Interpretation of the temperature dependent g values of the Cu(H2O)2+ 6 ion inseveral host lattices using a dynamic vibronic coupling modelMark J. Riley, Michael A. Hitchman, and Amisa Wan Mohammed Citation: The Journal of Chemical Physics 87, 3766 (1987); doi: 10.1063/1.452932 View online: http://dx.doi.org/10.1063/1.452932 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/87/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Theoretical study of the Cu(H2O) and Cu(NH3) complexes and their photolysis products J. Chem. Phys. 103, 1860 (1995); 10.1063/1.469710 Temperature dependence of the g value of Ni2 + in ZnSiF6⋅6H2O J. Chem. Phys. 88, 705 (1988); 10.1063/1.454148 State selected ion–molecule reactions by a TESICO technique. VI. Vibronic‐state dependence of the crosssections in the reactions O+ 2(X  2Π g , v; a  4Π u , v)+H2→ O2H++H, H+ 2+O2 J. Chem. Phys. 77, 4441 (1982); 10.1063/1.444446 Thermal Dependence of Fluorescence and Lifetimes of Sm2+ in Several Host Lattices J. Chem. Phys. 47, 3790 (1967); 10.1063/1.1701535 Magnetic Resonance Studies on Copper(II) Complex Ions in Solution. I. Temperature Dependences of the17O NMR and Copper(II) EPR Linewidths of Cu(H2O)6 2+ J. Chem. Phys. 44, 2409 (1966); 10.1063/1.1727057

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Interpretation of the temperature dependent 9 values of the CU(H20)~+ ion in several host lattices using a dynamic vibronic coupling model

Mark J. Riley, Michael A. Hitchman,a) and Amisa Wan Mohammed Chemistry Department, University a/Tasmania, Box 252C, Hobart, Tas 7001, Australia

(Received 14 May 1987; accepted 5 June 1987)

The causes of the previously reported temperature dependence of the g values of the CU(H20)~ + ions in Cu2+ doped Zn(H20)6(GeF6) and the Tutton's salts M2Zn(H20)6(S04)2' where M = K+, Rb+, NH4+, and Cs+, supplemented by new experimental measurements on the K + salt, have been investigated. The ground state dynamics of the complexes have been modeled on the cubic E X E J ahn-Teller Hamiltonian perturbed by an orthorhombic lattice strain. For each compound, the vibronic energy levels and associated wave functions were calculated numerically, the overall g values at any temperature being given by a thermal average of the g values of the individual vibronic energy levels, because of rapid exchange between the levels. For the Tutton's salts it was found that the low temperature g values are strongly influenced by the tetragonal component of the lattice strain, with this corresponding to an axial compression of the ligand field. The temperature dependence of the g tensors, on the other hand, was found to depend largely on the orthorhombic component of the lattice strain. For the K + and NH4+ salts, where structural data are available, the strain parameters derived using the model are in good agreement with the geometries reported for the Zn (H20) ~ + host complexes. For Cu2+ doped Zn(H20)6(GeF6) the model implies a lattice strain of tetragonal symmetry corresponding to a slight elongation of the axial metal-ligand bonds. The results are compared with those reported previously by Silver and Getz, and other workers, who used a simple model involving temperature dependent equilibria between energy levels corresponding to different orientations of the Cu (H20) ~ + ions in the host lattices to interpret the temperature dependence of the g tensors.

INTRODUCTION

From the earliest period of magnetic resonance spec­troscopy the EPR parameters of the Cu (H20) ~ + ion have been the subject of numerous theoretical and experimental studies. \-10 Frequently, these have been used to test models of the potential surface of a six-coordinate copper(II) com­plex produced by Jahn-Teller coupling. Particular interest has been shown in the way the EPR spectra vary as a func­tion of temperature. The pioneering work of Bleaney et al. 10

on Cu2+ doped into the trigonal sites in Zn(H20)6(SiF6 )

provided direct evidence for the so-called "warped Mexican hat" potential surface of the Cu (H20) ~ + ion. In this host lattice the x, y, and z molecular axes of the Zn (H20) ~ + ion are equivalent by symmetry and, at low temperature, spectra indicative of elongated tetragonal CU(H20)~ + guest com­plexes are observed, the axis of elongation being randomly distributed among the x, y, and z directions. However, above ~ 20 K the anisotropic spectra are replaced by an isotropic signal. Following the suggestion ofO'Brien,9 it is now recog­nized that the low temperature spectra are associated with CU(H20)~ + complexes each of which is localized in one of the three wells in the potential surface by random lattice strains. As the temperature is raised thermal population of higher, less localized levels allows rapid exchange among the three wells, so that an isotropic signal is observed. This type

a) To whom correspondence should be addressed.

of behavior has since been observed in many other sys­tems. 7 ,R

More recently, again following the pioneering work of Bleaney and co-workers,2 attention has focused on the spec­trum of Cu2+ doped into systems with lower symmetry. In an important paper, Silver and Getz5 discussed the behavior ofCu2+ doped into the Tutton's salt K2Zn(H20)6(S04)2' In this monoclinic lattice the x, y, and z molecular axes are not symmetry related, and at low temperature the guest CU(H20)~ + complex in each site has rhombicg and A ten­sors. On warming, it was observed that while the parameters related to one molecular axis were essentially temperature independent, those along the other two axes approached one another in value. This occurred without any change in the orientation of the principal magnetic axes, and Silver and Getz were able to explain the behavior of the g and A values using a very simple model in which the energies of the three wells of the Mexican hat potential surface are perturbed by a lattice strain of orthorhombic symmetry. It was assumed that the magnetic properties are dominated by just the low­est level in each well, the temperature dependence of the parameters being given by a weighted average of the Boltz­mann population of these three levels. It is implicit in this model that the rate of the transitions between the wells is sufficiently fast to produce motional narrowing of the EPR spectra. II The observed behavior suggested that just two wells were significantly populated over the temperature range of the experiment, and it was further assumed that the

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Riley, Hitchman, and Mohammed: 9 values of Cu(H20)~ t 3767

magnetic parameters of these are identical except for an in­terchange of the x and y axes. Excellent agreement with ex­periment was obtained for both the g and A values assuming an energy difference of 75 cm -I between the two wells, and from an analysis of the temperature dependence of the spec­tral linewidths the authors were able to estimate that the energy barrier to interconversion between the wells is ~ 120 cm -I. Later studies have shown quite similar behavior, both for Cu2+ doped into other Tutton's salts6 and various other host lattices,4 as well as for several pure copper(II) com­pounds. 6,13 The above model has been used to interpret not only the EPR measurements of these systems, but in some cases also the temperature dependence of x-ray diffraction data. 12- 14

It was recognized by Silver and Getz that their model is simplistic, in particular in its neglect of the higher vibronic levels in each well, and indeed their paper ended with the hope that their work would ..... stimulate further theoretical work ... to elucidate the role of the excited vibronic states." We recently developed a model to interpret the temperature dependent g values of Cu2+ doped K2ZnF 4. 15 This involved the numerical solution of the strain perturbed E X E vibronic Hamiltonian for the tetragonal CuF! - guest complex, and it was found that both the low temperature g values, and the temperature dependence of the g tensor could be explained in terms of a potential surface with a single minimum corre­sponding to a compressed tetragonal ligand geometry. In this model, therefore, in marked contrast to that used to explain the behavior of Cu2+ doped K2Zn(HzO)6(S04h the temperature dependence of the EPR parameters is due not to exchange between different static distortions of the copper(II) complex, but to a Boltzmann distribution over the vibronic energy levels of the potential surface, with vi­bronic coupling causing these to differ in their electronic compositions. This approach has also been successfully ap­plied to the interpretation of the temperature dependent EPR parameters of the tetragonal CuCl4 (HP) ~ - and CuCI4(NH3)~ - guest complexes present in Cu2+ doped NH4Cl. I6 In order to investigate the limitations of the Silver and Getz approach, and to derive optimum values for the parameters defining the potential surface of the CU(H20)~ + ion, we have extended the vibronic coupling model to allow the treatment of a six-coordinate copper (II) complex perturbed by an orthorhombic lattice strain. The model has been used to interpret the temperature dependent EPR parameters of Cu2+ doped M 2Zn(HzO)6(S04)2' M = NH4+, K+, Rb+, and Cs+, and Zn(H20)6(GeF6 ),

with the data being supplemented by new measurements for the K + Tutton's salt between 110 and 295 K.

