1693
initio calculations lead to 5x states at 60" with energies about 3 eV above the comparable states at 117", the INDO calculations lead to 5x states at 60" with energies about 0 to 1 eV below the comparable states at 117". 111. Discussion
The most grievious fault of INDO apparent in our calculations on ozone is the strong bias toward closed g e o m e t r i e ~ ,even ~ ~ when unfavorable electron interactions should make small bond angles strictly repulsive, e.g., for the lA1(5x), 3A2(5a), 'B1(5x), 3B1(5x), and 3B2(6x) states. Similar problems with INDO have been found p r e v i o ~ s l y . ~It~ appears as if INDO does not properly represent the repulsion involved (50) Subsequent to submission of this paper, A. K. Q. Siu and E. F. Hayes, Chem. Phys. Lett., 21, 573 (1973). published semiempirical H F calculations on the open (1 'AI) and ring (2lA,) states of ozone, in which they reported that the CNDO/2, INDO, and M I N D 0 approximations all favored the ring state by 5 to 10 eV over the open state, in agreement with our results. Siu and Hayes also reported ab initio Hartree-Fock calculations, leading to the ring state about 0.36 eV above the open ground state. However, as shown earlier from ab initio GVB and CI calculations, H F wave functions (which exclude electron correlations) are biased in favor of the ring state by 1 eV or more, so that the ab initio relative energy (0.39 eV) of the ring and open states obtained by Siu and Hayes is much smaller than the real spacing between these states. Extensive DZ-CI calculationsec indicate that the ring state is 1.57 eV above the open ground state. (51) M. Froimowitz and P. J. Gans, J. Amer. Chem. Soc., 94, 8020 (1972); see also T. Morton, Ph.D. Thesis, California Institute of Technology, 1972.
when triplet-coupled electrons are forced into close proximity. In addition it appears that INDO gives rise to x bonds that are far too strong. The latter explanation would be consistent with the short bond lengths observed for lA1(4x) and the large transition energies observed for the 4 n + 5x and 4 x + 6n transitions (at the calculated equilibrium geometry). Consequently, the use of INDO for calculating equilibrium geometries as in conformational studies or reaction pathways is very risky, even if correlation effects are included. Our calculations show that INDO treats the electron correlations involved in the GVB(1) wave function fairly well, so that using such correlated wave functions will cure some of the gross errors encountered when INDO is used with the H F method. However, introduction of CI need not improve the energy spectrum obtained with INDO and, in fact, may make it worse. (See, for example, the reordering of the 57r and 67r states after CI.) Nevertheless, despite certain significant errors in describing the overall energy spectrum, INDO does reproduce many of the energy separations properly, e.g., the singlet-triplet splittings of states arising from the same configurations. This indicates to us that it may be possible to develop a method on the order of INDO in complexity that would yield reliable results (comparable at least to ab initio MBS calculations). Work is in progress along these lines.
Symmetry Adapted Functions and Normalized Spherical Harmonic (NSH) Hamiltonians for the Point Groups o h y T d J D 4 h J D 2 d J C4,y D z h y C,, J. C. Hempel, J. C. Donini, B. R. Hollebone, and A. B. P. Lever* Contribution from the Department of Chemistry, York University, Downsview, Ontario, Canada. Received September 13, 1973 Abstract: A general method is introduced for the projection of normalized spherical harmonic (NSH) Hamiltonians. Subgroups of Ohwhich subduce CZvare specifically considered, and dn basis functions symmetry adapted to the various groups are derived. The advantages of a symmetry-adapted basis are emphasized in the interpretation of absorption spectra of three nickel(I1) systems characterized by CZu effective symmetry. Symmetry-adapted representations of HO for G = Oh, Td,Dah,C4y,D2d,D2h,and CZvare tabulated for the d1 (and dg) configuration, the spin triplet states of the d*(and d8)configuration, and the spin quartet states of the d3(and d7)configuration.
A
lthough the absorption spectra of many transition metal systems have been reproduced using a ligand field Hamiltonian and In basis set,1-3 calculations assuming D2d,D2h,or C2,symmetry are npt Hamiltonians for DZhor C, symmetry and a d" basis (1) R.Finkelstein and J. H. Van Vleck, J . Chem. PhYs., 8,790 (1940). (2) C. J. Ballhausen, "Introduction to Ligand Field Theory," McGraw-Hill, New York, N. Y., 1962. (3) J. S. Griffith, "The Theory of Transition Metal Ions," Cambridge University Press, London, 1961. (4) M. Gerloch and R. C. Slade, J . Chem. Soc. A , 1022 (1969). ( 5 ) A. Flamini, L. Sestili, and C. Furlani, Inorg. Chim. Acta, 5 , 241 (1971). (6) P. L. Meredith and R. A. Palmer, Inorg. Chem., 10,1049 (1971). (7) N. S. Hush and R. J. M. Hobbs, Progr. Inorg. Chem., 10, 259 (1968).
set incorporate a maximum of five empirical parameters (excluding interelectronic repulsion parameters), in contrast to a maximum of three such parameters for symmetries with a fourfold axis. Spectra are therefore often interpreted assuming the symmetry of a higher point group for calculation purposes. Only in certain can the assumption be justified.8 we present a method for the projection of normalized spherical harmonic (NSH) Hamiltonians which is applicable to all point group symmetries and offers not only the possibility of straightforward calculations for noncubic as well as cubic symmetries but also the possibility of (8) J. S. Griffith, Mol. Phys., 8,217(1964).
