Evaluating Magnetic Parameters of Polynuclear Complexes
Theory The spin-pin interaction in a dinuclear complex is governed by the Heisenberg spin-exchange Hamiltonian (3).
Fitting Experimental Data to Theoretical Equations on a Personal Computer where SMand SW are the spin states of the individual metal ions (SM > Sw); and J is the coupling constant or magnetic-exchange parameter.
Greg Brewer Catholic University Washington, DC 20064
Spin States
One of the most important experimental techniques in transition metal chemistrv is the determination of maenetic properties ( I ) . The mass susceptibility of a species, which is usually measured by the Guoy or Faraday technique, is used to calculate the molar susceptibility XM, which in turn gives the magnetic moment.
The magnetic moment is related to the spin state S by p =g(s(s+l))ln
whereg, the gyromagnetic ratio of the electron, has a value of 2.00. The number of unpaired electrons allows one to determine the oxidation state and the electronic ground state of the metal ion or ions in a complex. Introducing Magnetic Measurements There has been renewed interest in magnetic measurements in recent years because many current topics place considerable emphasis on magnetic properties. Examples are the study of superconductors, new materials, and metalloenzymes (2). This article focuses on polynuclear and cluster complexes. Unfortunately, the introduction of magnetic measurements in advanced inorganic chemistry courses is oRen ignored or confined to a lecture discussion of theory. Work up of experimental data for magnetically interacting systems is generally time-consuming and requires large computing facilities. The examples discussed in this article illustrate the determination of magnetic parameters for several polynuclear complexes by fitting data obtained from the literature to the appropriate theoretical equations. These examples can be used as a n introduction to the field of polynuclear transition metal complexes in advanced undereraduate inoreanic courses. Thev also acauaint the student with powerf;l data-fitting 'fhe interested reader will find many other examples in the enclosed references. Software for Evaluating the Parameters
This article describes a fast and convenient method for fitting experimental magnetic data to theoretical equations using either of two commercially available software packages that can be operated on a personal computer. The following two programs have been used to evaluate magnetic parameters for representative polynuclear complexes. They have been found to give remarkably good fits. GraFit (ErithacusSoftware, London,UK) (Availablein the U.S. from Sigma Chemical, $430.00) Eureka (Borland,Scotts Valley, CA, $79.00) Using these programs to evaluate parameters should also be more convenient for researchers, without compromising accuracy. The programs should facilitate the introduction of this fascinating and often neglected area in inorganic courses. 1006
Journal of Chemical Education
Applying this Hamiltonian to a particular spin-coupled system gives a square matrix. On diagonalization, this matrix gives the energies of the resultant spin states (ST's)as a function of SM,Sw, and J, as shown in eq 2 (3). E(%) = -J(s~s, + 1)- sM(sM + 1)- sw(sw + 1))
STis allowed to take on the values SM+SV s M + S w - l ...
(2)
SM-sw
The SM+ SW state is the ground state and the SM- Su state is the excited state if J > 0 (ferromagnetic interaction). The reverse is true for J c 0 (antiferromagnetic interaction). The energies for the trinuclear case considered in this article Cu(IIkM~IIkCu(II) in which the interaction between the terminal copper(I1) ions is zero, are given by eq 3 (3). E(ST)=JWST + 1)+ JSdS, + 1)+ ScuxU(Scm~,, + IkJ (3) where the spin state for the two copper(I1)ions is
,s,
= 1(Vz+ 44)
or SC,~, = 0 ('15 - U) When S ",
=1
then & = S M +1, ST=SM, and ST=SM-1 When SC","
=0
then ST= SM Magnetic Susceptibility
Antiferromagnetic and ferromagnetic interactions are distinguished experimentally by plotting magnetic susceptibility or moment versus temperature. A plot of susceptibility versus temperature will fall below the Curie law values for a n antiferromagnetic interaction. The plot will lie above the Curie law values for a ferromagnetic interaction. In addition, the antifemmagnetic interaction results in a magnetic moment (1)that rises to the high-temperature limit
with increasing temperature. The ferromagnetic interaction falls to the high-temperature limit with increasing temperature. The magnetic susceptibility is obtained by solving the van Vleck equation using the energy levels from eqs 2 and 3. For any polynuclear system, the molar susceptibility has the general form shown in eq 4.
