High Magnetic Exchange Coupling Constants: A Density Functional

Aug 13, 2012 - Daeheum Cho , Kyoung Chul Ko , Jin Yong Lee. International Journal of Quantum Chemistry 2016 116 (10.1002/qua.v116.8), 578-597 ...
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High Magnetic Exchange Coupling Constants: A Density Functional Theory Based Study of Substituted Schlenk Diradicals Iqbal A. Latif, Shekhar Hansda, and Sambhu N. Datta* Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India S Supporting Information *

ABSTRACT: The Schlenk diradical has been known since 1915. After a detailed experimental work by Rajca, its magnetic nature has remained more or less unexplored. We have investigated by quantum chemical calculations the nature of magnetic coupling in 11 substituted Schlenk diradicals. Substitution has been considered at the fifth carbon atom of the meta-phenylene moiety. The UB3LYP method has been used to study 12 diradicals including the original one. The 6-311G(d,p) basis set has been employed for optimization of molecular geometry in both singlet and triplet states for each species. The singlet optimization has led to the optimization of the broken-symmetry structure for 10 species including the unsubstituted one. This development makes it possible to carry out further broken symmetry calculations in two ways. The triplet calculation has been done using 6311++G(d,p) basis set and the optimized triplet geometry in both procedures. The broken symmetry calculations have used the optimized geometries of either the triplet states or the broken symmetry solutions. The first method leads to the prediction of electron paramagnetic resonance (EPR) compatible magnetic exchange coupling constant (J) in the range 517−617 cm−1. A direct optimization of the broken symmetry geometry gives rise to a lower estimate of J, in the range of 411−525 cm−1 and compatible with macroscopic Curie studies. The calculated J for the unsubstituted Schlenk diradical is 512 cm−1 that can be compared with 455 cm−1 estimated by Rajca. In both cases, introduction of groups with +M and +I effects (Ingold’s notation) decreases the J value from that for the unsubstituted Schlenk diradical while −I and −M groups at the same position increases J. These trends have been explained in terms of Hammett constants, atomic spin densities, and dihedral angles.

I. INTRODUCTION There has been a large number of theoretical and experimental investigations on high-spin organic diradicals based on the mphenylene moiety.1−15 The latter group has been extensively used as an effective ferromagnetic coupler. When connected between radical sites such as carbon-centered1,8,16 and nitrogencentered17 spin carriers, carbenes,18 and nitrenes,19 the resulting diradical almost invariably has a triplet ground state. In some of the m-phenylene based diradicals, E(S)−E(T) is much greater than thermal energy (RT) at an ambient temperature.20,21 There are only a few reports of a singlet ground state.22,23 Meta-phenylene is generally very unstable owing to a facile dimerization and reaction with oxygen.24 Nevertheless, many substituted derivatives of m-phenylene have been successfully synthesized.7−11 One of the most famous examples is the sterically crowded Schlenk diradical (Figure 1). It was first synthesized in 1915.7 Various substituted Schlenk diradicals have been studied by Rajca et al.9 The latter authors synthesized different alkyl and halogen substituted Schlenk diradicals and showed that most of them have triplet ground states. The intramolecular coupling constants (J) were not quoted. Zhang et al.24 have studied the substitution at the fourth and sixth positions in m-phenylene that is attached to two (substituted) methylene radical centers at positions 1 and 3, and calculated the magnetic exchange coupling constants (J) by the UB3LYP method using the 6-31G(D) basis set. To the best of our knowledge, the magnetic nature of the substituted Schlenk diradicals has remained largely unexplored. © 2012 American Chemical Society

Figure 1. Molecules under investigations (5-substituted Schlenk diradicals).

