Hipps, Merrell, and Crosby
2232 (6) A. R. Reinberg and S. G. Parker, Phys. Rev. 6, 1, 2085 (1970). (7)P. B. Dorain and R. G. Wheeler, J. Chem. Phys., 45, 1172 (1966). (8) (a) H. H. Patterson, J. L. Nims, and C. M. Valencia, J. Mol. Spectrosc., 42, 567 (1972);(b) A. M. Black and C. D. Flint, J. Chem. SOC.,Faraday Trans. 2, 71, 1871 (1975). (9)P. B. Dorain in “Transition Metal Chemistry”, Vol. 4, R. L. Carlin, Ed., MarceCDekker, New York, N.Y., 1968,Chapter 4. (IO) (a) J. R. Dickinson, S.B. Piepho, J. A. Spencer, and P. N. Schatz, J. Chem. Phys., 56, 2668 (1972);(b) S.B. Piepho, J. R. Dickinson, J. A. Spencer, and P. N. Schatz, ibid., 57, 982 (1972);(c) R. W. Schwartz and P. N. Schatz, Phys. Rev. 6, 8,3229 (1973). (11) J. C. Collingwood, S. B. Piepho, R. W. Schwartz, P. A. Dobosh, J. R. Dickinson, and P. N. Schatz, Mol. Phys., 29,793(1975), particularly Figure 1. (12)C. K. Luk and F. S. Richardson, Chem. Phys. Lett., 25, 215 (1974). (13)C. K. Luk and F. S. Richardson, J. Am. Chem. SOC.,96,2006(1974). (14)W. C. Yeakel, et al., submitted for publication. (15) D. Durocher and P. 6. Dorain, J. Chem. Phys., 61,1361 (1974).
(16) P. B. Dorain and R. G. Wheeler, Phys. Rev. Lett., 15, 966 (1965). (17)Reference 1 1 , Figure 10. (18) Reference 7,Table Ill. (19)F. Bonati and F. A. Cotton, lnorg. Chem., 6,1353 (1967). (20)See discussion comment following this paper.
Discussion R. KOPELMAN.Could your “cluster”state be a “defect” state where the “defect” (generalizedterm) is induced by the higher concentration and thus is behaving superlinear with concentration? P. N. SCHATZ. This would be a possibility, but I am not convinced it would account for the observed concentration dependence of the emission.
Determination of Geometrical Parameters of Excited States. Application to d6 Transition Metal Complexes of 0 and D4 Symmetryla K. W. Hipps,lb G. A. Merrell, and G. A. Crosby“ Department of Chemistry and Chemical Physics Program, Washington State University, Pullman, Washington 99 163 (Received May 10, 1976) Publication costs assisted by the Air Force Office of Scientific Research, Directorate of Chemical Sciences
Analyses of luminescence emission band envelopes arising from ligand field transitions of transition metal complexes are discussed from both theoretical and practical points of view. Equations are developed that allow geometrical parameters of electronic excited states to be obtained from a full Franck-Condon analysis, direct numerical integration of the band envelopes, or Gaussian curve-fitting techniques. The usefulness, inherent complications, limitations, and accuracy of the several methods are pointed out. Emission spectra for potassium hexacyanocobaltate(IU), trans-dibromotetrakis(4-methylpyridine)rhodium(III)bromide, trans-dichlorotetrakis(pyridine)rhodium(III) chloride, and trans-dibromotetrakis(pyridine)rhodium(III) bromide a t 2 K are given and analyzed. Symmetry-preserving expansions occur in all cases. Franck-Condon analysis (FCA) of vibrational structure on electronic spectra to obtain information about excited state geometries of diatomic molecules was pioneered by Hutchison in 1930.2Coon et aL3 have applied the method to triatomics, and Craig4 determined the lBzu excited-state C-C distance of benzene by FCA. More recently, many conjugated organics have been subjected to FCAe5-7In contrast, FCA of electronic spectra of inorganic materials is just beginning to yield information (excited-state geometries, excited-state force constants, and ground-state anharmonicities)s-10 of importance to those concerned with radiationless transitions,l1>l2 photochemistry,13 and the redistribution of charge during an electronic transition.14J5 Excited-state parameters derived from FCA of inorganic molecules are a desideratum of modern chemistry. The lack of Franck-Condon investigations of inorganic complexes is due to the fact that most of the reported spectra ‘have been either so sharp that they were essentially atomic and FCA was unnecessary or so broad and diffuse that detailed analyses were precluded. In this study we address the analysis of complexes of the second type. First, we consider the origins of the diffuse emission spectra of inorganic complexes. Then we present a detailed mathematical machinery The Journal of Physical Chemistry, Vol. 80, No. 20, 1976
