J. Phys. Chem. 1083, 87.3300-3304
3300
assumption, which seems slightly more accurate, is that the dielectric constant and specific volume are constant during the charging of the hard-sphere ion at constant volume for the whole system. This assumption seems intuitively more accurate since at least the volume of the whole system is constant, even though the specific volume of the solvent near an ion will change. It is expected that the dielectric constant will be mainly a function of specific volume and temperature.21 With this latter assumption, one derives an equation which is exactly the same as the Born equation except that it applies to the Helmholtz free energy of solvation rather than the Gibb’s free energy. Taking the appropriate derivatives of this equation gives an equation for the heat capacity at constant volume that is identical with eq 1, except that the partial derivatives of dielectric constant with respect to temperature are taken at constant volume rather than at constant pressure. There is good experimental evidencez1that the dielectric constant is a well-behaved function of density and temperature, so that this modified Born equation predicts a well-behaved value for the heat capacity at constant volume. It is perhaps useful to specu_latehere about the large differences between Cpo2and Cvoz at high temperature. The Born equation predicts electrostatic free energies of hydration of an ion, and the present predictions of the heat capacities and volume are obtained from the temperature and pressure dependence of that electrostatic free energy. Thus, the reason for the very large and negative heat ca(21) Uematsu, M.; Franck, E. U. J . Phys. Chem. Ref. Data 1980, 9, 1291.
pacity at constant pressure must be that the enthalpy of hydration AH(hyd) increases rapidly with temperature as is shown in Table I. In contrast to this, the internal energy of hydration, AU(hyd), does not change rapidly with temperature is not large). Since AH(hyd) = AU(hyd) + A(PV), the difference in behavior must be due to the PV term.Values of Vozdecrease rapidly with temperature, indicating increasing electrostriction of the water around the added solute ions. This is consistent with the large negative values of the enthalpy of hydration, since a large amount of heat would be released as the water contracts around an ion. The reason for the large difference between G P O 2 and Cvo2is clearly the large volumetric effects which are predicted with reasonable accuracy by the Born equation. Since they are predicted by the Born equation, it appears that they are primarily electrostatic in origin. Finally, one notes that the Born equation radius, 0.327 nm, was chosen to fit the heat capacity near 600 K and 17.7 MPa. If one fits the Born equation instead to the free energy of hydration at room temperature, one obtains about 0.185 nm, which is much smaller. Other quantities also yield “Born radii” smaller than 0.327 nm. It may be possible to explain these differences by refinements of the Born equation but that is beyond the scope of this paper.
(cvoz
Acknowledgment. R.H.W. and D.S.-M. thank the National Science Foundation (Grant No. CHE77-10624) for support of this work. We thank Jeffrey Gates for performing the calculation of Cvo2for H+(aq)+ OH-(aq). The work at Berkeley was supported by the Department of Energy through contract DE-AC03-76SF00098. Registry No. NaCl, 7647-14-5.
Theory of Circularly Polarized Luminescence from Racemic Mixtures G.
L. Hllmes and J. P. Rlehl’
Department of Chemistry, University of Missoufi-St.
Louis, St. Louis, Missouri 63 121 (Received: September 29, 1982)
The theory of circularly polarized luminescence (CPL) from racemic mixtures excited by a circularly polarized incident beam is developed. A model system is considered in which the effects of chemical racemization reorientation and energy transfer are treated. Specific calculations are presented for the concentrationdependence of the dissymmetry factor for a range of assumed racemization rate constants. I. Introduction During the last several years the use of circularly polarized luminescence (CPL) spectroscopy has become a widely used technique in the study of a variety of optically active systems.’ In CPL one measures the difference in emission intensity for left (L) and right (R) circularly polarized light. This technique is then the emission analogue of circular dichroism (CD), and as such probes the chirality of molecular excited states in the same way that CD probes the chirality of molecular ground states. In most applications of CPL spectroscopy the polarization of the incident exciting light is unpolarized or linearly polarized. Just as in the measurement of linearly polarized luminescence: one must be careful to consider the effects of photoselection on circularly polarized emission i n t e n s i t i e ~ . ~ The ? ~ principal concern, in this (1) F. S. Richardson and J. P. Riehl, Chem. Reo., 77, 773 (1977). (2) See, for example, J. P.Riehl, and F. S. Richardson, J. Chem. Phys., 72, 2138 (1980).
