Magnetic inequivalency in the EPR spectra of small metal clusters

J. Phys. Chem. 1986, 90, 1355-1360

1355

Magnetic Inequivalency in the EPR Spectra of Small Metal Clusters Saba M. Mattar* and Geoffrey A. Ozin Lash Miller Chemical Laboratories, University of Toronto, Toronto, Canada M5S 1Al (Received: July 22, 1985)

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The effective spin Hamiltonian parameters and analytical expressions for the resonance field positions of trimetal clusters possessing D3*symmetry are derived. The (super)hyperfinecouplings of the three spatially equivalent metal atoms are shown, in general, to be magnetically inequivalent. This leads to extra resonances which may be well resolved in the EPR and ENDOR spectra of oriented samples or powder samples possessing narrow line widths. The electronic ground- and excited-state wave functions, eigenvalues, charge distributions,and first ionization potentials for Ag3* (q = 0, 1,2) are computed quasi-relativistically by the SCF-Xa-SW method. The spin Hamiltonian tensor components are numerically estimated from the computed Xa wave functions. These can be used to predict the EPR spectra for the Ag?’ cluster.

Introduction During the past decade an increasing interest in the EPR spectroscopy of metal clusters has emerged. As a consequence, clusters of the alkali metals such as lithium, sodium, and potassium trimers and heptamers,’ and transition-metal trimers: pentamers, and higher order clusters have been successfully ~ r e p a r e d . ~ However, very few theoretical computations have been performed to predict the EPR properties of these clusters. This is due to the complexity of the computations coupled with the multicenter nature of these clusters. We have recently calculated the effective spin Hamiltonian parameters for a variety of silver pentamers of trigonal-bipyramidal ~ y m m e t r y . ~In these calculations it was assumed that the silver hyperfine coupling constants of the three equatorial silver centers are the same. This was justified because the highest occupied molecular orbitals were mainly s in character. However, this paper considers the implications of the inequivalency of hyperfine interactions due to an external magnetic field. Here it is shown that the (super)hyperfine interactions from three spatially equivalent centers in the molecule, upon the application of an external magnetic field, are rendered magnetically inequivalent. The magnetic inequivalency is determined by a simple method which relies only on the symmetry of the m~lecules.~This inequivalency is also evident when the (super)hyperfine coupling constants are calculated by means of guage invariant formulas as a function of the molecular orbital coefficient^.^,^ Computational Details The electronic structure of the clusters was computed by the self-consistent field-Xa-scattered wave method described prev i ~ u s l y . ~The . ~ computer program used was a modified version of the program developed at Harvard University8 and provided by D. Case and M. Cook. The a value of 0.701 45, determined by Schwarz: was used for the silver centers, the intersphere and outer-sphere regions. The magnitudes of the sphere radii used in setting up the muffin-tin model were computed according to (1) Garland, D. A.; Lindsay, D. M. J. Chem. Phys. 1983 78, 2813. Lindsay, D. M.; Thompson, G. A. J . Chem. Phys. 1982, 77, 11 14 and references cited therein. (2) Howard, J. A,; Preston, K. F.; Mile, B. J. Am. Chem. SOC.1981,103, 6226. Howard, J. A,; Preston, K. F.; Sutcliffe, R.; Mile, B. J. Phys. Chem. 1983, 87, 536. Howard, J. A.; Sutcliffe, R.; Mile, B. J. Chem. SOC.,Chem. Commun. 1983, 1449. Symons, M. C. R.; Forbes, C. E. Mol. Phys. 1964, 27, 467. (3) Howard, J. A.; Sutcliffe, R.; Tse, J. S.; Mile, B. Phys. Lett. 1983, 94, 561. Baumann, C. A.; Van Zee, R. J.; Bhat, S. V.; Weltner Jr., W. J . Chem. Phys. 1983, 78, 190. Knight Jr., L. B.; Woodward, R. W.; Van Zee, R. J.; Weltner Jr., W. J . Chem. Phys. 1983, 79, 5820. (4) Ozin, G. A.; Mattar, S. M.; McIntosh, D. F. J . Am. Chem. SOC.1984, 106, 7765. (5) Mattar, S . M.; &in, G. A.; submitted for publication in J. Chem. Phys. (6) Keijzers, C. P.; de Vries, H. J. M.; van der Avoird, A. Inorg. Chem. 1972, 11, 1338. (7) Johnson, K.; Smith Jr., F. C. Phys. Rev. E: SolidState 1972,5, 831. ( 8 ) Case, D.; Cook, M., personal communication. (9) Schwarz, K. Phys. Rev. B Solid State 1972, 5, 2466. Schwarz, K. Theor. Chim. Acta 1974, 34, 225.