EXPERIMENT AL

Crystals of K2Zn(Hp)6(S04)2 doped with ~0.5% Cu2+ were grown by recrystallizing the double sulphates in the appropriate proportions. EPR spectra were measured at X-band frequency using a JEOL JES FE3X spectrometer, using a standard JEOL flow cryostat and temperature con­troller to vary and record the temperature. The crystals were mounted using the standard JEOL rotation accessory.

The molecular g tensor of the guest CU(H20)~ + ion was obtained at ~ 110 and 295 K by rotating a crystal about three orthogonal axes and treating the measured g values by a procedure described in detail elsewhere. 17 A speck of finely powdered a,a'-diphenyl-/3-picrylhydrazyl (DPPH, g = 2.0036) was used as a reference marker. It was found that within experimental error (~5°) the principal g axes coincided with the M-O bond directions, confirming the ob­servation of Silver and Getz that the orientation of the g tensors is indeed independent of temperature.5 In order to measure the temperature dependence of the g values, crys­tals were oriented with the magnetic field approximately along each molecular direction and the EPR spectrum was recorded at various temperatures. The molecular g values were obtained from the crystal g values by solving the set of three simultaneous equations involving the molecular pro­jections appropriate to each magnetic field direction. The crystal and molecular g values at various temperatures, and molecular projections for the different crystal orientations are given in Table 1.

The molecular g values of (NH4)zZn(H20)6(S04)2 doped with ~0.5% Cu2 + were also measured over the tem­perature range 110-200 K and results essentially identical to those reported by Petrashen et al. 6 for the system were ob­tained. The new data are included in Fig. 4.

RESULTS AND DISCUSSION

The present approach closely follows that described pre­viously 15 for the interpretation of the temperature depen­dentg values ofK2Zn [Cu] F4. Basically, this involves calcu­lating the g values from the vibronic wave functions that are solutions to a vibronic Hamiltonian perturbed by lattice strain. The overall molecular g values at any temperature are obtained from the weighted average of the g values of the individual vibronic energy levels as given by the Boltzmann population distribution. In effect, therefore, the experimen­tal observables are the low temperatureg values and the way in which these vary as a function of temperature, while the only variable parameters in the series of five systems are the two components of the strain induced by each host lattice. Implicit in the approach is the assumption that the basic parameters defining the vibronic wave functions ofa regular CU(H20)~ + ion are essentially unaffected by the nature of host. Clearly, an important test that the model is realistic will be provided by a comparison of the strain parameters with the crystal structure of each host when this is known.

Calculation of the potential surfaces and vibronic wave functions

The vibronic Hamiltonian is that of the usual E X E

Jahn-Teller coupling Un with the addition of terms repre­senting the lattice strain U ST ' which formally make this a pseudo-Jahn-Teller problem. IS The approximation is made that the strain term does not destroy the symmetry of the cubic part of the Hamiltonian, so that there will be only one first order and one second order electronic coupling con­stant, and one harmonic vibrational force constant, instead of the six, nine, and two independent constants which the

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3768 Riley, Hitchman, and Mohammed: 9 values of Cu(H20)~+

TABLE I. Molecular and crystal EPR parameters ofK2Zn[Cu) (H20)6(SO.)2 measured at various temperatures.

T(K) gl' A • gx b Ax b g2 • A2 a gy b Ay b g3' A • gz b A z b

I J

178 2.3234 56.75 2.3272 61.61 2.2075 51.50 2.2506 51.04 2.0547 62.25 2.0341 61.61 193 2.3194 56.17 2.3231 60.86 2.2094 52.17 2.2528 52.24 2.0553 61.50 2.0351 60.86 208 2.3169 55.33 2.3206 59.89 2.2109 52.67 2.2542 53.27 2.0566 60.50 2.0367 59.89 223 2.3078 54.67 2.3144 59.68 2.2142 55.00 2.2606 53.44 2.0489 60.25 2.0290 59.68 233 2.3064 54.17 2.3100 58.34 2.2198 58.33 2.2674 61.43 2.0489 59.50 2.0291 58.33 243 2.3060 53.17 2.3095 57.23 2.2197 56.83 2.2670 59.37 2.0502 59.75 2.0305 57.23 253 2.3056 52.33 2.3092 56.33 2.2205 56.33 2.2680 59.00 2.0502 58.75 2.0305 56.33 266 2.3048 52.00 2.3084 55.96 2.2213 54.67 2.2691 57.16 2.0495 51.50 2.0299 55.96

a Subscripts 1 and 2 refer to the two signals observed using a crystal orientation with the magnetic field at 450 to the a axis in the a b crystal plane; subscript 3 re­fers to the high-field signal when the magnetic field is at 300 to the b axis in the b c* crystal plane. The hyperfine values are in (cm - I X 104

).

bMolecular g values were obtained by solving the equations: ~ = 0.0131g; + 0.0026g; + 0.9843~; ~ = 0.2107g; + 0.7795g; + 0.0098~; ~ = 0.9328g;

+ 0.0054g; + 0.0618g;. The hyperfine values were obtained by solving the same set of equations with Al replacing gl' A, replacing g" etc.

low symmetry of the problem actually requires. 18 The Ham­iltonian to be used is

H = Ho + Hrr + HST '

Ho = [0.5hv(P~ +P~ +Q~ +Q~)

+K3Qe(Q~ -3Q;)]I,

HJT =AI(QeO"z -Q€O"x) +Az[(Q~ -Q;kz

+ 2QeQ€O"x],

H ST = SeO"z - S€O"x,

where

o 1= [1 0] o l' (1)

Here A I' A 2 , and K3 are the first and second order coupling constants, and the cubic anharmonicity of the tetragonal component of the lahn-Teller active €g vibration, respec­tively. The energy of this vibration would be hv in the com­plete absence of the lahn-Teller effect, while Se, S€ are the tetragonal and orthorhombic components of the strain, re­spectively. All parameters in Eq. (1) are in units of cm -I.

The desired energy levels and wave functions of the eu (H20) ~ + complex are obtained by diagonalizing the ma­trix obtained by applying the above Hamiltonian (1) to a truncated set ofvibronic basis functions. This basis consisted of the two Eg (dx ' _ Y" d;? ) metal orbital wave functions and the N [= 1I2(nu + 1 )(nu + 2)] two-dimensional har­monic oscillator wave functions of the €g vibration up to the level nu' Because of the strong coupling in these systems, the displacement inp is large, and a sizable basis set of vibration­al wave functions is needed. In addition, since the factor group symmetry of the vibronic Hamiltonian is C I , no sym­metry blocking of the secular equation is possible. 15 In the present calculations a value of nu = 30, corresponding to a basis size of992, was found to provide a satisfactory conver­gence (this was tested by monitoring the effect of varying the basis size). Due to the large size of these matrices, a diagona­lization routine was adopted which incorporated condensed storage techniques using the Lanczos algorithm. 19 Neglect­ing the kinetic energy terms, and in the absence of lattice strain, vibrational anharmonicity, and second order cou­pling, the adiabatic potential described by the above equa­tion takes the form of the well-known "Mexican hat" surface

[Fig. 1 (a) ]. Here the potential energy minimum may be considered to take the form of a circular well, with a substan­tial radial displacement in the €g mode p but no dependence of ¢y, i.e., with the two components Qe and Q€ remaining equivalent:

E

z z

Q8 mode QE mode

FIG. I. (a) The adiabatic Mexican hat potential surface of the linear E Xe Iahn-Teller problem. (b) The Q, and Q, components of the Iahn-Teller active Eg vibration of an octahedral complex.

