A. 1. Companion and M. A. Komorynskyl lllinois Institute of Technology
Chicago
II
crystal
The surge of interest in transition metal chemistry in recent years has aroused in many inorganic chemists a desire for education in the theory of crystallme fields. Witness the publication of four textbooks on the subject within the last two years (1-4) and the popularity of crystal field and ligand field articles with an educational slant (5-8). Most available treatments though are couched in the language of group theory, a tool not a t the fingertips of most chemists, and frequently obscure the fact that crystal field theory is simply an application of quantum mechanical perturbation theory, a topic discussed in most quantum chemistry courses or studied in introductory quantum mechanics texts. Group theory, though an extremely powerful tool for simplifying the problem, is not necessary for a fundamental understanding of the effects of crystal fields. Moreover, in problems where the perturbation of next-nearest neighbors of lower symmetry must be considered, application of group theory becomes quite complicated. I n this article we present a method for determining crystal field splitting patterns within the ionic model without the use of formal group theory. The formulas developed will he applicable to the dl configuration in fields containing any number of ligands in any type of geometrical arrangement. The results should be useful to chemists with very little formal training in quantum mechanics. Theory
Within the electrostatic crystal field approximation, the ligands, actingas pointcharges, perturb the d electron energy level (fivefold degenerate in the isolated transition metal ion), removing its degeneracy partially or wholly, depending upon the degree of symmetry of the ligand spatial arrangement. The extent and nature of this splitting may be determined through the applics, tion of first order perturbation theory for degenerate systems (Q), which states that if we can formulate the electrostatic perturbing potential due to the ligands, y, and if there is known a set of wave functions (a basis set) describing the unperturbed degenerate d electron level, then we may calculate energy corrections to this level, Ekby solving the determinantal equation, given in abbreviated form: If,, - S,,Exl
= 0
(1)
and
' At present a graduate student in the Department of Chem istry, Washington University, St. Louis, Missouri.
Fidd splitting Diagrams We have chosen the real d orbitals as our basic set, because many chemists are familiar with them and with qualitative crystal field arguments based on their shapes. I n terms of a normalized radial function for a transition metal 3d electron, Rad(r)and the normalized spherical harmonics Y , , (8, +) the real 3d wave functions are : $1
=
+,
= d,,
*r =
& 2 - ~
dd
+, = d,, J.6
=
d,.
=
+
R~~(l/d~)lYs~ Ym'l
~~~(i/d\/2)[~~, + y2,*i R8.4Yl0 = R3&i/d\/2)[Y>,- Y .,*I = R8&i/d?)lY2, - Ym*l =
(4)
=
Since the real d orbitals are an orthonormal set, the integrals S,, will be 1 when p = q and 0 otherwise. Thus our determinautal equation (equation (1)) will be:
The problem thus reduces to evaluation of the integrals H,, and solution of equation (5) for the five roots, E,. I n group theoretical treatments one searches for a basis set of wave functions (always expressahle as linear combinations of members of our set) which, for the particular symmetry of the chemical system considered, minimizes the number of nonzero elements H,, in equation (5), and consequently reduces it from a 5 by 5 problem to a set of smaller ones. I11 our approach we will always use the same basis set, the real d orbitals, regardless of the symmetry of the system, and, if necessary, will solve equation ( 5 ) by numerical methods. There are now over 10,000 electronic digital computers distributed around the United States ; secular equation soluers, usually a part of their program repertoire, may be easily used to solve equations such as equation (5) for the energies E,. Many of these programs moreover yield the linear combination of unperturbed wave functions describing each energy level, i.e., the basis set which group theorists search for right a t the beginning. We will illustrate this point in a later section. The problem then is evaluation of integrals of type H , and the formation of the perturbing potential Y due to the ligands. If a perturbing ligand, numbered i, may he regarded as a point negative charge, then a d electron labeledj on Volume 47, Number 5, Moy 1964
/
257
the metal ion experiences a repulsive interaction in the vicinity of this ligand given by V6 = Ziea/r,;
(6)
where Z,e is the negative charge on the ligand, and r,] is the distance between ligand and the electron (Fig. 1). The potential function l/ri, is one frequently encountered in chemical physics, and can be easily expanded about some arbitrary origin in terms of spherical harmonics Y,, (8, +) centered a t that origin (10). Since the d electron is described by a wave function containing spherical harmonics centered a t the metal nucleus, this same origin is a convenient one for the expansion of the potential. The formal expansion of the potential takes the form:
(7)
trate the origin of Tables 1 and 2. Consider the integral H33,which from the definition of fi3 (equation (4)) becomes:
Substituting the expressions for Y,, given by Pauling and Wilson (11) in equation (lo), one can show by brute force integration that the only Y,, which give nonzero contributions to H33are YoO,Y20,and Ylo and that these contribute 1/24?, 7 and 3/74;, respectively. Table 1.
