Lanthanlde 4f
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3579
J . Phys. Chem. 1984, 88, 3579-3586
4f Electrlc Dipole Intensity Theory
Michael F. Reid and F. S. Richardson* Department of Chemistry, University of Virginia, Charlottesville, Virginia 22901 (Received: November 7, 1983)
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A general parametrization scheme is proposed for the electric dipole intensities of one-photon, no-phonon and one-photon, one-phonon 4f 4f crystal-field transitions in lanthanide complexes. It is shown how the parameters in this scheme may be used to characterize the relative line intensities observed in the 4f 4f vibronic spectra of lanthanide systems, and how certain parameters may be useful as diagnostic probes of lanthanide ligand interaction details. It is further shown how these general, phenomenological intensity parameters can be calculated or rationalized in terms of two specific models for the lanthanide-ligand-radiation field interactions. On the basis of these models, the relative signs and magnitudes of the intensity parameters may be correlated with various aspects of ligand structure, ligand-field geometry, and lanthanideligand interaction mechanisms. These correlations lead to a number of spectra-structure relationships important to predicting or rationalizing the optical properties of lanthanide complexes. Special attention is given to intensity effects which may arise from ligand polarizability anisotropy. -+
I. Introduction The optical properties of most trivalent lanthanide complexes in the near-ultraviolet, visible, and near-infrared spectral regions 4f can be accounted for in terms of intraconfigurational 4f radiative transitions.’** The magnetic dipole strengths of these transitions generally can be calculated or rationalized entirely in terms of spectroscopic states constructed within the 4fN electronic configuration of the lanthanide ion and chosen to be eigenstates of the even-parity crystal-field Hamiltonian. On the other hand, the electric dipole strengths of these transitions can be rationalized only by considering spectroscopic states of mixed parity, thus requiring a basis that extends outside the 4fN configuration and the inclusion of odd-parity terms in the crystal-field Hamiltonian. In centrosymmetric systems, the odd-parity crystal-field perturbations are provided by fluctuating crystal-field potentials created by the excitation of odd-parity vibrational modes in the ligand environment. For these systems, all of the 4f 4f electric dipole intensity is found in vibronic lines displaced from electronic origins by the frequencies of the promoting (odd-parity) vibrational modes, whereas the magnetic dipole intensity is generally found only in origin lines. For A J = 0, *1 (excluding J = J’ = 0) transitions, the magnetic dipole origin lines are generally more intense than the electric dipole vibronic lines, whereas the opposite generally holds true for AJ = 2, 4, and 6 transition^.^-^ This reflects the strong influence of the “free-ion” intermediate-coupling selection rules ( A J = 0, f l ; J = J’ # 0) on the magnetic dipole allowedness of transitions between multiplets. In noncentrosymmetric systems, the most important contributions to the odd-parity crystal-field perturbations generally arise from the charge distributions of the ligands fixed in their equilibrium positions. The contributions arising from ligand positional fluctuations (Le., vibrations) will, in these cases, generally be much 4f electric and magnetic dipole intensities smaller. The 4f observed in the spectra of these systems are distributed almost entirely among electronic origin lines. Correlations between 4f 4f electric dipole intensity and ligand structure are of significant importance in lanthanide coordination chemistry and in the design of lanthanide optical materials. On
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(1) B. G. Wybourne, “Spectroscopic Properties of Rare Earths”, Interscience, New York, 1965. (2) S. Hufner, “Optical Spectra of Transparent Rare Earth Compounds”, Academic Press, New York, 1978. (3) T. R. Faulkner and F. S. Richardson, Mol. Phys., 35, 1141 (1978). (4) T. R. Faulkner and F. S. Richardson, Mol. Phys., 36, 193 (1978). (5) J. P. Morley, T. R. Faulkner, F. S. Richardson, and R. W. Schwartz, J . Chem. Phys., 75, 539 (1981). (6) Z. Hasan and F. S. Richardson, Mol. Phys., 45, 1299 (1982). (7) J. P.Morley, T. R. Faulkner, and F. S. Richardson. J. Chem. Phvs..
77, 1710 (1982). (8) J. P. Morley, T. R. Faulkner, F. S.Richardson, and R. W. Schwartz, J . Chem. Phys., 77, 1734 (1982).
the one hand, they can be used to determine coordination numbers, coordination geometry, and the extent of complexation under various conditions (e.g., solvent medium, solution pH, or crystalline host). On the other hand, they can aid in the systematic search for complexes which will satisfy specific requirements for optical opacity or transparency in a given material medium. They are also important to the systematic use of lanthanide ions as optical probes in a variety of molecular and solid-state systems. There is one class of lanthanide 4f 4f transitions in which the electric dipole intensities exhibit extraordinary sensitivity to the ligand environment. These are generally referred to as hypersensitive transitions and, as a class, they all follow electric quadrupolar selection rules with respect to AJ (and, less faithfully, AL and AS). There has been considerable effort devoted to elucidating the intensity-ligand correlations associated with these transition~.~-’~ The first step in developing intensity-ligand structure correlations for lanthanide optical spectra is to parametrize the observed intensity data with a scheme sufficiently general to accommodate all of the detailed lanthanide-ligand-radiation field interaction mechanisms which might be operative. The phenomenological parameters obtained from such a general parametrization treatment can provide gross structural correlations, applicable to large classes of systems, which are free of mechanistic prejudices. For the isotropic, one-photon spectra associated with J multiplet to J multiplet lanthanide transitions, the 0, (A = 2, 4, 6) parameters derived from the J ~ d d - O f e l t ’ ~intensity ~ ’ ~ theory provide a completely general parametrization for electric dipole intensity (including vibronically induced contributions). We have recently described general intensity parametrization schemes for both the one-photon,no-phonon and one-photon,one-phonon electric dipole spectra associated with transitions between crystal-field l e ~ e l s . ‘ ~ J ~ This work is an elaboration of a general parametrization proposed earlier by Newman and Bala~ubramanian.’~ The QA parametrization of lanthanide intensity data has been used extensively, and many useful generalizations regarding spectra-structure correlations have been obtained from such ana lyse^.^ However, there is a paucity of detailed parametrizations of intensity data obtained on crystal-field transitions, even though the parameters derivable from these data carry significantly more
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(9) R. D. Peacock, Struct. Bonding (Berlin), 22, 83 (1975). (10) D. E. Henrie, R. L. Fellows, and G. R. choppin, Coord. Chem. Reu., 18, 199 (1976). (1 1) F. S. Richardson, J. D. Saxe, S. A. Davis, and T. R. Faulkner, Mol. Phys., 42, 1401 (1981). (12) B. R. Judd, J . Chem. Phys., 70, 4830 (1979). (13) B. R. Judd, Phys. Rev., 127, 750 (1962). (14) G. S. Ofelt, J . Chem. Phys., 37, 511 (1962). (15) M. F. Reid and F. S. Richardson, J. Chem. Phys., 79, 5735 (1983). (16) M. F. Reid and F. S. Richardson, Mol. Phys., 51, 1077 (1984). (17) D. J. Newman and G. Balasubramanian, J . Phys. C, 8, 37 (1975). (18) J. D. Axe, J. Chem. Phys., 39, 1154 (1963).
