J. Phys. Chem. 1989, 93, 179-184 The work presented here is illustrative of the profound effect that a small number of solvent molecules can have on the course of a chemical reaction. Considerable experimental and theoretical refinements can be expected in the near future to supplant the qualitative dependences reported here. Experimentally, the techniques are at hand to probe state-selective and time-resolved properties of size-specificclusters. We may therefore look forward to more fundamental insights into solvated reaction mechanisms on a molecular level. Acknowledgment. I am grateful for the assistance of R. A. Hertz and M. L. Homer in the operation of the hydrazine source. This work was funded by the Aerospace Sponsored Research Program. Appendix The expressions for the solvent size-specific reaction enthalpies that appear in eq 13 and 17 are formulated by adding additional terms to the gas-phase thermodynamic cycles to account for stepwise solvation. As an example, reaction 8 A+.M,
* AH+.M,I + M(-HJ
can be broken down to the form
179
A+ + e- * A M * M(-HJ + H H * H+ + eA
+ H+ ==+ AH+
AH+ + M AH'M AH+M,2
=+
AH'Ml
+ M * AH+M2
+M
-Ip(A) D,(M(-HJ) Ip(H) -DH+(A) -AEl,o(AH+MI) -AE2,1(AH+M2)
AH+M,1
-AEP1,,2(AH+M,1) It is straightforward to show that by summing the solvation terms, one obtains eq 13. The expression given by eq 17 for reaction 9 follows in a similar manner. Registry No. NH3, 7664-41-7; N2H4, 302-01-2; C H 3 0 H , 67-56-1; H20,7732-18-5; C6HSNH2+, 34475-46-2. ==+
Information Theoretic Prior Functions for Large Molecular Systemst J. T. Muckerman Chemistry Department, Brookhaven National Laboratory, Upton. New York I 1 973 (Received: June 21, 1988; In Final Form: September 26, 1988)
Several general cases of information theoretic prior functions for large molecular systems are derived. These are intended to serve as examples for the deviation of prior functions appropriate for still other systems. General procedures are outlitled for obtaining closed-form expressions for the desired prior functions in the continuous rigid-rotor, harmonic oscillator (RRHO) approximationand in the absence of angular momentum constraints, and these are applied to several cases of current experimental interest. Finally, a few cases of angular momentum deficient systems are treated.
Introduction The use of information theory has become widespread in the analysis and description of molecular collision phenomena.' An integral part of such an analysis is the definition of a reference distribution, or "prior" function, which represents a completely statistical distribution of the energy available to asymptotic products among their various degrees of freedom. Any deviation of a measured or calculated product state distribution from the prior distribution represents a dynamical effect. Recent experimental studies of atom-polyatomic molecule collisions, photodissociation of polyatomic molecules, and collisions involving polyatomic molecules of differing sizes and degrees of excitation have elicited a need not only for prior functions for systems of greater complexity but for simple approximations to those prior functions similar to those that have been derived for the atom-diatomic molecule case. We present here some prior functions based on the rigid-rotor, harmonic-oscillator approximation for several systems of recent experimental interest. The first of these corresponds to collisions of an atom with a linear triatomic molecule (e.g., H + C02).2 The second corresponds either to collisions of an atom with a general nonlinear polyatomic molecule or to the photodissociation 'This research was carried out at Brookhaven National Laboratory under Contract DE-AC02-76H00016 with the U S . Department of Energy and supported by its Division of Chemical Sciences, Office of Basic Energy Sciences.
0022-3654/89/2093-Ol79$01.50/0
of a general nonlinear polyatomic molectlle into an atom and a nonlinear polyatomic fragment. An example of this would be the photodissociation of CH31 to form C H 3 I*.3 The third case corresponds to collisions of a linear triatomic molecule with a general nonlinear molecule or to the photodissociation of a geneial nonlinear molecule into linear and nonlinbr polyatomic fragments. This case was motivated by recent experimental studies of collisions of C 0 2 with vibrationally excited a z ~ l e n e . Finally, ~ we present for two cases of interest the prior functions in the limit of zero total angular momentum. These limiting-case ptior functions are expected to be more appropriate for products of photodissociation from rotationally cold molecules in free jet expansions and seeded nozzle beams.
