Density functional theory study of the magnetic

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Density functional theory study of the magnetic coupling interaction in a series of binuclear oxalate complexes Marko Perić, Matija Zlatar, Maja Gruden-Pavlović & Sonja Grubišić

Monatshefte für Chemie - Chemical Monthly An International Journal of Chemistry ISSN 0026-9247 Volume 143 Number 4 Monatsh Chem (2012) 143:569-577 DOI 10.1007/s00706-011-0705-1

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Author's personal copy Monatsh Chem (2012) 143:569–577 DOI 10.1007/s00706-011-0705-1

ORIGINAL PAPER

Density functional theory study of the magnetic coupling interaction in a series of binuclear oxalate complexes Marko Peric´ • Matija Zlatar • Maja Gruden-Pavlovic´ Sonja Grubisˇic´

•

Received: 5 August 2011 / Accepted: 13 December 2011 / Published online: 11 January 2012 Ó Springer-Verlag 2012

Abstract Magnetic couplings in oxalate-bridged binuclear complexes, namely five isomers of [(VO)2(ox) (SCN)6]4-, trans-(equatorial, equatorial), cis-(equatorial, equatorial), trans-(axial, axial), cis-(axial, axial), and (axial, equatorial), as well as [Cr2(ox)(SCN)8]4-, [Fe2(ox) (SCN)8]4-, [CrFe(ox)(SCN)8]4-, [Fe2(ox)5]4-, [Cr2(ox)5]4-, [Ni2(ox)5]6-, and [Cu2(ox)(C12H8N2)2]2?, were calculated with the broken symmetry approach. Predominant antiferromagnetic coupling is found in almost all investigated complexes, except in [CrFe(ox)(SCN)8]4-. The best agreement with experimental values for the exchange coupling constants were obtained at the B3LYP level of theory, whereas the non-hybrid functionals gave the best trend for the investigated vanadium complexes. The linear relationship between coupling constant and (e2 - e1)2 as well as linear dependence of J and the square of overlap integral of magnetic orbitals was estimated.

topic in computational chemistry since the first half of the twentieth century [1–10]. The idea of using molecules, rather than the ionic and metallic lattices of typical magnets, stems from the rapid development of functional molecular materials, and gives a better insight into the nature of coupling between paramagnetic centers. According to the Heisenberg–Dirac–Van Vleck Hamiltonian the exchange coupling constant (J) can be obtained through the energy difference between highest and lower spin states (Eq. 1). _

_

HEX ¼ JSA SB _

ð1Þ

_

where SA and SB represent the effective spin operators for each of the two metal centers and J is the exchange coupling constant. The effective spin operators are related to the total spin operator (Eq. 2). _

_

_

S ¼ SA þ SB

Keywords Density functional theory  Binuclear oxalate complexes  Magnetic couplings  Broken symmetry Introduction A deeper understanding of the microscopic processes related to magnetic interactions has been a very interesting M. Peric´  M. Zlatar  S. Grubisˇic´ (&) Center for Chemistry, Institute of Chemistry, Technology and Metallurgy, University of Belgrade, Njegosˇeva 12, P.O. Box 815, 11001 Belgrade, Serbia e-mail: [email protected] M. Gruden-Pavlovic´ Faculty of Chemistry, University of Belgrade, Studentski trg 16, P.O. Box 158, 11001 Belgrade, Serbia

ð2Þ

Using the previous two equations, one can rewrite the Heisenberg–Dirac–Van Vleck Hamiltonian in the final form (Eq. 3).   J _2 _2 _2 HEX ¼  S  SA  SB ð3Þ 2 Hence, the expression for the eigenvalues of HEX for different spin states is presented in Eq. 4. J E2Sþ1 ¼  ½SðS þ 1Þ  SA ðSA þ 1Þ  SB ðSB þ 1Þ 2

ð4Þ

S, SA, and SB are the quantum numbers associated with the spin operators and E2S?1 represents the energy of a spin multiplet. In the case of two interacting centers, with one unpaired electron at each center, there are only two states that have to be considered, i.e., singlet and triplet. The energies of

