Vibronic intensities in centrosymmetric coordination compounds of the rare earths Part II. A vibronic crystal field-closure-ligand polarisation model and applications to the PrCl3−6

ELSEVIER Journal of Molecular Structure (Theochem) 390 (1997) 109-119

_____ __~~__~

THEO CHEM

Vibronic intensities in centrosymmetric coordination compounds of the rare earths

Part II. A vibronic crystal field-closure-ligand polarisation model and applications to the PrCli- and UBr% complex ions in octahedral

symmetry

R. Acevedo”‘“, C.D. Flintb, T. Meruane”, G. Muiioza, M. Passmand, V. Pobletee

aDepartment of Basic Chemistry, Faculty of Physical and Mathematical Sciences, University of Chile, Tupper 2069, PO Box 2777, Santiago, Chile

‘Department of Chemistry Birkbeck College, University of London, Gordon House, 29 Gordon Square, London WClH OPP, UK ‘Department of Chemistry, Metropolitan University of Educational Sciences, Av. J.P. Alessandri 774, PO Box 147-C Santiago, Chile

‘Department of Mathematics, City University, Northampton Square, London EC/V OHB, UK ‘Nuclear Energy Chilean Commission, Amunategui 95, PO Box 188-D, Santiago, Chile

Received 26 September 1995; accepted 16 January 1996

Abstract

A symmetry adapted formalism to evaluate the vibronic intensities induced by the ungerade vibrational modes in centrosymmetric coordination compounds of the rare earths is put forward and applied to several selected electronic transitions of the PrClk and UClp complex ions in octahedral symmetry. This current model is based upon a modified symmetry adapted version of the combined vibronic crystal field-closure-ligand polarisation approach. This model differs from that developed in Part I of this series, in that for the vibronic crystal field contribution to the total transition dipole moment, the closure procedure is employed rather than the utilisation of a truncated basis set for the central metal intermediate electronic states. A criterion is introduced to choose an appropriate set of phases for both the electronic and the vibrational coordinates so that to ensure the right sign for the interference term (which couples together both the vibronic crystal field and the vibronic ligand polarisation contributions to the total transition dipole moment). We have focused our attention on the modulation of the intermolecular force field and a version of a modified general valence force field has been adopted. The reasons for using this formalism rather than the superposition model (SM) are fully discussed in the text. Finally, it is shown that the agreement with experiment is satisfactory for most of the components of the transitions studied, despite the apparent simplicity of our model calculation. General master equations applicable to any fN electronic configurations are derived to show the utility and flexibility of this current formalism.

Keywords: Vibronic intensity; Crystal field; Closure; Rare earth complex ions

* Corresponding author.

0166-1280/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved PH SO166-1280(96)04765-3

110 R. Acevedo et al.lJournal of Molecular Structure (Theochem) 390 (1997) 109-119

1. Introduction

In Part I of this series [l], we presented a general formalism to account for the observed vibronic intensities associated with f - f electronic transitions in centrosymmetric coordination compounds of the lanthanide complex ions in octahedral symmetry.

The spectral intensities were rationalised on the basis of a symmetry adapted vibronic crystal field- without invoking closure-ligand polarisation approach. Both the crystal field and the ligand polarisation contributions to the total transition dipole moment, were included and we focused our attention upon the choice of the phases for both the electronic and the vibrational coordinates. Within the framework of the independent system model, the total transition dipole moment is made up of three components, derived from the vibronic crystal field, the vibronic ligand polarisation and a cross term (interference term), which couples together both the crystal field and the ligand polarisation contributions. The interference term, plays a crucial role in the theory since it may either add or subtract intensity, from each individual false origin for the various excitations, so that special care is needed to ensure the right sign for this quantity [l-6]. Our earlier approach [l], included the evaluation of the crystal field contribution jiCF, by considering the vibronic interactions of the initial and final states involved in the transition with a single 5d state. This procedure is, indeed similar to that of the Liehr and Ballahausen method [7], that we have successfully applied to centrosymmetric coordination compounds of the transition metal ions [l-6,8], although the greater complexity of the f-electron configurations required a more formal approach.

