Molecular theory of transition energy correlations of chromophores in liquids or glasses

for pairs

H. M. Sevian and J. L. Skinner Department of Chemistry and Theoretical Chemistry Institute, University of Wisconsin, Madison, Wisconsin 53 706

(Received 5 February 1992; accepted 20 March 1992) The absorption spectrum of an optical transition of a dilute solute in a glassy or liquid solvent is usually inhomogeneously broadened. In a concentrated solution, the question arises as to whether or not the transition energy distributions of nearby solutes are correlated. Such correlation has important implications for coherent or incoherent transport and optical dephasing experiments. We present a molecular theory of this correlation. For a simple model of Lennard-Jones solutes in a Lennard-Jones liquid solvent, we compare our theory to Monte Carlo simulations, finding reasonable agreement. For a model with longer range solute-solvent interactions, where the excited state solute is ionized, the theory predicts very significant correlation effects. This suggests that for more realistic models with dipolar interactions, significant correlation effects will also be present.

I. INTRODUCTION The absorption spectrum for some optical transition of a dilute solute in a crystalline, glassy, or liquid solvent is often dominated by inhomogeneous broadening.14 This means that each solute molecule (or atom) absorbs light at a distinct and well-defined frequency that is determined by its environment, which does not change on the time scale defined by the inverse of the absorption linewidth. Thus, the transition frequency of a particular solute is due to its interaction with nearby solvent molecules (or atoms). The amount of inhomogeneous broadening reflects the extent to which solvent environments differ, and hence is related to the local disorder around the solute. A microscopic theory of inhomogeneous broadening in crystals was introduced by Stoneham,’ and elaborated on by Kikas and Ratsep’ and Davies.6 In this approach the transition frequency of a particular solute is taken to arise from a superposition of contributions from crystal defects. A related microscopic theory of inhomogeneous broadening in structurally disordered solvents (liquids or glasses) was discussed by Messing et a1.,2s3 and later by Laird and Skinner,’ Loring,’ Simon et a1.,9 and Kador.” These authors assume that the transition frequency of a particular solute can be described by a sum of pairwise interactions with each solvent molecule. This approach has been generalized to include solvent dynamics, and hence departures from strict inhomogeneous broadening, by Walsh and Loring.” In addition, the connection between this approach and liquid state solvation theories has recently been demonstrated.‘2V’3 Experimental absorption line shapes in liquids or glasses are typically broad and featureless,24 and can be described by only a few parameters such as the low-order moments. Therefore, since a microscopic theory typically has a larger number of parameters, it is difficult to glean much microscopic information about solute-solvent structure and/or interactions from the line shape alone. Recently, some more sophisticated experiments have been performed, which 8

J. Chem. Phys. 97 (l), 1 July 1992

0021-9606/92/l

probe correlations between pairs of inhomogeneously broadened line shapes, and which can provide more of the desired microscopic information. For example, Sesselmann et aLi have studied the spectroscopy of dye molecules in polymer glasses at low temperatures, the absorption line shape of which is broad and featureless. They then subject the sample to relatively modest pressures, such that the overall line shape is hardly changed. More interesting and informative than this overall change is the frequency change of individual isochromats, which are collections of solute molecules that all absorb at the same frequency. This change was measured by hole burning. Thus, a photochemical hole was burnt, the pressure was changed, and the resulting hole shape was recorded. The breadth of these holes, which were originally very narrow, is a result of lifting the degeneracy of the original isochromat by the pressure change. If it had turned out that the holes broadened very little with pressure, this would have indicated that an isochromat consists of many solutes all with very similar environments, which are all perturbed the same way by the pressure change. In fact, the holes broadened substantially, showing that each isochromat consists of solutes whose environments are not similar, and therefore each solute’s frequency changes by a different amount. These experiments were subsequently analyzed quantitatively by Laird and Skinner” with a generalization of the statistical theories mentioned above. (See, also, the analysis by Kador.16 ) Furthermore, a quantitative prediction” of the burn-frequency dependence of the pressure shift of the hole was verified experimentally.” These pressure effects have also been used’87’9 to determine the local solvent compressibility (for example, in a protein” ) and the gas-phase transition energy of the solute. Electric field effects were also observed and analyzed with the same theoretical approach.” These experiments can all be interpreted as probing the correlation of two different inhomogeneous line shapes, before and after an external perturbation. A related effect, discovered several years ago,2’-28 in30008-l i$OS.OO