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corresponding states are E1 ¼ 3J4 and E3 ¼  J4. The exchange coupling constant is thus equal to the singlet– triplet separation. When J is negative, the singlet state is lower in energy and the coupling is antiferromagnetic. If J has a positive value, the triplet is ground state and the interactions are ferromagnetic. From that point of view the total coupling constant can be expressed as a sum of ferroand antiferromagnetic contributions (Eq. 5). J ¼ 2Kab 

ðe2  e1 Þ2 Jaa  Jab

ð5Þ

where e1 and e2 are the energies of the semi-occupied molecular orbitals (SOMOs) in the triplet state, Kab stands for the exchange integral, and Jaa and Jab are the Coulomb integrals. The Kab contribution will always favor the ferromagnetic interactions, whereas the second contribution will favor the antiferromagnetic interactions. Density functional theory (DFT) methods are widely used for the computation of exchange coupling constants (J). The broken symmetry (BS) approach developed by Noodleman [11–14] is certainly the most frequently applied one [1–10, 15, 16]. The spin states with highest multiplicity can be correctly described by a single determinant, which is not the case for the states with lower multiplicity. The energy of the BS single determinant, which is not the eigenstate of S2, is a weighted average of the energies of pure spin states, so the BS wavefunction can be expressed (Eq. 6) as follows: X WBS ¼ AðSÞWS ð6Þ S

The BS state is constructed in order to obtain antiferromagnetic state where the a and b spin densities are localized on different paramagnetic metals. The two different metal-based d orbitals, bearing electrons with different spins, located on each center with maximum local ms, are coupled together to have minimum total Ms. If we consider a binuclear system of interacting centers A and B, bearing magnetic orbitals a and b with one unpaired electron, two spatial nonorthogonal orbitals can be constructed, i.e., a0 = a ? cb and b0 = b ? ca. The corresponding BS determinant (Eq. 7) can be constructed as follows:

where M is the normalization factor and a and b stand for spins. The BS determinant given in Eq. 7 can be expressed as the sum of four different terms (Eq. 8).  1  ð8Þ jWBS i ¼ pffiffiffiffiffi /1 þ cð/2 þ /3 Þ þ c2 /4 M The /4 term can be neglected owing to the small value of c2, so the remaining three terms correspond to totally covalent A–B state (/1) and ionic states A?–B- and A-– B? (/2, /3). Any normalized state / a can be projected onto its S, Ms _ component using the MSS O operator. Therefore, each state can be expressed (Eq. 9) as: XX _ S /b ¼ ð9Þ MS O/a S

MS

According to Eq. 9, the relation between pure spin state and BS wavefunction is given in Eq. 10. _

WS ¼ ðAÞS0 OWBS

Finally the exchange coupling constant can be estimated according to the Yamaguchi approach (Eq. 11) [17, 18]. J¼

ðEHS  EBS Þ hS2 iHS hS2 iBS

ð11Þ

where EHS is the energy of the high   spin, EBS  isthe energy of the broken symmetry, and S2 HS and S2 BS are the expectation values of the high-spin and broken symmetry spin operators. A significant amount of magnetostructural research work during the past two decades has been devoted to analyzing the remarkable ability of the oxalate bridge to mediate exchange coupling between first-row transition metal ions in both homo- and heteropolynuclear compounds [19–22]. The most important ability of oxalate ion is the formation of homo- and heterometallic two- [23] and three-dimensional [24] networks that have applications as molecular-based magnetic materials. Thus, the DFT study of magnetic properties of a series of binuclear oxalatebridged complexes, namely five isomers of [(VO)2(ox) (SCN)6]4-, trans-(equatorial, equatorial) (1), cis-(equatorial, equatorial) (2), trans-(axial, axial) (3), cis-(axial, axial) (4), and (axial, equatorial) (5), and [Cr2(ox)(SCN)8]4- (6), [Fe2(ox)(SCN)8]4- (7), [CrFe(ox)(SCN)8]4- (8), [Fe2(ox)5]4-