An alternative procedure to work out the crystal field contribution is to employ the closure approximation over the intermediate central metal ion’s electronic states from which the intensity (with the cooperation of the odd parity vibrational modes of the complex ion) is acquired. When this model is adopted, the explicit nature of these intermediate states become to a certain extent irrelevant. These electronic states are employed to determine the choice of an effective energy gap, corresponding to some kind of weighted average over the intermediate states which are the terminal states of parity and spin allowed virtual electronic transitions, which originate

from the states connected by the transition of interest. This approach is somehow analogous to the Judd [9] and Ofelt [lo] formalism to account for the total intensity of the f - f electronic transitions in non- centrosymmetric coordination compounds of the lanthanide complex ions. The parameterisation scheme in the Judd-Ofelt procedure was not intended to and can not treat the intensities of individual vibronic origins associated with specific normal modes of the complex ion. In recent years, however, a substantial body of experimental data on the intensities of vibronic origins in the electronic spectra of octahedral complexes of the type ReXk and ReXi- (Re: Rare earth metal ion), which as example occur in the cubic hexahaloelpasolites, CszNaLnXd and also in cubic doped systems such as Cs*ZrBr, : UBri- and CszTeBrb : UBr& has become available, though most of this experimental information is at best semi- quantitative ([ll-191 and reference therein).

It is interesting though, to observe from the experimental data, that the relative intensities of the three vibronic origins due to the y3 (TV”), y4 (TV”) and vg (72”) vibrational modes are, observed to vary over several order of magnitudes, from one electronic transition to another in the same crystal. It is therefore essential to extend the Judd-Ofelt procedure to deal with this additional data and to attempt to understand and interpret the relative vibronic intensities of these false origins.

There are some previous theoretical studies on these coordination compounds of the rare earths, by Morrison and Crown [20], Richardson and co-workers ([14-H] and references therein). Morrison and Crown achieved, using a crystal field model, some success with the experimental relative intensity of the vg (T*“) vibrational mode of the ErClk complex ion in the CSzNaErC16 lattice. Richardson and co-workers developed a vibronic model to account for the vibronic intensities for the EuCl$ ion in cubic hosts, using a combined crystal field-ligand polarisation formalism. Unfortunately, this calculation suffers from some inadequacies, such as an incorrect assignment of the vibrational modes (which implies a rather inaccurate description of the intermolecular force field) and also that a numerical differentiation of the potential energy with respect to the symmetry coordinates of the system was employed. Nevertheless, despite these inadequacies,

R. Acevedo et al.iJournal of Molecular Structure (Theochem) 390 (1997) 109-119 111

the model employed by Richardson et al., may significantly be improved and applied, as we have shown with considerable success, to centrosymmetric and non-centrosymmetric coordination compounds of the lanthanide, actinide and curide complex ions.

A recent calculation of vibronic intensities has been performed by Reid and Richardson [18], employing the superposition model (SM) developed by Newman [21] and recently reviewed by Newman et al. [22]. This formalism may be regarded as a full parameterisation scheme to account for the observed spectral intensities in coordination compounds. This model has been employed by Richardson et al. [18], to rationalise the vibronic intensity induced solely by the V~ (rt,)-stretching mode of the UBrg- complex ion in the octahedral group. In this calculation, the electronic wavefunctions of the terminal electronic states connected by the excitation, were derived using the energy parameters reported by Satten et al. [23]. It is shown that within the framework of the SM, nine different intensity parameters are needed to be fitted from experiment to reproduce the observed vibronic intensity for a series of selected electronic transitions. Generally, speaking, for a Rex: type complex ion in the octahedral symmetry, the energy levels fitting requires a total of 22 parameters [24]. Also, for the v3 (r,,)-stretching and the ~4 (r,,)-bending modes, nine intensity parameters are required and as for the v6 (r,,)-bending mode, six additional intensity parameters are needed. Thus, a full parameterisation model to undertake vibronic intensity calculations, in principle would need to fit a total of 46 parameters from the experimental data. This is without mentioning the parameters needed to modulate the intermolecular force field and therefore the normal modes of vibration for the system. The experimental data that are most readily available for the ReXf, complexes are the relative intensities of the ~3, v4 and vg false origins. There will therefore, rarely, if ever be sufficient data available to fix the values of the parameters unambiguously.