+
(1) See, e.&: Bernstein, R. B.; Levine, R. D. Ado. At. Mol. Phys. 1975, 1 1 , 215. Levine, R. D.; Kinsey, J. L. In Atom-Molecule Collision Theory: A Guide for the Experimenfalist;Bernstein, R. B., Ed.; Plenum: New York, 1979, and references therein. (2) Flynn, G. W.; Weston, R. E., Jr. Ann. Reu. Phys. Chem. 1986,37,551. ( 3 ) Riley, S.J.; Wilson, K. R. Discuss. Faraday SOC. 1972, 53, 132. Sparks, R. K.; Shobatake, K.; Carlson, L. R.; Lee, Y. T. J. Chem. Phys. 1981, 75,3838. Hermann, H. W.; Leone, S.R. J. Chem. Phys. 1982,76,4766. Van Veen, G. N. A,; Baller, T.; De Vries, A. E.; Van Veen, N. J. A. J. Chem. Phys. 1984, 87, 405. Barry, M. D.; Gorry, P. A. Mol. Phys. 1984, 52, 461. Ogorzalek Loo,R.; Hall, G. E.; Haerri, H.-P.; Houston, P. L. J. Phys. Chem. 1988, 92, 5. (4) Barker, J. R. J . Phys. Chem. 1984, 88, 11. Jalenak, W.; Weston, R. E., Jr.; Sears, T. J.; Flynn, G. W. J . Chem. Phys. 1985,83, 6049; 1988,89, 2015.
0 1989 American Chemical Society
180 The Journal of Physical Chemistry, Vol. 93, No. 1, 1989
Muckerman
r(n + 1) = n!,
Many aspects of the present work are implicit in previous contributions to the literature by other^.^-^ The aim here is to provide an instruction manual to aid the experimentalist in constructing prior functions for more complicated systems.
Theory The theoretical approach to constructing a prior function has much in common with the application of statistical theories for unimolecular dissociation8 and chemical r e a ~ t i o n : ~define a "dividing surface" in phase space that separates one arrangement from another, and on that surface assume that all energetically allowed states of the system are equally probable. The key feature of an information theoretic prior function is that the dividing surface is placed in the region of asymptotic products. For a system containing N = n + k atoms distributed in two fragments containing n and k atoms, the general form of the (normalized) prior probability distribution is
+
Atom Linear Triatomic Molecule. Within the RRHO approximation, we may write the vibrational energy of a linear triatomic molecule as
EV=
(U1
+ X ) W +~ (Uza + u2b + I ) w ~+ (U3 + y2)~d3
5
+
The denominator of the fraction on the right-hand side of eq 1 is then
xxxxx(2j + 1)[E - E v 4
Here g(a,,ak) represents the total degeneracy associated with the two internal states with quantum number sets a, and ak, the internal energy associated with the two sets of internal states, E - e the relative translational energy, and E the total energy. The approach we take here is to employ simplifying assumptions regarding these degeneracy factors and to employ the continuous rigid-rotor, harmonic-oscillator (RRHO) approximation for the energy levels, thereby allowing the sums in eq 1 to be replaced by integrals. Apart from the straightforward application of the abovementioned statistical treatment, two useful mathematical tools deserve special attention. The first is the technique of a variable transformation in which one of the new variables is "redundant", Le., it does not appear in the integrand of the multidimensional integral to be evaluated. This was used by previous workers in the information theoretic context" to transform the probability density function [vide infra] po(fvfR) into pocfl), w h e r e h = fv + fR. Given that in the continuous R R H O approximation
+
El
+ E26 + E3
(7) where u2,, and u2b are the two components of the degenerate bending vibration, and the rotational energy is E, = j('j l ) B (8) E2a
D= %at
r ( 1 / 2 ) = d2
ha"%
u3
(9)
J
Invoking the additional approximation of continuous quantum numbers and changing variables from uito Ei and from j to ER, we have
D=