The Ligand Position Functions Dz,"nd
Since the Y,, (O,, 4;) describe all angular positions of the d electrons, i.e., are identical to the angular portion of the d electron wave functions, we may henceforth drop the subscript j. Eventually the Y,,* (&. 4;) will be numbers evaluated from the known 0 and .$ of the ligand i. The two radial distances entering the expansion, r< and r>, are the shorter and the longer of the radial vectors connecting the origin to the electron and to the ligand. Although the ligand distance is fixed, the electron radial distance varies from 0 to a, and as a consequence the symbol r, may refer sometimes to the ligand, sometimes to the electron. Figure 1 illustrates the latter case.
GI,"
sin +i
Table 2.
The Integrals H,. in Terms of Dz, and GL,
HI, = Hs2 =
Har = H4r = Hs = H!. =
Figure 1.
Coordinate system for rnetol ion end point-charge ligonds.
The total perturbing potential contributions from all N ligands:
Y is the sum of such
Combining equations (2), (6), (7), and (a), we arrive a t the following expression for the integral H,,:
If we introduce the following abbreviation for the radial integral:
then equation (10) reduces to:
The three spherical harmonics involved have the form: Evaluation of this formidable-appearing expression is simplified by the fact that ford orbitals the only nonzero terms in the infinite series occur for 1 = 0,2, and 4. We will consider evaluation of a sample H,,to illus258
/
Journal o f Chemical Education
yw*(ei,+ i )
=
1/.j6
3 ei - 1) ~ , * ( e ; ,C )= ( d \ / j / 4 d \ / * ) (cosa I%'(&, +$)
=
( 9 / 1 6 , / \ / * ) ( ( 3 5 / 3c0s4 ) ai - 10 cne2 0;
(13)
+ 11
As we substitute these in equation (12) we find it convenient to define the ligand position functions of Tahle 1, namely Doe$ DD20i, and Dpoi. In terms of these, equation (12) for Ha% falls into the form given in Table 2, provided that we introduce the additional definition
All the D,,' and the G,,' listed in Tahle 1 and the H,, of Table 2 were derived in much the same manner. Each D,,' and G,,' is characteristic of one ligand and its coordinates, and the sums yielding D,, and G,, are taken over all ligands in the chemical system considered. Application of these formulas to specific examples is discussed in a later section. Theoretical evaluation of the radial integrals a,, as defined in equation (II), requires knowledge of good radial wave functions R,, for 3d electrons. Two sets of easily used self-consistent-field wave functions appear in the literature: Watson's eight-parameter HartreeFock functions (1.2) and the four parameter approximations to these proposed by Richardson, Nieuwpoort, Powell, and Edge11 (13). With either set we may calculate a, as:
since, procceding from the origin, we find that r, = r and r, = R, (the metal-ligand distance) from 0 to R, whereafter rc = R, and r , = r. If it is assumed that metal-ligand orbitals do not overlap appreciably, then the second integral in equation (15) is negligible, and a,varies inversely as RW1. All integrands involved in equation (15) are intrinsicly positive, and hence a, is positive when we are discussing d-electron-ligand repulsion. Crystal field calculations thus far employing theoretical a, have not been particularly successful. In general the integrals are too small to account for experimentally observed splitting (14). Even the ratio m/an predicted by the theoretical integrals does not seem always realistic; McClure (15), for example, favors use of empirically determined ratios. In our examples in later sections we employ the approximation LYZ = 3a4, one roughly compatible with the SCF ratios computed by Piper and Carlin (14) and the experimental results of Hougen, Leroi, and James (16). In practice the a, and also frequently their ratios are determined experimentally through study of optical absorption spectra, usually in the form of the parameters defined by group theorists, Dq, Ds, and Dt. In the next section we illustrate the relation between a&and Dq. The equations relating a s and apto DS and Dt are given by Piper and Carlin (14) for some particular systems. The Perfect Octahedral Field
The six ligands characteristic of this symmetry, assumed of charge Ze and equidistant from the origin, have the angular coordinates listed in Figure 2. Armed with these, we first evaluate the parameters D,, and G,,. Consider Ddo = 2 D40i (from equation (14)),
where i denotcs the ligands labeled A, B, C, D, E, and F. From Table 1 we have:
Since the ligands are equidistant from the origin, all a , h i t h the same 1 are equal and renamed simply a,. Thus D4c = 28/3 a d . Similarly, one obtains Doo = 6a0 and Da4= 4ar. All other D,, and G,, are zero. Hence from Table 2: H I , = 6ao
+
a 4
HW
=
6au - 2/3 ar
Hu
=
6aa
+
(17)
ar
UU = fieo - 213 a* Ifsi = 6ao - 2 / 3
a4
All other H,, (all off-diagonal terms of equation ( 5 ) ) are zero, and consequently the diagonal terms above are directly the roots E, to equation (5). Thus for this particular example the real d orbitals diagonalize the determinant, a fact which group theory would have told us immediately, with no need of examining off-diagonal H That these latter are zero may be easily seen from inspection of Tables 1 and 2. Since the offdiagonal H,, are zero there is no mixing of the real d orbitals and we may associate in a one-to-one manner the His of equation (17) with the corresponding $; of equation (4). Hz$, H+ and Has are equal and constitute a triply degenerate energy level described by $s, +4, and $s (d,,, d,,, and d,J, the tss orbitals of group theory. The equality of H,, and Has indicates a doubly degenerate level described by +I and h (d,.-,. and d,.), the e, orbitals.