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Qe =pcostjJ; Q" =psintjJ. (2)

The effect of the second order electronic and anharmonic terms is to discriminate between Qe and Q", and hence pro­duce the "warped" Mexican hat potential surface, with ener­gy minima occurring at tjJ = 0°, 120°, and 240° for A2 > 0, corresponding to tetragonal elongations of the ligands along Z, x, and y, and three saddle points at tjJ = 60°, 180°, and 300° corresponding to tetragonal compressions. As has been not­ed elsewhere,20 because of the cubic dependence on p, the inclusion of the anharmonicity parameter K3 in the calcula­tions causes computational problems when, as in the present case, the displacement in p is large. This difficulty could be circumvented by extending the anharmonicity to a fourth order term, or alternatively the warping of the potential sur­face may be represented simply by the second order elec­tronic term A 2• Although it is more likely that the cause of the warping is in fact anharmonicity, probably supplement­ed by configuration interaction between the metal a lg (dz ' )

and a Ig (4s) orbitals,21 the correct form of the potential sur­face may be represented quite satisfactorily using just the parameter A 2 , and for the sake of simplicity this procedure was adopted in the present calculations.

Choice of the values of the parameters used in the calculations

Energy of the eg vibration

An energy of 300 cm ~ 1 was used for this vibration, this being the value assumed in other similar studies on Cu(H20)~+ ,8(b),9 and intermediate between the energies observed experimentally for this mode for Ni(H20)~ + (305 cm~ I) and Zn(H20)~ + (278 cm~I).22

Linear coupling constant A1

The linear coupling constant has previously been given values of 812 cm ~ I by Williams et al. 8(b) and 1300 cm ~ I by O'Brien.9 The latter value is probably too large, as it was based upon an incorrect assignment of the electronic spec­trum. A value of 960 cm ~ I has been suggested by Bill,23 based upon an average of the structural data of 15 pure com­pounds. A value of A 1 may be estimated from the optical transition to the upper Jahn~Teller surface, which has been observed at ~ 7500 cm ~ I in several pure copper (II) Tut­ton's salts.24 Neglecting orthorhombic strain and second or­der coupling the energy of this transition is given by

IlE=-4EJT + 21Se I, where the Jahn~ Teller stabilization energy EJT = A i / (2hv). Anticipating values of Se =- - 500 cm~l, this sug­gests a value A I =- 990 cm ~ I, Cooperative effects may make this somewhat larger than the value expected for an isolated complex, and taking all the above considerations into ac­count a value of AI=- 900 cm ~ I was used in the present study.

Second order coupling constant Az

In the present approach this parameter defines the bar­rier height 2/3 between the equivalent wells of the warped

Mexican hat potential surface along the Jahn~Teller radius Po:

/3= IA2Ip~· when A2 takes positive values in Eq. (1), then the potential surface is characterized by minima at 00, 120°, and 2400, cor­responding to tetragonal elongations. It is generally recog­nized that the value of /3 is hard to define, Previous estimates have been/3=335,21 450,8(b) and 600 cm~ 1.9 Assuming the first, most recent, estimate to be the most reliable, an initial value /3 = 300 cm ~ 1 (A 2 = 33 cm ~ 1 ) was used in the present calculations. However, an important purpose of the present work was to attempt to define the warping parameter more precisely by testing the sensitivity of the temperature depen­dence of the g values to the magnitude of /3.

Lattice strain parameters

Ham has shown that the effect of a strain induced by a host lattice can be viewed in terms of the displacement (Q~, Q ~) along the cg coordinates in the absence of the Jahn~ Teller effect. 25 For strong linear coupling this yields the rela­tionship

(3)

for the contribution of the strain to the cubic Hamiltonian. A comparisonwithEq. (1) gives the relationshipsSe =AIQ~, S" = A 1 Q ~, where A 1 is the linear coupling constant. In the present calculations the strain parameters S(! and S" were treated as variables in fitting the temperature dependent g values, with the optimum values being compared with esti­mates from the host crystal structures when these were known.

The molecular coordinate system

In the present calculations, the molecular coordinate system is defined by the symmetry of the lattice strain, since the Cu (H20) ~ + ion is assumed to have cubic symmetry when the strain is absent, with x,y, and z equivalent. Because the lattice strain in Zn [ Cu 1 (H20) 6 (GeF 6) reinforces the natural tendency of a copper( II) complex to elongate along one Cartesian axis, the ground state at low temperature is just that normally observed for such a complex, with the unpaired electron predominantly in dx ' _ yO, with a small amount of dz' mixed in by vibronic coupling. Here, there­fore, the g values are defined in the conventional manner, with gz > gx = gy. However, in the Tutton's salts the princi­pal component of the strain tends to oppose the natural ten­dency ofa copper(II) complex to elongate along the tetra­gonal symmetry axis, and this causes the potential to have two equivalent wells in the absence of an orthorhombic strain component. Static electronic wave functions at the minima of these wells would take the unconventional form dy'_z' or dx ' ~ z" The effect of the orthorhombic strain is to lower the energy of the former well with respect to. the latter, and also to mix a contribution from the dx ' orbital into this wave function. It follows that the low temperature principal g values are also defined in an unconventional manner, with gx > g y =gz· It also follows that the change from a near te­tragonal spectrum with gl <gil at low temperature to the

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3770 Riley, Hitchman, and Mohammed: g values of Cu(H20)~+

near tetragonal spectrum with gil <gl at room temperature does not correspond to a rotation of the principal axes, as was originally thought. I

The vibronic wave functions

Each eigenfunction of the vibronic Hamiltonian in Eq. (1) will be a linear combination of the electronic and vibra­tional basis wave functions ¢i and tPj' respectively, taking the general form

IIJ = L LCij l¢itP) i j

(4)

The electronic and vibrational properties of these vibronic functions are obtained by integrating over the vibrational and electronic coordinates, respectively. For example, the vibronic probability functions can be shown in the vibration­al and electronic subs paces by

(IIJI lIJ)vib (Qe,Q<) = ~(~Cij l¢itPj) r = [~ajtPj (Qe,Q<) r

+ [~bjtPj (Qe,Q<) r (IIJI lIJ)electr (x,y,z) = ~(~CI) l¢itP) r

(5)

By using the explicit form of the basis functions tPj' ¢e' ¢ < in Eqs. (5) and (6), the vibrational and electronic parts of the vibronic wave functions can be plotted separately in­stead of in the six dimensions (x, y, z, Qe, Q<, amplitude) which would be needed for the total vibronic function of Eq. (4). Plots of this nature are shown in Fig. 2 for the first four vibronic functions calculated for the K2Zn[Cu] (H20)6(S04)2 system.

The vibronic wave functions in Eq. (4) are very differ­ent from a static electronic wave function:

(7)

In fact, it is impossible to express the vibronic wave functions in this form. The electronic properties of the vibronic wave functions are only accessible as the expectation values of the product of electronic coefficients after integrating over the vibrational coordinates. From Eq. (6) we have

(A 2) = Ia]; (B 2) = Ib]; (AB) = IaA. (8) J J J

Equations (8) obey the Schwarz relationship26

(AB )2<, (A 2) (B 2),

which would be an equality if the expectation values of the coefficients were replaced by the static coefficients of Eq. (7). It is only the expectation value of the electronic coeffi­cients which are available for calculating theg values in these dynamic systems.

a)

a:

k x

FIG. 2. (a) The lower adiabatic potential ofK,Zn[Cuj (H,O)6(S04)' giv­ing optimum agreement with the experimental EPR data. The potential pa­rameters are hv= 300 em -I: AI = 900 em-I; A, = 33 em-I; So = - 1000 em-I; S, = 55 em-I. (b) The vibrational and electronic parts of the four lowest vibronic probability functions are shown on the left and right, respectively. The axes in the (Q8 and Q, ) vibrational, and the (x, y, and z) molecular spaces are shown.