The continuous analogue to the prior probability distribution function of eq 1 is the normalized prior probability density function
or Pocflf2af2bf3fR;E) =
10395
-fl - f 2 0
- f 2 b - f 3 -fRI1"
(12) wheref;, Ei/E, etc. Equation 12 may be used to obtain various one-dimensional prior distributions. Integrating eq 12 over all variables except f l (or, similarly, f 3 or f R ) , we obtain the prior probability density for the nondegenerate vibrations and the rotation of the linear triatomic molecule
we first define the distribution
in which fv is a redundant variable. To obtain the desired distribution function, we simply integrate over the redundant variable:
pocfi)
= 'y2(l
-fi)9/2,
i = 1, 3, R
(13)
Replacing any one of the vibrational variables f;, in eq 12 by fv and integrating over all the others gives the prior probability density for the total fraction of the available energy in vibration: POUv) =
The second tool is an extremely useful definite integral that will be employed repeatedly in the following discussion. For k and v with real parts greater than zero, the B (beta) function (14)
where for integer or half-integer
Replacing eitherfia or f Z bby f 2 f 2 , +f2b and integrating over all variables except f 2 gives the prior probability density for the fraction of the total available energy in the degenerate bending mode:
and v ~~
(5) Bullitt, M. K.; Fisher, C. H.; Kinsey, J. L. J . Chem. Phys. 1974, 60, 478. ( 6 ) Nordholm, S.; Freasier, B. C.; Jolly, D. L. Chem. Phys. 1977, 25,433. ( 7 ) Zamir, E.; Levine, R. D. Chem. Phys. 1980, 52, 253. (8) Marcus, R. A. J . Chem. Phys. 1952,20,359. Wider, G . M.; Marcus, R. A . J . Chem. Phys. 1962, 37, 1835. (9) Pechukas, P.; Light, J. J . Chem. Phys. 1%5,42, 3281. Pechukas, P.; Light, J.; Rankin, C. J . Chem. Phys. 1966, 44, 794. Light, J. C. Discuss. Faraday SOC.1967, 44, 14. (10) Ben-Shaul, A.; Levine, R. D.; Bernstein, R. B. J . Chem. Phys. 1972, 57, 5427
p0Cf2) =
9%f2(1- f 2 I 7 l 2
(15)
Similarly, we obtain the prior probability distribution for the fraction of the total energy appearing as internal energy in the triatomic molecule: ~ O c f i )=
3465/5&f;'(1- W 2
(16)
It should be kept in mind that eq 13-16 refer to the distribution of one variable irrespective of the values of the others in eq 12,
The Journal of Physical Chemistry, Vol. 93, No. 1, 1989 181
Prior Functions for Large Molecular Systems
Transforming to continuous energy variables and using the second (approximate) equality in eq 19, we obtain
I 1 .o
0.5
0.0
f,
or pocfl,
Figure 1. Single-variableprior probability density functions, p o ( f ; ) , for the atom-linear triatomic molecule system. The curves are labeled by i = 1, 2, 3, R, V, and I (see text).
Le., the prior is summed (integrated) over all values of the other variables. Figure 1 shows the various prior probability densities for the atom-linear triatomic molecule case. It is readily demonstrated that the prior functions satisfy the equipartition theorem by computing the average values of the variables according to
...A&;E )
22 = ($ ,
+ 2)!fR’/’[1
-fl
- ”’ -f, -fR]’/’
(23)
Equation 23, like eq 12, is independent of the total energy E . It is instructive to examine a particular case: consider the dissociation of CH31into CH3 and I. The prior probability density function is then (for s = 6) pocfl,. . . f s f R )
2’8! = -fR1/2[1
-fi - *** -f6 - fR1’”
P
(24)
Equation 24 may immediately be integrated over all vibrational degrees of freedom to obtain
This gives If we had not integrated over the variablefi corresponding to the fraction of available energy going into the nondegenerate “umbrella” mode of the CH3 radical, we would have obtained
