S. BJORKLUND, N. FILIPESCU, N. MCAVOY,AND J. DEGNAN
970
considering the error limits of the methods used and the difference in temperature. Although the thermodynamics of the equilibrium could not be established, complications in future vaporphase uv studies of iodine complexes do not seem serious. First, by using iodine concentrations of 2 M , the formation of 1 4 can be completely ignored. Second, for larger concentrations, in those cases where the complexes are weak so that the per cent of iodine complexed is small, valid correction of the total ab-
sorbance can be made by subtracting the absorbance curve of iodine of the same concentration as is used in complex formation. For these reasons, the analysis of Lang and Strong' on the benzene-iodine system is valid. Only in the case of high iodine concentration and strong complexation would special consideration be necessary.
Acknowledgment. This work was supported by grants from the National Science Foundation, NSF GP3691 and GP-6429 Research.
Correlation of Molecular Structure with Fluorescence Spectra in Rare Earth Chelates. I.
Internal Stark Splitting in Tetraethylammonium
Tetrakis(dibenzoylmethido) europate(111) by Sven Bjorklund, Department of Physics, The George Washington University, Washington, D.C.
Nicolae Filipescu, Department of Chemistry, The George washington University, Washington, D . C.
N. McAvoy, and J. Degnan Goddard Space Flight Center, N A S A , Greenbelt, Maryland
(Receioed August bo, 1967)
The internal Stark splitting in the emission spectrum of tetraethylammonium tetrakis(dibenzoylmethido)europate(III) in microcrystals at 77'K was analyzed in detail treating the ligand field as a perturbation on the free ion levels. The complete equivalence between crystalline field and a molecular (ligand) field in chelates with respect to the splitting of the intra-4f levels allows the use of equivalent operator techniques in the calculation of ligand field parameters. These parameters were derived from the splitting in the 'F1 and 'F2 levels of the Euaf ion without assuming a known molecular geometry, and their values establish the configuration of the emitting species. In addition, the ligand field parameters are used to calculate interatomic distances and bond angles. The method developed for deriving and verifying the geometrical arrangement, interatomic distances, and bond angles from fluorescence spectra proves to be a powerful tool.
Introduction Narrow-line fluorescence is observed in certain Eu complexes when excited with light absorbed by the organic ligand as a result of intramolecular energy transfer. The absence of broad-line fluorescence characteristic of rare earth ions incorporated in nonhomogeneous surroundings (e.g., glasses) indicates the presence of the same environment for all Eu ions similar to that present in doped single crystals. The Journal o j Physical Chemistry
The present work shows that the splitting of the 'F levels of Eu3+ ion observed in the fluorescence spectra of organic chelates can be effectively explained in terms of a first-order perturbation produced by the electrostatic effect of the ligand field on the complexed europium ion. Furthermore, we propose to determine the ligand field parameters in tetrakis-p-ketoenolate chelates from experimental data without assuming any one of the possible octacoordinated models as being
FLUORESCENCE: SPECTRA OF RAREEARTH CHELATES 0 023 0 008
0 035 0024-
I
0 059*
97 1
,
7100
7000
6900
6800
6700
6600
6500
6400
6300
6200
6100
WAVELENGTH
Figure 1. Fluorescence spectrum of microcrystalline EuDbT a t 77°K.
the emitting species. The only assumption made is that the emitting stereoisomer has Cz symmetry or higher, a condition which is satisfied by the two dodecahedrons and the two antiprisms configurations required by the use of either d4sp3or d6p3hybrid orbitals of the lanthanide ion in bond formation.'
Experimental Section Preparation of Chelates. The tetraethylammonium tetrakis(dibenaoylmethido)europate(III) (EuD4T) was prepared by adding 8 mmoles of 1 N sodium hydroxide to a hot solution of 8 mmoles (1.79 g) of dibenzoylmethane, 8 rnmoles (1.68 g) of tetraethylammonium bromide, and 2 mmoles (0.52 g) anhydrous europic chloride in 25 ml of absolute ethanol. The precipitated crystals were filtered, washed repeatedly with water-alcohol mixture, and dried in air. After recrystallization from o-dichlorobenzene, the crystals were washed again several times with absolute ethanol and dried under vacuum at 110". The chelate decomposes at 222-225'. The analogous pyrrolidinium chelate of EuD4 (EuD4Pyr) was synthesized from dibenzoylmethane, EuC13, and the free base by the usual procedure. Fluorescence Spectrum. The crystals were sealed with nonfluorescent tape between two quartz slides and lowered into a quartz dewar containing liquid nitrogen. Light from a 1000-W water-cooled, high-pressure AH-6 mercury arc was filtered through a Corning (357-54filter and directed into a metal box constructed to minimize stray light. The clear bottom part of the dewar was positioned in the box such that the exciting light was focused on the sample. The fluorescence emission was directed into a Cary Model 14 spectrophotometer and detected by a R136 red-sensitive
0 1~030001a
00200085002000900095 0005~00M-.fL-$'
0 l W ~ 0 0 7 0 145+0065* ~ 0
I
6000
i\
?: '
5900
5800
5700
I
400
0 400*0
5600
5503
5400
5300
5200
Slit widths in millimeters are given a t the top.