regard, is to recognize that, if the emitting molecules do not rotate rapidly compared to the emission lifetime, then the orientational distribution that one sees in emission will not be the same as in absorption. There are, in fact, other experimental problems associated with the measurement of circularly polarized light in photoselected systems; however, ail of these effects can be understood or eliminated such that the experimental measurement accurately probes molecular chirality. There are situations, however, in which circularly polarized exciting light is of some utility. Perhaps the most interesting application is in studies of racemic mixtures. If racemization is slow compared to the absorption/ emission process, then the light emitted will be partially circularly polarized. Dekkers, Emeis, and OosterhofP were J. P.Riehl and F. S. Richardson, J. Chem. Phys., 65,1011 (1976). J. P. Riehl, J . Phys. Chem., 84, 94 (1980). (5) H.P. J. M. Dekkers, C. A. Emeis, and L. J. Oosterhoff, J . Am. (3) (4)
Chem. Soc., 91, 4589 (1969).
0022-365418312087-3300$0 1.5010 @ 1983 American Chemical Society
CPL from Racemic Mixtures
-[jly-“T
The Journal of Physical Chemistry, Vol. 87, No. 17, 1983 3301
the first to demonstrate this phenomenon by observing the circularly polarized emission of racemic trans-0-hydrindanone in isooctane. More recently Schippers and Dekkers6have measured circularly polarized luminescence from a number of different racemic mixtures. Schippers and Dekkers7have also been interested in determining the optical purity of partially resolved systems. Part of their procedure in doing so requires the measurement of CPL upon irradiation with circularly polarized light. Proper interpretation of these kinds of experiments requires that one be able to describe all of the processes occurring between absorption and emission that influences the degree of circular polarization in the emission. Clearly, if racemization occurs in the same time scale as emission, then the net circular polarization will reflect the competition between these two processes. Even if the system does not racemize chemically, it is important to allow for energy transfer in the excited state. Although transfer between identical enantiomers will have no effect on the differential excited-state population, quite obviously, energy transfer between different enantiomers will tend to decrease any differential population generated by the circularly polarized excitation beam. In this work we formally treat circularly polarized emission from a model racemic mixture in which we allow for excited-state racemization and excited-state energy transfer. In section I1 we introduce the model system, develop the formal expressions for CPL, and discuss the differential time-dependent excited-state population, as well as the orientational dependence of the polarized emission intensity. In section I11 we specifically treat energy transfer and calculate results for a specific model system. 11. Theory and Model
A. Racemic Solutions. We consider a solution of an equal number of opposite enantiomers (so-called racemic mixture), and formally allow for the possibility of racemization and energy transfer in the emitting state. The various processes are as follows: absorption
De
kd
kll
kg
D,
Flgure 1. Energy-level diagram for a model racemic mixture. Absorption is to an intermediate state, e; emission is from the state n. k , and k , denote respectively racemization and energy transfer in the emitting state.
A I
INCIDENT BEAM
Figure 2. Schematic diagram of the CPL experiment. The incident circularly polarized beam is directed toward the sample which is situated at the origin of the laboratory (1,2,3) coordinate system. The detector is capable of analyzing for the net circularly polarized component of the emission.
In the above g, e, and n refer, respectively, to isomer d (or 1) in the ground state, intermediate excited state, or emitting state. For simplicity we have not allowed for racemization in the ground state. The rate constant k5 describes both emissive and radiationless quenching processes. In Figure 1we illustrate the various energy pathways in a simple energy-level diagram. Not included explicitly in the scheme above is the self-transfer of energy between identical isomers. For example dn(Q) + dg(Q’)
relaxation
kz
de
dn
racemization
energy transfer
1,
k
+ d, .& 1, + d, k,
(IV)
emission/relaxation
(6) P. H. Schippers and H. P. J. M. Dekkers, Chem. Phys. Lett., 88, 512 (1982). (7) P. H. Schippers and H. P. J. M. Dekkers, Tetrahedron, 38, 2089 (1982).
