R. Krishnamurthv and Ward B. Schaapl lnd~anoUn~vers~ty
Bloornington, 47401
1
I
Computing Ligand Field Potentials and Relative Energies of d Orbitals Theory
In our previous paper on this subject ( I ) , a simple, general approach is presented for calculating the relative energies of the d orbitals in ligand fields of a variety of geometrical configurations. This approach is based on the principle of additivity of the electrostatic crystal field perturbations generated by an assembly of ligands acting as point charges. Three simple primary arrangements of ligands, ML (axial), AIL2 (planar), and NILp (non-planar), are defined and used as building blocks in the construction of more complex geometrical structures. Similarly, the Dq values of the d orbitals associated with the ligand fields of the primary groups are additively combined to give the Dq values due to the fields of the larger structures. I n this paper we present the theoretical basis for this approach in as clear and explicit a manner as possible. First a general expression for the electrostatic potential is derived in terms of the charge and positions of the ligands surrounding the central metal ion. Next, the procedure for evaluating the matrix elements resulting from the action of the ligand field potential on the real d orbitals is considered. Wc then illustrate how the potentials of the primary groups are derived and evaluate the d orbital matrix elements in Dq units in the fields of the three primary ligand groups. Finally, the simple addition of both the potentials and the Dq values of the d orbitals associated with the primary groups to give values appropriate to more complex structures is discussed and illustrated. The mathematical procedures involved in these calculations are easy to follow and to carry out, yet the results obtained for large, complex ligand arrangements are rigorously correct. For convenience in making calculations, all useful quantities such as the d orbital basis set, expansions of products of d orbital combinations, potentials of the primary groups, and relative energies of each of the d orbitals expressed in terms of radial parameters and in Dq units are presented in tabular form. General Theory Expression for Potential Field Perturbation
If ligand or crystal field effects are treated as a perturbation on the wave functions and energy levels of the free ion, the problem reduces to finding the Hamiltonian to describe the perturbation and the corresponding matrix elements. The wave functions and their energy levels in the crystal field can then be found by standard perturbation theory. The determination of the energy levels requires the solution of the Schroedinger equation
Figure 1. Diagrmm showing generalized locotionr of on electron lr,%l ond a ligond lRj.Ej,+jJ with respect to the central met01 ion located at the origin of the coordinate axes.
where H=HO+Ht
and
Ho is the usual one-electron, spin-independent Hamiltonian operator representing the kinetic and potential energies of interaction of the free ion and H' denotes the perturbation arising from the electrostatic potential due to the ions of the entire crystal lattice or due to the ligands. If the simple point charge ionic model is used, the determination of the perturbing crystal field Hamiltonian H' involves primarily the evaluation of the electrostatic potential V appropriate to the symmetry of the ligand arrangement around the metal ion in question. The contributions to the energies of the electronic orbitals are given by integrals of the type H',, = ($,IVj$,), where the $ functions (basis set) describe the unperturbed degenerate electronic level of the free ion. This work was supported in part by the U.S. Atomic Energy Commission under Contract AT(11-1)-256 (Document. No. COO256-94). Presented in part before the Division of Inorganic Chemistry a t the 155th National Meeting of the American Society, San Francisco, Calif., April 1968, and before the XIth International Conference on Coordination Chemistry, Naifa, Israel, September, 1968. ' Author to whom correspondence may be addressed. Volume 47, Number 6, June 1970
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Although it is possible to calculate the crystal field potential in cartesian coordinates, it is more convenient to express this potential directly as a power series in spherical harmonics (2). Using this approach, we can derive a general formula for the potential in terms of the charge and position of the ligands giving rise to the field. Because the expressions for the d orbital wave functions are also conveniently expressed in spherical harmonics, subsequent integrations involved in the evaluation of the matrix elements are greatly facilitated. Figure 1 is a schematic diagram showing the arbitrary locations in spherical polar coordinates of a single ligand of negative charge q, a t the point (Rj,B,,+,) and of an electron of charge e at the point (r,9,+) with respect to the metal ion a t the coordinate origin. The potential V , at the point (r,B,+) due to the charge q, at (R,,9j,$j)is given by the relation
where R, and r are the radial vectors whose angular positions are specified by (B,,+,) and ( w ) , respectively, and lRj - rl is the distance between the ligand and the electron. If o is the angle between the two vectors R , and r, the portion of the potential function, I
,lRd ,-
rl'
may be expanded about any chosen origin in
terms df Legendre polynomials Pk(cos w ) centered a t that origin, where cos w = cos 9, cos 0 sin 9, sin 9 cos ( 4 , - +), i.e.
