Chemical Physics Letters 505 (2011) 11–15

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Chemical Physics Letters

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Broken-symmetry natural orbital (BSNO)–Mk-MRCC study on the exchangecoupling in the binuclear copper(II) compounds

Toru Saito a,⇑, Natsumi Yasuda a, Satomichi Nishihara a, Shusuke Yamanaka a, Yasutaka Kitagawa a,Takashi Kawakami a, Mitsutaka Okumura a, Kizashi Yamaguchi b

a Graduate School of Science, Osaka University, 1–1 Machikaneyama, Toyonaka, Osaka 560-0043, Japanb NanoScience Design Center, Osaka University, 1–3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan

a r t i c l e i n f o

Article history:Received 29 November 2010In final form 8 February 2011Available online 12 February 2011

0009-2614/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.cplett.2011.02.018

⇑ Corresponding author. Fax: +81 6 6850 5550.E-mail address: tsaito@chem.sci.osaka-u.ac.jp (T. S

a b s t r a c t

We have investigated the scope and applicability of Mukherjee’s state-specific multireference coupledcluster singles and doubles (Mk-MRCCSD) method on the magnetic interactions in dichloro-, oxo-, andperoxo-bridged binuclear Cu(II) complexes. Several reference orbitals including broken-symmetry natu-ral orbitals (BSNOs) are examined for the Mk-MRCCSD computations. We compare the effective exchangeintegral (J) values calculated by Mk-MRCCSD with those calculated by conventional complete activespace self-consistent field (CASSCF) and MRMP2. The BS methods with approximate spin-projection(AP) method are also performed for comparison. It is found that the Mk-MRCCSD method can be appli-cable to the prediction of exchange couplings in binuclear copper(II) containing systems.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

The binuclear copper(II) complexes play important roles in bio-inorganic chemistry and molecular magnetism [1–3]. The descrip-tion of their electronic structures is one of the most challengingissues in quantum chemistry due to their multireference (MR)characters. The standard single-reference (SR) spin-restricted Har-tree–Fock (RHF) based coupled cluster (CC) approaches describeclosed-shell molecules well, but these conventional methods arenot applicable to the description of bond breaking, open-shell sin-glet diradical species, and transition metal compounds [4]. Variousapproaches based on CC theories have been proposed to overcomethe difficulty in such MR problems [5]. In particular, Mukherjee’sstate-specific multireference coupled cluster (Mk-MRCC) theory,which is size-extensive and intruder free, is a promising approachto treat MR systems with high accuracy [6,7]. Although recentstudies have shown that the Mk-MRCC method provides accurateresults for organic diradical systems [8–10], no Mk-MRCC calcula-tion has been examined on transition metal containing compoundsand its scope and applicability on the systems is still unclear.

As for chemical reactions in binuclear copper(II) complexes, theprediction of relative energy for the conversion of the l-g2:g2 per-oxo Cu2O2þ

2 core to the bis(l-oxo) one is a challenging problem forboth experimental and theoretical aspects [11]. The relative ener-gies between two isomers have been evaluated by a variety ofSR- and MR-based methods [12–15]. Very recently, Yanai et al.

ll rights reserved.

aito).

investigated the isomerization of the Cu2O2þ2 molecule by perform-

ing the density matrix renormalization group and canonical trans-formation (DMRG-CT) calculation with a (28e, 32o) active space(28 electrons in 32 orbitals) [14]. According to their conclusions,any complete active space (CAS) treatment can fail unless a verylarge active space is used even for such model system, while theMk-MRCC computations are limited to small active spaces. In thisway, at present, it is impossible to investigate chemical reactions ofthe transition metal system with the use of the Mk-MRCC method.In contrast to chemical reactions, magnetic interactions betweentwo copper(II) ions have been widely investigated by the broken-symmetry (BS) density functional theory (UDFT) and sophisticateddifference dedicated configuration interaction (DDCI) methods[16]. The nondynamical correlation effects between the magneticCu(d9)–Cu(d9) orbitals corresponding to the minimal (2e,2o) activespace should be crucial to evaluate the effective exchange integral(J) values.