for FCA and restrict its application to complexes of 0 and Dd microsymmetry. For spectra that are too diffuse for FCA, we propose two methods for determining excited-state parameters. Finally, we present the results of applying FCA and the approximate methods to several complexes of the d6 type that are of photochemical interest. Experimental Section All complexes were prepared in our laboratory by methods given elsewhere.16J7Each substance was recrystallized several times and thoroughly dried. Distillation of the pyridine (py) and 4-methylpyridine (4-Mepy) was required or the resultant compounds did not produce well-structured spectra. All emission spectra were obtained from exciting the pure solids. The samples were mounted in the bore of an Andonian Associates helium cryostat; -2 K temperatures were obtained by pumping directly on the helium liquid in which the sample was totally immersed. Excitation was accomplished by directing the output of a 1000-W Hg-Xe lamp through 5 cm of aqueous CuSO4 solution (100 gh.) and a Corning 7-60 glass filter. Scattering exciting light was blocked by two 500-nm cutoff filters located before the entrance slit of a 0.5-m Jarrell-Ash monochromator equipped with curved bilateral slits
Geometrical Parameters of Excited States for the K~[CO(CN)G] measurements. For the others only a 550-nm cutoff filter was placed directly before the exit slit. Resolution was sample limited. The signal from a dry icemethanol cooled RCA 7102 photomultiplier was amplified by a Keithley microammeter and displayed on an X-Y recorder. Recorded signals were corrected to give the relative photon intensity sec-l cm-2 per cm-l bandwidth by comparison with a calibrated NBS standard lamp.l* Photon intensities (quantumyields) per unit area per k K (or cm-1) bandwidth are reported here rather t h a n the true intensity. The FCA, moments calculation, and least-squares Gaussian fits were all performed on the same data set for a given molecule.
Theoretical Development Origin of Broad Luminescence Bands of Complexes. As first pointed out by Orgel,15the expected width of a dd transition is related to the relative slopes of the energy vs. Dq plots of the terms involved in the transition. Since Kasha’s rulelg as modified by Demas and CrosbyZ0is usually satisfied for inorganic complexes, perusal of the Tanabe and Sugano diagrams21leads to the conclusion that only broad emission will be seen from all hexacoordinate dd emitters with a d s configuration, provided that s = 5,6, or 7. Additionally, it could occur for suitable ranges of A/B from metal complexes with d k configurations where k = 2,3,4, or 8. For many complexes, however, no emission may occur a t all. Emission bandwidths are a clear indication of a difference between ground- and excited-state geometry and binding. This can be easily visualized provided one posits that most of the bond strength in a complex does not derive from d electrons. In this vein, the tzgorbitals are nonbonding or weakly antibonding, whereas the eg set is antibonding. Changes in the electron populations of the eg and tag orbitals would then lead to a geometrical rearrangement. This argument, first given by Orgel,15may be stated in an alternate way. The slopes of the term energies represent the first derivative of the crystal field potential (d electrons only) with respect to the totally symmetric breathing mode. Two terms having different slopes are therefore required to have different geometries but the same symmetry. Information concerning symmetry changes is not contained in these diagrams. For complexes of 0 4 symmetry, the lines of the octahedral Tanabe-Sugano diagrams become surfaces. Two totally symmetric coordinates (Qa, axial; Qeq,equatorial) are required to specify the energy. We may relate the amount of extension in direction i (i = a or eq) to where E‘ is the en_ergyof the initial term, E is the energy of the final term, and QM is the mean of the equilibrium nuclear positions of the two terms. This relation is a simple generalization of Orgel’s treatment. As for the octahedral case, the Tanabe-Sugano surfaces provide no direct information about possible symmetry changes for transitons between terms. Besides the band structure generated by a difference in geometry, there are other factors that can, and often do, affect the band shape. Since virtually every emission that is predicted to be diffuse has either a degenerate ground or excited term, the influence of spin-orbit coupling cannot be ignored. For first-row transition metals such as C O ~ +a , term really represents a set of levels spread over as much as 500 cm-l, or perhaps more.9a Further, second-order and higher order spin-orbit coupling will cause the equilibrium position and potential surface of each state arising from the same term to
2233
