F. S. RICHARDSON
692
The Optical Activity of Trigonally Distorted Cubic Systems F, S. Richardson DepGFtment of Chemistry, University of Virginia,Charlottesville, Virginia 22901 and Department of Chemistry, Princeton Uniaersity, Princeton, New Jersey 08640
(Received J u l y $2 1970) ~
Publication costs borne completely by The Journal of Physical Chemistry
The optical rotatory properties associated with the ligand-field transitions of trigonal dihedral transition metal complexes are examined on a one-electron model. The chromophoric electrons localized on the octahedral MLO cluster are perturbed by a static trigonal field which originates with the nonligating parts of the ligand environment and with distortions of the MLa cluster from a regular octahedral geometry. The perturbation expansion of the wave functions is carried out to second order in the trigonal field ( P ) . The net rotatory strength associated with the complete manifold of ligand-field transitions vanishes to first order on the oneelectron model but is nonvanishing if second-order contributions are considered. The second-order contributions to the rotatory strength arise from the simultaneous actions of a trigonal perturbation (P,) with gerade octahedral parentage and a trigonal perturbation (Pu)with ungerade octahedral parentage. The contributions due to d-p and d-f orbital mixing under the influence of P,, and d-d mixing under the influence of P,, are cmsidered in detail. The influence exerted on the optical rotatory properties by chelate ring conformation, ring size, ring substituents, and distortions of the octahedral ML6 cluster, are examined and predictions are made. Vjbronic effects are discussed, but they are not explicitly treated in our model. The results obtained to first order in perturbation theory are similar to those previously obtained by Moffitt, Piper, and Karipedes, and Poulet on a static, one-electron model. In carrying the static, one-electron model to second order in perturbation theory, we obtain results which are in close agreement with the experimental CD data and which are easily interpreted in terms of specific structural features of trigonal dihedral metal complexes, Reduction of the general model to a crystal-field representation of the electronic states of the complex ion further aids in elucidating the essential symmetry-determined aspects of the optical rotatory properties. -4lthough the static, one-electron model is not appropriate for making accurate quantitative calculations of the rotatory strength, it provides a simple and correct representation of the nodal structure in the electronic states of the ML6 chromophore and in the interaction potential between the MLs chromophore and the ligand environment. The simplicity of this representation and the facility with which it can be used in making correlations between GD (data and molecular structure make it particularly attractive.
I. ~ t r o d ~ c ~ i ~ ~ Considerable research effort has been focused on the optical rotatory and circular dichroic properties of transition metal complexes which possess trigonal dihedral (Da)symmetry. The simplest examples of this class of metal complexes are M(en)3 and R/l(OX)a in which three identical bidentate ligands (en = ethylenediamine and ox = oxalate anion) are coordinated to the metal ion. The ML6 clusters (where L = ligating atoms) in these compounds deviate only slightly from octahedral (Oh) symmetry. The d + d type transitions are of particular interest for developing viable theoretical models of optical activity because: (a) their sbsorption bands lie in the experimentally accessible near-infrared, visible, and near-ultraviolet regions of the spectrum; (b) they are localized in a group with high iinherent symmetry; and, (e) their optical abisorption properties indicate that they are only weakly influenced by the ligand environment beyond the MLG cluster. For example, the visible absorption spectra of [ C ~ ( e n ) ~and ] ~ +[ C O ( O X ) ~are ]~nearly identical with those of [ C O ( N H ~ ) ~arid ] ~ + [Cof,Hz0)3l3+, respectively, which suggests that the nonligating chlehte ring atoms assert only a weak influence on the chiroiinophoric electrons responsible for the ligThe Journal of t'hysical Chemistry, VoZ. 75, No. 5, 1971
and-field transitions. Furthermore, the observed intensities (e -40-60) indicate that the electric-dipole selection rules are still principally determined by a molecular field of Oh symmetry. These results suggest that a perturbation model should e appropriate for treating any optical property which depends upon interactions between the d electrons localized in the 31L6 cluster and the nonligating parts of the ligand environment. We restrict our attention in this paper to transition metal ions whose orbital ground states are nondegenerate and whose orbital excited states are triply degenerate in an octahedral ligand field, Most of the experimental circular dichroism studies with trigonal dihedral systems have been done on the tris complexes of Ni(II), Cr(III), Rh(lII), and Ir(III), all of which have orbitally nondegenerate ground states and triply degenerate orbital excited states. Furthermore, we shall neglect spin-orbit interactions. According to the quantum-mechanical theory of optical activity both the magnitude and the sign of the Cotton effect associated with a n electronic transition i + j are determined by the rotatory strength, Rtg,of the transition. This quantity is defined by
PTZCAL
AcTrvrw
OF
TRIGONALLY DISTORTED CUBIC