photomultiplier. The entrance slit to the Cary was adjusted manually for different spectral regions to control the intensity of the light entering the monochromator. Dry air was caused to circulate inside the box to prevent possible frosting on the dewar. The wavoelength measurement is estimated to be within 2 A. The recorded fluorescence spectra of EuD4T and EuD4Pyr are shown in Figures 1 and 2, respectively. (Also see Table I.)
Equivalence of Ligand and Crystal Fields2 Let r, e, cp be the coordinates of a point in the electric field produced by a charge-distribution of density p(r') originating in the ligand. We can consider p(r') as being the time and space average over the entire ketoenolate structure as represented by both u bonds frame and delocalized T electrons. The potential is given by P
1
integrated over the source points with coordinates r'. In terms of spherical harmonics this is
For a crystal field with point charges q i at the points rt, the charge density p ( r ' ) would be given by (1) (a) C. Brecher, H. Samelson, and A. Lempioki, J . Chem. Phys., 42, 1081 (1965); (b) H. Bauer, J. Blanc, and D. L. Ross, J . A m , Chem. SOC.,86, 6125 (1964);(c) L. R. Melby, N. J. Rose, E. Abramson, and J. C. Caris, ibid., 86, 5117 (1964); (d) L. J. Nugent, M. I,, Bhaumik, S. George, and S. M. Lee, J . Chem. Phus.,41, 1305 (1964). (2) For [I somewhat similar treatment, see J. S. Griffith, "The Theory of Transition Metal Ions," Cambridge University Press, New York, N. Y., 1961,Chapter 8.
Volume 78, Number 3 March 1968
S. BJORKLUND, N. FILIPESCU, N. MCAVOY,AND J. DEGNAN
972
(3) and the integrals in (2) would reduce to
The equivalent integrals for the ligand field are
qrYfrn*(B',cp')G(r - rod7
A p = 47r
(21
i
+ 1)Q'lfl
(4) Thus the potential produced by the ligand field is given by eq 7, where
or 0 230
0 280
1-
0 640
500
00003
390
I I
'Do
I I
'FI 0)
'DI--'F,VI
I
0 q50
0000 00400014
-'-
0240
r --r--r-r-.
1
1
0 I20
0 150
'
0 360
0 300
4230
%o-'F,
'D0-'F4
7
5 0 4
TFQ
4 I 7300
7200
71W
7000
6900
6800
6700
6600
6500
6400
6300
6200
6100
6000
5900
5800
5700
5600
5500
5400
5300
0
WAVELENGTH A
Figure 2.
Fluorescence spectrum of microcrystalline EuDdPyr a t 77'K.
Slit widths in millimeters are given a t the top.
Table I : Fluorescence Lines in the Emission Spectra of EuD4T and EuDdPyr --------------Wave
6Do-* 7Fo 6Do-* "F1 'Do
-*
'Fz
'Do + 7Fs 'Do
+ 7F4
6Dl ?Fo 6D1+ 'F1 6D1 'Fz 'D1+ "Fs
17 ,248 17,007 16,819 Eo 375 cm-l above 'Fo 16 ,792 16 ,306 16 ,298 16,199 Eo 1,038 om-' above 'Fo 16,129 16,119 15,230 15,388 15,232 (?) 15,363 14,370 (?) 14 ,465 14 ,260 14,441 (1) 18,982 18,765 18,547 18,035 17,827 17 , 168 16,923 ( 1 ) 17,127 16,677 (1) 17,092 16,642 (1) 15,986 15,788 15,861 15,688 15,834 15,625
The Journal of Physical Chemistry
EuD4Pyr-
r
1
I
-----
numbers, cm-1
EuDR --
Transition
17,234 16,926 16,882 Eo 364 cm-l above ?Fo 16,802 16,333 16,299 EO1,036 cm-l above 'Fo 16,157 16,017 15 ,365 15 , 162 14,912 15,023 (1) 15,313 14,811 (?) 14,233 14 ,476 14,341 19 ,003 18,661 18,636 18 ,555 18,074 17 ,940 17 ,775 17,092 16,736 17,014 16,563 (1) 16,973 16,460(?) 15,849 15,688 15,519
1
15,024 ( 1 ) 14,910 (?)