0022-3654/86/2090-1355$01.50/0

Norman’s criteria.’O Structural details for the trisilver clusters are shown in Figure 1 and tabulated in Table I. The silver-silver distances chosen were those of metallic silver. First, the electronic structure of the neutral trisilver molecule was determined. This was followed by the systematic removal of half an electron and reconverging the resulting cation to self-consistency until the doubly charged cation was finally obtained. The one-electron eigenvalues of the HOMO of the molecules possessing half integral charge yield the first ionization potentials of the parent clusters through the use of the transition-state model.]] The ionization potentials are listed in Table 11. They are somewhat dependent on the sphere sizes chosen and only their relative magnitudes are of qualitative importance. In addition the predicted spin and dipole allowed lowest energy electronic transitions are included in Table

111. The resulting molecular orbital energy diagram for the three trisilver clusters Ag3q+ (q = 0, 1, 2) is shown in Figure 2. As the valence electrons are systematically removed from the clusters their respective eigenvalues drop in energy and there is a general relaxation effect. This effect was also observed for the positively charged silver pentamers studied p r e v i ~ u s l y . In ~ all cases the ordering of the energy levels did not change and, except for a few empty virtual orbitals, the energy gaps between the orbitals remained relatively constant. In our previous computations of the neutral and charged silver pen tamer^,^ a simple reallocation of the intersphere charge proportional to the existing charge within each sphere was performed. In this study the intersphere charge is reallocated to its parental spheres by expanding the radial part of the wave functions beyond the original sphere boundaries. This method, originally introduced by Case and Karplus,’* yields new wave functions that describe the molecular properties with the same order of accuracy as the scattered wave approximations themselves.I3 The program was also modified to yield the ( r W 3values ) for each atomic silver center in the Ag3@ clusters.

Theory, Results, and Discussion In general, the irreducible tensor operators for a molecule possessing a specific point group symmetry may be derived from the O3group and its Wigner rotation matrices by s u b d ~ c t i o n . ~ ~ ’ ~ These multicenter symmetry adapted linear combination (SALC) operators, representing the magnetic field, electronic and nuclear spins for the D3hM3 system are listed in Table IV. Here the notation of Koster et al.I5 for the D3hdouble group is used. In (10) Norman Jr., J. G. J. Chem. Phys. 1974, 61, 4630. Norman Jr., J. G. Mol. Phys. 1976, 31, 1191. (1 1) Slater, J. C.; Johnson, K. H. Phys. Reu. B Solid State 1972, 5, 844. Rosch, N. NATO Adu. Study Inst. Ser., Ser. B 1977, 24, 1. Slater, J. C. “The Calculation of Molecular Orbitals”; Wiley: New York, 1979. (12) Case, D. A. Karplus, M. Chem. Phys. L e f f .1976, 39, 33. (13) Cook, M. Karplus, M. J. Chem. Phys., 1980, 72, 7. (14) Kibler, M. In ‘Recent Advances in Group Theory and Their Application to Spectroscopy”; Donini, J. C., Ed.; Plenum Press: New York, 1979; pp 1-96.

0 1986 American Chemical Society

The Journal of Physical Chemistry,Vol. 90,No. 7, 1986

1356

Mattar and Ozin

TABLE I: Geometrical Muffin-Tin Parameters for Ag,"' position 1 2 3 4

atom

no.

X

OUT

1 47 47 41

0.0000 0.0000 -2.721 3 2.7213

Ag Ag

Ag

Y 0.0000 3.1422 -1.57 1 1 -1.571 1

Z 0.0000 0.0000 0.0000 0.0000

radius 5.8648 2.7226 2.7226 2.7226

eq 0 0 2 2

a 0.70145 0.70145 0.70145 0.70145

"Number of centers = 4; outer sphere at center 1; constant potential = -0.388561

TABLE 11: Predicted First Ionization Energies (eV) of Trisilver Clusters first ionization potential molecule

HOMO

term symbol

spin polzd

spin restrd

Ag3O

(4e')' (3a1'I2 (3al')'

ZE'

5.740 13.536 19.204

5.635 13.622 19.060

Ah2+

]A,' ZA,'

-2.0-

-


TALBE 111 Predicted Spin and Dipole Allowed Excitation Energies transition

energy /cm+

3e' to 3al' 2e' to 3al' le' to 3al' laz" to 3a1'

23 280 24 778 29 184 28318

"Energies obtained by the spin unrestricted transition-state method

of Slater."

n

>a -

W

-10.0-

2

t

- 6.0-

>,

0

P a c

W

-14.0J X

Figure 1. Diagram defining the Cartesian coordinates of the M j cluster, the external magnetic field, B , and the angles 0 and $.

