Ligand-field Theory Metal Complexes (1)

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ELSEVIER

THEOCHEM

Journal of Molecular Structure (Theochem) 43 I ( 199X) 97- 107

Dynamic ligand-field theory for square planar transitionmetal complexes

K. Wissing, J. Degen*

Imrrtur iir Theoreti.whP Chrmw Hrinrich-Hrrnr-Unrvertitdt D-40225 LXisseldorf German.~

Rcccivcd 19 August 1997: revised 6 November 1997; accepted 18 November 1997

Abstract

The electron-phonon coupling of square planar transition metal complexes is analysed by a perturbative model based on theelectrostatic ligand-field theory. Excited-state potential energy surfaces are characterized by taking into account linear vibroniccoupling within the point group D4t,. Results do not depend on new adjustable parameters and can be generalized to a greatnumber of complexes. The applicability of the model is demonstrated by comparing the results of the model with those obtainedfrom the optical spectra of [CuC14] *- and [MX,] - (where M = Pd, Pt and X = Cl, Br). In particular, the influence of E x (b ,a +bz, coupling, which is the subject of some controversy in the literature, is found to be of low importance. 0 1998 ElsevierScience B.V.

Keywords: Vibronic couplin g; Square planar complexes; Band profiles; Transition metals

1. Introduction

Transition metal complexes have been of greatinterest in the past because of their rich spectroscopyin the optical region of the electromagnetic spectrum,which is mainly due to the open d-electron shell.Currently, the search for new materials with specialoptical and magnetical properties pushes the investi-gation in this area. This paper deals with the nucleargeometry of square planar complexes in differentd-electron states. Geometry changes following opticalexcitation are caused by changes in the metal-ligandbonding strength, since valence electrons formingthe bonds are excited. The electrostatic ligand-heldtheory has provided an explanation of band positionsin optical spectra. The width and vibronic fine

* Corresponding author.

structure of these bands, if visible, have been treatedto date in most cases by symmetry rules or by argu-ments concerning orbital occupancy changes in goingfrom one electronic state to another. A sufficientqualitative understanding of the totally symmetricdisplacements in excited electronic states is possibleby those arguments. In the case of orbital-degenerateelectronic states, a distortion along a non-totallysymmetric mode is demanded by the Jahn-Tellertheorem. The strength of such couplings and the ques-tion of which mode dominates the coupling and deter-mines the distortion often remain unclear. For squareplanar complexes there is still ambiguity concerningthe question of the importance of couplings of unsym-metrical modes with d-electron states and the relativeimportance compared with the coupling for the totallysymmetric mode.

A model for the computation of spectroscopic band

0166-1280/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved.PII SO166-1280(97)00433-8


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9 K. Wissing, .I. Degn/Journal c Molrculcrr Structure (Theochem) 431 (1998) 97-107

profiles and nuclear distortions is developed here,which is as easy in application as the electrostaticligand-field model (LFM) for the computation ofband positions. It is an extension of the dynamic

ligand-field model for octahedral complexes [ 1,2] tothe lower, square planar geometry. There is no needto introduce new free adjustable parameters thenecessary information comes from vibrational spectra(frequencies), the metal-ligand distance and staticligand-held parameters. The mixing effect of thend,? orbital with the (n + 1)s orbital on the metal sdmixing), both having a,, symmetry in square planarcomplexes, is taken into account by its influence onelectron-phonon coupling.

The basic magnitudes in vibronic coupling theories[3] are the vibronic coupling constants, which quan-tify the force tending to distort the complex from ahigh-symmetry reference geometry along a normalcoordinate, depending on the electronic state of thecomplex. Many attempts have been undertaken inthe past to determine these constants by very differentapproaches. Determination of first-order Jahn-Tellercoupling constants is enabled by knowing the amountof splitting of the electronic energy levels caused bysymmetry lowering owing to nuclear distortionsalong non-totally symmetric modes. Therefore, bymeans of electronic structure calculations of arbitrarytype varying the nuclear geometry, these constantsare derived and, from the experimental viewpoint,vibrational progressions in optical spectra are usuallycorrelated by vibronic coupling constants to nucleardistortions. The theoretical foundation of vibroniccoupling theory is found in [3] and references therein.A parametrization scheme of vibronic coupling con-stants in terms of the angular overlap model (AOM)established by Bacci [4,5], is related to the presentapproach as discussed below.