In this work, we have considered 11 different substitutions at the fifth position of the m-phenylene group in the Schlenk diradical (Figure 1). Substitution has been done with groups that have ±M and ±I effects. To obtain J values that are comparable to the coupling constants from macroscopic Curie studies, we have relied on the strategy for optimizing the geometry of the broken symmetry state. It is shown that the exchange coupling constants are generally large and positive, in the range of 411 to 525 cm−1. The calculated J value for the unsubstituted radical is more or less in agreement with the Received: April 9, 2012 Revised: July 17, 2012 Published: August 13, 2012 8599

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coupling constant estimated by Rajca. The broken symmetry calculations have been done with the triplet state geometries to predict J values that can be observed from electron paramagnetic resonance (EPR) spectroscopy. In both cases, it transpires that electron pushing groups at the fifth position lowers the J value compared to that of the parent unsubstituted Schlenk diradical whereas introduction of electron withdrawing groups at the same position enhances the J value. These trends have been explained by simple concepts of electronic structure.

JY =

(1)

(2)

Therefore, one needs to compare the singlet and triplet ground state energies for a diradical, but the task is far from simple. Borden, Davidson, and Feller25 demonstrated that restricted quantum chemical methods can easily give qualitatively correct wave functions but fail to give a correct molecular geometry unless the basis set is very large, whereas the unrestricted methods easily yield reasonably correct energy and optimized molecular geometry. Therefore, it becomes difficult to determine a reliable S−T energy difference for a diradical. The latter quantity varies with different correlation methodologies and with basis sets. This has been well-documented by the early investigations of Nachtigall and Jordan,26 Cramer and Smith,27 and Mitani et al.12−15 To avoid this difficulty we relied on the well-known broken symmetry (BS) approach proposed by Noodleman.28 In this approach, a BS solution with ⟨S2⟩ = 1 is sought. The BS wave function is an approximately equal mixture of the singlet and triplet wave functions. Noodleman established the following expression for the magnetic exchange coupling constant, J=

⟨S ⟩T − ⟨S ⟩BS

(4)

Table 1. Computed Total Energies for Single-Determinant Singlet and Triplet Wave Functionsa substituents

ET (a.u.) (⟨S2⟩)

ES (a.u.) (⟨S2⟩)b

ES − ET (cm−1)

−F

−1333.371226 (2.057578) −3807.649115 (2.057543)

−1333.345895 (0.000002) −3807.623572 (0.000010)

5559.5

−Br

5606.0

a

The molecular geometry has been optimized in every case by the UB3LYP method using the 6-311G (d, p) basis set. We have used 1 a.u. = 219474.6 cm−1. bComputed α and β orbital energy values are same in Singlet. Thus the Singlet calculation has reduced to a restricted calculation.

Coupling Constant. The large singlet−triplet energy difference in Table 1 is understandable as the calculated singlet has a single determinant wave function (S1) with much higher energy. It is a pointer to the fact that the similar energetics would hold for other diradicals under study here, the only difference being that the process of optimization starting from default input led to singlet functions only in these two cases. Otherwise, these are not directly of any interest here, as the calculation of J involves an estimation of the energy of the twodeterminant singlet (S2) or the single determinant BS wave function. The two-determinant singlet S2 is in general very close to the triplet (T) in energy. For instance, for the Schlenk diradical (R = H) the following results have been obtained from CASSCF calculations using the cc-pVDZ basis set. A CAS(2,1) calculation on the singlet with symmetry 1 has given one configuration state function and the RHF singlet energy, −1225.726276 au. A CAS(2,2) calculation on the singlet with symmetry 2 has yielded one configuration and ROHF singlet energy −1225.728521 au. A CAS(2,2) calculation on the singlet with symmetry 1 has led to two configuration state functions and to a TCSCF singlet energy of −1225.811231 au. The best singlet energy has been obtained from the third step. Finally, a CAS(2,2) calculation on the triplet with symmetry 2 has given one configuration state function and the ROHF triplet energy −1225.813648 au. The

(E BS − E T̃ ) 1 + Sab 2

E T)

2

III. RESULTS AND DISCUSSION The geometries of the 12 molecules in both singlet and triplet states have been optimized by the UB3LYP method using the 6-311G(d,p) basis set. The optimized geometry for the singlet has been obtained only for the two species that have fluorine and bromine as substituents. For the rest of the diradicals, the optimization process has led to broken symmetry solutions with ⟨S2⟩ nearly equal to 1. These results are given in Table 1 and Table 2 respectively.