1 1 WBS ¼ pffiffiffiffiffi pffiffiffiffiffi ½ða þ cbÞaa1 aa2 a. . .an a. . .an aðb þ caÞbb1 bb2 b. . .bn b N! M ( ) ½aaa1 aa2 a. . .an abbb1 bb2 b. . .bn b þ cð½baa1 aa2 a. . .an abbb1 bb2 b. . .bn bÞ 1 1 ¼ pffiffiffiffiffi pffiffiffiffiffi N! M þ cð½aaa1 aa2 a. . .an aabb1 bb2 b. . .bn bÞ þ c2 ð½baa1 aa2 a. . .an aabb1 bb2 b. . .bn bÞ

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ð10Þ

ð7Þ

Author's personal copy DFT study of the magnetic coupling interaction

(9), [Cr2(ox)5]4- (10), [Ni2(ox)5]6- (11), and [Cu2(ox) (C12H8N2)2]2? (12), was performed in order to get a better insight into the mechanism of couplings through oxalate bridge (Fig. 1). Different [(VO)2(ox)(SCN)6]4- isomers were explored in an effort to determine relations between J and the various positions of M=O double bonds [25, 26]. Furthermore, we analyzed the influence of different functionals on the geometries and magnetic properties of the investigated binuclear species.

Results and discussion The structures of the binuclear complexes were optimized using BS optimization in order to obtain geometries of antiferromagnetically coupled states. All the optimized structures were ascertained to be minima on the potential energy surface, and are characterized by the absence of vibrational modes with negative force constants. The structural parameters for the energy-minimized structures and comparison with the available crystallographic data for the investigated binuclear complexes are presented in Table 1. For all molecules, the LDA functional gives shorter bond lengths than OPBE and PW91 functionals, except for the M=O double bond in vanadium complexes 1–5. Better agreement in M–Oox and M=O bond distances between the optimized and X-ray structure of 1, as well as for M–Oox in 6 was obtained with the LDA functional. The M–Oox bond distances in binuclear compounds 1 and 2

571

appointed trans to M=O bond are longer than other M–Oox bonds due to the strong trans effect. Looking at the optimized structures of 3 and 4, it can be seen that the M–Oox bond distances are shorter than M–Oox trans to M=O and slightly longer than M–Oox cis to M=O in binuclear compounds 1, 2, and 5. In addition, the longer bond distances for binuclear compounds 7, 9, and 11 can be expected owing to two extra electrons in the antibonding M–L r* orbitals. The shortest bond distances were obtained for compound 12 as a consequence of covalency of the metal– ligand bond. In general, bond angles were reproduced with good accuracy with all functionals (Table 1). Calculations of magnetic couplings were carried out with the B3LYP functional at the LDA, OPBE, and PW91 optimized geometries of all the binuclear compounds under study and the available X-ray structures of binuclear complexes 1, 6, and 12 [27–29]. All calculated exchange couplings are in good agreement with the experimentally obtained values (Table 2) [27, 28, 30, 31]. Model systems 2, 3, 4, and 5 are not yet synthesized, so there are no available experimental values for comparison. As already mentioned, very important data can be obtained by analyzing coupling constants of the various isomers. We also performed calculations with LDA, OPBE, and PW91 non-hybrid functionals, to get a better insight into the trends of change through a series of binuclear oxalate complexes (Table 3). With the exception of complex 8, in which ferromagnetic interactions were found, in all other complexes

Fig. 1 Structures of investigated oxalate bridged binuclear complexes: trans(equatorial, equatorial) [(VO)2(ox)(SCN)6]4- (1), cis(equatorial, equatorial) [(VO)2(ox)(SCN)6]4- (2), trans(axial, axial) [(VO)2(ox)(SCN)6]4- (3), cis(axial, axial) [(VO)2(ox)(SCN)6]4- (4) (axial, equatorial) [(VO)2(ox)(SCN)6]4- (5), [Cr2(ox)(SCN)8]4- (6), [Fe2(ox)(SCN)8]4- (7), [CrFe(ox)(SCN)8]4- (8), [Fe2(ox)5]4- (9), [Cr2(ox)5]4(10), [Ni2(ox)5]6- (11), [Cu2(ox)(C12H8N2)2]2? (12)

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˚ and angles /° for the resulting DFT optimized structures of oxalate-bridged binuclear complexes and Table 1 Selected average bond lengths /A comparison with the available crystallographic data M–O