0022-3654/84/2088-3579.$01.50/0 0 1984 American Chemical Society
3580 The Journal of Physical Chemistry, Vol. 88, No. 16, 1984
information about structure and lanthanide-ligand interactions than do the 0, parameters. Furthermore, in those cases where parametrizations of crystal-field intensity data have been reported, they have been based on the (A,E(t,X)) parameter set introduced by Axe.’* This parameter set is less general than those proposed and by Reid and by Newman and Balasubramanian (AJKM),I7 Richardson (A1Xp),15and it provides a complete parametrization basis (within the one-electron, one-photon approximation) only when all lanthanide-ligand (Ln-L) pairwise interactions in a system can be considered cylindrically symmetric and independ) the J = K f 1 subset ent.l5.l7 The t = X f 1 subset of ( A A (or of (AJKM)) encompasses the entire (AfZ(t,X)}parameter set. The “extra”, t = X (or J = K ) parameters are needed when either of the above-mentioned conditions on Ln-L interactions (cylindrical symmetry and independence) is not ~atisfied.’~,’’We reiterate again that, in their phenomenological form, the 0, and A& (or AJKM)intensity parameters are independent of the details of the Ln-L-radiation field interactions (except that they be of a oneelectron, one-photon nature). Detailed interpretation of the general intensity parameters in terms of specific ligand properties (chemical or structural) requires models for describing the lanthanide-ligand-radiation field in4f electric dipole absorption and teractions essential to 4f emission processes. The development of such models represents the most important, and most difficult, step toward constructing useful and reliable spectra-structure relationships for lanthanide optical properties. At the most general level of consideration, the models proposed so far can be divided into two classes: those which include effects due to Ln-L orbital overlap explicitly, and those which do not. Newman has been the leading proponent of the former, and he has presented forceful (and compelling) evidence for the importance of overlap-dependent contributions to 4f 4f electric dipole i n t e n ~ i t y . ’ ~However, most quantitative cal4f intensity culations (and qualitative rationalizations) of 4f spectra have been based on models in which the lanthanide 4fN and ligand electronic charge distributions are assumed to be nonoverlapping. The latter are generally referred to as electrostatic models and they fall into two distinct classes: 11~12,20 (1) those in which the charge distributions on the ligands are unresponsive to (i.e., not coupled to) the radiation field; and (2) those in which the ligands’ electronic charge distributions are assumed to be perturbed by the radiation field. The former (1) are often referred to as static-coupling models, while the latter (2) are often called dynamic-coupling models. [Note that dynamic coupling in this context implies ligand electron-radiation field coupling and not 4f electron-ligand vibration (Le., vibronic) coupling.] The point-charge crystal-field model employed in the original JuddOfelt13J4intensitystudies is an example of a static-coupling model, whereas the ligand polarization model introduced by Mason, Peacock, and Stewart9q21-22 is an example of a dynamic-coupling electrostatic model. From most studies reported so far in the literature, it appears necessary to include both static-coupling and dynamic-coupling mechanisms in carrying out intensity calculations based on electrostatic (nonoverlap) models. For negatively charged, spherically symmetric ligands, the point-charge (static-coupling) and dipolar polarizability (dynamic-coupling) contributions to the A : , , , ~(or A , + ~ , E ( A + ~ , X ) intensity ) parameters are of opposite signs, and only the static-coupling mechanism can contribute to the A;-,# p a r a m e t e r ~ . l ~ * ~ ~Therefore, ,*~ the static-coupling mechanism is essential to rationalizing the A;-,,p parameters, and
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Reid and Richardson both mechanisms must be considered in order to rationalize (at least) the signs of the parameters. For systems containing ligands in which Ln-L cylindrical symmetry cannot be assumed, the A& parameters may become nonvanishing and the staticcoupling and dynamic-coupling mechanisms may make either additive or subtractive contributions to the A:,,,? intensity parameters. Effectively, this means that the electric dipole transition moments induced via the static-coupling and dynamic-coupling mechanisms may interfere either constructively or destructively in producing electric dipole intensity. 15,20 In the present paper, we give the expressions necessary for analyzing lanthanide 4f 4f electric dipole intensity data in terms of the A; parametrization scheme (and its vibronic analogue). We also give expressions showing how the Ah parameters may be interpreted or calculated within the framework of an electrostatic (no-overlap) model which includes both static-coupling and dynamiecoupling mechanisms for the lanthanide-ligand-radiation field interactions. In the latter, we restrict our consideration to effects due to ligand point charges and induced dipolar polarizations (static and dynamic). In the paper which immediately follows, we report intensity analyses and calculations on two systems with complex ligand structures.
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11. General Parametrization Scheme A. Electric Dipole Transition Moments. Consider a transition,
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Bo, between the a-th vibrational level of electronic state A, A and the P-th vibrational level of electronic state B. The electric dipole moment associated with this transition may be written as where -eD’- is the electric dipole moment operator, eA and are electronic state functions, and xA, and x B are vibrational wave functions. In the consideration of 4f 4! transitions, it will be convenient to partition each 0 function into a part that depends only on the 4f-electron angular coordinates and a part that includes everything else which is relevant to the transition process (e.g., the 4f-electron radial functions, lanthanide wave functions outside the 4fN configuration, ligand-localized electronic wave functions, and the nuclear coordinate dependence of the 0 functions). Denoting the former parts as \kAand *B, and the latter parts as @ A and aB,we may rewrite eq 1 as
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where ( @Ax,JD1-I@BxBB) is evaluated over all but the 4f-electron angular coordinates of the system (including nuclear positional and vibrational displacement coordinates). Following the usual practice, we replace (@Ax&ID’-I@BxB,)with an “effective”electric dipole transition operator which we denote here by V. This operator acts only within the 4fNconfigurational manifold of the lanthanide ion. The q-polarized component of MA,Bs may now be written as (3)
In general, the dependence bf V, on nuclear displacement (vibrational) coordinates can be quite complicated. However, we assume here that this dependence can be satisfactorily accounted for by expanding V, about the equilibrium nuclear geometry to terms linear in the normal coordinates of the system. Performing this expansion, we may rewrite eq 3 as
(ed,,)
MA,BB,q
(19) Y. M. Poon and D. J. Newman, J. Phys. C, in press. (20) F. S. Richardson, Chem. Phys. Lett., 86, 47 (1982). (21) S. F. Mason, R. D. Peacock, and B. Stewart, Chem. Phys. Left.,29, 149 (1974). (22) S. F. Mason, R. D. Peacock, and B. Stewart, Mol. Phys., 30, 1829 (1975). (23) M. F. Reid and F. S. Richardson, J . Less-Common Met., 93, 1 1 3 (1983). (24) M. F. Reid, J. J. Dallara, and F. S. Richardson, J . Chem. Phys., 79, 5743 (1983). (25) J . J. Dallara, M. F. Reid, and F. S. Richardson, J . Phys. Chem., following paper in this issue.