where p is the Bohr mageton, 9.273 x lo-" erglgauss; N is Avogadro's number, 6.023 x 1V3part/mol; k is Boltzmann's constant, 1.38 x 10-16erg/K (preexponential term) or 0.695 em-% (exponential term); TIP is the temperature-independent paramagnetism; and the summation is over the allowed values of ST. Determing g and J
The procedure for determiningg and J is to fit the measured molar susceptibility to the appropriate theoretical equation. The value ofg in the complex is allowed to deviate from the free electron value of 2.00. The TIP term is normally very small relative to the other term, and it can usually be estimated from other measurements. It is also possible to treat this as another parameter to be determined. hut this should be done with caution to insure that it does not take on a chemically unreasonable value. It is instructive for students to show that ea 5 reduces to the well-known law, x = CIT when J = 0. Equations for the magnetic susceptibility are available in the literature for the following complexes.
'any hetero- or homodinuclear system (4) trinuelear complexes Cu(I1)-M(IIFCu(I1)(5)and Ni(IIFNi(IIFNi(I1)(6) tetranuclear copper(I1)complexes (7) Calculations and Results PC and Sofhvare
GraFit and AT The calculations described below were performed by GraFit on an AT computer with a math coprocessor. Equation entry is easily accomplished using the program's equation editor in which the equation is typed in a format analagous to BASIC. An example of a n equation file is given for eq 5 and shown just below it. Equation files are saved, which allows for easy future editing or modification. Data entry is done to a spreadsheet. Solution of the file requires only a few seconds even with 50 data points. The program produces the fitted parameters their standard deviations t h e reduced c h squarrd a tahle of fitted versus actual values The program can also display the theoretical function and experimental points produce publication-qualityplots
Eureka andXT Eureka was also used to solve for the magnetic parameters in these examples. It gave nearly identical results. However, the data entry is more cumbersome than it is with GraFit, and Eureka does not calculate the standard deviations of the fitted parameters. Calculation times are longer, and Eureka cannot display the theoretical function with the experimental data. Comparison calculations were done on an XT without a math coprocessor. The agreement between the two computers is quite close, but the XT cannot quickly handle complex fits with many data points.
Figure 1. Line Drawings of [Cu(OAc), .H20I2, CuVO(fsa)2en . CH30H,and [Co(C~salen)~(H~0)2](CIO~)~. Examples Using Transition Metal Complexes
Three examples were selected to illustrate the determination of magnetic parameters from susceptibility data (see Fig. 1). the homodinuclear [CU(OAC)~H~OI~ the heterodinuclear CuVO(f~a)~en CH,OH the heterotrinuclear [CdCusalen)z(HzO)zl(C104)2
.
Table 1shows the excellent agreement between the values of g and J obtained by GraFit on a PC and the literature values obtained on mainframe systems. Table 1. Comparison of Literature and GraFit Values of g and J (cm-')
literature
GraFit
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1007
0
1 w 2 0 0 3 M ) 4 M ) 5 M )
Temperature Temperature H20I2 as a function of Figure 2. Molar susceptibility of [CU(OAC)~. temperature. The solid curve is the best fit of the experimental points to the Bleany-Bowers equation (eq 5).
Homodinuclear
Figure 3.Plot of molar susceptibilityvs. temperature forthe BleanyBowers equation with g= 2.00and TIP = 60 x lod/cu.J = 0 (x),J = -25 (*), J = -50 R ). J = -75 (A)and J= -1 00 cm" (0).