Furthermore, and bearing in mind that the treatment of the nuclear dynamics of these systems in terms of the modes of an isolated Rex: complex is itself a modest approximation and in some cases the eigenvectors of the crystal field states are not well known, it does seem that generalised parameterisation approaches will have a rather limited applicability.

We have therefore focused our attention upon the combined vibronic crystal field-ligand polarisation approach, since we strongly believe that an excessive parameterisation of the model calculation will obscure both the physics and the chemistry of the problem.

In the present paper, we present a method to calcu- late the relative vibronic intensity distributions for a series of selected electronic transitions in systems such as PrClz- and UBri-, within the framework of a combined vibronic crystal field-closure-ligand polarisation approach. For this system, there are detailed experimental studies to test our theoretical predictions against the experimental data.

2. Vibronic model

The vibronic model used in this paper is identical to that employed in [ 11, where a more detailed account is given. We define a vibronic perturbation to the zero-th order static crystal field model to give a first order corrected Hamiltonian as given below

(1)

The potential energy V, may be partitioned into pure crystal field (CF) and pure ligand polarisation (LP) contributions, as follows [25,26]

(24

(2b) Here, the indices i stand for the rank of the central metal ion’s multipoles and r is a repeated representation label. Mt is the central metal ion’s multipoles transforming as the 7th component of the Pth irreducible representation and pDL corresponds to the oth vector component of the induced transient electric dipole associated with the Lth ligand subsystem. The symmetry adapted crystal field G~~JA(i,7) and ligand polarisation GbF.:(i,7) geometric factors have

112 R. Acevedo et al.IJournal of Molecular Structure (Theochem) 390 (1997) 109-119

been tabulated for most transitions of interest [3,25,26] and complete tabulations are available upon request to R.A. The phases of the symmetry coordinates Skr have been chosen according to the criterion given in Ref. [6].

3. The vibronic crystal field contribution (the closure procedure)

Within the framework of the vibronic crystal field model, the 6th component of the transition dipole moment associated with the IcY~(L~SJ~U$‘~~,) - Icx~(L~S~~U~)I’~Y~) electronic transition can be written

CL::! =

where the b.LSJuI’y) intermediate states at the energy E are abbreviated as I*).

The utilisation of the closure approximation should be handled with care for two main reasons: firstly, it is assumed that all the intermediate electronic states occur at the same energy and also that the summation is performed over the complete set of intermediate central metal ion’s electronic states. The first of these assumptions allow us to replace the energy denominators by some kind of effective energy gap, say L\E and the second statement implies that the condition &, I\k)(ql =E, where E stands for the identity matrix, is fulfilled.

For the sake of simplicity, it is currently assumed that the above condition is always met, nevertheless, in practice we normally use a restricted expansion over the intermediate electronic states. When a restricted set of intermediate states, say {a}, is used, then the closure should be re-written as: Co I+)(+1 =cE, where 0 < c 5 1. (That is, c is some kind of modulation factor.) The actual evaluation of this parameter is not easy and it does depend on the extent of the basis set to be used in the calculation. We can, though, anticipate that when moving along

the lanthanide series the number of intermediate electronic states available (most likely to be employed as intensity source) increases, so that the values of this parameter should approach unity. Furthermore, it is clear that when only the vibronic crystal field- closure model is utilised, then the calculated vibronic intensity distribution will be independent of these c values. A different situation arises, when a combined vibronic crystal field-closure-ligand polarisation method is employed. (Here the choice of the c values may strongly influence the calculated relative vibronic intensity distribution for each electronic transition.)

Thus, Eq. (3) becomes as follows

(4)

In the above expression, AE represents an effective energy gap, corresponding to a parity and spin allowed electronic transition. These matrix elements of the tensor product operators may be handled in two ways, depending on the extent to which their symmetry properties are to be exploited. The two methods are equivalent, though our “non-symmetry adapted approach” is perhaps more conceptually transparent, whilst a “symmetry adapted approach” is quicker if several electronic transitions are to be studied from a theoretical viewpoint.

3.1. Non-symmetry adapted approach

The point group basis states k+SJu)I’y) can be expressed as linear combinations of the free ion basis set IRAQI), using the transformation

la(u.h)ry) = 3 c(.u4k2ry)la(~syhf) (5) where, in octahedral symmetry, the expansion coefficients are tabulated by Griffith [27] and Golding [28]. The expansion coefficients can always be chosen in a way that they do not depend on the quantum numbers cr. In our current notation a denotes a branching label,

A detailed account on symmetry considerations is given by Kibler et al. ([29-311 and references therein), thus the reader is referred to this collection of articles.