@ 1992 American Institute of Physics

H. M. Sevian and J. L. Skinner: Chromophores in liquids or glasses

volves the correlation of inhomogeneous line shapes of two different vibronic transitions of the same solute. Particularly nice phosphorescence-line-narrowing experiments along these lines were performed by Suter et al.” on chromophores (solutes) in molecular glasses at low temperatures. The absorption spectra of both the S,-S, and the So-r, transitions are inhomogeneously broadened in these systems. In the experiments, S, was excited by either narrow- or broad-band radiation, and the ensuing phosphorescence from T, was frequency resolved. If the two inhomogeneous line shapes were completely uncorrelated, then the phosphorescence emission spectrum would be broad and independent of the excitation method. This was observed for some systems. For other systems, however, the phosphorescence spectrum was narrowed considerably upon narrow-band excitation, showing that isochromats of the So-S, transition correspond to a subset of frequencies for the So-T, transition, and hence the two inhomogeneous line shapes are correlated. The extreme case of very narrow phosphorescence spectra following narrow-band excitation, corresponding to complete correlation of the two inhomogeneous line shapes, was not observed. Although a quantitative theory of these experiments has not been put forth, a general molecularlevel theory for understanding these types of experiments has been developed by the present authors,29 and it seems clear that these effects can be understood within this framework. In particular, the extent of correlation between the line shapes has to do with the nature of the pairwise solutesolvent interaction for each solute vibronic state.29 In this paper we want to consider the correlation between yet another pair of inhomogeneous transition frequency distributions-those of the same vibronic transition of two identical nearby solutes. First consider the situation of a dilute solution (in any condensed phase). In this case each solute interacts with a different set of solvent molecules (or defects, in a crystal), and so the frequency probability distributions ofeach solute are uncorrelated. That is, the probability of finding a transition frequency on one solute is independent of what the transition frequency is on any other solute. Now, consider a concentrated solution of chromophores, where, for example, a pair of chromophores interacts with the same set of solvent molecules. In general, we now expect the probability,distributions ofthe two solutes to be correlated, in that the probability of finding a particular transition frequency on one chromophore does depend on the transition frequency of its near neighbor. Saying it yet another way, the joint probability distribution for the pair of chromophores does not factor into a product of individual distributions. This correlation of the transition energies of nearby chromophores has some important implications for at least three types of experiments. The first of these, involving exciton transport at very low temperatures in molecular or ionic crystals,“+‘* and electron transport near the metal-insulator transition in doped crystalline or amorphous semiconductors,J9*W have been studied with models of Anderson localization. The simplest models4’*42 involve a tight-binding Hamiltonian defined on a regular three-dimensional lattice, where the transfer matrix elements are nonzero (with a val-

9

ue of& only for nearest neighbors on the lattice, and the site energies are taken to be uncorrelated random variables whose probability distribution has a characteristic width W. One can also define a root-mean-square energy difference between nearest neighbor site energies, X, which for the uncorrelated case is on the order of W. The localization transition can be thought of as a competition between the transfer strength, J, which tends to delocalize the eigenstates, and the energy mismatch, X, which tends to localize them. Indeed, the localization transition occurs at some value of order 1 of the dimensionless ratio ofX/J. The origin of the random site energies in this problem is the same as that discussed above for the inhomogeneous line shapes. And since in real systems the localization threshold typically occurs at solute (dopant) densities where the mean distance between solutes is these site energies should, in principle, be quite sma11,30*32~39 correlated. Despite this, as far as we are aware there have been relatively few studies4345 concerned with how the localization transition might be affected by this correlation. (In a somewhat different context, however, Phillips and coworkers46’47 have shown how the presence of correlation can radically alter the nature of the localization transition, producing extended states in a one-dimensional disordered problem.) The important point is that if the site energies are correlated, then X can be very different from W, and the value of the dimensionless ratio W/J at the localization threshold will depend strongly on the amount of correlation. Similarly, for models involving both substitutional and diagonal disorder,48 for fixed W one would expect the critical concentration to be strongly dependent on this correlation. The second important consequence of this transition energy correlation involves optical dephasing in concentrated solutions at low temperatures, in particular, as measured by photon echoes. In several different crystalline systems one can measure the photon echo decay time as a function of position within the inhomogeneous line shape, finding, in general, a shorter decay time near the center of the line.49-s’ This has been interpreted as resulting from more near-resonant near neighbors at the center of the line. A quantitative theory of this effect was worked out several years ago by Root and Skinner.52 It was found that the dephasing rate was proportional to the joint probability of finding two nearest-neighbor chromophores both in resonance with the excitation laser. If the transition energies are correlated, this is reflected in the joint probability, and so the dephasing rate again depends strongly on this correlation. Root and Skinner introduced a phenomenological model of this correlation, assuming that it decays exponentially with distance, and from analyzing the experiments concluded that the relevant correlation length was on the order of a few lattice spacings or less. The third area where this transition energy correlation may be important involves concentrated solutions in the hopping or incoherent transport regime (typically at higher temperatures). Incoherent exciton transfer can occur between two nonresonant chromophores as long as the energy mismatch is made up by excitations of the medium, for example, phonons, and the transfer rate can depend strongly on the energy mismatch.53p54 If acoustic phonons are the

