Audrey L. Companion1 and Dennis J. Trevor Illinois lnstltute of Technology Chicago, Illinois 60616
I I
Crystal Field Splitting Diagrams 11. Many-electron weak-field systems
Ten years ago we presented in this Journal ( I ) a quite general method for determining d l crystal field splitting patterns within the ionic model without the use of group theory. Response to the paper indicates that i t has heen useful not only to educators, hut also to those whose research involves interpretation of dl(dg) transition metal spectra (24). In the past few years, elementary group theory has become part of the normal undergraduate diet in most universities and is no longer regarded as a foreign language by most graduating B.S. chemists. Although elegant group theoretical generalizations of many-electron crystal field theory (5) may perhaps he mastered easily by the topmost students, we have found in our own ex~eriencethat an understanding gap still exists between the physical phenomena described bv quantum mechanics and the more abstract results of gro"p-theory. Accordingly we present in this paper, a supplement to I, methods by which the d l f o m u las previously derived may he used, with a minimum of group theory, as an aid in understanding many-electron splitting and the origin of low symmetry crystal field parameters. In paper I the matrix elements Hp, describing the per-, turbation of a single d electron by a collection of point charge ligands (in any number and in any geometry) were developed as sums of coefficients bl,pq, obtained through hrute force integration over electron coordinates, times total ligand position parameters Dl, (or GI,) (Table 1).
The real d orbitals with unspecified radial portions were taken as the basis set for the ~erturhationcalculation, with V representing the total elecirostatic perturbing potential of all ligands. Each Dl, or GI, was further expressed as a sum over all i liwnds of individual lieand nosition oarameters Dl,' ( ~ a h l e 2 )involving . angular coordinates Gi and Bi and radial integrals, a l i defined, for practical purposes, by
i
F 180
'Y
-
-
' Present address: Department of Chemistry, University of Ken-
tucky, Lexington, 40506.
710 / Journal of ChernicalEducation
me
ure 1. The ligands ABCDEF are restricted to positions along the x, y, and z axes, hut each may he chemically different or may he the same atom a t different distances from the central metal M. With the methods of paper I and Table 2 we find that only the following ligand position parameters are non-zero for this complex
From Tahle 1we see that without terms D22 and D42 the secular equation for the prohlem has no off-diagonal elements, that matrix elements Hi; yield d electron energy states directly. Terms D22 and 0 4 2 vanish (eqn. (3)) when ligands A = B and C = D, in which case we have the still Table 1.
The
Integrals Hpq in Terms of Dl,,, and Gym
- .
These radial integrals. , a function of lieand charee - Z;.and ligand distance from the origin Rj, are generally treated as em~iricalnarameters. Onlv radial Darameters with 1 = 2 or 4 appear in d electron perturhatibns, and, to reduce the number of parameters further in the calculations of paper I, we employed the approximation a n = 3a4. (Since then, evidence increases in favor of az ar (6)J Using Tables 1 and 2, we illustrated the origin of ionic model splitting diagrams for several common d l systems. For those who have worked through paper I, we will show (1)how the d l energies calculated with Tahles 1and 2 can be used along with V to derive many-electron state energies (in a weak field), and (2) how the results obtained for a sample tetragonal complex may he applied to several related systems. We consider first the very general complex shown in Fig-
---
FQure I. Oewneby and coordinates of a general MABCDEF complex (I). metal ion is st the origin.
"..-
'"
5/28
-.
.,"
D.. + 5/12 D
Has = 317 G2,-5128 G4, + 5/12 G,. H- = 4 / 7 G2,+ 5fi114 G., H,, = - 4 / 7 G,, + 5 6 / 8 4 GI? H.. = 317 D.. - 5/28 D.. - 5/12 D..
Table 2.
The Ligand
Position Functions ~~~i and Gl,i
Figwe 8. Complex VIII. MAC$. Figure 2. Complex 11. MAzCzEF.
Figure 9. Complex IX. MAGS.
E Flgue 3. Complex Ill. MA2Cg2.
Figure 10. Complex X. M u .
E Figure 4. Complex IV. trankMkEn.
Figure 11. Complex XI. MAr.
"/"/ AA-
A Figure 5. Complex V. CikMA&.
Figwe 12. Complex XU. cis-MA2C2.
