Calculation of Exchange Coupling Constants in Triply-Bridged

Article pubs.acs.org/JPCA

Calculation of Exchange Coupling Constants in Triply-Bridged Dinuclear Cu(II) Compounds Based on Spin-Flip Constricted Variational Density Functional Theory Issaka Seidu, Hristina R. Zhekova, Michael Seth, and Tom Ziegler* Department of Chemistry, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 S Supporting Information *

ABSTRACT: The performance of the second-order spin-flip constricted variational density functional theory (SF-CV(2)-DFT) for the calculation of the exchange coupling constant (J) is assessed by application to a series of triply bridged Cu(II) dinuclear complexes. A comparison of the J values based on SF-CV(2)-DFT with those obtained by the broken symmetry (BS) DFT method and experiment is provided. It is demonstrated that our methodology constitutes a viable alternative to the BS-DFT method. The strong dependence of the calculated exchange coupling constants on the applied functionals is demonstrated. Both SF-CV(2)-DFT and BS-DFT affords the best agreement with experiment for hybrid functionals.

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INTRODUCTION Dinuclear copper complexes have been studied extensively in connection with molecular magnetism,1,2 high-temperature superconductivity,3 and modeling of metalloenzymes.4−9 Depending on the coordination of the copper centers and the identity of the bridging ligands, these systems can be either ferro- or antiferromagnetic. Magnetism in metal clusters can in general be considered as an interaction of the localized spin angular momentum on different metal centers.1 The weak interaction between the adjacent spins is conveniently described by a Heisenberg Hamiltonian,10,11

There have been a number of attempts based on empirical, magneto-structural, and theoretical considerations to rationalize trends in the observed J values of these complexes.26−41 The broken symmetry approach of Noodleman42−44 based on density functional theory (BS-DFT) is currently the most used scheme for the first principle calculation of exchange coupling constants. It has been employed to a variety of bi- and polynuclear metal complexes with a large variety of approximate functionals,12,13,16,17,39−41,45−54 including LDA,55 GGA,56−58 hybrid,59 hybrid meta-GGA,60−62 and long-range separated63−65 functionals. Other DFT-based methods employed in magnetic studies are the restricted ensemble Kohn− Sham (REKS) and restricted open shell Kohn−Sham (ROKS) schemes.12,13,53,66−69 Use has also been made of high level ab initio schemes14,17,36,38,39,46,52,53,70−73 including multireference wave function approaches.74−77 We have recently78 introduced second order spin-flip constricted variational DFT (SF-CV(2)-DFT)79 as an alternative scheme for the evaluation of exchange coupling constants in polynuclear metal complexes. Our method has so far been tested on two trinuclear copper compounds78 and a number of binuclear copper complexes with two bridging ligands.80 Our objective with the current study is to extend the test to a series of dinuclear complexes with three bridging ligands. This series has previously been studied by a number of theoretical methods as discussed in a recent study.54 The spinflip approach was originally introduced by Krylov in ab initio wave function theory.81

Ĥ = −∑ Jij Sî Sĵ i,j

(1)

In eq 1, Jij is the exchange coupling constant between two spins Sî and Ŝj of adjacent metal centers. In dinuclear copper complexes where each center carries a spin of Si = 1/2, the exchange coupling constant Jij represents the energy difference between the lowest triplet and singlet states of the dimer:12 J = Esinglet − E triplet

(2)

Thus, an antiferromagnetic singlet ground state has a negative J value, whereas a positive J value indicates a ferromagnetic triplet ground state. The exchange coupling constants can be experimentally determined from magnetic susceptibility measurements or neutron dispersion spectroscopy. Copper dinuclear systems have received considerable experimental and theoretical attention. These complexes contain both doubly or triply bridged copper atoms and cover a large range of positive and negative13−25 J values. Their structures are determined from X-ray spectroscopy and the J constants are measured from magnetic susceptibility experiments. © 2012 American Chemical Society

Received: October 2, 2011 Revised: January 24, 2012 Published: February 9, 2012 2268

dx.doi.org/10.1021/jp209507n | J. Phys. Chem. A 2012, 116, 2268−2277

The Journal of Physical Chemistry A

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Article

where f(ρ0,s0) is the second functional derivative of EXC with respect to the electronic density ρ and the spin-density s evaluated at the reference (ρ0,s0). Finally, εi0,εa0 are the reference state orbital energies of, respectively, ϕi(1) and ϕa(1). Assuming that our KS orbitals as well as all elements Uai are real allows us to write eq 5 in compact form as

COMPUTATIONAL METHOD AND DETAILS SF-CV(2)-DFT. In the simple second-order constricted variational DFT method79 (CV(2)-DFT), we start with a nondegenerate reference state that can be represented by a single Slater determinant Ψ0 = |ϕ1ϕ2ϕ3···ϕiϕj···ϕn| with the occupied reference orbitals {ϕi(1); i = 1,occ} obtained from a Kohn−Sham calculation.82 The reference state need not be the ground state of the system. We allow next the occupied reference state orbitals {ϕi(1); i = 1,occ} to vary by mixing into each ϕi(1) a fraction of every virtual reference state orbital {ϕa(1); a = 1,vir} according to δφi ∝ ∑aUaiφa. Such a variation affords a new set of occupied orbitals79 vir