18,518 16,564 ( 1 ) 16,482 (?) 15,534 15,536
7
FLUORESCENCE SPECTRA OF RAREEARTH CHELATES
973
v(r,e,d = C a z m r z ~ L m ( ~ , ~ ) (7)
be discussed in a forthcoming paper. Presently, Figure 2 is shown only to illustrate the identification of the individual transitions and the dependence of the splitting on the organic cation. For the tetrakis(dibenzoy1methide) chelates the ligand field parameters can be calculated from the fluorescence spectrum by using only transitions from the excited 6Do state to ?Fo,T I , and ?Fzlevels in the ground multiplet of Eu ion. In less symmetrical systems such as chelates from differently substituted pdiketones, transitions terminating at the 'F3 level may prove to be necessary. However, in this case, the solutions of the secular determinant are so complex that very probably they cannot be correlated to experimental data. The assignment of the emission lines to the individual jD -t 'F transitions in Figures 1 and 2 is well verified with the exception of several weaker lines marked by (?) which are only tentatively assigned because of lack of clearly defined structure in the 6700-7500-8 region. Although Blanc and Ross5 have suggested the possibility of spurious emission due to surface contamination, we find that the emission lines are reproducible even after recrystallization from different solvents or mechanical pulverization. Other authors' propose to explain the extra lines by the presence of two stereoisomers. The relatively few lines in Figures 1 and 2 which were assigned tentatively to 5D1 4 'F3,4 could presumably originate in another stereoisomer present in very small amounts. The assignment of these few lines is not essential to the discussion of the "predominant species." The orbitals available for bonding in the Eu ion are the empty 5d, 6s, and 6p orbitals. The fact that the partially occupied 4f orbitals are not involved in bonding to any great extent is indicated by the discrete structure of both absorption and emission spectra of the chelated ion and their similarity with those of inorganic ionic salts. There are four possibilities for the lanthanide ion to accommodate eight coordinating oxygen atoms's6 (see Figure 3) : two dodecahedral arrangements for the use of d4sp3 hybrid orbitals (models I and 11) and two antiprismatic configurations for d4sp3 and d6p3 (models I11 and IV). From the values of the ligand field parameters calculated below from experimental data, we propose to verify the geometry of the emitting stereoisomer.
Lm
whereas the expression for the potential for a crystal field is
V(r,e,cp)=
Z,m
AzmrzYz"(~,cp)
(8)
Consequently, a ligand field has the same potential as a crystal field, in spite of the delocalized electron density associated with the anion-ligand configuration, and will produce the same internal Stark splitting of the energy levels ai3 the equivalent crystal field, provided a,m =
A,"
It may well be that to produce the actual field, one needs an infinite number of point charges to get all the parameters almequal to A This, however, is not the point. What one needs in the calculations of internal Stark splitting is not the total field but only a small number of parameters aim, and these can always be found from a small number of point charges. Therefore, one can always find a hypothetical field produced by point charges which will have the same effects on the spectrum as the molecular (ligand) field. In firstorder perturbation theory, the energy levels are given by the roots of the secular determinant in terms of the parameters a," and A,", respectively, and both of these will give discrete solutions to the secular equation. Having established the equivalence between the ligand field and a crystal field, we can employ the operator equivalent methods developed for crystal field problems by Judd3 and st even^.^ Selection of Chelate and Fluorescence Spectra EuD4T was specifically selected for analysis in this first paper for three reasons. First, the symmetry of the enolate anion renders the two coordinating oxygen atoms equivalent ; second, all possible arrangements of four ligand bridges about the Eu3+ ion' yield a small number of ligand field parameters aim; and third, the very bulky tetraethylammonium cation is less likely to influence the symmetry of the ligand field-through penetration into the vicinity of the lanthanide ionthan other priimary, secondary, or tertiary ammonium cations. We will assume this type of influence to be negligible for the (C2H5)4N+ion. The fluorescence spectra of a large number of crystalline europium tetrakis(dibenzoy1methide) chelates with different organic cations were examined in detail to ensure proper identification of the individual transitions. We found that the splitting patterns of the fluorescence transitions in these chelates are remarkably sensitive to the nature of the cation. As an example, the emission spectrum of crystalline EuDIPyr a t 77°K is illustrated in Figure 2 for comparison with that of EuD4T in Figure 1. The cation effects in the splitting of the 7F levels of Eu3+ in octacoordinate chelates will
