2126
A. Bencini and D. Gatteschi
Ligand Field Spin-Orbit Coupling Calculations for d7, d8, d9 (d3, d2, dl) Five Coordinated Complexes of CSvSymmetry A. Benclnl and D. Gatteschl” Laboratorio CNR and lstituto di Chimica Generale, Universita di Firenze, 50 132 Florence, Italy (Received February 9, 1976)
The complete matrices for ligand field spin-orbit coupling perturbations are provided for d l , d2, and d3 ions in CsUsymmetry. The importance of using such matrices for the interpretation of the electronic properties of trigonal bipyramidal five-coordinated complexes is stressed, in contrast with previous uses in the literature of D3h matrices.
Introduction L1 The electronic properties of coordination compounds of I transition elements have been extensively investigated in the last years and an increasing amount of experimental information has been obtained by the use of single crystal electronic spectroscopy in linearly polarized light a t very low temperature and under high resolution conditions1,2of magnetic circular dichroism ~pectroscopy,~ electron spin resonance ~ p e c t r o s c o p y , ~etc. - ~ For the interpretation of these experimental data it is necessary to use a method of calculation which can keep pace with the more sophisticated experimental techniques, allowing for small perturbations such as the low symmetry components of ligand field and spin-orbit coupling. These, in fact, determine the nature of the ground electronic L2 level and are responsible for the large number of electronic Figure 1. The most general configuration of donor atoms for a five transitions observed experimentally. coordinate complex of C3”symmetry. Ligand field calculations have been generally used, because they provide a simple way of parameterization of energy levels, coupling scheme, however, is unimportant in the ligand field although, in some instances, also more sophisticated models formalism, when we perform “complete” calculations, i.e., were employed. Complete ligand field spin-orbit coupling when we consider all the interactions among the levels origimatrices, however, are easily available only for cubic symnating from different states of the dn manifold. Eigenvalues metries,&l0 as a consequence of the intial impetus in the study and eigenvectors of the spin-orbit coupling operator were of octahedral or tetrahedral complexes, while there is not obtained by direct diagonalization of its perturbation matrices much in the literature for low symmetry chromophores. To in the (23‘ symmetry adapted ( L S M L M ~basis } functions. have complete matrices in low symmetries, however, is parThe ( L S M L M ~ basis J functions for dl, d2, and d3 configuticularly important for five-coordinated trigonal bipyramidal rations correspond to those reported by Slater,17 with some complexes, for which the cubic matrices cannot provide even corrections in order to have them correctly connected by phase a first approximation to the energy levels. through the vector coupling coefficients.l8 C3” symmetry In that class of complexes, molecules having the full idealadapted linear combinations of these functions were written ized Dsh symmetry are very rare, the main distortions being by considering their transformation properties under symtoward either C3” or C2” configurations of donor a t o m ~ . ’ l - ~ ~ metry operations through the rotation matrices D J ( a p, , y).15,16 It has been often assumed, however, that the electronic levels Actually, in order to classify the LS functions according to the of a complex of C3” symmetry must be close to those calcuirreducible representations of the CsU symmetry group it is lated in Dsh symmetry, and all the theoretical treatments have sufficient to know their behavior under the group generators been restrained to this latter syrnmetry.l4-l6 In the following C&) and, say, a,.19 In Table 120 we report the symmetry sections we will show that the effect of a full C3” perturbation classification of the LS functions with L varying from 5 to can determine a dramatic deviation of the levels from those 0. of D3h symmetry and that it can also cause, in some instances, The symmetry functions for the double group Cs,* were a change in the spin-orbit split ground level. Since the interobtained by taking the direct product of the orbital and spin pretation of the fine structures of the electronic spectra defunctionsz1 through the coupling coefficients reported by pends strongly on the nature of the ground level, this result Koster et a1.22 They are reported in Tables II-IV,zo together is of paramount interest in correctly assigning the experiwith the appropriate symmetry labels. The perturbation mentally observed electronic transitions and we intend to matrices were constructed with a computer program which report here the complete matrices for dl, dz, and d3 ions in C3” expressed the matrix elements as a function of the one electron symmetry. spin-orbit coupling constant {and of monoelectronic ligand The Model of Calculation field parameters. In order to achieve the maximum of generOur ligand field calculations were performed in the wellality of our treatment, we have used the ligand field parameknown “weak-field coupling scheme”.* The choice of the ters:V22= (zt21VI & 2 ) , V 1 1 = (=tllVl j=t),VOO=(OlV(O),
I
The Journal of Physical Chemistry, Vol. 80, No. 19. 1976
2127
Ligand Field Spin-Orbit Calculations
I2E
/J I
' I I
'. '
0
1
2
~q [cm-:103)
. 8 0 1 2 3 0 1 2 3 .5 1 1.5 0 .4 Dq,, ~cm-k10~)9 ~cm-410~1 ( Iz/14)cq ( 12/14)ax
r4
r5
Figure 2. Energy level diagram for a dg ion in a five coordinate chromophore of CsVsymmetry: (-) levels; (- - - -) levels. Left diagram: Dq,, = D9 ; (/2//4)ax = (/2//4)eq = 1; { = 660 cm-'. From the left to the right the effect of changing Dqax,{, (/2//4)eq, and (/2//4)ax with Dqeq = 1000 cm?