φi′ = φi +

∑ Uai φa − a

+ EKS[ρ′] = EKS[ρ(0)] + U⃗ (AKS + BKS)U⃗ KS

Here (A + B ) represents the electronic ground state Hessian79 with elements defined as KS = δ δ (ε0 − ε0) + K KS ; BKS = K KS Aai ab ij a i , bj ai , bj ai , bj ai , jb

occ vir

1 2

a

(3)

that are orthonormal to second order in U. Associated with the new set {ϕ′i(1); i = 1,vir} we have the Slater determinant Ψ′ = |ϕ′1ϕ′2ϕ′3···ϕ′iϕ′j···ϕ′n|. It has to second order in U the density ρ′(1, 1′) = ρ(0)(1, 1′) + Δρ′(1, 1′) occ vir

= ρ(0)(1, 1′) +

∑ ∑ Uai φa(1′)φ*i (1) i

(I ) (I ) (AKS + BKS)U⃗ = λ(I )U⃗

a

occ vir

+

∑ ∑ ∑ Uai*Ubi φa(1′)φ*b(1) i a b occ occ vir

−

AKSU⃗

∑ ∑ ∑ UaiUaj* φi(1′)φ*j (1) i

j

a

where ρ(0) is the reference state density. The total Kohn−Sham energy with respect to the density matrix ρ(1,1′) affords the following energy expression to second order in U

∑ UaiUai* ](εa0 − εi0) ai

+

∑ ∑ UaiUbjKaiKS, bj ai bj

1 + 2 +

1 2

∑ ∑ UaiUbjKaiKS, jb ai bj

∑ ∑ Uai*Ubj*KaiKS, jb + O[U (3)]

Ψ(I ) =

ai bj (5)

∫

∫ φ*r (1)φu(1) r1

*

φt (2) 12 φ*r ( r1⃗ )φu( r1⃗ )[f (ρ0, s 0)]

× φ*t ( r2⃗ )φq( r2⃗ )d⇀ r1d r2⃗

(I ) (10)

vir occ

∑ ∑ Uai(I̅ )Ψi → a ̅ i

(11)

where Ψi→a is a determinantal wave function constructed from |ϕrϕs| by substituting an occupied reference orbital ϕi of α-spin by a virtual reference orbital of β-spin. Some of the states of the form Ψ(I) that can be obtained from eq 11 are the antiferromagnetic singlet state of lowest energy 11Ψ0, which will have a leading contribution from Urs̅ |ϕrφ̅ r| + Usr̅ |ϕsφ̅ | as well as the ferromagnetic triplet state 13Ψ0 with ms = 0, which will have a major contribution from Uss̅ |ϕrφ̅ s| + Urr̅ |φ̅ rϕs|. We can subsequently calculate J according to eq 2 as

Here EKS[ ρ ] is the reference state energy and “a,b” run over virtual reference state orbitals, whereas “i,j” run over occupied reference state orbitals. Further

× φq(2)dτ1dτ2 +

= λ(I )U⃗

a

0

KS C KS,XC K ru , tq = K ru , tq + K ru , tq +

(I )

which is identical in form to the equation one obtains from TDDFT84,85 in its adiabatic formulation after applying the same Tamm−Dancoff approximation. The results provided by the two methods SF-CV(2)-DFT and SF-TDDFT in its adiabatic formulation after the Tamm−Dancoff approximation have been applied to both are the same when the reference state for the SF-CV(2)-DFT calculation is the ground state. However, in contrast to SF-TDDFT, in the SF-CV(2)-DFT method, the reference state need not be the ground state. For the copper complexes studied here a natural reference would be the ferromagnetic triplet state 3Ψ1 with two electrons of the same spin in two different orbitals ϕr,ϕs which we can write as 13Ψ1 = |ϕrϕs|. A CV(2)-DFT calculation now affords state functions of the general form

(4)

EKS[ρ′(1, 1′)] = EKS[ρ0] +

(9)

from which we can determine the sets of mixing coefficients {U(I); I = (occ × vir) × (occ × vir)} that represent excited states. The corresponding excitation energies are given by λ(I). Within the Tamm−Dancoff approximation83 eq 11 reduces to

∑ ∑ Uai* φ*a(1)φi(1′) i a occ vir vir

+

(8)

Also, in line with standard convention1,2 U⃗ in eq 7 is now a column vector of length (occ + vir) with the running index “ai”. In CV(2)-DFT we seek points on the energy surface EKS[ρ′] such that ΔEKS[Δρ′] = EKS[ρ′] − EKS[ρ0] represents a transition energy from the reference. Demanding that such a transition must represent a transfer of one electron from the density matrix space spanned by {ϕi(1); i = 1,occ} to the density matrix space spanned by {ϕi(1); i = 1,vir} lead to the eigenvalue equation:

∑ ∑ UaiUajφj j

(7)

KS

J = E(11Ψ 0) − E(13Ψ 0)

(6) 2269

(12)