J. Chem. Phys., Vol. 97, No. I,1 July 1992

H. M. Sevian and J. L. Skinner: Chromophores in liquids or glasses

10

vehicle for incoherent hopping, if the energy mismatch is small, in a one-phonon theory the hopping rate will be low because the phonon density of states is small. For a larger energy mismatch the rate will increase with the increasing density of states. For an energy mismatch larger than the Debye energy of the phonons, the rate will again decrease since now multiphonon processes must be invoked. Thus, the rate depends strongly on the root-mean-square transition energy difference of nearby chromophores, which can be similar or quite different from the site energy distribution width, depending ‘on the amount of correlation. In this paper our goal is to propose a theoretical framework within which the microscopic origin of this transition energy correlation of nearby chromophores can be understood, and various quantities relating to this correlation can be obtained. In this paper we focus on the case of liquid or glassy solvents. In Sec. II we develop the general theory. In Sec. III we consider the specific case of Lennard-Jones solutes in a Lennard-Jones solvent, and compare our theory to Monte Carlo computer simulations. We also consider the case where the solute in the excited state is ionized, and hence has quite a long-range interaction with the solvent. The theory shows that for this case the transition energy correlation between two solutes is substantial. In Sec. IV we present a few concluding remarks.

vent interactions when the solute is in the excited and ground states. To obtain the configurational average of some quantity that depends on the solvent positions, we integrate over the solvent distribution function P(R, ,...,RNIO), which is the conditional probability that solvent 1 is at R, , solvent 2 is at R,, etc., given that the solute (in its ground state) is at the origin. Thus (j-U%

v...,

R,))

=

dR, .**dR,P(R

s

,,..., R,IO)

xf(R, ,...,R, 1,

(2.2)

where

dR,. ..dR.P(R

,,..., R,IO) = 1.

(2.3)

For example, the average transition energy, defined to be A, is A = (E(R I,***,RN)) =E”+

(T u(Ri))

(2.4)

=E”+~j-dR,4R,P(R ,,..., RNIO)u(Ri). (2.5) 1 Since all solvent molecules are equivalent this becomes dR, -*.dR,P(R ,,..., R,IO)u(R, s Defining a reduced conditional probability by

A = E” + N

II. MICROSCOPIC THEORY OF TRANSITION ENERGY DISTRIBUTIONS In this section we present a general theory of the transition energy probability distributions of both a single solute and a pair of nearby solutes in a liquid or glassy solvent, which we will take to be static. We have in mind the situation where the particular vibronic transition of interest of the solute is off resonant from any vibronic transitions of the solvent, and so (excitonic) delocalization of the wave function of an isolated solute is negligible. In fact, we also assume that the solvent does not even significantly perturb the solute electronic wave function, and so the solute transition dipole moment is independent of solvent configuration. Therefore, the only effect of the solvent is to perturb the solute vibronic energy levels. We will further assume that this perturbation is pairwise decomposable. Finally, we will take this solutesolvent interaction to be isotropic, although this last assumption can be relaxed quite easily. Let us first consider a single solute molecule or atom in a static solvent composed of N molecules or atoms. If the solute is located at the origin, then, for a particular configuration of the solvent, the transition energy of a solute transition between the ground state and some vibronic excited state is given by i

u(R;),

(2.6)

P(R, IO) =

dR,...dR,P(R,,...,R,IO), (2.7) s which is the probability of finding solvent 1 at R, (irrespective of the positions of the other solvent molecules) given that the solute is at the origin, this becomes A = E” + ii’

s

dR,P(R,

IO)u(R,

).

(2.8)

In fact, since this reduced conditional probability is simply the solute-solvent radial distribution function, g(R), divided by the volume, RR, IO) = g(R, l/v,

(2.9)

we find that

)u(R, 1, A=E” fpsd&&R,

(2.10)

where p = N/V is the solvent density. In a similar manner we can calculate the mean-square fluctuations of the transition energy, defined by F((E(R

E(R ,,..., RN) =E”+

).

,,...,Rv)

--A)‘)

(2.1) (2.11)

i=l

where R, is the position of the ith solvent molecule, R, is its magnitude, E” is the unperturbed (gas phase) energy of this transition, and v(R) is the perturbation of the transition energy due to a single solvent molecule a distance R from the solute. This perturbation is the difference of the solute-sol-

=N

s

dR, ***dR,P(R

+ N(N-

J. Chem. Phys., Vol. 97, No. 1,l July 1992

1)

s

, ,...,%lO)~(R,

dR, s-0dR,P(R

1’

,,..., RN/O)

H. M. Sevian and J. L. Skinner: Chromophores in liquids or

glasses

11

2

Xv(R,

Iv@,)

--p2

[s

d&M,

1

)v(R,)

(2.12)

where the second line follows from Eqs. (2.1) and (2.4). Defining the two-solvent reduced conditional probability by P(R,,R,

IO) =

dR,*.*dR,P(R

,,..., R,IO),

-

R,I),

(2.14) where gs (R) is the solvent-solvent radial distribution function, leads to

n(l+Bj) (

(2.21)

7

i

where Bj = exp[ - iv(R,)t

] -

(2.22)

1.