L---1 in1
....!......... Ibl
(01
Figure 2. The perfect octahedral fleld: (ol d electron level d isolated metal ion; ib) the spherically symmetric perturbalion; (c) octahedral splitting componenh.
In general, the integrals a. are contained in the parameters Do,' which have no angular dependence on ligand pmition, i.e., are spherically symmetric. Their equal contribution to all E , means that all five orbitals are raised in energy in an octahedral field by the amount 6a0, with no removal of degeneracy. In combination with large attractive interactions between ligands and the positively charged metal nucleus, this repulsive energy is contained in the lattice energy of a crystal or the ligation energy of a metal complex. Superimposed on these larger energy effectsis the smaller splitting effect, determined by the on integrals, relative to the spherically symmetric perturbed level. The common Volume 41, Number 5, Moy 1964
/
259
octahedral splitting parameter A,,, is thus 5/3 aa The parameter equivalent to A,,,. 10 Dq, is related to ours by Dq
or from equation (11). . . . Dp = 1/6 Ze2
=
1/G ad,
(Rsd)a[r6]r?dr
(19)
All other D,,and G,, are zero. As before, only diagonal elements appear in the determinantal equation ( 5 ) , and we may identify the H,, directly with a set of two doubly degenerate levels and a nondegenerate one:
+ 2asL +
3/7 axB - 4/7 a 8
E,,
=
Hxl = 114* = 3ao"
E,z
=
H M = RaaS
+ 3/56 + 2/21 aaL ads
+ 2aoL-
+ 2/7 azL - 3/14 ars - 8 / 2 1 a r L
3/14
Eza-ua, sy
(20)
+ 3aP
+ 2wlL - 3/7 a? +
4 / 7 asL
(22)
If all the ligands in the original system are equidistant from the origin, such that a? = azS = a, in equation (21), we have: E,.,
D*o = 4apL- 3asS
=
+ 3/7 azS $ 3/56 as
One can see that the relative stability of d., and the d,,, d,, pair depsnds upon the relative magnitudes of a 2 and aP. For both a% = 4 a and ~ a. = 304.~.d... is most stable. The splitting diagrams in Figure 3 are drawn roughly to scale under the approximation az = 3aa.
+ 3aoS
EZlLyl= EZV= HI1 = H6$ = 3aoS
3w6
+
For this system the five ligands effecting the field are assumed identical and of charge Ze, with the coordinates enumerated in Figure 3. Metal-ligand distances along the z axis, R,, are assumed larger than the planar ones, R,. Accordingly, for a given 1 there will be two typ2s of radial integrals, designated a,' and a:. Computing the I),, and G,, as in the last section, we find:
D4"= 16/3 rrrL
=
E,,, ,,= 3aoS - 3/14 a 8 - 3/14 a? Ed = 3 d - 3/7 azS 9/28 aP
The Trigonal Bipyramidal Field
E,,
E Z * - ~ *zy,
(18)
SOm
Don= 2mL
limit in the levels characteristic of trigonal planar symmetry:
+ 25/168 + 1/14 us - 25/42 ad + 1/7 a? + 25/28 ar
E,? = 5-0
.,
E12-Y1, 2g
The relative stability of these levels will depend on the magnitudes of the integrals at. We consider some special cases. Since a , varies inversely as R'f ', als will be larger than elL. As the axial distances R, become infinitely large (corresponding to removal of ligands D and E), all aiL integrals will approach zero, resulting in this
a4
(23)
Clearly from the formulas d,. is least stable, and intuition leads us toward choice of the xz, yz pair as most stable. If a, = 3a4, this is indeed true. However, if a% should be even so large as 3.5 an, then the d,.-,., d,, pair becomes lowest in energy. If the three planar ligands are removed, a; integrals vanish, and we have the energy levels for a linear of twoligands:
(21)
+ 9 / 2 8 adB+ 4 / 7 arL
= 5ao - 1/7 as
,,= 5-0
E,,,
E,,
=
2aoL
+ 2/21 a,= + 2/7 uzL - 8/21 a 8
= 2 0 8 - 4/7 a*=
= 2aoL
+ 4/7 axL+ 4/7 a,"
(24)
with the splitting diagram shown in Figure 3. Note that in all of the symmetrical distortions discussed for trigonal bipyramidal fields,no further removal of degeneracy occurs. Fields of Other Symmetry