In the present study, as a first step, we would like to investigatewhether the Mk-MRCC method is applicable to magnetic propertiesof transition metal systems. We examine Mk-MRCC with singlesand doubles (Mk-MRCCSD) approximation [15,16] on the magneticinteractions in four small binuclear copper(II) compounds. First, weinvestigate antiferromagnetic (AF) and ferromagnetic (F) couplingand diradical character (Y) in the dichloro-bridged Cu(II) complex,[Cu2Cl6]2�, since there have been many experimental and DDCIstudies [17–25]. As shown in Figure 1, the planar structure foundin dibenzotetratiafulvalenic cation (DBTTF) (1a) [19] shows AF cou-pling, whereas F coupling is observed in the non-planar structurefound in (Ph4As)2 (1b) [17] (for details, see Table 1 in Ref. [23]).

Figure 1. Illustration of model 1–3.Figure 2. Computational scheme of the Mk-MRCCSD method.

12 T. Saito et al. / Chemical Physics Letters 505 (2011) 11–15

The J and Y values calculated by the BS and MR methods are com-pared to the experimental values. In the case of BS methods, anapproximate spin-projection (AP) method is used to remove thespin contamination [26]. Then, we examine the J and Y values ofbinuclear copper-oxo and copper-peroxo model compounds, i.e.[HCu(II)(O2�)Cu(II)H] (2) and f½CuðIIÞðNH3Þ2�2O2�

2 g2þ (3). To our

knowledge, this is the first quantitative study to demonstrate theperformance of the Mk-MRCCSD method for magnetic behaviorsin transition metal containing systems.

2. Theoretical background

2.1. Approximate spin-projection (AP) method

The AP method is based on a local spin expression of HeisenbergHamiltonian [26]. In the case of two-site model, the Hamiltoniancan be expressed as

H ¼ �2JSa � Sb; ð1Þ

where Sa and Sb represent spin operators at sites a and b, respec-tively. J represents an effective exchange integral. By assuming thatthe spin contamination in the high spin (HS, S = 1) state is negligi-ble, we can readily derive the J value as follows:

J ¼ EBS � EHS

S2D EHS

� S2D EBS ; ð2Þ

where BS denotes the open-shell singlet (S = 0) state. Eq. (2) yieldsthe total energy of the BS state after the AP method

EAP ¼ EBS þ JhS2iBS: ð3Þ

In the case of post-spin-unrestricted Hartree–Fock (post-UHF)methods such as UHF-CC (UCC), we employed the followingPurvis–Sekino–Bartlett (PSB) scheme to calculate <S2>BS and<S2>HS values [27].

S2D E

UCCSD�

WUHFjS2jWUCCSD

D E

WUHFjWUCCSDh i : ð4Þ

2.2. Broken-symmetry natural orbital (BSNO) based Mk-MRCCSDmethod

As Evangelista et al. pointed out, the use of the localized activeorbitals is requisite to predict dissociation energies, while the erroris not so crucial for the energy differences such as singlet–tripletenergy gap (2J) and activation barrier heights [8–10,28]. For themagnetic behaviors in binuclear copper(II) systems, the nondy-namical correlation effects can mainly be derived from the ex-change interactions between the local S = 1/2 spins (2 electronsin 2 magnetic orbitals) [20–22]. We use the BSNO as the referenceorbital for the Mk-MRCCSD(2e,2o) computations like the UHF NO

complete active space self-consistent field (UNO–CASSCF) ap-proach [29]. The highest occupied NO (HONO) and lowest unoccu-pied NO (LUNO) pair, which corresponds to the magnetic naturalorbitals, is chosen as the (2e,2o) active space. In addition to UNO,the NO of the spin-unrestricted Brueckner reference determinant(UB-ref) of the Brueckner CC doubles (BD) calculation [30,31],namely BNO is examined. The UB-ref is defined by the requirementthat it has the maximum overlap with exact wave function [30].When the BSNO (UNO and BNO) is used, we can substitute totaldensity of UHF (or UB-ref) for averaged Fock operator, and diago-nalize the block of occupied orbitals except for the HONO–LUNOpair (magnetic natural orbitals) to extract the frozen and inactivecore orbitals in the same way as the MR second-order perturbationtheory (CASPT2) [32]. The CASSCF(2e,2o) NO (CNO) and semica-nonical spin-restricted open-shell HF (ROHF) orbitals are alsoexamined for comparison. The computational scheme of the BSNObased Mk-MRCC method is depicted in Figure 2.

