Energy Levels of d' and dg Ions in Chemically Significant Symmetry Sites
Allen J. Kassman
Philip Morris Research Center
Richmond, Virginia 23261
The introduction of ligand field theory to advanced undergraduate and first-year graduate students has become commonplace in most chemistry curricula. Although many texts are available which present elegant and rigorous expositions of the subject, much effort has been expended in attempting to simplify the presentation and forego much of the mathematical formalism. Such effort has concentrated on the dl and d9 electronic configurations because of the absence of mutual electronic repulsion effects in these systems. One particularly innovative approach is that of Krishnamurthy and Schaap ( I ) . Their treatment allows the student to calculate relative d-band splittings without any reference to group theoretical principles. They first evaluated the splittings caused hy three primary ligand clusters, allowing the student to determine the splittings for any other cluster by taking advantage of the principle of additivity of perturbations in the point charge approximation. The treatment is limited, however, by its inapplicability to several important low symmetry point groups. A more general approach to the d l and d9 ligand field calculation was made by Companion and Komarynsky (2). They also endeavored to avoid bringing group theory into the teaching matter by evaluating all the elements of the secular determinant in closed form,
dependent only on the ligand angular coordinates and the two radial parameters. Thus, the elements of the secular determinant would all he calculable by direct substitution and the roots of the determinant could then be found. While this is true in principle, and could be easily implemented with the computer, the solution by hand of any determinant larger than 2 X 2 is not a fruitful exercise. In isolated instances it bas been shown that closed form solutions can be written for specific geometric configurations (3, 4). In this article i t will be shown that the energy levels of .. .-. ~- a d l or ds electronic svstem can be written in closed form for almost every chemically significant geometrv. These solutions can be used without the student's having to either expand any determinants or be familiar with irreducible representations of group theory. It is necessary only that the concepts of elementary symmetry operations be understood. ~
Theory
With only slight modification the procedure of Companion and Komarynsky (2) will be followed. Letting the basis functions be written as the real d-orbitals in the order $1 = d(z2), iCz = d(x2 - y2), $3 = d,,, $4 = dz.. $ 5 = d,,, the secular determinant is as follows
Table 1. Elements of the Secular Determinant
+
H I , = 1/2 Dno 1/28 D M = -1/7 Dm 1/168 Dm 35/168 Du H.. = -117 Dqn 11168 Dan - 351168 Dad
Hm
+ +
+
Table 2. Nonzero Ligand Position Functions
DSO' = P ? ~ O eiS ~ 1) D~~~= sln2ei C O S ~ + ~ D 4oi - - P , ~ ( ~ ~ C O ei S ' - 3ocos2ei 3) D 42' -- pli sin2ei(7cos2ei - l)c082+~ D d = PA' sin3R i cosei cos3+i D d = pZ sin4 e; cos4+i
+
G4.' = Gd =
p.' pai
sina Bi cos& sin3+i sin4 R i sin4mi
Companion and Komarynsky (2) have written each integral of this determinant as an explicit function of two ligand position functions, D m and-&, where k = 2,4 and m = 1, . . . , k - 1,k. By making use of the character tables provided in Cotton's text (51 these definitions can be greatly simplified. If the nonaxial groups CI, C, and C, are excluded from consideration, or if we require at least a two-fold rotation axis coincident with the z-axis, then all ligand position terms with m = 1 vanish. Hlq and Hlg become identically zero. In addition, if we exclude the groups Cz and C a r then i t is no longer possible for any interaction to exist between d(z2) and d,,. For all the remaining point groups His and H45 are identically zero. The remaining terms in the definitions of the Hi, are given in Table 1. Dkm and Gkm are defined as the sums, over all ligands, of the Dam' and Gkmi given in Table 2. The polar coordinates of each ligand are specified by 0 and 6 and the radial parameters (3) are written as pz and pr, where04 = 6Dq for a simple octahedral complex.