A determination of the magnetic nature of the ground state of the coupled system and the strength of the magnetic nature requires the knowledge of J. In particular, we note E(S = 1) − E(S = 0) = −2J

2

DFT

As usual, we have relied on unrestricted Becke 3-parameter exchange Lee−Yang−Parr correlation hybrid density functional methodology (UB3LYP) for both geometry optimization and single point calculation. The BS calculation requires a good initial guess for ⟨S2⟩. Therefore, we have used a ROHF wave function of the molecules as the initial guess, and carried out a single-point calculation on the BS state using a larger basis set and the optimized geometry of the triplet as well as that of the BS state. All calculations have been performed using Gaussian 03 (G03) suite of electronic structure programs.31

II. METHODOLOGY The interaction between two magnetic sites is expressed by the well-known Heisenberg effective spin Hamiltonian Ĥ = −2JS1̂ ·S2̂

(DFTE BS −

(3)

where EBS is the total energy corresponding to the BS wave function and Sab is the overlap integral between the two magnetically active orbitals a and b. The quantity ET̃ stands for the energy of the triplet formed from the BS orbitals. Because of the very less spin contamination in the triplet, ET̃ can be approximated to the computed energy ET for the triplet wave function. The BS method has several advantages. Built in the framework of density functional theory (DFT), it can account for exchange and correlation energies with relative ease and requires very less computing time. It avoids the computational difficulties associated with a two-determinant (singlet) state, as the BS wave function is a single determinant. While calculating the singlet ground state for a diradical, it is quite possible that the calculation would lead to a BS solution when the twodeterminant singlet is higher lying but the single-determinant singlet is still higher in energy.29 A number of broken symmetry formulas for J are available. We have found that the most useful is the one put forward by Yamaguchi et al.30 8600

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Table 2. Computed Total Energies for the SingleDeterminant BS and Triplet Wave Functionsa,b substituents −H

ET (a.u.) (⟨S2⟩)

EBS (a.u.) (⟨S2⟩)

−1234.106693 −1234.104318 (2.056844) (1.040921) Ortho-Para Position Directing Groups (+M effect) −NMe2 −1368.107139 −1368.105147 (2.05695) (1.041119) −OH −1309.349887 −1309.347627 (2.057834) (1.038729) −OMed −1348.659734 (2.057605) Groups with I effect −Me −1273.4340586 −1273.431735 (2.058712) (1.040002) −CF3 −1571.246014 −1571.243568 (2.059169) (1.038296) Meta Position Directing Groups (−M effect) −COOH −1422.736836 −1422.734458 (2.059816) (1.040044) −CHO −1347.460732 −1347.458294 (2.061173) (1.039541) −CN −1326.371283 −1326.368854 (2.060417) (1.036954) −NO2 −1438.662629 −1438.660165 (2.059900) (1.039788)

Table 3. Total Energies for Triplet and Broken Symmetry States and Coupling Constants of Substituted Schlenk Diradicals from Single Point UB3LYP Calculations Using Optimized Geometry of the Tripleta,b

JY (cm−1) 513.1c

substituents

ET (a.u.) (⟨S2⟩)

JY (cm−1)

EBS (a.u.) (⟨S2⟩)