Lever, et al. 1 Projection of Normalized Spherical Harmonic Hamiltonians
1694
standardizing correlation procedures. The empirical parameters which arise are independent of the coordinate system used for calculations and may be compared with parameters from the crystal fieldg or angular overlap model'O to determine restrictions on parameter values. The approach parallels recent developments by Schaffer lo and Ellzey. As illustrated in the interpretation of spectra of three nickel(I1) systems characterized by CZbeffective symmetry, a representation of H on a basis symmetry adapted to the point group defines relationships between empirical parameters and experimental observables thus minimizing the difficulty of the parameter fitting procedure. Symmetry-adapted representations also minimize computation and provide a point group quantum number for identification of calculated energy states. A derivation of d" basis functions symmetry adapted to point group chains12 terminating with C2, is included as are normalized spherical harmonic Hamiltonians for all point groups in the chains. Symmetryadapted representations for d" configurations with n = I , 2, and 3 and G = Oh,T d , D 4 h , Cdt, D2d, D2h, and C, are tabulated (Appendix). Theory A transition metal complex with n valence electrons is represented by an n-electron Hamiltonian of the form H
=
H"
+ HG
(1)
where H a is the free ion Hamiltonian and
0 ; Gu E [R(3)1" (2) [R(3)]"is the nth rank inner direct product of the rotation grnup in three dimensions, R(3). Integers L = 0, 1, 2, . , characterize irreducible representations of [R(3)]"and are quantum numbers for H a . HG represents the ligand field, where [Ho,GuI
=
[HG,Gu] = 0 ; Gu E [GI"
(3)
and [GI" is the nth rank inner direct product of a point group G. Irreducible representations of [GI", denoted a(G), are quantum numbers for H . For the octahedral group, a(0) = A,, Az, E, T1,and TP. The potential HG is expanded in terms of tensor operators,'O where w
Hff = c P i i= 1
(4)
and the CMLtensors with M = L, L - 1, . . . , -L are a basis for an irreducible representation of R(3). Any vanishing of B,wL coefficients is determined not only by the symmetry of the environment of the transition metal but also by the choice of coordinate system.10s13 Non(9) M. T. Hutchings, SolidSratePhys., 16,227 (1964). (10) S. E. Harnung and C. E. Schaffer, Srruct. Bonding (Berlin), 12, 201 (1972), and references therein; C. E. Schaffer, "Wave Mechanics, the First Fifty Years," W. C. Price, S. S . Chissick, and T. Ravensdale, Ed., Butterworths, London, 1973, p 174; C. F. Schaffer, Struct. Bonding (Berlin), 14, 69 (1973). (11) M. L. Ellzey,Inf.J . Quantum. Chem., 7,253 (1973). (12) F. A . Matsen and 0. R. Plummer, "Group Theory and Its Ap-
plications," E. M. Loebl, Ed., Academic Press, New York, N. Y . , 1968, p 221. (13) J. L. Prather, Nat. Bur. Stand. (U.S.)Monogr., No. 19 (1961).
vanishing BMLcoefficients are directly related to ligand field parameters to be fitted from experiment. We choose to expand P in terms of linear combinations of tensor operators which transform according to an irreducible representation of G. The linear combinations
are orthonormal. A degeneracy index R identifies components of the irreducible representation &(G) and 7 distinguishes a ( G ) if more than one Q. of the same kind is subduced by L. Symmetry adaptation coefficients (LMi7C?,R)ff for the octahedral group have been tabulated by Griffith. l 4 After expansion L
T
since only those linear combinations of tensor operators which transform as A, (or A,,) can have nonl6 The coefficients AL;' serve vanishing coefficients. as empirical parameters to be evaluated from experiment.'? Their magnitudes are independent of coordinate system. In addition His a spin free Hamiltonian where 11v15~
[H,PUSF] = 0 ; PaSFE SnSF
(7 1
and SnSFis the group of permutations on the spatial coordinates of the n electrons. The partitions [A] of SnSF are exact quantum numbers for H and yield th'e multiplicity quantum number 3n = 2 s 1. Basis Functions. A representation of H in the 5" dimensional spin free vector space' VSF(dn)is a function of the ligand field parameters AL;' and the Racah parameters A , B, and C. Although eigenvalues and eigenkets can be obtained from a representation of H on the I d";3nLM) basis of VsF(d"),19-22 a representation of k on a basis symmetry adapted to [GI" has certain advantages. 1$1 7 , 2 3 Namely, the representation is block diagonal since
+
(d";31ZL;73nORIH1dn;3n'L';7'3n'a'R') = s(sn,3n'>s(a,a')s(R,R') (8) (d" ;"L;T% Rl Hi d";3nL' ;7'%R) and each block is identified by an irreducible representation of the point group of interest. Symmetry adaptation coefficients suitable for the projection of Id"; 3 ' Z L ; ~ m ~basis R ) functions for Oh or D l k symmetry are readily available.14 Making use of the previously tabulated functions we project bases for d" configura(14) Reference 3 , Appendix 2, Table A19. ( 1 5) Reference 3, p 202. (16) Y .Tanabe and H. Kamimura, J . Phys. SOC.Jup., 13,394 (1958). (17) B. R. Hollebone, A. B. P. Lever, and I. C. Donini, Mol. Phys., 22,155 (1971). (18) (a) F. A. Matsen and M. L. Ellzey, J . Phys. Chem., 73, 2495 (1969); (b) J. C. Hempel and F. A. Matsen, ibid., 73,2502 (1969). (19) For a discussion of calculation techniques see, for example, (a)
B. R. Judd, "Operator Techniques in Atomic Spectroscopy," McGrawHill, New York, N. Y., 1963; and (b)ref3. (20) C. W. Nielson and G. F. Koster, "Spectroscopic Coefficients for the pn. d", and fn Configurations," Technical Press, Cambridge, Mass., 1963. (211 J. P. Jesson. J . Chem. Phvs.. 48.161 (1968). (22) J. C. Hempel, D. Klassen,'W.'E. Hatfield, and H . H. Dearman, J. Chem. Phys., 58,1487 (1973). (23) J. R. Perumareddi, Coord. Chem. Reu., 4,73 (1969).
Journal of the American Chemical Society J 96:6 J March 20, 1974
1695 Table I. Basis Functions and Correlation Table for the Gerade Irreducible Representations of to Chains 9, 10, and 11 Terminating with &(I)
oh
-
Ddh
cd,
+
Or
&(I)‘
0 I = { E ,2S4, c2(z), 2cz’, 2 ~ ~ b)I .= { E , 3cz,i, 3 0 h ) . transforms like z but does not change sign under inversion.
c
Or
=
Dzh(I)*
cz~(I)’
+
{ E , c z ( z ) ,b h ( x z ) ,
oh Relevant
Reference 27. S, denotes a function which
gh(YZ)}.
Table 11. Basis Functions and Correlation Table for the Gerade Irreducible Representations of Terminating with Cz,,(II) or Chain 11 Terminating with CZ~(III) o h
-
-,
Ddh
or
C4v
--
Dzh(II)O
tions which are symmetry adapted12~17~24 to the following point group chains.25-27 o h
----f
D4h
0, + Dih o h
+ Dqh
Crt
CZU
(9)
+ D2d -+-
CZO
(10)
Dzh
CZZ
(1 1)
-
+
0, --j T h + Dzh Oh + T d
Dzd
.--)
--j
cz,
(12)
+
cz,
(13)
A chain is an ordering of groups in which each group contains the elements of every group lower in the chain and is a proper subgroup of every group higher in the chain. In these examples o h is the head and Cz, the tail of the chain. Many noncubic molecules of interest to transition metal chemists fall into one of these chains. The threefold groups are not discussed here but can be treated in an exactly analogous manner. Each group, a member of chains 9-13, can be denoted G(S) where S represents the set of octahedral operations retained by G. For example, three sets of octahedral operations can be retained to generate CZu. I
=
Cz.(II)b
+
{ E,C,’(Z),Uh(Xz>,ah(yZ))
(24) Reference 2, Chapter 1. (25) M. Hamermesh, “Group Theory and Its Application to Physical Problems,” Addison-Wesley, Reading, Mass., 1962, (26) L. Jansen and M. Boon, “Theory of Finite Groups,” Interscience, New York, N. Y., 1967. (27) G . F. Koster, J. 0. Dimmock, R . G . Wheeler, and H. Statz, “Properties of the Thirty-Two Point Groups,” MIT Press, Cambridge, Mass., 1963.