and V2M1 = (2lVl - 1) = -(-2lVll) where V i s t h e a p propriate Hamiltonian operator referring to the perturbation due to the ligands and I f 2 ) , I f l ) ,10) represent the one electron d orbitals. In fact, in the last years, it has become that all the most common ligand field parameterizations, including the angular overlap model, can be considered to be equivalent, and that, therefore, it is desirable to be able to pass from one to another in order to make the best use of the method according to the problem which is being studied. In Table V20 are reported the relations between our parameters and the most common ligand field parameters for five coordinate complexes, in the nonadditive scheme25(which in C3" symmetry requires three parameters Dq, Ds, and D t ) , in
the additive scheme25 (which requires up to six parameters, Dq and I2 for each set of independent donor atoms), and the angular overlap mode125 (which, for linearly ligating donors and neglecting 6 bonds, requires up to six parameters, e,' and e,' for each set of independent donor atoms). The angle 8 is the angle LX-M-L~(Figure 1).The spin-orbit coupling energy has been evaluated through the hamiltonian operator %, = {
li
*
i
si
where the sum is over all the electrons of the configuration, 1; and si are the orbital and spin angular momentum operators and {,is the spin-orbit coupling constant for a d electron.
TABLE v: Relations betweenthe Mosl Common Parameteriratlons01 the Llgand Fieldlor Flve Coordinate Complexes (For every fi, the first equatlon refers to the crystal tleld model In the addltlve scheme, the second to Ihe angular overlap model, and the thlrd to the crystal tleld model In the nonaddltlve scheme. The parameters are detlnedas in re1 27.)
Tabis VI - T h e matrix elements d i f f e r e n t from s e i 0 of the perturbation hvnlltonian f m a d' &on! %be ~ o l u m nnumber
corresponds
t o the J indsx o f the
a n d tne
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mafrix element.
r4 2
1
,
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"11
- 0.5 7
I 3
vz2
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3
The Journal of Physical Chemistry, Vol. 80, No 19, 1976
2128
A. Bencini and D.Gatteschi
TABLE VII: Matrlx Elements Dlfferent from Zero of the PerturballonHamlltonlan for a d2 Ion (The column number corresponds io the 1 and the row number lo the 1index of the H,, matrix element.)