−H

−1234.118434 −1234.115634 (2.057800) (1.036781) Ortho-Para Position Directing Groups (+M effect) −NMe2 −1368.120476 −1368.118074 (2.056412) (1.037170) −OH −1309.365808 −1309.363101 (2.057332) (1.035098) −OMe −1348.674376 −1348.671687 (2.057183) (1.035803) Groups with I effect −Me −1273.445780 −1273.443062 (2.057970) (1.036938) −Br −3807.661126 −3807.658255 (2.058682) (1.034404) −CF3 −1571.268384 −1571.265513 (2.058680) (1.034390) −F −1333.387424 −1333.384548 (2.057731) (1.033745) Meta Position Directing Groups (−R effect) −COOH −1422.754634 −1422.751806 (2.059500) (1.036310) −CHO −1347.475978 −1347.473090 (2.060460) (1.035665) −CN −1326.384962 −1326.382076 (2.060211) (1.033833)

430.4 486.7

500.6 525.8

509.4 522.8 520.9 530.1

a

The molecular geometry has been optimized in every case by the UB3LYP method using the 6-311G(d,p) basis set. We have used 1 a.u. = 219474.6 cm−1. bSinglet molecular geometry optimization has reduced to a BS solution. cEstimated J = 455 cm−1, ref 32. dBS convergence failure for −OMe.

601.8c

516.8 581.1 577.8

584.7 615.0 615.2 616.4

607.0 616.8 617.1

a

Calculated with the 6-311++G(d,p) basis set. bWe have used 1 a.u. = 219474.6 cm−1. cEstimated J = 455 cm−1, ref 32.

TCSCF calculation yields J = 265.2 cm−1, about half of the J calculated by the BS method. The triplet here remains uncorrelated, and therefore, not on par with the singlet that has become stabilized by correlation. The DF calculation involves more extensive correlation for the triplet as well as the BS state. Therefore, the J values calculated by BS method are quite reliable. Table 2, however, gives total energy for broken symmetry solutions with optimized geometries. Thus, the magnetic exchange coupling constant (J) calculated here should be close to those experimental values that are either obtained from the measurement of macroscopic magnetic moment (Curie studies) or estimated from a comparison of the calculated singlet and triplet energies of another species, as done in ref 32. Indeed, we obtain JY = 513.1 cm−1 for the unsubstituted Schlenk diradical. This is very close to the value of 455 cm−1 estimated by Rajca et al.33 This Table also contains data for R = OMe, though the BS calculation has failed to converge in this case. Nevertheless, the triplet energy is shown as it can be compared with the data in Table 3. Tables 3 and 4 show J values calculated with a larger basis set. As the J values in Table 3 have been calculated with the optimized geometry of the triplet, these would compare with the coupling constants that are directly measurable from EPR spectra of diradicals with a triplet ground state, while one assumes that the EPR transitions occur at a too small time scale to significantly alter the molecular geometry and correspond to vertical transitions. In the absence of experimental data, the calculated coupling constants in Table 3 have only predictive values. Also, single point BS calculations on the species with Br

Table 4. Broken Symmetry Energies from Single Point UB3LYP Calculationsa and the Optimized BS Geometries, and the Estimated Coupling Constants of Substituted Schlenk Diradicalsb substituents

EBS (a.u.) (⟨S2⟩)

JY (cm−1)

−H

−1234.116061 511.8c (1.040172) Ortho-Para Position Directing Groups (+M effect) −NMe2 −1368.118505 411.0 (1.040211) −OH −1309.3635202 492.6 (1.038063) Groups with −I effectd −Me −1273.443381 517.3 (1.040064) −CF3 −1571.265940 525.3 (1.037556) Meta Position Directing Groupsd −COOH −1422.752274 507.8 (1.039542) −CHO −1347.473562 519.4 (1.038885) −CN −1326.382528 522.0 (1.036809) a

Calculated with the 6-311++G(d,p) basis set. bWe have used 1 a.u. = 219474.6 cm−1. cEstimated J = 455 cm−1, ref 32. dBS convergence failure for −NO2 substituted diradical.