In Figures 4 and 5 we illustrate splitting diagrams easily computed from our formulas for cubic, twisted cubic, trigonal, and trigonalrelated fields. Figure 5 deserves more comment. A trigonal field may be derived from an octahedron of point charges by squeezing or stretching along an axis (labeled z in Fig. 5) which is perpendicular to the center of one face, with retention of metal-ligand distances. For compression the azimuthal angle, 8, is larger than arc cos 1/& (corresponding to a perfect octahedron): and for stretching, smaller. . ........ Figure 5 illustrates, under the approximation cuz= 3a4, changes in the -litting pattern decreased from 90°, in which ........ .................................................. I1. . . ............................................. . . . .............................................. case the six ligands form a IL, ,", , . . , ~ Y I IL. , IVI 5 ~ ) 101 IDI ICI hexagon in the xy plane, to 0", TRIGONAL PLANAR LINEAR FIELD TRIGONAL BIPYRAMIDAL where two groups, comprising Figure 3. Trigonol bipyramidal and related fields ( 0 ) d electron level of isolated metal ion; (bl spherically three ligands each, form a field (c) splitting component for the symmetries noted. ,-\
.L\
,.\
260 / Journal of Chernicol Education
,L\
of linear symmetry. The resulting splitting is 3 times that shown in Figure 3. For all angles but O0 and 90°, the elements HI?and H,, are nonzero, which signifies that the real d orbitals represented by J., and J.z,J.6 and $6 are "mixed" by the perturbation, and can no longer necessarily be identified in a one-to-one manner with energy levels. Only The extent of this mixing can J.3 (d,,) remains pure. be easily computed by perturbation methods (9). As an example, we consider the octahedral case of Figure 5, interesting in that it illustrates the arbitrariness of locating the cartesian axes; the splitting diagram in Figure 5 differs from that in Figure 2 only in the identification of orbitals with energy levels. The choice of axes in Figure 2 permits one-to-one identification of real d orbitals with energy levels. I n Figure 5 one of the
lower levels is associated with pure d,., a second, with a mixture of J.P and J.s,and a third with a mixture of J., and $2. TO compute the weighting factors for the latter we employ the results HI, = -1/9 or4, Hzz = +4/9 or4, and HI% = +.5&/9 a,computed with the formulas in Tables 1 and 2 for the octahedral case. These H,, form a small 2 X 2 determinantal equation with roots E' = a4,-2/3 a4. The coefficients C, and Cz describing the proper linear combination of J.Iand \CZ for the lower energy are found by substituting E = -2/3 a4 in either of the equations C,(H,, - E ) C1Hn
+ CzH,, = 0
+ C2(Hr2- E ) = 0
(25)
along with HII,Has,and H12values above. This yields the result: C, = - -\/ZCz. The requirement that the linear combination must he normalized means that CIS = 1, SO that C1 = - d 2 / 3 a n d C2 = 4%. Thus one linear combination of real.-d orbitals belongiug to E = -2/3 a,is ~ 1 / 3d,. -.\/2/3 d,l-,r. All other wave functions may be computed similarly.
+
Conclusion
TWiSTED CUBIC Figure 4. Cubic and twi3ted cvbic fields. All ligandr are equidistant from the origin. la) electron level of i d a t e d metal ion; Ib) spherically rymmetric perturbotion; Ic) and Ic'l rpiitlina som~anentsfor cvbic and twisted
We hope that in the preceding sections we have brought to those not inclined to approach the subject through group theory some inkling of the origin of common ionic model splitting diagrams and a working means of constructing less common ones. For practice we recommend study of square planar and tetrahedral symmetry splitting diagrams [illustrated correctly by Orgel ( I ) ] and verification of the rule that the splitting due to a tetrahedron of point charges is 4/9 of that of an octahedron of charges for equivalent metal-ligand distances, all easily done with our formulas. We do not intend to belittle the power of group theory in simplifying crystal field problems, but rather hope that after studying our simplified presentation readers may be encouraged to examine the more advanced treatments. Acknowledgment
The authors wish to thank the donors of the ACS Petroleum Research Fund for partial support of this work, the NSF for its support of one of us (M. A. I