Table IV the symbol 6 (where 6 = S, B, or I> represents either the electronic spin operator, S, or the external magnetic field, B, while I is the nuclear spin operator. In addition F,, is the nth irreducible representation of the SALC operators. As a clarifi1),1) represents the first cation the SALC operator T(2,3,4,1,r5( linear combination of nuclear spin operators from atomic centers 2, 3, and 4 that transforms according to the r5(l)irreducible representation. The notation for the individual atomic tensors is that of Buckmaster et The spin Hamiltonian is constructed by coupling the SALC operators of the same irreducible representation by means of the spin Hamiltonian second-rank tensors (magnetogyric, (super)hyperfine, quadrupole tensors, etc.). It takes the form = ~~~~)(2,3,4,~,r,,l)g(2,3,4,2,3,4,~,~,r~,i,1) x T(')(2,3,4,S,r2, 1)

+

2

C ~')(2,3,4,~,r,(i)j)g(2,3,4,2,3,4,B,S,r,(i)) x lJ=l

2

C ~')(2,3,4,s,r5(i),k)l +

k= I

C,,,( 7 W , 3 , 4 , 1 , r12)a(2,3,4,2,3,4,z,S,r,, , 1,1) x V(2,3,4,s,r2,i) +

(1 5 ) Koster, G. F.; Dimmok, J. 0.;Wheeler, R. G.; Statz, H. "Properties of the Thirty-Two Point Groups"; M.I.T. Press: Cambridge, MA, 1963. (16) Buckmaster, H. A,: Chatterjee, R.: Shing, Y . H. Phys. Stat. Solidi A 1972, 1 3 . 9.

As;

\ \

\r

.-I'

I

Figure 2. Molecular orbital energy level diagram (expressed in eV units) for Agln+,where n = 0, 1 , and 2.

The notation describing the second-rank spin Hamiltonian tensors is best explained by an example. The term g(2,3,4,2,3,4,B,S,r5(i)) is the magnetogyric tensor coupling the magnetic field and the electronic spin SALC operators that transform as the ith degenerate I', irreducible representation involving the 2, 3, and 4 atomic centers. The Magnetogyric Tensor Componentsfor M3 Clusters. From the Hamiltonian in eq 1 and the explicit use of the SALC orbitals in Table IV it is found that the Cartesian form of the g tensor has no off-diagonal terms (gi,= 0 if i # j). The diagonal terms are given by g,, = g,

3g(2,3,4,2,3,4,~,~,r,(l))

(2)

= 3g(2,3,4,2,3,4,~,~,r,(2)) g,, = 3g(2,3,4,2,3,4,~,~,r~)

Since the g,, and gYv components are due to the coupling from the doubly degenerate I', representation, then they are equal and the g tensor is axial. This is in accordance with the requirement symmetry of the M, molecuie because the g tensor of axial D3,, is a molecular property representing the entire molecule. The g,, is the unique diagonal tensor component indicating that the unique g principal axis coincides with the C, molecular symmetry axis, again as necessitated by the molecular symmetry of M3 (Figure 1).

EPR Spectra of Small Metal Clusters

The Journal of Physical Chemistry, Vol. 90, No. 7, 1986 1357

TABLE IV: Symmetry Adapted Linear Combinations of Pseudo-First-Order Operators Representing the Magnetic Field and Electronic and Nuclear SDins


TABLE V Relevant Spatial Symmetry Adapted Linear Combination of Atomic Orbitals

It should be noted at this stage that this method is general and is independent of the charge and ground state of the M3 cluster. Thus the form of all the spin Hamiltonian tensors will depend only on the geometry of the molecule. We have calculated the configurations of the trisilver clusters of D3hsymmetry Ag30, Ag3I+, and Ag32+. From Figure 2 it is seen that the Ag30 is a 2E’ ground-state molecule. Since it is not linear then, as a result of the Jahn-Teller effect, it will either dynamically or statically distort to the lower C,, geometry. Thus the theoretical derivation for the g tensor presented here does not apply to the Ag? cluster. In addition the Ag,’+ has a lAl’ ground state, is diamagnetic, and shall not be considered further. Ag32+is paramagnetic and has a 2Al‘ground state with one unpaired electron in the 3al’ orbital. The guage invariant formula for the g tensor4,” is ($Olt(k)li(k)ln)( nllj(k ?I+o) go = ge&j- 2 (3) n#Ok,k‘ E ( n ) - E(+,) where I+o)and In) are many-electron Slater determinants. In conjunction with the spatial symmetry adapted linear combination of atomic orbitals in Table V, it may be used to express the tensor components in terms of their molecular orbital coefficients. The notation used to describe the spatially adapted molecular orbitals of Table V has been fully described p r e v i ~ u s l y .It~ is seen that only the n a i and the ne” are coupled to the 3al’ orbital via the orbital angular momentum operator. This leads to the g tensor components: if i # j gi, = 0

cc

Here A( ne”) = XiN( 3a ’)N(ne”)(-cz( 3a 1’) c1(ne”) 3 /2c3(3a 1’) X [c2(ne”) - c3(ne”)] + c4(3a,’)[c2(ne”) + c3(ne”)]} (5)