+
real representation of the potential. The real and complex harmonics (4) are related according to the equations z,.= Y,*
where a 1 0 and the superscripts cos and sin refer to cosine and sine combinations which will be described later. Using the general symbol Zk, to denote any of the expressions in eqn. (5), we can express eqn. (4) in the following form
where the terms Z,(0,,4,) represent a set of numbers evaluated for the angles (B,,+,), which describe the location of the jth ligand, and Zk,(B,4) describes the angular position of the electron of the metal ion. All of the fourteen possible second- and fourth-order normalized harmonics, ZbCos and Z,e'n, expressed as functions of the angles 9 and 4 , are given in Table 1. This table is useful in evaluating Zk,(B,,+,) terms with respect to the d orbitals. Table 1.
Normalized Harmonics, Zn,, as Functions of the Angles B and @
4. ( d d Z
(ZP.)"
5. ( d d a
(ZS,~,")~
15. ( L ) ( d u J )
Expansion of harmonic product in terms of 2&,Zxo
1
; ;44
44,
-44;
7
[-4 7i ~ Z ~ p i , , + 7~ G z , ~ . j . ]
Expansions 1-5 are used in the evaluation of the angular integrals associated with onlv the diagonal matrix elements of eqn. (11). Expansions 6 1 5 are those associated with the non-diagonal elemenls.
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Journal of Chemical Education
(3) The radial integral C ( = (pk(r)))is equal to the average value of the kth power of the electronic radius r, multiplied by the factor ( q / R k + ' )and is treated as nn empirical parameter in this paper. By combining step (3) with steps (1) and (2), the matrix elements H'ij of eqn. (12) can he written in the following. simplified form H'ij =
Each one of the fifteen) possihle matrix elements of the 5 X 5 secular determinant (eqo. (11)) cau be expressed in t,erms of a sum of second- and fourth-order radial parameters, (pdr)) and (pr(r)),the coefiicient,~of t,he sum being the prodnets of Part A (ligand field coefficients) and part B (valne8 of the expansion The product? of the coefficients of part A and coefficients &). part B have non-zero vah~esfor only those t,erms in each part which contain the same Zka term. The determinantal equation (eqn. (11)) is then solved for the five roots of the perturbation leaving the rt~dialintegrals as empirical parameters. energy, Et,
The evaluation of the H',, matrix elements is illustrated and discussed in detail for the cases of the three primary ligand groups in the next section. Application of Theory lo Primary Groups
We will now consider the simpler crystal field environments around the metal ion in the three primary geometries ML, ML2 and ML4 discussed previously. The ligand arrangements in these primary groups have been illustrated in Figure 3 of our previous paper ( I ) . The coordinates of the ligand positions, with respect to the metal ion, are shown in Table 3. Note the choice Table 3. The Coordinates of the Ligand Positions in the Three Primary Groups with Respect to the Metal Ion
Primary Ligand group Lj
ligand distance Ri
..
anglefrom +Z axis)
+j
(asimuthal anplef~.om+ X axis)
By substituting these values into eqn. (9), the effective crystal field potential V1 takes the final form
I n eqn. (16), as well as in subsequent expressions for crystal field potentials throughout this paper, the spherically symmetrical Zwterm is omitted. This term describes a spherically symmetrical potential which causes an equal and uniform shift in energy of all the d orbitals and does not contribute to the splitting of the orbitals. Because we are interested here only in the velative separations of these energy - levels, the Zooterm will be ignored. If the potential V1 of the first primary group is used as a crvstal field ~erturhationoDerator over the basis set of real d orbitals given previously, the one-electron matrix elements corresponding to the potential field of the first primary group can be evaluated in term of the radial parameters ( a ( r ) ) and ( p ) ) To illustrate, we shall compute the diagonal matrix element which represents the action of the H'II (= ($I~VI~$I)) potential V1 on the d,. orbital (Zto) of the basis set. The appropriate expressions for V1 (eqn. (16)) and the expansion coefficients for the d,, (Z,,) orbital are substituted into the general expression for the matrix element H',, (eqn. (15)) to give
I n order to complete the computation of the matrix element H'u, we refer to Table 2 for the values of the coefficients &Oand pnacorresponding to the expansion of the (d,S2 orbital combination. Ignoring the spherically symmetrical Zoo term, this expansion is
By substituting the values of 8.