4
her’
dg(Q) + d,(Q’)
(1)
where Q and Q‘ denote respectively the orientation of the donor and acceptor molecules. Although this self-transfer does not change the total number of d (or 1) isomers in the excited state, it may affect the polarization of the emission through the scrambling of the orientational distribution. This is discussed in more detail below. The experimental technique is depided in Figure 2. An incident circularly polarized beam is directed toward the sample and the emitted beam is analyzed for the circularly polarized component. It is convenient to define a laboratory coordinate system denoted by 1, 2, and 3 (as illustrated) in which the detector is situated in the positive 3 direction, and the incident absorption beam lies in the (1,2) plane and makes an angle 0 with the 3-axis. The circularly polarized differential emission intensity, AI = Ileft - Iright, for a d, molecule with orientation Q at time t is given by the following expression: AI(u,Q,t)= huNdn(Q,t)Aw(Q)f ( u ) (2) where N d , is the number of d molecules in the emitting state n and orientation Q at time t ; AW(Q) is the differential quantum-mechanical transition probability, and f(u) is a normalized line-shape function. If the incident excitation beam is, for instance, left circularly polarized, we may write eq 2 as follows:
3302 The Journal of Physical Chemistry, Vol. 87, No. 17, 1983 M(W,Q,t)
= hUNd:(Q,t)AWd(Q)
f(W)
Hilmes and Riehl
t3
(3)
l y
where we explicitly show the dependence of the population of the emitting state on the incident polarization. Since our concern here is with racemic mixtures, we need also consider the contribution of 1 molecules to U
AI(w,Q ,t ) = h~N1:(0,t ) A W (Q ) f (W)
ii
(4)
Adding eq 3 and 4, we have hl(W,Q,t)
=
hWf(w)[Ni:(Q,t)AWQ)
+ Ndb(Q,t)Awd(Q)l(5)
The differential transition rates are defined as follows: - WR~(Q) AWd(Q)= WL~(Q)
AW(Q) =
w,l(Q) -
WR'(~)
(64 (6b)
where, for example, WLi(n) is the probability that a 1 molecule with orientation Q will emit a left circularly polarized photon. For enantiomers the following relations hold WLd(Q ) = wRd(Q)
=
wR1( a)
(74
wi(Q)
(7b)
so that AW(Q) AW(Q) = -AWd(Q) Equation 5 may then be rewritten as M ( w , Q , t ) = hd(W)AW(Q)[Ni:(Q,t)
- Nd:(a,t)l
= h wf(w) A W(Q )ANnL( 0,t )
(8)
\ I
Flgure 3. Relatiship between laboratory (1,2,3) and mdecular (x,y,z) coordinate systems.
probability. We have neglected pure magnetic dipole and higher multipole terms in eq 13. The orientation dependence of this equation is displayed by rewriting this result in the molecular coordinate system.
