using the easyspin toolbox in matlab to analyze the ...€¦ · to simulate the...

USING THE EASYSPIN TOOLBOX IN MATLAB TO ANALYZE THE PRESSURE DEPENDENT MAGNETISM OF AN S = 1, ONE-DIMENSIONAL SPIN CHAIN

By

Orlando Trejo

Department of Physics Submitted as an Undergraduate Thesis

April 8, 2019

UNIVERSITY OF FLORIDA

2019

3

To my mother, Carla Trejo

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ACKNOWLEDGEMENTS

Throughout my undergraduate research career I have received plenty of

support and advice from many people. Here I would like to acknowledge some of the

individuals and institutions who have helped me in various ways.

First, I would like to thank my advisor, Mark W. Meisel. His guidance was

valuable in helping me understand and complete this work. He shared his knowledge of

scientific concepts and his experimental expertise all while challenging me to solve and

ponder on the most fundamental aspects of the project. I would also like to

acknowledge the graduate student in our laboratory, John M. Cain, who trained me on

using several lab instruments. John never hesitated to answer any of my questions and

often expanded on the science at work, so a thanks goes to him for all of his insight on

much of the phenomena I encountered in the lab and on this project.

I would also like to thank Jared Singleton and Dr. Engelhardt at Francis

Marion University in Florence, South Carolina, who provided an analysis of the

Quantum Monte Carlo simulations on S = 1, Heisenberg chains. A thanks also goes to

Erik Čižmár, our colleague at P.J. Šafárik University in Košice, Slovakia, who I did not

have the pleasure of meeting in person, but nonetheless helped me with several coding

issues with the EasySpin toolbox.

Finally, aspects of this work and research experience were made possible by

funding from the National Science Foundation (NSF) supporting the National High

Magnetic Field Laboratory (DMR-1644779) and the single-investigator research and

training activities of Meisel Group (DMR-1708410).

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TABLE OF CONTENTS page

ACKNOWLEDGEMENTS ................................................................................................ 4

LIST OF FIGURES ........................................................................................................... 6

ABSTRACT ...................................................................................................................... 7

CHAPTER

1 INTRODUCTION ....................................................................................................... 8

NBCT and the Quantum Critical Point ....................................................................... 8EasySpin and the interacting Hamiltonian ............................................................... 10 Test Systems ........................................................................................................... 12 Summary ................................................................................................................. 14

2 TESTING THE CAPABILITIES OF EASYSPIN ....................................................... 15

Chain length dependence of QMC simulations ....................................................... 15 The Padé Approximations ....................................................................................... 16 Impurities ................................................................................................................. 17 Summary ................................................................................................................. 19

3 RESULTS ................................................................................................................ 20

Simulation results of D and J ................................................................................... 20

4 Conclusions and Future ........................................................................................... 22

APPENDIX

A Analayzing the E term in the Zero-Field splitting interaction .................................... 23

B Comparing simulations of the new library functions with previous algorithms of the EasySpin toolbox ............................................................................................... 24

C Nearest-neighbor interaction dependence on EasySpin versus the Pade approximation .......................................................................................................... 25

REFERENCES ............................................................................................................... 26

6

LIST OF FIGURES

Figure page 1-1 Nearest neighbor interactions for the NBCT complex ......................................... 10

1-2 NBCT magnetic suscpetibility as a function of temperature taken at various pressures. ............................................................................................................ 11

1-3 Microscopic comparison of spin exchange interaction of trimer system of the EasySpin simulations against the data. ............................................................... 14

1-4 Giant Spin analysis of the single-ion anisotropy and comparing the EasySpin simulations to the data. ....................................................................................... 15

2-1 Chain length dependence of QMC simulations ................................................... 16

2-2 Padé Approximations compared to the EasySpin simulations. ........................... 18

2-3 Evaluating Simulations and the NBCT susceptibilities at ambient pressure. ...... 19

3-1 Magnetic susceptibility sesults of the simulations compared to the NBCT complex at several pressures. ............................................................................. 21

A-1 Susceptibility dependence of E term ................................................................... 24