The idea of the model presented is to calculate

vibronic coupling constants starting from the electro-static LFM by computing the first-order term in theHerzberg-Teller expansion [3] and treat this operator(linear vibronic coupling operator) as a perturbationon the d-electron states:

3N-6

H = iz (~H/~Qi),Qi

Ho is the electronic Hamiltonian of the system atthe nuclear reference geometry, Qo, consisting of the

Fig. I. Repulsive interaction of an electron in an ep orbital on theligands tending to distort the complex in the direction of the h ,e ((3 ,)

nuclear coordinate (e, x /3, coupling case), which can be quantified

by a dynamic LFM calculation of the 0; coupling constant.

kinetic energy (T,) and the potential energy contribu-tions of electrons (e), ligands (L) and the metal ion

(M):

~=T,+v,,+v,,+v,,(+Hso) (2)

Hso is the spin-orbit coupling operator. Apart fromthe electrostatic ligand-field potential, VeL, the otherterms in Eq. (2) depend only weakly on nuclear geo-metry changes. The vibronic coupling operator istherefore computed as the derivative of the electro-static ligand-field potential with respect to the nuclearcoordinates at the square planar reference geometry.Fig. 1 provides a visual example of the repulsiveforce imposed by a d electron on the ligands in aneg x /3, coupling case, taken into consideration bythe dynamic ligand-field model.

Although electrostatic ligand-held theory approxi-mates the first ligand sphere only as point chargesand limits the basis to pure d-electron functions onthe central atom, semi-empirical treatment of theradial parts of the integrals has been applied to awide range of systems with great success [6]. In thefollowing, the orbital and many-electron d* and dcoupling cases as well as the influence of sd mixingon vibronic coupling constants are treated. Then, acomparison of the vibronic coupling predicted bythe model and that obtained from well-resolved

vibronic spectra shows that the semi-empirical treat-ment of the (same number of) parameters validatesthe model assumptions also in this extension of theLF model direction or to electron-phonon coupling.

2. Orbital coupling case

In Fig. 2 the splitting of the d orbitals in a squareplanar D4,, complex, including the influence of sd


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K. Wissing, J. DegdJoumal ojkfolecular Structure (Themhem) 431 (199X) 97-107 99

mixing, is shown. The relative energy of the dLzorbitaldepends strongly on the extent of sd mixing. We startwith the pure d orbitals, coupling with the modes ofthe square planar D4,, complex. Only vibrations with

even symmetry are able to couple to states originatingfrom a d configuration. The symmetrized Cartesiandisplacement coordinates of the vibrations in questionare [3]:

b,,(P,) Qp, =;iW2-X?+Yd

8 and 4 are electronic coordinates. 2 is the effectivecharge on the ligand. Since the ligand-field potentialdepends on polar coordinates, coordinate transfor-mations are necessary [l] (aV,L/aQi)o(aV,L/aX,)o,

aveL/aY,) O~ aveL/azjh + aveLlaRjkJ3 aveLia@j)O,

(IFIV,L/C~@,)Oi = 0 ,,/32; j = 1...4). The couplingoperators are calculated at the high-symmetry (Da,,)nuclear configuration up to the fourth multipole(X 5 4) of the electrostatic ligand-field expansion.Higher multipoles are of no interest here, becausethey give no contributions to matrix elements over dfunctions.

The derivatives of the ligand-field potential withrespect to the coordinates in Eq. (3) are:

x=0 A=2

+6

6 (3- = --

5+...