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At this point the secular determinant is still tw difficult to manipulate by hand. Closed form solutions are possible only by making use of several features of the character tables and the behavior of the Dami and Gkm' under operations of the particular point groups. For example, if the axis of symmetry collinear with the z-axis is nfold then only those Dam and Gkm with m an integer multiple of n are nonzero. The various combinations of m and n which are allowed, in d-electron systems, are given in Table 3. Further breakdowns for each n are made on the basis of the individual character tables. Cm and Dm are distinguished from the other three-fold point groups because the horizontal mirror planes cause D a and G43 to vanish. C4 and Cm are the only point groups with a fourfold axis which have a possible interaction between d(x2 yZ) and d,,. Similar considerations lead to the final separation as given in Table 4, where the energy in a d l configuration for each d-orbital (or linear combination of orbitals) is given as a function of the nonzero Dkm and Gkm under the appropriate point group symmetry designation. It is apparent from the closed form solutions which orbitals are degenerate because of symmetry considerations. These degeneracies are indicated by solutions of identical functional form, and are distinct from accidental degeneracies. Orbitals which interact to give mixed d-orbital states are shown by enclosing the solutions in brackets. Evaluation of the DamLand Gami are greatly simplified if the ligands are placed coincident with, or directly between, the axes. The exact placement of the W n d s will of course determine the symmetry point group of each molecule. The student can be expected to assign the point group designation of a structure with a few simple examinations for symmetry operations as outlined in the flow chart of Figure 1. It should be emphasized that the strict definitions of the symmetry point groups require that the two-fold rotation axes in the zy-plane or the vertical mirTable 3. Allowable Values of m and n
m
n 2 3 4 25
0,2,4 0,s 0.4 0
"*I
r
S P E C I A L bROUV1
O h . Td
NO
ti^^ oat in T.~I=1
ROTATION A X I S ?
solution n o t I n Table 4
ID
11RROR PLAITS?
111 V E R T I C ~ L
Figure 1. FIOW chart for assignment of point gmup symmetry
ror planes be preferentially located on the x- and y-axes, especially in the case of two- and four-fold symmetry point groups. This condition is usually fulfilled (although not always, as with Dad) if the student places as many ligands as possible on the coordinate axes. All the symmetry groups listed in the character tables of Cotton's text (5), with the exception of cL c,, c,, ca and c m , are mm. pletely represented in the closed form solutions of Table
4.
Applications
Two examples will be examined here to illustrate the usefulness of the closed form solutions in Table 4 for demonstrating the effects of symmetry lowering in transition metal clusters. First we consider an abstract case of a d l ion in a four-coordinated trans-substituted complex, two ligands being placed symmetrically on the x-axis and two
Table 4. Closed Form Solutions for dl Ions. in Arbitrary Symmetries
> 5,
Symmetry
AU with n
Designation
D e , Sa, Cm. Dlh
Nonzero Ligsnd position FunetioNl
!D,
7
?.
Dm, D.0. Du, G.,
Dm. D.o
Ed,'
Ed,Z
CI,ah,&
+ 2s
1
-'D.
1 + -Du 168
- !D.
+ -D.. 168
7
'0. 7
-0.0
+ -0, 2s 1
Dl, Ck,D I , Dw, (Oh. Tdb
Dm, D.0. D.,
LD, + 2s
7
1 -D,,
G,DI. C.8, D S , Sa
D s , Dm, Da, G o
+'"
1 D.o
Dz,
c%,D2h
Dm,DI?,D.o, D.. Du 5 + 4s-Du
1 48D.~
1 + 16s
-+Il -Dm o
46
Edzu
7
1
+ c,st/,
*%~"2
168
- I-o, 7
1 + 168 -0.0
--35 DM 168
Ed,.
Ed,.
1 -Do 42
'D, 14
1
1 I14D ,- -D. 42
'D,
-
LD.
- -DM
14
14
42
+ D143 9 " 5 -Dao + -D.1 42 42
1
1
- -0,
-DM
42
14
14
42
1 -Dl# 14
For de ions change s i p of all D t d and C h i .
-
LSetD?o = 0 and D.' = D40/7. "62 - " 3 = Ed., snd Edi. EdY'.
606
/
Journal of Chemical Education
3 14
- - Dn 5 ID,,- 42 -D..