Expanding the product yields

f(~)=‘+(~Bj)+f(~~ja,B,)+....(2.23)

(2.13)

and making the Kirkwood superposition approximation

P(R,,R,lO)--V -‘g(R, MR,ks(lR,

fCf)=

,

Taking the logarithm of both sides, expanding in a Taylor series, and using the Kirkwood superposition approximation as before, gives fct) = e4cr) +A,(r) + ..‘, (2.24) where

F =P d&g@, )v(R,I2 and

+P’

s

Xu(R,

dR,dR,g(R,)g(R,)h,(IR, MR,

-R,l) (2.15)

1,

where h, (R ) = g, (R) - 1, and we have taken the thermodynamic limit (N-t ~0, V-+ 00, N/V=I)). A formal expression for the transition energy distribution function itself is (2.16) p(E) = ME - E(R, ,..., R,))), which, with the integral representation of the delta function and Eq. (2.1)) can be written

m

(2.17)

.

(2.18)

>)

A simple approximation to this generating function involves making a cumulant expansion and truncating at second order,9*29which with Eqs. (2.4) and (2.11) gives ir(A

-E”)

=~ldR,dR,g(R,)g(R,)h,(lR, -Rzl)

-

iu(R,)t

x(e

_

l)(e-i”‘RZ”-

1).

(2.26)

One can show that A, (t) ap”, and so this is an “exponential density expansion.” For low densities one need keep only the first term. At high densities one hopes that truncating at second order will be a good approximation. This will be the case if for times when A, (t) becomes appreciable compared to A, (t), f(t) is already approximately zero. This, in fact, turns out to be true in several cases of interest (see below, for example). It is also interesting to note that if Eqs. (2.24)(2.26) are expanded for short times, one recovers the cumulant result of Eq. (2.19). Thus, the Gaussian expression for the line shape, arising from the second-order cumulant truncation, and the asymmetric line shape coming from Fourier transformation of Eq. (2.24) (truncated after second order), will have the same first and second moments.2’3 Next let us consider the case of two solute molecules, located at positions rl and r2, with (r, - r2 1=r. The transition energy of thejth solute is now given by

dteicE-E”‘ff(t),

where

f(t) z-e -

[(7AjBjBk) - (T‘I)‘]

A2(t)E+

- f*F/Z

(2.19)

Substituting this expression back into Eq. (2.17) and integrating gives a Gaussian expression for the transition energy distribution,

1 .

Ej (RI ,***y %r,,r,)

(2.20)

Since we have assumed that the transition dipole moment is independent of solvent configuration, and that the absorption spectrum is completely dominated by inhomogeneous broadening, this transition energy distribution is also the normalized absorption line shape. A more general result for this transition energy distribution that can produce asymmetric line shapes was actually given some time ago by Messing et a1.2s3Instead of performing a cumulant expansion, one can obtain their results by writing the generating function in a different way,

=E’+u(r)

+ i v(lR, -rjl), i=l (2.27)

where U(I) is the direct perturbation of the solute transition energy due to the presence of another solute a distance r away. We note that in writing the above we are not assuming that excitonic delocalization effects between the two solutes can be neglected. We are simply calculating the excitation energy of one of the solutes, which, in a two-site tight-binding Hamiltonian appropriate for describing delocalization, would become one of the site energies. If there were significant off-diagonal interactions, for example, of the transition dipole-transition dipole type, then diagonalizing the twosite Hamiltonian would produce delocalized eigenstates.

J. Chem. Phys., Vol. 97, No. 1,i July 1992

H. M. Sevian and J. L. Skinner: Chromophores in liquids or glasses

12

The average of some quantity that depends on solvent positions is now ,..., R,))

(JR,

=

s

dR, ***dR,P(R T

d&M,, 1

2

xg(R,, Iv@,, 11,

,,..., RNlr,,r2)

(2.28) xf(R ,,...&v), where P( R , ,...,RN Ir, ,r2 ) is the conditional probability distribution for the solvent coordinates given that solute 1 is at r, and solute 2 is at r2, and which is again normalized to 1. We can now calculate the average transition energy of solute 1 given that solute 2 is a distance raway. Defining this quantity to be A ( r), which is, of course, identical to the analogous quantity for solute 2, we have -409 = (4 0% ,...Jb;r,

Xv(R,,MR,, 1-p2

(2.37)

where P(R,,R21r,,r2)

=

s

dR,***R,P(R

,,..., R,Ir,,r,). (2.38)

In order to make further analytic progress we next make the simple four-body Kirkwood-like superposition approximation,‘5’56 RR, 3, Ir, ,r2 I=: V -‘gW,,

,r2 1)

MR,,

MR2,

Xg(&,ks(IR, (2.29)

=E”+W+(~u(Ri,)),

where R, = IRi - rj I. Using the definitions of the average and the reduced conditional probability, P(R,Ir,,r,)

=

dR,***dR,P(R,

s

F(r)

=P

(2.30)

J

dR,g(R,,

dR,P(R,

JYR, lr, x2 1~ v - ID,,

Ir,,r,)v(R,,

1.