+

and B(na2’) = XiN( 3al’)N(nai) [c2(3al’)c, (na2’)

+ 2c4(3a,’)c,(na,’)] ~

(17) Keijzers,

~~

C.P.;De Boer, E.J . Chem. Phys. 1972, 57, 1277.

(6)

In accordance with eq 2 the g tensor is axially symmetric and its principal axes are coincident with the molecular symmetry axes. However, eq 4-6 provide a more detailed picture of the relation between the electronic spin distribution and the g tensor anisotropies. A brief description of these relationships is presented. From a qualitative point of view, the three 5s orbitals of the silver atoms form three molecular orbitals that are mainly 5s in character. These are the 3al’ and the doubly degenerate 4e’ orbitals. From Table V it is seen that the rial' represents the in-plane s, p, and d bonding combinations of the atomic orbitals and the out-of-plane bonding contributions from the d,Z orbitals. In contrast the ne’ orbitals represent the corresponding antibonding combinations of the rial' orbital. The calculated molecular orbital coefficients of the 3al’ are cI(3a,’) = 0.9401, c2(3al’) = 0.280, c3(3a1’)= -0.122, and c4(3al’) = -0.1 13. These coefficients imply that the 3a1’ is mainly 5s in character, has bonding contributions from its s and p components, and has an antibonding contribution from its d atomic orbitals. On the other hand the 4e’ orbital has its first five molecular orbital coefficients positive while c6 (4e’) is negative. This indicates that, except for the bonding from the in-plane d atomic orbitals, this is an antibonding orbital which is significantly destabilized. The 3al’-4e’ HOMO-LUMO gap is computed to be approximately 24 670 cm-’. This suggests that the bonding of the Ag3*+cluster is mainly due to the one electron in the 3al’ orbital with minimal contribution from those electrons in orbitals that are mainly 4d in character. The 5p character in the 3al’ orbital is 8.9% compared to 13.3% in the 4e’ orbital. The latter has more 5p character because it is closer in energy to the 5p orbital of the original free silver atom. The g,, anisotropy is due to the coupling of the 3al’ and the la; orbital. The la; represents the in-plane antibonding combinations from the atomic px, pv, d,, and d,z+ orbitals. Its molecular orbital coefficients are cl(la;) = 0.0759 and c2(1a2’) = 0.997 indicating that it is almost totally 4d in character. Not surprisingly it has the highest energy of the molecular orbitals that are 4d in character and are grouped in brackets in Figure 2. Because the 3al’ orbital has only 0.9% d character its coupling with the d components of the la; results in a small contribution to the g,, anisotropy. By use of eq 3 and 6, the computed values of E(3al’)-E( la;), and the molecular orbital coefficients the value of the g,, tensor component is predicted to be 2.0094. The anisotropy of the g,, and gUucomponents arises from the coupling of the 3a1’ orbital with the le” and 2e” orbitals. The ne”(1) and ne”(2) represent the out-of-plane p and d orbitals

1358 The Journal of Physical Chemistry, Vol. 90, No. 7, 1986

bonding along the y and x axes, respectively. These orbitals are mainly 4d in character with minor contributions from p atomic orbitals. Their molecular orbital coefficients are listed below. cl(le”) = 0.0132 cl(2e”) = -0.026 (7)

where i = 2, 3, and 4 a,,(2)

= ~(2,3,4,2,3,4,1,~,r~(i),l)

~ ~ 4 =2 ~(2,3,4,2,3,4,z,~,r~(2),2) )

+ 0.75aJ2) = ayY(4)= 0.75aX,(2) + 0.25aYy(2)

c2(le”) = -0.839

cZ(2e”) = 0.542

(8)

a,,(3) = a,,(4) = 0.25aX,(2)

c3(le”) = 0.543

c3(2e”) = 0.840

(9)

aJ3)

The energy differences between the le”, 2e”, and the 3al’ orbitals, calculated by the transition-state method,” are 26 191 and 23 344 cm-I, respectively. By using this information in conjunction with eq 3 and 5 the perpendicular components are computed to be g,, = gYJ= 2.0132 (10) Since the le’’ and 2e” orbitals lie below the 3al’, then they both contribute to the increase of the perpendicular g tensor components compared to the free electron value. The molecular ”isotropic” g value is computed as + g A / 3 = 2.01 190 ( 1 1) g,,, = (gxx +