M–Oox

M–N

OoxMOox

transNMOox

transOMOox

trans-(equatorial, equatorial) [(VO)2(ox)(SCN)6]4- (1) X-ray

2.053/2.237a

1.582

2.054

76.20

164.50

171.60

LDA

2.023/2.237a

1.586

2.004

75.50

165.30

170.20

OPBE

2.077/2.368a

1.579

2.085

73.20

164.70

168.70

PW91

2.075/2.308a

1.599

2.067

74.50

164.80

169.60

cis-(equatorial, equatorial) [(VO)2(ox)(SCN)6]4- (2) LDA OPBE

2.026/2.222a 2.109/2.313a

1.587 1.580

2.005 2.085

75.60 73.60

166.50 167.00

169.40 166.80

PW91

2.092/2.278a

1.600

2.067

74.60

166.35

168.25

trans-(axial, axial) [(VO)2(ox)(SCN)6]4- (3) LDA

2.066

1.588

2.054

79.10

167.50

–

OPBE

2.129

1.578

2.158

77.50

168.00

–

PW91

2.115

1.600

2.124

78.30

167.80

–

cis-(axial, axial) [(VO)2(ox)(SCN)6]4- (4) LDA

2.069

1.590

2.052

79.00

168.10

–

OPBE

2.134

1.581

2.152

77.40

168.92

–

PW91

2.119

1.602

2.119

78.20

168.40

–

(axial, equatorial) [(VO)2(ox)(SCN)6]

4-

(5)

LDA

2.051/2.242a

1.587

2.031

77.25

167.00

169.50

OPBE

2.110/2.375a

1.579

2.124

75.40

167.05

167.30

PW91

2.101/2.312a

1.599

2.097

76.35

166.75

168.60

[Cr2(ox)(SCN)8]4- (6) X-ray 2.017

–

2.007

82.30

173.40

–

LDA

2.016

–

1.925

80.80

173.90

–

OPBE

2.084

–

1.989

79.20

173.10

–

PW91

2.069

–

1.983

79.90

173.30

–

[Fe2(ox)(SCN)8]4- (7) LDA

2.105

–

2.005

77.20

169.80

–

OPBE

2.172

–

2.080

75.50

168.70

–

PW91

2.159

–

2.061

76.30

169.20

–

[CrFe(ox)(SCN)8]4- (8) X-ray

2.177/2.056b

–

2.070/1.992b

–

168.00/173.05b

–

LDA

2.135/1.992b

–

2.000/1.928b

76.80/80.90b

169.40/174.00b

–

OPBE

2.205/2.059

b

–

b

2.069/1.996

b

74.90/79.60

b

168.20/173.50

–

PW91

2.190/2.045b

–

2.053/1.989b

75.80/80.30b

168.70/173.60b

–

X-ray

2.092c/1.977

–

–

80.80/168.70

–

–

LDA OPBE

2.101c/1.989 2.195c/2.047

– –

– –

79.00/166.60 77.00/164.70

– –

– –

2.165c/2.037

–

–

78.10/165.50

–

–

LDA

2.019c/1.931

–

–

81.60/172.70

–

–

OPBE

2.124c/1.990

–

–

77.43/171.23

–

–

PW91

c

2.089 /1.980

–

–

80.80/171.90

–

–

LDA

2.118c/2.054

–

–

78.30/169.70

–

–

OPBE

2.304c/2.169

–

–

74.03/164.90

–

–

PW91

2.218c/2.135

–

–

76.30/167.40

–

–

[Fe2(ox)5]4- (9)

PW91 [Cr2(ox)5]

4-

(10)

[Ni2(ox)5]6- (11)

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573

Table 1 continued M–Oox

M–O

M–N

OoxMOox

transNMOox

transOMOox

[Cu2(ox)(C12H8N2)2]2? (12)

a

X-ray

1.973

–

1.985

85.20

168.00

–

LDA

1.943

–

1.933

85.90

179.80

–

OPBE

2.009

–

1.999

83.40

179.80

–

PW91

1.995

–

1.989

84.50

179.80

–

M–Oox bond appointed trans to M=O bond

b

Cr–Oox, Cr–N bond distances and corresponding angles

c

M–Oox bonds in the bridge region

Table 2 Exchange coupling constants J/cm-1 calculated with the B3LYP functional (LDA, OPBE, PW91, and X-ray geometries) and comparison with available experimental values Complex Jexp