(2.32)

kUG2 1,

dR,g(R,,

MRn

)v(R,, 1. (2.33)

dR,g(R,,

)v(R,,

=EO+p

s

dR,g(R,)v(R,)

F( co) =p

dR,g(R,

s +p2

=A,

(2.35)

,r2 1 - A(r))2)

-R,l)

(2.41)

)v(R,

)’

dR,dR,g(R,)g(R,)h,(IR,

-R,l)

s

Xv(R,

which was introduced earlier for the single solute problem. This, of course, is what is expected on physical grounds. We can also calculate the mean-square fluctuations defined by F(r) = ((E, (Ii , ,...,b;r,

(2.40)

which after a change of origin gives, as it must,

(2.34)

),

which after changing the origin to the position of solute 1 becomes A(a)

1.

)v(R,,

+p2I dR,dR,g(R,,)g(R,,)h,(IR, XG,, )vW,, 1,

As r--r co, u(r) +O, andg(R,,) -+ 1 for all solvent positions R, such that v( R , , ) is nonzero. Therefore, s

I I@,,

R,

MR,, I2 F(m) =P sdR,g(R,,

then yields

A( 00) = E” +p

)2

v(R,,

Making the Kirkwood superposition approximation

s

)v(R,,

MR,, M&, k(&z 1 +P2 sdR,dR,g(R,, X&C IR, -

(2.31)

A(r) = E” + u(r) +p

MR,z

AS r+ 00, g( R ,2 ) + 1 for all solvent positions such that ) is nonzero, and g( R,, ) -+ 1 for all solvent positions such that v(R,, ) is nonzero. Therefore,

,. A(r) = E” + u(r) + N

(2.39)

R,I),

which gives

,..., &lr,,r,),

gives

-

1

)v(R2)

(2.42)

= F.

Another important quantity is the correlation function C(r), C(r) = ((E, (R, ,...,&;r, xW2

CR,

,...,

RN;rl,r2

,r2 1 - A(r)) (2.43)

1 -A(r))),

which will show how the correlation of the fluctuations of the transition energy of solute 1 and the transition energy of solute 2 decays as the distance between them increases. A calculation similar to that above gives

(2.36)

=((~v(R~,)>‘)-(~v(R~,))~

C(r) =P

d&g%,

k(R,2 Iv@,,

)uW,, 1

MR,, k(R2, ML) +P2 sdR,dR,g(R,, s

= N

s

dR,P(R,

Ir, ,r2 )v(R,,

)’

+NW- 1) ~R,~R2~(R,,R21r,,r2)

Xb(IR, -

J. Chem. Phys., Vol. 97, No. 1,l July 1992

R,

IbUG,

)v(R,,

).

(2.44)

H. M. Sevian and J. L. Skinner: Chromophores in liquids or glasses

Note that as r+ CO,for all solvent positions either u(R,, ) or u( R,, ) -0 in the first term, and in the second term either v(R,,),v(R,,),orh,(]R, -R,])-tO.Therefore,itisclear that C( CO) = 0, as must be true on physical grounds. To find the joint probability distribution, p( E, ,E2;r), which is the probability of finding solute 1 with transition energy E, and solute 2 with transition energy E2, given that the solutes are a distance r apart, we write PW, ,E2 ;r) = (SE, xS(E,

- El (R, ,...Jb;r, - E2

p(E, ,E,;r)

g(f, ,t, 1 = (e

g(t, ,t, 1 = e-

dt2eitE’ - E”-uu(r)‘tl

- i[x,u(R,,)r, + Z,u(&)r,]

(2.46)

(2.47)

>.

i(rl

+ c*)[A(r)

-E’-

U(I)]

-

(t:

+ t;)F(r)/l

-

t,t#i-)

(2.48) Performing the t, and t, integrations then gives the bivariate Gaussian distribution

Inserting integral representations of the delta functions gives

- 2C(r)(E,

fm J-cc

Performing a cumulant expansion of the above and truncating at second order gives [see the definitions of Eqs. (2.29), (2.36), and (2.43)]

,r2 1)

- A(r))2

dt,

Xei(E~-Eo--(~))f~g(f,,f2),

(R, ,...,%r, ,r, 1)). (2.45)

F(r)(E,

= 1 fm 47~‘J-co

13

- A(r))(E, 2(F(r)2

-A(r))

+ F(r)(E2

- A(r)J2

- C(r)‘)

I.