Thus, the 0th vector component of the transition

R. Acevedo et al.lJoumal of Molecular Structure (Theochem) 390 (1997) 109-119 113

dipole moment becomes

x C(J*M2~2a*rzY*)(al(LISI) x JIM, Ip~H%&S2~*MJ CF (6)

where Sr = S2 = S, since the product operator p’H@ is spin independent. This tensor product may still be simplified further on, under the following considerations.

The crystal field potential, in terms of the symmetry coordinates Sk is given by Eq. (2), and this identity may equally be expressed in terms of the nuclear Cartesian coordinates to give:

where the crystal field geometric factors G& (O,, &,) = ( - 1)4’ + ‘@’ + ‘k!‘,, (O,, 4~) are easily found in terms of the ligand subsystem nuclear Cartesian coordinates. A complete tabulation of the Slater tensor operator C$O, 4) = mk(O, f$), in terms of the Cartesian displacement coordinates may be found in Ref. [32].

Thus, tensor product p’H& may be ultimately be simplified using the identity

/fD$,(M)- kT(-l)kl+h-ldm

x(; :: _y p In our current notation, ~1’ = Db (0 = 0, k l), where Di(e,$) = - erkCjj(8,cp) are the standard Garstang tensor operators [33] and the tensor operators Tick’) are obtained from the identity

x Cp(ielk, - 4cfe+op, 44 (8)

where the coefficients c p = (YrO 1Ct lY&, _ ,> are tabulated by Condon and Shortley [34], and the 3j- symbols are tabulated by Rotenberg et al. [35].

The transition dipole moment becomes

x C*(.J& ll,alr,y,)C(J2M2lJ2a2rZY2)

x &$ -l)k’-h-lJ@z&(P+kr,M, -M2)

(9)

The reduced matrix element, in the above identity may be successively further reduced [36] to give an expression for the reduced matrix elements on the RHS of the above identity, in terms of monoelectronic reduced matrix elements of the general form: (311Tk(k’)l13), h’ h w rc are available upon request to R.A.

3.2. Symmetry adapted approach

In our symmetry adapted approach, the vibronic Hamiltonian is written as given in Eq. (2). For identical monoatomic ligand subsystems it is convenient to introduce the crystal field magnitudes, defined as given below

! 10)

In our current notation, the crystal field geometric factors have been symmetrised [25,26]. Furthermore, since the direct product T, x I’ is simply reducible, we have

$Mt(i, 7) = C X(F)“‘( - l)‘+?+ V F.? (Z c 6)

+

flT, I‘II.7, x 0, (11)

114 R. Acevedo et aLlJournal of Molecular Structure (Theochem) 390 (1997) 109-119

and hence from [lo] and [ 111, Eq. (4) becomes

x & X(iy2( - l)‘+T+ v .Y (5 f ,‘)

x ( - l)r’ +%+ FVr (C; 1; 6) -+

x (atL,Sl,alrl Ilo’“““,“l10(2L*S12a2r2)1( 12)

The reduced matrix elements on the RHS of the above identity, are obtained from the general expression

x ((Y1rIYIl~~(Tlrii,r)lLY2r2Y2) (13)

The tensor operators Of(Tlr’iYr) can be written as linear combinations of the R&ah tensor operators, using the identity

oFcTlriiJ) = kyf,=plrli, 7)~@, 4) (14)

A complete tabulation of the symmetry adapted expansion coefficients is available upon request to R.A.