Calculation of the 9 values

In the present systems, the degeneracies of all the vi­bronic levels (excepting Kramers degeneracy) are removed by the lattice strains. As the energy separations between the levels (~100 cm - I) are large compared with the Zeeman splitting, the Zeeman operator will not mix the levels. This means that, in effect, each level will have its own set of g values, and if vibronic relaxation between the levels is rapid on the EPR time scale, a Boltzmann average over the levels will yield the singleg tensor which is observed experimental­ly.11 The temperature dependence of the EPR parameters is explained by the change in the thermal population of the levels.

In order to calculate the g values the admixture of the excited states derived from the t 2g set of orbitals ofthe parent octahedral complex by spin-orbit coupling must be taken into account. The relative energies of these may be described by the equations

Exz =JlE+A](T)( - (Qe)/2-{3(Q<)/2),

E yz = JlE + A] (T)( - (Qe)12 + {3(Q<)/2),

Exy = JlE +A](T)(Qe)' (9)

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Riley, Hitchman, and Mohammed: g values of Cu(H20)~+ 3771

Here !::.E is the energy baricenter of the excited (xy, xz, yz) electronic states and Al (T) can be viewed as a linear constant of a T Xc Jahn-Teller problem. The experimentally observed splitting of the orbital triplet state is ~ 1500 cm - 1

in a number of pure copper(lI) Tutton's salts24 and, using the present set of parameters [Po = A I/(hv) = 3], this sug­gests Al (T) "'" - 330 cm -I. This value is also consistent with the ratio of the E X c and T X c coupling constants pre­dicted by the angular overlap modeI21

(b) for the present sys­tem: A II A I (T) = 3. It may be noted that although the met­al-ligand 1T bonding might be expected to be anisotropic, which would complicate the above treatment, in practice no significant anisotropy was detected in the analysis of the op­tical spectra. 24 A value of !::.E"", 11 900 cm - I, as observed in the electronic spectra of several copper( II) Tutton's salts, 24

was used in the calculations. Note that the energies of the dxy , dxz> dyz orbitals, as given by Eqs. (9), are slightly differ­ent for each ground state vibronic level because of the differ­ent values of (Qe) and (QE)' These expectation values may be calculated directly from the vibronic wave functions in Eq. (4).

The effect of spin-orbit coupling on the vibronic wave functions is estimated by first calculating the effects on the two possible spin states of each of the "pure" basis functions Id; ) and Id J _ y')' This is done by evaluating two separate 8 X 8 matrices since there is no spin-orbit matrix element connecting the I d f ) and I d J _ y' ) functions and the effect of the orthorhombic ligand field has already been included as strain terms in the vibronic Hamiltonian. The effect of the spin-orbit coupling is to "contaminate" the pure spin func­tions IdJ), Id z-;-), IdJ- y')' Idx~_y')' with the excited states:

It,be+) =c1IdJ) +iC2Idy~) -c3Idx~),

It,be-) = clld z-;-) + ic2 1d y~) + c3 1d x~),

It,bE+) =c4Id J_ y') +iC5Idy~) +c6 Id x-;) -ic7Id x;),

It,bE-) =c4Idx~_T) +iC5Idy~) -c6dlx~) +iC7Idx~)' (10)

If the static electronic wave functions as expressed by Eq. (7) were available, then the 1'1' ± ) spin functions could be found simply by substituting Eqs. (10) into Eqs. (7), yield­ing the pair of Kramers doublet wave functions:

t,b+ =A It,be+) +BIt,bE+); t,b- =A It,b;) +BI'I'E-)' (11 )

The principal g values could then be found by solving the matrix equation obtained by applying the Zeeman operator f-lj = (k,lj + 2.0023 Sj )(3 B between the above wave func­tions in the standard way; here (3 is the Bohr magneton, B is the magnetic field vector, and k is an orbital reduction pa­rameter.

However, since only the dynamic coefficients ofEq. (8) are available, the following circuitous path must be taken. To obtain a particular g-tensor element, gxx for instance, Eq. (11) yields

gxx = 2('1'+ I (klx + 2.0023sx ) 11P-)

= 2A 2( t,b: If-lx I t,be- ) + 2AB (t,be+ If-lx 1'1' E- )

+ 2BA (t,bE+ If-lx It,be-) + 2B 2 (t,bE+ If-lx 1'1';-)· (12)

Similar expressions occur for gyy andgzz . The product coeffi­cients are now all known and Eq. (12) may be evaluated for the dynamic vibronic wave functions. In applying the spin­orbit and Zeeman operators covalency was taken into ac­count by using an orbital reduction parameter k of 0.88. It was found that the same value could be used for each com­plex, with a spin-orbit constant of A = - 829 cm -I. In the present case, because of the choice of the molecular coordi­nate system, the diagonal g-tensor elements are also the prin­cipal values.

In summary, the steps in the calculation are as follows: ( 1) The vibronic Hamiltonian in Eq. (1) is solved to

yield the vibronic energy levels and wave functions, and for each level the following procedure is taken.

(2) The required properties of the wave function: (A 2), (AB), (B 2), (Qe), (QE)' are calculated.

(3) The values of (Q() and (Q<) are used to find the energies of the excited electronic states using Eq. (9) and these are employed to deduce the Kramers doublet wave functions [Eqs. (10)], by applying spin-orbit coupling.

( 4) The Zeeman operator is applied to these wave func­tions to calculate theg values of the level using Eq. (12), and the coefficients (A 2), (AB), (B2).

Finally, the energies of the vibronic levels are used to estimate the thermal average oftheg values at each tempera­ture.

It may be noted that for a potential surface of tetragonal or higher symmetry the term (AB) vanishes. Under these circumstances, although dz' and dx ' _ y' are mixed by vi­bronic coupling, the g tensor will be axially symmetric (gx = g y ), as is indeed required by symmetry. This has sometimes caused confusion in the literature,27 where it has been erroneously concluded that such a vibronic mixture will lead to orthorhombic g values, as would be the case for a static mixture of dz' and dx ' _ y"

In the original paper of Silver and Getz the temperature dependence of the copper( II) hyperfine parameters was suc­cessfully interpreted in a manner exactly analogous to the molecular g values. 5 No attempt has been made to treat the hyperfine parameters using the present model because these are related to the electronic part of the wave function in a more complicated manner than are the g values. In particu­lar, the isotropic contribution to the hyperfine interaction, conventionally represented by the parameter K, 2~ is known to have a different value when the electron occupies the dz '

orbital in a complex of tetragonal symmetry, from that of a complex with a normal d x ' _ v' ground state. 29 This is be­cause the metal 4s orbital m~y participate directly in the ground state wave function in the former situation. As the vibronic wave functions in the present complexes are consid­ered to be composed of different mixtures of d x ' _ y' and d z"

each will have a different value of K, and this would greatly complicate any interpretation of the temperature depen­dence of the hyperfine values.

Potential surfaces, vibronic wave functions, and g values of the complexes

As described above, it was assumed in the calculations that the dominant cause of the difference in the behavior of

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3772 Riley, Hitchman, and Mohammed: g values of Cu(H20)~+

the g tensors of the complexes is the lattice strain. Identical parameters, as given in the preceding section, were therefore used to define the basic unperturbed eu (H20) ~ + ion, and the orthorhombic lattice strain parameters were varied and the resulting molecular g values were compared with those observed experimentally. It quickly became apparent that the g tensors were rather sensitive to the strain parameters, and that the axial and orthorhombic components had quite different effects on the EPR spectra. Thus, for the four Tut­ton's salts, the deviation oftheg tensor from axial symmetry was found to depend almost entirely on the tetragonal strain parameter Se, with this being required to be negative, corre­sponding to an axial compression of the ligand coordination geometry. The temperature dependence of the g values of these compounds, on the other hand, was caused almost en­tirely by the orthorhombic strain parameter Se- However, the behaviorofZn[ eu] (H20)6(GeF6) as a function oftem­perature could only be explained by a rather small positive tetragonal strain, while the axial symmetry of the g tensor in this system implies a negligible orthorhombic contribution to the strain.