+
Basesd
o h
Relevant to Chains 9 and 11
Cz,(III)c
Basesd
Each Czu(S),where S = I, 11, or 111, retains a C a zor C?’ operation from the 3C2 or 6Cz’ classes of Oh, two mutually perpendicular reflection planes whose line of intersection is the twofold axis, and the identity operation. Symmetry operations of Oh are defined with respect to the symmetry axes of an octahedron which pass through the ligands of a six coordinate Oh system. A right-handed coordinate system is assumed. When the basis of an irreducible representationz6 of Oh, denoted @(Oh),is symmetry adapted to chain 10, for example, it is also a basis for one or more irreducible representations of DJh, DLd, and C2,. The matrices D(G,) E @(Oh)for the operations of the octahedral group retained by G = D I h ,DZd,or Cz, have a block diagonal form with each irreducible representation of G subduced corresponding to one such block. Basis functions and correlation tables for the gerade irreducible representations of Oh symmetry adapted to chains 9, 10, or 11 terminating with Cz1(I)are given in Table I. These bases are also symmetry adapted to chain 12. Correlation tables are included in Tables I1 and 111 for a basis symmetry adapted to chain 9 terminating with GC(JI),chain 11 terminating with C2,[11) or Czo(III),and chain 13 terminating with C2L(II), Also included in Tables 1-111 are definitions of S for DSd(S), S = I and 11, and DSh(S),S = I and 11. Symmetry properties of basis elements are summarized as functions of x , y , and z. The D(G,) E &(Oh)for @ = AI,, AZg,E,, TI,,or Tzgcan therefore be reconstructed since by definitionz6
Lever, et al.
N
Ga6t =
where B,
=
j=l
DjdGaMi
{4*;i= 1, N ] is the basis of
(14) and
@(oh)
Projection of Normalized Spherical Harmonic Hamiltonians
1696 Table 111. Basis Functions and Correlation Table for the Relevant to Chain 13 Gerade Irreducible Representations of 01, o h
+
Td
+
D~d(11)~ + CZ,(II)~
Bases"
by the point group symmetry of a member of the chains terminating with Cz, require ligand field Hamiltonians of the following form = V
(17)
= V + V'
(18)
HG
when G
=
0, and Td H G
when G = D4h, C4a, &(I),
HG when G
=
=
and D2d(II)
+ V' + V"
V
(19)
Dz,(I) and CzC.(I>
Ha
+ V' + V"'
V
=
(20)
when G = Dzh(II), Cza(II),and Cz,(III) D1,(G,) is an element of the matrix D(G,). Tables 1-111 were obtained by bringing a(oh) for a previously reported basisz7 to the appropriate block diagonal form and relating any similarity transformations required to basis transformations. 2 6 Linear combinations of tensor components symmetry adapted to Oh and characterized by the symmetry of the basis functions given in Table I have been re, 6 . The ported previously by Griffithi4for L = 0, 1. symmetry adaptation coefficients tabulated yield either gerade or ungerade functions since the inversion symmetry of the projected function reflects the inversion symmetry of its tensor components. One-electron tensor operators are g if L is even and u if L is odd. Components of a Id"smLM) basis are always g. Given L = 2 functions tabulated by Griffith and Table I , we deduce /Ego; Aig; Ai; Ad Big;
Bz; Ai)
=
=
120)
1
+
~ ? ( / 2 2 ) 122))
+ 121)J
ITzdyz); Eg(yz); Ebz); Bz)
=
i/V'?(/2T)
Eg(xz); E ( x z ) ; B,)
=
l/V'Z{ 12i) - 121))
ITzg(XZ);
where V V' V" V"'
=
=
(21)
DQlAlg/Oh4
+ D T I E , L ~ ~ ~ , ~ (22) = DU/E,e10h2+ DV1Ege/Oh4 (23)
=
DSIEgOIoh2
+ DN1Tzg(xy)loh4
DMITzg(xy)loh2
(24)
o h quantum numbers identify linear combinations of tensor operators which transform as A, (or AI,) of G (see Table I ) . AL;' coefficients (eq 6 ) are called D Q , D S , DT, DU, and D V by analogy to previously definedz8 parameters Dq, Ds, and Dt. Although AZp: of Oh subduces A, of Dzh(I)and &(I), no functions of this form appear in eq 19 since Hamiltonians for d" configurations can be projected from even tensors of rank four or less,2s3and irreducible representations of R ( 3 ) with L = 0, 2, 4 do not subduce Az, of Oh.3,27 Symmetry adapted expansions are given in Table IV. l 4
Table IV. Ooerator Exoansions Symmetry Adapteda to
o h
(15)
ITn,(xy); B z g ; Bi; An) = -i/V'z{122) - 122)) where the functions are identified by an irreducible representation and degeneracy index for the groups Oh, D4h, Dnd(I),and CnZ(I),respectively. Symmetryadapted functions for a chain terminating with'G,(II) are obtained as linear combinations of the tabulated functions. F o r L = 2 IEgo; Ai,; Ag; Ai) /ERE; Big; Big; Ad
ITzpn; E g n ; Bzg; Bd /Tng7z';E,Tz';
B3g;
Bz)
ITz,(xy); BZg; A,; AI)
=
=
120)
+ 122)) '/z(a121) + b121)) '/z{bi2T) + 4 2 1 ) )
1/.\/2{122)
=
=
=
(16)
i/z/Z{/22)- 122))
+
where a = i 1 and b = i - 1, and functions are identified by irreducible representations of oh, D4h, DZh(II), and CzL(II),respectively. Therefore, given the information in Tables 1-111 and previously tabulated l 4 symmetry adaptation coefficients for Oh, ld";"L;r"aR) basis functions and ligand field operators are readily generated for the point group symmetries of interest. Hamiltonians. Those d" compounds characterized Journal of the American Chemical Society
Reference 14.