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Figure 1. Structures of the eleven complexes studied. The complexes are numbered in order of increasing experimental J values (see Table 14). KS,XC where K KS,XC are well-defined integrals from reguai,bj and K ai,ij 84,85 The expression in eq 14 is correct to (s0)3 and lar TDDFT. has no singularities for s0 = 0. It can thus be used for small values of s0, where eq 13 becomes singular. In practice, we have found that eq 13 can be used for VWN.55 For functionals based on the generalized gradient approximation (GGA)56−58 where GGA VWN = E XC + we can write the exchange-correlation energy as E XC GGA VWN ΔE XC , we calculate the contribution from E XC according to eq 14 and the contribution from ΔE GGA XC in line with eq 14. For the hybrid functionals B3LYP59 and BH&HLYP,59 we need in HF,XC = addition to calculate the regular exchange integral K ai,bj −∫ ∫ ϕ − a * (r1)ϕ − b(r2)1/r12φi(r1)φ*j(r2)dr1dr2 already implemented in ADF.92 Computational Details. All SF-CV(2)-DFT calculations were performed with an all-electron TZ2P92 basis set within the unrestricted Kohn−Sham formalism,82 as implemented in ADF2010.92 The used functionals include LDA-VWN,55 BP86,56,57 BLYP,56,58 B3LYP,59 and BH&HLYP.59 Computational Model. The 11 studied dinuclear triply bridged Cu(II) systems are arranged and displayed according to increasing experimental exchange coupling constants (J) in Figure 1. The penta-coordinated complexes have the following structural characteristics. Complex 1 with the chemical formula [Cu2(μ-O2CC2H5)(μ-OCOC2H5)(μ-OH)(dpyam)2] (ClO4) and a Cambridge Crystallographic Data Centre (CCDC) refcode YAHZEN25 has no symmetry. Both Cu(II) centers have a square pyramidal coordination geometry. The coppers are bridged by two carboxylato ligands and a hydroxo group. The two oxygen atoms of the carboxylato ligand occupy, respectively, the equatorial position on one center and the apical position on the other center. The remaining equatorial sites are taken up by OH and nitrogens on the noncoplanar dpyam bases. Complex 2 with the chemical formula [Cu2(μ-O2CCH3)(μOH)(μ-OH2)(bpy)2](ClO4)2 and CCDC refcode JEJCIK1093 is of Cs symmetry with two Cu(II) centers bridged by carboxylato, hydroxo, and water ligands. The bidentate 2,2′bipyridine(bpy), the water, hydroxo, and carboxylato ligands

We could alternatively have determined J from J = E(11Ψ0) − E(13Ψ1). We prefer eq 12, as it gives a more balanced description where both states are described on the same footing in the form of eq 11, according to CV(2)-DFT. This is as opposed to the alternative where 13Ψ1 is described by a single determinantal function and 11Ψ0 according to CV(2)-DFT on the form eq 11. Such a practice is common in studies based on multiconfiguration wave function theory.86 It should be noted that a and i, as well as b and j in eq 8 are of different spins as all microstates Ψi→a̅ are generated by a spin-flip i → a,̅ where i is of α-spin and a̅ is of β-spin. The formula for KaKSi,d̅ required in eq 10 were first derived by Wang ̅ j and Ziegler, 78−89 see also refs 90 and 91. We have K KS,XC = ai̅ , b ̅ j

∫ [ϕ̅ *a(r1)ϕi(r1)ϕ̅ b(r1)ϕ*j (r1) ⎡⎛ ⎤ ⎛ KS KS ⎞⎞ ⎥ ∂EXC 1 ⎢⎜ 1 ⎜ ∂EXC ⎟⎟ × ⎢ 0⎜ − ⎥dr1 ⎟ ⎜ ⎟ ∂ρβ ⎠⎠ 0 0 ⎥ 2 ⎢⎝ s ⎝ ∂ρα ⎣ (ρ , s )⎦

(13)

,where integration over spin already has taken place so that ϕ̅ a*(r1),ϕ̅ b (r1) represent the spatial part of the two KS,XC by virtual orbitals of β-spin. The evaluation of Kai,b̅ ̅ j numerical integration might lead to numerical instabilities if s0 = ρα − ρβ ≈ 0. We can in that case carry out a Taylor expansion of ∂EXCKS/∂ρa,∂EXCKS/∂ρβ from ρ = ρα + ρβ and s0 = 0. Thus K −KS,XC = − ai , bj

−*

−

∫ [ϕa(r1)ϕi(r1)ϕb(r1)ϕ*j (r1)] ⎤ ⎡⎛ 2KS 2KS 2KS ⎞ ⎥ ∂EXC ∂EXC 1 ⎢⎜ ∂EXC ⎟ × ⎢ 2 + 2 −2 ⎥dr1 ⎜ ⎟ 2⎢ ∂ ρ ∂ρα∂ρα ∂ ρβ ⎥ ⎝ ⎠ α 0 0 ⎣ (ρ , s = 0)⎦

KS,XC − K KS,XC = Kai − , bj ab , ij

(14) 2270



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occupy the basal positions. A water molecule is at the apical site. Complex 11 with the chemical formula [Cu2(μ-O2CCH2CH3)(μ-OH)(μ-OH2)(phen)2](NO3)2 and a CCDC refcode YAFZUA0122 has two square pyramidal Cu(II) centers bridged by carboxylato, hydroxo, and water ligands. No modifications were made to the complexes except that the counterions were removed and in complex 4, the water of hydration was removed as well. The complexes studied in this work have also been used in ref 54.