2.3. Definition of diradical character

The NOON (n) obtained from the BS molecular orbital is relatedto an orbital overlap (T) between magnetic orbitals

n�i ¼ 1� Ti; ð5Þ

where nþi and n�i denote the occupation numbers of the correspond-ing occupied NO and the unoccupied NO, respectively. The Ti valuecontains important information to estimate the magneticinteraction, i.e. Ti = 0 for pure biradical and Ti = 1 for the closed-shellpair [33]. The diradical character (Y) is introduced by using NOON ofthe BS solution as follows;

Y ¼ 2WD ¼nþ2 � 4nþ þ 4nþ2 � 2nþ þ 2

; ð6Þ

where WD denotes the weight of double excitation [34]. In the caseof Mk-MRCCSD(2e,2o) calculations, the wave function of the delo-calized singlet state is

Wj i ¼ c1eT1 ðcoreÞ2ðvalenceÞ2x�x���

Eþ c2eT2 ðcoreÞ2ðvalenceÞ2y�y

���E

ðc1 P c2Þ: ð7Þ

As in the case of CASSCF(2e,2o), the Y value with the use of thedressed double excitation is defined on condition that open-shellsinglet contribution (c3) [12] is negligibly small,

Y ¼ 2c22: ð8Þ

3. Computational details

The def2-SV(P) basis sets were used for all atoms of 1a, 1b, and 3,while for 2 the TZV basis set and aug-cc-pVDZ basis sets were usedfor Cu and the other atoms, respectively [35–37]. In all calculations

Table 2Mulliken spin population on the Cu2Cl2 core for the BS (S = 0) and HS (S = 1) states of1a and 1b. Values in parentheses represent the results of the HS (S = 1) state.

Model Method Cu1 Cu2 Cl (bridge)

1a UBLYP 0.42 (0.43) �0.42 (0.43) 0.00 (0.22)UB3LYP 0.50 (0.49) �0.50 (0.49) 0.00 (0.20)UBHandHLYP 0.66 (0.65) �0.66 (0.65) 0.00 (0.14)UB2PLYP 0.68 (0.67) �0.68 (0.67) 0.00 (0.13)ULC-xPBE 0.52 (0.51) �0.52 (0.51) 0.00 (0.18)UHF 0.85 (0.84) �0.85 (0.84) 0.00 (0.07)UCCSD 0.66 (0.65) �0.66 (0.65) 0.00 (0.14)UB-ref 0.67 (0.66) �0.67 (0.66) 0.00 (0.13)

1b UBLYP 0.48 (0.48) �0.48 (0.48) 0.00 (0.19)UB3LYP 0.55 (0.55) �0.55 (0.55) 0.00 (0.16)UBHandHLYP 0.73 (0.72) �0.73 (0.72) 0.00 (0.10)UB2PLYP 0.75 (0.75) �0.75 (0.75) 0.00 (0.09)ULC-xPBE 0.58 (0.58) �0.58 (0.58) 0.00 (0.14)UHF 0.90 (0.89) �0.90 (0.89) 0.00 (0.04)UCCSD 0.73 (0.72) �0.73 (0.72) 0.00 (0.10)UB-ref 0.74 (0.74) �0.74 (0.74) 0.00 (0.09)

T. Saito et al. / Chemical Physics Letters 505 (2011) 11–15 13

Cartesian functions were used. Full geometry optimizations wereperformed by UB3LYP for 2 and 3 in the BS state without symmetryconstraint (C1 point group) [38]. Then frequency analyzes were car-ried out to check whether the optimized structures were the localminimum. The optimized Cartesian coordinates are provided inthe Supplementary Data. For 1a and 1b, X-ray crystal structureswithout counterions were used (see the Supplementary Data)[17,19]. For single-point energy calculations, UHF, UCCSD,UCCSD(T), UBDandUBD(T) were employed as the ab initio BS meth-ods [31,39]. The UBLYP, UB3LYP, and UBHandHLYP methods wereemployed as the pure and global hybrid exchange–correlation(XC) functional sets [38]. The long-range corrected ULC-xPBE anddouble hybrid UB2PLYP methods were also performed [40,41]. Allthese calculations were performed by GAUSSIAN 09 program packages,which was modified to calculate Eq. (4) [42]. The UNO–CASS-CF(2e,2o) and the following MRMP2(2e,2o) calculations were per-formed by GAMESS program package [43,44]. We prepared fourschemes as the reference orbitals for the Mk-MRCC computations:(i) ROHF, (ii) CNO, (iii) UNO, and (iv) BNO as shown in Figure 2.The Mk-MRCCSD(2e,2o) computations were employed by PSI3 pro-gram package [45], which was modified to use several referenceorbitals. The symmetry constraints were not used for all computa-tions. The Cu orbitals up to 3p, O and N 1s orbitals, and Cl orbitals upto 2p were frozen except for UHF and UDFT computations.