where Im indicates that the imaginary part of the scalar product between the electric-dipole transitionmoment lijCj and the magnetic-dipole transition moment is taken. According to the elassical electron theory of optics, electrons in an optically active system are constrained to move along helical paths in their response to an incident ligltt beam. This helical motion of the electrons (circulation t- linear displacement of charge) generates the parallel electric and magnetic transition dipoles which sppear in eq 1. I n optically active molecules this consti-aint placed on the motion of the electrons arises from a net dissymmetry in the internal electric fields L f the molecule. That is, the electrons move in an electrostatic potential field which lacks an inversion center, B mirror plane, and/or an alternating rotation-refleciion axis. In eq 1, g t and # j are the total electronic m a v e functions for molecular states i and j . Since these many-electron functions cannot, in general, be obtained with sufficient accuracy for calcula,ting spectroscopic properties, the usual procedure i s to partition the molecule into several simpler parts. To zeroth order the molecule is represented as a eompositc OS independent, noninteracting subsystems and interactions between the subsystems are then treated by perturbation methods. In most cases of general inr erest the individual chromophoric groups or subsystems are not dissymmetric so that the molecular optical activity is obtained only from the intergroup interactions. ‘The general methods and approximations conventionally employed in the independent systems, perturbation approach to optical activity have been presented 4xewhere and need not be rediscovered The final expression for the rotatory strength obtained from the independent systems model includes four types of irrnis: (a) 6‘tmio-electron” terms due to the pairwise couplings between electric transition dipoles loeated on ttvo different groups; (b) “twoelectron” terrns ciue to the coupling between an electric transition dipole on one group and a magnetic transition dipole on another group; (e) i‘one-electron” terms which arise when the electric- and magneticdipole transition monients of a one-electron transition localized in a single, inherently symmetric group have parallel components due to the static perturbing influence of ~ ~ i s ~ ~ m m e t r i c adisposed lly surrounding groups; and (d) coiitributions made by inherently dissymmetric groups considered in isolation. The “oneelectron” terms are so designated because they involve the dynamiccll behavior of only one electron; the rest of the electrons Ln the molecule are assumed to provide an average or static field in which the “oneelectron” must move when it responds to the incident light. ‘The (‘iwo -elect ron” terms involve the coupled
aJC
693
SYSTEMS
motions of two electrons, each localized in a different group, when they respond to the incident light. In considering only the ligand-field transitions of trigonal dihedral metal complexes we can negl.ect terms of type (a) and terms of type (d) if we take as our chromophoric group an octahedral 1 4 L cluster. This is possible since all pure d -+ d electronic transitions in this centrosymmetric system are electric-dipole forbidden. Certain of these transitions are, however, magnetic dipole allowed so that both (b) and (e) types of terms can contribute to the total rotatory strength. The relative importance of the “one-electron” and the “two-electron,” electric-magnetic coupled oscillator terms is determined principally by the chromophoric properties of %he ligand groups. I n order for the electric-magnetic coupled oscillator mechanism to be effective three conditions must be satisfied: (a) the transition energy of the electric-dipole transition in the perturber group (ligand) must not be greatly different from the transition energy of the magnetic dipole ligand field transition to which it is coupled; (b) the coupled electric and magnetic oscillators must have large parallel components; and, (c) the distance between the coupled chromophores must not be large since the interaction, in general, has a steep inverse dependence on R, the group separation distance. The one-electron terms also have strong dependence on the geometrical dispositions and radial. distances of the perturber groups with respect to the chromophoric group of interest. Although they are not directly dependent upon the energies of pertu~berstates, they are strongly influenced by the sign and magnitude of the average, net charge density on each perturber group or atom. I n the limit of completely nonoverlapping charge distributions between groups, the oneelectron terms receive contributions only from those perturbers that carry a “net” charge. Furthermore, the magnitude of the one-electron terms depends upon the differences in transition energies betxeen the electric-dipole allowed transitions and magnetie-dipole allowed transitions, centered in the NIL6 cluster, which mix in the presence of the dissymmetric ligand field. It is fairly eaay to speculate about the relative importance of various types of bidentate ligands in promoting the electric-magnetic coupled oscillator (e-m CO) mechanism. For example, diamino ligands such as ethylenediamine (en) and propylenediamine (pn) have intense, electric-dipole allowed transitions which only appear in the vacuum ultraviolet region of the spectrum, far away from the visible ligand-field Iransitions. The e-m CO contributions to the ligand-field rotatory strengths should be weak in this case. On the other hand, conjugated ligands such as the acetylacetonate ion jacac), o r t h o p h e n a n ~ ~ ~ o l(o-phen), ~ne (1) I. Tinoco, Advan. Chern. Phys., 4, 113 (1962). (2) J. A. Schellman, J. Chern. Phys., 44, 55 (19BG).
The Journal of Physical Chmktry, Vol. 76?No. 6,1971