8601

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and F as substituents have been successful, and the corresponding results have been included here. Table 4 shows improvement of the J values over those given in Table 2 because they are based on the optimized geometries of both triplet and BS solutions. As triplet energy values are already given in Table 3, Table 4 contains only the BS energy and the calculated coupling constant. For the unsubstituted species, the calculated J improves to 511.8 cm−1. In fact, very nominal changes are observed for most of the substituted radicals, except for 2 (with NMe2 substituent, J decreases by 19.4 cm−1) and 5 (with Me substituent, J increases by 16.7 cm−1), on going from 6-311G(d,p) basis to 6-311++G(d,p) basis. All the diradicals have intramolecular ferromagnetic coupling, that is, the triplet total energy is lower than the BS one. This is in agreement with the observations in refs 9. Also, the BS total energy at BS-optimized geometry is lower than the BS energy at triplet-optimized geometry. Therefore, the predicted EPR J values in Table 3 are necessarily (by about 100 cm−1) greater than the calculated Curie studies J values given in Table 4. General Trends. The ferromagnetic nature of the intramolecular magnetic coupling in these diradicals is in agreement with the spin alternation rule.34 In the systems under study, spin alternation can propagate through two possible paths: either through C2 (shorter route) or through C5 (longer route). Both paths significantly contribute as can be seen from the spin density distribution in Figure 2 where the spin densities on the ligand atoms have not been summed with those on the phenylene ring. Figure 2 and Table 5 reveal that in each species, total spin density is positive at positions 2, 4, and 6 whereas it is negative at positions 1, 3, and 5. Calculated Mulliken atomic charges in the triplet state are shown in Figure 3 where the atomic charges on the ligand atoms have not been summed with those on the phenylene ring. These are also given in Table 5. Figure 3 and Table 5 show that in each species, there is additional electron density in the ortho-para positions (2, 4 and 6), and overall hole density in the meta positions (1, 3, and 5) except when R is H or CF3. Meta positions along with the substituent (1, 3, 5 and substituent) always contain a net hole density as shown in Table 6. Although it is well-known that Mulliken charges and spin densities are not unique, in this work we make an analysis of our data in terms of these quantities. Because of the complexity of the systems under investigation, an natural bond orbital (NBO) perturbation analysis remains inconclusive. Instead, we seek here simple explanations based on conventional concepts of chemistry. Inductive Effect. The monoradical center Ph2C• has a strong +I effect (Ingold’s notation) and indeed there is a total charge of −0.476 on the coupler (m-phenylene ring) when R is H. See Table 6. There is a net electron density at ortho-para positions, the sum of carbon and hydrogen atomic charges being −0.632 (Table 6), a charge −0.029 on the carbon atoms at meta positions (Table 5), and a positive charge 0.185 on the hydrogen atom at position 5 (Figure 3). From Table 6 one can notice that there is a net migration of electron density (about 0.5) and positive spin density (about 0.3) to the coupler in all 12 diradicals. The spin polarization at ortho-para versus meta positions (Tables 5 and 6) is associated with stability as embodied by spin alternation rule.34 There is an accompanying separation of charges at ortho-para vis-à-vis meta positions. The strong

Figure 2. Mulliken atomic spin density distribution in the triplet state of diradicals from calculations using the 6-311++G(d,p) basis set: (a) Schlenk diradical, (b) for o-p-positioning substituent with +M effect, (c) for groups with I effect, and (d) diradicals with m-position directing groups (−R effect).

resonance shown in Figure 4a imparts a free radical character to the atoms at ortho-para positions and would lead to a further dramatic increase in the ortho-para electronic population. Of course, the four phenyl rings also participate in the delocalization of the nonbonding electrons, and limit the net electron flow to the coupler to about 0.5. Substitution Effect. On substitution, total coupler electron density remains more or less same, except when R is NMe2 or NO2 for which a large increase is observed, and R is CN for which a large decrease is found from Table 6. The reason for the large increase is the coupling of the resonance involving monoradical centers with strong mesomeric effects. See Figure 4b and Figure 4c. The large decrease is likely to arise from the −M effect of CN dominating the coupling of resonances. The 8602