JX-ray/

JLDA/

JOPBE/

JPW91/

B3LYP

B3LYP

B3LYP

B3LYP

Table 3 Exchange coupling constants J/cm-1 calculated with nonhybrid functionals on corresponding geometries and comparison with available experimental values Complex

Jexp

JLDA

JOPBE

JPW91

1

-3.85

-11.97

-4.19

-6.51

1

-3.85

-2.71

-8.34

-3.95

-4.67

2

–

-229.30

-111.92

-138.59

2

–

–

-55.47

-28.07

-31.47

3

–

-7.16

-6.77

-4.18

3

–

–

-0.96

-1.40

-1.21

4

–

–

0.37

-0.68

-0.44

4 5

– –

-1.54 3.85

-1.97 1.07

0.85 1.42

5

–

–

-0.95

-0.74

-0.95

6

-3.23

-15.78

-11.34

-11.53

6

-3.23

-3.94

-5.68

-3.86

-4.14

7

-3.84

-16.34

-10.73

-13.37

7

-3.84

–

-5.47

-4.80

-4.86

8

1.10

3.63

6.92

3.37

8

1.10

–

0.89

1.45

1.40

9

-10.04

-7.08

-9.24

9

-6.60

–

-4.50

-3.60

-3.84

10

-6.2

-14.97

-8.40

-10.51

10

-6.20

–

-6.82

-3.39

-4.15

11

-22.8

-129.02

-56.78

-59.85

11

-22.80

–

-17.45

-13.10

-10.55

12

–

-1747.66

-1267.36

-1253.85

12

–

-351.98

-348.79

-337.97

-330.92

antiferromagnetic couplings are dominant (Table 2). However, very weak ferromagnetic interactions were obtained for the LDA optimized structure of 4, whereas weak antiferromagnetic couplings were obtained for the OPBE and PW91 optimized geometries. The best agreement with experimental values for binuclear compounds 8, 9, 10, and 11 was achieved at the LDA optimized geometries, whereas for other complexes better agreement was obtained at the OPBE and PW91 geometries. The results obtained with non-hybrid functionals are too negative for complexes 2, 11, and 12 and slightly overestimated for other complexes (Table 3). However, quite good agreement was achieved for complexes 1, 8, 9, and 10. In order to gain a better insight into the coupling mechanism for complexes 1–5, detailed molecular orbital (MO) analysis was performed. A perspective drawing of the SOMO in the triplet state is shown in Fig. 2. A decrease of antiferromagnetic interactions in binuclear complexes 3 and 4, in comparison to isomeric compounds 1 and 2, can be considered as a consequence of

-6.6

moving the equatorial oxygen to the axial position. In the case of equatorial orientation of M=O double bonds in structures 1 and 2, d-metal orbitals are oriented perpendicular to the plane of the oxalate bridge. This orientation gives rise to the favorable overlap with the pz orbitals of bridging atoms. The contribution of oxalate carbon (Cox) pz orbital in the SOMO of 1 of 2 is large in comparison to other structures. The coupling of two metal centers in complex 2 is much easier, because it spreads over five atoms, in contrast to compound 1 where the path is longer. In the case of axial positions of M=O double bonds, d-metal orbitals are found in the plane of the oxalate ring. The in-plane overlapping with the bridge orbitals is poor, giving rise to significant reduction of antiferromagnetic couplings. For the equatorial–axial position in complex 5, d-metal orbitals are spatially orthogonal and the overlapping is disabled. As a consequence of orthogonality, we can expect significant ferromagnetic couplings. However, calculated constants at the B3LYP level of theory are slightly negative, whereas appreciable ferromagnetic couplings were obtained with non-hybrid functionals. Although

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Fig. 2 A perspective drawing of the semi-occupied molecular orbitals (SOMO) in the triplet state for a trans-(equatorial, equatorial) [(VO)2(ox)(SCN)6]4- (1), b cis-(equatorial,equatorial) [(VO)2 (ox)(SCN)6]4- (2), c trans-(axial, axial) [(VO)2(ox)(SCN)6]4- (3), d cis-(axial, axial) [(VO)2(ox)(SCN)6]4- (4), e (axial, equatorial) [(VO)2(ox)(SCN)6]4- (5)