694 and 2,2‘-bipyridyl (bipy) exhibit very intense, electricdipole allowed transitions .( 4 .*) in the near-ultraviolet region. Furthermore, in the tris complexes of these ligands with transition metal ions, these large transition dipoles are properly oriented to dissymmetrically coiiple with the magnetic dipole transitions on the metal. In these complexes, the e-m CO mechanism should makc large contributions to the ligandfield rota,tory strengths. The oxalate ligand represents an intermediate case since its low energy, electricdipole allowed transitions occur in the far-ultraviolet region (210-180 nm). For the tris complexes of conjugated ligands the greatest practical interest lies in the CD bands of the ligand tramitions rather than in the d --t d type metal transitions. These bands can be theoretically understood or ana1,yzed in terms of the electric dipoleelectric dipole coupled oscillator mechanism. Dipole- dipole iiiterae tions between the large transition densities on the three ligands account for nearly all the sign and intensity properties of the observed C D bands. Formally this mechanism is relatively easy to treat and the use of absorption frequency and intensity data can provide the necessary data input for making seminmpirical computations. The absorption frequencies give the necessary energy denominators in the perturbation expressions and the observed intensities give some measure of the magnitude of the interacting transition dipoles. The geometrical factors in the interactjon potential can then be deduced and the desired structural and stereochemical information obtained. It is not so straightforward, formally or computationzi2ly, to extract structural information from GD which aidises firom one-electron or e-m CO mechanisms. I n these cases, the accompanying absorption intensities are weak and usually arise from vibronic ratlior than static perturbations. Furthermore, neither the exact nor the approximate nature of the interaction potential functionsi are as clearly understood in these mechanisms BP chey are in the dipole-dipole coupling case. I n a previms paper,4 both the one-electron and the e-m 60 mechanisms were considered in deriving sector rules for the (3D of optically active complexes of the pseudotetragonal class. Sector rules relate the “relative” signs and mEtgnitudes of rotatory strengths (and, therefore, CD bands) to the spatial configuration of perturber groups about the chromophoric center of interest. Since thLey are based strictly on group theoretical arguments they can tell nothing about “absolute” signs and magnitudes. It was shown in ref 4 that the sector rules for the “net” rotatory strength assolciated with It11e ligaxid-field transitions of pseudotetragonal ccamplexes were identical for the one-electron and the e-tn CO mechanisms. This result implies that the CD spectral properties predicted on the basis of only a ontxlectron treatment should not be qualitaThe Journal of Physica:l Chemistry, Vol. 76,No. 6, 1971
3‘. S. RICHARDSON tively altered if e-m CO effects were subsequently introduced. I n the present treatment of trigonal dihedral complexes only the one-electron mechanism will be considered. Our reasons for this choice are threefold: (a) all of the essential, symmetry-controlled, qualitative aspects of the problem are displayed by this model; (b) the one-electron model permits a straightforward use of the crystal-field and simple ligand-field formalism; and, (c) vibronic effects can be easily represented (if not solved) on the one-electron model, whereas these effects introduce considerable complications on the e-m CQ model,
11. Previous Theories MoffittE treated the problem of optical rotatory power in trigonal dihedral metal complexes by: (1) assuming that to zeroth order the relevant electronic wave functions of the complex could be constructed from a cubic (0,) basis set composed of 3d metal orbitals; (2) applying an ungerade ligand-field potential of D3 symmetry to the zeroth-order electronic states; (3) permitting the 4p metal orbitals to mix with the 3d orbitals under the influence of the D3 ungerade potential. Perturbation of the zeroth-order states was carried out to first order in this model. Moffitt applied the model to the ‘Alg --9 lTT,transition in trigonal dihedral Co(II1) complexes and to the *A?, T z g transition of Cr(II1) trigonal dihedral complexes. Sugano,6 working within the framework of MofIitt’s physical model, gave an elegant treatment of the problem in which he deduced, contrary to MofEtt’s conclusions, that an ungerade trigonal field of Tzuoctahedral parentage acting on an octahedral complex cannot give rise to a net rotatory strength for the degenerate d --c d transitions. Sugano’s conclusions were based on the uniquely deduced arguments of group theory and therefore depended on the physical model only insofar as the zeroth-order states were assumed to be cubic and the ligand-field perturbation was of D3 symmetry with Tzu octahedral parentage. Sugano further showed that, to first order, only a ligand field of AI, symmetry can induce a net rotatory strength in the d -+ d transitions of octahedral complexes. If the perturbing ligand field potential is expressed as a multipole expansion, the first term which transforms as AI, in 0, is of the ninth order wit,h respect to electronic coordinates. Even if the 3d metal orbitals were mixed with 4f metal orbitals the direct product of the two sets spans in0 representation of the rotation group of order higher than five. Hamer has generalized Sugano’s conclusions to show that, for any transition metal complex whose symmetry includes an inversion -)r
(3) B. Bosnioh, Accounts Chem. Res., 2 , 266 (1969). (4) F. S. Richardson, J. Chem. Phys., in press. (5) W. Moffitt, ibdd., 25, 1189 (1956). (6) S. Sugano, ibid., 33, 1883 (1960).
FTIGAL
ACTIVITY O~F TRIGONALLY DISTORTED CUBICSYSTEMS
center, only a ligan Id potential which transforms as a pseudoscalar or irreducible representation can give rise to a met rotatory strength to first order.7 Poulet modified Moffitt's model to allow some trigonal splitting to occur within the degenerate octahedral ' n other words, it was postulated electronic states.* E that a component o f the DBligand field transforming gerade in Oh lifted the degeneracy of the zeroth-order octahedral st stes. Of course the "net" rotatory strength of a given d -+d transition was still computed to he zero (Sugano's result), but in Poulet's model the two components of the transition could be observed as separate Cotton effects in the ORD spectrum. The Cotton effects wouid be of equal magnitude, opposite in sign, anti separated by an energy interval equal to the trigonc 1 splitting. The experimental circular ichroism dlat a obtained for [Co(en)3I3+ show two ands, one ncgntivo and one po~itive.~-'l Furthermore, by measuring, the @D of the crystal, 2(k)D[Co(en)r]Clo.B-aC1' f 0, perpendicular to the trigonal axis of [ C ~ ( e n ) ? ] ~ - + ' , CD band associated with the lAA1 -+ 'E component of the 'Al, 3 lT1, transition was identified. l 1 Thew iexperimental results confirmed the presence of trigonal. splitting in the degenerate transition; however, it was found that the CD of the trigonal -.