__ z _ + G r4i(Y~ + Y4m2) + . . .

b2 32) Qp2 = ;iyl +x2 - y3 -x4) (3)

The numbering of the ligands is 1,2 on thepositive x,y axes and 3,4 on the negative x,y axes,respectively.

The orbital coupling operators are calculated asthe derivative of the electrostatic ligand-field potentialfor localized d electrons (given as an expansion inspherical harmonics) [6]

A=0 2x+ 1 r=X

(4)

with respect to the symmetrized Cartesian displace-ments in Eq. (3) at the square planar nucleargeometry. R, 0 and + are nuclear coordinates and r,

The indices of V are dropped. For reasons of symme-try, the following coupling cases are possible in D4,,symmetry:

Ulg x (2 b,, x Q (a,,+&,) x01h, XQ eg x (cu+P, +P2) (a,,+b2p) XL

The perturbation matrix elements with the couplingoperator are related by the Wigner-Eckart theorem

[3] for a given coupling case. Only one of the matrixelements has to be computed in each coupling caseto give the reduced matrix element, which is propor-tional to the vibronic coupling constant. When theligand-field parameters ~0, r~ and Dq are introduced

[61

Dq= A (r) (6)

the orbital coupling constants for a square planar


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complex are:

K. Wissing, J. DegdJmrnal of Moleculur Structure (Theochem) 431 (1998) 97-107

(7)

u,, for example, equals the matrix element(d;l tlV/6Q,),jd,2), the upper index (a ,) denotingthe symmetry of the electronic function d,z and thelower one denoting the type of coupling mode,which is the totally symmetric stretching vibrationin this case. For vibronic coupling with the doubly-degenerate electronic e state and for the higher-ordercoupling cases, the corresponding coupling matricesare also given. Since the LF potential, VeL, containsonly the repulsive interaction between ligands and d

electrons and no metal-ligand attraction, the ground-state (Y coupling constant is negative and not zeroas demanded. To compute excited-state 01 couplingconstants, the computed value for the ground-statecoupling constant has to be subtracted from theexcited-state values. This has the same influence asthe introduction of an attractive metal-ligand poten-tial in VeL. The terms with co, representing thespherical symmetric part of the ligand-field potential,vanish as a consequence. Stabilization energies andnuclear distortions due to linear (Y, 3, and fi2 couplingcases are (a are the corrected constants):

AE = - a3*/2k, and AQ, = -aL/k,

with F=u,~, b,,, bZg, e (8)

ki being the force constant of the vibration i. In thelinear e x /31 + 01) coupling case AQ, can have bothsigns and there are two kinds of possible minima,either two e x /3, or two e x /Z2 minima. According

to 6pik and Pryce [7], the deeper of the two kinds ofminima are absolute minima whereas the other twoare saddle points. The force constants can easily becalculated from the vibrational frequencies:

k; = 47r2c2piif

X ~58 991 X IO-.~i [g/mol].# [cmm2] kg/s2)

(9)

The reduced masses, pI, are the ligand masses

considering the choice of coordinates in Eq. (3).For calculation of the coupling constants in Eq. (7),the formal equivalence between the AOM and thedynamic LFM is exploited by relating the parametersDq and 9 to the well-known AOM parameters e, and

e, F3 :

lODq=3e,-4e,,3 3-4e,/e,

= 5 1 +e,/e,(10)

By this means, treatment of n (which is unknown inmagnitude) as an adjustable parameter is avoided.Vibronic coupling constants in terms of AOM para-

meters (e,, e,, Ge hR, Ge,JAR) have been derived byBacci [4,5]. For bending vibrations these constantsare equivalent to the constants in Eq. (7) obeyingEq. (10) and can be regarded as the counterpartof AOM coupling constants in terms of the LFM.Coupling constants for stretching vibrations requiretwo more parameters in the AOM (Ge 6R, &e@R)and differ from the constants given here becauseof diverging model assumptions concerning the dis-tance dependence of the metal-ligand interaction. In


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n+1s-

h d+2 ;Ig:

b,, dXz-+

R D4h sd-Mixing

Fig. 2. Term scheme for a d system, showing the influence of sdmixing.