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The Journal of Physical Chemistry, Vol. 80, No. 19, 1976
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TheJournal of Physical Chemistry, Vol. 80,No. 19, 1976
2130
A. Bencini and D. Gatteschi
1:
1(
C
P (1
0
... c X I
E
.I:C > (3 a W Z
W -c
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1 0 ,601 2 3 0 1 2 3 .f(crn410? O2/I4leqCI,/i41ax
Figure 3. Energy level diagrams for a de ion in a five coordinate chromophore of C3, symmetry: (-) rl levels; (. . . .) r2levels; (- - - -) r3levels, Lower: Tanabe-Sugano type diagrams for the lowest energy levels. Upper left diagram: Dq,, = Dq,,; 0 = S/Bo = C/Co = 0.8;(/2//4)ax = (/2//4)eq = 1 ; ( = 500 cm-"; a = 70 cm-'. From the leftto the right the effectof changing Dq,,, ,!3, {, (/2//4)eq, and (/2//4)ax with Dqeq = 1000 cm-'. B and Cas in ref 8, p 437. The nonzero matrix elements of the perturbation matrices, calculated according to the illustrated procedure, are reported in Tables VI-VIII.20 The electronic repulsion parameters appear only on the diagonal elements, according to the chosen formalism, The Trees correction has been considered in the form a L ( L 1).1
+
Energy Levels a n d P a r a m e t e r s The C3u matrices were checked by performing calculations with ligand configurations corresponding to T d , o h , and D3h I
The Journal of Physical Chemistry, Vol. 80. No. 19, 1976
symmetries. In all cases a perfect agreement was found with the values previously reported. Many five coordinate complexes are known for d7, ds, and d9 ions, having actual or approximate CsUsymmetry,l1>l2and therefore we have performed our calculations for such ions. Many of these complexes show large deviations of the angle d from 90". We have chosen a value of BO" in order to show the effect of a large C3" perturbation on the levels of D s sym~ metry. In Figure 2 are reported the energy level diagrams for a d9
2131
Ligand Field Spin-Orbit Calculations
-1 (
1I 1.5 .5 1 1.5 0.5 Dq Ccm-k103) Dq,$cm-:103) /3
0
.5
1
1
L a
0 05 3 Ccm-:lQ3)
0 1 2 3 0 1 2 3 r i2/ijeq II ~ / I ~ ) ~ ~
Figure 4. Energy level diagrams for a d7 ion in a five coordinate chromophore of C,, symmetry: (-) r4levels; (. . . .) rslevels. Lower: Tanabe= 1; @ = BIB0 = C/Co = 0.8;{ = 420 Sugano type diagrams for the lowest energy levels. Upper left diagram: Dq,, = Dq,,; (/2//4)ax = cm-'; LY = 70 cm-'. From the left to the right the effect of changing Dqax,0, {, (/2//4),,, and (/2//4)ax with Dq,, = 1000 cm-'. Band Cas in ref
8, p 437.
the e,'le,' ratios of the angular overlap model) and cast many ion. The first diagram is calculated for five equivalent ligands, and the other ones correspond to a change of one parameter doubts on the values of the parameters obtained by making a t a time, as indicated. The parameters reported are those of guesses of the I 2 1 1 4 ratios. the crystal field formalism. The most dramatic effects are In Figure 3 are reported the energy level diagrams for a d8 determined by the variation of the ratio between the quadratic ion. Again we observe a strong dependence of many levels on and quartic radial integrals, (Iz/I~),~~ the I 2 / 1 4 ratios. The ground orbital level is in the present case which can also cause a change in the ground level. Further, the effect of varying such 3E,and i t is split by spin-orbit coupling in such a way that in a parameter for the axial ligands is largely counterbalanced general rl state is the lowest in energy, I'2 being quite close by the effect of varying the same parameter for the equatorial to it. The variations of the (Z2/14)ax ratio can alter dramatically ligands. As a result the effect of changing both of them at the this pattern, allowing also a r3 ground level. same time and of the same amount is very small. These results In Figure 4 are reported the energy level diagrams for a d7 point out the necessity to design experiments which can allow ion. Also in this case the same considerations hold as for the the determination of the radial integral ratios (or conversely previous diagrams. In the present case the spin-orbit split The Journal of Physical Chemistry, Vol. 80, No. 19, 1976
2132
A. Bencini and D. Gatteschi
based on an insufficient number of experimental data are to be considered meaningless.
Acknowledgment. Thanks are due to Professor L. Sacconi, for his encouragement in the present work and to Professor I. Bertini for helpful discussions. Supplementary and Miniprint Material Available: Table I, reporting the symmetry classification of the LML functions, Tables 11-IV, reporting the basis functions for d1, d2,and d3 ions in C3u symmetry (3 pages), 5nd full-size copies of Tables V-VI11 (6 pages). Ordering information is available on any current masthead page. References and Notes (1)J. Ferguson, Prog. Inorg. Chem., 12, 159 (1970). (2) N. S.Hush and R. J. M. Hobbs, Prog. Inorg. Chem., 10, 259 (1968). (3) See, for instance, D. J. Hamm and A. F. Schreiner, Inorg. Chem., 14,519 (1975);M. J. Harding and B. Briat, Mol. Phys., 27, 1153 (1974);J. C. Col1
I
90'
85'
80'
I
I
75'
70'
7p Flgure 5. The dependence on the angle d7 ion in a five coordinate chromophore.