Perturbation Calculation Assume that the Hamiltonian can be broken up into H' where Ho is the Hamiltonian two parts H = Ho
+
(3) (a) B. R. Judd, Mol. Phys., 2, 407 (1959); (b) B.R. Judd, Proc. Roy. SOC.(London), A228, 120 (1955). (4) K. W.H. Stevens, Proc. Phys. SOC.(London), 65, 209 (1952). (5) J. Blanc and D. L. Ross, J . Chem. Phys., 43, 1286 (1965). (6) (a) A. F. Wells, "Structural Inorganic Chemistry," Oxford
University Press, New York, N. Y., 1962; (b) G. E. Kimball, J . Chem. Phys., 8, 188 (1940). Volume 78, Number 3 March 1.968
974
S. BJORKLUND, N. FILIPESCU, N. MCAVOY,AND J. DEGNAN combined7 to yield two new parameters pl" and plm(s) which have factors of cos mp and sin mp, respectively. Since for all the models under consideration a proper choice of axis will make the sine term zero, the part of (7) which we will need becomes
+ p4'(35X2 - 30z2r2+ 3r4) +
-leiV(r)
=
p20(3x2
-
r2)
@d2(7Z2- r 2 ) ( z 2- y2)
I
+
p22((z2
- y2)
+
+
- 6z2y2 p4) (9)
p44(~4
with P2O
=
m Figure 3.
of the 4f electrons of the free ion and H' is eV with V the electrostatic potential produced by the ligand and given by eq 8. H' is assumed small in comparison with Ho or [(r,)li.si >> HI. H o is invariant with respect to rotation around the origin, which means that the total angular momentum J of the 4f electrons in the unperturbed ion is a good quantum number and energies of the degenerate levels 7 F are ~ the eigenvalues of Ho. We need to calculate the splitting of the 7F0,7F1, and 7F2levels caused by eV. The energy levels E can be found from first-order perturbation theory by solving the secular determinants for the 7 . ? ?levels. ~ The secular determinant has the form
Detl(M'J1eVIMJj - Eb,Ml,M/= 0 If one expands the vectors I M J ) and (M'JI for felectron states in terms of spherical harmonics with 11 = 12 = 3, the matrix elements are proportional to integrals of the product of three spherical harmonics. One of the spherical harmonics originates in the expansion of the potential. Since the integral of the product of three spherical harmonics is proportional to the Clebsch-Gordan coefficient (Z1OIOl I20) and these coefficients have to satisfy a triangular condition for 11, 12, and I , we see that the maximum value for 1 that will give a contribution to the matrix elements is I = 6. Since the Clebsch-Gordan coefficient is zero unless 11 l2 1 is even, only terms with even 1 contribute. The first part of the Stevens and Judd method consists of expressing the rzYlm terms as polynomials in powers of r, (z, y, and x . The parameters al-lml and aL+lmlcan be
+
The Journal of Physical Chemistry
+
We will now show that only those parameters pl" listed above are needed. All four possible stereoisomers for the tetrakis species have a twofold rotational symmetry. If the axis of symmetry is taken to be the z axis, then eimv = e W v + v ) or eimn - 1 thusm = 0 , 2 , 4 , . , . . For the states 7Fo- 7F2,the coefficient of proportionality between the point-charge field and the equivalent angular momentum operators is zero for 1 = 6. Thus, since m and 1 have to be even, m S I , and there is no contribution for 1 = 6, one is left with the following parameters contributing to the splitting: pZ0, @do, p22, p2, and p44(the constant Poo has been omitted). In the dodecahedral configuration I, the eight equal charges are such that cp1 = cp2 = 0, (p3 = (p4 = a/2, (PS = (Pa = a, pl = pa = 3 ~ / 2 ,and el = e6, e3 = e7, el O4 = e3 ea = 6% Os = e7 e2 = a. The terms Y2 and Y42 add up to zero upon taking into account the relation
+
+
+
+
P,"(-(z) = (-l)"+lPt"(s) thus, only the parameters pZ0,p40, and pd4contribute t o the splitting of the 'F2level for the dodecahedron I. The geometry of dodecahedron I1 is such that all of the five possible parameters p20, p40, p22, p42, and p44can be nonzero. For antiprism 111, the equivalent point charges lie at the corners of two parallel rectangles, one rotated around the x axis by a/4 with respect to the other. The planes of both rectangles are perpendicular to the z (7) M. T. Hutchings, Solid State Phys., 16, 264 (1964).