The Ai, component arising from L = 0 contributes equally to the energy of each state of a d" configuration and is not included in the ligand field operator. and Dzd As indicated, the Hamiltonians for Ddh,C4C., are indistinguishable. Similarly, the Hamiltonian for Dz,(I) is indistinguishable from that for Cz,(I) (see Table I). Nonzero matrix elements of V , V', V", and yl'r on a d orbital basis symmetry adapted to the appropriate chain, eq 15 or 16, are given in Table V . The matrix elements for V and V' relate O S , DT, and D Q parameters to the previously definedz8 parameters Ds, Dt, and Dq where D Q = 6(21)'"Dq - 7/z(21)1"Dt (25)
1 96.6 1 March 20, 1974
DT
=
7/2(15)1'2Dt
D S = -7Ds (28) Reference 2, p 101.
(26) (27)
1697 Table V. Nonzero Matrix Elements for Empirical Ligand Field Operators on the d1 Basis Symmetry Adapted to Czv(I), GdII), and C Z ~ I I I ) a(Czv(S)) --S@--
I
~
I1 111 a ( o h ) R
~
0
~
~
Operatorb
a'(0h)R'
DQ
(dl ;la(Oh)i?;'a(CZv(S))loperatorb1d 1 ;la '( 0h)R' ;la(Czv(s)))---------. DS DT DU DV DM DN
~~~~~~
Defined in Tables I and 11.
b
Defined by eq 20 and 24.
Note that in the octahedral limit DQ = 6(21)'I2Dq. Note also that for a Hamiltonian of the form V V' the center of gravity rule holds within the eg and tzg levels of a d system for D S and D T energy components, an observation which is independent of the d basis used for calculations. See Appendix.
M
+
Six-Coordinate Systems Ligand substitutions and angular distortions leading t o a first coordination sphere characterized by C2, symmetry are summarized in Table VI and Figure 1.
I
.-
---
O O: + ;
---at
XL:
qj (xz)
+-
%
/ - 0
--
-
-pd
(fy)
---Ay
6 I
Table VI. Mixed Ligand Six-Coordinate Systems Which Can Exhibit CzQSymmetry Highest available point group symmetrya
Possible Czv distortion patternsb ~
oh Trans Trans Cis Trans Trans Cis Eauatorial Trans Trans Trans
C4 v Dih
cz, cz
Y
Dzh
CZ"
cz" czv cz Y
C,
2.
~
~~
I, 11, 111 I, I1 I, 11, 111 111
I I I11 I I1 11, I11 I11
Corresponds to the symmetry of an octahedral angular configuration. As defined by Figure 1.
Although mixed ligand systems may exhibit angular distortions, the highest symmetry available to each corresponds to the symmetry of an octahedral angular configuration. For example, the limiting symmetry for trans-MA4B2 is Dlh while that for cis-MA4B2 is Czs (Table VI). Any distortion of the octahedral angular configuration lifts the D4h symmetry of the trans
Figure 1. Definitions of the angular configuration for (1) oh (axes pass through ligand positions; a1 = az = 90"; 0 , = 90"; $I = +z = 90'); (2) CzdI) (I = { E , Cdz), U h (xz), U h ( y z ) ] ; ligands 5 and 6 lie on the z axis; o1 = oZ = 90"; O1 = ea, O2 = Bq, and 0" < el and ez < 180"; $1 = $z = 90"); (3) cz,(II) (11) = ( E , Cdz), u d x y ) , ad(fy) 1 ; ligands 5 and 6 lie on the z axis; 0" < a1= cyz 5 go", ei = 0 for 0" < e < 180"; $l = +bZ = 90"); (4) czV(~1I)(111 = ( E , cz'(xy), uh(XY), Ud(xy)); ligands 1, 2, 3, and 4 lie in the xy plane; 0 0 < ai,az, 51800; ei = 900;0 0 < +1 = ii/2 I 900).
system, while the cis system retains Cz, symmetry for a Cz,(III)angular configuration, as defined by Figure 1. Parameters suitable for empirical ligand field calculations on a d" basis are specified in eq 17-24. The number of parameter: required for each symmetry is dictated by the symmetry and the choice to restrict consideration to states arising from a d" configuration. That is to say
where is the number of times L subduces A1 (or Alg) of the point group of interest. However, assuming ligand additivity and an average electrostatic effect for fL;*l
Lever, et al. / Projection of Normalized Spherical Harmonic Hamiltonians
1698 Table VII. Nonzero Matrix Elements of the G ( I ) Hamiltonian for the Triplet States of a d* Basis Symmetry Adapted to CzY(I)
~~
~
Ai Az
F;l P;1 P;1 P; 1 F;la F;1 F;2 P; 1 P;1 P; 1 F;1 F; 1 F;2
Bib
~
F;1 P;1 F;1 F;2 F; 1 F;2 F;2 P;1 F; 1 F;2 F; 1 F;2 F;2
~~
~
0.43644
15 0.14548 -0.21822
-0.4oooO -0.34286 0.11429
0.14548 -0,21822
-0.25555
0.16496
-0.12778
0.08248
0.18443 -0.43033
0.07274 15
-0.12295
0.20000 0.17143 -0.22131* -0.05714 -0.11066*
0.07274
0.06148 0.14286* -0.09221 0.07143* 0.21517
0.34641* 0.29692* 0.12778 -0.09897* 0.06389
0.10648* -0.08248 -0.15972* -0.04124 0.37268*
The ld2;3F;la(Cz,(I)))for a = As, B1, B2 arise from @ ( o h ) = Ti,. The nonzero mdtrix elements for the 3Bzstatesarederivedfrom the matrix elements tabulated for the 3B1states using the negative of the starred (*) multipliers and the other multipliers as tabulated.
