J . Phys. Chem. 1994,98, 5627-5631
5627
Calculated Paramagnetic Hyperfine Structure of the Cz, Isomers of Ag3 J. Pablo Bravo-Vhsquez and Ramiro Arratia-PCrez'J Facultad de Quimica. Pontificia Universidad Catblica de Chile, Casilla 306, Santiago 22, Chile Received: January 26, 1994"
Dirac spin-restricted molecular orbital ( D S W - X a ) calculations on the two C b isomers of the matrix-isolated Ag3 clusters using the Hedin-Lundqvist-VoskeMacDonald relativistic local density exchange-correlation potential a r e reported. T h e resulting ground-state cluster dirac wave functions a r e used through a relativistic first-order perturbation algorithm to model the paramagnetic hyperfine interactions of these bent isomers. The calculated isotropic and anistropic spin populations, and paramagnetic hyperfine interactions of the acute and obtuse isomers are in reasonable agreement with empirical and resolved paramagnetic resonance data of the Ag3 clusters isolated on t h e C6D6 and N2 solid matrices, respectively.
I. Introduction The study of small metallic clusters using electron paramagnetic resonance (EPR) spectroscopy has been useful in elucidating cluster electronic structure and possible cluster geometries, electronic states, and spin distributions.'-5 Furthermore, highly resolved EPR spectra of metallic cluster radicals give detailed information about the isotropic and anistropic character of the magnetic tensors.6~7 It has been customary to assume that in most light radicals the observed hyperfine tensors could be decomposed into an isotropic term and traceless spin-dipolar contributions.I4 However, the EPR spectral interpretation of cluster radicals containing heavy atoms is often complicated by orbital contributions to the hyperfine (&) and Zeeman (gii) tensors arising from spin-orbit mixings with various excited states, thus inducing cluster anisotropic magnetic effe~ts.6~~~9-Il About a decade ago, naked Ag3 clusters were produced by cocondensation of 107Ag atoms and C6D6 on the cold surface of a rotating cryostat.3 Also, Ag3 molecules were generated by codepositing atomic silver with excess nitrogen at temperatures close to 4.2 K.5 Their recorded EPR spectra (at 77 and 4.2 K, respectively) suggested the existence of two Cbisomers,j5namely, ~ , ~ is in an obtuse Ag3 cluster entrapped in a C6D6m a t r i ~ ,which contrast with the acute Ag3 cluster isolated in an N2 m a t r i ~ . ~ Based on the EPR analysis, the two isomers were predicted to have different electronic states (2Al and 2Bz for the acute and obtuse isomers, respectively), radically different spin populations, and, hence, quite different resolved cluster g and hyperfine interactions (hfi).3-5 From a theoretical point of view, the electronic structure and the molecular stabilities of the Ag3 clusters have been the subject of several nonrelativistic ( N R ) Schrodinger or quasirelativistic Pauli-type theoretical studies,l"l among others, the N R semiempirical diatomics in molecules (DIM),I4 relativistic (Pauli) effective core potential (RECP),15 N R local density pseudopotentia1,lb complete active space multiconfigurational S C F (CASSCF) followed by multireference configuration interaction (MRSDCI) calculations (including about 400 000 configurations), and modeling spin-orbit coupling via two-component (Pauli) relativistic C I interaction.17vl8 Also, NR-single reference SCF plus singles and doubles configuration interactions (SDCI)'9v20 and a NR-multireference configuration interaction (MRCI) calculations on the stability of Ag3have been reported.21 All these calculations predicted the Cb structure to be of the lowest energy. The complete optimization of the Ag-Ag bond length for all angles B on selected regions of the potential energy E-mail:
[email protected]. Fax: 562-552-5692. *Abstract published in Advance ACS Abstracts, May 1, 1994.