FLUORESCENCE SPECTRA OF RAREEARTH CHELATES axis. Thus en+4 8, =
+ a/4, n = 1, 2, 3, 4, and
pn+4 = pn
+
975
p44is proportional to
T.
8
P44
-
p44(cos er)ei4vi i-1
sin4 0; .P44(cose),
=
+
p44(cos on+,) ei49n+4+ ei4Vp,= ei(49nfn) ei4vn
(1
= e4ivn
+ ein) = o
(11)
Consequently, for this antiprism /3d4 is zero. The other parameters can have nonzero values. We are thus left with Pzo, p40,pzz,and p2. For antiprism IV the equivalent point charges lie a t the corners of two parallel squares whose planes are perpendicular to the x axis. For each plane Pl"(cos e,) is the same for each charge.
pzz
-
8
4
C p22(cos e1)ei2Vi+ i = 5 pZ2(cose5)ei2qc; i=l with = e2iq1(1
e2i(sif(3x/2))
- 1 + 1 - 1)
x =
o
Thus pz2 is zer'o. I n the same way, one can show that fi42 is zero. We are thus left with Pzo, P4O, and Substituting the angular momentum operators J , J,,J+, and J- into (9), -lelV(r,e,p) becomes
+ + fij(r4)P40[35J,52- 30J(J + 1)Jz2 + 25Jz2 -- 6 J ( J + 1) + 3J2(J + I)'] + 1 (J+2 + J-? +
-leiV(r) = Q ~ ( ~ ~ ) -P J~( J~ [ 1)l ~ J ~ ~
QJ(T2)P2
(J.+2
+
J-2)
+ (J+Z +
[7Jz2- J ( J
J-2)
x
+ 1) - 511 + 1
BJ(T4)P442
+
(J+4
(12)
J-4)
where3 (yo
=
po = p1
:=
0;
Q1
=
-
1 -* 5'
pz =
2 - --
189'
=
-
11 315 -
For any geometry having a twofold rotational axis, the secular determinant for the 7F1 level is
=o
=
+
[(84P~(r~)P4~12pz(r4)P44)2-
+
24( - QZ(~')PZ~ 12Pz(r4)P40) X (6~2(r')PzO
+ 12P2(r4)P40+ 12P2(~4)~44) + 4 8 ( ~ z ( ~ ~ )4P z3pz(r4)P42)2]1/2 ~ (15)
Molecular Geometry Selection of the Model. The energy levels in 7Fzwere fed into an IBM 7094 computer which tested all possible assignments of these levels to the secular equations (14) as outlined below. The parameters (rz))pzO and (rz))pzzare known from the splitting of the 7F1level. The remaining four parameters (r4)>p40,(r4))p4z,(r4))p,4, and Eo are the unknown quantities in the five equations (14). From eq 14 one can see that E4 - E5 = x with the expression for 2 given by eq 15. For each assignment of different experimental energy values to the levels in eq 14 a set of values is obtained for each of the four parameters mentioned above. For each of these sets of values a value for x is obtained from eq 15. That value 5 is chosen which gives the best agreement with the experimental value of E4 - E5, In this way, the parameters (rc))plmand a unique assignment of the energy levels are obtained. The values are given in Table 11. The splitting of the 'F1 is due only to (rz))p20and (rz)@. If (rz))pzzis zero, two 'F1 sublevels would result, one of which is doubly degenerate. In the octacoordinated dodecahedron I, @t2is zero. In addition, (r4)Pd4 should have a definite nonzero value for both dodecahedrons. One can show from the e and cp dependence of P d 4 that the only case in which dodecahedron I1 can have a small p44is if the equivalent point charges were greatly displaced from the oxygen atoms (more than 20') and if the two oxygen atoms were nonequivalent. This is very unlikely. Volume 78,Number 9 March 1968
S. BJORKLUND, N. FILIPESCU, N. MCAVOY,AND J. DEGNAN
976
Table 11 : Energy Levels and Ligand Field Parameters Expected epectral data
Predicted by model
El = Ea = Ea = El = E, =
E1
241 cm-1 456 cm-la 429 cm-10 1119 cm-1 942 cm-la E3 = 950 cm-1" E4 = 1129 cm-1 Es = 1049 cm-l
= 241 cm-1 E2 = 456 cm-1 E3 = 429 cm-1 E1 = 1122 cm-1 E2 = 940 cm-1 E3 = 952 cm-1 Ea = 1124 cm-1 E5 = 1051 cm-1
Ligand Field Parameters -335.85 ( ~ 2 ) p ~=a F 6 7 . 5 0 (y4)p40 = -111.46 ( ~ 4 ) p=~ ~$24.18 ( ~ ~ =) p 26.16 ~ ~ (~2)p~O =
-
cm-l cm-l cm-l cm-l cm-l
+)p20
=
- 335.85 cm-1
(.')ha= $67.50 om-1 (v4)p40
= -111.46 cm-1
(r4)pb2= (r4)p44
=
$3.42 cm-l 0 cm-1
0 Ez and Es can be interchanged without affecting agreement with theory. Only the signs of the 822 and pa2 parameters are affected.