4. Results and discussions

4.1. [Cu(II)2Cl6]2 � (1a, 1b)

The calculated Y and J values for 1a and 1b are summarized inTable 1 (for details, see Supplementary Data Table S1 and S2). TheMulliken spin populations on Cu and bridging Cl atoms are summa-rized in Table 2. The HONO and LUNO of UNO, BNO, and CNO to-gether with their NOONs are depicted in Figure 3. Note that theseorbitals correspond to two single occupied MOs (SOMOs) obtainedwith ROHF. For 1a, the Cu(d9)–Cu(d9) superexchange interaction[46,47] via the bridging Cl atoms exhibit AF coupling withJ = �47 cm�1 from the definition in Eq. (1) [19,23]. The obtained Jvalues differ with the calculated methods, ranging from �219 to

Table 1Calculated diradical character and effective exchange integral values for 1a and 1b.

Method 1a 1b

Ya Jb Ya Jb

UBLYP 40.4 �219 95.9 85UB3LYP 73.1 �72 97.9 69UBHandHLYP 89.0 5 99.5 42UB2PLYP 90.2 �24 99.7 62ULC-xPBE 83.1 �25 99.4 58UHF 97.3 30 97.9 16UCCSD 87.0 �1 97.4 39UCCSD(T) – �16 – 41UBD 88.8c 3 99.8c 39UBD(T) – �18 – 43UNO–CASSCF(2e,2o) 97.2 2 99.6 7MRMP2(2e,2o) 97.2 �4 99.6 21ROHF–Mk-MRCCSD(2e,2o) 88.9 �33 99.8 31CNO–Mk-MRCCSD(2e,2o) 88.5 �34 99.9 32UNO–Mk-MRCCSD(2e,2o) 88.5 �36 100.0 32BNO–Mk-MRCCSD(2e,2o) 85.3 �42 99.6 36Expt. �47d 23e

a In %. Y values are evaluated by using Eqs. (6) and (8) for BS and MR methods,respectively.

b In cm�1. Eq. (2) is used.c The Y values for UB-ref are presented.d Ref. [19]. From the definition in Eq. (1).e Ref. [18]. From the definition in Eq. (1).

+5 cm�1. The superexchange interaction decreases with increasingHF exchange. As for UBHandHLYP, which contains 50% HF exchange,the spin delocalization effect stabilizing the HS (S = 1) state is dom-inant than the exchange interaction in the BS state, leading the po-sitive J value. The UHF method also shows the positive J value as wellas UBHandHLYP. On the other hand, the ULC-xPBE (with 100% long-range HF exchange) and double hybrid UB2PLYP method (with 53%HF exchange) are less sensitive to the amount of HF exchange thanthe conventional ones [24,25]. The UCCSD and UBD methods that in-clude dynamical electron correlation effects drastically improve theUHF methods. However, the magnitude of calculated J values signif-icantly underestimate the experimental one even at the UCCSD(T)and UBD(T) levels. The UNO–CASSCF(2e,2o) method gives the Y va-lue quite similar to UHF, and it also provides the positive J value.Although the subsequent MRMP2(2e,2o) calculation reproducesthe AF coupling qualitatively, its magnitude is smaller than theexperimental data. From the result, the second-order perturbativecorrections to the CASSCF(2e,2o) wave function do not provide aqualitatively reliable result for the exchange coupling of 1a. In con-trast to MRMP2(2e,2o), the Mk-MRCCSD(2e,2o) results rangingfrom �42 to �33 cm�1 are close in value to the experiment. There-fore, it is found that the minimal (2e,2o) active space in combinationwith the exponential ansatz performs well. Although there is notmuch difference in the NOONs of HONO–LUNO pair, the totalBNO–Mk-MRCCSD(2e,2o) energies for both spin states lie about11 mEh above the other Mk-MRCCSD(2e,2o) ones (see Supplemen-tary Data Table S1).