42
(63, although the impurity ion problem is usually treated as a C ~ Vcase. It is an interesting problem to calculate how much distortion has to exist before the difference in symmetry is quantitatively distinguishable. The substitutional impurity, a Cu2+ (d9) ion in a six-coordinate site, will have three oxygen ions located a t @ = 0, 2 ~ 1 3 4, ~ 1 3 , and 0 = cos-I 1 / 4 3 and three oxygen ions located a t @ II. = r / 3 , r , 5 a / 3 , and 0 = r - c o s l 1 / 4 3 , We have assumed that the impurity ion is located midway between the two oxygen planes and that the only distortion is a rotation of one plane with respect to the other of an amount $. The condition II. = 0 represents the pure octahedral configuration with a double degenerate ground state a t -pq and a triply degenerate excited state a t 21 3p,, wherepa = 6Dq. Looking a t CQin Table 4, we have the following contributions D2"= 0 D,,= 5 6 1 3 ~ ~ Figure 2 . Energy levels of a d ' ion in Ddn and Dm symmetry.
along the y-axis. Designating terms from the y-axis ligands with primes, we will first assume that p = p ~ / p a= pz'lpa'. We will also call a = pz'lpz = p4'/p4 Using Figure 1 the cluster is designated as belonging to point group D m ,and the nonzero ligand position functions are determined from Table 4. Substituting the ligand positions and the above definitions in Table 2 we find
D,,= 6(l+a)p,
The exercise that might arise at this point is that of finding the maximum distortion such that the octahedral approximation would not cause an error of more than 2% in the magnitude of the crystal field splitting.
This can be readily solved to show that II. 5 8". In corundum this distortion is actually known to be on the order of onlv 2". In realitv. the nosition of the impurity . . ion betwwn the two oxvgen groups gives rise to a more aerious distortion, and presents a srill more difficult exercise in the use of the closedform solutions Summary
If a = 1,then the ligands are identical and
for all values of p, as should be the case for Du, symmetry. Figure 2 illustrates the dependence of the d-orbital splittings on the value of p for a = 1 on the left and a = 1.5 on the right. The orbitals labeled d(z2) and d(xZ - y Z ) are of course not pure in the case a = 1.5, hut are designated in the same fashion as in the Du, diagram. The important distinctions between these two particular cases are the breaking of the degeneracy of the d,, and d,, orbitals and the general increase in the relative level separations. Also, whereas d(z2) is a possible ground state for large p in Du, symmetry, it becomes a less likely candidate for ground state for large p in D m symmetry. This analysis might profitably be extended to determine the d-orbital splittings and crystal field stabilization during the entire course of a ligand substitution reaction. A second application, one of direct concern to this author, is the effect of distortions on transition metal ions in solids or on solid surfaces. In corundum, A1203, it has long been known that the symmetry a t the A13+ site is only C3
It has been demonstrated that closed form solutions for the energy levels of d l and d9 electronic systems can be written for almost every chemically significant symmetry group. The excluded groups are of such low symmetry as to preclude meaningful analysis under almost any circumstances. The student can follow a simple, straightforward analysis of the symmetry elements of a transition metal cluster to assign the proper symmetry classification. No detailed understandine.~of m o w. theoretical principles is necessary since the analytic solutions already display confirmrational degeneracies. As an instructional tool, this treatment could be extended to any @ system where the strong field approximation is valid, and the energy of each configuration could he obtained by summing the one-electron energies. Literature Cited (1) Ktishnamurthy, R., and Sehaap.
W. 8.. J. CHEM. EDUC.. 46, 799 11968); J. CHEM EDUC.. 41.433l19701. (2) Companion. A. L..andKomsrynsky,M. A , J.CHEM.EDUC..11.257 (19611. 131 Sneer. 5. T.. Jr., Perumaroddi. J. R., and Adamson. A. Mi.. J. Phys (Bern.. 72. ,*?",,a,%S,
(4) Wasron.J. R.. andStoklosa. H. J.. J.CHEM.EDUC.. 50. 18611973). (51 Carton. F. A.. "Chemical Applicationr o l Group Them".'' Wiloy-Inforscience. New
Yurk. 1963. (61 Weakliem. H. A , and McClure, D.
S..J. Appi
S . J . Chem Phys.. 36. 2757 (1M21.
Phys.. 33.347s 119621: McClure. D.
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