Thus, on this h i s of the splitting in the 'F1 and the negligible value of (r4)pd4 (which is small enough to be of the order of experimental accuracy) , both dodecahedrons I and I1 are ruled out as possible configuration for the emitting species. Furthermore, the parameter (rZ)>pz2is large enough to indicate that antiprism 111 should be selected as the emitting species since the only remaining alternative, antiprism IV, predicts a zero value fo, (r2,\S22 even after considering possible displacement of the point charges along the ligand bridges. The outcome is consistent with the results of Matkovic and Grdenic,* who have determined from diffraction experiments that the tetrakis(acety1acetonato)cerium(IV) has the configuration of antiprism 111. Both Eu3+ and Ce4+ would be expected to use the same bonding orbitals in forming octacoordinated p-diketonate complexes, because their ionic radii are 0.98 and 0.94 A, respectively, and their electronic configurations are identical except for the number of inner 4f electrons. Interatomic Distances and Angles. The values of the ligand field parameters listed in Table I1 can be used to calculate molecular parameters such as interatomic distances, bond angles, and mutual orientations of particles. The signs on the parameters (r2)>pzo and (r4)p40, when compared with the graphs of the e functions in Yzoand Y40, obviously restrict the angle e from the x axis to a point charge t o the range between 55 and 70". For antiprism I11 we also have the following relationship for the ratio of the parameters pZoand p? (7-2)P20
(r4))P40
-
i 6 ~ 2 ( COS 3 e - 1) 35 cos4 e - 30 cos2 e
+3
(2) (T~)
where R is the distance to an equivalent point charge The Journal of Physical Chemistry
0
35 COS^
e - 30
COS^
= [ ( r 2 ) p 2(r4)p40(16) ( (3 cosz e
Energy Levels
'Fi
and 0 is its angle with respect to the z axis. Solving for R, we obtain
e + 3)(r4)
- i)(r2)
]
'Iz
(16)
Substituting our experimental values for the parameters (rz)>pzoand (r4)p40 and setting (r4) and (r2)equal to 1.706 uO4and 0.834aO2,respectively (obtained from interpolation of data reported by Freeman and Watsong),leaves an expression for R in terms of 0. By definition, R is the distance from the Eu3+ ion to an equivalent point charge. We are now going to assume that the point charge is located along the Eu-0 bond,l0 so that the angle 0 to the point charge is the same as that to the oxygen. Since the 0 -+ Eu bond involves electron donation from the oxygens in the anion to the Eu3+ ion, this approximation is a very good one. The value for the angle 6 calculated below leads to very satisfactory results for interatomic distances and bond angles which in turn are verified by the recalculated energy levels predicted by the derived molecular geometry. The distance between the two oxxgen atoms in a ketoenolate ring is taken to be 2.81 A as reported in ref 8. This distance does not change significantly upon changing the complexed ion. 1 1 , 1 2 The relationship between the Eu-0 bond length, REu-0, and the 0Eu-0 bond angle y in a chelate ring is
The geometry of antiprism I11 allows y t o be expressed in terms of the 0 and (o angles of the'two oxygens in a given ring. Since both oxygen atoms have the same 8, the expression for y is y
=
COS-^ [sin2e cos
(cpl
- v2)
+ cos2 e]
(18)
Furthermore, the ratio of the parameters (r2)p20 and (r2)>pz2 yields
i=l
which for an antiprism reduces to (8) (a) B. Matkovic and D. Grdenic, Acta Cryst., 16, 456 (1963); (b) Nature, 182, 465 (1958). (9) A. J. Freeman and R. E. Watson, Phys. Rev., 127, 2058 (1962). (10) Such an assumption is not necessary if the splitting in the 'Fz levels is large enough to determine the value of the (r2)842 parameter with reasonable accuracy. I n such a case R and 8 can be determined using only the ratios of (r2)Pao to (r2)822 and (rn)Sdo to ( r 4 ) h % . (11) See, for example, G. J. Bullen, R. Mason, and P. Pauling, Nature, 189, 291 (1961). (12) The constancy of the ligand "bite" was also used by W. G. Perkins and G. A. Crosby, J. Chem. Phys., 42, 407, 2621 (1965), in crystal field calculations on Yb3+ and Tm3 + tris-p-ketoeiiolates.