Most known square planar transition metal com-plexes have a d8 or d electron configuration. Withoutregarding sd mixing, a d9 complex has the oppositeterm scheme from that depicted in Fig. 2, owing to

the electron hole formalism. Likewise, the orbitalcoupling constants derived above are directly applic-able to d complexes when all signs of the constantsare changed. It should be noted that, in the linearcoupling case, the signs of 0, and p2 coupling con-stants are of minor importance, because distortionsalong both directions of one coordinate are equivalent.This is different to the situation for an octahedralcomplex coupling with an E vibration, for example,where the sign decides whether the complex is tetra-gonally elongated or compressed.

fact, the indirect semi-empirical treatment of 7 assuggested by Eq. (10) is preferable, since it is wellknown that the calculation of ligand-field parameterssuch as D q from first principles gives results that arean order of magnitude too small. It has been shown,however, that the dependence of D q on metal-liganddistance - predicted by ligand-field theory to beproportional to Rm5 compares fairly well withexperimental observations for various transitionmetal complexes and corresponds to other types ofcomputational method [9,10], which is a conditionfor successful application of the dynamic LFM forcomputation of the vibronic coupling constants ofstretching modes.

Regarding vibronic coupling with the totally sym-metric mode, the CY oupling constant of the groundstate (GS) is subtracted from excited-state (ES) 01coupling constants as in the orbital coupling case.The constants depend only on the occupation of thed orbitals:

A: = ,; (d:"a;), AzS = ;g (&z;)

The sum runs over the 10 d spin orbitals and d, is theoccupation number of the spin orbital i. Eq. (11) isalso valid for configuration interaction (CI) basefunctions in the d + s) basis.

3. Many-electron coupling cases for d8 and d9configurations

The coupling constants for multiplets resultingfrom many electrons in the open d shell are a linear

combination of orbital coupling constants, since thevibronic coupling operator is a one-electron operator.

The construction of linear vibronic coupling con-stants for many-electron strong-field states with Piand p2 modes is limited to electronic E states, becausenon-degenerate states cannot couple to non-totallysymmetric modes in the linear case. The E x PI andE x 3? coupling constants for the six occurring Estates in a d* configuration are computed by Griffithsirreducible tensor method [ 1 l] and listed in Table 1.Non-diagonal matrix elements between different Estrong-field states are all zero, as can be seen from

Griffiths formulas. Regarding CI in the d + s) basis,owing to non-diagonal elements of the electron

Table 1Many-electron coupling constants for d8 strong-field E states and non-totally symmetric modes reduced to orbital coupling constants

(11)


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102 K. Wissing, J. DegedJoumal ofMolecular tructure (Theochem) 431 (1998) 97-107

repulsion and the spin-orbit coupling operator inEq. (1) (static problem), a coupling constant for a CIbasis function, I = cicl\kE,i (i = 1...6), is:

A;;,, 42 is the coupling constant for matrix elementsover strong-field states qEE.,, which can be simplifiedto orbital coupling constants with Table 1.

In d8 complexes non-diagonal coupling constantsbetween different strong-field states only occur fornon-degenerate states. The effect should be significantfor large coupling constants between states, whichare close in energy (if kAE/4a2 < 1, the lower poten-

tial has two minima). The second-order coupling case]3&g(argee zg igb b )+B,,(al,e~b&blJ] X f12 involvingthe two lowest-lying non-degenerate triplet statesmay be of interest:

A,,B,Al%-

A,,jB, _ a,.hz A&- -a@: (13)

4. Mixing of nd and n + 1)s functions at the centralatom (sd mixing)

The effect of sd mixing has been found to be impor-tant in the interpretation of the electronic energies insquare planar complexes. It is therefore necessary todiscuss its influence on the linear vibronic coupling.sd mixing causes an enlargement of the dz2 orbitalin the z direction and a decrease in the x and y direc-tions. Other orbitals are not influenced for symmetryreasons. Excited-state energies relative to the groundstate are changed when the occupation of the di2

orbital is changed. Considering linear couplingcases, electrons in a d,z orbital are not able to contri-bute to couplings with non-totally symmetricmodes. The influence of sd mixing is then limited tothe a: coupling constant and causes it to be lessnegative.