=
-e(qAIV,(o)lqB)
e
(XA,IXBB)
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c (*AIV,(’)(dv*j*)t*B) dvj
(XA,IQdvjlXBB)
(4)
where Vq(0) is just V, evaluated with the nuclei of the complex fixed in their equilibrium positions, and VJ’)(dv*j*)= (aV,/ c ~ Q ~Assuming ~ ~ ) ~ identical . harmonic force fields and vertically disposed potential surfaces for states A and B, eq 4 reduces to MAmBs,q
= -e(*AlVq(o)lqB)8~fi e (*AlVq(’)(dV*j*)lqB)(Qdvj)G,(d”j).p(duj)+l ( 5 ) dvl
Lanthanide 4f
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The Journal of Physical Chemistry, Vol. 88, No. 16, 1984 3581
4f Electric Dipole Intensity Theory
where a(duj) and P(dvj) denote occupation numbers for the duj mode in the a and P vibrational states, respectively. (In our notation, u is an irrep label for a vibrational mode, j labels the components of a degenerate mode, and the label d is used to distinguish between modes having the same irrep.) The first term in eq 5 is applicable to an A B (4f 4f) no-phonon transition, whereas the second term applies to the one-phonon vibronic transitions associated with an A B excitation (or deexcitation). B. Effective Transition Operator. The “effective” transition operator, V, introduced above is defined to operate only on the angle-dependent parts of the 4f-electron wave functions, and it acts only within the 4fNconfigurational manifold. It is convenient, therefore, to expand this operator in terms of a set of intraconfigurational unit tensor operators which span all 4fN states, and to treat the associated expansion coefficients as a set of generalized parameters for V. The detailed physics of the lanthanide-ligand-radiation field interactions represented by V are then contained in these parameters. Following our previous work,15the VJo)operator of eq 4 may be expressed (in the J M basis) as follows:
where the A:-:.,,*i. are obtained as coefficients in the expansion
- -+
(-l)”’f(2t
+ 1)1’2
(11)
(Substituting v = 0 into eq 11 recovers eq 7.) The (A:-:,,*(du*)} may be used to parametrize the VJ1)(du*j*)vibronic transition operator and, therefore, the (A!-&dv*) ( Q d , ) ) provides a parametrization basis for the one-phonon, du-vibronic transition moments (see eq 5). Note that whereas the A:->o nonuibronic parameters transform as the identity irrep in the symmetry group of the system, the A$cc,,,,(du*)uibronic parameters transform as the conjugate irrep of the active vibrational mode (dv). The maximum number of A:-&du*) parameters needed to characterize the du-vibronic line intensities in the 4f 4f spectra of a lanthanide complex is equal to the number of times u is contained in t-, for each value of X ( = 2 , 4, 6). For example, in a system of Oh symmetry a total of 10 parameters may be no!vanishing by symmetry for the l-(tzu) vibrational mode, since 1occurs once in t- = 2-, 3-, 4-, and 5-, and twice in both t- = 6and t- = 7-. Thus, there are six t = X 1 parameters and four t = X parameters in this case.I6 We now have a set of parameters, {A;-;} or {A;->o),for the nonvibronic (zero-phonon) electric dipole transition moments which is completely general within the one-electron, one-photon approximation for lanthanide-ligand-radiation field interactions. We also have defined a set of parameters, A;-cev*(dv*), for the electric dipole transition moments associated with one-phonon vibronic (4f 40 transitions. This set of parameters is completely general within the one-electron, one-phonon, one-photon approximation. The completeness (or generality) of each parameter set requires that all t = A, X f 1 (A = 2 , 4, 6) members of the set be included (consistent with crystal-field or vibrational mode symmetry constraints). However, in the special case where all Ln-L interactions can be assumed to be cylindrically symmetric (Cmulocal symmetry) and independent, it can be shown that thy t = X parameters vanish (see section 111). In this case, our A:parameters have a one-to-one correspondence with the A,Z(t,Xf parameters of the Judd-Ofelt intensity theory.I5 C. Electric Dipole Strengths. The electric dipole strength of Bo, may be expressed as (from eq 3 ) the transition, A,
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where X = 2 , 4, 6, t = X, f 1, p = 0, f l , ..., f t , and the Ut’ are unit tensor operators which operate on the 4f-electron wave functions (P). [Note that conjugation of the A$ coefficients in It will be convenient eq 6 is given by (A$)* = (-l)t+l+J’A:-:pJ for further development to rewrite eq 6 in a point-group basis. Following the notation of Butler,26 we obtain
(-l)A+t(2r
+ 1)”2
(7)
where a, 6, and c are branching multiplicity labels, i is an additional basis label which is necessary if q is not a one-dimensional irrep of the symmetry group, 0 denotes the identity irrep, and the superscripts f are parity labels which convert the SO3 irrep labels into O3irrep labels. The (A$] or (A:->,] coefficients of eq 6 and 7 may now be used as general parametrization bases for the effective transition operator, V i o ) ,and for the nonvibronic (zero-phonon) electric dipole transition moments (see eq 5). Ignoring mechanism-dependent constraints, the number of nonvanishing A:-; parameters which may occur for each ( X , t ) pair of values is equal to the number of occurrences of the identity irrep in 1- (which is determined by the point-group symmetry of the system). For centrosymmetric systems, V i 0 )vanishes, since the identity irrep cannot be contained in t- for these systems. Occurrences of the identity in t- (=1--7-) are listed in Table 1 of ref 15 for 22 different noncentrosymmetric symmetry groups. The VJ’)(du*j*) operator of eq 4 may be expressed (in a point-group basis) as follows: I 6
where, as before, X = 2, 4, 6 and t = A, X A 1 . Recalling that the A:-:,y,(du*) parameters are given by
(26) P. H. Butler, “Point Group Symmetry Applications: Methods and Tables”, Plenum Press, New York, 1981.