are ordered in a square pyramidal geometry with water at the apical position. Complex 3 with the chemical formula [Cu2(μ-O2CH)(μ-OH)(μ-OMe)(dpyam)2](ClO4) and a CCDC refcode EBEFIB24 is of Cs symmetry. Each Cu(II) center has a trigonal bipyramidal ligand environment. The carboxylato and the methoxo ligands occupy the equatorial position. The O atom at one of the axial positions belongs to the hydroxo ligand and the N atom at the other axial position belongs to the bidentate 2,2′bipyridylamine(dpyam) ligand. The position of the H atom of the hydroxo ligand was adjusted to correspond to the remaining series of complexes, the justification for this is given in ref 54. The modified structure is provided as supporting document. Complex 4 with the chemical formula [Cu2(μ-O2CH)(μ-OH)(μ-Cl)(dpyam)2](ClO4)·0.5H2O and the refcode RUXDIX0124 is of Cs symmetry and features noncoplanar bases with a trigonal bipyramidal geometry around each Cu(II) center. The two Cu(II) centers are bridged by carboxylato, hydroxo and chloro ligands. The O atom on one of the axial positions belongs to the hydroxo ligand and the N atom in the other axial position belongs to the bidentate dpyam ligand. Complex 5 with the chemical formula [Cu2(μ-O2CH)(μ-OH)(μ-Cl)(dpyam)2](PF6) and a CCDC refcode YAHYUC25 is of Cs symmetry and the two Cu(II) centers bridged through the carboxylato, hydroxo, and chloro ligands. Each Cu(II) center has a trigonal bipyramidal geometry. The axial position is occupied by the O atom of the hydroxo ligand and N atom of the bidentate dpyam ligand. Complex 6 with the chemical formula [Cu2(μ-O2CCH3)(μ-OH)(μ-OH2)(phen)2](ClO4)2 and a CCDC refcode YEMNIO23 has the bidentate 1,10-phenanthroline(phen) at each Cu(II) center. The complex has no symmetry and the two Cu(II) centers are bridged by the carboxylato, hydroxo and water ligands. Each Cu(II) center has a square pyramidal coordination. The phen, carboxylato and the hydroxo ligands occupy the basal position with the water at the apical site. Complex 7 with the chemical formula [Cu2(μ-O2CCH3)(μ-OH)(μ-OH2)(phen)2](BF4)2 and a CCDC refcode CITLOH22 has no symmetry and has two Cu(II) ions bridged by carboxylato, hydroxo, and water ligands. Each Cu(II) ion is five coordinated and features a square pyramidal geometry with noncoplanar bases where the bidentate phen, hydroxo, and carboxylato ligands at the basal position and the water ligand at the apical position. Complex 8 with the chemical formula [Cu2(μ-O2CCH2CH3)(μ-OH)(μ-OH2)(bpy)2](ClO4)2 and a CCDC refcode YEMNEK23 has no symmetry and two Cu(II) centers bridged by the carboxylato, hydroxo, and the water ligands. The complex has noncoplanar bases with a square pyramidal coordination geometry at each Cu(II) center. Complex 9 with the chemical formula [Cu2(μ-O2CH)(μ-OH)(μ-OH2)(dpyam)2](S2O8) and a CCDC refcode CITLEX22 has two dpyam ligands and three bridging groups (hydroxo, carboxylato, and water). The complex is of Cs symmetry featuring noncoplanar bases with a square pyramidal coordination geometry around each Cu(II) center. The dpyam, hydroxo, and carboxylato ligands at the basal position and the water at the apical site. Complex 10 with the chemical formula [Cu2(μ-O2CCH3)(μ-OH) (μ-OH2)(bpy)2](NO3)2 and a CCDC refcode CITLIB22 is of Cs symmetry with a square pyramidal coordination geometry around each Cu(II) center. The two bidentate bpy bases, a hydroxo group, and a carboxylato ligand

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RESULTS AND DISCUSSION Electronic Structure of Complexes 1−11. Figures 2−4 display the frontier orbitals of complexes 1−11 as well as the corresponding orbital energies. The orbitals were obtained from unrestricted Kohn−Sham SCF calculations on the reference triplet state in its optimized geometry using the LDA-VWN functional. The triplet ground state has two unpaired electrons of α-spin in 1a and 2a or a′ and a″. These orbitals are referred to as SOMOs (singly occupied molecular orbitals), while the lowest vacant orbitals of β-spin are referred to as SUMOs (singly unoccupied molecular orbitals). The SUMOs have almost the same composition as the corresponding SOMOs; however, their energies are higher as the smaller number of β-electrons lead to fewer stabilizing exchange interactions compared to the α-manifold. The two unpaired electrons can be distributed in the four frontier orbitals to give three possible electronic configurations: (1a)2(2a)0, (1a)1(2a)1, and (1a)0(2a)2 in the case of no symmetry and (a′)2(a″)0, (a′)1(a″)1, and (a′)0(a″)2 in the case of Cs symmetry. These configurations give rise to microstates of different multiplicity and symmetry. The construction of the symmetrized microstates was carried out with the clebeschegordan.exe program.94 The number of microstates is calculated to be six. Tables 1 and 2 display all six singlet and triplet microstates which can be built with the two unpaired electrons and the four SUMOs and SOMOs of Cs symmetry or no symmetry, respectively. Tables 3−13 display results from SF-CV(2)-DFT calculations based on the unrestricted LDA KS-DFT geometries of the reference state a(3A′)1 for the Cs symmetry and a(3A)1 for no symmetry. The ligands around the Cu(II) ions in the studied complexes 1, 2, 6, 7, 8, 9, 10, and 11 adopt a square pyramidal coordination geometry54 and those around the complexes 3, 4, and 5 adopt a trigonal bipyramidal geometry. The frontier orbital for the complexes that adopt the square pyramidal geometry can be described as an out-of-phase combination of the dx2−y2-orbitals on the two Cu centers and the ligand p-orbitals at the basal/equatorial sites. Here dx2−y2 refers to a local coordinate system on each copper site with the z-axis pointing toward the apical ligand. In complexes 1, 2, 6, 7, 9, 10 the two dx2−y2 orbitals contribute to both SOMOs (and SUMOs) with ++ and +− contributions, respectively. In complexes 8 and 11, the SOMOs (and SUMOs) are symmetry broken with contributing orbitals on either one of the two Cu(II) centers. In complex 1 there is a small antibonding contribution from the p-orbital of the apical O atom of the carboxylato ligand on one center and an additional small antibonding contribution from the O atom of the apical monatomic carboxylato ligand. For the complexes that adopt a trigonal bipyramidal geometry, the frontier orbitals can be described as an out-of-phase 2271