(=&r) of two alternate orientations shown for the third group, ML4. I n one orientation (A), the projections of the staggered groups are fmed a t angles of 45' between the * X and Y axes in the X-Y plane, while in the other (B), the projections fall on the axes. The orientations A and B thus differ by a rotation of 45" about the Z axis. First Primary Group, Ml (ligond on +Z Axis)
I n the case of the first primary group with its single axial ligand, if the coordinates of the ligand given in Table 3 are substituted into the Zk,(R,,+,) functions listed in T a u e 1, it can be shown that the non-vanishing Z&,+,) terms occurring in the expansion of the expression for the potential are only Z20(8,,+~)and Za (8,,+,), all other terms being equal to zero. The values of these terms are
(=7)2d &4r. and pna
into e q n (17), the matrix element H'II
turns out to he
Following precisely the same procedure, we can compute the remaining four diagonal matrix elements of and HrSS,in terms the potential VI, i.e., HIz2,Hfa3, of the same radial parameters, remembering again that it is necessary only to multiply the values of the coefficients preceding the 2 2 0 and Z40 harmonic terms of the potential by the values of 020 and 040 coefficients associated with the same harmonic Z, terms occurring in the expansions of (Z~Z"""~, (ZZZ"")~, (Z2BOP)% and (Z2Pin)z (see Table 2). All other combinations can be ignored. This simplification is the result of the orthogonality and normalization conditions mentioned previously. The solutions are easily derived and are as follows Volume 47, Number 6, June 1970
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437
Because only the second- and fourth-order harmonic terms are needed to calculate the d orbital splittings, the matrix elements of the potential Vl within the d manifold are functions of only the two parameters, MT)) and (pdr)). The perturbation operator V1 gives rise to only diagonal matrix elements. Each one of the remaining ten possible non-diagonal matrix elements H',, (i Z j) in the field of the first primary ligand gronp becomes equal to zero. Because the non-diagonal elements are eliminated, the need for setting up and separately solving the "blocked-out" or smaller (e.g., 2 X 2 or 3 X 3) factored secular determinants along the diagonal does not arise. The entire determinant of order 5 X 5 (eqn. (11)) thus reduces to the simple case of a set of five diagonal 1 X 1 determinants, IH',, - E,I = 0, in which i = j. Consequently, the values of the diagonal terms HI,< (i = 1,2, . . ., 5) become directly equal to the five roots E, of eqn. (11). It becomes possible therefore to associate uniquely the relative energies of the d orbitals represented by the diagonal matrix elements HIn, . . ., HIS6in a one-to-one manner with the real d wave functions of our basis set, i.e.
A further discussion of the conditions necessary for the appearance of only diagonal matrix elements is given below in connection with the second primary group. The evaluation of the matrix elements in terms of Dq values will also be considered in the next section. Second Primary Group,
MLz (Ligands on + X and +Y Axes)
If we substitute the coordinates (BI,+I) and (OZ,+Z) given in Table 3 for the ligands LI and L of the second primary group into the Z,(e,,+,) functions of Table 1, the only non-zero Z,(O,,+,) terms remaining are ZZO (8,,+,), Z4~(Oj,+,)and Z4;0.(8,,+,). The valnes of these three terms are
After substituting these values in eqn. (9) the expression for the effective crystal field potential Vafor M L becomes
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The potential VZfor the second primary group differs from that for the first primary gronp in the values of the coefficientsof the terms and in that it contains an additional fourth-order harmonic term Zd4-. Because the Z 4 P term appears only in the expansions of the d+,2 and d, orbitals (Table 2), the appearance of this additional term affects only the relative energies of these two orbitals. The matrix elements corresponding to the effect of the crystal field perturbation V2 on the d orbital basis set are obtained by the same procedure described above for VI. The only non-vanishing terms in each matrix element are those which are products of terms in V2 and in the orbital expansions (Table 2) which contain identical Z, harmonic terms. As before, only secondand fourth-order harmonic terms occur in the potential field, so that the results can he expressed in terms of only two parameters (pZ(r)) and (p4(r)). The resulting matrix elements arising from each term of the potential V2 acting on each d