+ 'E) was quite a bit more intense component (ljp,li than that of the "4, -=+ IAAzcomponent. The net rotatory strength of the degenerate transition is not zero. These results do not negate the correctness of Sugano's fornzal :reatment, but point up the inadecies of 1 iof5ft's and Poulet's physical models. iper and Haripides h a w developed both a crystal field model12 arid a molecular orbital modeli3 for optical activity in irigonnl dihedral metal complexes. Their crystal field treatment is an extension Qf Mofitt's model to include trigonal splitting of the excited deenerate statea and the mixing of 4f metal orbitals with the 3d orbitals. I n their molecular orbital model the electric-dipole orbidden transitions between u antibonding orbitals wow electric-dipole intensity from the metal-to-ligand charge transfer transitions. In calculating the rotatory strength for [Co(en)313+, they assume that the ligating nitrogen atoms have been angularly displaced from their octahedral reference positions by 4" in a plane defined by a presumedly planar chelate ring. This physical model results in a net rotatory strength for the 'AI, 3 lT1, transition, although the relative magnitudes of the component I A I --p 'E and ' A I -+ 'A2 rotatory strengths are not in agreement ~sithexperimental data. Furthermore, the sign of the net rotatory strength depends upon whether the h - G o - N angles in the chelate rings are 990' or 4%> 30" > 4%> 270" cos :3& < 0
2toe !$Oe' 333''
A 30" > (bi 150" @.I 270" >
=-
> 330" > 90" > 210"
cos 34.1 > 0 cos 34( > 0 cos 34p, > 0
If we further :tssume that the signs of both qe and qj are identical for all pairs of perturbers (i, j ) , then AZ0A3*> 0 for the A isomer and < 0 for the A isomer. Therefore, h-Rnet(Tzg) > 0, A-R,,t(Tz,) < 0, A-RnOt(Tlg) > 0, and A-Rnet(T1,) < 0. Note that the signs of the net rotatory strengths are independent of the absolute signs of the group charges, qe and qj, so long as they (the group-charge signs) are identical. The "net" rotst,ory strength of the Az, -+ T,,transition is determined by second-order terms. However, if the trigonal compoiients of this transition are split sufficiently then the CD spectrum will exhibit two bands, one due to the Az --9 AI, component and the E, component. Since other arising from the Az the first-order ro tatory strengths are considerably larger than the secoiid-order contributions, the signs of these individual CD konnds will be determined by the signs of the first-oi*der results given in Table XI. Since the products of radial integrals in PQpd and in F Q f d are >O and the cwrgy differences, Ed and Ef, are > 0, the signs of the first-order terms are determined by the sign of the function eAS3 (eq 32). Furthermore, sin > 0 for all n o ~ ~ ~ ~ g a trturbers ~ n g and cos 3& > 0 for the b isomer and cos < 0 for the A isomer. If we assign a "net"' positive charge to each of the nongating perturbcs sites, then eql < 0 for all sites i
705
and the signs of the component first-order rotatory strengths of the Az, 4 Tz, transition are as follows A isomer R'(A1)
< 0;
R'(E,)
A isomer %(AI) 3 0; .&?'(]E,)
>0 130° and simultaneously elongating the complex :]long Ibe C,(Z) axis would give the functions (3 cos2 el, -- 1) and cos 3 eL signs opposite to those for (3 cos2Or -.- 1 and cos 3 I $ ~for a given A or A isomer. In this case, 3 g~ is negative and pi is positive, then the splitting induced b y the distorted L atoms is of the same sign as that caused by the i atoms. Again, the sign of the second-order rotatory strength is uniquely determined by the arrangement A or A of the chelate rings. (See Figure 5 €or definitions of distortion operations on the RTL8 cluster.)
'
c. POLQ
CWPRESSIO~
j
/
POLApl ELOKOATIOR
Figure 5. Distortion operations on ME8 cluster.
Only two types of trigonal distortion involving the ligating atoms can effectively scramble the signs of the second-order rotatory strengths associated with the h and A isomcrs. If, in a A isomer, the azimuthal angles were expanded and the complex was simultaneously h compressed along the C,(Z) axis, the sign of cos q would be positive and the sign of (3 cos2 OL - 1) would be negative. These signs are identical with those obtained by elongating a A isomer along the C3(Z) axis while simultaneously expanding the azimuthal chelate ring angles to >60°. Likewise, if both isomers were compressed and one has its L-R/I-L azimuthal ring angles expanded while in the other these angles are contracted, the two isomers would have second-order rotatory strengths of the same sign. In Table XI11 we have summarized the effects expected to occur if the ligating atoms are distorted from their octahedral positions and if the influences of the nonligating atoms are neglected. We refer to changes in the L-14-L chelate ring azimuthal angles as expansion or contraction from 60°, and to changes in the size of the ME6 cluster as elongation or compression of the complex along the C,(Z) axis. The isomers A(a), A(d), A(a), and A(d) are those most likely to occur in real systems. If a complex were compressed and its L-M-L angles were greater than 60" the geometry of the MLe cluster would deviate significantly from that of a regular octahedron. Likewise i f a complex were expanded and the L-M-L chelate ring angles were The Journal of Physical Chemietrgs Vol. 76,No. 6,lQ7l
S. RICHARDSON
708
the bridging structure will distort the 4 geometry of MLBsuch as to make the L-M-L chelate ring angles
Table XIU: Dintortions of MLs Cluster Iaomer
Distortion operations
Aa Ab
Expansion elongation Eixpansion compression Contraction elongation Contractioin compression
AG A
A&
Ab AG
Ad
+ +
-+ -+
+ -++
Expansion elongation Expansion compression Contmctio n elongation Chtractio n compression
-+
K
Rnet(Td
O O
>O
O O
O