B3LYP is generally the most accurate functional for the calculation of magnetic properties [32], better trends were obtained with non-hybrid functionals. To introduce a quantitative measure of the delocalization, Mulliken population analysis has been performed. Table 4 shows Mulliken spin densities at vanadium metal centers and oxalate carbon atoms. The extent of the delocalization of the magnetic orbitals on the ligands is not only crucial for the J values but is related to the spin density distributions, as evident from the single-determinant description of the triplet state. It is expected that a larger delocalization of the magnetic MOs on the ligand is

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associated with a smaller spin density on the vanadium atoms. In the case of binuclear complexes 1 and 2 spin densities at the bridging carbon atom are higher than for complexes 3 and 4. Furthermore, the smaller spin densities at vanadium metal centers, obtained for the equatorial position of the M=O double bond, indicate the larger delocalization. The smallest spin densities at the bridging carbon atom were obtained for compound 4, with smallest exchange coupling constants. The calculated and experimentally obtained constants for chromium binuclear compounds are negative, revealing dominant antiferromagnetic couplings. Better agreement with experimental values was obtained with the B3LYP functional, whereas the non-hybrid functionals slightly overestimate J. However, a very reasonable result for 10 was achieved with the OPBE functional. In comparison to vanadium complexes with one unpaired electron per metal site, the addition of two more electrons does not lead to significant changes in J. With an increased number of unpaired electrons per metal site, both the ferromagnetic and antiferromagnetic interactions increase at the same time. The antiferromagnetic pathways originate from dxz– dxz, dyz–dyz, and dxy–dxy interactions, whereas the ferromagnetic pathways are present due to dxz–dyz, dxz–dxy, and dxy–dyz interactions, because these orbitals are orthogonal. The Fe(III) ion in complexes 7–9 has high spin configuration, and two more electrons are placed in dx2 y2 and dz2 orbitals. The additional aniferromagnetic (dx2 y2  dx2 y2 , dz2  dz2 ) and ferromagnetic (dx2 y2  dz2 , dx2 y2  dðxz; yz; xyÞ , dz2  dðxz;yz;xyÞ ) interactions lead to moderate negative coupling constants in 7 and 9, similar to chromium compounds. The schematic representation of possible antiferromagnetic and ferromagnetic pathways is sketched in Fig. 3. With the exception of the complex 2, which is not yet synthesized, the strongest antiferromagnetic couplings are present in complexes 11 and 12. In the heterobinuclear compound 8 an additional ferromagnetic pathway operates between occupied dx2 y2 and dz2 orbitals of Fe(III) and empty dx2 y2 and dz2 orbitals of Cr(III) (SOMO–LUMO interactions). The net effect is total ferromagnetic coupling across the oxalate bridge. The unpaired electrons in compound 12 are placed in dx2 y2 orbitals, thus the only possible pathway is dx2 y2  dx2 y2 . Very strong interactions in copper binuclear compounds can be explained through r overlapping of x2–y2 d-metal orbitals with symmetry-adapted p orbitals of bridging ligand. The corresponding magnetic orbitals of the BS state for 12 are depicted in Fig. 4. The lobes of metal orbitals are pointed directly toward ligand orbitals, leading to substantial overlap. In the case of nickel complexes, antiferromagnetic interactions arise from r overlapping of dz2 and dx2 y2

Author's personal copy DFT study of the magnetic coupling interaction Table 4 Mulliken spin densities for binuclear compounds 1–4 calculated at B3LYP level of theory at LDA, OPBE, and PW91 optimized geometries

Complex

575

LDA/B3LYP q/Cox

1

OPBE/B3LYP q/V

q/Cox

0.008137 0.008138 0.020930

3

0.002921

0.012729

Fig. 3 Schematic representation of antiferromagnetic (left) and ferromagnetic (right) pathways

metal orbitals with p orbitals of the ligand. In addition, ferromagnetic interactions between dz2 and dx2 y2 metal orbitals are also present, leading to a reduction of antiferromagnetic couplings compared to copper complexes. Estimation of linear relationship between coupling constant and (e2 – e1)2 and S2 parameters In addition to the simple MO analysis and Mulliken population analysis, magnetic couplings in binuclear complexes with one unpaired electron per metal site can be analyzed by the use of Hoffmann’s [33] and Kahn’s [34] approaches. The application of these two approaches is shown for the complex compound 1 as an illustration. As mentioned in the theoretical part the exchange interactions can be expressed as a sum of ferromagnetic and antiferromagnetic interactions (Eq. 5). In Hoffmann’s approach, the two-electron integrals are approximately constant for slight changes in structural