be different. At temperatures above about 20 K, the observed spectrum usually becomes a superposition of different electronic t r a n s i t i o n ~ . ~Furthermore, J~~~~ each of the consequent bands may have one or more origins.lo If, after spin-orbit coupling is accounted for, degeneracies still remain, then electronic-nuclear coupling must be considered. The symmetry changes associated with the JahnTeller effect will reveal themselves as progressions in nontotally symmetric vibrational frequencies. In addition, interaction between potential surfaces can lead to multiple minima of complex forms. It is only for the lowest excited state (electronic and vibrational) that a harmonic oscillator type potential seems reasonable. In all that follows, we assume that the Born-Oppenheimer approximation ( B O A ) is valid for the lowest excited level. At a sufficiently low temperature (4 K), only the lowest excited level will be populated; we then have only to contend with the emission spectrum from a single excited level. Having circumvented the. bulk of the excited-state contributions to the diffuseness of a spectrum, we must now consider the ground-term contribution. An appealing simplification is to consider only those systems in which the ground term is nondegenerate, for which the -0 K spectrum should be relatively uncomplicated. This condition obtains for all of the complexes studied here. Therefore we limit our discussion to transitions from a single electronic excited state in its lowest (n’ = 0’) vibrational state to a single vibrational level of a nondegenerate electronic state. The terminal vibrational state is determined by the selection rules. Unfortunately, even from systems obeying these tight restrictions, sharply resolved spectra may not necessarily arise. In condensed phases, molecules are not isolated, but for dd transitions we may often treat a complex as a nearly independent entity. Nevertheless the total specification of the molecular state must include the quasi-continuum of the phonon bath. States having the same electronic and vibrational quantum numbers can have different phonon quantum numbers, and transitions will therefore be “blurred.” Factor group splittings and nonuniformity of sites in the crystal can also lead to further reductions in the resolution of vibrational components of a band. Quantitaive treatment of these factors is beyond our capability a t this time, and with a posteriori justification we assume that lattice contributions to band structure are identical for every vibronic transition. We therefore introduce the same shape function, h ( w ) , for each intrinsic band (vide infra). Franck-Condon Analysis. We now turn our attention to the predicted band pattern and FCA when the special situation of -0 K and a nondegenerate ground state apply. Consider a molecule having p degrees of vibrational freedom. Adopt the BOA23and assume only two electronic states to be involved in the transition. The excited state is then specified by
IE) = le)la‘) (2) where the excited electronic function l e ) is a function both of the electronic coordizates, and, parametrically, of the nuclear coordinates, 2. IO’) is taken as the p-factored product of n’ = 0 harmonic oscillator functions of the excited normal displacement coordinates, 8’. The possible terminal states, I F ) , are given as
e,
IF) = If)lri) (3) where we again use the harmonic approximation to specify lfi) as a p-factored product of harmonic oscillator functions of the The Journal of Physical Chemistry, Vol. 80, No. 20, 1976
Hipps, Merrell, and Crosby
2234
ground ( I f ) ) state normal coordinates, &. The p-fold collection of n values specifies a single terminal level. In order to investigate the overall band shape for the le) I f ) transition, we first evaluate the intensity and energy of the individual IE) I F ) transitions and then superpose the results for all IF). If mk is the kth component of an electronic transition moment operator, the transition probability between states IE) and I F ) for k polarization depends on the square of the integral
-
-
(ElmklF) =
($1
( e l m k l f )16)
(4)
where the central integral is over electronic coordinates only. Because of the parametric dependence of the electronic functions on 8, th_e electronic integral is a function of 8. Calling it Mk (8 X O ) where , 20is the ground-state nuclear geometry, we have
t. Substitution into eq 7 provides us with the result t
(ElmklF) =
II (os’lns)[ M k ( X o )r=t+l fi s(O,nr) s=l + 1 =5t + l Mk’(Oj’IQJIni) r=t+l R
d(O,nr)}
(8)
#I
The probability of a transition depends on the square of eq 8 times the cube of the energy difference between states I E ) and IF). Assuming
MkJ(o,’(&,lnJ)= M k J 6 ( n J , l )
(9)
we have
+
The usual procedyre a t this point is to expand Mk in a power series of & about XO.Terms higher than linear are neglected,23 i.e.
( E l m k l F ) =Mk(20)(6‘1fi)-t C M k J ( @ l Q J I c ) (6) I
where MkJ = (8Mk/aQJ)Q= xoand the sum is over all p values of j . If Mk (20) is nonvanishing, it usually dominates, and the transition is said to be equilibrium geometry allowed (EGA). If Mk ( 2 0 ) vanishes, the transition is said to be vibronically allowed (VA). A transition is usually VA only if M k ( 8 0 ) vanishes by symmetry; in this case, MkJ also vanishes if j labels a totally symmetric mode. For t totally symmetric modes, we choose to label their occupation numbers consecutively as nl, n2, . . . , nt.