surface of Ag3 predicted the 2B2 electronic obtuse ground state as the bent minima, which is nearly degenerate with the lowlying 2Al acute excited state.17-20 In particular, the geometry optimization via the MRSDCI method17J8 predicted an apex angle B = 54.2O for the acute (2Al) isomer, while an apex angle of 0 = 63.7O was predicted for the obtuse (zB2) isomer. The SDCI calculations predicted the angles 0 = 5 5 . 3 O and 69.1' for the acute and obtuse isomers, r e s p e ~ t i v e l y . ' ~However, ~~~ the semiempirical NR-DIM calculations predicted an angle B = 76.6' for the obtuse isomer.14 In the present study, we examine the role of spin-orbit coupling on the electronic structure and paramagnetic properties of these Ag3 isomers. We report here relativistic spin populations and hyperfine tensors calculations by using four-component Dirac molecular wave functions and relativistic first-order perturbation theory previously developed by U S . ~ - ~ O This method has been successful in recognizing spin-orbit effects on the electronic structure and optical, magnetic, and photoelectron properties of heavy metallic cluster^.^-^^
11. Method of Calculation In the self-consistent Dirac scattered wave (SCF-DSW-Xa) formalism the four-component (vectorial) molecular wave function is approximated as a Slater determinant by an effective Coulomb and exchange-correlation p ~ t e n t i a l . ~ - 'For ~ the exchangecorrelation potential we used the Hedin-Lundqvist (HL) local density potential22 modified according to MacDonald and V o ~ k o ~ ~ to include relativistic effects. This modified relativistic local density potential adds Coulomb correlations, not included in the original local Slater's X a potential, and it exhibits important differences from the nonrelativistic counterparts in the high electron density regions of thevalence orbitals of heavy atom^.^^-*^ Thus, spin-orbit interaction plus some correlations effects as well as the scalar nonquantum electrodynamic (NQED) relativistic corrections are then implicitly included at the S C F stage.8-13.2s,26 The calculation of the molecular hyperfine interactions uses the Dirac molecular wave function as the starting point. Since these wave functions already include spin-orbit effects (and transform according to the molecular double point group), only the nuclear hyperfine interaction need be included in the firstorder perturbation scheme, so that the effects of magnetic fields I~ are described by the perturbation operator H I , ~ - where
In eq 1 a is the vector of 4 X 4 Dirac matrices and A is the electromagnetic vector potential. The a matrices are off-diagonal,
0022-365419412098-5627%04.50/0 0 1994 American Chemical Society
Bravo-Vgsquez and Arratia-Ptrez
5628 The Journal of Physical Chemistry, Vol. 98, No. 22, 1994
TABLE 1: Total Valence Poopulations of Ag3 acute obtuse DSWDSWatom t? j Xa MRSDCIb X a MRSDCIb -1 1
-2
112 112 312
total p 2 -3
312 512
total d total Ag(1) -1 1 -2
112 112 312
total p 2 -3
312 512
1.006 0.082 0.122 0.204 3.977 5.941 9.918 11.128 0.839 0.117 0.173 0.290 3.912 5.895 9.807 10.936
0.971 0.167 9.950 11.088 0.804 0.193
0.765 0.126 0.195 0.252 3.894 5.901 9.795 10.812 0.985 0.094 0.147 0.241 3.937 5.931 9.868 11.094
=%
A 4
I I II I / I111111I I
I
0.602 0.227 9.979 10.808 0.919 0.219
total d 9.959 9.959 total Ag(2) 10.956 11.097 The K quantum number defines both I and j through K = -a6 + l / 2 ) with a = 1 for j = I + 112 and a = -1 for j = I - 1 1 2 . b Mulliken gross population analysis; see ref 17.