Mattar and Ozin

Forbes and Symons have trapped Ag32+by subjecting silver salts in aqueous toluene glasses to y-irradiation. Although they could not experimentally determine the g tensor components they do report an average or “isotropic” g value.2 However, both theory and experiment predict opposite deviations from the value of the free electron. We believe the positive deviation of the computed g tensor from the free electron value for Ag32+is more realistic than the negative deviation assigned experimentally. In the equilateral triangular Ag3’ there are three electrons occupying three molecular orbitals that are mainly 5 s in character. In this case there is appreciable contribution to the g tensor from the molecular orbitals that are 5p in character. For example, the energy difference between the singly occupied 4e’ molecular orbital and the lowest lying 2az” orbital is orbital is only about 10000 cm-l. The close proximity of the molecular orbitals with 5p character causes the g tensor components to be slightly less than the value of the free electron.2 The contribution of the 5p orbitals to the g tensor of Ag30 will be even more pronounced if the molecule is subjected to a Jahn-Teller distortion and possesses C, symmetry. In this case, all the virtual molecular orbitals with empty 5p character will contribute to the reduction of the value of the g tensor components. However, for the Ag3z+cation tl. : singly occupied molecular orbital (SOMO) is the 3a,’, as shown in Figure 2. In this case, the SOMO and the 2 a F orbital are now approximately 33 000 cm-’ apart. In addition, the coupling of the 3al’ to 2a2” via orbital angular momenta is symmetry forbidden. This greatly decreases the contribution from the orbitals with 5p character. At the same time the 3al’ S O M O lies only about 15 000 cm-I higher than the molecular orbitals that are almost entirely 4d in character. This causes the d orbitals of the AS+ : to have a greater positive contribution to the g tensor relative to the case of the Ag30 cluster. Thus it is quite conceivable that the g tensor for Ag3,+ should be greater than the g value of the free electron. In the estimation of the g tensor components the spin restricted form of the SCF-Xa-SW method was used. We are presently investigating a method that will yield more refined results starting from the spin polarized wave functions. It is hoped that this method will be used in future investigations similar to the one presented here. Finally, we note that in spite of the different SALC orbitals chosen for Ag3@ and Ag5@ the contribution of the equatorial atoms in the Ag54f clusters is exactly the same as that of Ag32+ m o l e c ~ l e .This ~ proves that the g,, expression (eq 3) used in both cases is indeed guage invariant. The (Super)hyperfine Tensors for an M3 Cluster. The expansion of the (super)hyperfine terms of the spin Hamiltonian (eq 1) followed by transforming the coupling (super)hyperfine tensors into their Cartesian form yields azz(i)= a(2,3,4,2,3,4,Z,S,r2) aJi) = a,(i) = u2,(i) = ax2(i)= 0

(12) (13)

= aJ3)

aJ3)

= -a,(4)

= -a Y* (4)

= 0.443[~,(2) - ~,,(2)]

(15)

The above equations indicate that the (super)hyperfine tensor for M(2) center is diagonal. However, in contrast to the diagonal g tensor its diagonal elements are not equal. The a tensor components for M(3) and M(4) possess off-diagonal xy and y x elements. This means that the three (super)hyperfine tensors do not have aligned principal axes. As a result they are magnetically inequivalent in spite of their spatial equivalency. In general the inequality of the diagonal elements a,(i)

#

uJi) Z a,,(i)

( i = 2, 3, 4)

is a result of the low symmetry at the apices of the triangular molecule. Although the axyand ayxof M(3) and M(4) differ from one another only by a negative sign, this is enough to cause their superhyperfine and resonance field positions to be different. A similar case was also encountered in the analysis of the 170 resonance field positions of metal superoxides possessing C, symmetry.s The effects of magnetic inequivalency are not new and have been noticed by a number of workers previously.ls It has been shown that magnetic inequivalency can affect the shape of the resulting EPR The molecular orbital energy diagram (Figure 2) shows that the 3a,’ H O M O is quite well separated from the la2’ and lower orbitals. Hence the possibility of low-lying excited quartet states is very small. It is therefore safe to assume that the Zeeman interaction is the largest spin interaction and at least 1 order of magnitude larger than the (super)hyperfine terms. In the following analysis of the resonance field positions it is considered as the unperturbed part of the spin Hamiltonian while the (super)hyperfine interactions are assumed to be adequately handled by perturbation theory. A detailed analysis of the resonance field positions for a central Cu atom in a D4hcomplex and for metal . ’ ~ folsuperoxides of C, symmetry has been ~ n d e r t a k e n . ~The lowing analysis is similar to the two previous cases. In essence, the Hamiltonian (eq 1) is expanded explicitly and a new basis set is chosen that renders the Zeeman Hamiltonian diagonal. Once this has been done, similar transformations of the nuclear spins of the three metal centers are independently carried out. This leads to the following g and (super)hyperfine tensor expressions. gz = gZz2 cos2 8 + [gXXz cos2 4