(ElmklF)= M k ( z o ) ( @ l f i -t )
5
J=t+l
MkJ(G’lQJ[fi) ( 7 )
For the EGA case the terms in the summation can be neglected. For the VA case only the terms in the summation contribute to the matrix element. For an intermediate case [Mk(Xo)= M k J ] ,eq 7 requires modification, since the set of M k J 6 5 t ) does not vanish by symmetry. We consider this ‘interesting case no further in what follows, and we assume the validity of eq 7 henceforth. In the general case of a symmetry change during the transition, the vibrational overlap calculation is very complicated. For example, if an octahedral molecule is tetragonally distorted in the excited state, we would expect progressions in both a1 and eg ground-state modes, both of these appearing totally symmetric to the excited state. If the excited-state symmetry is subgroup related to the ground-state symmetry, partial factorization of the overlap integral is possible. Here, coordinates transforming as the irreducible representations of the lowest symmetry group are independent. Fortunately, it appears that most broad dd luminescence is principally due to a symmetry-conserving expansion. Certainly the data presented here demand a vanishingly small symmetry change. Because of the localized nature of a dd transition, we expect that the internal ligand vibrations should be unaffected by the transition. W e therefore restrict the analysis to the case where the ligands behave as point masses and symmetry is preserved. The above restriction allows us to place t = 1 for 0 symmetry and t = 2 for 0 4 symmetry. We may now simplify eq 7 considerably by setting ( O l ’ I n k ) = 6G,h)6(n,O)provided j > The Journal of Physical Chemistry, Vol. 80, No. 20, 1976
in the VA case. wm is the ground-state oscillator frequency of the mth mode, and y is a collection of constants. The quantities I (Os’lns)I are the totally symmetric vibration (al) FC factors. The modes j , fcr which Mkj is nonzero, are called allowing modes. a is the 0’ 0 transition energy. , For a VA transition with a single allowing mode, eq 10a and lob are formally identical. The only distinction is that the predicted intensity distribution for a VA transition begins a t a* = 01 - w j rather than a. For either case, the 0 K emission spectrum is therefore
-
I(w)
= A C 1(01’ln~)121(0~’ln~)12.. . I(0t’lnt)12 lntl
t
w
- a* + C nsws) w 3 (11) s=l
where h ( w ) is the shape function for a single vibronic line and a* is the effective origin for the transition. h ( w - 6) maximizes at w = 6 and has unit area. If more than one allowing mode contributes to the band, Z(w) will be a sum of terms like eq 11, one term for each allowing mode. For a single a1 progression, eq 11may be simplified to
I(w) =A n
I (O’ln)I 2 h ( -~ a* + nwO)w3
(12)
The FC factors can be calculated. In Appendix A we geperate their values for a harmonic oscillator model where excitedstate geometry (Q’ = Q - a ) and frequency (w’/wo = k 2 )differ from those of the ground state. By comparing the calculated Z ( w ) to that observed, we may use “best fit” criteria to arrive at values of a and k2. The difficulty, of course, is in the choice of h ( w ) . As discussed earlier, it has been introduced in an ad hoc manner. A normalized Gaussian form was chosen for convenience. For cases where more than one progression is observed, eq 10b is consistent with a superposition of equations of the form of eq 12. Moments Analysis. Although eq 12 can be used effectively for well-resolved spectra by the utilization of numerical analysis, it leaves much to be desired as a qualitative or semiquantitative formulation. It is essentially useless for bands
2235
Geometrical Parameters of Excited States
with unresolved vibrational structure. In an attempt to circumvent these difficulties, we resort to the method of moments.Z4p25We define the s t h moment of the (photon) emission spectrum as [I]”s=
Jband I ( w ) [ ( w - v)3+s/w3]dw
(13)
In Appendix B, we show that (consistent with eq 12) [I]n0-2/[1]0-3
gj
= a*
- Tiwo
(14)
+ A(2E + l)] (15) [I],O/[I]O-~ = -~o~[Ti(8A +~8A + 1)+ A(3 + 4A)] (16) [1]a-1/[1]o-3 = ~o’[Ti
where ii=A+b2/2 A = ( k 2 - 1)’/4k2 b2 = Mwoa2/h
(17)
For comparison, a “Gaussian” band with intensity distribution G(w) = Aw3 exp[-c(w - i3)2] has moments [G]o-2/[G]o-3 = G [Gla-l/[G]~-~ =1/2~ [GIao = 0
(18)
Several qualitative features become immediately apparent. Both the Stokes shift (-2Tiwo) and the l/e width depend on the displacement, a , and the difference in ground and excited oscillator frequencies, The l/e width varies linearly with the displacement and therefore is proportional to Ti,while the Stokes shift goes as Ti2.Moreover, a Gaussian shape function is too symmetrical. The actual intensity distribution has more intensity a t lower frequencies than does G(w). In principle, all of the information contained in eq 1 2 is available from algebraic manipulation of the moments, m 1moments being required to determine m parameters, if, as usual, only relative rather than absolute intensities are available. Unfortunately, the imprecision in our measurements of relative intensities renders this procedure impractical for more than two parameters. Furthermore, for just thbse cases where moments fitting procedures are required, wo is also an unknown. In order to extract information under these circumstances, further approximations are required. For the systems under consideration, b2 is usually -20. On the other hand, A is seldom larger than 0.1 (k2 = 0.5). T o a good approximation we may take Ti = b2/2. With less certainty, the terms in A for the [lla-l moment can also be neglected to give
+
[1].-1/([1]0-3(a*
- G ) ] 1 wo
Pb d2[1]~-1/([1]o-3~02)
(19a) (1%)
Or, if we fit a Gaussian to the data, we obtain [2c((r* - 41-1
P wo
(204
(w&)-~’~ P b @Ob) These approximate equations have significance only when a single Progression is responsible for the observed spectrum. Estimation of a*, made reasonable by the relative sharpness of the high-energy side of the band, allows estimation of b and wo from either the calculated moments (eq 19a, 19b) or the Gaussian paramcters (eq 20a, 20b). Both the moments and the Gaussian procedures for estimating b are expected to function best when a reliable estimate of wo is available without recourse to eq 19a or 20a. This is possible when the emission is
Figure 1. Solid-state emission spectra of D4complexes at 2 K: XXX, experimental data points; -, computer-generated “best fit” FranckCondon envelope as defined by least squares (see eq 12 and 22); (a) tran~-[RhBr~(4-Mepy)~]Br, “best fit” of truncated data set; (b) trans[RhBr2(4-Mepy)4]Br; (c) trans- [RhC12(py)4]CI; (d) trans- [RhBr2(PYMB~.