different ligands, assumptions implicit in most applications of empirical crystal field certain systems require fewer than N Dparameters. By relating empirical crystal field parametersg for the six-coordinate molecules in Table VI to the empirical AL;‘ parameters, one finds that
+ R 3 )sin28, - (R2 + R4) sin2 O2 = (R,+ R4) sin (90 - a,)sin2 0 + (29) (R2 + R 3 ) sin (90 - a2)sin2 0
DU DM
=
(R,
where R , is an empirical radial parameter for ligand i, and the angles e,, a,,and a2 are defined in Figure 1. Therefore, D M (and D N ) always go to zero when a C2,(II) or C2,(III) system (Table VI) retains an octahedral angular configuration (0 = Q! = 90’) and ligand additivity is assumed. Similarly, DU (and D V ) go to zero when a trans-MA4XY C241) system is charac6. In such a terized by 8, = 90 - 6 and OZ = 90 case H,,, has exactly the same form as HG where G = Dqh,C,, and DZd. These compounds may therefore be described with a Hamiltonian of higher symmetry than C2, and reflect a property previously termed intermediate symmetry by Griffith.* The equatorial (meridional) configuration of a MA3BS system (CzU(I)symmetry) can also display a type of intermediate symmetry. When the system retains an octahedral angular configuration, D S and DT go to zero and the system can be described with the parameters DU, D Q , and DV To demonstrate the utility of this procedure, it is applied in the analysis of three nickel complexes whose polarized crystal spectra have been published. It is a feature of the spectra of noncubic complexes of the twofold groups that the selection rules are in general not so well obeyed as those for representative examples from the fourfold groups. For this reason the assignments cannot be considered so secure; application of the tensor Hamiltonian technique will serve to provide an additional means of verifying assignments through correlation of parameter values. Octahedral nickel(I1) systems (dS) have an orbitally nondegenerate 3A2, ground state. Three relatively intense bands are observed in the absorption spectrum of an octahedral complex and may be assigned t o the 3A2g(F)+ 3T2g(F),3T1g(F), and ”,,(P) transitions of
+
(29) G. Maki, J. Chem. Phys., 28,651 (1958).
increasing energy, respectively. In a complex of C z 0 symmetry, the orbital degeneracy of the excited triplet states is lifted and there are nine possible spin-allowed transitions. The x, y , and z components of the electric dipole vector transform as B1,B2, and A,, respectively, for C2, symmetry where x, y, and z refer to the symmetry axes for Go.30 DZh(I)and C2,(I). Polarized single crystal electronic spectra of bis(diethylenetriamine)nickel(II) chloride monohydrate have been reported by Fereday and Hathaway. The tridentate ligand diethylenetriamine (den) coordinates equatorially. 3 2 Secondary nitrogens, which lie trans to one another, define a Cz axis and the N-Ni-N angle made by the primary nitrogens Although the nickel(I1) within a den ring is -162”. cation as a whole does not have a C2 axis, polarized absorption spectra indicate an effective symmetry higher than the C1 crystallographic site symmetry. Three absorption bands are observed in each of three crystal orientations with the electric field vector essentially along x, y , and z symmetry axes of C2, or D2h.30 An assumption of D,, symmetry, suggested by Fereday and H a t h a ~ a y , ~ ’ the observation of nine triplet-triplet bands. H D Z h (is~ ) indistinguishable in form from HC2.(1)(eq 19) and the x , y , and z components of the electric dipole vector transform as b3,, bZu,and bl, for QZh symmetry. 30 The representation of H (eq 1) for &(I) symmetry on a basis symmetry adapted to CZv(I)is given in Table VI1 for the triplet states of a d 2 configuration. See Appendix. Since the trace of a matrix is invariant under diagonalization, various combinations of the experimentally observed transition energies are directly related to traces of blocks of the representation and can be used to establish relationships between B, D Q , DS, DT, D U , and DV. For example 2 4 -&DU % d D V = T T ( ~ B ~, ) T T ( ~ B ~ , (30) ) 7
+
where T I - ( ~ ~ ( D ~refers * ) ) to the sum of the energies of transitions from the ground state to excited triplet (30) For a definition of these symmetry axes see (a) character tables given in ref 27 or (b) A. B. P. Lever, “Inorganic Electronic Spectroscopy,” Elsevier, Amsterdam, 1968. (31) R. J. Fereday and B. J. Hathaway, J. Chem. Soc., Dalton Trans., 197 (1972). (32) S. Biagini and M. Cannas, J . Chem. Soc. A , 2398 (1970). (33) Note that ref 31 incorrectly assumes the 3A2,(Oh) state to subduce a %,(D2h) state.
Journal of the American Chemical Society / 96:6 / March 20, 1974
1699
states characterized by the quantum number a( DZh). Similarly DT
+
1 2
= - Tr(3B2g)
1
- Tr( 3B3g)-
2
15B
-
wn 20
Tr( 3B1,) (31)
1 1 D Q =-Tr(3B1g) 3Tr(3B2E) 3 1 - Tr(3B3g)- 3Tr(3Ag) (32) 3
+
+
Therefore, when nine triplet-triplet transitions are observed, there are three' independent variables to be fitted from experiment: B (or De), DU (or D V ) , and D S (or DT). Transitions calculated with B = 780, D S = -435, DU = -100, D Q = -32,215, DT = 370, DV = 350 cm-l, and a d 2 basis are compared with experimental values for [Ni(den)z]2+ in Table VIII. The parameters reported exactly reproTable VIII. Calculated and Experimental Spin-Allowed Transitions for [Ni(den)z]2+ Exptl'--PolarizaEnergy tion 7 -
11,200 11,600 11,600 18,100 18,300 18,600 28,700 28,700 28,900
--
&(I)
Assignment-C2,(1)
Calcdb energy 11,659 11,556 11,925 18,077 18,343 18,152 28,688 28,700 28,597
Reference 31. B = 780 cm-l, D S = -435, D U = -100, D Q = -32,215,DT = 370,andDV = 350cm-landad2basis.
duce the 3A, + 3B1gtransition energies as well as the trace (or sum) of the 3Ag + 3B3gtransition energies and the trace of the 3A, + 3Bzgtransition energies. Although the first excited triplet state is calculated to have the same point group quantum number as the highest excited triplet state for all parameter sets which reproduce the traces, this is not observed experimentally. An interpretation of the polarized single crystal spectra of [Ni(den)z]2+assuming DZhsymmetry implies that the angular distortion revealed by the X-ray structure study32 does not determine the effective symmetry of the system. Assumption of Cz,(I)symmetry is consistent with the angular distortion and implies only that the primary nitrogens of the den ligand are distinct, which may reflect the small bond length difference obtained in the X-ray structure determination and/or a small deviation in the N-Ni-N angles within the two den rings. Cz,(III). Polarized single crystal absorption spectra for bis(DL-histidinato)nickel(II) monohydrate have been reported by Meredith and Palmer.6 The tridentate histidine molecules bond through three inequivalent functional groups and coordinate facially. 3 4 Oxygens (34) K. A. Fraser and M. M. Harding, J . Chem. SOC.A , 415 (1967).