FLUORESCEXCE~ SPECTRA OF RAREEARTH CHELATES
977
(r2))pz0- 8(3 cos2 e - 1) -8
3 sin2 e
(r2)'z2
i=l
cos 2 9 ,
8
Solving for
cos2ptyields i =1
I n antiprism III, however, all of the vi's can be expressed in terms of one of them. Thus, we find that 8
cos 2pf i=l
=
COS 2pl - sin 2p1)
(20)
Substituting eq 20 and the values for (r2)820 and into eq 19 we obtain an expression for cp1 in terms of e. Thus y, as given by eq 18, can be plotted against the angle e. Furthermore, since RE"-o in eq 17 depends on y only, it too can be plotted against the same angle 8. The Eu-0 distance, REu-O, and the distance from the Eu3+ ion to the equivalent point charge, R, were plotted against e in Figure 4. The two RE"-o os. 0 graphs in the diagram correspond to the positive and negative values of (r2)p22, At the intersection point of the two functions, the equivalent point charge coincides with the oxygen site. If we assume that this is the cas:, we obtain the values e = 55'53' and REu-0 = 2.40 A. The stereochemistry of octacoordination was theoretically investigated by Kepert, l 3 who showed that a considerable distortion from the hard-sphere model should be expected when coulombic repulsions are taken into consideration. Thus, for the square antiprism the hard-sphere model predicts an angle 0 of 59' 15', whereas with coulombic interaction the most favorable positions of the ligands are predicted by theory to be at e = 55'48'. This is in remarkable agreement with 2ur value. In addition, the Eu-0 bond distance of 2.40 A obtained from ligand field parameters in the present ,work is equal to the Ce-0 distance in the tetrakis(acetylacetonato)cerium(IV)determined by diffraction experiments.* From eq 19 and 20 we find that the point charges are displaced from the corners of a perfectly square Archimedean antiprism by an angle of only 0'14' in the projection of the zy plane. If the ligand field had a D d d symmetry corresponding to equivalent point changes at the corners of a perfectly square Archimedean antiprism, one would expect two Stark components in ?F1and three in 'Fz levels. Examination of the 5 D ~ 7F1,2lines in Figure 1 shows two 7F1 levels very close to each {other and two pairs of very finely split 7F2 levels, approaching the t ~ o - ~ three-7F2 f levels pattern. The 0-Eu-0 angle in a ketoenolate ring is found from eq 18 to be either 71'20' or 71'59' depending on the sign of (r2)>pZ2.The same angle in the Ce4f acetylacetonate was found to be 72 f 1 O . 8 (r2)>p22
-
rae 1.801.60-
1.40-
I
! r9-55'53' ir
&
e.DEGREES
i 7
118
Figure 4. Dependence of R(0) and R E u - O ( e ) on the angle e.