0; if t,he:y are attractive, K 6 < 0. 6 . IncreastxI Bing Size. If we assume that the MLs cluster retains its Lan geometry and that the chelate rings are planmL',six-membered rings should induce a lrtrger rotatory strength than de, five-membered rings so long as the radial distances, R, are not significantly increased. This is simply because the number of perturbilng sites is hciaased. Assuming ring pucker and the le1 form For trjs complexes, the rotatory strength should still be higher than for the five-membered ring analogs. NOWif we introduce the likely condition that
greater than go", we find that the trigonal splitting should be increased significantly. This follows from the fact that now the ligating atoms and the nonligating atoms both contribute negative splitting factors. I n general, the distance between the nonligating atoms and the metal ion in six-membered chelate systems will be slightly greater. This will tend to decrease the trigonal splitting and the rotatory strength. 6. The Sign and Magnitude of the At present there is considerable controversy concerning the sign and magnitude of the trigonal splitting in the first excited singlet (ITxg) state of [Co(ea>,I3+. From the polarized crystal spectra of (A 1- [Rh(en)aCL]z. NaCl.6H20 doped with ~1 mol. yoof (+)-[Co(en)gl8++, Denning determined K to be 4-3.5 cm-1.23 Dingle analyzed the polarized crystal spectrum of (+)- [Co(en)aCls]* NaCl.6Hz0 and reported the splitting to be approximately zero (K = 0 i2 cm--1)e24 spectrum of (+)-[Co(en)3ja+ in solution exh bands in the region of the IAIg -+ ITre: absorption.10 These bands are of unequal intensity and opposite in sign. The lower energy CD band (493 nm) is positive and somewhat more intense than the higher energy CD band (428 nm) of negative sign. T h are separated by about 3100 cm-'. Mason also measured the CD of the uniaxial crystal, (+)-[Co(en)&1,1z.NaCl.6H&, with the radiation propagated along the Ca rotation axis of the [ C o ( e n ) ~ ] ~ + complex ion.1o I n this case only the CD of the 'AI 4 'E component of the 'AI, 'TI, t r a n ~ i t i ois~observed; the measurements show a single positive at 475 nm which is much more intense t gous band found in solution. These res cited as evidence that the positive band in the solution due to the 'Al CD spectrum of ( + ) - [ C ~ ( e n ) ~ ]is~ + 'E component of the 'AI, transition and that the less intense negative band arises from the AI 'A2 component. If this interpretation is correct, then the sign of the trigonal splitting is negative; that is, the 'E component of the lT,, state lies lower in energy than the lAAz component in the presence of the trigonal environment. It is difficult to assess the magnitude of the splitting from the CD spectrum since considerable overlap of the two CD ba The results of Denning and Dingle, mentioned above, indicate that the trigonal splitting is very small in the crystal. The CD results indicate that the trigonal splitting is of finite magnitude in solution and is consistent with the possibility that it is small. A negative value of K is consistent with the CD results in solution. According to our model the vicinal effects originating with the ligating atoms and with the non-
-
-
(23) R. G. Denning, Chem. Commun., 120 (1967). (24) R. Dingle, db&., 304 (1965).
-
--+
OPTICAL .kCTIV.I'GY
01F
TRIGONALLY DISTORTED CUBIC SYSTEMS
ligating members of the chelate rings are additive. If the CoNGcluster is (distorted from an octahedral configuration sa that 125.3" > BL > 54.7", then the individual contributions of each ligating atom L to K will be positive. This corresponds to compression of the complex along the P8 axis. The contributions of the nonligating atoms to K will, in general, be of negative sign. The net value of the splitting parameter K is given by a sum of tmms. The individual terms originate with the various perturbing sites which constitute the trigonal en'sironnnent. If the effective charge q on the perturbing site is negative its contribution to K will be positive (when 125.3" > Bp > 54.7"); if q is positive then the contribution to K i s negative. If the nonPigating as well as the ligating atoms are presumed to be effective in pert 11rbing the electronic distribution at the metal ion, then the net value of K could be quite small due to cancellat ion of the oppositely signed terms in the expression fcr K * This could possibly account for the small magnitude of K determined from the polarized crystal spectra studies, A4ccordingto our model, a very small net value of K does not neoesaarily lead to a vanishing second-order rotatory strength or to a small net rotatory strength within the I A I e -+ lTlg transition. The net rotatory strength is obtained from the sum of contributions made by the individual perturbing sites in the trigonal environment. The sign of each term is determined by the sign of pp2 cos ?qp sin3 0,(3 cos2 8, - I), where p labels the perturbing s i k s . If the complex is compressed along the trigonal axis and if the N-M-N chelate ring angles are contracted from 9Qo,the terms arising from the ligating atoms and those arising from the nonligating atoms arp of the same sign in the expression for the second-order rotatory strength. The signs and magnitudes of the individual contributions to K are gaged by qp(3 (v)s2 8, - 1). The sign of this function depends upon the sign of qp, as has been explained previously. Although the individual contributions to K may sum to zero QP to a very small net value, the sum over the individual contributions to the second-order rotatory strength will not vanish so long as there are perturbing siteis not situated on the nodal surfaces of the function cob 3& sm20,(3 cosz Bp - 1). All the terms in the sum that determines the rotatory strengths are of the game sign SO long as both the ligating and nonligating atoms are in 1 he same sector. In order, the:i, to rationalize the observed CD spectra of [ C ~ ( e n ) ~i ]i ~solution + and in a crystal, it is necessary be o f finite magnitude and negative in sign. According to the static crystal-field model, a small trigonal isplittjng does not preclude a large net rotatory strength wiihhin the lA,, .--t 'TI, transition, if the nonligating atoms are effective perturbing sources. I n section I'd we dealt only with the static part
7Q9