which when integrated over f R gives This is the same result that one would obtain by assigning “1/2ki”’ to each translational and rotational degree of freedom, and “kT“ to each vibrational degree of freedom. Atom General Nonlinear Molecule. We turn now to the slightly more complicated case of a system fragmenting into (or forming as products of a chemical reaction) an atom and a general nonlinear molecule. In this case there are s 3 N - 6 vibrational degrees of freedom, where N is the number of atoms in the product molecule. There are also three rotational degrees of freedom in the molecule, and (even in the rigid rotor approximation) the precise characterization of the rotational energy levels depends upon whether the molecule is a symmetric or asymmetric top, etc. Now, in the spirit of eq 1 , we are more concerned with the number of levels that are energetically allowed than with their energies. If we are willing to sacrifice the high-resolution structure in our prior function, we may, for example, suppress the K dependence of a symmetric top molecule and consider each J level to be ( 2 J 1)2 degenerate. The results in this section and subsequent sections are accordingly based on the additional approximation that all nonlinear molecules are spherical tops. This approximation takes proper account of the number of levels associated with each J state and simplifies the mathematics by treating all finer structure as additional degeneracy of the parent state. We may therefore write
+
+
(18)
i= 1
ER
= J(J
+ l)B
(J
+ X)’B
VI
J
(27)
Equation 27 would apply equally well tofl, the fraction of the total energy in the nondegenerate symmetric stretch mode of the methyl radical. The prior probability density for relative translational energy is
Several single-variable prior probability density functions for the case of CH3 I are shown in Figure 2. Linear Triatomic Molecule General Nonlinear Molecule. Following the procedure outlined above, one could derive probability density functions appropriate for increasingly complicated cases, e.g., diatomic moleculelinear triatomic molecule, etc. We will skip those and focus now on a system with two “large” fragments, namely, the case of a linear triatomic molecule and a general nonlinear molecule. Let t label the degrees of freedom associated with the former, and g those with the latter. The parameter s will once again denote the number of vibrational degrees of freedom in the general nonlinear molecule. On the basis of the discussion above, we may write
+
+
+ Ez: + E26’ + Ejl = jIGl + l)Bl, (2j, + 1 ) degenerate E:
E,‘
= E,‘
S
E# = ZEig i= 1
(19)
where we have used the semiclassical Langer approximation. The denominator of eq 1 is then D = c-~c(2J + 1)2[E- E v - E R ] ’ / ~
= 8(1 -f2)’
PO&)
(20)
ERg = Jg(Jg
+ 1)B,
(2Jg + 1)2 degenerate
+ E,‘ = E# + ERg
E{ = E: Ef
(29)
182 The Journal of Physical Chemistry, Vol. 93, No. 1, 1989
Muckerman
Using these relations and eq 5 , we find
D=
0 0.0
0.5
or
!
.0
f,
22
+ 7)!vRg)'/2[1 -fr'-ff]'/'
-(s x
(32)
wherefRf is the fraction of the total energy (i.e., both fragments and their relative translation) appearing as rotation in the triatomic molecule, etc. Once again the RRHO prior probability density, when expressed in terms of the fraction of available energy in the various degrees of freedom, is independent of the available energy. To obtain any required single-variable prior probability density function from eq 32, we may integrate it over all other variables. For example, if we wanted the prior function for total vibrational energy in the linear triatomic molecule, we would replace any triatomic vibrational variable by fv' and integrate over all the other &' and fg and find that
(s + 7)! pocfv') = -1 (1 6 (s
+ 3)!
-fv')"3vv')3
4
Vv')= s+8
V+)
(3
+ 7)(1 -f1,3,R')s+6
(s + 7)!
P0V2')= ( S + i u pocf[) =
(s + 7)! -1 4! (s + 2)!