The simplest systems are the homodinuclear complexes, which contain two identical metal atoms. The salt copper(I1) acetate monohydrate
tion, shown below, can be used to evaluate theg and J parameters for a pair of interacting S = 112 ions.
is a prominent historical example. In this compound the d9 S = 112 copper ions are square pyramidal with apical bonds to water. The four basal positions are oceupied by four carboxylate oxygen atoms from the four bridging acetates. The bridging acetates provide the magnetic exchange pathway. A unique feature of antiferromagnetic homodinuclear complexes is a maximum in the plot of susceptibility versus temperature (Fig. 2). which for copper(I1)acetate monohydrate occurs a t 255 K (8).The Bleany-Bowers equaTable 2. GraFit Output for [CU(OAC)ZHZ~]Z Reduced Chi souared = 1.305e-010 Variable
Value
9 COUDIIW
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1008
2.163 -146.8
constant X
Y
Temperature
Molar Susc.
93.5 100.1 120.2 143.5 160.5 181 200.9 220 239.7 258.6 277.9 294.2 300.3 321.3 349.5 396.5
0.000532 0.000604 0.000902 0.001224 0.001424 0.001574 0.001674 0.001746 0.001786 0.001796 0.00179 0.001778 0.001756 0.001752 0.001708 0.001594
Journal of Chemical Education
SM. Err. 0.004514 0.4403
Calculated 0.00051 58 0.0006127 0.0009166 0.001231 0.00141 2 0.001574 0.00168 0.001743 0.001777 0.001789 0.001786 0.001775 0.001769 0.001 743 0.001 7 0.001619
The TIP term is 60.0 x lo4 cm31molper copper ion. The following equation file is the form in which eq 5 is input to GraFit. .375*p2*2*exp(2*J/(k*T))/(T*(1+3*eyp(2*J/(k*T))))+a where the constants k and a are provided by the user, and g and J are determined by the program. Table 2 gives the experimental and calculated values of the molar susceptibility. The solid curve shown in Figure 2 is the theoretical function superimposed on the data. It is possible to generate a series of such curves with GraFit simply by manually selecting different values for J (see Fig. 3). This helps students get a feel for how the molar susceptibility function changes from simple Curie law behavior (J = 0) to an extreme case in which there is a susceptibility maximum (J is large and negative), as in the present example. Heterodinuclear ACu(I1)-V(N) complex, is selected to illustrate a heterodinuclear complex (9).(See Table 3 for the data.) . CH,OH CuVO(f~a)~en The ligand, ( f ~ a ) ~ e nis4 ;a dinucleating ligand, which holds two metal ions in a coplanar arrangement. The ions are linked by two bridging phenolic oxygen atoms. This complex can be fitted to the same theoretical equation as copper(I1) acetate because it consists of two interacting S = 112 ions. (Cu(II), which is a d9 ion, and V(IV), which is a dl ion, both have S = 112). This was the first reported heterodinuclear ferromagnet, ending the belief that fernmametism was confined to homodinuclear complexes. The f&omagnetism in this complex, as well as in many other molecular ferromagnets, is due to the orthogonality of the magnetic o r b i t a l s ? ~11). ~, Compared to the agreement found between analogous values for copper(I1) acetate, the J values determined for this compound differed slightly more, when comparing GraFit values and literature values. This is attributed to
.
