (see, e.g., refs 98−103). In this study, we perform ZFS calculations using the UB3LYP/6-31G(d,p) level of theory, reproducing the same value of axial parameter D for the parent structure, i.e., maminyl diradical, just as reported by Rajca and co-workers12 (Table 6). In addition, we calculated axial parameter D values Table 6. Static Zero-Field Splitting (ZFS) Axial Parameter D Values (cm−1) Obtained at the UB3LYP/6-31G(d,p) Level of Theorya diradical 1 2 3 4 5 a


D (cm−1) 3.42 3.11 3.20 −0.60 −0.93

× × × × ×

10−2 10−2 10−2 10−2 10−2

Here, “parent” structure implies m-aminyl diradical.

for anion-containing complexes 2, 3, 4, and 5 to investigate the extent of triplet-state delocalization and shape of overall spin density in these, as yet unexplored, systems. The ZFS interaction being approximated by a point-dipole model is useful in providing an estimate of the average interelectron distance between unpaired spins in the triplet state and, consequently, the extent of triplet-state delocalization. On the one hand, the magnitude of the axial ZFS parameter D is indicative of how strongly the unpaired spins interact in a triplet state. On the other hand, the sign of D indicates the shape of the overall spin density (oblate shape when D > 0 and prolate shape when D < 0). For the triplet (S = 1), the three magnetic sublevels MS = −1, 0, +1, split into 0 and ±1 levels due to the inclusion of spin−spin interaction between unpaired spins. If the zero-field-splitting axial parameter D has a negative sign, then the degeneracy in magnetic sublevels MS is removed by placing higher magnitude MS levels lower in energy, whereas the situation is reversed for the positive values of D. Thus, if D > 0, then MS = ±1 sublevels will be higher in energy compared to MS = 0; and if D < 0, the MS = ±1 sublevels will be lower in I


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design of single-molecule or single-chain organic magnets giving rise to a magnetically bistable ground state.101 The Br−/ m-aminyl diradical/Br− complex exhibits the negative D value of the largest magnitude. This result would suggest that the development of organic single molecule magnets can make use of atoms/ions having larger (more diffused) atomic orbitals in their electronic valence shells, e.g., Br− ions. An even more promising candidate to yield negative D values of large magnitude is the m-aminyl diradical−halide complex with two iodide (I− ions) sandwiching the m-aminyl “parent” structure. Furthermore, all five m-aminyl diradical-derived complexes display large and positive exchange coupling coefficient J values. An increase of J values is observed going from the neutral maminyl diradical (“parent”) structure to anionic m-aminyl diradical complexes involving the halides F−, Cl−, and Br−. A further reflection of high-spin (i.e., triplet) ground states of our diradical complexes was found in the alternating total spindensity distribution pattern, which is characteristic of spinpolarization and triplet ground states. A reasonable correlation has been observed between the exchange coupling coefficient J and aromaticity indices NICSzz(+1) and HOMA, which is consistent with our earlier study.18 Although studies of magnetic properties of organic aromatic molecules have a very long history,150 it is very satisfying to observe the manifestation of a relationship between magnetism and aromaticity in such diverse complexes as investigated in this study. This study reveals a somewhat different trend for J and D quantities when we compare our five diradicals derived from the m-aminyl “parent” structure. In particular, although the magnitude of the exchange coefficient J for the neutral m-aminyl diradical is much smaller compared to the anionic m-aminyl− halide complexes, by contrast, the calculated axial ZFS parameter D values display a very different trend. For the neutral m-aminyl diradical as well as the anionic m-aminyl diradical−halide complexes involving F− and Cl− ions, the ZFS parameter D values are almost identical and positive. However, the anionic m-aminyl diradical−halide complexes involving Br− ions exhibit negative and smaller D values. The extent of tripletstate spin-density distribution pattern changes considerably going from neutral m-aminyl diradical as well as m-aminyl diradical−halide complexes with F− and Cl− ions, to m-aminyl diradical/Br− and Br−/m-aminyl diradical/Br− complexes, highlighting the oblate-to-prolate spin-density transition with a reorientation of the ZFS tensor. Overall, the results presented in this study are expected to be useful in the rational design of high-spin organic molecular magnets.