(9) (10)
This equation is averaged over all orientations to obtain the result u ( o , t ) = hWf(W)(AW(Q)ANnL(Q,t))
\
(11)
where the brackets denote the ensemble spatial average and we have assumed that the line shape and energy of the transition are independent of orientation. B. Time-Dependent Orientational Distribution. The orientation dependence of eq 11 is most easily treated by first defining a function q(Q,t) as the probability that a molecule with orientation Qo = (e0,r$,,$,) at time t = 0 has an orientation Q at time t. The angles (e, 4, and $) are defined with respect to the laboratory 1,2,3-coordinate system as shown in Figure 3. It follows that
dQ,t)= J 20 T d 4 0 0~ 2 f f d $0 o ~ f Bof sdeo i n dQo) G(QolQ,t) (12) where G(Qojs2,t)is a function describing the time evolution of the molecular orientation. The initial probability of excitation is calculated from the direction and polarization of the incident light and molecular transition matrix elements. The probability that a molecule will absorb a left circularly polarized photon is given by the following: s(Qo) = Icz[Ipleg12+ Ipzeg12- 2i(plegmleg+ pzegmZeg)+ higher order terms] (13)
where
where Rli and RZ are elements of the Euler rotation matrix relating the molecular and laboratory coordinate systems. The important implication of eq 16 is that, though the magnitude of the probability depends on the values of the matrix elements, the orientational dependence of the initial absorption probability is the same for both enantiomers. If the multipolar expansion of the interaction Hamiltonian is carried out to higher orders, then this is not necessarily true, but we do not consider those effects in this work. Thus, from the discussion above mnL(Q,t) = Nl:(fl,t)
- Nd>(Q,t)
= NI>(t) ?IL(a,t)- Nd:(t)
= (elpllg)
(144
mZeg= (elm&) (14b) p and m denote, respectively, components of the electric and magnetic dipole moment operators, and Icz denotes the non-geometry-dependent parts of the absorption
(18)
where
(19) In a racemic mixture the function G1(QoJQ,t)is rigorously identical with Gd(QolQ,t)and we thus drop the subscript. We have therefore the result that VlL(Q,t)
= qdL(n,t)E qL(Q,t)
(20)
Therefore WnL(Q,t)= AN,%) ??IL(Q,t)
= WnL(t) VL(Q,t) pzeg
qdL(Q,t)
(17)
(21) (22)
and we need only be concerned with the differential time-dependent population. I t should be noted that, in a partially resolved system, the enantiomers will have different environments and, thus, different evolution functions, G(Qo(O,t),although in many cases this difference may be neglected.
The Journal of Physical Chemistry, Vol. 87, No. 17, 1983
CPL from Racemic Mixtures
Rgn = Im (glZln)(nlslg)
Equation 11 may be rewritten as follows: hl(w,t) = hwfb)
a,,%) (AW(Q2) vL(Q,t))
(23)
Within these applications the spatial average, as implied by brackets in eq 23, contains no information on molecular chirality. In other words, the orientational distributions of both enantiomers are identical. The term in brackets can be considered, therefore, as a multiplicative factor and does not contain any discriminatory information. C. Differential Excited-State Population. From section IIA it follows that min(t)/dt = +(k3 + k5)Nd, - k3Nin + k4NdnN4- kdNdpN1, (244 md,,(t)/dt = +(k3
k5)Nln- h a d , ,
+ k4NlnNd,- k4NdnN4 (24b)
We have assumed above that the absorption/relaxation process is rapid compared to all other processes. Combining eq 24a and 24b, we obtain d D ( t ) / d t = d[N,,(t) - Nd,,(t)l/dt = -(2k3
+ k5)AN(t) + 2k4NdnNlK - 2k4NlnN4 (25)
In most applications, the number of excited molecules is small compared to the total number present. Thus Nd, = Nb N/2 (26) where N is the total number of molecules present in the illuminated sample volume. It is important to note that, in the model presented here, the rate constant, k4, is time dependent. This is due to the nature of the energy-transfer process as will be discussed below. We now write dAN(t)/dt = -[2k3 + k5 k4(t)]AN(t) (27)
+