orbital are given in Table 4 together with the corresponding results from V1. I n the case of Vz, it should he noted that only the diagonal matrix elements have non-zero values and that it is again possible to associate the values of the five roots of the determinantal equation in a one-to-one manner with the relative energies of the five d orbitals of the tetragonal basis set. This simplification arises whenever (1) the harmonic terms Z,, occurring in the ligand field potential are limited to Zzo, Z 4 ~and/or , Z4?, and (2) the tetragonally oriented real d orbital basis set defined earlier is used. It can be seen in Table 2 that the first five expansions, (Z,o)z, (Z,,m"2, (Z22"'")Z, (Z2;w)2,and (Z2r'")2, which correspond to our d orbital basis set h(d,S, $2(d214, 4b(dZy), h(dZ2), $2(d,,), respectively, are the only expansions which contain the ZZO,Z40, and/or Z4b"" terms. These terms give rise to non-zero values of the coefficients of the angular integrals, 620,PN and/or PI$"",which contribute only to the diagonal elements whenever the potential field also contains the same harmonic terms acting as operators. If the ligand arrangements and their corresponding ligand field potentials are restricted to those which contain only 2 2 0 , ZN, and/or Z4;Od terms and no others, then all the non-diagonal matrix elements will he equal to zero, and only the diagonal terms need be considered. I n other words, the real d orbitals of our basis set do not mix or form new combinations within the basis set under such ligand field potentials. For these restricted potentials, we can deduce the following relationships among the diagonal matrix elements of eqn. (11). 1. Because the magnitude and sign of the coefficients preceding the harmonic terms ZZ0and Zlo are identical in the expansions of both (Z2;")2 and (Z2r'")2,the matrix elements HI44and HIgi of the potential are equal to each other. This equality implies that the corresponding d orbitals J.4(d,) and h(d,,) are always degenerate in these fields, which they are in the cases of the first two primary groups. 2. I n the expansions of both (Z2z'"')2and (Z22'in)2, the magnitude and sign of the coefficients of the Z30 and Za terms are identical. If the harmonic terms of the potential contain only the Z,, and Zloterms, the matrix elements HIz2and HIS3become equal to each other and thus constitute a doubly degenerate level described by h(d,.+) and fi3(d,,) orbitals, as in the case of the
axial ML group (see Table 4). The d orbital splitting falls in a 2,2,1-pattern in such fields. 3. The &;"" term is a differentiating term with respect to the d,.-,. and d,, orbitals and these orbitals represented by HIn and HIa3 become non-degenerate only when the Z4,"" term is present in the potential, as in the case of the second primary group ML. Note that if the differentiating term drops out from the potential, the d,.,. and d , orbitals become degenerate and their energies turn out to be equal to the average value of
HIzz and HIra, i.e., E(d,,,,)
=
E(d,,)
=
I - [Hrn 2
+
H',,] (See Table 3 ) . Evaluafion o f Matrix Elements in Terms o f Dq
The two independent one-electron parameters, (,a (r)) and ( P 4 ( r ) )are sufficient to completely specify the relative energies of all the d orbitals in the fields of the primary groups ML and ML2. Of these two radial integrals, the fourth-order (p4(r))parameter is related to the familiar Dq parameter of the octahedral poten-
tial field by the equation Dq = 1/6(pn(r)). The second-order (,a(r)) parameter, on the other hand, has no exact equivalent in cubic symmetry. For calculations within the crystal field model, therefore, it is most advantageous to relate the (&)) parameter of non-cubic symmetry to the octahedral Dq parameter and to introduce the symbol p to denote the ratio of the radial parameters, i.e., p = (,a(r))/(p&)). Using these relationships, the relative energies of the d orbitals given in Table 4 can be expressed as linear functions of the parameters, p and Dq, as shown in Table 5. For values of p = 1 and p = 2, which are the most commonly used values of this parameter, the results obtained in terms of Dq units for the first two primary groups are given in the lower part of Table 5. As an example, the relative energy of the d,. o~bital in Dq units in the field of the second primary group M L is evaluated as follows (From Table 4 )
Table 4. d Orbital Matrix Elements (Expressed in Terms of the Radial Integrals)= Arising from the Individual Harmonic Terms of the Crystal Field Potentiolr of the First Two Primory Groups
Diagonal d orbit,al matrix elements
ML (first primary group) IZd
H'.
