2502
J. C.HEMPEL AND F. A. MAT~EN
6(L, L ’ ) ~ ( u u’)(dN;[h];yLIIR1lIdN;[X];y‘L) , (A.2) and the crystal field term
(dN;[A];vL;u@,[ Irr211dNi’[X];v’L’;u’a) (A. 3) The electrostatic terms can be evaluated using fractional percentage coefficients which have been tabula,ted.28 The crystal field terms are evaluated from the formula
be calculated with fractional percentage coefficients and have been tabulated (Table V). We display in Table VI the db2T2 matrix obtained by this technique. Crystal Field Matrix for d6. 2T2(10 X 10). H 2 matrix elements are (dN;[X];vL;ul IH21Id‘”; [A] ;v’L’;u’) = 6-
(dN;[X];vl;;uC%jlH2\ ldN;[A];v’L’;u’~)
VL 4 v’L’ (dN; [X];vLIIU4(IdN; [X];v’L’) (A.4) = [dA1
ut&]
where
lclr
m
L’
1 v
L
u
l=l 2=2 3=3 4=1 5=2
D D D F
6=1
G G
1 1 1 1 1 1 1 1 1 2
E
F
7=2 8 s 1 H 9=1 I 10=1 I
M-m
and the second factor on the right of (A.4) is the reduced matrix element of Racah’s spin-free unit tensor operator of rank 4.22 A tabulation of the coefficients defined by (A.5) sufficient for all dN octahedral problems is to be published. The reduced matrix elements of U4 may
4 v’L‘ (d6;v2L~IU411d6;y’2Lf) [ U T , A1 u’Tz Indices: r
X 62/315A
I, 4 (L’M- w z u‘aR)(- l ) L - M
vL
(23) C. W. Nielson and G. F. Koster, “Spectroscopic Coefficients For the pn, dn, and fn Configurations,” The M. I. T,Press, Cambridge, Mass., 1963.
Luminescence of Chromium (111) Compounds by J. C. Hempel and F. A. Matsen Department of Chemistry, The Un&&ty
of Texas, Au8t-h Teaxa 78718 (Received February 84, 1060)
Crystal field energy level calculations,including spin-orbit interaction, provide a basis for the understanding of intersystem crossover in chromium(II1) compounds. I n particular, they provide a scheme for predicting the ratio of fluorescence to total luminescence for effectively octahedral chromiurn(II1) systems.
I. Introduction Crystal field calculstions are an effective basis for the interpretation of absorption spectra of transition metal Complexes. We will show that crystal field theory also provides a model for predicting the luminescence of some of these systems. Transition metal complexes appear to possess double point group symmetry so that the exact quantum numbers which characterize electronic states are irreducible representations of the double group. These irreducible representations are expected to be conserved in an intersystem crossover. We shall consider a radiationless transition which The Journal of Physical Chemistry
occurs on an energy hypersurface of an electronic state. If K is an exact quantum number which characterizes the state, K is said to be conserved during the process. Different geometrical configurations on a hypersurface are often approximately characterized by different spin quantum numbers, SI, SZ, etc. An intersystem crossover is a transition in the Kth state from a configuration characterized by spin SI to a second configuration characterized by Sz. The process is denoted by SI -+ s z I n this process I - is conserved but S is not. The rate of intersystem crossover is a function of the shape of the hypersurface and the energy of the nuclei.
2503
LUMINESCENCE OF CHROMIUM(III) COMPOUNDS
In this paper we make a formal analysis of the intersystem crossover process and the factors which affect phosphorescent intensity. We take as examples effectively octahedral chromium(II1) complexes. That intersystem crossovers occur in transition metal complexes is evidenced by a study of the luminescence of chromium(II1) complexes.' Under favorable conditions the rate of intersystem crossover for chromium (111) compounds2 may be as high as lo9 sec-l. We make extensive use of data and theories of Schlafer and coworkers.s 11. Electronic Energies We represent a transition metal complex with N valence electrons by an N-electron Hamiltonian of the form
H = HsF
+Q
(1)
HSFis a spin-free Hamiltonian4
HSF= H 0
+v
(2)
(4)
where [GI" is the Nth rank inner direct product of the point group G . For the octahedral group, the irreducible representations are denoted by cy = AI, Az, E , T1, and T2and are quantum numbers for HSF. Finally [HSF,PaSF] =
0; P2F E
SNSF
(5)
where SNsF is the group of permutations on the spa,tial coordinates of the N electrons. It follows that the partitions [ X I of SNSFare also exact quantum numbers for HSF. The states so characterized are called permutation states. Of these, only the permutation states labeled [ A ] = [2",1N-2"]
0 _< p _
Q ( 4 A ~ )
(22)
This axiom, which is due to Orgel,' is supported by Tischer's recent pressure studies on chromium(II1) doped silicate glasses which show Y ( ~ Eis) roughly ~) with independent of pressure while Y ( ~ Tincreases increasing pressure? We set Q(4A2)= 0 so =
'/JcQ2
+ '/21cQ2
c e
&('E) = &"('E)
+
E0(4T2) ' / z l c { & - & ( 4 T 2 ) ) 2 (23) Following Schlafer and coworkers* we make the following association with spectra. Absorption =
+
v(*T2)= &0(4T2) v Y(")
=
E0(2E)
(24)
P
Qeq(4A2)
Figure 4. Potential energy curves for t h e low-lying states of the VsF(da)configuration for a Q (f42) corresponding to Dq/B 31 2.1.9 (7) L.Orgel, J . Chem. Phys., 23, 1824 (1955). (8) R. E. Tischer, ibid., 48,4291 (1968).