A-2 Comparing library functions with previous homemade algorithms ...................... 25

A-3 Difference plot between EasySpin simulation and the Padé Approximation with the parameters shown ................................................................................. 26

7

USING THE EASYSPIN TOOLBOX IN MATLAB TO ANALYZE THE PRESSURE DEPENDENT MAGNETISM OF AN S = 1, ONE-DIMENSIONAL SPIN CHAIN

By

Orlando Trejo

Department of Physics Undergraduate Thesis

April 8, 2019

The EasySpin toolbox, which operates within MATLAB software, was originally

designed to analyze electron paramagnetic resonance spectra. Yet, previous versions

of the toolbox (<v4.5) have been used to generate algorithms which allowed the

isothermal magnetization and temperature-dependent magnetic susceptibility to be

modelled. Now, EasySpin (>v5.2) contains library functions which allow these magnetic

properties to be calculated. The first step was to compare the simulation results of the

new version against the previous algorithms. Next, the magnetism of the spin trimer,

[Mn3O(O2PPh2)3(mpko)3]ClO4, was simulated and these results are discussed.

Ultimately, the goal was to analyze the magnetic behavior of the S = 1, one-dimensional

spin chain [Ni(HF4)(3-Cpy)4]BF4 at several pressures. The results indicate a trend that

the nearest-neighbor interactions weaken and the single ion anisotropy remains at

D ≈ 4.5 K, as the pressure rises. The implications of these results are discussed.

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CHAPTER 1 INTRODUCTION

1.1 NBCT and the Quantum Critical Point

The magnetic properties of [Ni(HF4)(3-Cpy)4]BF4 (NBCT) have previously been

studied at ambient pressure [1] and were characterized with the Hamiltonian

ℋ = 𝐽 𝑆! ∙ 𝑆!!!"!

+ 𝐽´ 𝑆! ∙ 𝑆![!"]

+ 𝐷 (𝑆!!)!!

− 𝜇!𝐵 ∙ 𝑔 ∙ 𝑆 (1-0)

where the interactions and notation are discussed later. Manson et al. [1] report on the

nearest-neighbor coupling and single-ion anisotropy of NBCT which are interpreted in

section 1.2. When considering the two-dimensional array of chains with nearest

neighbor interactions, as seen in figure 1-1, the interchain coupling terms, J´, are

negligible, indicating that NBCT consists of nearly isolated S = 1, one-dimensional

magnetic chains. This observation allowed the susceptibility of the system to be fitted to

that of an S = 1 Heisenberg chain with J = 4.86 K. Furthermore, Manson et al. report1 a

single ion anisotropy of D ≈ 4.3 K. These results place the D/J ratio near unity, where a

quantum critical point (QCP) in energy space is found, separating the Haldane phase

from the Large-D phase [2].

1 The single-ion anisotropy (D) was studied using the Angle Overlap Model with UV-vis data [1].

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Figure 1-1: Cartoon of Ni chain array showing the intrachain interactions, J, and interchain interactions J´, used in Equation (1-0).

Cain et al. [3, 4] have used pressure in an attempt to drive NBCT through the

QCP, near D/J ~ 1. Using a SQUID magnetometer and a measurement probe equipped

with a pressure cell, the temperature-dependent magnetic susceptibility was measured

at various pressures (figure 1-2).

The experimental and theoretical aspects of the QCP and the associated

quantum phase transition go beyond the scope of this work. Here the goal is to use the

EasySpin toolbox in MATLAB software to simulate and predict the pressure effects on

the low-field magnetic interactions. In the following section the methods and interacting

Hamiltonian that are used in the onset of the analysis are described.

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Figure 1-2: Plot of susceptibility vs. temperature of NBCT (randomly oriented single crystals) measured at various pressures [3, 4].

1.2 EasySpin and the Interacting Hamiltonian

The EasySpin toolbox is a free to use program that operates within the

framework of the MATLAB software [5]. It was originally developed by electron

paramagnetic resonance (EPR) spectroscopists with the main function to simulate and

fit a wide range of EPR spectra. However, the newest versions of EasySpin (>v5.2.23)

now contain library functions which calculate a number of magnetometry parameters

such as the temperature-dependent magnetic susceptibility and the isothermal

magnetization of clusters and chains.