4.1. Quantitative treatment

The interaction between the nd,z and the (n + 1)s

orbital is a 2 x 2 problem for a square planar complex[ 12,131. The perturbation matrix of the static problemis:

(14)

Hsd equals (sl V,Jdzz), where Vex is the static ligand-field operator given in Eq. (4), and AEYd s the energydifference between the s and d.1 orbitals. Most com-monly in AOM treatments the parameter (J,d is intro-duced, which describes sd mixing between the s anddz: orbitals caused by the potential VLL of one ligandlying in the z direction:

(15)

Since there are four ligands lying in the .xy plane, theHSd matrix element is:

F,,dcZ(8, 4) is a geOIIEtriC f Or [14] and HJd iSnegative as can be seen from Eq. (19). Assumingthat AE,Yd >> ksd 1, the perturbation energy forthe dzz orbital can be approximated byEsd = -40,~~ [ESd d ) = E(d,z) -4axd]. The eigen-vector for the perturbed dz2 orbital depends onlyon the ratio p = AEld/I?rsdt and can be given analyti-cally ( - m < p I 0):

d;; = cl d,z + c2s

and

(17)

This function, used as basis function for the pertur-bation treatment with the vibronic coupling operator


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(aV/aQ,),, yields:

with:

To derive an expression for p in Eq. (17) in terms ofcommon ligand-field parameters, we can write, withEq. (15): p = Hsdlusd. The integral Hsd is computedby usual ligand-field theory to be

so that p is:

12 D q

p=-JJijTg

(19)

(20)

If Eqs. (16) and (19) are used, an expression for ?Ican be given as:

(21)

which depends only weakly on the unknown para-meter AEsd. If we assume typical values for AE [ 151( = 80000 cm-) we are able to include the influenceof sd mixing in dynamic ligand-field theory, avoiding

The Pd* and Pt+ complexes have a d8 configura-tion and the term splittings under the influence ofdifferent interactions are depicted in Fig. 3. CarefulAOM analysis of these complexes has been performedin [28], and the values for e, and e, are overtaken

Table 2Comparison of computed results and parameters obtained from the xy-polarized absorption spectrum of the (metH)zCuClj complex

the treatment of 11 as a free adjustable parameter. Thecorrected (Y coupling constant of the dzz orbital with sadmixtures is computed from Eqs. (17), (18) and(20). Common AOM parameters such as e,, e, and

(T,dare sufficient to account also for the influence of sdmixing on potential energy shifts.

5. Comparison with vibronic spectra

We shall apply the theory developed so far to thesquare planar tetrachloride of Cu*+ [ 15,161 and tothe tetrachlorides and tetrabromides of Pt2+ and Pd*[ 17-261, comparing it with well-resolved vibronicspectra reported in the literature. The ground-state

vibrational frequencies [ 16,271 of the chromophoresare taken for computation of the force constants withEq. (9) which are assumed to be valid also for excitedstates.

The (metH)2CuC14 (metH = bis(methadonium))complex has a d9 configuration giving rise to threeligand-field transitions, which are clearly seen in thexy-polarized spectrum [16]. Adaptation of the AOMparameters to the positions of these bands yields e, =5270 cm-, e, = 920 cm- and aBd = 1500 cm-. Withthe vibrational frequencies [ 161 (V, = 275 cm-, VP, =

195 cm-, 3p, = 181 cm-) and R = 227 pm (accordingto [ 161 there is a very slight deviation from the squareplanar geometry, resulting in small differencesbetween R values; 227 pm is an average value), theresults given in Table 2 are obtained and comparedwith experimental values.