Figure 1. Calculated valence (total and partial) local density of states (using u = 0.4 eV) of the obtuse Ag3 (2B2) cluster. The a and b label the Ag( 1) and Ag(2) projected spinor contributions, respectively. The upper section indicates the cluster valence energy levels that belongs to the ys irreducible representation of the group Cb.
and the evaluation of the matrix elements involves products of the “large” and “small” components of the Dirac four-component wave function.sb For the hyperfine term,
A = (k X r)/r3
I
T - T O W Mdecule
(2)
a1
i
1112 Aglll
.4
i
43/2 kg(1)
a5
i
4 9 2 Ag(1)
b l = .1/2 Ag(2)
wherep is the nuclear magneticmoment ( p = -0.1 135 for Io7Ag).l Matrix elements of these operators are evaluated in the basis spanning the “two” rows of the double valued irreducible representations of the molecular orbital holding the unpaired electron. The details of the evaluation of the angular and the radial integrals have been described earlier.8J7 The resulting perturbation energies are then fitted to the usual spin Hamiltonian for hyperfine interactions,
H~~~~ =
CS.A,.I,
b4
b5
.0
.7
.0
.5
E.
(3)
n
where a value of S = ‘ / 2 is used to describe the Kramers doublet, In is the nuclear spin (I = l / 2 for lo7Ag), A, is the hyperfine coupling constant, and n = Ag( 1) or Ag(2) denotes the apical or basal silver atoms, respectively. Thus, the matrix elements of H1 may be directly identified with those of Hspin.8-1L,27 The Ag-Ag bond distances d(Ag-Ag) and the apex bond angle 0 for both isomers were adopted from the MRSDCI calculations, since these included two-component spin-orbit correlated functions which are important for clusters containing heavy atoms.I7J8 The selected optimized geometrical parameters for both isomers are d(Ag-Ag) = 2.88 A’ and 0 = 54.2’ for the acute ( 2 A ~ ) isomer, while d(Ag-Ag) = 2.72 A’ and 0 = 63.7O for the obtuse (2B2) isomer, respectively.17 The symmetrized basis functions for the row 1 (mj = + I / * ) and row 2 (m,= J / 2 ) of the double group C,* were generated by the usual procedures?Jl and truncated at 1 = 2 for each Ag atom. 111. Results and Discussion
The total valence populations of both isomers are given in Table 1. These are reported in terms of the atomic-like spinors as calculated by a relativistic population analysis algorithm! and are also given in terms of a Mulliken gross population analysis, as calculated by the MRSDCI method.17 The calculated total valence populations suggest that the formal atomic charges for the acute (2Al) isomer could be assigned as Ag( 1)-0.128Ag2+o.064 or Ag( 1)-OJJ88Ag2+0JJ44, according to the DSW-XCYor the MRSDCI17 methods, respectively. The obtuse (2B2) isomer could be formulated as Ag( 1)+0J88Ag2-0,w4or Ag( 1)+0.192Ag2-0,097as
.3
.4
5
-
Ag(2)
I
6 1 2ANI
-2
.1
0
1
1.V)
Figure 2. Calculated valence (total and partial) local density of states ) The a and b label the (using u = 0.4 eV) of the acute Ag3 ( 2 A ~cluster. Ag( 1) and Ag(2) projected spinor contributions,respectively. The upper section indicates the cluster valence energy levels that belongs to the ys irreducible representation of the group Cb*.