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EasySpin has the capability of incorporating various interactions and system

conditions2 into the computations, and therefore, the most dominant interactions that

describe NBCT should be considered. It is important to specify that in the following

analysis, the magnetism is characterized for randomly oriented crystals in the low-field

limit, as 𝐵⟶ 0, where the temperature is varied from 2 K to room temperature.

When a weak external magnetic field, 𝐵, is applied to a spin system the Zeeman

effect accounts for energy splitting according to

ℋ!""#$% = −𝜇!𝐵 ∙ 𝑔 ∙ 𝑆 , (1-1)

where 𝜇! is the Bohr magneton, 𝑔 is the Landé g-tensor and 𝑆 is the spin vector.

Unpaired electrons also give rise to unquenched angular momentums at low

temperatures, which create sufficiently strong crystal fields that may lift the degeneracy

and produce more energy states. Specifically, Ni(II) chains often show a splitting of the

S = 1 triplet state given by a gap D, the so called Single Ion Anisotropy (SIA) [6]. This

splitting can be characterized by the contribution to the Hamiltonian

ℋ!"# = 𝐷 (𝑆!!)!!

+ 𝐸 (𝑆!!)!!

− (𝑆!!)! .

(1-2)

The E term is left out of the simulations as its effects are small and most apparent at

temperatures below the region of applicability (shown in figure A-1). Setting E = 0,

ℋ!"# = 𝐷 (𝑆!!)!!

. (1-3)

2 EasySpin allows the user to apply system conditions such as indicating an open chain or closed chain with periodic boundary conditions (PBC). If the sample is a single-crystal, one can also set the crystal orientation of the crystal [5].

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Combining Equations (1-1) and (1-3) with the Heisenberg Hamiltonian which only

takes into account the intrachain interactions, J, we arrive at the Hamiltonian assumed

to describe the magnetic behavior of the NBCT system,

ℋ = 𝐽 𝑆! ∙ 𝑆!!!"!

+ 𝐷 (𝑆!!)!!

− 𝜇!𝐵 ∙ 𝑔 ∙ 𝑆 , (1-4)

and when interactions to a neighboring chain are included as J´, Equation (1-0) is

realized.

EasySpin can therefore be used to simulate and extract the nearest-neighbor

interaction term, J, and a single-ion anisotropy term, D. In the following section simple

test systems are considered in order to demonstrate the capabilities of the EasySpin

toolbox.

1.3 Test Systems

As a first approach, the library functions in EasySpin (v5.2.23) were compared

with previous “homemade” algorithms which were used to report on the magnetism of

Mn(II) and Fe(II) trimers [7]. The Murray group used these Easyspin (<v4.2) algorithms

to simulate the temperature-dependent magnetic susceptibility and isothermal

magnetization. This work was successfully reproduced when using the new library

functions in EasySpin (v5.2.23) (Appendix B, figure A-2).

Next, the magnetic susceptibility of the spin trimer, [Mn3O(O2PPh2)3(mpko)3]ClO4,

was simulated and compared to previous fitting results obtained by Olajuyigbe et al. [8].

When determining the nearest-neighbor coupling, an equilateral model was used where

all nearest-neighbor interactions were assumed equal and no single-ion anisotropy was

considered (D = 0). According to the report, an exchange term of J = 16.0(1) cm-1 and

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g = 1.94(1) are expected. The simulation, shown in figure 1-3, indicates a J term and

Landé g-factor in agreement with the literature.

Figure 1-3: Simulations results using the EasySpin toolbox (blue) and the Christou group fit compared to the manganese data.

Olajuyigbe et al. then used a Giant Spin model with a total ground state spin,

S = 6 and J = 0 in order to determine the single ion anisotropy. The reduced

magnetization (M/𝑁!!) was measured at several field strengths and the literature gives

D = -0.29(2) cm-1 and g = 1.94 [8]. Accordingly, the EasySpin simulations are consistent

with D = -0.29 cm-1 and g = 1.94, as shown in figure 1-4.