(2.49)

I

We could also have performed an exponential density expansion of Eq. (2.47), leading to an asymmetric joint probability distribution function. However, as we will see in Sec. III, at least for the single solute distribution function the result from this procedure does not differ greatly from the Gaussian approximation that follows from the cumulant expansion. More importantly, for the exponential density expansion of g( 1, ,t, ) the numerical calculations required to evaluate the spatial integrals for each value of 1, and t2 would be prohibitively time consuming. In the limit of large solute separation (r--t a ), from our previous discussion we see that p(E, ,G;m

(E, --A12+ (E, -Al2

1=

2F

=p(E,

1

1

) = d2n-(F(r)

ce p(E, ;r)=s-cc dE,pW, ,E2;r),

[E2 --A(r)

p(E,;r) =

l

A quantitative dimensionless measure of the correlation between the two solute transition energy distributions as a function of solute separation is given by

-w,

-‘-l(r)Y/2F(r)

(2.52)

@zme

Clearlyp(E,;co) =p(E, ). Finally, we can define the conditional probability of finding a transition energy E2 on solute 2 given that solute 1 has transition energy E, (and is a distance r away) by p(E,;rlE,

) = pziELr)

(2.53)

. 1;

From Eqs. (2.49) and (2.52) we see that it is a Gaussian in E2,

- (C(r)/F(r))(E, 2(F(r)

- C(r)‘/F(r))

(2.51)

which from the above gives

(2.50)

)p(E,),

as defined in Eq. (2.20), and so the joint distribution factors into a product of singlet distributions.

p(&;rlE,

More generally, we can define the transition energy distribution for a single solute, but in the presence of another solute a distance r away, by

- A(r))12

- C(r)2/F(r))

III. SIMPLE LENNARD-JONES

I.

(2.54)

MODELS

Note that from the Schwartz inequality IF(r) I < 1. Also, as T(r) -0, p(E,;rlE, ) does not depend on E,, and the two solute distributions are completely uncorrelated. In the limit

As an example of the formalism derived above, in this section we first consider the specific case of two LennardJones solutes in an equilibrium Lennard-Jones liquid solvent. In principle, we must specify five interaction potentials: solute-solvent, solvent-solvent, solute-solute, solute*-solvent, and solute*-solute (solute* denotes the ex-

r-r COwe find, of course,that r( 00) = 0.

cited stateof the solute). The first threedeterminethe equi-

r(r)

= C(r)/F(r).

(2.55)

J. Chem. Phys., Vol. 97, No. iI1 July 1992

14

H. M. Sevian and J. L. Skinner: Chromophores in liquids or glasses

librium configurations, while the last two lead to the transition energy perturbations. In what follows, however, we will simplify this situation considerably. While this simplification is not particularly realistic, it does facilitate a quantitative comparison of our theory with numerically exact Monte Carlo simulation. Thus, we will assume that the first three interaction potentials are all identical, and are given by

u,(R) =4t[(g2-(g].

(3.1)

Therefore, ground state solute and solvent molecules all interact the same way. This also means that g,(R) = g(R). We will also take the last two interaction potentials to be identical, and to have a Lennard-Jones form but with a different (Tparameter (other models are, of course, possible” ) , (3.2) Thus, the solute transition energy perturbation due to a solvent or another solute molecule is u(R)=u(R)

=/I,(R)-uv,(R).

(3.3)

g(R) could be obtained from integral equation theories or from computer simulation. Since we are also interested in making a direct comparison of our theory with simulation, in this study we take the latter route. The comparison is particularly straightforward, since we can simply simulate a one-component Lennard-Jones fluid, and any pair of molecules can be taken to be solutes. We consider a canonical ensemble of 8 10 particles in a three-dimensional cubic box with periodic boundaries, and we choose thermodynamic parameters so that the solvent is a liquid near the triple point? po3 = 0.81408 and /CT/E = 0.786 666 7. Potential interactions are truncated at half the box length ( =: 50). The average transition energy for a single solute, A, was calculated from Eq. (2. lo), setting c1 /a = 1.1, E" = 0, and using u(R) from Eq. (3.3) and g(R) from the simulation. The one-dimensional integral was performed numerically using Gaussian quadrature, but u(R) was truncated at R /a = 5 to agree with the potential used in the simulation. We find that A = 6.16. (In this and what follows, all energies are reported in units of E.) This can be compared with a result obtained completely from simulation by simply averaging the solute transition energy from Eq. (2.1) over many configurations. In our simulation we used 1252 uncorrelated configurations, and could also average over all 8 10 particles as solutes. The result is A = 6.14. These two results should, in principle, agree exactly. In practice, however, g( R ) obtained from the simulation was first binned and then smoothed. In addition, we have not worried about corrections on the order of l/N in the definitions of the solvent density and g( R ) . These effects account for this small difference. The value of A z 6 indicates a blue shift from the gasphase absorption. This is because the perturbation u(R) is positive for the first solvation shell, which makes a dominant contribution to the transition energy. We can also calculate the mean-square fluctuations of the transition energy for a single solute, F, from Eq. (2.15).