sufficiently structured to allow direct observation of wo or, less accurately, when the estimate from eq 19a or 20a identifies a particular known vibrational frequency. Results Figures 1 and 2 represent the results of a least-squares comparison of eq 12 to the emission spectra of four complexes at 2 K. The physical quantities determined from the computer-generated fits are presented in Table I. Seven parameters were adjusted to give the minimum square deviation between eq 12 and the observed spectra (vide infra). The anThe Journal of Physical Chemistry, Vol. 80, No. 20, 1976
Hipps, Merrell, and Crosby
2236
TABLE I: Geometrical Parameters of 0 and D4 Complexes Derived from Franck-Condon Analyses a*,e
a
w0,
X,
Complex
k2a
bb
kK
kK
trans- [RhBr2(4-Mepy)4]Br (Figure la) trans- [RhBr2(4-Mepy)d]Br (Figure lb) trans- [RhClz(py)d]Cl (Figure IC) trans- [RhBrs(py)4]Br (Figure Id) K3[Co(CN)d (Figure 2)
0.70
4.43
16.753
0.200
0.0001
0.57
4.57
16.676
0.180
0.0007
0.95
4.36
17.809
0.350
0.0021
0.73
4.84
16.700
0.184
0.0002
0.65
4.49
17.071
0.423
0.0026
See eq 17. a* is the effective origin of the transition. See eq 21; 00 = wL
k 2 = w'/wo.
kK
+ x;see eq 22.
'1 * t; In In
W z
w 4
Zl t-
f, 5 w k(r
W
17
15
16
111
12
13
11
10
kK
Figure 2. Solid-state emission spectrum of Ks[Co(CN)6]at 2 K: X X X , experimental data points; -, computer-generated "best fit" FranckCondon envelope as defined by least squares (see eq 12 and 22).
if I-
Y z
\
w -
I
590
610
I
630
I
I
650
I
I
670
I
I
690
,
I
710
I
,
nm
Figure 3. Emission spectrum of solid tran~-[RhBr~(py)~]Br at 2 K. Phonon structure is superposed on a single totally symmetric progression. Compare Figure I d .
harmonicity term for the ground state was introduced to account for the location of each vibrational peak, but no corresponding corrections were made to the overlaps. wo and x satisfy the ground vibrational energy relation For &[Co(CN)6] and [RhBrz(py),]Br, the fitting procedure converged rapidly and uniquely to give the rather good predicted curves shown. For [RhBr2(4-Mepy)d]Br, however, slightly different results were obtained from very closely spaced data taken only from the structured part of the band (Figure l a ) , as opposed to the same number of points spread over the entire spectrum (Figure lb). In the latter case the predicted curve falls significantly below the observed spectrum on the low-energy tail. We believe that the parameters associated with Figure la'are the more accurate ones. Our The Journal of Physical Chemistry, Vol. 80, No. 20, 1976
Figure 4. Gaussian representation of electronic emission spectrum of XXX, experimental data points; -, computer-generated KS[C~(CN)s]: "best fit" of data to q w ) = A exp[-c(o - 3'1.A = 0.386 67 x o = 13.290,and c = 0.271 96.