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Figure 1-4: The reduced magnetization, using the GSM with S = 6, plotted as a function of field per temperature unit.

1.4 Summary

The magnetic properties of NBCT at ambient pressure place the system near a

QCP separating the Haldane phase from the Large-D phase [2]. These results have

motivated the method of using pressure to drive NBCT through a quantum phase

transition [3, 4]. In this section, the spin chain Hamiltonian was introduced and the

dominant interactions were presented.

It was demonstrated that the new library functions are consistent with the

“homemade” algorithms implemented on previous versions of the toolbox to analyze

simple clusters. Furthermore, comparisons show that EasySpin reproduces the results

for the low-field, temperature-dependent, D = 0 analysis of the susceptibility and the

isothermal, high-field Giant Spin magnetization analysis of a spin trimer reported in the

literature. In the next section the precision in the simulations is established by

comparing them to results obtained from the Quantum Monte Carlo (QMC) method.


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CHAPTER 2 TESTING THE CAPABILITIES OF EASYSPIN

2.1 Chain length dependence of QMC simulations

One of the main restrictions in using the EasySpin toolbox is the limitation on the

number of spin sites. The EasySpin simulations were restricted to a maximum of eight

spins on a standard laboratory computer, as it was not feasible to include more sites

due to the excessive memory requirement. Yet, the Quantum Monte Carlo method has

the ability to simulate thousands of sites. The following analysis considers the scaling of

the thermodynamic limit with chain-length for QMC results of Heisenberg

antiferromagnetic chains, which will later be used to determine the precision of the

EasySpin simulations.

Figure 2-1: Plot showing the magnetic susceptibility per spin versus temperature with

varying chain length of the QMC simulations. Work by Jared Singleton and Larry Engelhardt, FMU, Florence, South Carolina.


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Jared Singleton and Dr. Engelhardt at Francis Marion University (FMU) in

Florence, South Carolina, produced QMC simulations of the magnetic susceptibility per

spin as a function of temperature for S = 1 chains of varying length, N. It was observed

that for a closed chain with periodic boundary conditions, the thermodynamic limit is

approached quickly, such that N = 60 is effectively the same as N = ∞. Comparatively,

figure 2-1 shows the results for an open chain. At low temperature the thermodynamic

limit is approached slowly and N = 60 is close to the N = ∞ limit for T/J > 0.5. Hence,

assuming J ≈ 5 K, as predicted by Manson at al. [1], the QMC simulations will suffice for

temperatures T > 2.5 K.

2.2 The Padé Approximations and the T/J ≥ 1 rule

The Quantum Monte Carlo (QMC) method can be used to fit the temperature-

dependent magnetic susceptibility of Heisenberg antiferromagnetic chains (HAFCs) to a

high precision. However, the QMC method is a computationally intensive task and in

2013, Law et al. decided to use the well-known Padé Approximations (PA) to

approximate the QMC results of HAFCs for spin susceptibilities of S = 1, 3/2, 2, 5/2 and

7/2 [9].1 In the published work all calculations were performed for N = 2000 spin sites

and the deviations between the PA and fitted data were reported to be of the same

order of magnitude as the QMC simulation error. Hence, the precision of the EasySpin

simulations can be analyzed by comparing them to the PA established by Law et al. [9].

As described in section 1.1, the nearly isolated magnetic chains of NBCT allow

the susceptibility to be modelled by an S = 1 HAFC with J ≈ 5.0 K, g ≈ 2.10 and D = 0.

Hence, a PA was constructed using these parameters and the magnetic susceptibility

1 The Padé approximations were completed on closed chains using periodic boundary conditions using D = 0 and an antiferromagnetic spin-exchange term J [9].

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was simulated in EasySpin. Figure 2-2 shows the comparison, where the inset indicates

the difference. Further analysis on the J dependence reveal that the simulations stay

within 1% of the PA for temperature values when T/J ≥ 1 (figure A-3). Therefore, the

N = 8 simulations scale as N = 2000 for T/J ≥ 1.