The second term reduces to a three-dimensional integral over R, , R, and one angle, which was performed by Gaussian quadrature. For this calculation, as above we truncated the potential at R = 5u, and also set g(R) = 1 for R>5a. Our result is F = 27.8 (in units of 2). This can also be compared with the exact result from simulation, which is F = 26.7. This difference is presumably due to the Kirkwood approximation used in deriving Eq. (2.15). In Fig. 1 is shown the transition energy distribution, or absorption spectrum, of a single solute in the solvent. The solid line is the Gaussian distribution calculated from Eq. (2.20), which is a result of the cumulant expansion. Also shown is the exponential density expansion results of Eq. (2.17) with Eqs. (2.24)-(2.26). The integrals in Eqs. (2.25) and (2.26) were performed by Gaussian quadrature for each value oft. The data points are from the Monte Carlo simulation, and the error bars in this and all subsequent figures are two standard deviations. As discussed in Sec. II, both approximations have the same first and second moments, which we just showed are reasonably accurate. Therefore, it is no surprise that both distributions describe the exact data pretty well. However, the simulation results give an asymmetric line shape, and so we see that the exponential density expansion provides a more accurate description. The large inhomogeneous width, compared to the shift, corresponds to large density fluctuations-high density configurations produce a blue shift while low density configurations lead to a red shift. Now moving on to the case of two solutes, the average transition energy of one solute as a function of its separation from another solute, A ( r), is shown in Fig. 2. The solid line is obtained from Eq. (2.33). This can be compared with exact results from the simulation by binning all pairs of molecules with interparticle spacing r, and averaging the transition energies of Eq. (2.27). The results are the data points in Fig. 2. The simulation data and the theory are not in exact corre-

0.07

0.0

P(E)

0.02

C

FIG. 1.p(E) vs E for a single solute. (In this figure and the ones that follow, all energies are in units of 6.) The circles with error bars are the simulation data, the solid line is the Gaussian approximation, and the dashed line is the exponential density expansion result.

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H. M. Sevian and J. L. Skinner: Chromophores in liquids or glasses

15

6.5

-4(r)

6

F(r)

5.5

:

26

24

/,

08 00 Pp 0

1

1 2

3

, 4

0

5

1

2

,

,

]

3

4

5

r/a

r/u

FIG. 2. A(r) vs r/u. The circles are from the simulation and the solid line is from theory.

spondence because of the superposition approximation, but the agreement is nonetheless quite good. Both results show characteristic oscillations due to shells of solvent molecules between the solutes. On the scale of the overall distribution width, however, these oscillations are quite small. Also, note that at the largest solute separations of r = 5a, the values of the theory and simulation are in good agreement with the values of A discussed above. This, of course, is to be expected, since for large distances between solutes the average transition energy should not depend on the other solute’s presence. Next we can calculate the mean-square fluctuations as a function of solute separation, F(r), from Eq. (2.40). The evaluation of the second term is highly computationally intensive. The integrals reduce only to five dimensions, and in making the computer program as efficient as possible, we find it advantageous to use elliptic coordinates. We find reasonable convergence using 90-point Gaussian quadrature. We are, however, just barely within the confines of our computational power. For example, in order to compute F( r) for one value of r, a typical run time on a DecStation 5000 is 24 hours. The results are shown in Fig. 3. The data points come from the Monte Carlo simulation. As seen, the agreement is not quantitative, but the difference in the results is at most a few percent. Both again show characteristic oscillations. At the largest values of r both theory and simulation approach the values of F discussed above. In Fig. 4 is shown the correlation function C(r). The solid circles are from Eq. (2.44), and the open circles are from the simulation. Both theory and simulation show oscillations that have decayed nearly to 0 at r/a = 5. Especially interesting is a negative correlation of the solute transition energies at r = 2a, presumably corresponding to configurations with exactly one solvent molecule between the solutes. In some configurations this solvent molecule is closer to one solute than the other, thereby raising one solute’s transition energy while lowering the other’s, which would account for this negative correlation. This interesting structure in C(r)

FIG. 3. F(r) vs r/a. The open circles are from the simulation and the solid circles (with straight line segments connecting the calculated points) are from theory.

is clearly not well represented by the phenomenological exponentially decaying model introduced by Root and Skinner.52 Finally, we can calculate the conditional probability, p( E2 ;rl E, ), that for two solutes a distance r apart, if one solute has transition energy E, then the other one has E, . In Fig. 5 the solid line is from Eq. (2.54), with E, = 11.94 and r/o = 1.1. The corresponding data points are from the simulation. The agreement is not too bad. Thus, for this simple model the theoretical calculations are quite reliable, and there do exist interesting correlations between the transition energies of two nearby solutes. It is important to realize, however, that the effects are quite small. The magnitude of the dimensionless quantity

C(p)

FIG. 4. C(r) vs r/o: The open circles are from the simulation and the solid circles are from theory.

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16

H. M. Sevian and J. L. Skinner: Chromophores in liquids or glasses

0.1

0.04

PC&; r I Ed P(E)

0.05

0.02

0

-

0

-150

- 100

-125

-75

E FIG. S.p(E,;r[E, ) vs E2 at r/o = 1.1 and for E, = 11.94.Theopen circles are from the simulation and the solid line is from theory.