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Table 5. Sum of Atomic Charges and Spin Densities Only on Carbon Atoms at ortho-para and meta Positions of Coupler in Triplet State of Diradicala positions 2, 4, 6

a

positions 1, 3, 5

R

charge

spin density

charge

spin density

H NMe2 OH OMe

−1.035 −1.767 −1.269 −1.115

0.747 0.672 0.728 0.721

−0.029 0.541 0.332 0.241

−0.450 −0.403 −0.423 −0.425

Me Br CF3 F

−1.359 −1.285 −1.594 −0.956

0.732 0.757 0.752 0.750

0.524 0.602 −0.072 0.196

−0.453 −0.444 −0.447 −0.425

COOH CHO CN NO2

−1.898 −1.662 −0.825 −2.589

0.744 0.757 0.766 0.735

0.758 0.924 1.485 1.138

−0.452 −0.451 −0.456 −0.428

Calculated with the 6-311++G(d,p) basis set.

total spin density on the spacer always decreases on substitution. Figure 3 and Table 5 make it clear that when hydrogen is replaced by a substituent, electron density at ortho-para positions increases further (except for F and CN) and it decreases further in the meta positions (except for CF3). Along with the attached hydrogen atoms and substituents, ortho-para positions show an increased electron density and the meta positions reveal an increased hole density, except when R is F (−I effect) and CN (−I, −M effects). See Table 6. This happens regardless of the substituent having +M (NMe2, OH, OMe), −M (COOH, CHO, NO2), +I (Me) or −I (Br, CF3) effect. The groups CN and CF3 acquire a large negative charge of −1.422 and a large positive charge of 1.030, respectively. Aided by −I influence, CN shows a consistent −M effect, but the same effect is masked for other −M groups. Fluorine has a very strong −I influence that tends to increase the net electron density at meta positions at the cost of the net electron density at ortho-para positions. Table 5 reveals that ortho-para spin density decreases for +I and +M, and increases for −I and −M except for COOH and NO2. Meta spin density decreases for −I and +M, and remains almost unchanged for +I and −M (except for NO2). From Table 6 it is evident that the net ortho-para spin density always decreases on substitution, and the net meta spin density decreases for +M, F(−I), and NO2(−M). Physical Organic Chemistry. The substitution effect considered here for the J values is different from the traditional substitution effects considered in physical organic chemistry. The traditional effects arise from the studies of Hammett constants, isodesmic reactions, and so forth, and have a bearing on reactivities. Magnetic properties are rather physical characteristics. Hammett constants are a direct measure of relative acidity, and a positive value holds for electron withdrawing groups. Therefore, they appear to be the most promising factors, and we first consider the variation of the calculated J with Hammett constants σm and σp. We have observed two types of variation. The variation of the Curie law compatible coupling constants gives a linear plot of the form J = J0 + aσJ where for σm, J0 = 486 ± 19 cm−1 and a = 71.3 while for σp, J0 = 496 ± 12 cm−1 and a

Figure 3. Mulliken atomic charges in the triplet state of diradicals from calculations using the 6-311++G(d,p) basis set: (a) Schlenk diradical, (b) for o-p-positioning substituent with +M effect, (c) for groups with I effect, and (d) diradicals with m-position directing groups.