calculated by the DSW-Xa or the MRSDCI17 methods, respectively. Thus, both calculational methods (Pauli and Dirac types) agree reasonably well in assigning the formal charges to the apical and terminal atoms of the corresponding isomeric species. However, we should bear in mind that the small differences seen in the two calculational procedures could lie in the fact that the orbitals of the MRSDCI method extends to infinity,l7J8 while the partial wave in the multiple scattering description is confined to the finite cellular region about the silver nuclei. Nevertheless, the agreement between the two calculational methods is meaningful, although both methods are essentially different. The valence electronic structure of both isomers are represented in Figures 1 and 2 in terms of the calculated total and partial density of states (DOS).l2J3 It can be seen from these figures that in both isomers the higher intensity valence d-bands are quite separated (-2.0 eV) from the Fermi level ( E F ) . Also, the d-bands arising from the d3/2 and ds/2 spinor components of the apical and terminal atoms are more mixed in the obtuse Ag3 cluster. Moreover, the effects of the spin-orbit coupling on the d-band for each apical and terminal silver atoms could be qualitatively estimated from the maximum partial-DOS of each peak shown in these figures. For example, in the obtuse (2B2) trimeric silver cluster (see Figure 1) the d3/2-d5/2 spinor-like energetic separation is larger for the apical atom (21.0 eV) than the terminal atoms (20.8 eV), while the opposite trend is true for the acute (2A1) trimeric silver cluster (see Figure 2), where
Structure of the Cz, Isomers of Ag3
The Journal of Physical Chemistry, Vol. 98, No. 22, 1994 5629
TABLE 2: Spin Population (SOMO) (a) In Terms of Atomic Spinors atom
spinor
acute
obtuse
obtuse (Walch et ai.).
SI/2
0.008 0.048 0.063
b
total p
0.535 0.024 0.016 0.040 0.005 0.002 0.007 0.108 0.041 0.028 0.079
0.010
0.053 0.005
0.050
d3/2
0.012
0.006 0.022
0.011
0.005
PI12 P3/2
total p d3/2 d5/2
totald s1/2
PI12 p3/2
d5/2
0.038
totald
0.111
0.11
0.013 0.020 0.033 0.360 0.025
0.00 0.375
(b) In Terms of Pauli Decomposition
acute atom Ag( 1)
I 0 1 1 1 2 2 2 2 2 0 1
1 1 2 2 2 2 2
m
a
O
b
-1 0 1 -1 0 1 2 -2 0 -1 0 1 -1 0 1 2 -2
0.004 0.002
0.001 -
0.001 0.022 0.003 0,000 0.002 0.000 0.000 0.000 0.000 0.000 0.062
total MO
obtuse
8
a
8
0.535 0.035 0.005 0.000 0.000 0.086 0.002 0.069 0.002 0.004
0.059 0.051 0.017 0.016 0.358 0.026 0.001 0.021 0.000 0.004 0.000 0.003 0.003 0.975
0.008 -
0.012 0.004 0.001
0.001 0.938
0.010 -
0.000
0.000 0.000 0.002 0.000
0.001 0.000 0.000
0.000 0.000 0.000
0.000 0.025
In terms of nonrelativisticorbitals;see ref 20. Dashed line indicates contributions forbidden by single point group symmetry. the calculated d-spin-orbit splitting values are 0.8 and 0.5 eV for the terminal and apical atoms, respectively. Obviously, these calculated d3/2-d5/2energy separations in both the trimeric silver clusters are different from the observed spinorbit coupling of the free Ag atom,28 which is indicative of the quenching of the d-spin+xbit coupling due to d-bonding interactions occurring in both the clusters. Moreover, the DSW calculated values of the d-band splitting for the Ag32+, A g P , and Ag5q+ clusters are 0.8,0.9, and 1.2 eV, respecti~ely.~-~l Thus, it becomes apparent that the splitting of the valence d-band increases with increasing cluster size, and the same trend has been observed in photoemission experiments on a low coverage of silver clusters on carbon s u b ~ t r a t e . ~ ~ . ~ ~ In the present DSW spin-restricted calculations, all the cluster spin densities arise from the single occupied molecular orbitals (SOMO). Table 2 gives the spin populations for the partially occupied orbitals of the acute and obtuse Ag3 clusters. These are given in terms of atomic-like spinors for j = l/2,3/2, and s / 2 , and they are also reported in terms of a Pauli description consisting of spherical harmonics multiplied by spin functions.8 In the latter charge decomposition procedure we consider only the two large components of the relativistic wave functions and assume that the radial wave functions inside each atomic nuclei are the same for 1 = j - 1 1 2 and 1 = j ' 1 2 . The sum of two such spinors can then be interpreted as a nonrelativistic function of mixed spin, with spin a corresponding to column 1 and spin p to column 2. One measureof theextent of spin-orbit mixings into the relativistic molecular orbitals is to determine the amount of minority spin. The SOMO levels of the acute and obutse silver clusters contain