I(r) = C( r)/F( r), discussed earlier, always remains 0.05 or smaller, showing that to a reasonable approximation the transition energy distributions for the two solutes are uncorrelated, even at rzo. The reason the effects are small has to do with the shortrange nature of the solute-solvent potential. Therefore, it is impossible for the two solutes to interact simultaneously with a large number of solvent molecules, which is what is needed for a large effect. For longer range (such as dipolar) interactions, on the other hand, one expects these correlations to be much more significant. To test this hypothesis, one could generalize the theory to include dipolar interactions, and run a dipolar simulation to extract the spatial distribution functions. However, the extra orientational degrees of freedom makes this approach nontrivial. A simpler approach is to again consider Lennard-Jones solutes in a Lennard-Jones solvent, but now where the solute in the excited state is singly ionized. In the excited state there will be a long-range charge-induced-dipole solute*-solvent interaction of the form59 - e2a/2R 4, where a is the polarizability of a solvent molecule. If we assume that the solute*solvent interaction also has a Lennard-Jones contribution u. (r), then we see that v(R)

=

- e2a/2R 4 =

- y~(u/R)~,

FIG. 6. p(E) vs E for the charge-induced-dipole model.

In Figs. 6-9 are shown the Gaussian approximation for and our results for A(r), F(r), and C(r) for this charge-induced-dipole model. The spectrum shows a large red shift, due to the large negative perturbation u(R). A(r) and F(r) are qualitatively similar to the pure Lennard-Jones case, except the effect of the second solute is more pronounced. Unlike the pure Lennard-Jones case, C(r) is always positive, although the steep drop at r = 2a presumably is due to the effect described earlier. We see that the decay of C(r) is more or less exponential, as suggested by Root and Skinner.52 In Fig. 10 we have plotted I’(r) as a function of r. Since the magnitude of I(r) is of order 1 for small r, it follows that the inhomogeneous transition energy distributions of the two solutes are highly correlated.

p(E),

IV. CONCLUDING REMARKS We have argued that, in principle, the transition energy distributions of two nearby chromophores in a liquid or

(3.4)

where y = e2a/2eo 4. Taking the values for argon of a = 1.66X 1O-24 cm3,6o dkB = 119 K5* and 0 = 3.4 &58 gives y = 8.7172, which is what we used in our calculation. Thus the only difference between this calculation and the previous one is the above form of v( R ) = u (R ) . Because the potential is longer range, a simulation with periodic boundaries would require a much larger box, making the simulation much more time consuming. Since we have already compared simulation to theory in the previous section, for this problem we present only theory results. As the ground state of the solutes (and solvents) is unaltered, we may still use the g(R) from before.

A(r) -111

1

2

3

4

5

r/u

FIG. 7. A(r) vs r/o for the charge-induced-dipole model.

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H. M. Sevian and J. L. Skinner: Chromophores in liquids or glasses

a0 F(r)

r(r)

70

0.25

8

60”^

I

‘L

I

3

4

0 5

6

I

I

1

2

r/o

4

5

6

f-In

’ FIG. 10. r(r)

FIG. 8. F(r) vs r/u for the charge-induced-dipole model.

glassy solvent are correlated, and we have presented a molecular theory of these correlations. For the pure LennardJones model our theory is in good agreement with computer simulation. For the longer range charge-induced-dipole model our theoretical results show significant correlation. Most of the relevant experimental results are for disordered crystals. In this case the inhomogeneous broadening arises from chemical or physical defects in the lattice, and the transition energy perturbation due to a single defect is typically dipolar,’ which is even longer range than our charge-induced-dipole model. Thus, we suspect that in these systems this correlation will also be significant. In the near future we hope to study more realistic models of liquid and glassy solvents, and also to consider the case of disordered crystals. As discussed in Sec. I, transition energy correlations manifest themselves indirectly in coherent transport and de-

I

I

I

3

I

,

vs r/u for the charge-induced-dipole model.

phasing experiments. Thus, one application of these correlated distributions would use them as input for quantummechanical theories of interacting chromophores (with, for example, transition dipole-transition dipole interactions) that can describe delocalization and/or dephasing. One possibility for measuring these correlations more directly involves a time-resolved fluorescence-line-narrowing experiment on a concentrated chromophore solution at low temperatures but in the incoherent transport regime.6’ Suppose the chromophores were excited with pulsed narrowband radiation. At the shortest times the fluorescence would be direct, coming from the molecules originally excited. As time evolves exciton transport would occur, and the fluorescence would emanate from near neighbor, and then further removed chromophores. From time and frequency resolving this fluorescence, one might be able to extract the conditional probability as a function of distance that if one chromophore has a transition energy E its neighbor has E’. This would enable a direct comparison of a theory such as the one described herein with experiment.

ACKNOWLEDGMENTS The authors are grateful to Professor Bruce Berne for suggesting the charge-induced-dipole model, and to Professor Bob Silbey for an illuminating remark. We also thank the National Science Foundation for support from Grant No. CHE90-96272.

I

0' 0

I

2

3

4

5

r/u

FIG. 9. C(r) vs r/u for the charge-induced-dipole model,

6

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H. M. Sevian and J. L. Skinner: Chromophores in liquids or glasses

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