0.014374

1.159604 0.002560

0.002061

0.002263 1.149926

0.002235

1.140828

0.002583 1.49023

1.120923 0.002850

1.135605

0.002174

0.002592

1.136716 0.005717

0.002389

0.002770

q/V

0.005717

0.004948 1.098099

q/Cox

1.136328

1.121096 4

q/V

0.004948 1.095527

2

PW91/B3LYP

1.159615 0.002475

parameters for homobinuclear complexes with the same bridging ligand. Thus, a linear relationship between coupling constant and (e2 – e1)2 can be estimated. The calculated results for the oxalate-bridged compound 1 are shown in Fig. 5. The exchange coupling constants were calculated at the B3LYP level of theory on the X-ray structure of 1 and geometries obtained with different functionals. The linear fitting correlation coefficient R is 0.9968. In Kahn’s approach, an approximately linear relation between J and the square of the overlap integral of magnetic orbitals in the BS state (S2) can be also estimated (Fig. 6). The linear fitting correlation coefficient R is 0.9912. It is evident that the S2 parameter can be used to describe the antiferromagnetic couplings in binuclear complexes.

Conclusion The calculations revealed predominant antiferromagnetic coupling in most of the investigated complexes, except for [CrFe(ox)(SCN)8]4- in which ferromagnetic coupling was found. For the model complex isomer (axial, equatorial) [(VO)2(ox)(SCN)6]4- ferromagnetic couplings were estimated with non-hybrid functionals, whereas very weak antiferromagnetic couplings were obtained with the B3LYP functional. In comparison to B3LYP results, nonhybrid functionals afforded the better trends for the five

Fig. 4 Corresponding magnetic orbitals of broken symmetry state for complex compound [Cu2(ox)(C12H8N2)2]2? (12)

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Nusair (VWN) parametrization [39]. The obtained geometries are further optimized with OPBE and PW91 functionals in order to compare data. The all-electron Gaussian-type TZVPP basis set was used for metal atoms, and the split valence basis set with one set of first polarization functions (SVP) for nonmetallic atoms [40, 41]. The exchange coupling constants were calculated on X-ray and optimized structures, using B3LYP [42–44] hybrid functional. For comparison purposes, coupling constants were also calculated with LDA, OPBE [45], and PW91 [38] functionals. All calculations were performed with the Orca program package, version 2.8-20 [46]. Fig. 5 Calculated |J| values of complex 1 as a function of (e2 – e1)2. Calculations were done with the B3LYP functional on X-ray crystal structure and geometries obtained with LDA, OPBE, PW91, BP86 [35, 36], BLYP [35, 37], and PWP [38] functionals

Acknowledgments This work was financially supported by the Serbian Ministry of Education and Science through the Grant No. 172035 and is part of COST CMST Action CM1002 (‘‘COnvergent Distributed Environment for Computational Spectroscopy (CODECS)’’).

References

Fig. 6 Calculated |J| values of complex compound 1 as a function of S2. Calculations were done with the B3LYP functional on X-ray crystal structure and geometries obtained with LDA, OPBE, PW91, BP86, BKYP, and PWP functionals

isomers of vanadium complexes. Further, the addition of unpaired electrons increases both the ferromagnetic and antiferromagnetic interactions. When the interacting centers have the lobes of the magnetic orbitals directly pointing toward each other and when the overlap between them is substantial, then the interaction is strongly antiferromagnetic. If the magnetic orbitals are orthogonal and hence the overlap is zero, then the direct interaction is ferromagnetic. When a magnetic orbital of one center overlaps with an empty orbital of the other centers the direct interaction is also ferromagnetic. Computational details Optimization of binuclear complexes was performed using the LDA functional characterized by the Vosko–Wilk–

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