Figure 8. Complex VI. MkEF. Figure 13. Complex XIII. MA4
A Figure 7. Complex VII. MAsE
very general complex MA&EF, 11, of Figure 2. With the definitions of parameters DQ, DS, and DT given in Figure 2, the d l energies of complex I1 may be written
-
E(~,.L,~,B,J= 6DQ f ZDS DT E(d,,,A,) = 6DQ - 2DS - 6LYT E(d,,,B2,) = -4DQ E(d,,,d,,,E,) = -4DQ
+ 2DS -
DS
DT
(4)
+ 4DT
where the parameter DQ is the arithmetic mean of the octahedral parameters DqA and Dqc (DqA = 116 a d A ) characteristic of complexes MA6 and MCs, and DT and DS are difference parameters that we will later relate t o tetragons1 parameters Dt and Ds. In Figures 3-13 we picture 11 other complexes derived from that in Figure 2 by substitution or removal of ligands.
For all of these the d1 energies are given exactly by eqn. (4), provided that the parameters DQ, DS, and DT are defined as given in each diagram in terms of radial integrals. Since all of these may be easily derived from the parameter definitions for the general complex 11, we will comment only briefly on interrelations of possihle interest. In many of these complexes (IV, VI, VII, X, XI, and XIII) DQ is identical to the usual octahedral splitting parameter Dq; in all others DQ is an arithmetic mean of octahedral Dq values. For some cases (IV, V, VII, IX, and X) simultaneous solution of equations defining DQ and DT permits evaluation of two separate Dq values. In other cases (XI, XII, and XIII) DQ and DT are not independent parameters. The parameters DQ, DT, and DS for the trans-tetragonal complex IV are identical to those normally used by spectroscopists: Dq, Dt, and Ds. Comparison of these with the parameter definitions (Fig. 5) for the cis-complex V permits one easily to see the origin of the relationships frequently just stated in textbooks DQ(trans) = DQ(cis) - 7/4DT(cis) DT(trans) = -ZDT(cis) (5) DS(trans) = -2L)S(cis) Volume 52, Number 11, November 1975 / 711
In addition, the relation between parameters of complex IV and complex VII, derived from IV hy replacement of one ligand, is DQ(W = m v n , D T W ) = 2DT(lln) (6) DS(IV) 2DS(VII) Relations between parameters of other of the complexes shown may be deduced by the interested reader from examination of the radial integrals involved in the parameter definitions. We emphasize that the general formulas as given here do not describe complexes for which equatorial ligands alternate, such as square planar trans-MA& since the D22 and Dm .-terms in eon. . (3). do not vanish. We will. however, now show that a similar generalization holds for the energies of dn svstems ( n > 1 ) of all the complexes I1 co XIII. T; derive the splitting patternof the d 2 configuration of a typical tetragonal complex, trans-MA.&, IV, we first examine the crystal field potential V. For complexes I1 through XI11 we observed that only ligand parameters Dw, Dzo, 040, and D44 are non-zero. Since these parameters result from non-zero perturbations caused by terms YL, (0,b) in the general potential
-
tegrals representing ml values. Since the tetragonal potential of eqn. (8)has no terms involving ml = 2, matrix ele-. ments such as (+I14-1) or (q q 2 ) are zero and only tbose in eqn. (10) contribute to many-electron energies. With eqn. (lo), tetragonal d 2 energies may he easily derived from weak field octahedral wavefunctions such as tbose given by Ballhausen (7) and reproduced in Table 3. We illustrate this briefly with a simple example. The unique octahedral state 3A2g, arising from the ground state 3F term of d2, correlates with the 3B1, state in a D~I, complex. The wavefunction for this state is a linear combination of 3F components with ML values 2 and -2 (Table 3), where (3F,2) = /2,0/, a Slater determinant over two imaginary d orbitals both with alpha spin and with ml values +2 and 0. A simple perturbation calculation with the tetragonal crystal field operator, breaking down the many-electron energy into one-electron contributions, yields
E(%,,DL = IWA,,,OJ* V WA*OhM
D
=
the last three terms in this equation represent in^- the only non-zero tetragonal one-electron matrix elements that can result from permutation of electrons among orbitals in the d2 determinant. Substituting values of these integrals from eqn. (lo), we have
this means that a general form for the tetragonal potential must be V = a,Y,
+ a20PYla + a,d.*Ya + a&Y,,'
+
Y,,) (8j
Similar analysis of the third octahedral component of S T ~ g(F) in Table 3, which correlates with 3B2Kin D4h, yields
and the sum is taken over all N ligands. With knowledge of d l energies a general formulation of V such as eqn. (a), showing only which spherical harmonics occur, is sufficient to solve d n problems. We employ the d l energies in terms of the parameters given in eqn. (4), but i t is more convenient to have these expressed as energy matrix elements over imaginary d orbitals, R3d(r)Yfm(8,r$), since the latter, with well-defined ml values, are generally used to describe many-electron configurations. With the definitions of real d orbitals presented in paper I, we obtain, by expansion of the orbitals involved in eqn. (4)
The remaining two A2, states andbree E, states have energies given by solution of secular equations, elements of which are given in Table 4. Matrix elements for all diagonal and off-diagonal terms may be derived by the simple metbod illustrated above, using eqn. (10) and the fact that only integrals of the type ( ( 3 F , M ~ ) l q ( 3 P f l ~ ' ) ) or (PF,ML)I~ P F , M L ' ) )with M i - ML = 0,4 or -4 are non-
where
Table 3.