= 53.1. This is shown in Figure 5. Thus the coupling constant linearly increases with increase in acidity (electron withdrawing power). The variation of the EPR compatible coupling constants gives an exponential distribution of the form J = J0 + b[1 − exp(−cσJ)] where for σm, J0 = 575 ± 13 cm−1, b = 98.5, and c = 1.51, while for σp, J0 = 601 ± 6 cm−1, b = 38.2, and c = 1.55, as shown in Figure 6. At higher acidity the coupling constant shows an asymptotic behavior as it has been obtained for fixed (triplet) geometry. In both cases, the J0 values are nearly the same for variation with σm and σp, and close to the J values calculated for R = H (513 cm−1 and 602 cm−1, respectively). Another way to consider the net substituent effect is by using exothermicity or endothermicity of isodesmic reactions.35 For instance, the interaction energy of the two radical centers can be determined from the energetics of the isodesmic reaction 8603

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Table 6. Sum of Atomic Charges and Spin Densities Net of Carbon Atoms and Corresponding Substituents at orthopara and meta- Positions of Coupler in Triplet State of Diradicala positions 2, 4, 6 and associated hydrogen atoms

a

positions 1, 3, 5 and substituent

total coupler

R

charge

spin density

charge

spin density

charge

spin density

H NMe2 OH OMe

−0.632 −1.414 −0.842 −0.671

0.740 0.644 0.697 0.693

0.156 0.761 0.382 0.200

−0.443 −0.414 −0.431 −0.432

−0.476 −0.653 −0.460 −0.470

0.329 0.229 0.266 0.260

Me Br CF3 F

−0.940 −0.749 −1.094 −0.484

0.700 0.726 0.720 0.720

0.533 0.427 0.732 0.033

−0.447 −0.444 −0.450 −0.428

−0.407 −0.322 −0.362 −0.451

0.252 0.282 0.270 0.291

COOH CHO CN NO2

−1.319 −1.185 −0.274 −1.791

0.713 0.726 0.734 0.710

0.729 0.647 0.063 0.997

−0.459 −0.459 −0.451 −0.433

−0.591 −0.538 −0.211 −0.794

0.254 0.266 0.283 0.277

Figure 5. Variation of the Curie law compatible coupling constant with Hammett constant. Least square fitting is obtained for a function of the form J = J0 + aσ. The best fit for σm is obtained with J0 = 485.74, a = 71.26 and standard deviation 19.07. The best fit equation for σp is found for J0 = 496.10 and a = 53.14 with standard deviation 12.00.

Calculated total energy values (E) in atomic units using the UB3LYP method and the 6-311++G(d,p) basis set are − 23 2 .3 1 13 04 , − 73 3 . 21 46 7 3, −1 23 4. 1 1 84 35 , a n d −1234.116061 for singlet benzene, doublet C6H5(CPh2), and Schlenk diradical in its triplet and broken symmetry states, respectively. These give ΔE = 86.39 cm−1 when the diradical is in its triplet state, and ΔE = −434.65 cm−1 when it is in the BS state. The difference in the ΔE values exactly equals the energy difference 521.04 cm−1 between optimized BS (Table 4) and optimized triplet Schlenk radical (Table 3). As ⟨S2⟩T − ⟨S2⟩BS deviates from the ideal value of 1, the ΔΔE is approximately equal to J in Table 4 (511.8 cm−1). At a first glance one would expect a ΔE of 0.5J for the reaction of the triplet Schlenk radical and −0.5J for the reaction of the BS solution. The chemical

Calculated with the 6-311++G(d,p) basis set.

C6H6 + 1, 3−C6H4(CPh 2)2 → 2C6H5(CPh 2)

(5)

Figure 4. Possible resonances involving the monoradical centers: (a) general case, (b) with +M group, and (c) with −M group at position 5. 8604

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Table 7. Dihedral Angles at Various Orientations of the Schlenk Diradicals Investigated Here substituent

Figure 6. Variation of the EPR compatible coupling constant with Hammett constant. Least square fitting is obtained for a function of the form J = J0 + b[1 − exp(−cσ)] expanded up to the second order in σ while taking J as constant for large σ. The best fit equation for σm is J = −112.45σ2 + 148.86σ + 574.69 with a standard deviation of 13.12. The best fit equation for σp is J = −46.071σ2 + 59.337σ + 601.02 with standard deviation 6.41.