8 of the ground state for a
+
levels of the ground 4Az term are r(4 and rj r6. Which of them has lower energy depends fundamentally on the energy difference between the excited 4Azand 4A1 (4AZ'fand 4A1ffof D3h symmetry) and the ground term. In fact in C3" symmetry the ground 4Az term is represented by a linear combination of functions having ML equal to 0 and f 3 . The latter component has matrix elements of Yfs0different from zero with 4A1, a fact which is not operative in D3h symmetry, and had not been previously appreciated for CsU symmetry, where only simplified second-order spin-orbit coupling arguments had been ~ s e d . ~Further ~ - ~ l the coupling with the 4A1term is greater for the I'5 level than for the r4 one ((3i/2){and ( i / 2 ) {in absolute value, respectively), then the stabilization of the rj (4A2)level with respect to the r4 (4A2) one increases as the mixing of the excited 4Az term with the ground 4Az term increases. The admixture of the two 4Az terms into the ground level is determined by the value of 0 and, as it is shown in Figure 5 , as 8 is lowered from the value of 90' the energy separation between r4 and r5 r6 states decreases, until finally a cross over occurs. The present results show, therefore, how it is necessary to make use of complete matrices in order to evaluate the properties, such as magnetic moments, g tensors, etc., which depend largely on the nature of the ground level. In fact for low symmetry complexes the parameters required are many and to fix arbitrarily some of them is dangerous. Diagrams of the type of Figures 2-5 are therefore the best means of investigating the effect of the parameters, and sample calculations
+
The Journal of Physical Chemistry, Vol. 80, No. 19, 1976
lingwood, P. Day, R. G. Denning, P. N. Quested, and T. R. Snellgrove, J. Phys. E7, 991 (1974);J. Ferguson, H. J. Guggenheim, and E. R. Krausz, Mol. Phys., 27,577 (1974). (4)B. McGarvey, Transition Metal Chem., 3, 90 (1966). (5)T. F. Yen, Ed., "Electron Spin Resonance of Metal Complexes", Hilger, London, 1969. (6)K. D.J. Root and M. T. Rogers, Spectrosc. Inorg. Chem., 2, 116 (1971). (7)B. A. Goodman and J. B. Raynor, Adv. Inorg. Chem. Radiochem., 13,242
(1971). (8)J. S.Griffith, "The Theory of Transition-Metal Ions", University Press, Cambridge, 1961. (9)J. Ferguson, Aust. J. Chem., 23, 635 (1970). (IO) A. D. Liehr and C. J. Ballhausen, Ann. Phys., 6, 174 (1959). (11) P. L. Orioli, Coord. Chem. Rev., 6,285 (1971). (12)R. Morassi. I. Bertini, and L. Sacconi, Coord. Chem. Rev., 13,343 (1973); L. Sacconi, ibid., 8,351 (1972). (13)J. S.Wood, Prog. Inorg. Chem., 16,227 (1972). (14)M. J. Norgett, J. H. M. Thornley, and L. M. Venanzi, J. Chem. SOC.A, 540 (1967). (15)C. A. L. Becker, D W. Meek, and T. M. Dunn, J. Phys. Chem., 72,3588 (1968). (16)C. A. L. Becker, D. W. Meek, and T. M. Dunn, J. Phys. Chem., 74, 1568 (1970). (17)J. C. Slater, "Quantum Theory of Atomic Structure", Vol. 11, McGraw-Hill, New York, N.Y., 1960. (18)For d3 ions the fractional parentage coefficients of Racah (G. Racah, Phys. Rev., 63,367 (1943))were used. (19)Owing to the intrinsic parity of the LS basis functions this operation is equivalent to a rotation by A around yaxis, C2(y).
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