+ gyy2sinz 41 sinz 8

(1 6)

+

g2A2(2) = arr2(2)gzzZ cos2 8 [aX,2(2)g,,2 cos2 4

+ a,2(2)gYY2sinZ41 sin2 8 (1 7 ) gZA2(3)= azr2(3)gzZ2 cos2 8 + [(ax,(3)g,, cos 4 + axY(3)gYY sin $)2 + (aY,(3)g,, cos 4 + uyYgypsin $)2] sinZ6 (18)

+

g2A2(4) = ~ , , ~ ( 4 )cos2 g ~8~ [(aX,(4)g,, cos 4 + axy(4)gyy sin 4IZ+ (ay,(4)g,x cos 4 + ayygyy sin 4123 sin2 8 (19) It is necessary to include the superhyperfine interactions only as first-order perturbants. This Hamiltonian then takes the form

H,,,, = m 4 ( 2 ) 1 2 z + A(3)13z+ 4 4 1 ~ 1

(20)

(18) Kasai, P. H.; Hedaya, E.; Whipple, E. J . Am. Chem. SOC.1969, 91, 4364. Abkowitz, M.; Chen, I.; Sharp, J. H. J . Chem. Phys. 1968,48,4561. Kasai, P. H. J . A m . Chem. SOC.1972, 94, 5950. (19) Mattar, S. M. Ph.D. Thesis, McGill University, Montreal, Canada, 1982.

The Journal of Physical Chemistry, Vol. 90, No. 7, 1986 1359

EPR Spectra of Small Metal Clusters When it is added to the Zeeman term and the composite energy matrix elements are determined the resulting energy difference gives the shifts in the resonance field positions as &UHF

= -[m2A(2) + m3A(3)

\

+ mJ(4)l/gP

(21)

Here m2, m,, and m4 are the nuclear magnetic quantum numbers for the centers. From eq 8-19 and 21 it is seen that the (super)hyperfine splitting from all three metal centers, for a given orientation, will not be the same and thus are magnetically inequivalent. Such inequivalency is expected to manifest itself in single-crystal samples of definite orientation with respect to the external magnetic field. The three (super)hyperfine components will only be equivalent when the external magnetic field, B, is coincident with the molecular z axis. The shifts in the resonance field positions would then be


&.UHF

= - d i ) [m2 + m3

+ mil /&#

(22)

where i is either 1, 2, or 3. The shifts in the resonance field positions along the x and y axes will not be equal to one another or to that of eq 22. These expressions are obtained by substituting the appropriate values of 0 and 4 in eq 16-19 and 21. The specific values for these angles are 0 = 90.0, 4 = 0 when B is parallel to the gxxprincipal axis, and @ = 90.0, 4 = 90 when B is parallel to the gyyprincipal axis. Due to the different contributions from the three Ag centers to the resonance field positions the EPR and ENDOR single crystal and powder spectra of D3hsystems will display “extra” resonance lines in the x and y regions of the spectra. The (Super)hyperfne Tensor in Terms of MO Coefficients. By using the guage invariant anisotropic (super)hyperfine expression

together with the SALC orbitals in Table V it is possible to prove that the (super)hyperfine tensor components have the same form as those derived previously (eq 8-1 5). Since the isotropic part of the (super)hyperfine tensors adds equal weight to the diagonal elements it does not affect the final orientation of the principal tensor axes. However, the magnetic inequivalency should manifest itself in the anisotropic components at the various levels of approximation. In fact it can be seen that only the first-order dipolar contributions is sufficient to exhibit magnetic inequivalency in M, clusters of D3* symmetry. This is illustrated by determining analytically the guage invariant expressions for the (super)hyperfine tensors of the Ag32+cluster. Since the H O M O for Ag?’ is the 3al’ the first-order anisotropic dipolar contribution is

where the symbols have been defined p r e v i ~ u s l y . ~With ~ ’ ~ the use of eq 24 and the SALC orbitals of Table V we get

‘/s ( r-3) ,cz2(3a 1’) + y7( f 3

) d [c32(3a ’’)

+ c42(3a,’) ]

