Development of MagSaki(Tri) Software for the Magnetic Analysis of Trinuclear High-spin Cobalt(II) Complexes

Abstract

MagSaki(Tri) software was developed for the purpose of analyzing the magnetic data of trinuclear octahedral high-spin cobalt(II) complexes. The software enables analyses for five types of trinuclear cobalt(II) structures, including equilateral triangle shapes, isosceles triangle shapes, and linear shapes, to obtain magnetic parameters: the interaction parameters, J and J’, the spin-orbit coupling parameter, λ, the orbital reduction factor, κ, and the axial splitting parameter, Δ.

1 INTRODUCTION

The magnetic properties of coordination compounds have been extensively studied for more than half a century; however, it is still difficult to simulate magnetic data for some high-spin cobalt(II) complexes, because effects of the ligand field and the spin-orbit coupling work at the same time [1]. In the early studies by Lines [2] and Figgis [3], considering the axial ligand field turned out to be important. Then, Sakiyama developed magnetic susceptibility equations for dinuclear octahedral high-spin cobalt(II) complexes considering the axial ligand field [4,5,6,7], and software was also developed [8,9]. Regarding trinuclear octahedral high-spin cobalt(II) complexes, such software has not been reported, although the magnetic analysis was successfully done [10].

Therefore, in this paper, a MagSaki(Tri) software is reported for the purpose of analyzing the magnetic data of trinuclear cobalt(II) complexes, including triangle-shaped and linear structures (Figure 1).

Figure 1.

 Equilateral-triangle-shaped (a), isosceles-triangle- shaped(b), and line-shaped (c) trinuclear cobalt(II) structures.

2 MAGNETIC PARAMETERS

The main symbols for the magnetic parameters are summarized in Table 1.

Table 1. List of the main symbols [1,4,5,6,7,8,9,10]

Symbol	Unit	Meaning
J, J’, J”, J”’	cm–1	Interaction parameters
T	K	Absolute temperature
v	–	Distortion parameter defined as Δ/(κ λ)
Δ	cm–1	Axial splitting parameter
θ	K	Weiss constant
κ	–	Orbital reduction factor
λ	cm–1	Spin-orbit coupling parameter
μeff	μB	Effective magnetic moment
χA	cm3 mol–1	Atomic magnetic susceptibility
χM	cm3 mol–1	Molar magnetic susceptibility

3 METHOD

The software was developed using REALbasic software [11] on a SONY PCG-9G1N computer (OS: Windows XP Home edition). Magnetic susceptibility equations were obtained in the same way as described in reference [12].

4 FUNCTION OF MAGSAKI(TRI) SOFTWARE

The software calculates theoretical χ_A and μ_eff values for five types of trinuclear cobalt(II) structures, including triangle-shape and linear-shape structures, and displays the theoretical curves of χ_Aversus T and μ_effversus T. The software optimizes magnetic parameters to fit the theoretical curves to the observed data.

5 CALCULATION MODES

The software has five calculation modes for trinuclear high-spin cobalt(II) complexes. Mode 1 is for equilateral-triangle-shaped tricobalt complexes (Figure 1a), including three equivalent cobalt(II) ions (Co_A1 = Co_A2 = Co_A3). Here, equivalent cobalt(II) ions have equal parameter values. Four parameters, J, v, Δ, and κ, are used in mode 1, and parameter θ is optional. Mode 2 is for isosceles-triangle-shaped tricobalt complexes (Figure 1b), including three equivalent cobalt(II) ions (Co_A1 = Co_A2 = Co_B). Five parameters, J, J’, v, Δ, and κ, are used in mode 2, and parameter θ is optional. Mode 3 is for isosceles-triangle-shaped tricobalt complexes (Figure 1b), including two types of cobalt(II) ions (Co_A1 = Co_A2 ≠ Co_B). Eight parameters, J, J’, v_A, v_B, Δ_A, Δ_B, κ_A, and κ_B, are used in mode 3, and parameter θ is optional. Mode 4 is for line-shaped tricobalt complexes (Figure 1c), including three equivalent cobalt(II) ions (Co_A1 = Co_A2 = Co_B). Four parameters, J, v, Δ, and κ, are used in mode 4, and parameter θ is optional. Mode 5 is for line-shaped tricobalt complexes (Figure 1c), including two types of cobalt(II) ions (Co_A1 = Co_A2 ≠ Co_B). Seven parameters, J, v_A, v_B, Δ_A, Δ_B, κ_A, and κ_B, are used in mode 5, and parameter θ is optional (Table 2).