Volume 78, Number 8 August 1960
2506
J. C. HEMPEL AND F.A. MATSEN Energy
T = 130' K
I
. -6
Figure 5 . Potential energy curves for the low-lying states of the WF(d*)configuration for 8 Q (4A2) corresponding to D q / B > 2.1.8
fluorescent to phosphorescent intensity for a given compound is given by
where P is the probability of emission per molecule from the lowest vibrational state and N the number of molecules in that state. Now
P B__- P(4T2) P(P)
v(F>'
P(2E) Y(P)~Y(~T~) (30)
where P(2E)and P ( 4 T z are ) the probabilities for absorption to the 2E and 4T2 states, respectively, and v the frequency of the indicated process."J The ratio
is found experimentally to be always much greater than one. Among the processes following absorption, which affect N ( 2 E )and N ( 4 T z )are , (1)radiationless transitions from one excited state to another; (2) intersystem crossover in the first excited state (state I V';I)) ; ( 3 ) radiationless transitions from the first excited state to the ground state; (4) radiative transitions from the first excited state to the ground state. The Journal of Physical Chemistry
-4
-a
e
0
4
6
Z/T Figure 6. A plot of In (OF/@P) us. Z/T for a number of chromium(II1) systems. Experimental points are identified in Table I.
For effectively octahedral chromium(II1) complexes, process 1 is much faster than all others. Processes 2 and 3 are faster than process 4.11 Assuming that a pseudo-Boltzmann equilibrium exists between the 2E minimum and the 4T2minimum of U;I)
I
where W represents the statistical weight of the state. On substituting eq 26 and 28 into 32 and taking the natural log of both sides we obtain
Taking the quantum yields CPF and +p to be directly proportional to the intensity of the observed emission, the natural log of both sides of eq 29 yields +F
In-
@P
=A
+Z/T
(34)
(9) Potential energy figures after H. L. Schlafer and coworkers, ref 3a. (10) For a discussion of absoration and emission intensities see. for example, N. J. Turro, "Molecuiar Photochemistry,'' W. A. Benjamin, Ino.,NewYork, N. Y., 1967. (11) See ref 2 and H. L. Schlafer, J. Phys. Chem., 69,2201.(1966).
LUMINESCENCE OF CHROMIUM(III) COMPOUNDS
Table I : Ratios of Quantum Yields a t T
= 130°K
No. in Fipure 6
Compounda
Z/T
+
ipP/OPv
exptlc
F F F F F F F F F F S P
>lo00 >lo00 >loo0 >lo00
23.7 20.3 19.1 14.3 9.7 5.2 5.2 5.2 4.2 2.4 0.0 -0.4 -4.2
[CrC16Ia[ CrFe] (NH4MCrFd [C~F(HZO)~]S~FB KZ[CrFdHzO)] [CrFdH20)dH2O [CrFa(HzO)a]'/aHzO [CrFdHzO)i [CrFdHzO)d 2H20 1 [Cr(antip)e]3 + 2 [Cr(urea)e] [Cr(HzO )61 Fa 3 4 [cr(&o)d3+ NH, [CrFden)l 5
$B/QP,
oalodb
>loo0 652 652 652 240 39.5 3.61 2.46
3.61
+P F+P F
-
0.10
0.054
'
This is the See Table I1 for spectral data and references. a in a waterratio predicted with an A determined for [Cr(urea)~] glycerol glass. ' Observed luminescence indicated by F and P (see Table 11). Experimental quantum yields given in ref 2. a
+
2507 chromium(II1) complexes assuming A is constant. Calculated values of @ r / @ p and Z/T and experimental values of @F/@p where available are given in Table I. A is expected to vary somewhat from system to system12but remain a positive number since it is an In function and the experimentally observed intensity of related absorption curves, eq 31, indicates the argument will be greater than unity. A quantitative study of @F/@P vs. 1/T for a given chromium(II1) compound will yield both A and the quantity 2.13We predict (1) lowering the temperature enhances fluorescence for compounds with a positive 2; (2) lowering the temperature enhances phosphorescence for compounds with a negative 2. In addition and independently of the Boltzmann approximation, we can divide chromium(II1) complexes into three categories using absorption spectra data. (1) We can predict that when14
R v ( ~ T v~ ()~/ E ) (35) is less than 1 that &0(4T2)