(dv~lVl~lvs)
+ 71 ( d r ) )
ML. (second primary group)
IZd
4 -5 (~4~1)
IZm)
-
IZd
i1 (P~(V))
IZ4.9
- 71 (P&D
In these mat1.i~element?, the terms (p2(r)) and (p,(r)) represent radial integrals and have the following definit,ions: ( m ( 7 ) ) = m .Pdr and (p,(r)) = ~q [ ~ , ~ ( r ) ] $ . r ~ d r ,w h e ~ . e R ~ ~is( the r ) radios of the d orbital wave function, e and r are t,he [Rnl(r)]2 ep elect,ronie c h e w and radius, respectively, q is the point charge of the ligand, and R is the metal-to-ligand bond distance. The radial integrals (pn(r))and (p4(r))can also be expressed in terms of I)q, Ds, and Dl pxrxmeteta hy the following relationships: 1 1 1 Dq = 6- (p,(r)), DI= 7- (p*(r)), and Dl = 21 - (pr(r))(Ref. (15)). The mtatrix elements of this tahle as writ,ten apply to the dl and highspin dLeleetloaie configurations. By rhanging the sipnu of bhe radial integrnlu, the above matrix elemends can also he used to describe the dg and the high spin d4 con@yrxt,ions.
SO
Table 5. The Relative Energies of the d Orbitals in the Fields of the First Two Primary Groups Expressed as a Function of p Itool and in Units of D a for D = 1 and D = 2 (bottom1
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Dividing through by (pdr)) and letting (~4(79)gives
p =
Substituting (pa@)) = 6 Dq and choosing the value for this parameter yields
(p,(r))/
p =
Table 1 and the (Bj,4,) co.ordinatesof Table 3, we have
2 as (In the Z4;"" term, the minus and plus signs apply to *
Up to the present time, most discussions of the relative stabilities of d orbitals in crystal fields have assumed that p = 2.0 (8, 9). Some recent experimental studies via optical absorption spectra, on the other band, indicate that p = 1.0 may be a better choice for tbis parameter in most cases (10-15). We have therefore included Dp values derived for the choices of both p = 1 and p = 2 in Table 5. The evaluation of tbis parameter will he discussed in more detail in a later section.
the arrangements A and B, respectively, because
C 4), can be derived from the second primary group NIL2. The expressions for the potentials of these planar polygons include only the Z, and ZPDharmonic terms. The term Z 4 Y is eliminated in their potentials because of the loss of the four-fold axis of symmetry, the C4 axis. This loss occurs because it is not possible to have a value of a less than the value of n in the term Z,,""", where n is equal to the order of the principal rotatioual axis of symmetry. The potential for the planar trigonal structure ML3 (DgO)also contains only the same harmonic terms. For these planar structures, the overall potentials can therefor2 be derived from the primary poten-
tial Vz by dro;)ping the Z 4 Y term from Vz and then multiplying by the factor ( 4 2 ) to account for the change in coordinttion number from two in MLz to n in ML,. I n its general form, the potential derived from V2 for regular plmar polygonal structures ML, in the X-Y plane is given by the expression
The related mono- and bipyramidal structures can be built up from the planar polygons by using the first primary group ML to add one or two extra ligands along the Z axis of the polygon. Therefore, by summing the corresponding potentials for planar AIL, structures and those of the ML groups, the expressions for the potentials of the mono- and bipyramidal structures, designated ML,+I and ML,+z respectively, can be obtained. The general expressions for the crystal field potentials of polygon-based monopyramidal l\IL,+l (D,,) and for the polygon-based bipyramidal NIL,+% (Dnh) structures are given by
By following analogous procedures, the relative energies of each of the d orbitals in fields of planar polygonal geometries (ML,) can be obtained by multiplying the Dq values of the orbitals associated with the ML, primary group (Table 5) by the factor n/2 and then and d,, orbitals in order averaging the energies of d., to make them isoenergetic and degenerate. Furthermore, by summing the planar polygonal Dq values and the Dq values of the orbitals in the field of the axial primary group IML (Table 5), the relative d-orbital energies in crystal fields of monoand bipyramidal (ML.+,) structures are obtained. The general expressions for the d-orbital energies as functions of the parameter p and the coordination number n for these structures are listed in Tablc 9. 3. Pofenfiols and Dq Volues Derivable from the Third (General) Primory Group ML4
The third primary group can be used to build up complex geometrical structures related to polyhedra,
Table 9. Relative d-Orbital Energies Expressed in Terms of the Parameter p and the Coordination Number n in Crystal Fields of Plonar Polvaonol and Mono- and Bi~vramidolConfiaurations
.."