+
small but nonnegligible amounts of minority spin, accounting to 6.6% of spin a,and 2.5% of spin & respectively.. This is consistent with the smaller SOMO-LUMO energy separation (0.1 1 eV) of the acute cluster compared to 0.21 eV of the obtuse cluster. Also, applying the principles of density functional theory (DFT),31932 the obtuse Agj cluster would be harder than the corresponding acute cluster, and so more stable. This is in agreement with the calculations of Walch et aLzoand Balasubramanian et al.17 who predict the ZB2obtuse structure to be lower in energy than the corresponding acute 2A1 cluster. (a) Spin Populations. As shown in Table 2, the larger spin populations for the acute isomer correspond to the 5s1p spinors of the Ag(1) apical silver atom (53.5%), while the larger spin populations for the obtuse isomer correspond to the Ag( 2) terminal silver atoms (36.0%), which is similar to the value of 37.5% calculated by Walch et a1.20 It should be noticed that 5s contributions for the apical silver atom in the obtuse Ag, cluster are forbidden by single point group symmetry, but in relativistic theory these contributions are allowed by double point group symmetry. Furthermore, we should bear in mind that in relativistic theory the charge distribution arising from j = 112 state (either s1/2 or p1p) is spherically symmetric, and its contribution to the hyperfine tensors is isotropic. Similarly, the states will result in anisotropic contributions from j = 3 / 2 or character to the hyperfine tensors since these charge distributions are nonspherical.ll Following the arguments of the previous paragraph, we can estimate from Table 2 the isotropic (piso = p561/2 pspl/2= pip) and the anisotropic b a n i s o = p5p3/2 + p4d3/2 + = ~3/2,5/2)spin populations for each isomer. Thus, the calculated isotropic spin populations for the acute cluster are pl/z( 1) = 0.56 and p1/2(2) = 0.15. The isotropic spin populations deduced empirically by Kernisant et al.5 (in terms of nonrelativistic silver orbitals) are p5,(l) = 0.51 and pSs(2) = 0.12, for a total isotropic unit spin population of psS( 1) 2psS(2) = 0.75,5 while the calculated total isotropic spin population piso= 0.86. The calculated and empirical isotropic spin populations are in reasonable agreement, but it should be noticed that the empirical estimate were obtained from the ratio between the observed cluster hfi and the atomic gasphase hfi.5 Thus, the empirical model neglects orbital deformation effects due to cluster bond formation. The calculated anisotropic spin populations (see Table 2) for the acute cluster are p3/2,5/2(1) = 0.02 and p3/2,5/2(2) = 0.06, giving a total paniso= 0.14. Since the total spin populations should be close to 1.0, the empirical analysis assigned the remaining anistropic spin population psp to be between 0.10 and 0.15 of 5p character on each basal Ag(2) atom.5 Thus, the calculated and empirical anisotropic spin populations are in close agreement, but the empirical model neglects the small anisotropic contributions arising from the apical Ag(1) atom. The calculated isotropic spin populations (pim) for the corresponding obtuse silver trimer (see Table 2) are p l p ( 1) = 0.056 and p1/2(2) = 0.385; thus the total isotropic spin population amounts to piso = 0.826. These calculated values could be compared against the empirically determined isotropic spin populations by Howard et al., who used unmodified 107Ag oneelectron parameters to obtain psS( 1) = 0.060 and pSa(2)= 0.440 for a total unit spin population of 0.940.334 However, it should be noticed that the empirically determined piso(1) 0.06 was ascribed as due to core spin polarization effects since these contributions on the apical atom are forbidden by single group symmetry, but these contributions are allowed by double group symmetry, and there is no need to invoke core polarization effects.3~~ The calculated anisotropic spin populations are p3/2,5/2(1) = 0.096 and p3/2,5/2(2) = 0.039, giving a total paniPo= 0.174. The empirical determination of the remaining anisotropic spin populations were assigned as psP = 0.100.3-4 Our calculations
+
+
-
5630 The Journal of Physical Chemistry, Vol. 98, No. 22, 2
L,
Figure 3. Contours of the largest component (91) of the HOMO of the obruse Ag3 (*B2) cluster in the YZ plane. Contour values (electron/ b0hr3)'/~are between +0.121 and -0.121 with increments of 0.006. Negative contours are indicated by dashed lines.