Lever, et al.
from the carboxy group and nitrogens from the a-amino group of each ligand lie in a plane with an 0-Ni-0 angle of 100.3" and an O-Ni-NA angle of 79.7". The two remaining bonds, imidazole nitrogen-nickel bonds, deviate from colinearity by 2.4". The site symmetry of the nickel atoms is Cz.3 4 Eight of nine possible spin-allowed triplet-triplet transitions are observed in x , y , and z polarization^^^ with some of the transitions forbidden (not observed) in each polarization. Meredith and Palmer assumed CzD(I1I)symmetry, the symmetry of the first coordination sphere of the nickel ion, in an interpretation of the spectra. A full calculation for Cz, symmetry was carried out within the d 2 configuration using an empirical Hamiltonian in the form of eq 4. The coordinate system used by Meredith and Palmer to define the tensor operators lies with its z axis bisecting the 0-Ni-0 angle and its x axis bisecting the NA-Ni-0 angle. For this choice of coordinate system nonzero B.wL coefficients multiply only those C M Loperators with L = 2, M = 0, =t2 and L = 4, M = 0, *2, =t4,10s13It should be noted, however, that the coordinate systems with x , y , and z labels permuted correspond to Hamiltonians of the same form. The coordinate choice is fixed only by the assignment of the point group symmetry quantum number associated with each calculated energy state. That is to say, the form of the projection operators35 or symmetry adapted basis functions used to determine a(G) reflects the coordinate system choice. Therefore, a fitting procedure which reproduces energy level splittings by adjustment of BML parameters for L = 2, A4 = 0, =t2 and L = 4, M = 0, *2, A 4 can be expected to find a number of equivalent fits. B.wL coefficients reported by Meredith and Palmer convert to D Q = -31504, DS = -1955, DT = 2155, D M = 3409, and D N = -5149 cm-l. These AL;' parameters, defined by eq 23, reproduce the energy level splittings and assignments given in Table IX (with B = 783 cm-') in any coordinate system. Table IX. Calculated and Experimental Spin-Allowed Transitions for [ N ~ ( D L - ~ ~ s ) ~ ] ~~~~
_---_
Assignment
Cz,(III)
3B1 + 3 A ~ ( ~ ) d -,3B2
-
+
3A~(~) 3Bz
-.
33B1(Z) A~(~) 3Az 3B1(z) + 3Bz -P
+
Energy, cm-l-----
---Calcd-----6 variables"! 5 variables 10,706 11,300 11,884 16,472 18,848 19,244 27,213 28,751 29,608
10,615 11,456 11,649 16,500 18,881 18,692 27,404 28,863 29,588
-
ExptP 10,600 11,400(~) 16,500 ( x y ) 18,800 ( x z ) 19,OOO ( y ) 27,300 ( y ) 28,900 (2) 29,600 ( X Y )
F,B = 783, D M = 3409, D N = -5149, D S = a Reference 6. -1955, DT = 2155, and D Q = -31,504 cm-l. B = 820, D M = 3000(DN = -3959, D S = -3000, D T = 1859, and D Q = -31,069 cm-1. d Allowed polarization Czu symmetry. e Polarizations observed.
A representation of H for C2,(III) symmetry on a d 2 basis symmetry adapted50 C2,(III) is given in Table X (35) S. R. Polo, "National Technical Information Service," U. S . Dept. ofcommerce, AD282493, 1961.
Projection of Normalized Spherical Harmonic Hamiltonians
1700 Table X. Nonzero Matrix Elements of the C2@) Hamiltonian for the Triplet States of a dl Basis Symmetry Adapted to Czv(S),S = I1 and 111
A1 Bla
Ae
Bza
F;1 P;l P;l P; 1 F;lb F; 1 F:2 P; 1 P; 1 P; 1 F; 1 F; 1 F;2
A1
Aza
Bi
F; 1 P;1 F; 1 F;2 F; 1 F;2 F;2 P; 1 F; 1 F; 2 F; 1 F;2 F;2
0.07274 15 0.14548 -0.21822
-0.43033 0 . 2 m 0.17143 -0.22131* -0.05714 -0.11066*
0.06148 0.14286* -0.09221 0.07143* 0.21517
0.07274
15 0.14548 -0,21822
-0.4oooO -0.34286
-0.34641* 0.19795* 0.25555 -0,02474* -0.03194 0.12372*
-0.05324* 0.12372 -0.31944* 0.16496 0.10648*
-0.12295 -0.25555
0.16496
-0.12778
0.08248
0.18443
-0.11429
0.43644
BP
The nonzero matrix elements for the 3Bz states are derived from the matrix elements tabulated for the 3B1(or 3Az)states using the negative of the starred (*) multipliers and the other multipliers as tabulated. The ldz,3F;l~(Cz,(II)))for a = B,, At, and Bz arise from U ( 0 h ) = TI,. a
for the spin triplet states. See Appendix. The following relationships are deduced from Table X 4 + 21-(5)'/'DN
2-(3)'/2DM 7 2 - - 7D s
-(-) DT + 15B
8 5 7 3
3 1 -DS - - D T 7 21
=
Tr(3B2)- Tr(3A2) (33)
=
Tr(3B1) - 3Tr(3A1) (34)
5 32/TiDQ
~
=
1 1 Z T ~ ( ~ A ~T)T ~ ( ~ B ~Tr(3B1) ) (35)
+
where Tr(3a(C2,(III)) refers to the sum of the energies of transitions from the ground state to excited triplet states characterized by a(Czv(I1I)). When nine spinallowed triplet-triplet transitions are observed, the energies of the triplet states can be specified with four independent variables, for example, B, D M (or DN), D S (or DT), and D Q . Transitions calculated with B = 820, D M = 3000, DN = -3955, D S = -3000, D T = 1859, and D Q = -31069 cm-l are compared with experimental values for [ N ~ ( D L - ~ ~ins )Table * ] IX. This fit, which retains the C2, assignments made by Meredith and Palmer, exactly reproduces the 3B1 3B1and 3B1 + 3A1 transition energies as well as the trace of the 3B1 3A2transitions. Since one 3B1+ 3B2transition is not observed experimentally, five variables were allowed to vary independently in the fitting procedure. The fit compares favorably with that of Meredith and Palmer, in which six parameters were varied independently. A scissoring distortion in the xz plane30 can be measured in units of 2p, where LO-Ni-0 = 90" - 2p and LO-Ni-NA = 90" 2P. Relating AL;' parameters to empirical crystal field parameters one finds that
-
-
+
[COS(4P)lDN
=
[-sin (2p)]{ 1/5/21DQ
+ 1/1/3DT]
(36) where -20 cv 18" for Meredith and Palmer's fit6 of the experimental spectrum and -2p 'v 14" for the fit obtained with five independent variables. From the X-ray structure d e t e r m i n a t i ~ n one ~ ~ would expect -2P N 10.3". The AL;' parameters obtained for Journal of the American Chemical Society
Ni(DL-his)z therefore not only adequately reproduce the absorption spectrum but can be used to predict a distortion observed experimentally. Although electric dipole selection rules for C2,, symmetry are not strictly obeyed, all predicted transitions are observed.