The magnitude of the equivalent point charge was calculated from the value of the (rz))pzoparameter extracted from spectral data. We substituted our calculated values 0 = 55'53' and R = 2.40 8, and the interpolated value for (r2)from ref 9 in the following expression for (rz)pzoderived from eq 5 and 10
The value q = - 1.54]e[obtained from this expression indicates that the equivalent point charge lies closer to the Eu3+ ion than the oxygen atom along the Eu-0 bond, since q would not be expected to exceed -0.7 le1 at the oxygen ~ i t e . ' The ~ ~ magnitude ~ ~ of q from eq 21 is strongly dependent on 0 and R. Thus, since R(e) decreases rapidly with 8 in the 55-57' region, q takes values from -60.931eI to -0.261el for 0 = 55' and 0 = 57', respectively. Therefore one can restrict the magnitude of 0 within approximately a 1' range above the intersection value 55"53'. This in turn indicates that the R E ~ -iso between 2.36 and 2.40 8. These results are not significantly changed by including second-order corrections such as J-mixing or shielding effects since (1) these were estimated to be of the order of 10 cm-1 and (2) in eq 16 we use the square root of the ratio of two experimentally extracted parameters, each being more than an order of magnitude above secondorder corrections. The geometry of the emitting species was verified by (13) D.L. Kepert, J . Chem. Soc., 4736 (1965). (14) L. 13. Forster, J. A m . Chem. Soc., 86, 3001 (1964). Volume 72, Number 3 March 1068
978
J. 1,. WALKER,JR.,G. EISENMAN, A N D J. P. SANDBLOM
the calculation of the expected splitting of the energy levels of the Eu3+ ion due to a ligand field produced by an antiprismatic configuration having angles and distances as derived. The new values for (r'>pimobtained
from our calculated q, R, 0, and cpl were substituted into eq 13 and 14 to predict the energy levels listed in Table I1 next to the experimental values. The agreement js found to be within the experimental accuracy of 2 A.
Electrical Phenomena Associated with the Transport of Ions and Ion Pairs in Liquid Ion-Exchange Membranes, 111. Experimental Observations in a Model System1 by J. L. Walker, Jr., G . Eisenman, and J. P. Sandblom Department of Physiology, Uniuersity of Chicago, Chicago, Illinois
60667
(Received August 80, 1967)
An experimental model system, formally equivalent to a liquid ion-exchange membrane having incompletely dissociated sites and counterions, has been devised in order to test the steady-state properties deduced theoretically in a preceding paper. The membrane model consists of a convection-free column of HC1 in 2-propanol contained between AgCl boundaries. Since the AgCl boundaries are reversibly permeable to C1- but not to H+, C1- corresponds to the dissociated counterion, while H + is trapped and acts as the dissociated mobile site. The neutral HC1 molecules correspond to the associated ion pairs and it has been verified that the concentrations of these species are interrelated through a simple law of mass action. The dissociation constant of HC1 and the mobility of H + and C1- have been measured from conductance and transference number studies, and the mobility of the undissociated HC1 species has been measured from the value of the limiting currents at high applied voltages. These parameters suffice to define the theoretically expected instantaneous and steady-state current-voltage characteristics of the system as well as the steady-state concentration and electric-potential profiles within the membrane. The theoretically expected values are compared with those observed experimentally and found to be in satisfactory agreement.
Introduction I n the preceding theoretical paperj2 Sandblom, Eisenman, and Walker have extended Conti's and Eisenman's treatment of the steady-state properties of a mobile-site ion-exchange membrane having completely dissociated sites and counterions3 to include the effects of association of these species into electrically neutral pairs. From the extended treatment,2 the quantitative behavior of an incompletely dissociated mobile-site membrane is expected to depend on parameters which do not appear in the Conti and Eisenman treatment, such as the degree of dissociation and the mobility of the undissociated species. Despite certain simplifying assumptions, made chiefly for convenience of exposition, the theoretical The Journal of Physical Chemistry
treatment2 is sufficiently general to serve as a basis for a rigorous understanding of the properties of the usual liquid ion-exchange membra ne^^-^ in which sites and counterions are almost certain to be incompletely dissociated. Before using it for such a purpose, however, it (1) This work was supported by National Science Foundation Grant
GB-4039 and USPHS Grant GM-14404-01. It was assisted by USPHS General Research Support Grant FR-5367 and an NIH post-doctoral fellowship to J. L. Walker, Jr. (2) J. P. Sandblom, G. Eisenman, and J. L. Walker, Jr., J . Phys. Chem., 71, 3862 (1967). (3) F. Conti and G. Eisenman, Biophys. J., 6, 227 (1966). (4) G. M. Shean and K. Sollner, Ann. N . Y . Acad. Sci., 137, 759, 763 (1966). (5) 0. D. Bonner and J. Lunney, J . Phys. Chem., 70, 1140 (1966). (6) J. W. Ross, Science, 155, 1378 (1967). (7) G. Eisenman, Anal. Chem., in press.