AQZ (pm) AQyP[16] Ak$ (cm-)sc1

a S:p[16]

(% ( SP,)

17 15.9 - 1130 4.1 -416 - 1070 3.9 -4

(0.005, 0.07)11,13 21.2 - 450, - 690 1.7, 2.5 -6

Calculated with the inclusion of sd mixing (7 = 0.30).


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alg bl, , 1, Es.\ I

Al,

R3 1. D4,, , sd_mixing 2. Electr.-electr.interaction

Fig. 3. Section of the term scheme for a square planar dX complex,visualizing the different states of approximation in the perturbationtreatment. The configurations used at approximation level I areholes in the d shell.

(Table 3) for our calculation and assumed to be validfor all compounds containing the chromophore inquestion. For example, we cannot distinguish betweenK2PtCI, and the doped compound, Cs,ZrC& : PtCl$- ,in our calculation. Cl is neglected for the resultsin Table 4. No higher-order coupling between differ-ent strong-field states is regarded. At this level ofapproximation, results are equal for singlet andtriplet strong-field states with the same electronconfiguration.

Parameters from the static problem necessary forthe calculation and values derived from the equationsabove are listed in Table 3. In Table 4 calculated

The strong field states of approximation level 2 in Fig. 3 serve asbasis functions for the perturbation treatment with the vibroniccoupling operator in Eq. (4).

Huang-Rhys factors (S = AE/~v,,~) are comparedwith experimental values; this is a more sensible testfor the model than the comparison of nuclear distor-tions, because the first depend on the square of and

the latter depend linearly on the coupling constant.The determination of the Huang-Rhys factors

from the electronic spectra is confined to transitionsshowing well-resolved vibrational fine structure.Hence transitions to the E, and B,, states are notevaluated. The Huang-Rhys factor is identified withthe number of the most intensive progression side-band. If that number or the energy of the O-O transi-tion is not given in the reference, we have estimated it.In Table 4 the observed values are given and it isindicated whether the number is obtained from anemission or an absorption spectrum. If the reportedspectra are polarized, we only analysed the .ry-polar-ized spectrum because of its usually higher resolutionand owing to the fact that the transition to the Azgstate is only visible in xy polarization. The spectra in[20,22,23] are microphotometer tracings.

6. Discussion

The expressions for the vibronic coupling con-stants permit some general remarks to be made.From Eq. (5) it is seen that the relative magnitudesof the coupling constants depend only on the para-meter 7, which depends on the ratio eJe, throughEq. (10). Reasonable values of this ratio for transitionmetal complexes give 17 values between 0.7 and 2.5.In the strong-field approximation the lowest elec-tronic transitions for d8 and d9 complexes areone-electron jumps to the b,, orbital (big - b,,

Table 3Literature parameters for dX complexes necessary for the calculations (upper part) and values derived from the formulae given in the text (lower

part)

[PtClJ ?- [PtBr4]- [PdClq12- [PdBr,] -

e,, r, WI (cm-) 12 420, 2800 10920, 2200 IO 150, 2000 9510, 1780fi,, VP,, Cal ~271 (cm-) 329, 304, 196 205, 188, I28 307, 274, 198 190, 171, 127Dq (cm-). R [27] (A) 2606, 2.308 2396, 2.445 2245, 2.313 2141. 2.4444. v 1.03, 0.44 1.10, 0.44 I.1 I, 0.42 1.14. 0.42k,, k,#. kO> kg/s) 226, 193, 80 198, 167, 77 197, 157, 82 170, 138. 76

-56, -39, -46 -49, -32, -39 -48, -32, -38 -44, -28. -34-57, 3.6, -3.6 -48, 2.1, -3.3 48, 1.9, -3.3 -43, 1.3, -3.0


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Table 4Comparison of computed and experimental Huang-Rhys factors estimated from literature spectra of dX complexes

[PtCI,]- [PtBr,]- [PdCIJ- [PdBr4] -

Calc. Exp. Calc. Exp. Calc. Exp. Calc. Exp.