Table 2. Calculation modes

Mode	Shape	Cobalt(II) ions	Interaction (s)
1	Equilateral triangle	CoA1 = CoA2 = CoA3	J
2	Isosceles triangle	CoA1 = CoA2 = CoB	J ≠ J’
3	Isosceles triangle	CoA1 = CoA2 ≠ CoB	J ≠ J’
4	Linear	CoA1 = CoA2 = CoB	J
5	Linear	CoA1 = CoA2 ≠ CoB	J

6 MAGNETIC SUSCEPTIBILITY EQUATIONS

Magnetic susceptibility equations are shown below.

[Magnetic susceptibility equation for modes 1, 2, and 4]   

${\chi }_{\text{M}}=\frac{{\chi }_z+2{\chi }_{x }}{3}$

  

${\chi }_{z\left(x\right)}=N\frac{F_{1, z\left(x\right)}}{F_2}$

  

$\begin{array}{c}{F}_{1,z\left(x\right)}=\sum _{n=\pm 1}^{}\left[\frac{E_{z\left(x\right),n}^{\left(1\right)}{}^2}{k \left(T-\theta \right)}\left\{\frac{2}{8}\text{Exp}\left(\frac{-E_n^{\left(0\right)}-\frac{25}{9}A}{k T}\right)+\frac{2}{8}\text{Exp}\left(\frac{-E_n^{\left(0\right)}-\frac{25}{9}B}{k T}\right)+\frac{20}{8}\text{Exp}\left(\frac{-E_n^{\left(0\right)}-\frac{25}{9}C}{k T}\right)\right\}-6 E_{z\left(x\right),n}^{\left(2\right)} \text{Exp}\left(\frac{-E_n^{\left(0\right)}}{k T}\right)\right]\\ +3\sum _{n\ne \pm 1}\left[\left\{\frac{E_{z\left(x\right),n}^{\left(1\right)}{}^2}{k \left(T-\theta \right)}-2 E_{z\left(x\right),n}^{\left(2\right)}\right\}\text{Exp}\left(\frac{-E_n^{\left(0\right)}}{k T}\right)\right]\end{array}$

   (n = ±1, ±2, ±3, ±4, ±5, ±6)      

$\begin{array}{c}{F}_2=\sum _{n=\pm 1}\left[\frac{2}{8}\text{Exp}\left(\frac{-E_n^{\left(0\right)}-\frac{25}{9}A}{k T}\right)+\frac{2}{8}\text{Exp}\left(\frac{-E_n^{\left(0\right)}-\frac{25}{9}B}{k T}\right)+\frac{4}{8}\text{Exp}\left(\frac{-E_n^{\left(0\right)}-\frac{25}{9}C}{k T}\right)\right]\\ +\sum _{n\ne \pm 1}\left[\text{Exp}\left(\frac{-E_n^{\left(0\right)}}{k T}\right)\right]\end{array}$

   (n = ±1, ±2, ±3, ±4, ±5, ±6)       (A, B, C) = (3 J / 4, 3 J / 4, –3 J / 4) (mode 1)       (A, B, C) = (J – J’ / 4, 3 J’ / 4, –J / 2 – J’ / 4) (mode 2)       (A, B, C) = (J, 0, –J / 2) (mode 4)   

[Magnetic susceptibility equation for modes 3 and 5]   

${\chi }_{\text{M}}=\frac{{\chi }_z+2{\chi }_{x }}{3}$

  

${\chi }_{z\left(x\right)}=N\frac{2F_{\text{A},1, z\left(x\right)}}{F_{\text{A},2}}+N\frac{F_{\text{B},1, z\left(x\right)}}{F_{\text{B},2}}$

  

$\begin{array}{c}{F}_{\text{A},1, z\left(x\right)}=\sum _{n=\pm 1}\left[\frac{2 K^2}{24 k T}\left\{\text{Exp}\left(\frac{-\frac{25}{9}A}{k T}\right)\right\}+\frac{2 L^2}{24 k \left(T-\theta \right)}\left\{\text{Exp}\left(\frac{-\frac{25}{9}B}{k T}\right)\right\}+\frac{20 M^2}{24 k \left(T-\theta \right)}\left\{\text{Exp}\left(\frac{-\frac{25}{9}C}{k T}\right)\right\}-2 E_{\text{A}, z\left(x\right),n}^{\left(2\right)}\right]\\ +\sum _{n\ne \pm 1}\left[\left\{\frac{E_{\text{A},z\left(x\right),n}^{\left(1\right)}{}^2}{k \left(T-\theta \right)}-2 E_{\text{A}, z\left(x\right),n}^{\left(2\right)}\right\}\text{Exp}\left(\frac{-\left(E_{\text{A}, n}^{\left(0\right)}-E_{\text{A},0}^{\left(0\right)}\right)}{k T}\right)\right]\end{array}$