The energy minima of the potential surfaces of the com­plexes giving optimum agreement with experiment, plotted as a function of the relative displacement in Qe and Q. as defined by the angle ¢ (Fig. 1), are shown in Figs. 3 (a) and 3 ( c ) -3 (f). In effect, these picture the warping of the rim of the Mexican hat and of the eu (H20) ~ + ion calculated for each host. A sectional plot of the square of the vibrational part of the wave function associated with the lower levels is also shown on each plot. The energies and g values of the four lowest levels for each host are given in Table II.

It should be noted that the one-dimensional potentials shown in Fig. 3 are highly schematic, though they do allow a ready comparison of the systems. However, they are slightly misleading in that the path ofleast energy around the poten­tial is far from circular. This pathway will be a function of bothp and ¢, as shown in the two-dimensional contour plot of the potential of K 2Zn [ eu 1 (H20) 6 (SO 4) 2 shown in Fig. 2. Plots of the two-dimensional vibronic probability func­tions and the square of the metal part of the wave functions are also shown in this figure. It may be seen that the vibra­tional parts of the wave functions of the two levels of

0 a)

i a (J -1

0 0

::!

" 1;;;-2 .. " <l

r.l

-3

0 c)

i d)

e (J -1

0 0

::!

" ~-2 .. " <l

r.l

-3

0

~

e) f) I e

(J -1 0 0 0 ~

" 1;;;-2 .. " <l r.l

-3 0 120 240 360 0 120 240 360

t t

FIG. 3. The one-dimensional cross sections of the lower potential surfaces as a function of ¢ at constant Po. Schematic plots of the lower vibronie prob­ability functions are also shown. General potential parameters are hv = 300 em-I; AI = 900 em-I; A2 = 33 em-I. (a) K2Zn[Cu] (H20)6(S04)2; So = -1000 em-I, S. = 55 em-I. (b) K2Zn[Cu](H20)6(SO.)2 with warping parameter set to zero (A 2 = 0); So = 720 em-I, S. = 695 em-I. (c) Rb2Zn[Cu](H20)6(S04)2; So = - 800 em-I, S. = 110 em-I. (d) (NH4)2Zn[CU] (H20)6(S04)2; So = - 550 em-I, S. = 120 em-I. (e) Cs2Zn[Cu](H20)6(S04)2; So = - 650 em-I, S. = 200 em-I. (f) Zn[Cu](H,o)6(GeF6);So = l00em- l

.

K 2Zn [ eu 1 (H20) 6 (SO 4) 2 are indeed localized largely in the two lower wells, with the two associated electronic parts of the wave functions being predominantly dz' _ y' and dz' _ x',

For the higher levels, the vibrational wave functions become

TABLE II. Relative energies and molecular g values calculated for the lowest four levels of the CU(H20)~ + ion in various host lattices.

Levell Level 2 Leve13 Level 4

EI E2 E3 E. Lattice (em-I) gx gy g, (em-I) gx gy g, (em-I) gx gy gz (em-I) gx gy gz

K2Zn[Cuj (H2O)6(SO.)2 0 2.417 2.150 2.045 72 2.161 2.411 2.039 187 2.357 2.222 2.039 265 2.247 2.322 2.035 Rb2Zn[Cu] (H2O)6(SO.h 0 2.435 2.131 2.054 153 2.151 2.425 2.043 216 2.407 2.169 2.034 299 2.435 2.125 2.060 (NH4)2Zn [CU] (H2O)6(S04)2 0 2.427 2.119 2.067 181 2.134 2.422 2.057 235 2.414 2.143 2.074 291 2.426 2.112 2.074 Cs2ZN[Cu] (H2O)6(S04)2 0 2.447 2.117 2.062 235 2.434 2.139 2.051 292 2.269 2.280 2.060 293 2.304 2.243 2.060 Zn[Cu] (H2O)6(GeF6) 0 2.100 2.100 2.474 147 2.277 2.277 2.109 147 2.277 2.277 2.109 257 2.102 2.102 2.471

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Riley, Hitchman, and Mohammed: g values of Cu(H20)~+ 3773

progressively delocalized, with the electronic components of the wave functions containing an increasing admixture of the dx ' orbital. For the fourth level of this species the metal part of the wave function in fact consists of an almost equal mixture of dz' _ y' and dx" giving rise to an unconvential d function with substantial lobes along each of the Cartesian axes (Fig. 2).

Although potential surfaces are conventionally used to illustrate the E X € Jahn-Teller problem, it must be pointed out that strictly speaking the numerical solutions of the non­adiabatic vibronic Hamiltonian (1) presented here produce vibronic states that cannot be associated completely with either the lower (hat) or upper (cone) part of the Mexican hat potential surface. However, in the present case of strong coupling the lowest levels will be associated to a good ap­proximation with the lower surface of the Mexican hat, with the electronic part of the adiabatic vibronic functions given by

\{I = sin(Z /2) Idz') + cos(Z /2) Idx ' _ y')' (13)

where

Z = tan - 1 [ (A 1 P sin ¢ - A LfJ2 sin 2¢ + S E ) /

(A 1 P cos ¢ + A2 p2 cos 2¢ + So)]

with the phase convention of Eq. (1).

(14)

The expectation values of (sin2 Z /2), (cos2 Z /2), and (sin Z /2, cos Z /2) were calculated using 30-point Hermite quadrature. 30 These were compared directly with the coeffi­cients (A 2), (B2), and (AB), respectively (exact equality would occur in the absence of nonadiabatic coupling between the Mexican hat surfaces). Very small differences were observed, confirming that the association of the vi­bronic wave functions with the lower potential surface is quite realistic in the present cases.

It may therefore be noted that although the present treatment has used the dynamic E X € formalism, the effect

of the strain has been to localize the lowest vibronic wave functions in both an electronic and geometric sense. For this reason, it could be argued that the compounds discussed here should not be classed as Jahn-Teller systems, as the dynamics of the problem are to a large extent quenched by the low symmetry of the hosts. However, these rigid elec­tronic and geometric properties differ for the lowest vibronic states in each compound, due to their localization at differ­ent minima of the potential surface. The perturbation of the cubic E X € J ahn-Teller Hamiltonian by the low symmetry strain of the host remains a convenient way to study the adiabatic potentials and solve the vibronic equations. In ad­dition, as may be seen in Figs. 2 and 3, the higher vibronic levels become delocalized in the geometric sense, and this not only has some effect on the g values at higher tempera­tures, but also provides a relaxation pathway between the different minima.

The calculated temperature dependences oftheg values in each host, obtained using the strain parameters listed in Table III, are compared with those observed experimentally in Figs. 4(a)-4(f). Despite the constraint that identical pa­rameters were used to define the basic potential surface of the Cu (H20) ~ + ion in each host, the agreement between the calculated and experimental values is good in every case. In fact, the g values were found to be relatively insensitive to these basic parameters, so that varying A l' A 2, hv, or /3 by ± 10% while keeping all else constant caused only a small change both in the EPR parameters and in their temperature dependence. This is in marked contrast to the strain param­eters, where changing either So or SE by this amount had a significant effect on the behavior of the g tensors.

The four Tutton's salts form a series in which both the magnitude of the deviation of the g tensor from tetragonal symmetry at low temperature, and the temperature at which gx andgy start to converge, change progressively [Figs. 4(a) and 4(c)-4(e)]. The present model relates the increase in

TABLE III. Structural distortions and "best fit" parameters defining the potential surfaces.