S,(n,,,** I I 6 [25,29] L.p IS 8 [25] rG 10 I5.S,( B o) 5, 7* 4 [20] ,P 6. 9* 5 [2l] .h.p 4, 6* 6. 9* ,u ,241 h.P***

4 [20.22] C.dS,( A :,)** I I 7 [20] a.h.p I.5 8 [2l] r.h.p IO 6 [23] . I5 8 [24] a,~+

6 [26] W

S,(E&** I I I5 10 I4

SD,. S/3? 0.06, 0.2 0.03, 0.3 0.02, 0.2 0.02, 0.2

L Absorption spectrum. - emission spectrum. et - excitation spectrum.Estimated value (without band analysis).Calculated from the Stokes shift (if absorption and emission spectra are available): S = (A.Es, - 2~,,)/2v, (V n is the frequency of the enabling

mode).Pure potassium salt.

Doped compound - Cs2ZrCI,:MC$ (M = Pt or Pd).Calculated with the inclusion of sd mixing (AESd 1s set at 80000 cm- [IS], and erd = l/4e, [28]).*At this level of approximation (level 2 in Fig. 3) the same numbers are computed for states differing only in their multiplicity.**According to the calculations in [2X]. the spacing of the two spin orbit split states, Pj and PY of the B,, state, is about 430 cm-. The

observed long progression could therefore be due to two electronic transitions.

(1ODq jump), e, - h tp and a Ig - b g, ordered accord-ing to increasing energy). From Eq. (12) the Qczupling constants for these orbital transitions are

Ii&Y -a, r (I? = bsg, eE and alp) and Table 1 showsthat the /3, and pz constants for strong-field E states

of d8 ions correspond also to the orbital couplingconstants (the sign is of minor importance in a linearcoupling, because the direction of the distortions isequivalent). That is why a plot of the d couplingconstants against 7, setting DqlR = 1 (Fig. 4), revealsinformation concerning the relative strength ofdifferent coupling cases in ds and d complexes.The following results are obtained for complexeswith d8 and d configurations.

Result 1: Concerning electronic E states, theabsolute /3, and /32 coupling constants are at

most l/S and l/8 of the absolute cy constant forreasonable 11 values. Assuming Ila/t)a, = 1 and

WPz = 3/2 for the vibrational frequencies, onecan compute S, 2 25S,, and S, > 18S0, for theHuang-Rhys factors with Eq. (8). Consequently,in square planar complexes, linear vibroniccoupling with non-totally symmetric modes issupposed to be of minor importance with regardto its contribution to the Stokes shift.Result 2: The Q coupling constants of states

resulting from ulg - b transitions are smallerthan the ones from the other orbital transitions,although the inclusion of sd mixing reduces thedifference. This result is not in perfect agreementwith the spectrum of CU (see below).

Result 3: Higher-order couplings with the PI

d9 dXA; forDqlR = 1

Fig. 4. Plot of d coupling constants (d coupling constants withnegative sign) against the parameter 1, setting DqlR = I. The d8coupling constants indicated correspond to the dX strong-field statesin Fig. 3 (multiplicity is not important). The QI coupling constantof the ground state is subtracted from the other 01 constants and setto zero (see text).


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106 K. Wining, J. DegerdJournal of Molecular Structure (Theochem) 431 (1998) 97-107

mode should have a low influence because ofthe small coupling constant and higher-order /3,couplings are also not favourable, because in d8and d complexes there are no low-lying elec-

tronic states close in energy that are able tocouple to this mode.Result 4: The (Y coupling constant of the 1ODqjump does not depend on 17.

Regarding the results obtained from optical spectragiven in Table 2 for the Cu+ complex, a considerablecorrespondence between computed and experimentalnuclear displacements and Huang-Rhys factors forthe 2B2B and Eg states is found. The A lg band iscalculated to be too narrow even if the effect of sdmixing, which tends to broaden this band, is consid-ered. The width of the 2E, band should, in principle,arise from a coupling corresponding to Result 1, theunsymmetrical modes could be responsible forthe missing resolution of vibrational fine structure.