The distorted F complex (1b) synthesized by Willett et al. is alsoconsidered [17]. The twist angle (46.5�) causes the F coupling withJ = 23 cm�1 from the definition in Eq. (1) [18,23]. The large Y valuesfor 1b indicate that the unpaired electrons are almost completelylocalized at each copper site. The calculated J values exhibit F cou-pling regardless of level of theory as listed in Table 1. The J valuesobtained by the UDFT methods overestimate the experimental va-lue, and these values decrease with increasing HF exchange ratiofor the global hybrid functionals. The results of UB2PLYP andULC-xPBE lie between those of UB3LYP and UBHandHLYP. In con-trast to UDFT methods, the UHF calculation slightly underesti-mates the experiment. The inclusion of the dynamical electroncorrelation effects overshoot the experimental value, and the ob-tained values are close to the UBHandHLYP result. The UNO–CASS-CF(2e,2o) method gives the small positive J value (7 cm�1), whilethe result of the MRMP2(2e,2o) method (21 cm�1) is in good agree-ment with the experiment. The Mk-MRCCSD(2e,2o) results rangingbetween 31 and 36 cm�1 slightly overestimate the experimental Fcoupling (23 cm�1). This tendency is consistent with the very re-cent DDCI study of Calzado et al. (27 cm�1) at the same geometry

Figure 3. HONO and LUNO of 1–3. Values represent are NOONs of HONO and LUNOfor the UNO, BNO, and <CNO>.

Table 3Calculated diradical character and effective exchange integral values for 2 and 3.

Method 2 3

Ya Jb Ya Jb

UBLYP 93.7 2047 0.0 �2447UB3LYP 76.3 2989 10.7 �2234UBHandHLYP 86.7 �125 44.4 �1859UB2PLYP 87.6 �186 49.0 �3224ULC-xPBE 84.8 �72 18.8 �2411UHF 94.4 �26 84.2 �426UCCSD 87.6 �95 39.9 �2197UCCSD(T) – �101 – �2721UBD 87.7c �92 35.0c �1951UBD(T) – �102 – �2660UNO–CASSCF(2e,2o) 93.9 �42 83.7 �405MRMP2(2e,2o) 93.9 �156 83.7 �1545ROHF–Mk-MRCCSD(2e,2o) 83.7 �164 47.7 �2704CNO–Mk-MRCCSD(2e,2o) 82.7 �175 45.5 �2633UNO–Mk-MRCCSD(2e,2o) 83.3 �159 45.5 �2597BNO–Mk-MRCCSD(2e,2o) 84.1 �104 35.8 �2839

a In %. Y values are evaluated by using Eqs. (6) and (8) for BS and MR methods,respectively.

b In cm�1. Eq. (2) is used.c The Y values for UB-ref are presented.

Table 4Mulliken spin population on the Cu2O2 core for the BS (S = 0) and HS (S = 1) states of 2and 3. Values in parentheses represent the results of the HS (S = 1) state.

Model Method Cu1 Cu2 O1 O2

2 UBLYP 0.02(0.53)

�0.01(0.53)

�0.08(0.96)

–

UB3LYP 0.87(0.53)

�0.87(0.53)

0.00 (0.96) –

UBHandHLYP 0.97(0.97)

�0.97(0.97)

0.00 (0.05) –

UB2PLYP 0.98(0.98)

�0.98(0.98)

0.00 (0.04) –

ULC-xPBE 0.81(0.80)

�0.81(0.80)

0.00 (0.06) –

UHF 1.04(1.04)

�1.04(1.04)

0.00 (0.00) –

UCCSD 0.91(0.91)

�0.91(0.91)

0.00 (0.03) –

UB-ref 0.93(0.92)

�0.93(0.92)

0.00 (0.03) –

3 UBLYP 0.06(0.41)

�0.06(0.41)

0.00 (0.48) 0.00(0.48)

UB3LYP 0.42(0.47)

�0.42(0.47)

0.00 (0.42) 0.00(0.42)

UBH andHLYP

0.65(0.70)

�0.65(0.70)

0.00 (0.20) 0.00(0.20)

UB2PLYP 0.67(0.72)

�0.67(0.72)

0.00 (0.18) 0.00(0.18)

ULC-xPBE 0.47(0.49)

�0.47(0.49)

0.00 (0.40) 0.00(0.40)

UHF 0.86(0.87)

�0.86(0.87)

0.00 (0.08) 0.00(0.08)

UCCSD 0.63(0.74)

�0.63(0.74)

0.00 (0.15) 0.00(0.15)

UB-ref 0.63(0.71)

�0.63(0.71)

0.00 (0.19) 0.00(0.19)

14 T. Saito et al. / Chemical Physics Letters 505 (2011) 11–15

as this work [22]. Although the total BNO–Mk-MRCCSD(2e,2o)energies for both spin states lie about 9 mEh above the otherMk-MRCCSD(2e,2o) ones in accordance with 1a (see Supplemen-tary Data Table S1), its J value does not differ from the others. Inthis way, the Mk-MRCCSD method together with the minimal(2e,2o) active space yield reliable J values for 1a and 1b regardlessof the reference orbitals.