momentum. j because j must be antiparallel and equal in magnitude to I, the angular momentum of relative motion. Thus specifying j determines l, and of the 2j l possible projections of j onto l only one (Le., the antiparallel one) is allowed by the J = 0 constraint. Consider the simple case of a triatomic molecule dissociating into an atom and a diatomic molecule. A prototypical example is HDO OH D and O D H. The restriction of the total angular momentum, J , to zero makes the effective degeneracy of diatomic rotational levels equal to unity, Le., g, = 1 for all j . This affects the denominator in eq 1:
+
-
+
+
where we have gone immediately to continuous quantum numbers and have employed the Langer correction to the rotational energy. Defining x (j 1/z)B'/2and a ( E - EV)l12we may write
+
1
(34)
If the general nonlinear molecule were H 2 0 (corresponding to = 4 / 1 1 ; if it were azulene (for which s = 48), cfv') = Other useful results include P0cfi,3,R') =
Figure 2. Single-variable prior probability density functions for the case of CHI + I. The curves are labeled as in Figure 1. The solid curves correspond to prior functions,pocfl), constructed without consideration of the conservation of angular momentum. The dashed curves correspond to prior functions, p!cfl), appropriate for the case of zero total angular
(33)
Using eq 17, we find that the prior average fraction of the total energy imparted as vibrational energy of the triatomic molecule is
s = 3), then
1
PE2
Do= ~ ~ S , " , [ -a x2]1/2 2 du dx = - (37) 8B'I2w
The ratio of Do,which is proportional to the total number of energetically allowed states in the limit of zero angular momentum, to D,the same quantity without consideration of angular momentum, is
5- 1 5 ~ ( B ) ' / ~ D
-fzf)s+5 1 - hf)s+2
(35)
Prior Functions for Angular Momentum Deficient Systems. The discussion up to this point has assumed that the conservation of angular momentum is not a constraint on the 'system. It has focused entirely on the energy and degeneracy of levels, including those with very large rotational quantum numbers. It is of more than casual interest to determine how the picture might change when the conservation of total angular momentum places a severe constraint on the system. For example, in the photodissociation of molecules from a free-jet expansion, where rotational degrees of freedom are often "frozen out", it would seem appropriate to assume that the total angular momentum available to the products is limited to values quite close to zero. Accordingly, we consider here the limiting case of zero total angular momentum. The major modification that the consideration of zero total angular momentum places on the procedure followed above is in specifying the effective degeneracy of molecular rotational states. This constraint eliminates a factor of (2j 1) in the effective degeneracy of a state with principal rotational quantum number
+
32 E
which is seen to decrease as the available energy increases. The prior probability density is then p$(uj;E) =
8B% rE2
-[ E - (U + '/&
- (j
+ '/2)2B]'/2
(39)
or
Integrating eq 40 over Ev, we obtain the prior probability density for the J = 0 continuous rotational quantum number irrespective of vibration: p80';E) =
16B'l2 3aE2
-[ E - (j + j/2)2B]3/2
Here we see that prior probability density does depend upon the total available energy. In a similar fashion we can obtain the J = 0 diatomic prior probability density for vibrational energy irrespective of rotation:
The Journal of Physical Chemistry, Vol. 93, No. I, 1989 183
Prior Functions for Large Molecular Systems
or or Equation 43, which is once again independent of the total available energy, is to be compared with the comparable expression in the absence of the zero angular momentum constraint:
The effect of the angular momentum deficiency on the vibrational distribution is to make the 10w-f~states relatively less probable because some of the low-fvstates are associated with states of very high j which have had their effective degeneracy greatly reduced. We mentioned above that D and Do are proportional to the number of energetically accessible states with no angular momentum constraint and with the constraint of zero angular momentum, respectively. The proportionality constant, which includes the energy-independent factor from the translational density of states, depends on p3I2,where 1.1 is the reduced mass for relative motion. As a dramatic example of the effect of low total angular momentum, consider the prior branching ratio” in the photodissociation of HDO: (45) for these two cases. Now D is proportional to m3/2through the factor (Bw)-’, where m is the reduced mass of the diatomic product molecule. Therefore r:D,OH = 1 in the absence of an angular momentum constraint. However, eq 37 indicates that Do is proportional to m rather than m3I2. This implies that
Equation 46 reflects the fact that for a given amount of energy in product rotation, OD must have a larger quantum number j than OH. The lowering of the effective degeneracy of rotational states by a factor of 2 j + 1 therefore affects the number of available OD rotational states more than that of OH. As a final example, we consider the atom-general nonlinear molecule case in the limit of zero total angular momentum. The effective rotational degeneracy becomes 2Jg + 1 instead of (2Jg + 1)2,as before. This new degeneracy factor exactly cancels the Jacobian in the transformation from the variable Jg to ER in the computation of Do:
+
Do = C***CC(2Jg 1)[E - E# - ERg]’I2= 4
0 .
1=1
01
Jg
.