The spectra of Pd2+ and Pt2+ complexes reveal goodagreement between theory and experiment (Table 4).First, it is correctly predicted that vibrational pro-gressions with the totally symmetric mode, observa-ble in transitions involving the Big state (a Ig - b Igjump), are shorter than progressions involving Azgand Azg states (bzg - b,, jump). In contrast to theCu complex, Result 2 agrees with the progressionlength in the spectra. From Table 4 the Huang-Rhysfactors for E, states should be similar to those for Algstates. In fact, similar half-widths are observed forabsorption bands to these states. Upon changingfrom the chloride to the bromide ligand, itis correctly predicted that the progression becomeslonger. Going from Pt2+ to Pd2+ no significant effecton the band width is observed either in the calculationor in the spectra. Including the sd mixing effectyields an increase in the progression length calculated

for transitions between the ground state and the jBlgstate (Result 2). The calculated absolute values inTable 4 are a bit too large, but of the right order ofmagnitude. Configuration interactions due to electronrepulsion and the spin-orbit coupling operator resultonly in small changes of the computed results.Huang-Rhys factors in most cases differ by lessthan unity for the spin-orbit split states comparedwith the strong-field parent states. This is reasonable,because of the comparatively strong influence of the

ligand field in relation to octahedral and tetrahedralcomplexes. Corresponding to Result 1, the broadnessof the E, bands is supposed to be due to (Y couplings.

In [29,30] unusual spectroscopic features have

been revealed for K?PtCl, spectra. First, an energygap between the absorption and emission spectrumof about 1800 cm- with no spectral intensity isfound and, second, the vibrational frequency in theemission spectra lies between the ground-state fre-quencies of the CY nd PI stretching modes. By usingtime-dependent theory of electronic spectroscopythese features are explained by proposing anexcited-state potential in the Qp, direction dependingon the three adjustable parameters k,tfjw,rr), a and A:

v(Qp,) = VP;, +A exp( -aQi,) (22)Optimization of Eq. (22) yields a distortion along the/3, stretching mode of the same order of magnitudeas along the (Y mode (AQ, = 12 pm, AQp, = 12 pm in[29] and AQ, = 12 pm, AQa, = 16 pm in [30]). Theprogression frequency in the emission spectrumcan then be explained by the missing mode effect(MIME). The excited state has not been specified.

Although our simple model is not able to explainthe unusual spectroscopic features, such a strong 6,distortion seems unlikely on the basic assumption ofour model: i.e., that the excited-state distortion iscaused by a change in the electron distribution inthe d shell of the central ion. We use a limited basisand consider only the first ligand sphere, which how-ever is necessary in semi-empirical ligand-field theoryin order not to increase the number of parameterstoo much. In fact, the present approach allows forthe calculation of nuclear distortions, band shapesand stabilization energies on the basis of commonparameters derived from static ligand-field or AOManalysis of optical spectra with no additional free

adjustable parameters to be fitted to experiment. Onthe other hand, that is why quantitative correspond-ence cannot be expected.

7. Conclusion

A perturbative treatment of the electron-phononcoupling of square planar transition metal complexesallows for characterization of the potential energy


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K. Wissing, J. DegedJournul of Molecular Structure Theochem) 431 199X) 97-107 107

minima of electronic states within the open d shell.For square planar dx and d complexes, vibroniccoupling of degenerate electronic states to /31 and 01modes is expected to be negligible compared with

coupling to the totally symmetric vibration. Resultsobtained from analysis of optical spectra exhibitingvibronic structure are in good agreement with thepredictions of the model. Thus the method may beused to help interpret optical spectra with less well-resolved structure. It is not too elaborate to extendthe model to systems of other symmetry. Tetrahedralsymmetry, which makes the treatment of three Jahn-Teller-active degenerate modes necessary, might beof interest.

Acknowledgements

Financial support from the Deutsche Forschungsge-meinschaft and from the Fond der ChemischenIndustrie is gratefully acknowledged.

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Ligand-field Theory Metal Complexes (1)

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