Four-Coordinate Systems Many ML4 systims exhibit one of the two limiting symmetries available to a four-coordinate system: D4h (square planar) or Td (tetrahedral).' Equations 29 relate empirical crystal field parameters for fourcoordinate systems to parameters DU and D M defined by the chains incorporating D4h. With the assumption of ligand additivity, DM (and D N ) go to zero for the D4,,angular configuration of cis-MA2B2(8 = a = 90") and the system exhibits intermediate symmetry.8929 Similarly, one finds that the D4hangular configuration of trans-MA2B2can be described with D Q , DU, DV, and the Ddhangular conDS, and D T = 1/2(5/7)'/2D& figuration of MA, with DS, D Q , and DT = 1/2(5/7)'/'DQ. Projections of ligand position vectors are defined to lie on 45" diagonals in the xy plane for Td symmetry in contrast to the D4hsystem, where symmetry axes pass through ligand positions. For this reason a Czu(1I) distortion of the Td configuration is equivalent to a Cz,(I) distortion of the D4h configuration. Parameters which indicate the distortion from Td symmetry arise in the first approach while parameters which indicate the distortion from D,, symmetry arise in the second. DM for chain 13, incorporating Td,is given by
DM
=
~ R Sin2 A 81 - ~ R Sin2 B 82
(37)
where R A and RB are the empirical radial parameters for ligands in positions 1 and 3 and 2 and 4, respectively. Ligand positions are indexed by the spherical coordinates (ei,q5J with q51 = 45", = q5* 90°, and 81 = 03, O2 = 04. The T , angular configuration of MA2B2, characterized by CZv(11),can be described with D M , DN, and D Q when an assumption of ligand additivity holds. We have restricted consideration to MA4 and MAzBz systems, systems which can exhibit CZ0symmetry and which require Hamiltonians and basis functions symmetry adapted to chains 9, 10, 11, or 13. The theory and techniques outlined are applied to a four-coordi-
1 96.6 1 March 20, 1974
+
1701 Table XI. Calculated and Experimental Spin-Allowed Transitions for Ni(i-Pr-sal)z
Gerloch and Slade4 report an extensive oriented single crystal magnetic susceptibility study of Ni(i-Prsalh for T 2 80°K in which they conclude that the Obsd5 energy, Assignment Calcd energy, effective symmetry of the system is Dzd. Magnetic CZtiII) cm-l cmDzdII) susceptibilities were calculated using an empirical crystal field Hamiltonian incorporating spin-orbit and , 910" 38Zd 3A2 + 3E(xy)b 3Az -,3 B ~%(XY)~ 3,007 Zeeman terms with X 'v -100 cm-l and k = 0.4. 3B1,3Bz(xy) 5,796 6,000 WXY) 3,880 'Ai 9,078 3B2 Converting empirical crystal field parameters used in 6,917 3Aa(z) 14,208 14,280 3B1(~) the magnetic susceptibility fitting procedure to AL;r 3A~(z) 17,572 14,544 17,500 3A~ parameters, one obtains DS = 132, DT = -1350, 3B1, 3Bz(xy) 19,076 14,593 19,000 3E(x~) and DQ = 9947 cm-l. Triplet-triplet transition enerReferences 4 and 36. b Allowed polarization. c B = 835, gies calculated with these parameters (and B = 790 C = 2,700, DS = 2,200, DT = -4,982, and D Q = 20,000~m-~. cm-l) are compared with experimental values in Table dReference 4: B = 790, C = 3160, DS = 132, DT = -1350, and XI. It should be noted that the bands at -17,500 and D Q = 9947cm-l.
----
---
0
Table XII.
Empirical Ligand Field Parameters from Polarized Single Crystal Studies of Nickel(I1) Systems
--.
~~
Parameters,a cm-1 DS DT
I
System
5
Symmetry
Ni2+:Mg0 Ni(NHMNCS)2 Ni( DL-his)z
oh
[Ni(den)dClz.HzO Ni :ZnO Ni(i-Pr-sal)*.
CZdU Td Did
Dah
CZ"(II1)
B
DQ
815 847 783 820
-22,408 - 29,8 14 -31,504 -31,069
780 795 790 835
-32,215 11,135 9,947 20,000
DM
DN
Ref
-5149
41 42 6 This work
- 770 - 1955 - 3000
216 2155 1859
3409 3000
- 3955
-435
370
DU - 100
DV 350
132 2200
-1350 -4982
This work 41 4 This work
For energy levels calculated with a d2 basis.
19,000 cm-l are disregarded in the interpretation of data proposed by Gerloch and Slade. Triplet-triplet transitions calculated with B = 835, DS = 2200, DT = -4982, and DQ = 20,000 cm-l are also given in Table XI. These parameters reproduce all observed transitions and preliminary calculations 40 indicate that the parameters will reproduce the susceptibility data for T 2 80°K with X = 140 cm-1 and k 11 0.58. The assumption of Dzd symmetry requires that the imine nitrogens and hydroxy oxygens be indistinguishable for symmetry purposes. A full polarization study of the absorption spectrum has not been done due to the unfavorable placement of nickel centers in the single crystal. 37 Ni(i-Pr-sal)z could be characterized by a small splitting in the 3A2+ 3E bands. In such a case the assumption of Cz0 symmetry predicts the four observed triplet-triplet bands. One of the bands is forbidden in DZdsymmetry (see Table XI). The low-lying spin-forbidden transition observed experimentally is essentially independent of crystal field strength and a C value consistent with its observation is included in the footnote to Table XI for each interpretation of the absorption spectrum.