For the sake of notation and computational convenience, we introduce the symmetry determined II-coefficients as follows

x(-p”’ (TM, : Z2) (15) Thus, in terms of these symmetry adapted coefficients, the relevant reduced matrix elements may be

written as

k (16)

The above equation was derived in a equivalent form by Kibler [39]. The polyelectronic reduced matrix elements of the above identity, may be simplified using a standard reduction procedure as shown in Refs. [36-381. These matrix elements are then finally expressed in terms of monoelectronic reduced matrix elements as follows

4. The vibronic ligand polarisation contribution

For isotropic ligand subsystems, we have shown that the c&h vector component to the total transition dipole moment may be written in a symmetry adapted form as [l-8]

where the vibronic ligand polarisation magnitudes are as follows:

@(i, 7) = - 01L(~1-_,2) 8

(18)

A complete tabulation of these quantities may be obtained upon request to R.A. In [18], the magnitudes c~~(vt_~) represent the mean ligand subsystem polar- isabilities, measured at the frequency of the electronic transition. Furthermore, the G;:,,(i, 7) are the symmetry adapted ligand polarisation geometric factors [l-8,25,26]. The above equation may still be simplified further. Next, we express the symmetry adapted central metal ion’s multipoles in terms of the Racah tensor operators, as follows

(19)

R. Acevedo et al./Journal of Molecular Structure (Theochem) 390 (1997) 109-l 19 115

where, the expansion coefficients Riqry(i,7) are tabulated by Griffith [27] and Golding [28], for most cases of spectroscopic interest. Furthermore, the symmetry adapted central metal ion’s wavefunctions are written as given in Eq. (5), and therefore the matrix elements on the RHS of [17] are given as

For the sake of notation and convenience, we introduce the Z[, ,,:,(F iJlI’&) symmetry determined coefficients, defined as follows

x wf, Illulrlyl)c(J2MzCl,u2r2Y2)( - 1)J -MI

x (Jq : ;II,) (21)

and therefore, the relevant reduced matrix elements are expressed as

=z~,.):,(r,JIrdr)(~(LSVI(D’Ila’(LfSy’) (22)

and the polyelectronic reduced matrix elements on the RHS of the above identity may be further reduced and ultimately expressed in terms of the monoelectronic reduced matrix elements

3 i 3 (311Di1/3)=7e(r4)ff o o o

i 1 as given in Refs. [25] and [26].

5. Applications

Simple applications of this combined vibronic crystal field-closure-ligand polarisation formalism,

will be illustrated with regards to some selected electronic transitions for both the UBri- and the PrClk complex ions in perfect octahedral environments. For these systems, Flint and Tanner [19] and Tanner [12] have provided semi-quantitative data on the relative intensities of the vibronic origins associated with several electronic transitions, in cubic lattices. We emphasise that we do not seek to simply fit the experimental data and do not attempt to optimise the assumed values of the physical quantities (i.e. the effective energy gap A,?, the effective charge on the ligand subsystems, the mean ligand polarisability values and the metal radial wavefunctions) to get the best possible agreement with experiment. Our target is to improve our understanding of the complex intensity mechanisms for the observed spectral intensities, as well as to rationalise those factors (electronic and vibrational) upon which the intensity is mainly derived from.

Merely adjusting these quantities in what is an approximate model just to fit the experimental data would not necessarily be sound from a physical viewpoint.

5.1. Application to the 1 3P& Ig) - i3H4,1’) transitions of the PrCl$ complex ion

The static crystal field wavefunctions, were found in terms of the la(LSlu)I’y) wavefunctions by fitting the experimental energy levels to an approximate Hamiltonian of the form

to give the eigenvectors, given in Part I. For the electronic states considered here, the J-mixing is small.

The vibrational wavefunctions were calculated in the harmonic approximation, by fitting the observed vibrational frequencies for the crystal to a model for the isolated PrClz- complex ion. The vibrational L matrix, relating the set of the symmetry coordinates to the set of the normal coordinates for the system, have been reported in Part I of this series.

The need for caution in defining the phases of the vibrational wavefunctions has been emphasised in [1-S],,,$s,, for the vibronic intensity calculation, the 0, are calculated from Eq. (14) in terms of

116 R. Acevedo et aLlJournal of Molecular Structure (Theochem) 390 (1997) 109-119

Table 1 Calculated values of the non-zero reduced matrix elements of the effective operator for PrCl:-, in units of low3

(3P,,4,,Ii@(T,)ti3F~1) = 1.175 (3PaA ,liOE(T,)i13F&) = 9.309 (3PgI ,i10rz(T,)i13FF,T3 = - 0.608 (3Pgi &OE(T,)ii3F& - 1.332 (3Pd ,~~o~~(T~)~~~F,TJ - - 1.413 (3~~,lla41(T,)I13HqA,)= - 1.120