Figure 2-2: Padé approximation of QMC results vs. EasySpin simulations. Inset shows the difference plot which remains within 1% difference down to T = 5 K.

2.3 Impurities

In order to model the sharp rise at low temperature, an intrinsic contribution of

paramagnetic impurities was included so that the total susceptibility is given by

𝜒!"#$% = 1 − 𝑝 𝜒!"# + 𝑝𝜒!" (2-1)

where 𝑝 indicates the amount of impurities present in the sample. In the following

simulations an impurity level of 7% was assumed. The magnetic susceptibility data


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acquired at ambient pressure was then simulated using Equation (2-1) and with the

parameters given by Manson et al. The simulation stays within 1% of the data for

T/J ≥ 1 and the results are shown in figure 2-3.

Figure 2-3: Plot of the susceptibility vs Temperature. In red is the data at ambient pressure. In black is the simulation taken to a temperature of T = 4.86 K. The Landé g-factor is proportional to the magnetic moment and therefore scales

the high temperature saturation limit of the magnetic susceptibility, which remains fairly

constant as a function of pressure. Furthermore, the methods of synthesis indicate the

impurity factor should be the same in all data sets. Therefore, the 7% impurity

assumption and g = 2.10 are considered fixed for the rest of the analysis.


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2.4 Summary

The QMC results for HAFCs reaches the thermodynamic limit with a minimum of

60 spin sites for T/J > 0.5. In the case where QMC simulations of 2000 sites were used,

the PA constructed by Law et al. [9], effectively approximates the QMC results to within

the fitting error. Hence, the PA were implemented as a means to test the EasySpin

simulations to the QMC results.

The PA was then constructed with the parameters presented by Manson et al.

[1]. With D = 0 and J as the parameter, it was shown that EasySpin simulations

(8 spins) remain within 1% difference of the QMC results (2000 spin) for T/J ≥ 1. Finally,

the temperature-dependent magnetic susceptibility data acquired at ambient pressure

was simulated with D > 0. A small intrinsic paramagnetic contribution of 7% was

required to simulate the low temperature tail and achieve best agreement with the

experimental data.

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CHAPTER 3 RESULTS

3.1 Simulations results of D and J

The temperature-dependent magnetic susceptibility was simulated for several

pressures using Equation (1-4) with a 7% impurity contribution, as in Equation (2-1) and

g = 2.10. As the pressure was increased an overall trend of decreasing J terms was

observed. Subsequently, according to the T/J ≥ 1 rule, smaller J values correspond to

simulation curves with a larger temperature range. This observation allowed the

simulation program to give a better indication of the single-ion anisotropy, a low

temperature effect, for higher pressures. The curves were simulated invoking Equation

(1-4) and a summary of the extracted parameters is shown in table 3-1.

Figure 3-1: Plot of the susceptibility of the NBCT sample at several pressures. The black lines indicate EasySpin simulations using finite D and J. In blue are simulations with D = 0 and finite J.


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Table 3-1: Parameters extracted from the simulation in figure 3-1.

Pressure (GPa) J (K) D (K) g ~D/J

1.47 2.12(2) 4.5(8) 2.11 2.1 1.05 2.25(2) 4.6(7) 2.11 2.0 0.35 3.62(2) 0 – 5 2.11 ?? 0.00 4.83(2) 4.3

(Manson et al.) 2.11 0.9

For pressures up to 0.35 GPa, EasySpin gives no indication of the single-ion

anisotropy as D = 0 and D = 5 K remain within the same percent difference. At higher

pressure, the nearest-neighbor coupling term decreases. Imposing the T/J ≥ 1 rule, the

simulations at high pressure become valid through lower temperatures and as a result,

the D term was predicted at about D ≈ 4.5 K.