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Figure 2. Frontier SUMO orbitals a′ ̅, a′′ ̅ with Cs symmetry and 1a,̅ 2a̅ without symmetry for complexes 1−4, with the corresponding orbital energies. The SOMO orbitals a′, a″ and 1a, 2a are not shown because they have a spatial distribution similar to their SUMO counterparts.

Figure 3. Frontier SUMO orbitals a′ ̅, a′′ ̅ of Cs symmetry and 1a,̅ 2a̅ without symmetry for complexes 5−7, with the corresponding orbital energies. The SOMO orbitals a′, a″ and 1a, 2a are not shown because they have a spatial distribution similar to their SUMO counterparts.

combination of Cu dz2 orbitals and the ligand p-orbitals in the axial position with an antibonding contribution from the ligand p-orbitals in the equatorial sites. Here dz2 refers to a local coordinate system centered on each copper with the z-axis pointing toward the axial ligands.

In complexes 2, 3, 4, 5, 9, and 10, the Cs symmetry was imposed and we obtained orbitals of a′ and a″ symmetry with a′ slightly below a″ in energy. For complexes 1, 6, 7, 8, and 11 where symmetry was not imposed we obtained the orbitals 1a and 2a with 1a of slightly lower energy. We observe for the complexes two structural motifs, in 2272



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Table 3. Lower Excited States for Complex 1 Based on SFCV(2)-DFT Calculations Using LDA-VWN Functional no symmetry state

E, cm−1

contributing microstates

%

(3A) 1(1A)

0 144

2(1A)

4454

92 44 16 45

3(1A)

5454

b(3A) a(1A) c(1A) a(1A) c(1A) b(1A)

0

80

Table 4. Lower Excited States for Complex 2 Based on SFCV(2)-DFT Calculations Using LDA-VWN Functional Cs symmetry

a(1A′)

0

(1A″) a(3A″) b(3A″) c(3A″)

0 1 0 −1

b(1A′)

Cs symmetrized microstate functions (a′)2 configurationa,b |a′a′ ̅| (a′)1(a″)1 configuration 1/√2(|a′a″̅| − |a′ ̅a″|) |a′a″| 1/√2(|a′a″̅| + |a′ ̅a″|) |a′ ̅a″̅| (a″)2 configuration |a″a″̅|

0

Ms

C1 symmetrized microstate functions

0

(1a)2 configurationa,b |1a1a|̅ (1a)1(2a)1 configuration 1/√2(|1a2a|̅ − |1a2a|) ̅ |1a2a| 1/√2(|1a2a|̅ + |1a2a|) ̅ |1a2a ̅ |̅ (2a)2 configuration |2a2a|̅

1

b( A) a(3A) b(3A) c(3A) c(1A)

0 1 0 −1 0

(1A″) 2(1A′)

6183 6593

98 53 45 100 46 54

−1

state

E, cm


%

(3A″) 1(1A′)

0 358

(1A″) 2(1A′)

4993 5760

b(3A″) a(1A′) b(1A′) (1A″) a(1A′) b(1A′)

95 60 33 94 37 47

Cs symmetry 3

Table 2. Singlet and Triplet Microstates of the Binuclear Triply-Bridged Copper(II) Compounds without Symmetry a(1A)

%

b(3A″) a(1A′) b(1A′) (1A″) a(1A′) b(1A′)

state

b

nosym (C1) microstates


0 287

Table 6. Lower Excited States for Complex 4 Based on SFCV(2)-DFT Calculations Using LDA-VWN Functional

Electron configuration. a′ and a″ are the frontier orbitals of α-spin, whereas a′ ̅ and a″̅ are frontier orbitals of β-spin. a

E, cm

Cs symmetry

Table 1. Singlet and Triplet Microstates for the Binuclear Triply-Bridged Copper(II) Compounds in the Cs Symmetry Ms

state (3A″) 1(1A′)


Figure 4. Frontier SUMO orbitals a′ ̅, a′′ ̅ for Cs symmetry and 1a,̅ 2a̅ without symmetry for complexes 8−11, with the corresponding orbital energies. The SOMO orbitals a′, a″ and 1a, 2a are not shown because they have a spatial distribution similar to their SUMO counterparts.