(3P~,l10r~(T1)l13H4TJ = - 1.534 (3f’pDA ,~~07'(T,)~f3H4TJ = 0.580 (3~~,ll~T~(~Z)l13~~~,) = - 1.188 (3Pgi IiiOr2(Tz)i13HqT2) = 1.347 (3P,,A I~~OT~(T,)~~3FFzT~ = 33.66 (3P,,z4 lliOE(T,)Ii3F,E) = 1.787

(3P,,4,,1iO’1(T,)3F4T,) = 1.609 (3PpoA ,I1 0r1(T,)ti3F,T,) = 1.246 (3P,,A ,iiOE(TJ3H&= - 0.947 (3P,,A ,l10E(T,)l13H&) = - 1.270

(3P~,,l10E(T,)i13FzE) - - 31.48 (3PpoA ,t10Tz(T,)ii3F,T~ = 2.527

the Racah tensor operators. The electric multipoles for i = 5 and 7 make a negligible contribution to the intensity and have not been included in this calculation. Using (& = = 0.410 A” and (r”)rr = 0.418 A” [9] and evaluating the matrix elements as given by Eq. (16), the necessary non-zero reduced matrix elements are readily evaluated (Table 1).

The remaining steps in the calculation are to collect the contributions due to the various components of the electronic eigenfunctions, evaluate the crystal field and ligand polarisation components (this latter step has been worked out in detail in Part I of this series and does not need to be repeated here), and the cross term between them and then distribute the intensity amongst the normal modes (Tables 2 and 3). The resulting oscillator strengths are, for each transition

f(vs)=6.94.10-’ (AE)(U3L33+U4L43)2

f(v4)= 1.95*10-6 (AE)(U3L34 + u,L,)*

f(V6) = 2.39.10-6 (&?)(u&66)2

Inserting the experimental transition energies (AE) from [12], gives the calculated oscillator strengths in Table 4. For easy comparison with the experimental relative intensities in emission, we also give the relative intensity ratios of the vibronic origins within each electronic transition (Table 5).

Table 2 Calculated crystal field UcF.z/104 (closure approximation) electronic factors for the I 3P,yI 1) - I 3HqI’) electronic transitions of the PrCl:- complex ion

P/Sk, ‘41 E TI Tz

s3 -12.80 12.64 34.02 -29.42

54 4.21 -19.94 -12.75 - 6.57

Sg 20.85 12.75 -13.60

As it was shown in Part I, the agreement among the calculated vibronic intensity distribution and the experimental data was excellent. The most direct comparison of this calculation with that presented in [l] is provided by the crystal field electronic factors in Table 2 of this paper and Table 2 of [l]. This isolates the essential differences between the models without the common (but signed) contribution from the ligand polarisation term or the “smearing” of the contributions by the mixing of the symmetry coordinates in the normal modes and the small J-mixing due to the crystal field. It is satisfying to observe that the general order of magnitude of these factors is com- parable even though they depend on different linear combinations of different reduced matrix elements. Closer examination shows that 8 of the 11 non-zero electronic factors have the same sign and are within a factor of about 2.5, which we regard as good as might be expected for these simple models.

ThecaseoftheA1+E+SsandA,-T2+S3 electronic factors are situations of greater concern and interest since they are of opposite signs in the two models.

Examination of the individual terms that contribute to this result provides no obvious explanation. The result is that in both cases there is a cancellation of

Table 3 Calculated oscillator strengths f(vi)/10-9 (crystal field-closure- ligand polarisation model) of vibronic origins for the 1 ‘Pd ,) - 13H4r) electronic transitions of the PrCli- complex ion

Vibronic origin

Electronic state

41 E TI TI

“3 0.590 0.003 85.800 0.430 v.I 25.170 22.930 46.400 17.510 vf, 0.170 5.970 15.750

R. Acevedo et al.IJoumai of Molecular Structure (Theochem) 390 (1997) 109-119 117

Table 4 Calculated contributions to the dipole strengths D(v,)/lO“ D* of vibronic origins for the13Pd t) - 1 3H.J’) transitions of the PrClk complex ion