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CHAPTER 4 CONCLUSIONS AND FUTURE

The results predict an increasing D/J ratio with pressure that reaches a value of

D/J ≈ 2.1. This change may indicate that NBCT is indeed being driven through the QCP

at D/J ~ 1 [2]. NBCT is therefore a promising candidate as a complex that can be

pressure tuned through a quantum phase transition. At this point the parameters

calculated using EasySpin are only plausible due to the assumptions considered in

developing the model and more work remains to be done.

First, the paramagnetic contribution to the temperature-dependent magnetic

susceptibility was included as a method to account for the low-temperature tail, shown

figure 1-2. However, the actual content of impurities is not known and the effect seen at

low temperature may be quantum mechanically driven.

When analyzing the single ion anisotropy, it is apparent that the D term effect on

the susceptibility is more dominant at lower temperatures. Yet, the uncertainty in the

single ion anisotropy is still quite large even at 1.44 GPa. This may come as a result of

the low number of spin chains used in the simulations.

Single crystal synthesis is currently being discussed as a way to better analyze

the magnetism of NBCT. Additionally, integration of the EasySpin toolbox with the

HiPerGator2 computing system is forthcoming. The success of this endeavor will allow

NBCT to be simulated with more spin sites and open the possibilities for studying other

clusters and chains more effectively.

2 HiPerGator is a higher computing system part of The University of Florida.

23

APPENDIX A: Analyzing the E term in the Zero-Field splitting interaction

Figure A-1: Susceptibility versus temperature plotted for several values of the E term

shown in equation (1-2).

From Equation (1-2), the E term effect on the simulation was studied. Figure A-1

shows the susceptibility plotted for several valued of E. It is clear that the E term affects

the characteristic upturn of the data. However, this effect occurs at low enough

temperatures that for the purposes of simplifying the simulations the E term is

subsequently left out, i.e. E = 0.


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APPENDIX B: Comparing the new library functions with previous algorithms of the EasySpin

toolbox

Figure A-2: Simulations with new EasySpin library function. Plot shows the

reproducibility compared to the work by Guillet et al. [7].

Previously, the Murray group studied the magnetism of Fe(II) and Mn(II)

complexes with EasySpin (<v4.2) [7]. The magnetism was simulated by appropriating

“homemade” algorithms with the EasySpin version at the time. A first analysis involved

comparing the new library function of EasySpin (v5.2.23) to the previous algorithms.

The simulation in the new version were constructed with the same parameters as

previously predicted. In the plot shown above, the trimer consisted of two equal nearest

neighbor interaction, J’ and a third value J.

There is a small deviation in the manganese simulations that seems to grow as

the temperature increases. The effect may be due to thermal fluctuation which increase

with temperature, however at this time the deviation has not been explored further.


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APPENDIX C: Nearest-neighbor interaction dependence on EasySpin versus the

Padé approximation

Figure A-3: Difference plot between EasySpin simulation and the Padé Approximation

at different J values.

The J dependence was analyzed for J values above and below the predicted nearest-

neighbor interacting term of J ≈ 5.0 K for NBCT. The difference between the EasySpin

simulations and the Padé approximation with the same parameters was then plotted.

The difference remains below 1% for T/J ≥ 1, for all J terms considered.


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REFERENCES

[1] J.L. Manson et al., Inorg. Chem. 51 (2012) 7520.

[2] S. Hu, B. Normand, X. Wang and L. Yu, Phys. Rev. B 84 (2011) 220402(R).

[3] J. M. Cain et al., to be submitted (2019).

[4] M.K. Peprah et al., Bull. Am. Phys. Soc. MAR19 (2019) L38.04, http://meetings.aps.org/Meeting/MAR19/Session/L39.4

[5] S. Stoll, EasySpin – EPR Spectrum Simulation, www.easyspin.org/.

[6] C.P. Landee and M.M. Turnbull, J. Coord. Chem. 67 (2014) 375.

[7] G.L. Guillet et al., Chem. Commun. 49 (2013) 6635.

[8] O.A. Adebayo, K.A. Abboud, and G. Christou, Inorg. Chem. 56 (2017) 11352.

[9] J.M Law, H. Benner, and R.K. Kremer, J. Phys. Condens. Matter 25 (2013) 65601.

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