Table 3. GraFit Output for CuVO(fsa)zen CHaOH
Reduced Chi squared = 4.009e-010 Variable
Reduced Chi squared = 6.419e-009 Value 2.03
9
coupling constant
51.22
X
Y
Temperature
Molar Susc.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
55 62 69 79 87 100 109 118 130 140 149 161 167 175 189 198 213 220 229 239 252 262 272 280 293
Table 4. GraFit Output for [Co(Cusaien)z(Hz0)z](CIO4)z
0.01836 0.0161 0.01432 0.01238 0.01113 0.00959 0.008697 0.007975 0.007162 0.006614 0.006181 0.005677 0.005437 0.00516 0.00473 0.004495 0.004146 0.004 0.003834 0.003665 0.003448 0.003305 0.003173 0.003068 0.002918
Sld. Err,
Variable
0.002385
B
1.382
coupling constant
2.436
Calculated 0.01832 0.01612 0.01437 0.0124 0.01116 0.009573 0.008703 0.00797 0.007158 0.006593 0.006153 0.005646 0.005422 0.005149 0.004729 0.004493 0.004146 0.004001 0.003829 0.003654 0.003448 0.003305 0.003173 0.003075 0.002927
The heterotrinuclear case is illustrated by a Cu(I1)Co(I1)-Cu(I1) wmplex,
-1 6.49 X
Y
Temperature
Molar Susc.
Std. Err. 0.004004 0.145
Calculated
The TIP term for this complex is 500 x lo4 cm3/mol. The experimental data and calculated values are given in Table 4. The value ofg for this complex indicates a large orbital contribution, which is expected for cobalt(I1). Conclusion The examples discussed in this article introduce the important area of polynuclear and cluster wmplexes in transition metal chemistry A convenient method for fitting magnetic parameters is given. I t has several advantages.
l o w cost and availability of personal computers interactive nature of the technique simultaneous determination of standard deviations ability to quickly display theoretical and experimental susceptibility data ability to visually examine the effect on the magnetic properties of systematically varying the coupling constant
the relative insensitivitv of the function to variations in positive J values. ~ t u d g n t scan demonstrate this point to themselves bv olottine the theoretical molar susceotibilitv function wit6a'seriesif positive J values, as illusgated f& negative J values in Figure 3. Heterotrinuclear
Value
It is also possible to use these same programs to investigate other types of magnetic behavior of coordination complexes, such as high-spidow-spin equilibria (12). Literature Cited
which was prepared by the reaction of Cu(sa1en) with cobalt(I1) perchlorate (5). In this molecule the three metal ions form a n isosceles triangle with the cobalt ion that occupies the central position. The octahedral cobalt is wupled to each of the copper(I1) ions through both bridging phenolic oxygen atoms. The exchange between wbalt and copper is antiferromagnetic. There is no magnetic interaction between the copper atoms because there is no suitable exchange pathway. The equation for a n interacting S = 112 - 312 - 112 system is
Eamshaw,A.hlZaiuetion toMa@etaehemistry:Academic: Landon, 1968. Hay, J. P.; Thibeault.J. C.; Ho6mann,R J . A m Chem. Sor 1815,97,4884-4999. Sinn. E. C d Chem Rm 1910.5.313447. O'Connor, C. J . P m g m s in Inorganic Chemiaby las3,29,20%283. Gruber, S. J.;Harris,C.M.;Sinn, E. J. Chem. Phys. 1888,49,2183-2191. Ginsbrg, A P ; Martin,R. L.:Shemood.R. C.lnar#. Chem. 1988, 7,93%936. Rubenackr, G. Dmmheller, J. E.; Emerson, K; Willett, R. D. J Mogn. Mogn. Molt 198(l.54,1483-14S6. 8. Figgia, B. N.; Martin, R. L. J. C h . Soe. lsM1,38373846. 9 . Kahn,0.; Tala, P.; Galy. J.; Coudenne. H. J . A m Chem Sac 1818, IW,99314933. lo. Koch. C . A; Reed, C . A ; Brewer, G. A: Reth,N. P.; Seheidt, W. R ; Gupta, 0.; Lang, G. J Am. Chem Soe. las3.111.764~1647. 11. Pei,Y; Joumaux. J.; Kahn, 0. Inog. Chem. l88SS28.1W-103. 12. Kennedy, B. J.; Mffirsth, A. C.: Murray, K S.; Sheiton, B. W.; White, A. H. Inorg. Chem. 1887,26,48%95. 1. 2. 3. 4. 6. 6. 7.
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