4.2. [HCu(II)(O2�)Cu(II)H] (2)

It is well established that the linear Cu–O–Cu bond having anangle of 180� shows superexchange AF coupling [48,49]. The calcu-lated J and Y values are summarized in Table 3 (for details, see Sup-plementary Data Table S3). The Mulliken spin populations in the BSstates are localized on each Cu(II) ion except for the result of UBLYP

as summarized in Table 4. The UDFT methods with small amountof HF exchange (UBLYP and UB3LYP) predict the large spin popula-tion (�1.0) at the O atom, i.e. Cu1.5–O1�–Cu1.5 in each HS state. Itmeans that spin delocalization effect stabilizing the HS state isdominant than superexchange interaction, and the HS state is in-deed predicted to be the ground state, i.e. J > 0. It strongly indicatesthat the sign of the J value calculated by UDFT methods heavily de-pends on used XC functionals. Therefore, the UDFT methods do notalways give qualitatively reliable results for strongly correlatedsystems such as 1a and 2, not to mention quantitative predictions.In contrast to UDFT methods, CC methods based on both UHF andUB-ref yield AF coupling. The additional perturbative triple excita-tion corrections (T) to these methods have an insignificant effect onthe J values. The MRMP2(2e,2o) calculation based on the UNO–CASSCF(2e,2o) wave function enhances the magnitude of J valueby correcting the dynamical correlation effect. As shown in Figure3, the differences in NOONs among three methods are relativelysmall similar to 1a and 1b. Nevertheless, the total BNO–Mk-MRCCSD(2e,2o) energies for both spin states lie 7–9 mEh abovethe others (see Supplementary Data Table S3). The magnitude ofthe J value calculated by the BNO–Mk-MRCCSD(2e,2o) method isat least 52 cm�1 smaller than the other Mk-MRCCSD(2e,2o) andMRMP2(2e,2o) results.

4.3. f½CuðIIÞðNH3Þ2�2O2�2 g

2þ (3)

The superexchange interaction between two Cu(II) ions via per-oxo (O2�

2 ) is responsible for the strong AF coupling [11]. For example,the singlet–triplet energy gap (�2J) of oxyhemocyanin measured bySQUID magnetic susceptibility indicated that the AF coupling wasstronger than 600 cm–1 [50]. As listed in Table 4, the spin densityin the BS state is delocalized at the Cu2O2 core. As for UDFT methods,the correlation between the electronic structure of the Cu2O2 core

T. Saito et al. / Chemical Physics Letters 505 (2011) 11–15 15

and the HF exchange ratio is discussed elsewhere [12,33]. As far asconventional pure and global hybrid DFT methods are concerned,the absolute values of J decrease with the increase of Y values. TheULC-xPBE method provides the relatively small Y value (18.8%),while it enhances the localization of the spin density distributionat the Cu2O2 core as listed in Table 3. Although a large portion ofHF exchange in UB2PLYP (53% cf. 50% in UBHandHLYP) brings aboutthe largest Y value (49.0%) among the UDFT methods, the UB2PLYPmethod provides the strong AF coupling (�3224 cm�1) due to theperturbative correction. UBLYP and UB3LYP also reproduce the AFcoupling since the electronic structure corresponds to the interme-diate correlation regime. The Y value obtained by UHF is quite sim-ilar to those of UNO–CASSCF(2e,2o), but it is quite larger than thoseobtained by the UDFT methods. Unlike 1a, 1b, and 2, the Y valuesestimated by UCCSD and UB-ref are much smaller than the valueestimated by UHF. Since more or less dynamical correlations effectsare included in UB-ref [15], the NOON of BNO and the subsequent Yvalue also differ from those of UNO as depicted in Figure 3. Thesesignificant changes in the J values suggest that the dynamical corre-lation effects by highly accurate methods are crucial for the mag-netic interaction in the Cu2O2 core. The perturbative triplescorrections (T) also greatly stabilize the singlet state by 500–700 cm�1.