reasons for this conjecture are as follows: (a) for high n the calculation of overlaps via harmonic oscillator functions becomes questionable; (b) for large local distortions of the lattice, Le., large n, the assumption that every line should have the same shape, h(o),becomes less reasonable; (c) with our detector, uncertainty in the relative intensity is largest a t long wavelengths. Figure 3 shows the unusually well-resolved phonon structure that was recorded for [RhBrz(py)d]Br.The spectrum has been reproduced without correction in order to show its unique features. For the subsequent analyses the phonon structure was ignored by manually smoothing the curve. The smoothed experimental curve supplied the data points in Figure 5c. Figures 4 and 5 depict the best Gaussians [G(w)],as defined by least squares, to represent the emission spectra of K ~ [ C O ( C N ) ~[RhBr2(4-Mepy)41Br, ], [ R ~ C ~ ~ ( P Y ) and ~ICL [RhBrz(py)4]Br, respectively. We note that, in general, the observed intensity is higher at low frequencies than that predicted by G(w). Values of A , T j , and c are given in the figure captions. Rather than report numerical values for the integrated moments, we report certain extracted parameters in Table 11. Discussion The most obvious result is that a single vibrational progression can account for the observed bands in Figures 1and 2. For K3[Co(CN)6], the 420-cm-1 vibration is the totally symmetric one, just as predicted by the Tanabe-Sugano diagram. The values recorded in Table I allow us to assign the progressions in the 0 4 cases to metal-halogen (M-X) stretches rather than to the M-N stretching frequency. In Appendix
2237
Geometrical Parameters of Excited States TABLE 11:Comparison of Geometrical Parameters Derived from Franck-Condon, Direct-Integration, and Gaussian Analyses Complex
a
Parameter
FCAa
Integrationb
Gaussiahc
0.200 0.156 0.350 0.161 0.184 0.164 0.423 0.100
0.258 0.097 0.281 0.192 0.242 0.124 0.518 0.079
0.295 0.093 0.311 0.173 0.249 0.119 0.467 0.088
Obtained from Figures 1 and 2 and eq 12. Obtained from Figures 4 and 5 and eq 19. Obtained from Figures 4 and 5 and eq
20.
C we show that this domination by M-X stretching is predicted from the crystal field theory by eq 1. In order to compare the relative merits of the direct integration and Gaussian fitting procedures for estimating the degree of geometrical distortion and the energy of the vibration responsible for the progression, we have prepared Table 11.The values determined by FCA (eq 12) are the best values. Although both approximate methods give poor estimates of wo, the Gaussian approximation gives surprisingly good results relative to the more tedious direct integration. The inferior quality of the results from direct integration is probably due to the experimental uncertainty of the relative photon intensity at low energies. Also, any deviations from the model implicit in eq 12, such as anharmonicity, would tend to appear at lower energies. Any extra low-energy intensity would greatly affect the calculated moments. Either of the methods associated with eq 19 and 20 should be used with extreme caution when insufficient structure is available to ascertain the existence, and spacing, of a single progression. Arguments based on analogies with similar complexes are not satisfactory. For example, [RhC12(4Mepy)4]Cl shows two origins for the M-X stretching progression.26 Only when structure is observed and understood, may these methods be applied to give estimates of b and k 2 without resorting to FCA. One last point relates to the practical assessment of the time when the system is at an effective 0 K temperature in its excited state. For complexes of the type reported here, monitoring the emission lifetime as a function of temperature allows one to define an effective 0 K temperature. Some of the details of this procedure have been rep0rted.2~,27 Computational Details Substitution of the equation for I ( u o ~ u ,I) derived in Appendix A into eq 12 and insertion of the shape function h(w)produce the working expression
*
I ( w=), " - e - k 2 b 2 / ( l + k 2 )
w
1+k2
3 30
[h(w)/2"-ln!]
n=O
X
m=O
k2)(n+m)/2l2
16
1s
14
13
kK
-
[ 2 ICmnl(k2b)m(l - k2)(n-m)/2/(1+
17
Figure 5. Gaussian representations of electronic emission spectra of 0 4 complexes: XXX, experimental data points; -, corn uter-generqz7;(a) transated "best fit" of data to q w ) = A exp[-c(w [RhBr.#-Mepy)4]Br, A = 0.41831 X 23 = 14.887,c = 0.946 21; (b) trans-[RhCMpyk]CI, A = 0.45434 X lo+, W = 14.760,c = 0.527 66 (c) frans-[RhBr2(py)4]Br,A = 0.41739 X lo-*, W = 14.629,
(22)
where h ( w ) I,-[w+nwi-n2~-a*12/(w2/4), ai= wo - x, and w is the full width at l/e height of a single vibrational peak. S is a scaling factor. Since no progressions extending beyond n = 20 have been observed, the series was truncated at n = 30. To obtain a realistic fit of this equation to the experimental curves (and save money) good initial guesses of all seven pa-
c = 0.970 51.
rameters should be made. Our procedure was as follows: k 2 was assigned a value of 0.8, a physically reasonable number; b was chosen as 4.3,a value first determined for &[Co(CN),] (in general, an initial value for b can be best extracted from a Gaussian type fit of the intensity distribution); a* was chosen as the frequency of the first perceptible peak appearing in the The Journal of Physical Chemistry, Vol. 80, No. 20, 1976
Hipps, Merrell, and Crosby
spectrum on the high-energy side of the band; S was assigned the value of the maximum experimental photon intensity; w was chosen to be the l / e width of a typical vibrational peak. First estimates for ui and x were obtained from a rough fit of eq 21 to the observed vibrational progression on the band. All seven parameters were varied to minimize the least-squares deviation of the computed curve from the experimental intensities corrected for instrumental response. Typically, the total spectrum was divided into about 200 segments for data analysis. Moments were calculated using Simpson's rule.