reflecting the increase in an average distance between unpaired electron spins. As a result of the increase of an average distance between unpaired spins, the strength of the spin−spin interaction is reduced, which, in turn, narrows the splitting between MS = 0 and MS = ±1 components of the triplet state, as is reflected by the diminished absolute value of D in complexes containing bromide (Table 6 and Figure 3). The practical implications for design of single molecule magnets that involve the use of halide anions favor the use of atoms/ions having more diffused atomic orbitals in their electronic valence shells, namely, bromide ions. The Br−/m-aminyl diradical/Br− complex where the Br− ions are sandwiching the m-aminyl “parent” structure exhibits a negative D value of the largest magnitude. Because molecules having a negative D value of large magnitude are essential101 in the design of single-molecule or single-chain organic magnets, a very promising candidate is the m-aminyl diradical−halide complex with two iodide (I− ions) sandwiching the m-aminyl “parent” structure. Thus, it is beneficial to pursue a new strategy of building single molecule organic magnets having negative (and large magnitude) D values, and therefore large barrier heights for (slow) relaxation of magnetization by using heavier halide ions, for example, Br− or I− ions. Such external ionic species could be stabilized by inserting them in molecular crystals. It is worthwhile to discuss the overall spin-density distribution displayed in Figure 3. In particular, although large spin-density contributions can be seen on anionic species of Cl− and Br− in 3, 4, and 5, the spin-density contribution on F− in the complex 2 is rather small. The axial ZFS parameter D data (positive versus negative values) in Table 6 seem to reflect the fact that the spin−spin interactions are more similar in structures 1, 2, and 3 than in structures 4 and 5. In Figure 3 we observe that in m-aminyl diradical complexes with F−, Cl−, and Br−, the unpaired spin density on anion increases dramatically going from F− to Cl− and to Br−. We also observe that the unpaired spin-density type on halides is of the same type as on unpaired spin centers localized on the nitrogen atoms represented by blue-colored spheres in Figure 3. These results suggest that the presence of an external ion near aromatic coupler significantly affects the strength of the magnetic exchange interaction as can be seen in a steady increase of the magnitude of the magnetic exchange interaction constant J as we go from 1 to 5 structures, as shown in Table 2 (UB3LYP/6-31G(d,p)). This sequence correlates well with the steady increase of the unpaired-spin density on halide ions going from structures 1 to 5, as seen in Figure 3 (UB3LYP/6-31G(d,p)).

4. CONCLUSIONS In this investigation we explored the nature of noncovalent anion−π interaction and its impact on magnetic properties, specifically on magnetic anisotropy and on intramolecular magnetic exchange coupling. We find that noncovalent anion−π interaction strongly influences magnetic anisotropy in m-aminyl−halide diradical complexes, which is a significant new result, keeping in mind that for single molecule magnets magnetic anisotropy is the most influential factor. This study demonstrates that the sign and magnitude of the axial zerofield-splitting (ZFS) parameter D depends on type and geometrical arrangement of halide ions positioned above (or above and below) the central ring of the m-aminyl diradical. Only the anionic m-aminyl diradical−halide complexes involving Br− ions exhibit negative D values. This fact is important because molecules having D < 0 are essential in the

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.6b09666. Mulliken charges in the halide ions (in electron charge units, |e|) for selected DFT functionals, optimized XYZ coordinates and the respective optimized geometries of all the magnetic species in the gas phase, Mulliken charge distribution structures, complete reference to the Gaussian’09 W package of quantum chemistry programs (PDF) J


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AUTHOR INFORMATION

Corresponding Author

*L. Bytautas. E-mail: [email protected]. Phone number: 1-409944-1273. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Md. Ehesan Ali for useful discussions and interest in this work. D.B. and D.J.K. acknowledge the support (via grant BD-0894) of the Welch Foundation, of Houston, Texas, USA. A.M. is grateful to DST, India, for financial support.

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Across the Periodic Table; Frenking, G., Shaik, S., Ed.; Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim, Germany, 2014; doi 10.1002/ 9783527664658.ch13.

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