As for MR-based methods, the MRMP2(2e,2o) method signifi-cantly augments the AF coupling as compared to UNO–CASS-CF(2e,2o). The Mk-MRCCSD(2e,2o) method greatly suppress thediradical character. The Y values calculated by the BNO-Mk-MRCCSD(2e,2o) solutions (35.8%) is smaller than the others rang-ing from 45.5% to 47.7%. The difference can be attributed to the factthat the bond nature of the reference BNO is significantly differentfrom those of ROHF, CNO and UNO. The total BNO–Mk-MRCC ener-gies for both spin states lie about 14 mEh above the others, whilethe differences in the total energies among the others are within0.4 mEh (see Supplementary Data Table S4). As compared to 1aand 1b, in which the absolute values of J are small, the deviationsof the BNO-Mk-MRCCSD(2e,2o) results from the ROHF–, CNO–, andUNO–Mk-MRCCSD(2e,2o) ones are remarkable in 3.

5. Conclusions

We have investigated the scope and applicability of Mk-MRCCSDmethod on the magnetic interactions in four binuclear copper(II)containing compounds. As for F coupling (1b) and strongly AF cou-pling (3) systems, all methods used in this work perform well forpredicting qualitatively correct J values, although the bond nature(diradical character) varies with the methods. As compared to theconventional pure and global hybrid density functionals, the recentrange-separated and double hybrid DFT methods show significantimprovements for the prediction of exchange couplings. TheMRMP2(2e,2o) result of 1a indicates that the second-order pertur-bative treatments with the use of the minimal active space do notreproduce the experiment quantitatively as well as UCCSD(T) andUBD(T). It should be difficult to determine a proper active spacewithout empirical adjustments if it is larger than the minimal(2e,2o). The most straightforward way is performing the DMRG cal-culations with the full valence active space. Alternatively, we findthat the Mk-MRCCSD(2e,2o) calculations also yield quantitativelyreliable results due to the exponential ansatz accounting for thedynamical correlation. Therefore, the Mk-MRCCSD(2e,2o) approachhas potential to estimate the reliable magnetic interactions withhigh accuracy for binuclear copper(II) complexes.

Acknowledgements

T. S. is grateful for the Research Fellowships from Japan Societyfor the Promotion of Science for Young Scientists (JSPS). This work

has been supported by Grants-in-Aid for Scientific Research(KAKENHI) (Nos. 19750046, 19350070) from JSPS and that onGrants-in-Aid for Scientific Research on Innovative Areas (‘‘Coordi-nation Programming’’ area 2170, No. 22108515) from JSPS and thaton Priority Areas (No. 19029028) from the Ministry of Education,Culture, Sports, Science and Technology (MEXT).

Appendix A. Supplementary data

Supplementary data associated with this Letter can be found, inthe online version, at doi:10.1016/j.cplett.2011.02.018.

References

[1] R.H. Holm, P. Kennepohl, E.I. Solomon, Chem. Rev. 96 (1996) 2239.[2] L. Que Jr., W.B. Tolman, Nature 455 (2008) 333.[3] O. Kahn, Molecular Magnetism, VCH Pub, New York, 1993.[4] J. Paldus, X. Li, Adv. Chem. Phys. 110 (1999) 1.[5] P. Carsky, J. Paldus, J. Pitter, Recent Progress in Coupled Cluster Methods:

Theory and Applications (Challenging and Advances in ComputationalChemistry and Physics), Springer, New York, 2010.

[6] U.S. Mahapatra, B. Datta, D. Mukherjee, J. Chem. Phys. 110 (1999) 1822.[7] F.A. Evangelista, W.D. Allen, H.F. Schaefer III, J. Chem. Phys. 125 (2006) 154113.[8] F.A. Evangelista, W.D. Allen, H.F. Schaefer III, J. Chem. Phys. 127 (2007) 024102.[9] T. Saito, S. Nishihara, S. Yamanaka, Y. Kitagawa, T. Kawakami, M. Okumura, K.

Yamaguchi, Mol. Phys. 108 (2010) 2533.[10] S. Nishihara, T. Saito, S. Yamanaka, Y. Kitagawa, T. Kawakami, M. Okumura, K.

Yamaguchi, Mol. Phys. 108 (2010) 2559.[11] E.I. Solomon et al., Faraday Discuss 148 (2011) 11.[12] C.J. Cramer, M. Włoch, P. Piecuch, C. Puzzarini, L. Gaugliardi, J. Phys. Chem. A

110 (2006) 1991.[13] P.-Å. Malmqvist, K. Pierloot, A. Rehaman, M. Shahi, C.J. Cramer, L. Gagliardi, J.