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DA,BB.q
= e21(PA1Vq’41PB)12
(12)
For a = P = 0 (Le., a nonvibronic transition), we have DAB,q = e21(QAIV4(0)IQB)12
(13)
which, using eq 6, may be rewritten as
c A:-~C(X+l,l’qlt-p)(-1)q(~Al~~+l\k,)12
DAB,q = e21
A,w
I
(14)
where P A and P Bare eigenfunctionsof the even-parity crystal-field Hamiltonian. Expanding P A and P B in a JMj intermediate; coupling basis (within the fN configurational manifold), the Ut matrix elements in eq 14 may be expressed as linear combinations of matrix elements of the form (4fNJI[sL]JMJIu:+14f~~[s’~1J’MM’,). The latter are subject to the selection rules: IAJI 5 6, unless J or J’ = 0, in which case lAJl = 2, 4, 6. The overall selection rule for DAB,qis qTB 3 Y ~where , Y~ and Y~are the irreps of states A and B in the symmetry group of the system. Equation 14 gives the q-polarized component of the electric dipole strength associated with the purely electronic A B (4f 4f) crystal-field transition. The total isotropic dipole strength of this transition is given by
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DAB
=
(l/3)CDAB,q
(15)
where q = -1, 0, + l . Now consider a pure J multiplet to J’multiplet transition in which all the sublevels of the initial J state are assumed to be equally populated. If we further assume isotropic radiation and
3582 The Journal of Physical Chemistry, Vol. 88, No. 16, 1984
randomly oriented systems, the total nonvibronic electric dipole strength of this transition may be expressed as
D
= e%
JJ~
x
Qxl
+
( [SL]JI IUx’I IV [S2 1 1J’)1’
( 16 )
knowledge of the relevant Ux’bp‘ matrix elements in eq 19. Application of eq 16 to the parametrization of Jmultiplet-to-J’ multiplet transition intensities requires only that the SLJ compositions of the initial and final states be known. In most cases, this can be assumed to be given satisfactorily by the “free-ion” wave functions for the respective multiplet levels.
(17)
111. Intrinsic Intensity Parameters
where Oh
= (2X
+ l)-l c tfl
The Ox’s are the familiar Judd-Ofelt intensity parameters for multiplet-to-multiplet transitions. The q-polarized component of the electric dipole strength for the one-phonon vibronic transition, A, B l ( d v ) , is given by -+
DA,Bl(du),q
= eZCl( \kAIV,(’)(dv*j*)l\kB) J
121(edv)
1’
(’ 8,
For this one-phonon excitation process, in the harmonic approximation, I(QdY)[2 = ( h / 8 7 f 2 C ~ d , ) ,where gdv is the fundamental frequency of the du vibrational mode (expressed in wavenumbers). Using eq 8 for the V : ’ ) ( d v * j * ) vibronic operator, eq 18 may be rewritten as
The overall selection rule for DAoB,(dv),4is yB(qv*) 3 yA,where yA and y B are the irreps of the crystal-field states A and B , Y* is the conjugate irrep of the du activating mode, and q is the radiation polarization direction. For isotropic spectra, the total A, Bl(dv)electric dipole strength is given by -+
where q = -1, 0, +l. D. Intensity Parametrizations. For spectra resolved and assigned in terms of individual crystal-field transitions, eq 14 provides the basis for a general parametrization of electric dipole intensities observed in the zero-phonon origin lines. Of course, an important prerequisite to this intensit parametrization is an accurate knowledge of the relevant U, matrix elements which, in turn, requires accurate descriptions of the crystal-field wave functions (\kA and in eq 14). The latter are most commonly obtained as eigenfunctions of a parametrized even-parity crystal-field Hamiltonian in which the parameters are chosen to yield eigenvalues in optimum agreement with experimentally determined crystal-field energy levels. The quality of these wave functions may be further checked by using them to calculate Zeeman splittings and magnetic dipole transition strengths, recalling that these quantities, to first order, depend only on the angular parts (Le., the JMj compositions) of the 4f-electron wave functions.Iq2 Lack of accuracy in the U:’ matrix elements of eq 14 will not necessarily degrade the quality of the data ”fits”, However, it will tend to contaminate the empirically determined ”best-fit” A$ parameters with effects not intended in the basic parametrization scheme, and this will preclude the accurate calculation or rationalization of the parameters in terms of specific models. More specifically, the A;-p parameters have been defined to reflect only those aspects of electric dipole transition intensities which depend upon the odd-parity components of the lanthanide-ligand interactions, and they are not intended to absorb any effects due to the even-parity crystal-field potential. It is important, therefore, to have reasonably accurate U:’ matrix elements over the 4felectron crystal-field states of a system before attempting to obtain the A$ parameters from intensity data fits. Equation 19 provides the basis for a general parametrization of electric dipole intensity observed in the one-phonon vibronic lines of 4f 4f spectra. As in the case of the nonvibronic origin lines (discussed above), proper utilization of the parametrization scheme for vibronic line intensities requires accurate
I+
-
Reid and Richardson
For many lanthanide complexes comprised of structurally simple ligands (e.g., atoms or monatomic ions), all of the Ln-L pairwise interactions may, to a good approximation, be treated as cylindrically symmetric and independent. In this approximation, each intensity parameter A$, (and its vibronic analogue) can be expressed as a sum of independent contributions from each of the Ln-L pairs. Furthermore, since each Ln-L pair has C,, local symmetry (Le., cylindrical symmetry), all of the t = X parameters will vanish since the identity irrep is not contained in t- = 2-, 4-, or 6- for the C, symmetry g r o ~ p . ’ ~ , ’ ~ Cylindrically symmetric and independent Ln-L pairwise interactions are two of the basic assumptions inherent in the “superposition”model developed by Newman and his collaborators for parametrizing the even-parity lanthanide-ligand interactions in crystal^.*'^^* The other basic assumption is that the dependence of these interactions on Ln-L radial distances can be represented by simple power-law expansions of the form (R0/RL)‘,where RL is the Ln-L distance and Ro is defined as the value of RL for a particular (reference) ligand in the system of interest. This model leads to a rationalization of the even-parity (phenomenological) crystal-field parameters of a lanthanide system in terms a set of power-law exponents (7) and a set of “intrinsic” crystal-field parameters which are independent of coordination number and geometry. To apply Newman’s superposition model to the odd-parity lanthanide-ligand interactions responsible for generating 4f 4f electric dipole intensity, we define the intrinsic intensity parameters, A:, to be the parameters A:-: ( t = X f l ) for a single ligand at Ro on the Z axis. For an arbitrary arrangement of ligands, we obtain the following expression for the nonvibronic A$ intensity parameters: 15
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where the spherical tensor C!,(L) rotates ligand L from the Z axis of the system, RL is the Ln-L distance, and 7; is a power-law exponent which, in general, will depend upon both X and t . Expression 21 applies only to a system in which all of the ligands are chemically equivalent. For mixed-ligand systems, each chemically equivalent set of ligands will, in general, have a different set of A: and 7: parameters, reflecting differences in the details of the respective Ln-L interaction mechanisms. The intrinsic intensity parameters defined by eq 21 depend only on the nature and identity of the Ln-L pair. They are independent of coordination number and geometry, and they are free of mechanistic assumptions except that each Ln-L pairwise interaction be cylindrically symmetric and independent. The parameters derived for a given Ln-L pair should be transferable from one system to another, independent of any structural considerations. This transferability of the intrinsic intensity parameters has extremely important implications for predicting (or rationalizing) the 4f 4f intensity spectra of structurally complicated systems from data obtained on relatively simple systems (whose spectra are amenable to detailed analysis). The vibronic intensity parameters, A:-:*v.(dv*), may also be expressed in terms of intrinsic parameters which depend only on Ln-L pairwise interactions (with the atoms fixed in their equilibrium positions). To derive these expressions, the ~LC$(L)(RO/RL)~’~ terms in eq 21 must be differentiated with respe$ to Qdpj and evaluated at Qdri = 0. As an example, the A$-T-( 1-) intensity parameters for the 1-(tZu)mode of an octahedral
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(27) D. J. Newman, Adu. Phys., 20, 197 (1971). (28) D. J. Newman, Aust. J . Phys., 31, 79 (1978).