mnL(0) = qmeL(0) (33) where q denotes the fraction of initially excited molecules that relax to the emitting state. We thus have the result AIL(w ) = K(w ) ANeL (0)R,B (34) where K(w) is a constant, and B denotes the time integral. D. Dissymmetry Factors. The absolute differential excited-state population, AiVeL(0)depends obviously on the incident light intensity, concentration, and path length of light in the sample solution. It is convenient to eliminate this dependence by considering the absorption dissymmetry factor, gab0,which is defined as follows:
'
gabs = (e - fR)/{y2(€L -
A(t) = - J t k 4 ( t ) dt
A€/€
(35)
respectively extinction coefficients for left (L) and right (R)circularly polarized light. The number of molecules excited by the incident beam is proportional to the extinction coefficient. Therefore
However
N f ( 0 ) = N&O)
(37)
So that NLL(0)- Nd>(o)
(28)
(29)
E
tL and tR denote
gfibs
where
(32)
One last note is that, since we have assumed that absorption and relaation to state In) is rapid, the initial population of state In) is directly related to the population of initially excited state le). If we allow for other relaxation pathways, we have
This has the formal solution m r ( t ) = flN*(O)e-@h+k6)tAA(t)
3303
=
'/Z[NLL(O)+ NkL(O)l
= 2ANeL(0)/NeL(0)
(38) (39)
Additional simplification arises if we consider the emission dissymmetry factor, gem,which is defined in analogy with eq 35 as follows:
We have added the superscript T to denote the incident polarization. Substituting above we have then the result hlL(w,t) = hwf(w)ANnL(0)( AW(Q)qL(Q,t))e-(Pk3+k6)tAA(t) It is, in fact, the emission dissymmetry factor that is the (30) more common experimental result, since it measures the differential emission intensity relative to the total emission In most applications one observes a "steady-state signal". intensity.l This is obtained from above by integrating over long times. We write, therefore, from eq 349 h l L ( w ) = XmAIL(o,t)d t 2 h l ~ 2K(wU"(O)R,B gkm = - It is relevant to look at the limiting case in which the initial IL K(w)NeL(O)Dgn(k5)-' orientation of excited molecules is isotropic. This can be obtained either by choosing an excitation/ emission geometry such that this is so3or by considering systems in which the molecules reorient rapidly compared to emission. In = l/ZgfibsdmB /. (42) either of these two cases the orientational average is time independent, and thus we write where 7 ( =kc') is the emission lifetime, and D, denotes the dipole strength of the emissive transition. If one ML(w)= h wf(w)ANnL(0) CR, me-(2k3+k5)td(t) dt (31) neglects racemization and energy transfer (i.e., k5 >> k4N, k 3 ) , then B = l / k 5 , and eq 36 reduces to the result of where C is a constant and R, denotes the "rotatary Dekkers, Emeis, and Oo~terhoff,~ namely strength" for the emissive transition.s In the usual dipole &m = %db&km (k5 >> k3, k4N) (43) approximation this can be written ~~~~
(8) Thia definition of "rotatory strength"differs slightly from the usual isotropic definition. See, for example, ref 1.
~
(9)We have mode omitted here, for simplicity, any differences in line shape.
3304
The Journal of Physical Chemistry, Vol. 87, No. 17, 1983
Hilmes and Riehl
It can easily be seen that gFm = -1/&abedmB 1 /7
(44)
E . Energy Transfer. Radiationless energy transfer in the excited state between identical molecules has been the subject of a number of recent investigations. Gochaunaur and FayerlO have used a diagrammatic Greens function approach to the self-transfer problem and obtained a result that is applicable to both high and low concentrations. We have recently presented a Monte-Carlo calculation" that explicitly includes the geometry dependence of the energy-transfer process. Since the polarization retained after transfer is, in fact, relatively small, the two approaches described above give similar results. Using the basic notation of ref 10 we define G,(t) as the probability that an initially excited molecule will remain excited at time t. For the model presented by Gochanaur and_Fayer,'O G,(t) is calculated from the Laplace transform of G,(c) which is given by
6,(c)= ~ ( ( ~ ~ y ~ C -~ [1 / 4+) [~ 1~ / T ~ Y ~ -C ~ ] ( E T 0.1887y2C2)]1/2+
- O.1887y2C2]/4(c7- 0 . 1 8 8 7 ~ ~ C (45) ~)~]
~ [ C T
where T is the lifetime of the excited state; y is an orientational factor and equals 0.846; C is a unitless concentration given by C = Y33nRZp