environment in the Schlenk diradical is, however, different from that in C6H5(CPh2), and the additional delocalization of the nonbonded electrons in the former species stabilizes both triplet and BS states by an amount of 174.13 cm−1, which explains the calculated ΔE values. The net substituent effect is sometimes defined by the exothermicity or endothermicity of the reaction

dihedral angle, degree

−R

C19−C18−C32−C44 (φ1)

C14−C13−C12−C21 (φ2)

−H −NMe2 −OH −OMe −Me −F −Br −CHO −COOH −CN −CF3 −NO2

33.04 34.78 32.96 32.75 33.54 31.88 32.70 32.89 32.98 32.62 32.60 32.58

33.10 34.17 32.72 33.17 33.44 32.18 32.70 32.53 32.99 32.59 32.62 32.68

C6H5X + 1, 3−C6H4(CPh 2)2 → 5, 1, 3−XC6H3(CPh 2)2 + C6H6

(6)

Calculated total energy values for C6H5X at the UB3LYP/6311++G(d,p) level and ΔE for reaction 6 with triplet diradicals are given in the Supporting Information, Table S1. We add a caveat here. The ΔE values are only indicative of the relative stability of the products, which primarily relies on the change in charge distribution. Instead, the magnetic behavior is mainly determined from spin density, that is, differential electron density for up and down spins. Thus ΔE for reaction 6 is unlikely to offer any direct explanation for the magnitude of ΔJ. The same is witnessed by Supporting Information, Table S1, where we find quite arbitrary variation of ΔJ with ΔE. As discussed in the previous paragraph, the difference ΔE(BS reaction) − ΔE(triplet reaction) would be comparable to ΔJ, but this is obvious and no new information can be gleaned from it. A much better understanding is to be obtained from the extent of planarity of the species and spin density as discussed in the following. Dihedral Angle. The calculated dihedral angles φ1 and φ2 between m-phenylene and the two radical centers vary in the narrow range from 31.88° to 34.78° as shown in Table 7. This structural feature has limited the width of variation for all calculated coupling constants. The factor of planarity can be defined by xp = cos φ1 cos φ2. The species is completely planar when xp = 1 (φ1 = φ2 = 0) while it is completely nonplanar if φ1 = φ2 = 90° (xp = 0). In the latter case McConnell’s formula36 indicates a strong antiferromagnetic coupling, that is, a large negative value for J. Furthermore, the strength of the coupling would be linearly dependent on xp that describes the extent of conjugation between the spacer and the two radical centers. Variation of the calculated coupling constants from Tables 2−4 versus xp has been illustrated by Figure 7. This figure indeed shows linear plots with large positive slopes and large negative intercepts. Both the slope and the intercept are smaller in absolute magnitude for plot (c) corresponding to Table 3. The calculations in this table have involved increased J values with

Figure 7. Variation of coupling constant with factor of planarity, x = cos φ1 cos φ2. The least-squares fit is obtained with a linear equation of the form J = mx + J0: (a) 6-311G(d,p), Curie law compatible J from Table 2, m = 2824.8, J0 = −1480.5 and rms deviation =12.1568 ; (b) 6311++G(d,p), Curie law compatible J from Table 4, m = 3291.7, J0 = −1808.4 and rms deviation =16.7076; (c) 6-311++G(d,p), EPR compatible J from Table 3, m = 2709.6, J0 = −1311.8 and rms deviation =13.0336.

proportionately less spread due to single point BS calculations at triplet geometry, thereby making the plot flatter with decreased slope and smaller negative intercept. Spin Density. In an unrestricted triplet state of two additional positive spins, each conjugated atom with nearly one π electron can have either a positive or a negative net spin. Of course, the extent of positive net spin prevails over that of negative net spin. An increase in electron density on an electron rich atomic center (π electron population >1) with a net positive spin tends to add more contributions from orbitals of negative spin to the same center, thereby decreasing the net positive spin. Similarly, a decrease of electron density (addition of hole density) from an electron deficient (hole rich) atom (π electron population