of the more general model discussed in section 111. More specifically, we assumed that the behavior of the electrons responsible for the optical rotatory properties could be described in terms of the eigenfunctions of the wave equation (He -
HevM
=
E$
(38)
where H , is defined by eq 13 and H,, i s defined in eq 9. All the nuclei in the complex ion are clamped in their equilibrium positions and each electron is assumed to move in the electrostatic field created nuclei and the rest of the electrons. To obtain a more complete description of any electronic property of the complex, it is necessary to treat explicitly the coupIin@;s between the electrons and the vibrational motions of the nuclei. As stated earlier, a detailed treatment of the vibronic effects on the optical activity of m e t d complexes will not be given here. However, it is of some interest to consider in a very qualitative way how vibronic interactions of the Jahn-Teller type might influence the trigonal splitting parameter K , and our interpretation of the two CD bands observed in the region of the 'Alg +- lT1,transition of ib-Co(en)P-. If vibronic effects are to be included, an immediate problem is to determine a t what stage in the perturbation sequence they should be introduced. For example, in the systems of interest here, the low-lying electronic excited states are orbital triplets in an octahedral field. There is strong experimental evidence that in these cases the most important Jahn-Teller (J-T) interaction is between an eg y i b ~ a ~ ~mode ~ n aand l the orbital triplet. This interaction nally distorted excited state of M L ~ W level is triply degenerate. The wave functions associated with this energy level are eigenfunctions of the vibronic Hamiltonian. Although degenerate in energy, they are spatially nondegenerate an correspond to distortions along the three tetragonal axes of the cubic system. I n complexes with true trigonal dihedral symmetry, the static perturbations on ML6 will trigonal distortions. If, to zeroth or degenerate electronic states are expressed in terms of a cubic basis, the perturbation matrix due to the trigonal field has only OR-diagonal elements ( ~ A Athe ~ orbital components of the electronic triplet slate mix). I n the absence of vibronic interactions, the component electron-vibrational wave functions of the electronic triplet state can be written as, $&, $,&, ${&, where the vibrational functions # v are identical for each electronic orbital function 5, q , and t. The trigonal perturbation matrix elements depend, t electron coordinates since, ($t#v\Vtri = for each Y However, if the electronic state can effectively couple to an eg mode resulting in a tetragond J-T distortion, then the vibronic wave functions must be written as
$,>(#+l#v>
(+J~~,i,l+,)
The Journal of Physical ChemistrpJ, VQE.76, No. is, 1971
5. RICHARDSON
710 f, &.here 40 4,, and 4t are displaced oscillator functions located a t the three minima on the J-T distorted potential energy surface. Now the perturbation matrix dements due to the trigonal field must be written, for example, as (+e~clVeigl+,d,) = (+e/Veigl+,:(4,14,), where the overlap factor is not equal to unity. When the J-T effect is very strong then approaches aero and the trigonal perturbation is completely quenched. I n any case, the presence of e,-T, coupling \vi11 quench the effects of any operator which is off-diagonal In the cubic bases. This quenching (partial or complete) of an electronic trigonal perturbation on khe orbital triplet states of cubic systems by dynamical J-T coupling is just a specific example of the Ham effect. I t was pointed out by HamZ6 that profound changes in such observables as spin-orbit splittings, trigonal field splittings, and g factors in orbital tripiet states, can occur as a result of the dynamic J-T eff act. ‘These changes are collectively known as the Ham effect. It is also pomible for a triply degenerate electronic state in a regular octahedral system to suffer a J-T distortion by coupliing with the tPg vibrational mode. I n this case, however, the vibronic Hamiltonian is diagonal only in a trigonal (Dad) basis. Consequently, the matrix eltments for electronic operators which are also diagonal in a trigonal basis remain finite in the extreme J--T coupling limit. The Jahn-Teller effect due to T--tzgcoupliing lieads to a distorted potential energy surfac 3 for Ihe ground vibronic level which has four minima, m e along each trigonal axis. I n addition to aitering the potential energy surface (e, and tPgmodes) or mixing the orbital components (tzgmode) of a par1 icular orbital triplet state, vibronic coupling via the eg m d tzemodes can also induce mixing between two different triply degenerate electronic states Le.g., the ITI, and ITzgin Co(II1) and the 4T1, and 4Tz,in Cr(I1I) 1. These “interstate” interactions will, of course, be much weaker than the “intrastate” interactions since, in most systems of interest, the adjacent triplets are separated by 5000-8000 cm-’. Another typP of “interstate” interaction whose consideration i s eseential for understanding the optical absorption spectra of cubic or nearly cubic metal complexes originates with the coupling of the ungerade tl, and lzU vibrational modes with the electronic states. This coupling through the ungerade modes induces mixing between the gerade and ungerade electronic states and accounts for nearly all of the observed intensity in the ligand-field bands of cubic and nearlycubic transition me tal complexes. This coupling (commonly called ;he Nerzberg-Teller intensity mechanism) has little effect 011 the energies of the electronic states involved, but produces the electric transition dipoles essential f a r observable light absorption. The matrix elements for any operator whose orbital part is exclusively off-diagonal in the real representa-
(&1$,)
($El+, )
The S;?urnalof Phyricecl Chem&?try, Vol. 76,No. 6,1071
tion forced on the triply degenerate electronic term by vibronic coupling with the eg vibrational mode are reduced by the factor Y .- (4sj+,) = (4Cj4S) = (4,/#f)* This applies to the orbital angular momentum operator C whose nonvanishing matrix elements within the TI, vibronic state are -$ .