Chem. Phys. 128 (2008) 204109.[14] T. Yanai, Y. Kurashige, E. Neuscamman, G.K.L. Chan, J. Chem. Phys. 132 (2010)

024105.[15] L. Kong, M. Nooijen, Int. J. Quant. Chem. 108 (2008) 2097.[16] A. Bencini, Inorg. Chim. Acta. 361 (2008) 3820.[17] R.D. Willett, C. Chow, Acta Crystallogr. Sect. B 30 (1974) 207.[18] C. Chow, R. Caputo, R.D. Willett, B.C. Gerstein, J. Chem. Phys. 61 (1974) 271.[19] M. Honda, C. Katayama, J. Tanaka, M. Tanaka, Acta Crystallogr. Sect. C 41

(1985) 197.[20] O. Castell, J. Miralles, R. Caballol, Chem. Phys. Lett. 179 (1994) 377.[21] C.J. Calzado, J. Cabrero, J.P. Malrieu, R. Caballol, J. Chem. Phys. 116 (2002) 2728.[22] C.J. Calzado, C. Angeli, D. Taratiel, R. Caballol, J.P. Malrieu, J. Chem. Phys. 131

(2009) 044327.[23] M. Rodriguez, A. Llobet, M. Corbella, Polyhedron 19 (2000) 2483.[24] R. Valero, R. Costa, I. de P.R. Moreira, D.G. Truhlar, F. Illas, J. Chem. Phys. 128

(114 103).[25] P. Rivero, I. de P.R. Moreira, F. Illas, G.E. Scuseria, J. Chem. Phys. 129 (184 110).[26] K. Yamaguchi, Y. Takahara, T. Fueno, in: V.H. Smith, H.F. Schaefer III, K.

Morokuma (Eds.), Appl. Quant. Chem. D. Reidel, Boston, MA, 1986, p. 155.[27] G.D. Purvis III, H. Sekino, R.J. Bartlett, Coll. Czech. Chem. Commun. 53 (1988)

2203.[28] T. Saito, S. Nishihara, S. Yamanaka, Y. Kitagawa, T. Kawakami, M. Okumura, K.

Yamaguchi, Chem. Phys. Lett. 498 (2010) 253.[29] J.M. Bofill, P. Pulay, J. Chem. Phys. 90 (1989) 3637.[30] K.A. Brueckner, Phys. Rev. 96 (1954) 508.[31] N.C. Handy, J.A. Pople, M. Head-Gordon, K. Raghavachari, G.W. Trucks, Chem.

Phys. Lett. 164 (1989) 185.[32] K. Andersson, P.-Å. Malmqvist, B.O. Roos, J. Chem. Phys. 96 (1992) 1218.[33] T. Saito et al., Chem. Phys. 368 (2010) 1.[34] K. Yamaguchi, Chem. Phys. Lett. 33 (1975) 330.[35] F. Weigend, R. Ahlrichs, Phys. Chem. Chem. Phys. 7 (2005) 3297.[36] A. Schafer, C. Huber, R. Ahlrichs, J. Chem. Phys. 100 (1994) 5829.[37] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007.[38] A.D. Becke, J. Chem. Phys. 98 (1993) 1372.[39] K. Raghavachari, G.W. Trucks, J.A. Pople, M. Head-Gordon, Chem. Phys. Lett.

157 (1988) 479.[40] J. Heyd, G. Scuseria, M. Ernzerhof, J. Chem. Phys. 118 (2003) 8207.[41] T. Schwabe, S. Grimme, Phys. Chem. Chem. Phys. 9 (2007) 3397.[42] M.J. Frisch et al., GAUSSIAN 09, Revision A.02, Gaussian Inc., Wallingford CT, 2009.[43] K. Hirao, Chem. Phys. Lett. 190 (1992) 374.[44] M.W. Schmidt et al., J. Comput. Chem. 14 (1993) 1347.[45] T.D. Crawford et al., J. Comput. Chem. 28 (2007) 1610.[46] H.A. Kramers, Physica 1 (1934) 304.[47] P.J. Hay, J.C. Thibeault, R. Hoffmann, J. Am. Chem. Soc. 97 (1975) 4884.[48] K. Yamaguchi, Y. Kitagawa, T. Onishi, H. Isobe, T. Kawakami, H. Nagao, S.

Takamizawa, Coord. Chem. Rev. 226 (2002) 235.[49] J. Tahir-Kheli, W.A. Goddard III, Chem. Phys. Lett 472 (2009) 153. and

references therein.[50] D.M. Dooley, R.A. Scott, J. Ellinghaus, E.I. Solomon, H.B. Gray, Natl. Acad. Sci.

U.S.A. 75 (1978) 3019.