   (n = ±1, ±2, ±3, ±4, ±5, ±6)      

$\begin{array}{c}{F}_{\text{A},2}=\sum _{n=\pm 1}\left[\frac{2}{8}\text{Exp}\left(\frac{-\left(E_{\text{A}, n}^{\left(0\right)}-E_{\text{A},0}^{\left(0\right)}\right)-\frac{25}{9}A}{k T}\right)+\frac{2}{8}\text{Exp}\left(\frac{-\left(E_{\text{A}, n}^{\left(0\right)}-E_{\text{A},0}^{\left(0\right)}\right)-\frac{25}{9}B}{k T}\right)+\frac{4}{8}\text{Exp}\left(\frac{-\left(E_{\text{A}, n}^{\left(0\right)}-E_{\text{A},0}^{\left(0\right)}\right)-\frac{25}{9}C}{k T}\right)\right]\\ +\sum _{n\ne \pm 1}\left[\text{Exp}\left(\frac{-\left(E_{\text{A}, n}^{\left(0\right)}-E_{\text{A},0}^{\left(0\right)}\right)}{k T}\right)\right]\end{array}$

   (n = ±1, ±2, ±3, ±4, ±5, ±6)      

$\begin{array}{c}{F}_{\text{B},1, z\left(x\right)}=\sum _{n=\pm 1}\left[\frac{2 K^2}{24 k T}\left\{\text{Exp}\left(\frac{-\frac{25}{9}A}{k T}\right)\right\}+\frac{2 L^2}{24 k \left(T-\theta \right)}\left\{\text{Exp}\left(\frac{-\frac{25}{9}B}{k T}\right)\right\}+\frac{20 M^2}{24 k \left(T-\theta \right)}\left\{\text{Exp}\left(\frac{-\frac{25}{9}C}{k T}\right)\right\}-2 E_{\text{B}, z\left(x\right),n}^{\left(2\right)}\right]\\ +\sum _{n\ne \pm 1}\left[\left\{\frac{E_{\text{B},z\left(x\right),n}^{\left(1\right)}{}^2}{k \left(T-\theta \right)}-2 E_{\text{B}, z\left(x\right),n}^{\left(2\right)}\right\}\text{Exp}\left(\frac{-\left(E_{\text{B}, n}^{\left(0\right)}-E_{\text{B},0}^{\left(0\right)}\right)}{k T}\right)\right]\end{array}$

   (n = ±1, ±2, ±3, ±4, ±5, ±6)      

$\begin{array}{c}{F}_{\text{B},2}=\sum _{n=\pm 1}\left[\frac{2}{8}\text{Exp}\left(\frac{-\left(E_{\text{B}, n}^{\left(0\right)}-E_{\text{B},0}^{\left(0\right)}\right)-\frac{25}{9}A}{k T}\right)+\frac{2}{8}\text{Exp}\left(\frac{-\left(E_{\text{B}, n}^{\left(0\right)}-E_{\text{B},0}^{\left(0\right)}\right)-\frac{25}{9}B}{k T}\right)+\frac{4}{8}\text{Exp}\left(\frac{-\left(E_{\text{B}, n}^{\left(0\right)}-E_{\text{B},0}^{\left(0\right)}\right)-\frac{25}{9}C}{k T}\right)\right]\\ +\sum _{n\ne \pm 1}\left[\text{Exp}\left(\frac{-\left(E_{\text{B}, n}^{\left(0\right)}-E_{\text{B},0}^{\left(0\right)}\right)}{k T}\right)\right]\end{array}$

   (n = ±1, ±2, ±3, ±4, ±5, ±6)      

$K=\left(4 \left|E_{\text{A}, z\left(x\right),n}^{\left(1\right)}\right|-\left|E_{\text{B}.z\left(x\right),n}^{\left(1\right)}\right|\right) / 3$

  

$L=E_{\text{B}.z\left(x\right),n}^{\left(1\right)}$

  

$M=\left(2 \left|E_{\text{A}, z\left(x\right),n}^{\left(1\right)}\right|+\left|E_{\text{B}.z\left(x\right),n}^{\left(1\right)}\right|\right) / 3$

   (A, B, C) = (J – J’ / 4, 3 J’ / 4, –J / 2 – J’ / 4) (mode 3)       (A, B, C) = (J, 0, –J / 2) (mode 5)   

7 MAGNETIC SIMULATIONS

Here the temperature dependences of the χ_AT product are demonstrated for three cases, an equilateral-triangle-shaped case, a line-shaped case, and an isosceles-triangle-shaped case.