Octahedral Weak Field Wavefunmiom ford'
(0lVl0) = 6DQ - 2DS - GDT 4DT (+lIVI*l) =--4DQ - DS (10) (f 21~1f2) = DQ 2DS - DT (f2lV(T2) = 5DQ where (+lII/I+l) = I d l * V d l dv, the subscripts in the in-
+
Table 4.
+
Formulas and Matrices Darcribina Solittina of the ' d Confiauration
The notations along the rider of the matricer indicate the parent atomic and octahedral termr from which the interactions arise. For use ar secular equations, add (-E) t o each diagonal element; solution will yield, for example, for t h e A S gmatrix, two energies that will be associated with AZgterms in DAh. E ( P ) is the energy of 'P relative t o the ground state %. Rather than the isolated term energy, a reduced value, according to observed nephelauxattc relaxation (81, may be more appropriate in calculations.
712 / Journal of Chemical Education
zero for a V of the form of eqn. (8).Analogous formulas applicahle to the ds configuration are given in Table 5. Rather than dwell further on details of these derivations, we prefer to emphasize the power and generality of the energy expressions in Tables 4 and 5. Although derived for a D4h complex, the validity of these expressions extends through all complexes II through XIII, provided that the splitting parameters are defined as in the accompanying diagrams, these admittedly being sometimes artificial definitions. Complexes such as V or XII, not belonging rigorously to the D4h point group, possess in the 90" angle conformations described here what has been called intermediate symmetry. Their characteristic perturbing potentials V differ from that of a true D4h complex such as IV only in the constants a[,, and this difference is reflected in the parametric definitions of DQ, DS, and DT. All possible coupling of the many-electron states (identical to the coupling for a D4h complex) is contained in the matrix elements of Tables 4 and 5. Electronic states of complexes with Cdv symmetry or based on a pentagonal pyramidal geometry may be more properly renamed by deletion of the g subscript from the symbols in Tables 4 and 5. Classification of all states according to some lower symmetry molecular point group such as Cz, may be accomplished by examining the transformation properties of the real d orbitals in that point group and correlating the state symmetries with those of eqn. (4), or by application of the correlation tables given hy Wilson, Decius, and Cross (9). In addition to applicability of Tables 4 and 5 to d 2 and d6 configurations of all complexes I1 through XIII, because of holelectron eauivalence fdn = dlo-") and the d n = ds-" equivalence chaiacteristic of a weak field, the expressions in Table 4. as thev stand. also describe splitting of the 4F and 4P teims of d7 configurations of a l i compiexes, and, with change of sign of DQ, DS, and DT, splitting of 4F and 4P terms of d3 and 3F and 3P terms of dB. Similarly eqn. (4) describes not only d l and d6 splitting, but, with change of sign of all parameters, also d 4 and d9 configurations. T o apply these formulas to the spectral fitting of a complex such as MA4EF, VI, we would take as a first guess for DQ the Dq value for an octahedral complex M&. If Dq values are known for ME6 and MFs, then a first approxi-
mation for D T would be, from the relations given in Figure 6 DT = 2/7[2D~(MAd - DqCME,) - Dq(MF,)] Since a 2 and a 4 integrals are of the same order of magnitude (6), D S N 3DT. For a d8 complex, these three parameter values, with signs changed, may be used in the matrices of Table 4 to compute a first approximation to the exoected enerev level scheme. The matrices mav be easilv programmed to permit systematic variations around initial oarameter choices until a satisfactorv. spectral fit is ob. tained.