Relative enerev in Do unib in field of-
d Orbital
Plane polygon ML. (Dm*) ( n = 3, >4)
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Monopyramid ML.+I (C..)
Ripyramid
ML.+a ( D n d
such as the cube, square antiprism (twisted cube), pentagonal antiprism, icosahedron, and trigonal prism. For these applications, appropriate values must be chosen for the polar angle p in the general expression for Vg (eqn. (19)). The cubic and square antiprismatic structures of coordination number eight require that p be equal to one half the tetrahedral angle, i.e., p = 54'44' and cos p = ( 1 3 For this angle, the ZN term drops out of t,he expression for V8. If we choose orientation A (Table 3), for which the sign of the Z4$0.term is negative, the potential terms are identical to those for a regular tetrahedron. Multiplying the tetrahedral potential by two gives the crystal field potential for eight ligands arranged as a cube (ML,). Correspondingly, the Dq value of each d orbital in a regular cubic field, is just twice its value in a tetrahedral field. A square antiprism can be generated by rotating either the top or bottom half of a cube by 45' about the Z axis while maintaining p equal to one half the tetrahedral angle and keeping the metal-to-ligand bond distances unaltered. Rotation of four of the eight ligands by 45' gives, in essence, a total of four ligands with projections between the * X and + Y axes (negative Z44"0' term) and four ligands which project on the X and Y axes (positive Z44coB term). The net effect is that and d,, the Z 4 pterm cancels, out, causing the d., orbitals to become degenerate. The remainder of the expression, which contains only the Z , term, is the potential for the square antiprismatic ML, structure. The Dq values of the d,>, d,,, and d,, orbitals are the same as they are in the cubic (ML,) structure, but because the differentiating Z44m'term drops out it is necessary to average the Dq values of the d,*-,% and d,, orbitals calculated for MLs. The average Dq value is assigned to each of these orbitals in order to make them consistent with their symmetry representations ( e ) in the twisted cubic field (D4,). By combining orientations A and B of the third primary group, it is possible to build up more complex structures related to the symmetry of a cube. For example, the potential field expression for a structure of coordination number twelve in which the ligands are arranged a t the centers of the edges of a regular octahedron can be derived. If p = 45' in V g (orientation B) and if Vois multiplied by two, the potential due to the field of eight ligandri, four above and four below the XY plane and located at the centers of the edges of the octahedron, is obtained. If p = 90' in V g (orientation A), the potential due to the four ligands a t the centers of the edges a t an angle of 45' between the *X and i Y axes in the XY plane is obtained. Summing these two potentials gives the expression for the potential applicable to the 12-coordinate structure being considered
It can be seen that this potential is equal to -(1/2) of that of the regular octahedral potential. This relationship implies that the splitting of the d orbitals due to the field of twelve ligands in the ML12 complex is equal to one-half the octahedral splitting, but with in-
version of the tz, and e, levels of the octahedral field, providing that the metal-to-ligand bond di~tancesare the same in both structures. From the third primary group, it is also possible to derive all the regular polygonal prismatic structures of the formula NIL, in both the eclipsed (Dnn) and staggered (Dnd) orientations if the principal axis of symmetry is greater than four-fold. The potentials of these structures contain only the ZZoand Zlo harmonic terms. The Z,4"a term is not allowed in these potentials just as in the case of planar polygonal fields with n > 4. The potential of an eclipsed trigonal prism of Da, symmetry also contains only these same harmonic terms. By using the primary potential V g (eqn. (19)) term from it, we can derive a and dropping the Z4$OS general expression for the potential fields of regular polygonal prismatic structures ML,, where n is the total number of ligands.