Cs microstates

−1

E, cm−1

( A″) 1(1A′)

0 407

(1A″) 2(1A′)

5492 6066

contributing microstates 3

b( A″) a(1A′) b(1A′) (1A″) a(1A′) b(1A′)

% 97 49 49 100 50 48

Table 7. Lower Excited States for Complex 5 Based on SFCV(2)-DFT Calculations Using LDA-VWN Functional Cs symmetry

a Electron configuration. b1a and 2a are the frontier orbitals of α-spin, whereas 1a̅ and 2a̅ are frontier orbitals of β-spin.

state

E, cm−1


%

(3A″) 1(1A′)

0 332

(1A″) 2(1A′)

5644 6153

b(3A″) a(1A′) b(1A′) (1A″) a(1A′) b(1A′)

97 55 42 99 43 56

d-orbitals and the p-orbital of the hydroxo ligand. In the second structural motif (complexes 3−5), there is a sigma interaction between the Cu d-orbitals and the p-orbital on the chloro ligand

the first (complexes 1, 2, and 6−11), there is a sigma interaction between the Cu d-orbitals and the p-orbital on the hydroxo ligand for a′ or 1a whereas for a″ or 2a there is a π interaction between the Cu 2273



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no symmetry

no symmetry

state

E, cm−1


%

state

E, cm−1


%

(3A) 1(1A)

0 288

0 325 5853

5955

98 35 26 52 47 17

(3A) 1(1A) 2(1A)

2(1A)

b(3A) a(1A) c(1A) b(1A) a(1A) a(1A)

3(1A)

6677

b(3A) b(1A) a(1A) c(1A) a(1A) c(1A)

95 95 89 7 7 89

1

3( A)

6674

The spin states can be expressed as a linear combination of the symmetrized spin microstates defined by eq 11. The lowest singlet for the complexes consists mainly of two single determinantal microstates |a′a′ ̅| and |a″a′′ ̅| or |1a1a|̅ and |2a2a|̅ and has the form

Table 9. Lower Excited States for Complex 7 Based on SFCV(2)-DFT Calculations Using LDA-VWN Functional Cs symmetry state

E, cm

(3A) 1(1A)

0 283

−1

1

2( A)

6066

3(1A)

6595


%

b(3A) a(1A) c(1A) b(1A) a(1A) a(1A) c(1A)

98 40 32 67 32 27 66

11Ψ 0 = c1|a′a′| + c 2|a″a″|

The statistical weights vary from an even contribution to a contribution dominated by one determinantal microstate depending on the energy difference between a′ and a″ or 1a and 2a. The difference in energy between a′ or 1a and a″ or 2a is due to the extent of overlap between the Cu d-orbitals and ligand p-orbitals. The sigma interaction that exist in a′ or 1a is out-of-phase with a smaller overlap whereas the pi interaction in a″ and 2a is of relatively larger overlap in absolute terms, as a result we observe that a′ and 1a are of lower energy than a″ and 2a. Also, we observe that the complexes with trigonal bipyramidal geometry are of higher energy than those of the square pyramidal geometry as a result of the increase number of out-of-phase interaction between the Cu d-orbitals and the ligand p-orbitals. In complexes 2, 4, 5, and 10, where the pair a′ and a″ have similar energies, the singlet state is an almost 50−50% mixture of the two microstates. For complexes 3 and 9 where the orbital energy difference between the two frontier orbital pairs is significant, the singlet state is predominated (60−70%) by the |a′a′ ̅| microstate. Complexes 1, 6, and 7 have (30−40%) contribution from |1a1a|̅ and 16−30% from |2a2a|.̅ Complexes 8 and 11 of no symmetry consist predominantly (∼100%) of the open shell singlet composed of |1a2a|̅ and |1a2a|̅ as 1Ψ0 = 1/(2)1/2[|1a2a|̅ − |1a2a|]. ̅ The two orbitals 1a and 2a are localized on different sites of the complex and 1Ψ0 represent a symmetry broken solution. 1Ψ0 represents the lowest energy singlet in these complexes. The corresponding ms = 0 triplet solution of lower energy is given by 3Ψ0 = 1/(2)1/2[|1a2a|̅ + |1a2a|]. ̅ The triplet component ms = 0, which is of lower energy than the singlet is given primarily as

Table 10. Lower Excited States for Complex 8 Based on SFCV(2)-DFT Calculations Using LDA-VWN Functional no symmetry −1

state

E, cm

(3A) 1(1A) 2(1A)

0 315 5601

3(1A)

6978


%

b(3A) b(1A) a(1A) c(1A) a(1A) c(1A)

96 97 96 2 2 96

Table 11. Lower Excited States for Complex 9 Based on SFCV(2)-DFT Calculations Using LDA-VWN Functional Cs symmetry −1

state

E, cm

(3A″) 1(1A′)

0 166

(1A″) 2(1A′)

5730 6356


%

b(3A″) a(1A′) b(1A′) (1A″) b(1A′) a(1A′)

98 65 33 100 66 34


3Ψ = 1/ 2 [|1a2a | + |1a 2a|] 0 ̅ ̅

Cs symmetry state

E, cm−1


%

(3A″) 1(1A′)

0 304

(1A″) 2(1A′)

6074 6482

b(3A″) a(1A′) b(1A′) (1A″) a(1A′) b(1A′)

98 50 49 100 50 50

(15)

(17)

The complexes, thus, exhibit a ferromagnetic behavior due to the triplet ground state just as in the case of all the other systems. Further investigations must be done to explain why we get the symmetry broken solution for complexes 8 and 11 only. Calculated J Values Based on SF-CV(2)-DFT. Table 14 displays the J values obtained from SF-CV(2)-DFT calculation, the experimentally determined J values and those obtained by employing the broken symmetry approach (BS-DFT).54 The BS-DFT results were obtained for the hybrid functionals B3LYP and BH&HLYP. Displayed in Table 14 are the J values obtained as the singlet−triplet energy gap (ΔEST) based on

for a′, whereas for a″ there is a π interaction between the Cu d-orbitals and the p-orbital of the chloro ligand. 2274