where
Appendix A. Calculation of Franck-Condon Factors Throughout this appendix we drop the prime on 0'. First, we write (UOIun)
5
1:-
+ a)]e-a2(Q+a)2/2dQ
NoNne-a'2Q2/2Hn [a(&
(Al)
where H n ( x ) is the n t h Hermite polynomial of argument x . Let JC = Q a; then
+
(UOJun) E
NoNn
Km
e-a'2(x-a)2/2H
aJCe-a2x2/2
dx
Let b = a a , p = a x , and k = a'/a; then
J - m
Using the generating function for Hn(p)
we have
Appendix B If in eq 12 we assume h,(w)to be a 6 function, then =A
C I (O'ln)I 2(a* - nu0 - u ) ~ + ~ n
(Bl)
By carrying out the summation we find
[ I ] o -=~ A Define
-ik2bd(1
+ k2)(1- k 2 )
I (O'ln)l 2 = A -i
tB2)
t ~ Eo 3
033)
[I]G-l/A= wo2(f12 - Z2)
(B4)
[I]o-'/A =
+
s' = i d ( 1 - k2)/(1 k2)s y =
n
CY*
[I]G0/A= uO3(-?
+ 3 i t 3 - 2E3)
(B5)
Our problem is to evaluate 5 in terms of displacement and potential parameters. This is most easily done by finding a recurrence relation via the step-up (b+,a+) and step-down (b-,a-) operators for the excited and ground-state vibrational functions, respectively.28 Using
-
X [id(l k2)/(1
Equating powers of s, we obtain
Sn
b+ = a+a+
+ a-a- + a0
(B7)
where ah = (k/2)(1f k - 2 ) and a0 = k b / 2 ( h and b are as previously defined in the text), we have (O'lb+ln) =
The Journal of Physical Chemistry, Vol. 80, No. 20, 1976
036)
(O'lb+ = 0
+ k 2 ) I nn!a
+
m
(O'ln
+ 1)
+ a-&(O'ln
- 1) + ao(0'ln) (B8)
Geometrical Parameters of Excited States
2239
When this equation is squared in two ways and the (O’ln f 1)(0’1n ) terms are eliminated, we find
+
+
+
a+(a+ a-)(n 1)1(O’ln 1)12 = a-2[1+ (a-/a+)]nI(O’ln - 1 ) ( 2 + (k2b2/2)[1- (cY--/cY+)]I (O’ln)l2 (B9)
which can be algebraically manipulated to give
nl(O’ln))2=[4h2/(1+k2)2][A(n-1)1(0‘1n -2)12 + (b2/8)1(0’ln - 1)12] (B10) Summation of eq B10 gives (B11)
ii= A + b2/2
Multiplication of eq B10 by n followed by summation gives -
n2=E(ii+l)+A(2ii+1) Multiplication of eq B10 by yields
-
0312)
followed by summation
n2
+ + 1)+ A(6ii2 + llii + 3) + 4A2(25 + 1)
n3 = ii(ii2 3ii
(B13)
Substitution of eq B11, B12, and B13 into B3, B4, and B5 produces the results given earlier. Appendix C In ref 29 it is shown that the d-orbital energies for a 0 4 symmetry complex with axial (a) and equatorial (eq) ligands are E(bl) = 6Dq(eq) 2Ds - Dt
+
E(a1) = 6Dq(eq) - 2Ds - 6Dt E(b2) = -4Dq(eq) E(e) = -4Dq(eq) and Dt have
+ 2Ds - Dt - Ds + 4Dt
(CU
= (4/7)[Dq(eq)- Dq(a)]. Assuming Ds = 3Dt,30we E(b1) = 8.86Dq(eq) - 2.86Dq(a) E(a1) = -0.86Dq(eq)
+ 6.86Dq(a)
E(b2) = -1.14Dq(eq) - 2.86Dq(a) E(e) = -3.43Dq(eq)
-
(0)
0.57Dq(a)
where the ordering is as expected if Dq(a) < Dq(eq). If one assumes that the emitting level is predominantly represented by the configuration e4b2al,while the ground level is predominantly e4b22,the change in crystal field energy on transition, AVcf,is
+ 9.72Dq(a)
((23)
(aAvcf/aQa) 35(dAVcf/aQeq)
(C4)
AVcf
0.28Dq(eq)
This suggests that We predict (eq 1of text) that excitation of a 0 4 complex will produce a very large expansion of the M-X distance but a tiny
increase in the M-4-Mepy distance whenever Dq(a) < Dq(eq). For the case where Dq(a) > Dq(eq),the weaker ligand is still maximally displaced but now only by a factor of -5. This procedure may be used for any other configurational transition. It is more proper to use the actual energies; these are linear combinations of configurational energies. References and Notes (1) (a) Research supported by AFOSR(NCt0AR USAF Grant AFOSR-76-2932.