(46)
Ro is the so-called critical transfer distance, i.e., the molecule separation in which energy transfer and emission are equally probable; and p is the number density of absorbing/emitting molecules. Use of eq 45 for a racemic mixture is valid if we assume that d-d transfer and d-1 transfer occur at the same rate. This is not a bad approximation for most systems. The probability that excitation remains on molecules if an 1 molecule was initially excited, G / ( t ) ,is given by G / ( t ) = G,(t)
+ '/[I - G,(t)l
(47)
The probability that an 1 molecule has been excited by a d molecule is Gld(t) = 1/[1 - G,(t)l
(48)
We may therefore write Ni(t) = Ni(0) G,'(t) + Nd(0) Gld(t)
(49)
and correspondingly Nd(t) = Nd(0) Gdd(t)
Ni(0) G&t)
(50)
therefore ANL(t) = ANL(0)[G/(t) + G t ( t ) - Gdd(t) - GJ(t)] =
G,(t)
(51)
We can therefore identify G,(t) by comparison with eq 30
G,(t) = eA@)
(52)
(10) C. R. Gochanour and M. D. Fayer, J. Phys. Chem., 85, 1989 (1981). (11)G. L.Hilmes, H. H. Harris, and J. P. Riehl, submitted for publication. (12) See, for example, D. P. Craig in 'Optical Activity and Chiral Discrimination", S.F. Mason, Ed., NATO A.S.I., Reidel, Dordrecht, The Netherlands, 1978, Chapter 12.
E
5 '
x
+
c
0
0
-4
0
Red.
LO;
0
2
-3
0
0
0
I
i a - c = r t - e -
I
+
__ -
Flgure 4. Calculated diss mmetry factors (arbitrary units) vs. reduced concentration: k , = 10 ( ), lo7 ( ), 10' (A),lo9 (a),and 10'' (0)
I
S-1.
111. Results and Discussion The results of the previous section indicate that the effect is proportional to the product of the dissymmetry factors for absorption and emission. Since the sensitivity of most CPL spectrometers is one is limited to investigations of systems with fairly large gem and gabs. In Figure 4 we plot theoretical dissymmetry factors (in arbitrary units) vs. reduced concentration for a model racemic solution. B is obtained by numerical integration, and G,(t) is obtained by numerical inversion of eq 45. We have estimated an excited-state lifetime of 1ns and varied the racemization rate constant from 1 X 10l6and 1 X 1O1O S-1.
-
-
As mentioned above, the assumption that d d and d 1 transfer are equally probable is not a bad approximation. However, there is considerable interest in these types of chiral discriminatory forces.l' The diagrammatic approach of ref 10 is not directly applicable to systems containing two or more solute species. Not only can the magnitude of the energy-transferprobabilities be different, but also the pair distribution function for d-d (or 1-1) vs. d-1 pairs may be different. It is possible that a MonteCarlo approach as in ref 11 may be of some use. We note again that the results displayed in Figure 4 are for a situation in which the orientational distribution with respect to the emission geometry is time independent. This restriction is, in fact, more restrictive than it need be. As shown previowly,'J3 if the absorption and emission transitions are parallel than to the limit of what is described here, the differential emission and total emission have the some orientational dependence and, thus, cancel in gkm. Results for systems in which the orientational dependence of Al cannot be neglected can be modeled in the manner of ref 1. In summary, the measurement of circularly polarized luminescence from racemic mixtures may be an excellent way of determining racemization rate constants for certain optically active systems. It may also be possible to gain insight into the nature of chiral discriminatory forces in systems in which chemical racemization is not possible, but in which differential energy transfer may be observed. We are exploring the latter in our laboratory, from both theoretical and experimental perspectives. Acknowledgment. Acknowledgments are made to the Graduate School of the University of Missouri-St. Louis for award of a Summer Research Fellowship and to the Research Corp. for partial support of this work. We also acknowledge Professor Harold H. Harris for several discussions concerning various aspects of this work. (13) J. Snir and J. A. Schellman, J. Phys. Chem., 78, 378 (1974).