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Table 14. List of the Binuclear Triply-Bridged Copper(II) Compounds, the Calculated ΔEST Values (SF-CV(2)-DFT), the Values Obtained from the BS Approach and the Experimental Values in cm−1 ΔEa complexes

LDA

b

b

BLYP

1 2 3 4 5 6 7 8 9 10 11

143.7 286.2 357.7 407.3 332.1 287.8 283.0 315.1 166.1 303.5 324.6

83.0 208.5 212.1 297.3 227.8 208.7 208.1 230.9 115.4 224.5 239.6

BP86

b

101.0 216.2 242.4 317.6 242.6 217.9 216.6 237.8 129.5 232.0 246.0

PBE

b

98.6 214.8 235.8 307.5 241.6 215.8 214.5 236.7 128.2 230.7 244.6

B3LYPb

BH&HLYPb

B3LYPc

BH&HLYPc

expt.

85.1 150.0 132.2 194.2 163.9 156.6 154.8 162.4 111.2 153.2 166.2

50.5 66.9 50.0 75.4 70.0 68.0 67.6 71.8 40.4 65.8 70.7

99.2 169.0 162.6 185.7 161.3 169.7 166.6 176.7 120.9 166.2 181.4

57.6 83.2 82.4 83.6 78.3 80.5 80.4 87.0 53.2 79.6 85.6

24.1 38.6 62.5 79.1 79.7 120.0 120.8 148.9

a Note that positive values corresponds to a ferromagnetic ground state. bValues obtained from unrestricted SF-CV(2)-DFT. cBroken symmetry values obtained from ref 54.

Figure 5. Frontier SUMO orbitals a′ ̅, a′′ ̅ for Cs symmetry and 1a,̅ 2a̅ without symmetry for complexes 1−4 using the BH&HLYP functional, with the corresponding orbital energies. The SOMO orbitals a′, a″ and 1a, 2a are not shown because they have a spatial distribution similar to their SUMO counterparts.

B3LYP and BH&HLYP afford values closer to the experimental estimates. Where experimental values are available, it can be observed that the BH&HLYP is in good agreement with experiment for complexes 1−5 and the best agreement with experiment for complexes 6, 7, and 8 is the B3LYP calculation. For complexes 9−11, where the experimental J values are unavailable the calculations with the hybrid functionals compare well with the values obtained for hybrid functionals in the BS-DFT method. The hybrid SF-CV(2)-DFT calculations in general yield results that are comparable to those obtained with the BS-DFT method. We display in Figure 5 the SUMOs and SOMOs for complexes 1−4 generated by the BH&HLYP functional. A comparison with Figure 2 where the same orbitals are displayed

SF-CV(2)-DFT for the LDA-VWN, BP86, BLYP, PBE, B3LYP, and BH&HLYP functionals. We find for the complete series of complexes that the SFCV(2)-DFT schemes afford positive ΔEST and J values for all the functionals. Thus, our scheme predicts complexes 1−11 to be ferromagnetic, with the triplet (3Ψ0) having a lower energy than the singlet (1Ψ0), in agreement with experiment. Comparison between the J values obtained from the theoretical methods and the experimental estimates can be made through Table 14. As observed in previous studies,78,79 the SF-CV(2)-DFT results demonstrate a strong functional dependence. The LDA and the GGA’s strongly overestimate the magnitude of ΔEST for all complexes. On the other hand, the hybrid functionals 2275