Vibronic origin

Electronic state

Al E Tl T2

DCF(v,) 1.68 4.86 11.91 3.67

DLP(v3) 4.18 4.38 36.02 1.54 D’CFJP’(“s) -5.31 -9.23 41.44 -4.76

DcF(v4) 0.65 13.75 34.81 3.01

DLp(v.J 19.35 1.45 23.06 7.97

@cJ”‘) 7.14 8.94 -9.41 7.74 12.38 4.62 5.26

;&, 15.52 21.56 3.24

(Vh -27.73 -19.97 8.28

the crystal field and ligand polarisation terms for these transitions in model II, which does not occur in model I, and the calculated intensity from model II is then much lower than is experimentally observed for these two transitions. Further calculations and experiments will be required to determine whether this is a common feature of the two models or specific to this particular transition.

A detailed discussion of the comparison of experimental and calculated intensities was included in [l] and need not be repeated here. The results of the present paper do suggest however that any attempt to account for vibronic intensities in terms of purely crystal field or purely ligand polarisation contributions is unlikely to be realistic. A similar conclusion was reached by a different route in [18].

5.2. Applications to several electronic transitions of the UB& complex ion in cubic environments

In this section, we study some selected electronic transitions in both absorption and emission for doped lattices such as Cs2ZrBr6 : UBri- and

Table 5 Calculated relative vibronic intensities of vibronic origins for the 13Pd ,) - 13HJ’) electronic transitions of the PrCIz- complex ion. Experimental values are given in parentheses

Pivibronic A1 origin

E TI T2

“1 0.02 (0.25) 0.00 (0.04) 1.00 (1.00) 0.03 (0.04) uj 1.00 (1.00) 1.00 (0.00) 0.54 (0.04) 1.00 (0.60)

Vh 0.007 (1.00) 0.07 (0.07) 0.90 (1.00)

CszTeBrd : UBri-, based upon the detailed experimental studies carried out by Flint and Tanner in a series of four detailed papers ([19] and references therein). The eigenvectors appropriate to the U4’ in octahedral fields have been obtained using a truncated Hamiltonian as given below

H,I = F ham + ~ ~ + ~ ~(ti)(ls;) + V,,(O,) i<j rii I

+ d(L + 1) + @(G2) + yG(R7) + ns (28)

In the above expression for the Hamiltonian, 6(R7) corresponds to the eigenvalues of the Casimir operators of the rotational group R,, employed to classify the states of the fn electronic configuration and G(G2) is the eigenvalue of the Casimir operator of the group GZ, which serves to classify the state of the f n configuration [38]. The fitting procedure is given in detail in Part IV of [19] and hence some extra details are not needed to be repeated here. It is, though, worth mentioning some points about the least square fit performed by Flint and Tanner. For both the UCl$ and UBr% ions in these cubic lattices, only the o-Trees correction parameter was included in the energy level calculation. For the UBri- ion, 26 observed energy levels were included in the fitting procedure and as a result, we observe that the mean error deviation indicates that the least square fit was modest. The fitting procedure can significantly be improved introducing a full Hamiltonian as [24].

H = E,, + F

Fkfk + s&&, + d(L + 1) + @(G2)

+ y&R,) + 7 T$ + & pk@k + F M’$ + V&O, )

(29)

The above Hamiltonian is defined to incorporate the isotropic part (including the spherically symmetric crystal field interactions of the 4f- electrons) and the non-spherically symmetric components of the even parity crystal field (i.e. the crystal field potential). The atomic parameters included are: (a) Six two- body electrostatic parameters, that is the Slater parameters Fk (k = 2, 4 and 6) and the configuration interaction parameters (Y, /3 and y; (b) Six three-body electrostatic parameters T’ (I = 2, 3,4, 6, 7 and 8); (c) The spin orbit parameter t,,; (d) Three spin-other

118 R. Acevedo et al./Journal of Molecular Structure (Theochem) 390 (1997) 109-119

orbit parameters Mk (k = 0, 2 and 4); and (e) Three electrostatically correlated spin-orbit parameters Pk (k = 2, 4 and 6). Also a parameter E,, is used to shift the energy of the entire configuration. In addi- tion, two non-relativist crystal field parameters, namely Bb4) = 1.128 Bi and Bh6) = - 1.277 Bg [40]. It is then, obvious that a general procedure to improve the least square fitting would require an input of a substantial amount of experimental data. This rarely occurs and therefore we have decided to carry out the vibronic intensities calculation, employing the set of wavefunctions as reported by Flint and Tanner to test both the utility of our model and also the quality of the reported eigenvectors to reproduce the experimental relative vibronic intensity distributions for several selected electronic transitions. The vibronic intensity calculation has been done using a molecular model rather than a full lattice dynamics approach. Work in this direction is in progress in our laboratory.