The prior probability density is p!(
E f,. ..,E,g,ERg;E) =
Of particular interest among the single-variable prior probability density functions is that for rotational energy irrespective of vibration. The result is (49)
( 1 1) Levine, R. D.; Kosloff, R.Chem. Phys. Left. 1974, 28, 300. Levine, R. D.; Bernstein, R. B. Chem. Phys. Left. 1974, 29, 1.
where the label g is understood. Unlike eq 41, this rotational prior probability density for J = 0 is independent of the available energy. Owing to the structure of eq 48, eq 50 also applies to any nondegenerate vibrational mode in the general nonlinear molecule. The J = 0 prior for the total internal energy in the molecule is
For the case of methyl iodide pgcf,)
=
l5/(1
-fi)I3l2,
i = R, 1, 2
(52)
while in the absence of an angular momentum constraint (see eq 25) (53) and ~ O ( f i , ~ )is given by eq 27. The constrained prior probability density for the total internal energy is
The J = 0 rotational prior probability densities of eq 52 and 54 are also shown in Figure 2. Discussion and Conclusions Starting from the general expression for the information theoretic prior probability distribution of product states, we have employed the continuous rigid-rotor, harmonic oscillator (RRHO) approximation to derive closed-form single-variable prior probability density functions for several classes of systems involving large molecules. The simplest of these is the atom-linear triatomic molecule system, of which H C 0 2 is but one example. By suppressing the splitting of a parent J state in general nonlinear molecules arising from nonspherical symmetry, we have been able to derive similar expressions for the atom-general nonlinear molecule case. As a model for still more complicated cases, we treated the linear triatomic moleculegeneral nonlinear molecule system. Other cases, e.g., two general nonlinear molecules, may be treated by a straightforward extension of the present work. We have also explored the effect of low total angular momentum on the prior functions for certain situations such as photodissociation from “cold” molecular beams. By examining the limiting case of zero total angular momentum, we have demonstrated a dramatic effect in the prior branching ratio in atom-diatomic molecule systems and have also shown an effect on atom-general nonlinear molecule systems. The question of how useful these formulas are has yet to be addressed. An important aspect of this question is the way in which the classical-to-quantum correspondence is to be handled. While there is no unanimity of opinion on this matter, the most common practice is to derive the appropriate classical formula in terms of the fraction of energy available in the mode of interest, then interpret the result in a quantal sense (e.g., eq 13, which was derived by using the continuous implementation of the energy expression given by eq 7, which references the energies to the minimum of the potential, would be interpreted in terms of the energy available relative to the zero-point energy of the molecule). This seems a reasonable procedure except for branching ratio calculations similar to that of eq 45, where the energy available to the products in the two branches must be measured from the same reference point, e.g., the minimum of the product molecule potential. As an example of this classical-to-quantum correspondence and of the errors inherent in this procedure as well as those from using a harmonic model, we consider the prior probability for the fraction of energy in the v2 (umbrella) mode of CD3 in the CD3 I* channel from the photodissociationof CD31with 248-nm radiation.
+
+
184
J . Phys. Chem. 1989, 93, 184-190
-
-
5
\
c
E,(J,K) = J ( J
a
+ l)B + P ( C - E)
with w1 = 1787.2, w2 = 421.6, w2x2 = -18.1, w 3 = 2398.6, w4 = 870.9, B = 4.801, and C- B = -2.42 cm-'. The available energy is 12 905 cm-l and the rotational degeneracy is 0
0.0
0.5
(57)
1 .o
f2
Figure 3. Prior probability density function po(f2), where u2 is the umbrella mode of the methyl-d3radical, for the case of 248-nm photodissociation of CD31 into CD3 and I*. The solid line corresponds to the continuous RRHO prior given by eq 27; the points to a direct count of
+
where gJ = (2J 1 ) and gK = for K = 0, f 3 , f6, f 9 , ..., and 4 otherwise. To plot both priors on the same ordinate, the quantal probabilities P"(u2) have been multiplied by the discrete Jacobian:
anharmonic quantum states (see text). We compare in Figure 3 the result given by eq 27 with that from a direct count of the rovibrational states using an energy expression reflecting the large negative anharmonicity of the mode in question and taking proper account of the statistical weights of rotational states arising from permutation symmetry of the identical nuclei.