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(12) Moreira, I.; de, P. R.; Costa, R.; Filatov, M.; Illas, F. J. Chem. Theor. Comput. 2007, 3, 764−774. (13) Illas, F.; Moreira, I.; de, P. R.; de Graaf, C.; Barone, V. Theor. Chem. Acc. 2000, 104, 265−272. (14) Moreira, I.; de, P. R.; Illas, F. Phys. Chem. Chem. Phys. 2006, 8, 1645−1659. (15) Tokii, T.; Hamamura, N.; Nakashima, M.; Muto, Y. Bull. Chem. Soc. Jpn. 1992, 65, 1214. (16) López, C.; Costa, R.; Illas, F.; Molins, E.; Espinosa, E. Inorg. Chem. 2000, 39, 4560. (17) López, C.; Costa, R.; Illas, F.; de Graaf, C.; Turnbull, M. M.; Landee, C. P.; Espinosa, E.; Mata, I.; Molins, E. Dalton Trans. 2005, 2322−2330. (18) Sletten, J. Acta Chem. Scand. A 1983, 37, 569. (19) Julve, M.; Verdaguer, M.; Gleizes, A.; Philoche-Levisalles, M.; Kahn, O. Inorg. Chem. 1984, 23, 3808. (20) de Meester, P.; Fletcher, S. R.; Skapski, A. C. J. Chem. Soc., Dalton Trans. 1973, 2575. (21) Castillo, O.; Muga, I.; Luque, A.; Gutierrez-Zorrilla, J. M.; Sertucha, J.; Vitoria, P.; Roman, P. Polyhedron 1999, 18, 1235. (22) Youngme, S; Phatchimkun, J.; Wannarit, N.; Chaichit, N.; Meejoo, S.; Van Albada, G. A.; Reedijk, J. Polyhedron 2008, 27, 304. (23) Chailuecha, C.; Youngme, S.; Pakawatchai, C.; Chaichit, N.; Van Albada, G. A.; Reedijk, J. Inorg. Chim. Acta 2006, 359, 4168. (24) Youngme, S.; Chailuecha, C.; Van Albada, G. A.; Pakawatchai, C.; Chaichit, N.; Reedijk, J. Inorg. Chim. Acta 2004, 357, 2532. (25) Youngme, S.; Chailuecha, C.; Van Albada, G. A.; Pakawatchai, C.; Chaichit, N.; Reedijk, J. Inorg. Chim. Acta 2005, 358, 1068. (26) Willet, R. D.; Gatteshi, D.; Kahn, O., Eds. Magneto-Structural Correlations in Exchange Coupled Systems, NATO ASI Series C; Reidel: Dordrecht, The Netherlands, 1985; Vol. 140. (27) Rodriguez-Fortea, A.; Ruiz, E.; Alemany, P.; Alvarez, S. Chem. Monthly 2003, 134, 307−316. (28) Hodgson, D. J. Prog. Inorg. Chem. 1975, 19, 173. (29) Crawford, V. H.; Richardson, H. W.; Wasson, J. R.; Hodgson, D. J.; Hatfield, W. E. Inorg. Chem. 1976, 15, 2107. (30) Ruiz, E.; Alemany, P.; Alvarez, S.; Cano, J. J. Am. Chem. Soc. 1997, 119, 1297. (31) Ruiz, E.; Alemany, P.; Alvarez, S.; Cano, J. Inorg. Chem. 1997, 36, 3683. (32) Meenakumari, S.; Tiwary, S. K.; Chakravarty, A. R. J. Chem. Soc., Dalton Trans. 1993, 2175−2181. (33) Burger, K. S.; Chaudhurri, R.; Wieghardt, K. J. Chem. Soc., Dalton Trans. 1996, 247−248. (34) Nishida, Y.; Kida, S. J. Chem. Soc., Dalton Trans. 1986, 2633− 2640. (35) Kawata, T.; Yamanaka, M.; Ohba, S.; Nishida, Y.; Nagamatsu, M.; Tokii, M.; Kato, M.; Steward, O. W. Bull. Chem. Soc. Jpn. 1992, 65, 2739−2747. (36) Castell, O.; Miralles, J.; Caballol, R. Chem. Phys. 1994, 179, 377−384. (37) Bencini, A.; Gatteschi, D. J. Am. Chem. Soc. 1986, 108, 5763− 5771. (38) Miralles, J.; Daudey, J. −P.; Caballol, R. Chem. Phys. Lett. 1992, 198, 555−562. (39) Cabrero, J.; Ben Amor, N.; de Graaf, C.; Illas, F.; Caballol, R. J. Phys. Chem. A 2000, 104, 9983−9989. (40) Adamo, C.; Barone, V.; Bencini, A.; Totti, F.; Ciofini, I. Inorg. Chem. 1999, 38, 1996−2004. (41) de Biani, F. F.; Ruiz, E.; Cano, J.; Novoa, J. J.; Alvarez, S. Inorg. Chem. 2000, 39, 3221−3229. (42) Noodleman, L. J. Chem. Phys. 1981, 74, 5737−5743. (43) Noodleman, L.; Case, D. A.; Aizman, A. J. Am. Chem. Soc. 1988, 110, 1001−1005. (44) Noodleman, L.; Peng, C. Y.; Case, D. A.; Mouesca, J. −M. Coord. Chem. Rev. 1995, 144, 199−244. (45) Martin, R. I.; Illas, F. Phys. Rev. Lett. 1997, 79, 1539−1542. (46) Caballol, R.; Castell, O.; Illas, F.; Moreira, L.; de, P. R.; Malrieu, J. P. J. Phys. Chem. A 1997, 101, 7860−7866.

based on LDA reveals that the two sets are qualitatively similar. However, the contributions from the bridging ligands are reduced for BH&HLYP compared to LDA. As a result, interaction integrals between the two centers are not as large for BH&HLYP as LDA. As a result, the J constants for BH&HLYP are smaller than for LDA.

III. CONCLUDING REMARKS The SF-CV(2)-DFT methodology for the calculation of exchange coupling constants has previously been employed to trinuclear Cu(II) systems78 and doubly bridged dinuclear Cu(II) complexes.80 The performance of the same methodology is here tested on a series of dinuclear triply bridged Cu(II) systems. The comparison of the SF-CV(2)-DFT J values with coupling constants obtained from BS-DFT and experiment demonstrated that the unrestricted SF-CV(2)-DFT scheme in conjunction with hybrid functionals affords values that are comparable to those obtained by the BS-DFT method and experiment. Thus, SF-CV(2)-DFT is a viable alternative to the BS-DFT scheme. The SF-CV(2)-DFT methodology has further the advantage of providing a description of the different states in terms of a configuration interaction (CI) expansion of microstates. Also it is possible to optimize the geometry of the different spin-states separately.78 Similar features are not readily available in the BS-DFT scheme

■

ASSOCIATED CONTENT

S Supporting Information *

Additional supporting figure. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by NSERC. The computational resources of WESTGRID through Compute Canada were used for all calculations. T.Z. thanks the Canadian Government for a Canadian Research Chair. I.S. is grateful to Dr. Masood Parvez for the access to CCDC and Dr. Ramon Costa for the revised structure of Complex 3.

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REFERENCES

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Calculation of Exchange Coupling Constants in Triply-Bridged

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