nate nickel system characterized by an approximately tetrahedral configuration. A tetrahedral nickel(I1) system is expected to exhibit three spin-allowed absorptionnbands, assigned as 3T1(F) + 3T2(F), 3Az(F), and 3T,(P). In Cz, symmetry the orbital degeneracy of the states is removed and there exists the possibility of observing nine triplet-triplet transitions. However, if the distortion from Td symmetry is small, the two levels arising from the ground state lie at very low energies. C20(II). Polarized single crystal spectra for bis(N-isopropylsalicylaldiminato)nickel(II) have been reported by Basu and B e l f ~ r dand ~ ~ by Gerloch and Slade. The conformation of the first coordination sphere corresponds very nearly to the C2,(II) distortion of a tetrahedral configuration. 37 Planar isopropylsalicylaldimine (i-Pr-sal) ligands bond through an imine nitrogen and a hydroxy oxygen with an 0(1)-Ni-0(2) angle of -125" and a N(l)-Ni-N(2) angle of -121". The ligand planes intersect at an angle of 82". The system contains discrete molecules characterized by a triplet ground state.a7V38 Six bands are observed in the absorption spectrum of Ni(i-Pr-sal)2 with the electric field vector parallel to b and to c crystal axe^.^!^^ One band (-ll,OOO cm-l) is weak and sharp, characteristic of a spin-forbidden band. Another (-22,400 cm-l) is relatively very intense. The four remaining bands (at -6000, 14,280, 17,500, and 19,000 cm-l) are of comparable intensity, characteristic of spin-allowed d-d bands. 39
Discussion AL;' parameters for three nickel(I1) systems characterized by a Cz, effective symmetry are compared with parameters for 0 t h e r ~ l 1six-coordinate ~~ and four-coordinate nickel systems in Table XII. As indicated, the relative magnitudes of D Q parameters for the sixcoordinate species reflect an increase in ligand field
(36) G. Basu and R. L. Belford, J. Mol. Spectrosc., 17,167 (1965).
(37) M. R. Fox, P. L. Oriolo, E. C. Lingafelter, and L. Sacconi, Acta Crystallogr., 17,1159 (1964). (38) L. Sacconi, P. L. Orioli, P. Paoletti, and M. Ciampolini, Proc. Chem. SOC.,London, 7,255 (1962). (39) L. Sacconi, P. Paoletti, and M. Ciampolini, J. Amer. Chem. Soc., 85,411 (1963).
(40)J. C. Hempel, to be submitted for publication. (41) R. Pappalardo, D. L. Wood, and R. C. Linares, Jr., J. Chem. Phys., 35,1460(1961). (42) A. B. P. Lever, Coord. Chem. Rev., 3, 119 (1968).
Lever, et al.
1 Projection of Normalized Spherical Harmonic Hamiltonians
1702
“strength” for ligands which coordinate through nitrogen over that of an oxygen ligand in MgO. This observation tends to support the proposed interpretation of the Ni(i-Pr-sal)z spectrum which indicates an increase in ligand “strength,” as measured by D Q , over that of the oxygen ligand in the tetrahedral ZnO matrix. The magnitudes of parameters D S and DT for the various systems, as well as D M and D N or DU and D V , appear to reflect ligand substitution more sensitively than angular distortion. For example, D M for a sixcoordinate Cz,(II) molecule is found to be much larger in absolute magnitude than DU for a Cz,(I) molecule. D M is predicted to correspond to a sum and DU to a difference of empirical radial parameters (eq 29). If it is assumed that the potential due to the ligands is #he sum of a potential due to each ligand and that similar ligands are indistinguishable, certain deductions can be made concerning the symmetry of the various systems. For example, since D S and DT are nonzero for Ni(i-Pr-sal)z, the system is not characterized by a Td angular configuration and since D M and D N are nonzero for Ni(DL-his)z, the system is not characterized by an Oh angular configuration. These deductions do correlate with X-ray structure determinat i o n ~ . ~Further, ~ ~ ~ ’ the magnitude of a distortion in the first coordination sphere of Ni(DL-his)z can be deduced from AL;‘ parameters D Q , DT, and D N using a relation applicable to all six-coordinate systems charterized by Czu(IIJ)symmetry. The nine triplet-triplet transitions of a nickel(I1) system can be described with three independent parameters for Cz,(I) symmetry and with four independent parameters for C2,(II) or Cz,(III) symmetry. Nine triplet-triplet transitions observed for Ni(den)zz+were fitted with three independent parameters ( B , D S , and DU), while the eight triplet-triplet transitions observed for Ni(DL-his)z were fitted with five independent parameters ( B , D Q , OS, DT, and DM). The four triplettriplet bands observed for Ni(i-Pr-sal)%were fitted with B, D Q , D S , and DT (with WM = D N = 0). For each of these systems, all triplet-triplet transitions predicted assuming Cz, symmetry are observed. Results of this study imply that the angular configuration of the first coordination sphere determines the effective symmetry of the system and that predictions based on an assumption of ligand additivity hold. Summary and Conclusion The utility of normalized spherical harmonic Hamiltonians may be summarized as follows. (i) Their adoption would provide a standard and systematic procedure for generating ligand field Hamiltonians in the noncubic point groups.
(ii) The scalar parameters generated with these NSH Hamiltonians are truly spherical and are therefore independent of coordinate axis choice. We have chosen in this article to expand all the tensor components along the same axes, i.e., a right-hand coordinate system quantized along the C4(z) octahedral axis, irrespective of the symmetry axes of the molecule concerned. In the threefold groups the same choice may be made but it may be computationally more favorable to utilize a set quantized along C3(xyz) ( o h ) . In either event the magnitudes of the scalar parameters obtained will be the same. (iii) Clearly within a group of molecules belonging to the same point group, the variation of a given parameter, such as D S , as a function of the ligand or metal, may be compared and contrasted. It may also be useful to include, within this comparison, complexes of other stereochemistries whose point groups lie within the same subduction chain and whose Hamiltonians contain the same tensor component. In this way a systematic body of knowledge concerning the noncubic point groups can be amassed. Series of D S or DT paralleling the spectrochemical series of Dq could be generated. These would differ from the spectrochemical series in that the latter, for a given metal, is a function of only one ligand, while the former would be at least two dimensional being a function normally of at least two ligands. ‘For this reason, a melding of this tensor technique with the orbital angular overlap model will probably prove beneficial. lo This model provides an alternative t o that based on the Gerloch and Slade43C, parameter. Acknowledgment. We are indebted to the National Research Council of Canada for financial support. Appendix Tabulated Symmetry-Adapted Representations. Table V contains the nonzero matrix elements of a representation of H, eq 1, for G = On,T,, Ddh,DZh,DZd,and on d 1 bases symmetry adapted to Czu(I),Cz,(II), and Cz,(III), To determine which matrix elements are required for the point group of interest refer to eq 17-24 to determine the form of HG. Tables VI1 and X contain the nonzero matrix elements of H for the spin triplet states of a d2 basis symmetry adapted to C2,(I) and Cz,(III), respectively. Tables VI1 and X also apply to the spin quartet states of a d7 basis. The representation of HG for dlO-nis in each case the negative of the representation for d“. The representation of H odoes not change sign. (43) M. Gerloch and R. C . Slade, “Ligand Field Parameter,” Cambridge University Press, London, 1973.
Journal of the American Chemical Society / 96:6 1 March 20, 1974