.....
. .... ... ... . .. .
. .......................... . ..... . .... . ...... ............ . . . I . . . .
I..
I
.,
.'
Figure 4. Contours of the largest component ($9)of the HOMO of the acure Ag3 (2Al) cluster in the YZ plane. Contour values (electron/ bohr3)l/*are between +0.134 and -0.075 with increments of 0.006. Negative contours are indicated by dashed lines. suggest that the isotropic character of the spin distribution of the acute silver cluster are larger than those corresponding to the obtuse cluster. This would imply that, if the EPR spectrum of the obtuse cluster were recorded at 4.2 K, it would show larger anistropies than the acute cluster. In order to get a visual picture of the semioccupied molecular orbitals holding the unpaired electron, we plotted the largest component of each wave function contours for both the clusters, and these are shown in Figures 3 and 4. The antibonding character of both molecular orbitals is clearly seen, but the SOMO of the obtuse isomer possess a global ?r symmetry, while, the SOMO of the acute isomer present global u symmetry. Both contour plots show mainly silver 5 s contributions and small silver 5p contribu-
Bravo-VQsquez and Arratia-P%rez
TABLE 3: Hyperfine Coupling Constants (in C) of Ags Aii Al refs (a) Acute 432.8 421.7 a Io7Ag(l) calc expt 310.8 310.1 (5) 85.7 82.5 a 10'7Ag(2) calc expt 76.0 72.6 (5) (b) Obtuse "J7Ag(l) calc 12.8 0.3 a exptb 38.5 (3,4) 107Ag(2) calc (-)326.2 (-)332.2 (I exptb 295.0 (394) Present work. b Reported as the isotropic part of the cluster hfi. tions as well. Furthermore, there is some 4d character mixed in the SOMO of the obtuse cluster, which is present to a lesser extent in the acute cluster (see Table 2). (b) Cluster Hyperfine Interactions. The calculated and experimental hyperfine coupling constants (Ahfi) of the acute and obtuse silver trimers are listed in Table 3. It should be remarked that the calculated relativistic hyperfine constants include the Fermi contact term, the spin-dipolar term, and the orbital term, as could be deduced from eqs 1-3.8 However, the EPR analysis has been done by considering only the first two terms and neglecting those contributions arising from unquenched orbital angular momentum (due to spin-orbit coupling) to the hyperfine coupling c0nstants.~-5 Furthermore, the cluster EPR spectra show large negative g shifts, implying dominant contributions to the g-tensor anistropy arising from intramolecular spinorbit coupling to the lowest lying empty molecular 0rbitals.~-5 According to our calculated spin populations, we expect some degree of tensor anisotropy in each EPR spectrum, which may be experimentally unresolved. (i) Acute Ag3 Cluster. The calculated and experimental hyperfine coupling constant for this isomer are listed in Table 3. It can be seen from this table that for both apical and terminal Io7Agnuclei the calculated constants are slightly anisotropic, in reasonable agreement with those resolved in an N2 matrix. However, a better agreement with experiment will require a more complete relativistic wave function, such as a relativistic spinpolarized wave function to account for core spin polarization effects,33J4which may be significant for the cluster studied here. We have found earlier in silver tricarbonyl that core spin polarization effects (estimated via quasirelativistic spin-unrestricted calculations) account for nearly 30% of the total silver hyperfine coupling constants.35 However, both the calculated and resolved All - A 1 are greater than zero. Obviously, the observed anisotropic character of the hyperfine