Lanthanide 4f
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The Journal of Physical Chemistry, Vol. 88, No. 16, 1984 3583
4f Electric Dipole Intensity Theory
(Oh) LnL, system are given byI6 A$yi-(I-) = -(15)1/2d$/R~M~1/2(22) where X = 2 or 4,and MLis the mass of L. A complete listing of the A:,.,.(dv*) parameters, expressed in terms of the intrinsic parameters, are given in Table 3 of ref 16 for the l-(tlu) and l-(tzu) vibrational modes of an octahedral (Oh) LnL6 system.
8:
IV. Electrostatic Intensity Models In this section we show how the general intensity parameters described in section I1 may be calculated or rationalized in terms of two different models for the lanthanide-ligand-radiation field interactions. In both models the electronic charge distributions on the lanthanide ion and the ligands are assumed to be nonoverlapping, and the interactions between these charge distributions are described in terms of an electrostatic perturbation potential, H! The essential differences between the two models involve (1) differences in the basis state manifolds within which "operates, and (2) differences in the complex-radiation field interaction modes. In the model referred to here as the static-coupling (SC) model, H'is defined to operate within a basis set comprised only of electronic states localized on the lanthanide ion, and the electric dipolar components of the radiation field are presumed to interact directly with the 4f electrons. In the model referred to as the dynamic-coupling (DC) model, H' is defined to operate within a basis comprised of lanthanide 4fN states and a set of electronic excited states localized on the ligands. In this model, the electric dipole radiation interacts with the complex via the ligand-localized electronic charge distributions. Referring to eq 2, 4, and 6 of section 11, the central task for our model calculations of nonuibronic intensity is to find expressions for the V p )transition operator. Using the notation of section IIA, we have for the A B transition
In our treatment here, we assume that H'is comprised entirely of electrostatic interactions between nonouerlapping charge distributions on the lanthanide ion and the ligands. Furthermore, we+restrict our attention to the nonvibronic intensity parameters A;-,, defined in section IIA for the V,O)effective transition operator. A . Static Coupling. Following J ~ d d ' ~ .and " our previous workI5 the static-coupling VJo)operator may be expressed as V,(o)[SC]=
Z. (2h h,t,p
+ l)AtpB(t,h)C(-l)l 1
(25)
where X = 2, 4, 6, t = X i 1, and the E ( t , X ) are defined by eq 14 of Judd.I3 This equation may be reexpressed in terms of our A$ parameters as follows:
Therefore (2X+ 1) A:-;[SC]
= -A,,"(t,X)
(2t
+ 1)1/2
(27)
The E(t,X) quantities are entirely independent of ligand properties, depending only on the electronic structure of the lanthanide ion. The ligand dependence of the A:-;[SC] parameters is contained entirely in the odd-parity crystal-field parameters, A,. Representing the ligands as having isotropic and polarizable charge distributions with a net charge of eqLand a mean (static) polarizability of at, the contribution of ligand charge (chg) to the A , parameters is given by
-+
where the superscript 0 indicates evaluation with the complex fixed in its equilibrium geometry. Expanding both @: and to first order in H', VJ0)may be expressed (to first order in H') as
where qL denotes the charge on the L-th ligand (in units of e), (OL,$L,RL) are the positional coordinates of the L-th ligand, and the CLPare spherical tensors. The contribution from the dipolar polarization of the ligand charge distribution by the lanthanide ion is
Vio) = -C(RA&lDFle) ( e ( H I R B d g ) / A e ~ e
- C(RAdgIHle)(elDFIRB&)/A,A e
(24)
where 6 denotes the ligand electronic ground-state wave function (definecf as a product of individual ligand wave functions), and RA and RB denote the radially dependent parts of the zeroth-order 4fN wave functions. In the static-coupling model, the intermediate states e are constructed from 4fN-' nl configurations on the lanthanide ion (where nl must be an unoccupied orbital of even parity). In the dynamic-coupling model, the intermediate states e are ligand excited states accessible by & e electric dipole processes. The first-order perturbation expression 24 provides the starting point for most of the electrostatic intensity models described so far in the literature. In the Judd-Ofelt intensity theory,I3J4 VJo) is described entirely in terms of static-coupling contributions, whereas the ligand polarization intensity model of Mason, Peacock, and Stewart2'+2zis based entirely on the dynamic-coupling contributions to VJ0). If the nonoverlap condition (for lanthanide L electron-exchange and ligand orbitals) is relaxed, and Ln terms are included in HI,then the (e}basis set can be expanded to include Ln L charge-transfer states. The possible importance of including at least the lowest-energy Ln L charge-transfer states (4fN+lX,where A denotes the highest occupied ligand orbital) in 4f 4f intensity modeling has been argued by Henrie, Fellows, and Choppin,lo especially for Sm(II1) and Eu(III), which have relatively high optical electronegativities. This, however, would provide only a partial representation of the overlap- and covalency-related intensity effects suggested to be important by Ne~man.'~.~~
-
-
-
-
(29) D. J. Newman, personal communication.
where qLnis the charge on the lanthanide ion (in units of e ) . We note that for a single negatively charged ligand the A,[chg] and A,[pol] parameters will have the same sign. In our survey of intensity parameters for systems with isotropic l i g a n d ~ , *we ~%~~ calculated the contributions from both A,[chg] and A,[pol]. However, eq 29 is a rather poor approximation, because it does not include the polarizing effect of ions other than the lanthanide. Attempts to rigorously treat these effects, and higher-order polarizabilities, are fraught with convergence problems.30 In light of these complications, we shall include only the charge contribution in what follows. Our justification for this approach is that point-charge calculations of even-parity crystal-field parameters generally yield the correct signs, though not the correct magnitudes ( N e ~ m a n , ~Table ' 9). B . Dynamic Coupling. In the dynamic-coupling model, the intermediate states (e)of expression 24 are electronic excited states localized on the ligands, Df; operates only on ligand-localized electronic wave functions, and H' is an electric multipole (Ln)-electric dipole (L) interaction operator. Combining terms and evaluating the matrix elements in eq 24 over all ligand electronic coordinates, one obtains VJo)[DC] =
( RAIr$RB)RL-("+')( CA+(j)Cf'-(L)]iJ(-l)q'aq-ql(L) (30)
where X = 2,4, 6, t ' = X
+ 1,j labels the 4f electrons, CA'(j) and
(30) D. Garcia, M. Faucher, and 0. L. Malta, Phys. Reu. B, 27, 7386
(1983).