--
Conclusion
We have oresented here an examole of how our d l aoproach may b e extended to the stud; of many-electron systems. and in so doine have derived a set of matrices aonlicable to 12 weak-fie6 transition metal complex type; 'We leave as an exercise for the reader proof that these formulas also apply to linear complexes of type MEF, to Czv species such as MA2EF (both derived from Fig. 2) and possibly others that have not occurred to us. These must be limited to complexes with ligand combinations for which position parameters D22 and D42 are zero and for which one-electron energies may by judicial choice of parameter definitions be expressed in the form of eqn. (4). There is no reason (other than algebraic entanglement) why the method described here could not he applied to weak-field d" systems of lower symmetry such as Dzh trans-MAzBz, complexes symmetrically distorted from the rigid 90' angles that we have assumed, or even trigonal systems. For those interested in these applications we conclude with an outline of the procedure 1) Solve the dl energy problem as described in paper I (the results may of necessity be numerical), and in so doing, 2) Determine from non-zero Di, (or GI,) which spherical h a monies occur in the potential V for the problem, 3) Use non-zero H,, values over real orbitals to obtain d' energy matrix elements over imaginary orbitals, and with these, 4) Systematically break down integrals over many-electron
wavefunctions into one-electron contributions, examining all states that could possibly be coupled by the spherical harmonics occurring in V (the ML rule); 5) Solve the resulting secular equations by substituting appro-
Table 5. M a t r i m Describing Splitting of the d5Configuration
see notes below Table 4. The rymbols "G,'F. 'D, and 'prefer to term valuer above the ground state '5 which, since nondegenerate, i s not split by the field. Note that there is no mixing among A,g quartets, that AXgstater are not coupled by DS, D T , and that B,g states are coupled only by DS. D T .
VoIum 52. Number 11, November 1975 / 713
priate numerical values for the one-electron parameters in-
volved. Application of elementary group theory in step 1 may simplify the process considerably. Wasson and Stoklosa (4) and more recently Kassman (10) have presented useful adaptations of par I in which symmetry operations have been employed to derive solutions for dl energy levels for ions in virtually all chemically significant symmetry sites. Literature Cned (1) Companion, A. L.. and KomarynsLy, M. A,, J. CHEM. EDUC.. 41. 257 (1964). rrferredto hereas wper 1.
714 / Journal of ChemicalEd~~dtbn
12) i r w r . A. B. P.," h o ~ m i c~ ~ a c t m ns@-ow: ie ~ m e r i c a n~laevier~ub~iahing Company, Inc.. New York. 1968. 13Jamith.D. W..J Chem Soc (London). 1969A. 2529. 14) Wasson, J. R., and Sbklosa, H. J., J. CHEM. EDUC., TO, 186 (19731. Them authors inelude in their p a p r an extensive list a1 recent pepera and texts dealing with ny.tal field theory and applications of p u p theory. (5) Hempel, J. C., Donini, J. C., Hollcbone, B. R.. and irver. A. B. P.. J Amer. Chem. Sor., 96.169311974~. 16) Companion,A. L..J P h y Chem., 73,739(1969). 17) Ballhsuaen, C. J.. "Infrodudon to Ligand Field Thwri." MeCraw-Hill Bmk Co., Inc.. New York. 1962,Appendir 11, p. 95. 181 Jpgenren, C. K., "Absorption Spectra and C h e m i d Bonding In Complexes." Pergamon Presr, LU.,Londan and New York, 1961. Chap. 8. (9) Wilson, E. R. Jr.. Decius.J. C.. and C m , P. C.,"Molrmlar Vibrations," MeCrswHill Book Co.. Ine.. New York. 1955. AppendixX-8, pp. 333-340. (10) Karsman, A. J . , . CHEM. EDUC, 51.605 (19741.