tensors could be ascribed to the effects of spin-orbit coupling and nonzero contributions from spin-dipolar interactions. It has been shown by Pyper et al.36 that a heavy atom containing an unpaired s electron (like Ag and Au) has a significant contribution from spin-dipolar interactions which arises entirely from the small components of the relativistic wave function, thus giving rise to magnetic anisotropy.36 (ii) Obtuse Ag3 Cluster. The calculated and experimental hyperfine coupling for the obtuse silver trimer are listed in Table 3. It can be seen that the calculated constants for the apical silver atom are underestimated respect to the experimental values. This may well be the result of some density contributions due to core polarization effects that are neglected in the present calculations. However, theisotropiccontributions to the hyperfine tensors of the apical silver atom are forbidden by nonrelativistic symmetry, but the relativistic spinors only give a small value for this interaction. Thus, the observed isotropic contributions of the apical silver atom could be interpreted only in terms of relativistic orbital theory. The calculated and experimental hyperfine tensors for the basal atoms are in reasonable agreement, but the present calculation gives negative constants. It is known
Structure of the C2, Isomers of Ag3
The Journal of Physical Chemistry, Vol. 98, No. 22, 1994 5631
that negative signs of the hyperfine coupling constants are not determined by EPR spectroscopy. The calculated anisotropic contributions to the basal silver hfi are small and were not resolved in the experimental work. In a earlier work on Ag3*+we predicted anistropies of about 5.0 G,9which were recently detected by highresolution EPR ~pectroscopy.~
(3) Howard, J. A.; Preston, K. F.; Mile, B. J . Am. Chem. SOC.1981,103, 6226. (4) Howard, J. A.; Sutcliffe, R.; Mile, B. Surf. Sci. 1985, 156, 214. (5) Kernisant, K.; Thompson, G. A.; Lindsay, D. M. J . Chem. Phys. 1985,82, 4739. (6) van der Pol, A.; Reijerse, E. J.; de Boer, E. Mol. Phys. 1992,75, 37. (7) Michalik, J.; Wasowicz, T.; van der Pol, A.; Reijerse, E. J.; de Boer, E. J. Chem. SOC.,Chem. Commun. 1992, 29. (8) (a) Yang, C. Y.; Case, D. A. In Local Density Approximations in
IV. Concluding Remarks
Quantum Chemistry and Solid Stare Physics; Dahl, J., Avery, J. P., Eds.; Plenum: New York, 1983. (b) Arratia-Perez, R.; Case, D. A. J . Chem. Phys.
We have performed the first fully relativistic DSW calculations on both isomers of the Ag3 clusters that were produced on the C6Ds and N2 solid matrices at low temperatures. Our results suggest that extracting meaningful information from the EPR spectra of clusters containing heavy atoms certainly requires the use of relativistic operators and orbitals to give a more reliable EPR interpretation. We have shown that the reported empirical spin population distributions are hampered because the assumed empirical model neglects the contributions of the j = 3/z and j = 5 / 2 spinor which induces anistropic behavior. The calculated hyperfine paramagnetic parameters are in reasonable agreement with experiment, but a better agreement could be only reached by using a relativistic spin-polarized wave function33J4to account for core spin polarization effects. Calculations using a relativistic spin-polarized wave function are planned in the near future. However, the general trends in spin populations and hyperfine parameters are qualitatively well represented by the present work and support the previous experimental and theoretical identification of both trimeric silver clusters.