(b) Presented at the Michael Kasha Symposium on Energy Transfer in Organic, Inorganic, and Biological Systems, Florida State University, Jan 6-10, 1976, by K.W.H. during tenure of a National Science Foundation postdoctoral fellowship at the University of Michigan. (2) E. Hutchison, Phys. Rev., 36, 410 (1930). (3) J. B. Coon, R. E. DeWames, and C. M. Loyd, J. Mol. Spectrosc., 8, 285 (1962). (4) D. P. Craig, J. Chem. SOC.,2146 (1950). (5) G. R. Hunt, E. F. McCoy, and I. G. Ross, Aust. J. Chem., 18, 591 (1962). (6) K. Miller and J. N. Murrell, Theor. Chim. Acta, 3, 231 (1965). (7) Y. Fujimura, M. Onda, and T. Nakajima, Bull. Chem. SOC.Jpn., 48, 2034 (1974). (8) H. H. Patterson, J. J. Godfrey, and S.M. Khan, lnorg. Chem., 11, 2872 (1972). (9) (a) K. W. Hipps and G. A. Crosby, lnorg. Chem., 13, 1543 (1974); (b) G. A. Crosby, G. D. Hager, K. W. Hipps, and M. L. Stone, Chem. Phys. Lett., 28, 497 (1974). (IO) P. Hochmann, H. T. Wang, and S.P. McGlynn, Abstracts, 29th Southwest Regional Meeting of the American Chemical Scciety, El Paso, Tex., Dec 1973, No. 83. (11) G. Blasse and A. Bril, Philips Tech. Rev., 31,304 (1970). (12) Y. Haas and G. Stein, Cbem. Phys. Lett., 15, 12 (1972). (13) M. S. Wrighton, L. Pdungsap, and D. L. Morse, J. Phys. Chem., 79, 66 (1975). (14) T. M. Dunn in “Modern Coordination Chemistry”, J. Lewis and R. G. Wilkins, Ed., Interscience, New York, N.Y., 1960, Chapter 4. (15) L. E. Orgel, J. Chem. Phys., 23, 1824 (1955). (16) lnorg. Synth., 2, 225 (1946). (17) R . W. Gillard And R. Ugo, J. Chem. Soc., 549 (1963). (18) R . Stair, N. Schneider, and J. Jackson, Appl. Opt., 2, 1151 (1963). (19) M. Kasha, Discuss. Faraday SOC.,No. 9, 14 (1950). (20) J. N. Demas and G. A. Crosby, J. Am. Chem. SOC.,92, 7262 (1970). (21) For example, F. A. Cotton, “Chemical Applications of Group Theory”, 2d ed, Wiley-lnterscience, New York, N.Y., 1971, pp 266-267. (22) G. D. Hager and G. A. Crosby, J. Am. Chem. SOC.,97,7031 (1975); G. D. Hager, R. J. Watts, and G. A. Crosby, ibid., 97,7037 (1975); K. W. Hipps and G. A. Crosby, ibid., 97, 7042 (1975). (23) H. Sponer and E. Teller, Rev. Mod. Phys., 13, 75 (1941). (24) M. Lax, J. Chem. Phys., 20, 1752 (1952). (25) S.E. Schnatterly, C. H. Henry, and C. P. Slickter, Phys. Rev. [Sect] A, 137, 583 (1965). (26) Unpublished work, this laboratory. (27) R. W. Harrigan and G. A. Crosby, J. Chem. Phys., 59, 3468 (1973). (28) While this paper was in preparation, a general treatment of the overlap problem appeared: E. E. Bergman, Nuovo Cimento SOC./tal. Fis. 5,22b, 249 (1974). (29) M. Gerloch and R. C. Slade, “Ligand Field Parameters”, Cambridge University Press, London, 1973. (30) H. J. Clifford, Ph.D. Thesis, University of New Mexico, Albuquerque, N.M., 1970.
Discussion M. GOUTERMAN.What is the origin of these bands? Singlettriplet? How intense are they? K. W. HIPPS. The question is very well chosen. We are going to address it in another publication. For a discussion relative to K$o(CN)6, see K. W. Hipps and G. A. Crosby, Znorg. Chem., 13,1543 (1974).For an analogous situation in ruthenocene, see G. A. Crosby, G. D. Hager, K. W. Hipps, and M. L. Stone, Chem. Phys. Lett., 28, 497 (1974). We believe we are picking out a single electronic level derived from a ligand field term of triplet parentage.
The Journal of Physical Chemistry, Vol. 80,No. 20, 1976