Equation 27 is seen to give a reasonable estimate of the more elaborate direct count of states, which required approximately 30 min to compute on a MicroVax I1 computer.
Thermal History and Concentration Effects on Shpol'skii Spectra: Study of Acenaphthene in +Hexane J. W. Hofstraat,* I. L. Freriks, M. E. J. de Vreeze, C. Gooijer, and N. H. Velthorst Department of General and Analytical Chemistry, Free University, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands (Received: August 5, f987;In Final Form: May 2, 1988)
Thermal history and concentration effects on Shpol'skii fluorescence spectra have been studied for acenaphthene in n-hexane as a model system. This system yields a spectrum with both narrow lines and broad bands, whose intensity ratio is strongly dependent on freezing time and freezing direction of the sample and on the concentration of acenaphthene. The highest intensity of quasi-lines is obtained with fast freezing speeds for the sample locations that solidify first. The intensity of the quasi-lines strongly decreases in the direction of freezing. The broad bands disappear at high concentrations of guest molecules. The thermal history effects are explained by assigning the quasi-lines to molecules that occupy substitutional sites in the solid matrix, whereas the broad bands are due to isolated molecules in intercrystalline positions that are subject to an amorphous environment. The reduction of the broad-band intensity at high concentrations is attributed to the formation of aggregates and crystallites in the intercrystalline regions. Such systems give rise to a strongly shifted fluorescence emission with a low quantum yield.
Introduction It is generally agreed upon that the Shpol'skii effect is due to a reduction in the inhomogeneity of the environment of molecules trapped in low-temperature n-alkane However, whether or not highly resolved spectra are obtained for a particular combination of guest molecule and n-alkane depends on a number of experimental parameters, whose effect is not clearly understood. Especially the composition and the mode of preparation of the sample appear to be important in this respect. Concerning the composition of the sample, it is accepted as a rule of thumb that the best spectral resolution is obtained if there is a good match between the longest dimension of the guest molecule and the length of the n-alkane (the "key-and-hole r ~ l e " ~ - ~Many ) . exceptions to this rule, however, appear to exist.2 In addition, the concentration of the guest molecules also has a surprisingly strong effect on the spectra.6 Dekkers et al., for instance, observed quasi-linear fluorescence spectra for highly 'To whom correspondence should be addressed at Ministry of Transport and Public Works, Public Works Department, Tidal Waters Division, P.O. Box 20907, 2500 EX The Hague, The Netherlands.
0022-365418912093-0184$01.50/0
concentrated ( 10-2-10-3 M) solutions of acenaphthene in n-hexane and n-heptane, whereas dilute solutions ( 104-10-5 M) solutions yielded broad-banded spectra.6 The mode of preparation of the samples also has a profound influence on the appearance of the Shpol'skii spectra. Differences in cooling speed or application of thermal treatment (annealing) of the sample may lead to strongly differing spectra. In an effort to bring some structure to this complex matter, Rima et al. roughly discerned two types of aromatic compounds.' The first group (type A) consists of larger molecules (like coronene and perylene) whose dimensions require the substitution (1) Shpol'skii, E. V.; Bolotnikova, T. N. Pure Appl. Chem. 1974, 37, 183. (2) Hofstraat, J. W.; Gooijer, C.; Velthorst, N. H. In Methods in Molecular Spectroscopy; Schulman, S.G . , Ed.; Wiley: New York, 1988; Vol. 2, Chapter 4, pp 283-400. (3) Shpol'skii, E. V. Sou. Phys. Uspekhii 1962, 5 , 5 2 2 . (4) Shpol'skii, E. V. Sou. Phys. Uspekhii 1963, 6, 411. (5) Pfister, C. Chem. Phys. 1973, 2, 171. (6) Dekkers, J. J.; Hoornweg, G . Ph.; MacLean, C.; Velthorst, N. H. J . Mol. Spectrosc. 1977, 68, 5 6 . (7) Rima, J.; Nakhimovsky, L.; Lamotte, M.; Joussot-Dubien, J . J . Phys. Chem. 1984.83, 4302.
0 1989 American Chemical Society