(WsI4l+t4t) = (#,4,14I+E+O = (+f4tllzi$+,+,>
=
itfir (39)
where i t has been assumed that the electronic functions can be taken as pure metal 3d orbitals. This will result in a quenching of spin-orbit coupling and in a reduction of the magnetic moment of the ‘TI, state. Evidence that the ‘TI, excited state of [Co(en)aI3+undergoes a strong Jahn-Teller distortion has recently been reported by Denning.23 Using magnetic circular dichroism, Denning determined the magnetic moment of the excited ‘TI, state and subsequently deduced a reduction factor c0.05. The extraordinarily small value observed for the trigonal splitting in [ C ‘ O ( ~ I I ) ~ ] ~ + could also be attributed to a strong tetragonal distortion of the complex (a manifestation of the static 6-T effect). Denning has further suggested that the two CD bands observed in the IA1, --* T I g region arise from two different J-T vibronic states which are derived from the ‘TI, electronic state rather than from the two trigonal components ‘AI and ‘E. If we assume that the 7’1, electronic state is strongly coupled to an eg vibrational mode, the lowest vibronic level resulting from this coupling is an orbital triplet whose component wave functions may be written as: +&,OO, $q4q,o0, and $&,oo, .vc.here the subscripts 00 on the vibrational wave functions 4 indicate that both components of the e, mode are in their ground vibrational levels. The next lowest vibsonic level is sixfold degenerate and is composed of two orbital triplets, one transforming as TI, in On and the other as 2 1 2 , . The basis functions for these vibronic states are: +t4g,o~, *P4(,10, SRn,Ol, $rl+bn,lO, W f . 0 1 , and ‘I/sdr,10, where the subscripts 01 and 10 QXP q5 specify that one of the components of the e, vibrational mode is singly excited. The energy level scheme for the three lowest vibronic states that result from Tl,-e, coupling i s ”
,Td> T 2 g i q < < g -
-,
,
/
----
E’
.
/
/
TI,-%
,------El
/
-41
c-___-
A2
dep) I
-. - 7 - W
---
K - - - L E o
Tlg-eg Couplirig (25) F. S. Ham, Phys. Rev., 138, A1727 (19BS),
‘Frigonal Field
OPTICALACTIIITYOF TRIGONALLY DISTORTED CUBICSYSTEMS
711
where TIg1and Tzelrepresent vibronic states in which turbation potential, the ligand environment is reprethe e, mode i,j singly excited, and TlgOrepresents the sented as a constellation of classical point charges whose vibronic state in which the eg mode is unexcited. vo(e,) overall symmetry is D3. In such B representation, is the fundamatal frequency of the eg mode and K is overlaps between the charge distributions centered on the trigonal spl ihting,parameter the R/ILBcluster and those centered on other sites in the Expanding on Dlenning's suggestion, it might be ligand environment are completely neglected. This supposed that the two CD bands associated with the procedure of representing the ligand environment by a ]A,, ITzgeleiitronio transition in [ C ~ ( e n ) ~ ] ~ + afrom rise constellation of point charges is an acceptable approxiseparate trtmeitions from the ground state AIg to the mation so long as each atomic site is included and the two vibronic i~wels,Tle0and (TIgl, Tz,'). If this were expansion of the perturbation potential is carried t o a the case, then l%ieband maxima should be separated by sufficiently high order. the fumdamentd frequency of the e, mode, vo(eg) = It was shown that the net rotatory strength associated em-'. Furthermore, both bands should have with the complete manifold of ligand-field transitions components which are polarized Ito the 6 3 axis of the vanishes to first order on the one-electron model but is trigonal system Additi , the signs of the net nonvanishing if second-order contributions are conwith the two vibronic rotatory strengths R S S O ~ sidered. The second-order contributions to the rotatransitions shmld be the same. The CD data retory strength arise from the simultaneous actions of a ported by MoCaEFery and Masonlo on the uniaxial trigonal perturbation (P,) with gerade octahedral percrystal, (+)-[C0(en)~@1~] .NaCI.6H20, show that only centage and a trigonal perturbation (Pu)with ungerade the low-frequency C Y 3 band apparently has a compooctahedral percentage. The contributions due to d-p i the Cyaxis. Furthermore, a simple nent polarized -to and d-f orbital mixing under the influence of P,, an analysis does not reveal how the two CD bands obd-d mixing under the influence of P,, were considered in served in the sdution spectra can have opposite signs if detail. The influences exerted on the optical rotatory they are aasuaned to arise from transitions to the two properties by chelate ring conformation, ring size, vibroriic levels, ?'lgt' and @xgl, T,,'). substituents, and distortions of the octahedral At this point we must reject the suggestion put forth ctjons were anade. cluster, were also examined and p by Denning concerning the CD spectra of [ C ~ ( e n ) ~ ] ~ + . The results obtained to first I- in perturbation The interpretation of the CD data based on the static theory are similar to those previously obtained by modelLpresented in section PV appears to be qualitaR/loffitt,5Piper and Karipedes,l* an oulet8on a static, tively correct, and :t small trigonal splitting is not inone-electron model. I n carrying static, one-elecconsis tent with this interpretation. The reduced tron model to second order in pertu ation theory, we magnetic montent of the excited TI,state reported by obtained results which are in close eement with the Denning does, laomever, indicate the presence of a experimental CD data and which easily interpreted in strong J-T coupling; between the eg vibrational mode terms of specific structural features of trigonal dihedral and the TI, cAlectronic state. The influence of this metal complexes. Reduction of the general model to a coupling on the optical rotatory properties should be crystal-field representation of the electronic slates of greatrr detail than has been done here. the complex ion further aids in elucidating thp essential mary symmetry-determined aspects of the oplieal rotatory e primary purpose of this study was to examine properties. Although the static, one-electron model is the optical rotatory properties of trigonal dihedral not appropriate for making accurate quantitative caltransition metal complexes on a one-electron model in culations of the rotatory strength, it provides a simple which the chromophoric elec trons localized on an octaand correct representation of the nodal structure in the hedral M L cluster are subjected to a