[-cl(3al’)] [c3(3al’) - 31/2c4(3al’)]-

1

(27)

W3),

(r-,)d - ~ ~ ~ ( 3 a ~ ’ )- - [ ~ ~ ( 3 a ~ ’ ) 5 7

-

~~~(3a~’)l

2(r-3), +cz2(3al’)5

[c3(3al’) + 31/2c4(3al’)]

(r-3)d - [ ~ ~ ~ ( 3 a ~ ’ ) - ~ , ~ ( 3 a ~- ’2(31/2)c3(3al’)c4(3aI’)] ) 7 where (r-3)sdis zero for the same center. chyp

= %fl(3al‘)ggN@/?N

(30)

and i = 2, 3, and 4. azis0(3)= a F ( 4 ) = ‘/,[ags0(2)

+ 3u$O(2)]

(31)

uaniso(3) YY = u;p(4) = ‘/,[3aF(2)

+ a$”(2)]

(32)

aaniso(3) = a;isa(3) = -a$~0(4) = +?(4) XY

3112 = 3 4 3 2 ) - a;;’“(2)]

(33)

The expressions in eq 8-15 and 25-33 are identical in form and indicate that the symmetry technique and the guage invariant expressions lead to the same results. They may be individually used to detect the effects of magnetic inequivalency in multicentered systems. Equations 25-33 show that the off-diagonal components for the centers 3 and 4 in Ag32+may arise from the dipolar coupling of the bonding s components and the out-of-plane d,2 components of the 3al’ orbital. There are also contributions from the components that are proportional to ~ * ~ ( 3 a ~ ,’~) (, 3 a ’ ’ ) and c2(3al’), i.e., to the in-plane 5p, in-plane 4d, and out-of-plane 4d; character. However, since from the computed molecular orbital coefficients it is known that the p and d character in the 3al’ is small the (super)hyperfine anisotropies are also expected to be small. There are additional contributions to the anisotropic components from the coupling of the 3al’ orbital with the low-lying doublet states via orbital angular momentum and dipolar interactions. Such contributions are expected to be large only if the energy differences between E(3al’)-E(ne”) and E(3al’)-E(na2/) of eq 23 are small. Clearly from Figure 2 this is not the case. Consequently, these contributions will not be included in the numerical prediction of the magnitude of the overall (super)hyperfine tensors. The isotropic Fermi contact term is the same for all three centers; it is proportional to cI2(3a,’). Since the 3al’ is 58.5% s in character it is the dominant term of the (super)hyperfine splittings. From the spin polarized SCF-Xa-SW computation for the Ag32+one obtains the net spin density on Ag(2), Ag(3), and Ag(4). By using the hyperfine splitting for Ago reported by Forbes et aL2 as 68.6 mT and the computed net spin density, the isotropic splittings are found to be uis0(2) = aiS0(3)= aiS0(4)= 20.1 mT This is in excellent agreement with the values of 20.2 mT for Ag,” experimentally determined by Forbes et aL2 Since the values of the 3a,’ molecular orbital coefficients have been computed they may be substituted in eq 27-33 to give an estimate of the anisotropic (super)hyperfine tensors. Such a computation yields a i Y ( 2 ) = -0.804 mT, a y ( 2 ) = -0.274 mT, and 4?(2) = 1.077 mT, which yields the matrix elements a e ( 3 ) = a e ( 4 ) = 0.986 mT a;,”’””(3) = u$”(4)

= 0.256 mT

1360

J . Phys. Chem. 1986, 90, 1360-1365

aaniso(3) = aaniso(3) = -aaniso(4) = -aaniso l'x ( 4 ) = 0 . 2 9 3 mT XY

XY

YX

and the total superhyperfine matrix elements become


a(2)=

:

0 (,,.826

21.177

0 0

Conclusions

19.296

a(3)=

(t0.839 0.293

:.293 20.164

a(4)=

(,,,j9 -0.293

01)293 0 20.164 0 19.296

Consequently, if isotopically pure Ag species are used and the experiment is designed to minimize the spectral line widths, the extra lines due to magnetic inequivalency may ultimately be detected.

0 0 19.296

)

Although the anisotropic contribution is small compared to the isotropic one it is of the order of 1.0 mT. Such splittings are easily detectable in EPR spectra of narrow line widths and certainly detectable in ENDOR experiments. Inspection of the reported EPR spectra of Forbes et aL2 shows that the experimental line widths are large and at best of the order of magnitude of 1.5 mT. Thus the extra splittings due to magnetic inequivalency may go undetected. In addition to magnetic inequivalency, other possible reasons for these large experimental line widths in the spectra of Agj2+are the contribution from overlapping spectra from the silver isotopomers @ 'A'g3, '@'Ag2107Ag, l@'Ag lo' Ag,, and lo7Ag3and the coexistence of significant amounts of paramagnetic solvent cations. These cations may cause line broadening via spinspin interactions and cross-relaxation mechanisms.