Energy Bond length deviations induced Lattice strain Positions of separation

by lattice strain (pm) a parameters (em - I ) minima (em-I)

Compound 8z 8y 8x So S, -So/So (rPI>g (rP2>g El2 b 8 l2

c

K2Zn[ Cu) (H2O)6(S04)2 - 5.1 2.3 2.8 - 1 ()()() 55 18 129.0 228.7 72 75 K 2Zn(H,o)6(S04)' - 6.5(0.4)d 2.9(0.5)d 3.6(0.5)d - 1280e 80e 16 Rb2Zn[Cu) (H,o)6(S04)' -4.1 1.5 2.5 - 800 110 7.3 126.4 230.2 153 ~ 175 (NH4)2Zn[ Cu) (H,O)6(S04)2 - 2.8 0.9 1.9 - 550 120 4.6 124.2 233.5 181 ~230

(NH4)2Zn (H2O)6(S04)2 - 3.2( 1.2)' 1.0( 1.2)' 2.2( 1.2)' - 630' 140' 4.5 Cs2Zn[Cu) (H2O)6(S04)2 - 3.3 0.8 2.5 - 650 200 3.2 124.4 184.2h 292 ~290

Zn[Cu) (H,o)6(GeF6) 0.5 -0.3 -0.3 100 0 O{ilig' 147 154

a Deviations of the bond lengths of the host complex from a regular octahedron with Zn-O = 210 pm. Values are deduced from the "best-fit" EPR parameters for the Cu2 + doped systems using Eq. (3), and x-ray data for the pure Zn2 + compounds.

b Energy of the lowest level localized largely in the upper well from present calculations. 'Energy of the minimum of the second well estimated using the Silver and Getz model. d Reference 37 (standard deviation per bond length in parentheses). eEstimated from the bond length differences assuming an energy of the Eg modeof300cm- ' ; and a value of A I = 900 cm- I for the linear coupling constant. 1 Reference 38 (standard deviation per bond length in parentheses). g Expectation values of the angle calculated from the vibronic wave functions. hThere is no wave function localized about the upper minimum in this complex. i In tetragonal symmetry, the wave functions based in the two upper rb = 120,240 minima, will have (rP> = 0'. However, small orthorhombic strains in this monoclinic compound would localize these wave functions, and the values given were calculated with S, = 5 cm - I.

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3774 Riley, Hitchman, and Mohammed: g values of Cu(H20)~+

2.5 ,-----,---,-----,-, 2.5 ,-----,,----, __ --,

a) 2.4

fIl !:l 2.3

~ I 2.2 ..

2.1

2.0 L-__ '--__ '--_----"

o 100 2.5 r----,---,------,

2.4

fIl r.l 2.3

~ I 2.2 ..

2.1

2.0 L-_----L __ -:-::---_.,,-J o 100

2.5 ,-----,---,----,

fIl !:l 2.3

b) 2.4 ~IE::------~

~. 2.2

100 200 300

100 200 300

~ I 2.2 .. ." n 2.2

2.1 2.1

VVVV~ vvv V'

2.0 2.0 L-_----' __ -'-__ .,,-J o 100 200 300 0 100 200 300

Temperature (Kelvin) Temperature (Kelvin)

FIG. 4. The experimental and calculatedg values ofCu(H20)~ + in various host lattices as a function of temperature. The calculated g values in (a)- (f) correspond to the potential surfaces in Figs. 3(a)-(O, respectively. (a) K2Zn[Cu] (H20)6(S04)2' (b) K2Zn[Cu] (H20)6(S04)2' The full, dashed, and dotted curves are for f3 = 0, ISO, and 450 em -', respectively. (c) Rb2Zn[Cu] (H20)6(S04)2' (d) (NH4)2Zn[CU] (HP)6(S04lz. (e) Cs2Zn[Cu] (HP)6(S04)2' (0 Zn[Cu] (H20)6(GeF6)·

ogyz = (gy - gz) which occurs as the cation changes from NH4+ to Cs+ to Rb+ to K+ (0 gyz = 0.052,0.055,0.077, and 0.105 along this series at 4.2 K) to a progressive increase in the tetragonal strain parameter, which almost doubles on going from the ammonium to the potassium salt (Table III). This strain component produces a compression along the z molecular axis, raising the energy of the potential energy minimum at ¢ = 0° in the cubic warped Mexican hat poten­tial surface to such an extent that the levels associated with this are not significantly populated even by 300 K, and rotat­ing the other two minima away from their initial positions at ¢ = 120° and 240°. It is this latter rotation, which is associat­ed with a progressive increase in the participation of the dz '

orbital in the ground state wave function, which is the domi­nant cause of the orthorhombic component of the g tensor at 4.2 K. The positions of the two lower minima are indicated for each complex in Table III. The temperature at which gx and g y start to con verge increases from ~ 30 to ~ 60 to ~ 90 to ~ 120 K along the series M = K +, Rb+, NH4+, Cs+ (Fig. 4) and the model relates this to a progressive increase in the orthorhombic strain parameter (Table III). This tends to stabilize an elongation along the x direction, and has the effect of progressively raising the energy of the minimum which occurs at ¢ = 240° in the cubic warped Mexican hat

potential surface (Fig. 3). It should be noted that the ortho­rhombic strain will cause the two lowest minima to be shifted away from a symmetrical position about ¢ = 180°. This means thatgx andgy are not equivalent at these two minima, after interchange of the x andy axes (Table II), as was as­sumed in the Silver and Getz model.s

The temperature dependence of the g tensor of Zn[Cu] (H20)6(GeF6) is quite different from that of the Cu2+ doped Tutton's salts, with the two lower g values ris­ing, and the highest g value falling, as the temperature is raised above ~40 K [Fig. 4(f)]. The present model relates this to a modest axial strain which is opposite in sign to that in the Tutton's salts, tending to stabilize an elongated, rather than a compressed, tetragonal geometry. This lowers the en­ergy of the well centered at ¢ = 0° in the cubic warped Mexi­can hat potential surface [Fig. 3 (f) ] , and the change in the g values as the temperature is raised is due to the thermal pop­ulation of the two, equivalent higher energy wells.

Comparison with host crystal structures

In the case of the potassium and ammonium Tutton's salts the strain parameters producing optimum agreement with the EPR data of the guest copper(II) complex may be compared with the structure ofthe zinc(II) host compound. The differences between the Zn-O bond lengths, as deduced from x-ray analysis, are listed in Table III from which it may be seen that in each case not only are the short, medium, and long bonds deduced to occur in the same direction in the guest Cu (H20) ~ + ion as in the host complex, but the devia­tions of the copper(II) complex from tetragonal symmetry are quite similar to those of the zinc(II) complex from a regular octahedral geometry. The Zn-O bond length differ­ences imply the lattice strain parameters indicated in Table III, and it can be seen that these compare very favorably with those deduced from the EPR spectra of the copper (II) guest complexes, particularly as far as the ratio SolS. is con­cerned. The host strain parameters are formally subject to rather large uncertainties, if the standard deviations in the Zn-O bond lengths are taken into account (Table III). However, it should be noted that very similar trends in bond lengths, i.e., M-O(z) ~M-O(y) < M-O(x), have been ob­served31 in the crystal structures of the K + and NH/ Tut­ton's salts of Mg2+ and Nf+, so it would seem that the distortions are a true feature of these lattices. The close cor­respondence between the distortions deduced from the anal­ysis of the copper (II) g values with those in the coordination sphere of the zinc host complexes suggests not only that the model used to interpret the EPR parameters is realistic, but also that the study of the spectra of doped systems of this kind has the potential to provide semiquantitative informa­tion on distortions in the host lattice.