3584
The Journal of Physical Chemistry, Vol. 88, No. 16, 1984
C r ( L ) are spherical tensor operators, and ad(L) is a component of the L-th ligand’s dynamic electric dipolar polarizability tensor (measured at a frequency of v = ABA/h).The a J L ) is a measure of the ligand electrons’ dipolar response to the electric field of the incident radiation. More precisely
= U-1)9a94L)E9
P q m
9
(31)
where pqt(L) is the electric dipole moment induced in ligand L by the radiative electric field component E,. In what follows, it will be convenient to work with the polarizability tensors expressed in an irreducible basis. The transformation needed in going from the (qq? spherical basis to the (nm) irreducible basis is given by
a< =
C
(l-q,l-qln+rn)a,,,
(32)
994’
where n = 0, 1,2. Inverting eq 32 and substituting the result into eq 30 gives (setting t’ = X 1)
+
V$o)[DC] = (A
-cc C(
+ 1)(2X + 1)(2X + 3)
Reid and Richardson pendicular to the symmetry axis (al). In an irreducible basis we may write ((28)~’
+
= - ( 1 / 3 ) ’ ~ ’ ( ~ ~ ~ ’2 a l ’ ) ~= -(3)’/’&~
(38)
- al’)L
(39)
(a:),’ = (2/3)’/’(~i11’
where the primes denote reference to a local coordinate system on L, &L is the mean (isotropic) polarizability of L, and PL is the polarizability anisotropy. All other components, (a“,L’, of the local polarizability tensor vanish. To obtain the ak(L) required to evaluate the Zp(’+’.n)tparameters defined by eq 35, we have 4 X L ) = (~X)L’C“~(~L’~+L’) (40) where n = 0 or 2, and Cm(0L’,+L’) rotates the local coordinate system on L to one parallel to that defined for the overall complex. Evaluating eq 35 separately for n = 0 and n = 2, we may write
(rA) R ~ - ( X + Z ) (
=7
A:->[DC,Z]
(;
0“ i) + [(A
)$ x
( l - q , l - - q l n + m ) * ( - l ) p l a ~ ( L(33) ) where we have set (RAIrjhJRB)= (r’). Following the procedures given in ref 15 (eq 31-34), our eq 33 above can be reduced to the form
where @’ is the intraconfigurational unit tensor operator (acting on the 4fN wave functions), and L
(41)
1)(2A
+ 1111’2 x
(r*)(--I ~ P ~ ~E, CA ! +~ ’~ ) - ( L ) R ~ - ( ~ +(42) ~)~~
~(X+’)-(L)
Ayp[DC,p] = -7(5)’”
n,m
Zp(A“+n)’= Z[(2n
+ A,”i[DC,P]
A;i[DC] = A:-i[DC,&] where
3
A 9’ i.L
= (2/3)’/‘&
+ 1)(2t - l ) ( A + 1)(2A + 1)(2h + 3)/3]’”
X
(35)
wheret = X + 1 whenn = 0, t = X, A + 1 whenn = 1, t = X, X 1 when n = 2, and p is restricted to the components of t- that transform as the identity irrep in the symmetry group of the system. Finally, we can rewrite eq 34 in the form
q
A2
A1 + 1
‘3
t
(0
’)
0 0
[(A
+ 1)(2A + 1)(2A + 3)]’’2 X
(- l)pZ {C(A+”-(L) a2+(L)}!;RL-(*+2) L
(43)
where, from eq 39 and 40, a z ( L ) = ( 2 / 3 ) ’ / 2 P ~ C ~ ( e ~ ’ , In ~~’). expression 42, t = X + 1, whereas in expression 43 t = A, X f 1. Expressions 42 and 43 are less general than eq 37 insofar as they assume either spherical or cylindrically symmetric ligands (or ligand fragments) with S-type electronic ground states. However, in the great majority of applications, the ligand environment about the lanthanide ion can be partitioned in such a way so as to make these assumptions valid. C. Calculations of Electrostatic Intensity Parameters. The static-coupling and dynamic-coupling electrostatic mechanisms, as formulated here, make independent contributions to the “effective’! electric dipole transition operator Vi0). Therefore, we may write
v p = V9(o)[SC]+ V,(O)[DC]
(44)
and (considering only the ligand charge contributions to SC) A$ = A$[SC,chg]
+ A$,[DC,&] + A:-i[DC,P]
(45)
[ 3 / ( 2 t + 1 ) ] 1 ~ 2 ( v h ) ( - 1 ) P Z ! ~ ’ ~ n ) t ~ ~ U ~ + ( h t l , l ~ - q ~(36) t - p ) ( - 1 ) 4where A$[SC,chg] is given by eq 27 and 28, A$[DC,&] is given 1
which, from eq 6, leads to the following expression for the dynamic-coupling intensity parameters:
(37) Expressions 34-37 are completely general within the ligandpolarization dynamic-coupling model employed here, and they contain no restrictions regarding ligand structure and symmetry. It will be useful, however, to cast them in an alternative form which is more explicit with respect to their dependence on ligand structural properties of special interest. For most lanthanide complexes of chemical interest, the ligand environment can be conveniently partitioned into monatomic and/or diatomic fragments, and ligand charge and polarizability can then be distributed among atoms and diatom chemical bonds. Assuming an S-type ground state for each of the ligand fragments (L), each aL can be represented as a symmetric second-rank tensor quantity which may be brought to a diagonal form in a local (ligand) reference frame. Assuming that each ligand fragment has cylindrical symmetry, each local polarizability tensor can be expressed in terms of just two quantities: the polarizability measured parallel to the symmetry axis (all),and the polarizability measured per-
by eq 42, and A$[DC,P] is given by eq 43. The static-coupling contributions are restricted to the t = X f 1 parameters, the [DC,a] contributions are restricted to the t = X + 1 parameters, and the [DC,P] mechanism may contribute to each of the t = A, X & 1 parameters. Calculations of the static-coupling intensity parameters require one set of ligand parameters, (qL), and one set of lanthanide electronic parameters, {E(t,X)). The latter depend on the details of the (e) basis set chosen to evaluate the perturbation expression given in eq 24. Krupke3’ has calculated values for the E(t,X) parameters of six different Ln3+ ions, using both 4fN-’5d and 4fN-’n’g intermediate basis states. Calculations of the dynamic-coupling intensity parameters require two sets of ligand parameters, (aL) and (PL), and one set of lanthanide electronic parameters, (+). Proper use of the {PL)parameters requires detailed knowledge of both the atomic positions and the positions and orientations of the chemical bonds in the ligand environment. V. Qualitative Relationships
Direct calculations of the electrostatic intensity parameters according to the prescriptions given in section IV are relatively straightforward. However, the quantitative significance of the (31) W. F. Krupke, Phys. Rev., 145, 325 (1966).