static trigonal ~ o in r ethe electronic states of the NIL6 c ~ r o ~ o ~ and field provided by the ligands. The trigonal field origiinteraction potential between the I l k , ~chromophore nating with tlie ligand environment was assumed t o be and the ligand environment. The simplicity of this small comp:tred to the essentially octahedral (0,) field representation and the facility with which it can be exerted by the MLe oluster. The wave functions of the used in making correlations between G D data and chromophoric electrlons responsible for the ligand-field molecular structure make it particularly attractive. transitions were developed to second order in a perAb initio quantum mechanical calculations on the turbation exp:tasicm in which the static trigonal field class of complex ions considered in this paper are, a t potential was treated as the perturbation operator. present, out of the question. Even minimum basis set I n order to avoid complications introduced by exchange SCF wave functions for these systems cannot be obeffects and yE*t retain the essential symmetry-detertained with sufficient economy to justify the effort. mined aspects of the optical rotatory properties, the On the other hand, semiempirical xnolecular orbital trigonal field potential was represented by a singlemodels are not yet sufficiently accurate or reliable for center m ~ l ~ i ~ el ~ansion ~ a r (crystal-field expansion) making useful calculations of the eiectronie spectral out the metal atom. In this expansion of the perproperties of transition metal complexes except when ~
-
The Journal of Physical Chemistry, Vol. 76, No. 6,1971
7 12
NQTES
extensive calibration calculations can be performed. If the parameters of the molecular orbital model are calibrated for ont: or two spectral properties of a few systems belonging to a specific class of complex ions, then it is possible in some cases to compute the same properties for other members of this class. This procedure has been used with some success in determining the relative transition energies in certain complexes. However, its appljcatiori to the calculation of the rotatory strengths associated with d + d transitions appears to have little promise at this time. Optical activity can be classified as a second-order optical property and its extreme sensitivity to the details of the electronic structure of the overall ;system requires that very accurate wave functions be used in calculating the rotatory strengths. Ir the absence of accurate electronic wave functions, a theoretical analysis of the optical rotatory properties i s beat accomplished by determining the symmetry-~ontrolletl aspects of the problem. This type of analysis Pltm been of enormous importance in correlating the stereochemical properties of optically active systems with experimental ORD and CD spectra. Theoretically, the success of such an analysis depends upon the correctness with which the nodal surfaces of
Electron Spin Resonance Spectroscopy of the adlical (SOSF .) in Solution. Line Width and Its Temperature Dependence'
by Pa,ul 1\4. Nutkowitz and Gershon Vincow*12 Department of Chemistry, University of Washington, Seattle, Washiitgton 981196 (Received August IO, 1970)
the molecular electronic states are represented. If a one-electron perturbation model is adopted (as was done in this study), then the nodal surfaces of the predominant perturbation potential functions must also be correctly represented. So long as quantitative calculations of rotatory strengths are not our goal, it would appear that the optical rotatory properties of metal complexes can be most clearly analyzed and understood in terms of an independent systems model in which the metal atom and the ligands are treated as nonoverlapping sub=systems. For the d + d transitions, the crystal-field model becomes appropriate if the ligands have no strong electric-dipole allowed transitions in the visible part of the spectrum.
Acknowledgments. I wish to express my gratitude to Professor W. J. Kauzmann for his discussions and helpful suggestions concerning this work. I also gratefully acknowledge the support provided by the n'ational Science Foundation through a grant t o W. Kauamann, Princeton University, and a grant to F. R. administered through the Center for Advanced Studies, University of Virginia.
In this note we report measurements of the temperature-dependent line width of the fluorosulfate radical, S03F., in peroxydisulfuryl difluoride, S206F2,and in a fluorocarbon solvent. The contributions of a variety of mechanisms to the line width are discussed in order to establish which is the principal line-broadening effect.
Results
Publication coats assisted by the U.S. Army Research Ofice, Durham
Our results for the line width of 503F. in its neat liquid dimer SzOeFz are given in Figure 1. The width
There has been considerable recent experimental and theoretical interest in the esr line width of small inorganic free radicals and ions in liquid solution, and in the variation of these widths with temperature and solvent v i s ~ o s i t y . ~The experimental results have been explained on the basis of a variety of relaxation mechanisms such as motional modulation of the anisotropic electronic Zeeman interaction, the electron-nuclear hyperfine interaction, the spin-rotational interaction, and the Ohrbsch process. The possibility of a chemically shortened lifetime as the dominant line-broadening mechanism hao also been i n ~ o k e d . ~
(1) This work was supported by the U. S. Army Research Office, Durham. 12) Alfred P. Sloan Foundation Research Fellow; to whom inquiries should be addressed. (3) (a) P. W. Atkins, N Keen, and M. C. R . Symons, J. Chem. SOC.,2873 (1962); (b) 8 . Fujiwara and H. EIayashu, J . Chem. Phys., 43, 23 (1965); (c) R. Wilson and D. Kivelson, ibid., 44, 154 (1966); (d) P. W. Atkins and D. Kivelson, ibid , 44, I69 (1966); (e) W. B. Lewis, M . Alei, and L. 0. Morgan, ibid., 44, 2409 (1966); (f) D. Kivelson, ibid., 45, 1324 (1966); (9) J. Q. Adams, ibid., 45, 4167 (1966); (h) N. Vanderkooi, Jr., and R. T. Poole, Inorg. Chem., 5 , 1351 (1966); (i) J. R. Thomas, J. Amer. Chem. Soc., 88. 2064 (1966); (j) P. W. Atkins, Mol. Phys., 12, 201 (1967); (k) P. W. Atkins and M. T. Crofts, ibzd., 12, 211 (1967); (1) R. E. D. McClung and D. Kivelson, J. Chem. Phys., 49, 3380 (1968); (m) L. Burlamacchi, Mol. Phys., 16, 369 (1969). (4) D. M . Gardner and 6. M. Fraenkel, J. Amer. Chem. Soc., 78, 3279 (1956).
The Journal of Phy8ical t7hemistry, Vol. 76, No. 8, lQ7l