The compound Zn(H20)6(GeF6 ) crystallizes in a rhombohedral space group at room temperature, with the zinc(II) ion on a site of trigonal symmetry.32 The lattice strain on a guest copper(II) complex would therefore not remove the equivalence of the x, y, z Cartesian axes and the EPR spectrum should be similar to that observed for Zn(H20)6(SiF6 ). In agreement with this theg tensor above 200 K is reported to be isotropic.4 On cooling to 191 K the

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Riley, Hitchman, and Mohammed: g values of Cu(H20)~+ 3775

compound undergoes a phase change to a monoclinic space group,33 in which no symmetry restrictions apply to the lat­tice strain. The low temperature g tensor has been reported to have axial symmetry, implying that any orthorhomic strain component is insignificant (Ziatdinov et al. have esti­mated an upper limit of20 cm -1 for the in equivalence of the two upper wells, from the uncertainty in the measurement of gl ).4 The temperature dependence of the g values implies an axial strain component some 5-10 times smaller in magni­tude than those in the Tutton's salt hosts (Table III), sug­gesting that the deviation from octahedral symmetry of the Zn-O bond lengths in the low temperature form of Zn(H20)6(GeF6 ) is quite small. This seems reasonable in view of the fact that the similar compound Zn (H20) 6 (SiF 6)

maintains the trigonal symmetry from room temperature to 4.2 K. 8 (C)

Sensitivity of the calculations to the warping parameter f3

The height of the barrier between the minima of the Mexican hat potential surface of a copper(lI) complex of cubic symmetry 213 has been notoriously difficult to quanti­fy. It is therefore of some importance to investigate the sensi­tivity of the present treatment to this parameter. The effect of raising and lowering 13 by 50% on the calculated g values of K 2Zn[Cu] (H20)6(S04)2 is shown in Fig. 4(b). The dominant effect is to change the barrier height between the two lower wells. When 13 is altered, a corresponding change must be made in the tetragonal strain parameter so as to produce a minimum in the potential which gives low tem­perature g values in agreement with experiment. It can be seen that fair agreement with experiment can still be achieved using the higher and lower values off3 [Fig. 4(b) ], which is not surprising as this depends principally on the position and energy separation of the minima. However, these extreme values, 13 = 300 ± 150 cm - 1, would appear to represent the feasible range of the warping parameter in these systems. A lower value would require unreasonable values of the tetragonal strain compared with those estimat­ed from the host crystal structures (less than ~ - 500 cm- 1

, compared with - 1280 cm- 1 calculated from the structure of the potassium zinc salt), and also would not produce the two minima necessary to localize the lowest wave functions. A higher value of 13 would not provide an excited state ~ 150 cm - 1 above the ground vibronic state that is delocalized to some extent over both wells, as is re­quired to explain the temperature dependence of the hyper­fine linewidths in K 2Zn[Cu] (H20)6(S04)2 5 and Zn[Cu] (HzO)6(GeF6).4

It is interesting to consider the potential surface which would result if no warping at all were present (13 = 0), and this is illustrated in Fig. 3(b) for K 2Zn[Cu] (H20)6(S04)2 with the magnitude of the strain unaltered and the two com­ponents chosen to give agreement with the low temperature g values. What results is a surface which has a single, almost harmonic, potential minimum, centered at a value of ¢J cor­responding to a coordination geometry giving rise to the g values observed at low temperature. As may be seen from Fig. 4(b), such a situation produces ag tensor which is al-

most unchanged on raising the temperature to 300 K. In the absence of the warping term, temperature dependent g val­ues would occur if the strain terms were very much smaller, and the potential surface highly anharmonic. This would give rise to a situation in which the geometry and concomit­tant electronic wave functions of the Cu (H20) ~ + complex change gradually as a function of temperature (as far as is possible within the constraints of quantized vibrational ener­gy levels), as has sometimes been suggested34 as an explana­tion of the thermal behavior of copper(lI) EPR parameters. This is in marked contrast to the Silver and Getz model where a dynamic equilibrium exists between "isomers" of the same complex with identical geometries, but different orientations in the crystal lattice. The temperature depen­dent g values corresponding to potential surfaces calculated with 13 = 0 and a range of small values of the lattice strain parameters were investigated, but these invariably showed very poor agreement with experiment. This confirms the fact that the warping of the potential surface has a pronounced effect not only on the properties of copper (II) complexes in a cubic environment, but also on those of lower symmetry, which are not formally subject to the Jahn-Teller theorem.

Comparison of the present model with that proposed by Silver and Getz

The original treatment of the g values of K2Zn[Cu] (H20)6(S04)2 over the temperature range 10-350 K by Silver and Getz5 basically involved the dynamic equilibrium between two different levels of a Cu (H20) ~ +

complex, each having an identical geometry and g values, but different orientations of the x and y molecular axes (us­ing a coordinate system defined with respect to the bonds as in the present work). Excellent agreement with experiment was obtained with an energy difference between the states of 0 12 = 75 cm -1. The present calculations, which also includ­ed additional experimental data over the temperature range 170 to 298 K (Table I), suggest a very similar separation between the two lowest vibronic energy levels (72 cm - 1).

Moreover, to a good approximation, the g tensors of these levels differ largely in the effective interchange of gx and gy (Table II). In the original treatment, on the basis of a slight increase in gz observed at ~ 340 K, Silver and Getz5 inferred that the separation between the lowest and highest energy well (corresponding to an elongation along z in the present coordinate system) was 813 = 450 em - 1. However, in the later study of Petrashen et al.,6 an estimate of ~ 1125 cm- 1

was obtained from a consideration of the structure of the host Zn(H20)~ + complex in this salt. The present calcula­tion suggests an energy of ~ 1000 cm - 1 for the level local­ized in the highest well [Fig. 3 (a)], in good agreement with the latter study. It may be noted that the values of gz mea­sured at ~ 300 K in the present work do not differ signifi­cantly from those at lower temperatures [Fig. 4(a)]. We carried out a thermogravimetric analysis of K 2Zn(H20)6(S04)2 and observed that it commenced to lose water at ~ 330 K, so possibly the results of Silver and Getz obtained above this temperature involved a partially dehydrated sample.

On the basis of the change in linewidths as a function of

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3776 Riley, Hitchman, and Mohammed: g values of Cu(H20)~+

temperature Silver and Getz were able to estimate5 a value of _ 120 cm - 1 for the energy barrier to the interchange between the minima of the two lower wells, It is important to recognize that this does not directly represent the height of the potential maximum between the wells. Rather, it indi­cates the height of the lowest vibronic level which is suffi­ciently delocalized to allow rapid exchange between the minima. In the present calculations, this would presumably be the third level, - 187 cm -I above the ground level [Fig. 3 (a) ]. The agreement between this estimate and the value of -120 cm -I obtained by Silver and Getz would be improved slightly if /3 were somewhat lower than the value of 300 cm -I used in the calculations though this would produce a somewhat poorer fit with the experimental g values (see the preceding section). In fact, ifit is hard to see how a signifi­cantly delocalized level with an energy as low as 120 cm- I

could occur without this causing a deviation from the simple SG model by influencing the temperature dependence of the g values.

The SG model has been applied to the temperature de­pendeneeofthegvaluesoftheCu2 + doped RB+, NH4+, and Cs+ zinc(II) Tutton's salts by Petrashen and co-workers,6 who obtained estimates of the separation between the two lower energy levels 8 1•2 of - 175, 230, and 290 cm - I for these three salts, respectively. The present calculations sug­gest potential surfaces and wave functions which are in mod­erate agreement with these conclusions for the rubidium and ammonium salts [Figs. 3(c) and 3 (d)], for which the ener­gy differences between the lowest vibronic levels are estimat­ed to be - 153 and - 181 cm - I, respectively, with similar g values for the two levels in each case, except for the inter­change of the x and y axes. For the Cs + salt, however, the present model implies a picture which is rather different from the simple SG approach. The first excited vibronic state, at - 235 cm -I, has quite similar g values to the lowest energy state (Table II), and is localized largely in the same well as this [Fig. 3 (e) ]. The next pair of levels are very closely spaced at - 292 and 293 cm - I and, while this energy is similar to the value of 81•2 =290 cm- I obtained by Petra­shen et al. using the SG model, the current calculations indi­cate that both levels are delocalized approximately equally over the wells, each with gx =g y' It should be noted that the delocalization is a consequence of the near degeneracy of levels 3 and 4, so that a quite small difference in choice of parameters in the calculation would produce eigenfunctions largely localized one in each well. However, the fact remains that for a system such as Cs2Zn [Cu] (H20) 6 (SO 4) 2 a result such as that presented here is quite feasible, so that it may be misleading to use the simple SG approach, at least without stressing its limitations.

It seems probable that, if the relative energies and mag­netic properties of the species in equilibrium are left as free parameters, a model such as that proposed by Silver and Getz will be able to explain the temperature dependence of the EPR spectra of most copper (II) complexes. However, this does not mean that such an approach will always be physically meaningful, and an important aspect of the pres­ent study was to investigate under what conditions the SG model is likely to be realistic.