Lanthanide 4f
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The Journal of Physical Chemistry, Vol. 88, No. 16, 1984 3585
4f Electric Dipole Intensity Theory
results obtained from such calculations will depend quite sensitively on the choice of input parameters (e.g., qL, aL,PL, E(t,X), and (rX)). On the other hand, certain qualitative relationships between the intensity parameters can be deduced without resorting to detailed numerical calculations, and these relationships may have some utility in rationalizing empirically determined values of the parameters with respect to relative signs and magnitudes. A . Isotropic Ligands. When the ligands in a system are isotropic with respect to charge (qL) and polarizability (aL),then within the approximations of our electrostatic model each Ln-L pairwise interaction can be assumed to be cylindrically symmetric. The electrostatic model discussed in section IV also assumes independent Ln-L pairwise interactions. Therefore, in the case of isotropic ligands, it is valid (and convenient) to work with the intrinsic intensity parameters described in section 111. For each chemically equivalent set of ligands in a system, we can write
A: = A:[SC,chg]
+ A:[DC,a]
(46)
where, from eq 21, 27, 28, and 42 At[SC,chg] = (2X
+ 1)(2t + 1)-’/2e2qLRo(‘+l)Z(t,X)
(47)
(48)
+
In eq 47, t = X f 1, but in eq 48, t = X 1. Therefore, in the case of isotropic ligands, the t = X - 1 intensity parameters are determined entirely by the static coupling. Both terms in eq 46 will contribute to the t = X + 1 intensity parameters. The signs of the A”,,, [DC,a] parameters are determined by the signs of the 3-j symbol, which are positive for A = 2 and 6 and negative for X = 4. For negatively charged ligands, the signs of the A:+, [SC] parameters are determined by -E(X+l,X), which is negative for X = 2 and 6 and positive for A = 4. Therefore, in the case of negatively charged, isotropic ligands, the static-coupling and dynamic-coupling electrostatic mechanisms are predicted to make oppositely signed contributions to the t = X 1 intensity parameters. A summary of the signs predicted for the intrinsic parameters (associated with negatively charged isotropic ligands) is given as follows:
+
h= 2 , 6
h=4
+
4
These sign relationships suggest that empirically determined signs for parameter ratios may be useful in determining the predominance of one mechanism over the other in contributing to the Ax+, parameter^.^^,^^ The Ax+l -3 [DC,a]:A:+l [SC,chg] ratios, expressed in terms of ligand and Ln parameters, are given by
where N2 = (14/5), N4 = -(70)’12/3, and N6 = (70/13)(5/11)’/2. When one uses Krupke’s values for Z(X+l,X)31 and Freeman and 2 parameter ratios for the Eu3+ Watson’s values for ( r X ) , 3these ion are evaluated to be as follows: X = 2, 3.24(aL/qL)/A3; X = 4,1.28(aL/qL)/A3;X = 6, 1.01(aL/qL)/A3. These relationships suggest that the X = 2 intensity parameters should exhibit the strongest correlation with the aL/qL. ligand properties and that the dynamic-coupling mechanism will make the dominant contributions to the & parameters in Eu3+ systems. B. Anisotropic Ligands. In general, the anisotropic contributions to the intensity parameters, A:-;[DC,P] (see eq 43), cannot be analyzed in terms of the intrinsic parameters introduced in (32) A. J. Freeman and R. E. Watson, Phys. Rev., 127, 2058 (1962).
section 111. The A; parametrization assumes cylindrically symmetric Ln-L interactions and, in general, this condition is not satisfied by the interactions inherent in the [DC,p] mechanism. The one exception to this is the case in which the ligands are coordinated such that the prirlcipal axis of their polarizability ellipsoid is collinear with the RLn-L vector. In this special case, each Ln-L pair can be assumed to have the necessary C,, local symmetry. For this case, we obtain from eq 43 an expression for the intrinsic intensity parameters: AP[DC,P(iadial)] =
-7(5)l/’ 6 4
{;
(i 0”
;1 +
;)[(A
+ 1)(2h + 1)(2h t 3)]”’
(-lP(fj
+
; A)(?l+ (-l)’-’
1)”’
X
x
(2/3)”’&
(SO)
where t is restricted to X f 1. Note that PL may be either positive or negative, depending on the relative magnitudes of a,,and al. Simple predictive relationships regarding the relative signs and magnitudes of the [DC,P] contributions to the A$ intensity parameters are not immediately obvious for the general case in which the Ln-L interactions are not cylindrically symmetric. In this case, both the signs and the magnitudes of the A:-i[DC,P] parameters may be expected to be very sensitive to subtle details of internal ligand architecture (e.g., the locations of substituent groups and the relative orientations of chemical bonds) and overall coordination geometry. Such relationships may eventually emerge from semiempirical analyses of intensity data obtained on structurally complex systems; but, so far, only two such analyses have been carried out.2S The A$ empirical parameters can provide a unique probe for assessing the possible importance of the [DC,P] mechanism (within the context of the electrostatic intensity model described in section IV), since neither the static-coupling nor the [DC,&] mechanism can contribute to these parameters. It will be shown in the following paper25that these parameters are essential to rationalizing the intensity data obtained on two different europium(II1) systems, each having relatively complicated ligand structures. C. Frequency Dependence. Both the static-coupling and dynamic-coupling contributions to the calculated intensity parameters are expected to decrease with decreasing 4f 4f transition frequencies. The origins of this frequency dependence are readily apparent from the perturbation expression for the “effective” transition operator, V,(O)(see eq 24). The &B energy denominators in this expression will increase in magnitude as the A B (absorption) transition frequency is red shifted. Therefore the magnitude of Y,(O)(and hence the magnitudes of the A$ parameters) will be smaller for transitions lying in the infrared region vs. those occurring in the visible and ultraviolet spectral regions. In the static-coupling model, this frequency dependence is contained in the E(t,X) parameters, as originally defined by Judd.I3 In the dynamic-coupling model, this frequency dependence is reflected in the “dynamic” polarizability parameters, aLand PL. This expected frequency dependence of the intensity parameters suggests that different sets of A;-; may be needed to fit empirical intensity data obtained in different spectral regions. The infrared vs. visible region parameters should be related by relatively simple scaling factors. However, intensity parameters derived from near-ultraviolet and ultraviolet spectra may be expected to exhibit a more complicated frequency dependence, especially for complexes exhibiting relatively low-frequency electric dipole allowed transitions (localized either on the lanthanide ion or on the ligands). D. Lanthanide Dependence. For a given ligand system and total coordination number, the “lanthanide dependence” of the electrostatic intensity parameters is given by the values of E(t,X)RL-(‘+’)for static-coupling contributions and the values of ($)RL-(*+l) for dynamic-coupling contributions. Both the E(t,X) and (rX) decrease in magnitude across the lanthanide series (from Pr3+ to Yb3+),31*32 whereas the values of RL-(t+l)generally increase in magnitude (Le,, RL decreases across the series). Krupke’s analysis of intensity data obtained on lanthanide ions doped into
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3586
The Journal of Physical Chemistry, Vol. 88, No. 16, 1984
Reid and Richardson
-
the YzO3 lattice (a case in which the RLvalues probably do not change significantly with lanthanide dopant) reveals the expected decrease in the magnitudes of the intensity parameters across the series (from Pr3+ to Yb3+).31 It will be of significant interest to obtain accurate intensity data on a homologous series of systems which have identical ligands and ligand-field symmetries, but which differ with respect to the RL values. In this case, one would expect to observe smaller variations across the lanthanide series, compared to those observed in Krupke’s study, reflecting the countertrends of the RL-(,+I)vs. Z(t,X) and ( r X )effects.