The input data needed to perform the vibronic intensity calculations is as given below:

Vibrational frequencies: Ye = 194 cm-‘, yq = 81 cm-‘, Ye = 62 cm-’

Radial integrals [41]: Relative amplitudes of vibration:

(r’)f = 0.592 A’, (r4) = 0.483 A” La3 = 0.14349, L34 = 0.01808, Lq3 = - 0.14184, La = 0.19626 and Lo6 = 0.15800

The comparison among the observed and calculated vibronic intensity distributions for a series of electronic transitions is given in Table 6. As it is shown, this model though very approximate is able to reproduce most of the observed intensities for the vibronic origins. We could indeed improve our results by fitting some parameters in the vibronic intensity calculation. However, we believe that this latter approach is not always transparent, so we avoided this kind of procedure. The details of the calculations are not given here, since we feel that though the calculation itself is very complicated many of the most interesting points have already been covered in Part I and in the current manuscript.

6. Conclusions

A detailed formalism for the calculation of the intensities of individual vibronic origins in the electronic spectra of centrosymmetric coordination

compounds of the rare earths has been developed in a basis which is consistent with the classical Judd- Ofelt method for the whole electronic transition and applied to various excitations for both the PrClk and UBri- complex ions in octahedral environments. For most vibronic origins studied, the agreement between the theoretical predictions and the experimental data is as good as could be expected for such a simple physical model. In some cases, discrepancies are observed. Whilst this could be interpreted as support- ing the preferences of our earlier model (Part I), involving a single d state as the intensity source, we feel that further experimental and computational studies will be required to determine the generality of this result. Work in this direction is in progress, together with the application of this isolated ion model to other complex ions.

Table 6 Calculated relative vibronic intensities of vibronic origins for a series of selected electronic transitions of the UB$ complex ion in octahedral environments 6.1. The 13H, bTz) + 1’Hdr) electronic transitions

rivibronic A, Ea Tl T2

origin

y3 2.8 (2.4) 0.8 0.75 (0.7) 30.0 (8.0)

y4 6.0 (2.1) 1.9 1.1 (1.1) 14.5 (10.0) ye 1.0 (1.0) 1.0 1.00 (1.0) 1.0 (1.0)

6.2. The 1 ‘Gq A ,) -+ 1 3H4 A ,) electronic transition

r/vibronic origin

f(v&f(v&f(vs) = 1.0:3.0:0.0 exp:l.l:l,O:O.O

6.3. The II6 aTJ + 1 3H41’) electronic transitions

r/vibronic A, E Tl T2 origin

v3 1.0 (1.5) 1.0 (1.0) 1.7 (7.0) 1.6 (1.0)

y4 1.2 (1.0) 60.0 (2.0) 25.0 (3.0) 0.3 (0.4)

V6 0.2 (0.0) 12 (2.0) l.O(l.0) l.O(l.0)

6.4. The I ‘I, ‘T?) + I ‘F4r) electronic transitions

r/vibronic A, E Tl T2

origin

v3 1.0 (1.0) 1.0 (1.0) 1.0 (1.0) 1.9 (1.0)

v4 3.7 (4.0) 0.14 (1.2) 2.0 (3.0) 2.1 (...)

v6 0.0001 (0.05) 0.5 (1.4) 3.0 (0.0) 1.0 (1.0)

a Only the v, vibronic origin is visible.

R. Acevedo et al.iJournal of Molecular Structure (Theochem) 390 (1997) 109-119 119

Acknowledgements

R.A. would like to express his gratitude to both Fondecyt. grant 1950668 for financial support and the EC. DG-XII for the tenure of a bursary at the Chemistry Department, Birkbeck College, University of London, UK.

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Vibronic intensities in centrosymmetric coordination compounds of the rare earths Part II. A vibronic crystal field-closure-ligand polarisation model and applications to the PrCl3−6

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