A tensorial symmetry technique independent of the ground electronic state of a molecule has been used to derive the spin Hamiltonian of M, clusters having D3,,symmetry. The analytical expressions for these tensor components have also been derived using guage invariant molecular orbital theory. Both methods show that magnetic inequivalency, arising from the application of an external magnetic field, expresses itself naturally in the forms of these spin Hamiltonian tensor components. The results of this study draw attention to the fact that unless all resonances in an EPR spectrum are resolved and their line shapes simulated to an acceptable degree of accuracy, taking into consideration the effects of magnetic inequivalency, one is not able to unequivocally assign a specific spatial symmetry to a paramagnetic molecule in experiments where an external magnetic field is applied.

Acknowledgment. We acknowledge the generous financial assistance of the Natural Sciences and Engineering Research Council of Canada and the Connaught Foundation of the University of Toronto. Registry No. Ag3, 12595-26-5; Ag3+, 12595-27-6; Ag3'+, 52221-34-6.

Kinetics of CO Oxidation over Ru(0001) Charles H. F. Peden and D. Wayne Goodman* Surface Science Division, Sandia National Laboratories, Albuquerque, New Mexico 871 85 (Received: August 1, 1985; In Final Form: November 7 , 1985)

The oxidation of CO over a model Ru(0001) single-crystalcatalyst has been studied in a high-pressure reaction-low-pressure surface analysis apparatus. Steady-state catalytic activity as a function of temperature and partial pressure of CO and O2 was measured. Both the specific rates and the relative activity (Ru > Pd, Rh) obtained in this study compare very favorably with the results on high area supported catalysts. Surface concentrations of oxygen were monitored following reaction and found to be dependent on the partial pressures of the reactants. Further, the highest rates of reaction corresponded to reaction on a ruthenium surface covered with a monolayer of oxygen as detected subsequent to reaction by AES. The kinetics measured at various reactant partial pressures (leading to various surface oxygen coverages following reaction) suggest that a chemisorbed atomic oxygen species present at high oxygen coverages may be a crucial reaction intermediate, largely responsible for the optimum reaction rates. Reaction rates under less than optimum conditions (0, < 1) may be limited by different processes and/or involve a different reaction mechanism: notably, the reaction of adsorbed CO with a less active form of chemisorbed oxygen. These results possibly explain the altered relative activity (Pd, Rh > Ru) observed in ultrahigh vacuum measurements on clean surfaces.

Introduction

The heterogeneous catalytic oxidation of carbon monoxide has received a great amount of attention owing to the practical and ideal nature of this system. The removal of C O as COz from automotive exhaust is accomplished in a catalytic converter with the supported noble metals Pt, Pd, and Rh. In this regard, numerous studies of the kinetics as well as investigations into the effects of supports and additives on the kinetics of this reaction over supported catalysts have been reported.'-* Additionally, the (1) Cant, N. W.; Hicks, P. C.; Lemon, B. S. J . Catal. 1978, 54. 372 (2) Kiss, J. T.; Gonzalez, R. D. J . Phys. Chem. 1984, 88, 892. (3) Kiss, J. T.; Gonzalez, R. D. J . Phys. Chem. 1984, 88, 898. (4) Oh, S. H.; Carpenter, J. E. J . Catal. 1983, 80, 472. ( 5 ) Yao, Y.-F. Y. J . Cutal. 1984, 87. 152

0022-3654/86/2090- 1360$01.50/0

relative simplicity of this reaction on a metal surface make it an ideal model system of a heterogeneous catalytic process, a process involving molecular and dissociative (atomic) adsorption, surface reaction, and desorption of products. Thus, the full battery of surface science techniques have had an impact on our understanding of the elementary molecular processes occurring in this reaction.s22 Further, the lack of any reported structure sensitivity ( 6 ) Okamoto, H.; Kawamura, G.; Kudo, T. J . Catal. 1984, 87, 1. (7) Su, E. C.; Watkins, W. L. H.; Gandhi, H. S. Appl. Catal. 1984, 12, 59.

(8) Bennett, C. 0. Catalysis under Transient Conditions, Bell, A. T., Hegedus, L. L., Eds.; American Chemical Society: Washington, DC, 1982; ACS Symp. Ser. No. 178, p 1. (9) Engel, T.; Ertl, G. Adc. Cutul. 1979, 28, 1

0 1 9 8 6 American Chemical Society

Magnetic inequivalency in the EPR spectra of small metal clusters

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