due to lanthanide-ligand covalency or orbital overlap, so at best they provide an incomplete representation of the 4f 4f electric dipole intensity problem. However, it seems likely that each of the mechanisms contained in these models will make important contributions to 4f 4f intensity, and in fact these mechanisms provide the basis for nearly all calculations or interpretations reported so far in the literature for 4f 4f electric dipole intensity spectra (however, see ref 10 and 19 for important exceptions). The most important results given in section IV are those showing how, within the dynamic-coupling model, ligand polarizability anisotropy may contribute to the various t = A, X f 1 intensity VI. Conclusion parameters. It was also shown that for negatively charged, isoTwo of the major objectives in this paper were to describe tropic ligands the static-coupling and dynamic-coupling mechageneral parametrization schemes for the electric dipole intensities nisms are predicted to make oppositely signed contributions to 4f no-phonon and one-phonon (vibronic) associated with 4f the Ai:,,, intensity parameters. This result, combined with the transitions, and to show how these parametrizations may be useful finding that onZy the static-coupling mechanism may contribute in developing spectrastructure relationships and in assessing the to the parameters (in the case of isotropic ligands), provides relative importance of various intensity mechanisms. The a means for determining the predominance of one mechanism over A:-; intensity parameters described in section I1 are completely the other from the relative signs of the empirically determined general within the one-electron, one-photon approximation for intensity parameters (see section VA). We have previously lanthanide-ligand-radiation field interactions, and the A&v, s h o ~ that n the ~ relative ~ ~ signs ~ ~of ~ the A&,, ~ ~ vs. &,,, empirical intensity parameters for vibronic transitions are similarly general parameters reported for seven different systems can only be exwithin the one-electron, one-phonon, one-photon approximation. plained (within the context of the electrostatic intensity model) For systems in which all Ln-L pairwise interactions are cylinin terms of dominant dynamic-coupling contributions to the Aill,, drically symmetric and independent, these parameter sets have parameters. The observed sign relationships cannot be explained the restriction that t = X f 1 ( t # A). In this case, the Ai:,,, on the basis of the static-coupling model, no matter how it may parameters can be related to the A,,E(t,X) intensity parameters be varied or manipulated. In the paper that follows,25we shall defined by Axe.’* The latter provide the basis for most intensity show the importance of anisotropic dynamic coupling to rationalizing the intensity data obtained on two lanthanide systems 4f crystal-field analyses reported in the literature for 4f containing structurally complex polyatomic ligands. transitions. For systems in which Ln-L cylindrical symmetry cannot be assumed (or in which the Ln-L interactions cannot be In calculating intensity parameters from models, it is important assumed to be independent), the t # X restriction is lifted and to keep in mind that the contributions from different mechanisms a complete parametrization basis must include all t = X, X k 1 may, in general, be either additive or subtractive. Therefore, the parameters. The t = X parameters have no counterparts in the dipole strengths calculated for individual transitions will include “cross-term” contributions (interferences between different A$(t,X) scheme. mechanisms) which may be either positive or negative in signGZ0 The great majority of lanthanide complexes of interest in It is likely that these interference effects will be important to chemistry contain polyatomic ligands and multidentate chelate systems. For most of these complexes, Ln-L cylindrical symmetry explaining observed s. :ra-structure relationships in lanthanide is a poor approximation. In analyzing the intensity data obtained 4f spectra. 4f on these types of complexes, it is exceedingly important to include Finally, we point out the need for more 4f 4f intensity the t = X parameters m the general parametrization s ~ h e m e . * ~ , ~ ~analyses based on individual crystal-field line intensities (both The intrinsic intensity parameters discussed in section I11 have origin lines and vibronic lines). The A:-; parameters obtainable special appeal for developing systematic schemes for predicting from such analyses contain considerably more information than the Q A parmaeters obtained from analyses of total multiplet-tothe intensity properties of systems having widely different coormultiplet transition intensities. The relative magnitudes and dination geometries and coordination numbers, but identical Ln-L pairs. The idea is that any given Ln-L air has certain intrinsic especially the signs of the A;-> parameters are sensitive to structural intensity properties (embodied in the A , parameters) which are and mechanistic details which are lost (or combined and averaged) in the $2, parameters. Calculations of the latter from models do transferable from one system to another. The parameters are not provide very useful or stringent diagnostic tests for the models. defined to be entirely independent of overall coordination geometry, Unfortunately, very few A;->-type parametrizations (or A,,Z(t,h) but they do assume independent Ln-L pairwise interactions, each parametrizations) have been reported so far in the literature. of which has a local C,, symmetry. Their applicability, therefore, is likely to be restricted to systems containing monatomic ligands. Acknowledgment. This work was supported by the National In section IV we showed how the general intensity parameters Science Foundation (NSF Grant CHE-8215815). Several very can be calculated or rationalized on the basis of two different useful discussions with Professors D. J. Newman and S. F. Mason models or mechanisms. Neither of these models includes effects are also gratefully acknowledged.
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-
-
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-
-R
-
a:
(33) E. M. Stephens, M. F. Reid, and F. S. Richardson, Znorg. Chem., in press.
(34) M. F. Reid and F. S. Richardson, J. Chem. Phys., 80,3507 (1984).