Acknowledgment. We thank Prof. R. G. Parr for stimulating discussions during his recent visit to Chile, and Prof. N . C. Pyper for interesting e-mail discussions concerning his calculations of atomic hyperfine structure. We also thank Lucia HernlndezAcevedo for code programming. This work has been supported by Fondecyt (grant 92/0604). References and Notes (1) Weltner, Jr., W. In Magnetic Atoms and Molecules; Van Nostrand: New York, 1983. (2) Weltner, Jr., W.; Van Zee, R. J. Annu. Rev. Phys. Chem. 1984,35, 291.
1983, 79, 4939. (9) Arratia-Perez, R.; Malli, G. L. J . Chem. Phys. 1986, 84, 5891. (10) Arratia-Perez, R.; Malli, G. L. J . Magn. Reson. 1987, 73, 134. (11) Arratia-Perez, R.; Malli, G. L. J . Chem. Phys. 1986, 85, 6610. (12) Arratia-Perez, R.; Hernandez-Acevedo, L. J . Mol. Struct. THEOCHEM 1993, 282, 131. (13) Arratia-Ptrez, R. Chem. Phys. Lerr. 1993, 213, 547. (14) Richtsmeier, S.C.; Gole, J. L.; Dixon, D. A. Proc. Natl. Acad. Sci. U.S.A. 1980, 77, 5611. (15) Basch, H. J. Am. Chem. SOC.1981, 103,4657. (16) Flad, J.; Igel-Mann, G.; Preuss, H.; Stoll, H. Chem. Phys. 1984, 90, 257. (17) Balasubramanian, K.; Liao, M. Z. Chem. Phys. 1988, 127, 313. (18) (a) Balasubramanian, K.; Feng, P. Y. Chem. Phys. Letr. 1989, 159, 452. (b) Balasubramanian, K.; Das, K. K. Chem. Phys. Lett. 1991,186,577. (19) Bauschlicher, Jr., C. W.; Langhoff, S. R.; Patridge, H. J . Chem. Phvs. 1989. 91. 2412. 120) Walch; S. P.; Bauschlicher, Jr., C. W.; Langhoff, S. R. J . Chem. Phys. 1986,85, 5900. (21) Ramirez-Solis, A.; Daudey, J. P.; Novaro, 0. Z . Phys. D 1990, 15, 71. Hedin, L.; Lundqvist, B. I. J . Phys. C 1974, 4, 2064. MacDonald, A. H.; Vosko, S. H. J . Phys. C 1979, 12, 2977. Das, M. P.; Ramana, M. V.; Rajagopal, A. K. Phys. Rev. A 1980,
Arratia-Perez, R. Chem. Phys. Lett. 1993, 203, 409. Arratia-Pkrez. R. J. Phvs. Chem. 1991. 95. 7239. , Case, D. A.; Lopez, J. P. J. Chem. Phys. 1984, 80, 3270. (28) Moore, C. E. In Atomic Energy Levels,NBS, US GPO; Washington, DC _ _ 1971: Vnl 2._. (29) Apai, G.; Lee, S. T.; Mason, G. SolidState Commun.1981,37,213. (30) McLachlan, A. D.; Liesagan, J.; Leckey, R. G.; Jenkin, J. G. Phys. Rev. B 1915, 11, 2811. (31) Parr, R. G.; Zhou, Z. Acc. Chem. Res. 1993, 26, 256. (32) Yang, W.; Parr, R. G. Proc. Natl. Acad. Sci. U S A . 1985,82,6123. (33) Cortona, P.; Doniach, S.; Sommers, C. Phys. Rev. A 1985,31,2842. (34) Solovyev, I.; Leichtenstein, A. I.; Gubanov, V. A.; Andropov, V. P.; Andersen, 0. K. Phys. Rev. B 1991, 43, 14414. (35) Arratia-Perez, R.; Marynick, D. S.J . Chem. Phys. 1987,87, 4644. (36) (a) Pyper, N. C. Mol. Phys. 1988,64,933. (b) Zhang, Z. C.; Pyper, N. C. Mol. Phys. 1988,64, 963.
.~
_.