Theoretical χ_AT versus T curves for the equilateral-triangle-shaped trinuclear cobalt(II) complexes were calculated using mode 1, and the results are shown in Figure 2. When decreasing the temperature from 300 K to 100 K, the χ_AT product is not constant even in the cases with no interaction (J = 0 cm^–1) due to the single-ion spin-orbit coupling [12]. In the lower temperature range, below ~100 K, the χ_AT product increases when the cobalt(II) ions are ferromagnetically coupled, but decreases when the interaction is antiferromagnetic.

Figure 2.

 Theoretical χT versus T curves for trinuclear cobalt(II) complexes of equilateral triangle shape with the variation of J [The curves represent from the bottom J = −10, −5, −1, −0.2, 0, +0.2, +1, +5, and +10 cm^–1, respectively.] when λ = −170 cm^–1, κ = 0.9, and Δ = 0 cm^–1 (a) and when λ = −100 cm^–1, κ = 0.7, and Δ = 0 cm^–1 (b), where θ and TIP are not used.

Theoretical χ_AT versus T curves for the line-shaped trinuclear cobalt(II) complexes were calculated using mode 4, and the results are shown in Figure 3. The tendency is similar to those for the above equilateral-triangle case, and the characteristic temperature dependence above 100 K is due to single-ion spin-orbit coupling.

Figure 3.

 Theoretical χT versus T curves for trinuclear cobalt(II) complexes of equally-spaced linear shape with the variation of J [The curves represent from the bottom J = −10, −5, −1, −0.2, 0, +0.2, +1, +5, and +10 cm^–1, respectively.] when J’ = 0 cm^–1, λ = −170 cm^–1, κ = 0.9, and Δ = 0 cm^–1 (a) and when J’ = 0 cm^–1, λ = −100 cm^–1, κ = 0.7, and Δ = 0 cm^–1 (b), where θ and TIP are not used.

Theoretical χ_AT versus T curves for the isosceles-triangle-shaped trinuclear cobalt(II) complexes were calculated using mode 2, and the results are shown in Figure 4. When J is positive, the low-temperature data are affected by J’ (Figure 4a), but when J is negative, the curves are not affected so much by J’ (Figure 4b). The theoretical curves are more affected by J than J’.

Figure 4.

 Theoretical χT versus T curves for trinuclear cobalt(II) complexes of isosceles triangle shape with the variation of J’ when J > 0 (a) and when J < 0 (b). [The curves represent from the bottom (J/ cm^–1, J’/ cm^–1) = (0,0), (+10, −10), (+10, −5), (+10, 0), (+10, +5), and (+10, +10), respectively for (a) and (J/ cm^–1, J’/ cm^–1) = (−10, −10), (−10, −5), (−10, 0), (−10, +5), (−10, +10), and (0,0), respectively for (b), when λ = −170 cm^–1, κ = 0.9, and Δ = 0 cm^–1.]

The magnetic analysis for trinuclear cobalt(II) complexes of isosceles triangle shape were performed previously [10].

8 REQUIREMENTS

The software (MagSaki(Tri) 003W) will run on Windows computers. The Macintosh version (MagSakiTri7) will run on Macintosh computers.

This work was supported by JSPS KAKENHI Grant Number 15K05445. Financial support by Yamagata University is also acknowledged.

References

• [1]   O. Kahn, Molecular Magnetism, VCH (1993).
• [2]   M. E. Lines, Phys. Rev., 131, 546 (1963).
• [3]   B. N. Figgis, M. Gerloch, J. Lewis, F. E. Mabbs, G. A. Webb, J. Chem. Soc. A, 2086 (1968).
• [4]   H. Sakiyama, R. Ito, H. Kumagai, K. Inoue, M. Sakamoto, Y. Nishida, M. Yamasaki, Eur. J. Inorg. Chem., 2027 (2001).
• [5]   H. Sakiyama, R. Ito, H. Kumagai, K. Inoue, M. Sakamoto, Y. Nishida, M. Yamasaki, Eur. J. Inorg. Chem., 2705 (2001).
• [6]   H. Sakiyama, Inorg. Chim. Acta, 359, 2097 (2006).
• [7]   H. Sakiyama, Inorg. Chim. Acta, 360, 715 (2007).
• [8]   H. Sakiyama, J. Chem. Softw., 7, 171 (2001).
• [9]   H. Sakiyama, J. Comput. Chem. Jpn., 6, 123 (2007).
• [10]   H. Sakiyama, H. Adams, D. E. Fenton, L. R. Cummings, P. E. McHugh, H. Okawa, Open J., Inorg. Chem., 1, 33 (2011).
• [11]   REAL Software, Inc., 1996–2006http://www.realsoftware.com/.
• [12]   H. Sakiyama, A. K. Powell, Dalton Trans., 43, 14542 (2014).