The factor ( 4 4 ) is a multiplying factor to account for the change in coordination number from four in ML4 (third primary group) to n in the polygonal prism ML,. Using analogous procedures, the relative energies of the d orbitals in the ligand fields of the polygonal prisms represented by the above potential may be derived from the corresponding energies of the orbitals in the field of the third primary group (see Tables 7 and 8). Because the differentiating Z4rm' harmonic term is ahsent in these potentials, it is necessary to average the energies of the d,,-,, and d,, orbitals given in Table 7 or Table 8 in order to make them degenerate and consistent with their identical symmetry representations in polygonal prismatic fields. The Dq values for the prismatic structures listed in Table 3 of our previous paper (1) were obtained by this approach. One particularly interesting structure which is closely related to the pentagonal antiprism is the regular icosahedron, in which all five d orbitals are degenerate (h, symmetry) (24) and no splitting should be observed. This structure can be considered to consist of a pentagonal antiprism with the addition of two apical groups along the Z axis. Using eqn. (24) with n = 10 and p = 62"28' (one-half the icosahedral angle) and adding twice the potential field of one ligand on the Z axis (first primary group), we obtain the potential field due to twelve ligands in a regular icosahedral array, i.e.
where
Comparison of the potential of the pentagonal antiprism l\ILlogiven above with the potential of the first primary group ML (eqn. (16)) shows that VML,,is just equal to -2V1, so that the potential field of an icosahedral structure is zero, i.e., its potential has no second or fourth-order harmonic terms and only the spherically symmetrical Zoo term remains which does not give rise to splitting of the d orbitals. Volume 47, Number 6, June 1970
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445
Limitations
The limitations mentioned in Paper I apply also to this paper. The method described is a first-order perturbation treatment based on the point-charge electrostatic model. I t considers the action of the electrostatic field on only the d electrons and neglects possible interactions of the d electrons with s and p electrons of higher electronic levels. I t neglccts also the overlap of clcctron clouds, spin-orbit coupling and inter-electronic repulsion. The method is therefore applicable only to rl' and dnelectronic configurations and to weak field (high spin) d4 and d%omplexes. A further limitation arises from the choice and orientation of the primary groups which are used as building blocks in the construction of the larger, more complex structures. I n the three primary groups and in all of the structures built up from the primary groups, the only Z,, harmonic terms which appear in the potential field expressions are the Z2~, Z,O, and/or Z4p' terms, which give rise only to diagonal matrix elements if the tetragonally oriented real d orbitals (Fig. 1 of our previous paper ( 1 ) ) are used as the basis set: These tetragonally oriented orbitals do not mix or form new combinations in any of these potential fields, and it is possible to identify particular d orbitals of the usual basis set with the energies represented by the diagonal matrix elements and to combine their energies in an additive manner as the component primary groups are combined to build up the larger structures. Although this simple additive approach is applicable to most structures known for coordination compounds, it is not directly applicable in the cases of certain geometrical structures of low symmetry, such as those of C20,Ca0, D2&and Dl&symmetry, whose potentials contain Z,,
446 / Journal of Chemical Education
terms other than those listed above and therefore give rise to off-diagonal matrix elements. Acknowledgment
We wish to thank Prof. J. R. Perumareddi, Department of Chemistry, Florida Atlantic University, Boca Raton, Florida, and Prof. J. S. Griffith of Indiana University, Bloomington, Indiana for their helpful and stimulating discussions. Literature Cited (1) KRISHNAXURTHI, R.,
\."-",.
&ND
SCIIAAP, W. B., J. CHEM.EDUC.,46, 799
,lURO>
(2) Emriro, H.,WAGTER,J.. AND KIMBALL. G., "Quantum Chemistry," John Wilev & Sons. Inc., New York. 1944, A~pendixV, pp. 369-71. E. U..A N D SHORTLEI, G 11.. "The Theory of Atomic Spec. (3) CONDON. tra," Camhridee University Press, London and New York. 1953, ?hen+-- Y ~ T T
S.. "The T h w v of Transition Metal Ions." Cambridm "~ University Presa, London A d New York. 1961, p. 166: (5) PAULING. L.,A N D WILSON.E. B., "Intloduotion to Quhntum Meoha. nics." MeGrsw-Hill Book Company. Ine.. New York. 1935, pp.
(4) Gmmmlr. J.
.-.
.az-v" A"-
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