The present calculations suggest that for the Cu2+ doped Tutton's salts the simple SG model works very well for the K + salt, but becomes increasingly unrealistic along the series Rb +, NH/ , to Cs +. The basic reason for this is that the SG model depends upon the vibronic wave functions being strongly localized in separate minima. This depends on the height of the barrier between the wells, which in turn is decided by the relative magnitudes of the warping of the potential surface, as defined by 2/3, and the energy difference between the weB minima, which depends on the size of the orthorhombic strain component. As in the present calcula­tions 2/3 is taken to be - 600 cm - 1 for the Cu (H20) ~ + ion, the deviation from the simple SG approach along the above series is related to the progressive increase in the orthorhom­bic strain parameter from 55 cm -I for the K + salt to 200 cm -I for the Cs + salt (Table III). It is noteworthy that the magnitude of the strain parameters influence the height of the barrier between the wells, with the barrier decreasing as the strain starts to dominate the potential surface (Fig. 3).

In general terms, because of the participation of the ad­ditional vibronic levels besides those which are localized in the two lower wells, and because the two localized levels do not have identical magnetic parameters after allowing for the interchange ofthex andy axes, the SG model will tend to overestimate the separation between the two localized levels. This overestimation is 3, 22, 49 cm -I for the K +, Rb +, and NH/ salts, respectively, compared with the present calcula­tions (Table III). As discussed above, in the present calcula­tions the Cs + salt was found not to have a localized upper level for direct comparison with the SG model. It must be stressed that in systems involving other ligands, /3 might be quite different, so that the SG model might become unrealis­tic over a different range of barrier heights. The factors likely to influence /3 have been discussed elsewhere. 2 1

The EPR spectrum of Zn[Cu] (H20)6(GeF6) in the temperature range 4.2-190 K has been reported by Zait­dinov et al. 4 and interpreted using the SG model, though in this case with two equivalent wells having minima - 154 em - I above the lower minimum. The present calculations essentially confirm this view [Fig. 3 (f) ], with a pair oflevels -147 cm- I above the ground level (Table II). Again, it must be stressed that the reason that the SG model is realistic is because the warping parameter /3 is large. Were this not the case, the potential surface would rise only gently as a function of cp, and the vibronic levels associated with the lowest minimum would occur at low energy and with a large enough amplitude to significantly affect the g values. It is this kind of vibronic coupling mechanism which is thought to give rise to the temperature dependent g values in Cu2+ doped K2ZnF4 15; here, only a single minimum occurs in the potential surface, and /3 = 50 cm -I.

Recently, the SG model has been used to interpret not only the variation of the g values of several pure copper(II) complexes, but also the way in which the x-ray diffraction data of these change as a function of temperature. 12-14 Here, the model should be applied with particular caution, since the behavior of pure compounds is expected to be complicat­ed by the cooperative interactions which may occur between the complexes. This means that whereas in a doped system,

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Riley, Hitchman, and Mohammed: 9 values of Cu(H20)~+ 3777

such as those discussed here, the lattice strain parameters are essentially temperature independent, in a pure copper(II) compound the change in geometry associated with the exci­tation of one complex to a higher vibronic level will alter the the environment of its neighbors. It may therefore no longer be valid to use a fixed set of energy levels to describe the behavior of the system as a function of temperature. The problem is directly analogous to the treatment of spin-equili­bria in solid transition metal complexes, where it has often been found35 that it is impossible to use a simple Boltzmann population distribution over a fixed set of energy levels to describe the behavior of the system. We are currently ex­tending the model described above to include cooperative interactions, and intend to test this on the temperature de­pendence of the EPR parameters of a number of pure cop­per(II) complexes shortly.

GENERAL CONCLUSIONS AND COMPARISON WITH OTHER SYSTEMS

The present study confirms the basic suggestion of Sil­ver and Getz5 that the temperature dependence of the EPR parameters ofK2Zn [Cu] (H20)6(S04)2 is essentially due to an equilibrium between structural forms of the CU(H20)~ + ion which differ only in the directions of the long and inter­mediate Cu-O bonds in the crystal lattice. The lattice strain in this compound has a strong axial component tending to stabilize a compressed tetragonal ligand geometry, with only a small orthorhombic component removing the degeneracy of the two lower energy wells in the potential surface. It is interesting to compare this system with others having a te­tragonalligand strain and temperature dependent g values. As will be discussed in detail elsewhere,16 the species CU(HZO)2Cl~- present in Cu2+ doped into NH4Cl at low pH has a potential surface quite similar to that of the CU(H20)~+ ion in KzZn[Cuj(HzO)6(S04)2 except that the two lower wells are essentially equivalent (except for the effects of random lattice strain), because of the effective axi­al symmetry of the ligand field. At 4.2 K the complex is "locked" into one of the potential minima, and orthorhom­bicgvalues (gx = 2.4I,gy = 2.18,gz = 2.02) are observed. As the temperature is raised the intensity and line shape of the spectrum undergo changes that are characteristic of slow relaxation between the two wells. Above ~ 30 K rapid ex­change occurs between the wells and the gx and gy values become averaged.

As has been shown elsewhere by Reinen and Krause,36 orthorhombic potential energy minima occur when the strain tends to stabilize an axially compressed geometry, and is comparable in magnitude to the natural tendency of a cop­perOI) complex to adopt a tetragonally elongated geome­try, as parametrized by (3. Specifically, orthorhombic mini­ma are expected when Sf) < 9{3, and this condition is satisfied both for the systems studied here and for the Cu (H20) 2CI~­center in Cu2+ doped into NH4Cl. In other systems, the axial compression is sufficiently large (>9{3) that it com­pletely overcomes the tendency to elongate along one bond direction, and a single potential minimum occurs corre­sponding to a tetragonally compressed geometry. This is the

case of the Cu(NH3)2Cl~- center present in Cu2+ doped into NH4Cl at high pH, where the high ligand field strength of NH3 relative to CI- is sufficient to have this effect. 16 The CuF: - complex in K 2Zn [Cu] F 4 is interesting, because here the axial ligand compression is only just sufficient to force a single potential energy minimum (Sf) =- - 540 cm - \ {3 =- 50 cm -1).15 The shallow slope of this well means that low ener­gy "angular" vibrational levels are present and the thermal population of these causes temperature dependent g values. This is in marked contrast to the systems studied here, where the much more dramatic temperature dependence of the g values is due largely to the population oflevels that are local­ized in different wells. An important variable parameter in these systems is clearly the size of the warping parameter (3, which is apparently quite small in K2Zn [Cu] F 4. 15 The pos­sible factors which may be expected to influence{3 have been discussed previously.zl

When a uniaxial strain reinforces the natural tendency for copper (II) to adopt an elongated tetragonal geometry, this simply lowers one of the three minima in the cubic warped Mexican hat potential surface with respect to the other two. Temperature dependent g values would now be expected only in the unusual circumstance that the strain energy is comparable to thermal energies, as is indeed the case for Zn[Cu] (H20)6(GeF6 ), where Sf) =- 100 cm -I. Similar behavior has been reported for the related systems Mg[Cu] (H20)6(SiF6 )7(a) and Zn[Cu] (H20)6(TiF6).7(b) Usually, strain energies will be larger than this, so that higher wells will not be significantly populated at normal temperatures (this would be the case, for instance, if the sign of the strain were reversed in each of the other complexes discussed in this section). This is the reason why copper(II) complexes involving four relatively strong, and two weak ligands tend to have g values which are essentially temperature independent.

ACKNOWLEDGMENTS

The Humboldt Foundation and the Australian Re­search Grants Scheme are thanked for financial assistance, and M. Riley acknowledges the receipt of an Australian Commonwealth Research Scholarship. The help and hospi­tality of Professor D. Reinen, Fachbereich Chemie der Uni­versitat, Marburg, West Germany is gratefully acknowl­edged; Professor Reinen is also thanked for many stimulating discussions.

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3778 Riley, Hitchman, and Mohammed: 9 values of Cu(H20)~+

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