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A simple power-law approximation for the solvation time correlation function
riitz%JLAR LIQUID6 ELSEVIER
Journal of Molecular Liquids 73,74 (1997) 419432
A SIMPLE POWER-LAW APPROXIMATION FOR THE SOLVATION TIME CORRELATION FUNCTION Fernando 0. Raineri, Harold L. Friedman, and Baw-Ching Perng Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794-3400, USA Abstract In previous work we developed a new class of molecular theories of solvation, the “surrogate Hamiltonian” (SH) th eories, to elucidate the structural, energetic, and dynamical aspects of the solvation process relevant to ultrafast time-domain spectroscopy and charge transfer reactions in solution. In these theories we represent both the solute and the solvent molecules by interaction site models of the sort already used in many computer simulation studies of solutions. Of special interest to the characterization of the nonequilibrium solvation process is the solvation time correlation function 2(t), that can be obtained in time-dependent fluorescence Stokes shift experiments. The experimental and theoretical studies of 2(t) in polar solvents have revealed an important dependence of the solvation time correlation function on the nature of the solute, especially with respect to the multipolar order of the charge jump. Here we investigate a simplified version of our theory of solvation dynamics, which is based on the straightforward application of a method previously developed by our group! the reference frequency modulation approximation (RFMA). In the earlier paper we showed that when applied to the solvation tcf, the RFMA leads to a simple interpretation of the power law equation for 2(t) found empirically by Maroncelli and coworkers. Here we propose a new power-law approximation to the solvation time correlation function that retains the dependence of 2(t) on the features of the solute. More specifically, the new application of t&e RFMA discussed here enables us to exoress the solvation tcf Zv(tj of solute Y in terms of the solvation tcf 2R(t) of another solute R in the same solvent. The solutes R and Y may differ in size, shape, and charge distribution. The consistency of 2(t) calculated with more accurate methods and the simpler RFMA theory is quite satisfactory. I
1
\
I
I. Introduction 111the last few years a vast experimental’ and theoretical2 effort has been directed to uncover the molecular details of the dynamical solvent response triggered by a “jump” in the charge distribution of a solute molecule. Typically the charge jump (by photoexcitation) carries the solute from its ground (or precursor P) state to its first electronic excited (or successor S) state. The solvent, which is initially equilibrated under the field of the P-state solute, is abruptly destabilized by the field of the newly created S-state solute charge distribution. In suitable cases the solute fluoresces, and the ensuing process of nonequilibrium solvation is monitored by the evolution of the solvation time correlation 0167-7322/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PI1 SO167-7322(97)00085-8
420
function
(tcf)1>2
4t) - u(m)
2(t)= u(0)
- u(m)
’
(1.1)
where v(t) is the frequency of the solute fluorescence at time t after the charge jump. This description corresponds to the time-dependent fluorescence Stokes shift experiments. * Establishing the detailed molecular mechanism leading to 2(t) is crucially important for understanding the role of nonequilibrium solvation in the rate constants of electron and charge transfer reactions in solution.3 It has been found 1,2 that the solvation response in polar solvents is intimately related to the dielectric properties of the solvent, an area in which Josef Barthel’s contributions are essential. A useful discussion of the wide variety of theoretical methods that have been applied to analyze the solvent dynamics responsible for the features of 2(t) has recently appeared.2 Because of the amount of detailed information that they provide, molecular dynamics computer simulation studies of solvation dynamics have,been important aids in the development of additional approximate theories. More recently, however, the simulation effort has been directed toward the study of systems using realistic representations of the solute and solvent molecules4y5 Among the more recent approximate treatments are the instantaneous normal mode analysis6 (which is also capable of detailed representation of solute and solvent molecules); semianalytical methods that focus on the solvent response and regard the solute as the source of an external potential7 and a novel continuum dielectric formulation of the solvent that satisfies the dielectric boundary conditions for arbitrary shapes of the solute.* In the last few years our group has developed a molecular theory of 2(t) (and other equilibrium and nonequilibrium solvation processes) that applies to model solutions in which both the solute and solvent molecules are represented by rigid non-polarizable Interaction Site Models (ISM’S).~-‘~ We recall that the ISM are models in which the potential energy of interaction between two molecules is a sum of pairwise additive site-site terms including, in addition to Lennard-Jones or similar short range interactions,Coulombic interactions between partial charges located at the molecular sites. The theory is based on a simple renormalized linear response development that incorporates non-linear aspects of equilibrium solvation. The renormalization is carried over the solute-solvent potential energy of interaction; the outcome is a statistical mechanical theory based on a surrogate (renormalized) Hamiltonian. We refer to this approach as the surrogate Hamiltonian (SH) theory of solvation. In passing over less detailed models, the SH theory recognizes that typical solutes and solvents of chemical interest are comprised of polyatomic molecules that cannot be adequately represented by simple models. Like the computer simulation studies, the SH theory consistently accounts for the “size”, the “shape”, and the charge distribution of the actual molecules. There are several implementations of the SH theory that have been discussed in our previous reportsg-l5 In the case of solvation dynamics in polar solvents, the best results for the solvation tcf 2(t) are obtained with one of the versions of the SH theory, the Renormalized Dielectric Theory (SH-RDT). There has also been some effort in obtaining friendlier expressions for 2(t) that would be convenient to implement. For example. we reported” a simple new method for estimating time correlation functions of collective dynamical variables, the Reference Fre-
421
quency Modulation Approximation (RFMA). It is based on the convolutionless generalized Langevin equation reported by Tokuyama and Mori.17 By applying the RFMA approximation to a simplified version of the expression for the solvation tcf given by the SH-RDT. we recovered the “power-law” relation 2(t)
=
@l(V
,
(1.2)
found empirically by Maroncelli and coworkers.18~1g Both the pure-solvent single-dipole tcf @r(t) = (p(t) . /.J)/(P. P) and the exponent o on the’right hand side of this equation depend only on the solvent. Unfortunately, the approximations that lead to Eq. (1.2) are so drastic that any dependence of 2(t) on the solute features is lost. In contrast, experimental resultsa as well as detailed studies by computer simulation4 and various approximate theories al2 have revealed an important dependence of the solvation tcf on the nature of the solute, especially with respect to the multipolar order of the charge jump. In this paper we present a new and straightforward application of the RFMA to derive an approximate power-law form for the solvation time correlation function that depends realistically on the nature of the solute molecule. More specifically, we investigate the possibility of estimating the solvation tcf 2y(t) for the P -+ 5’transition of a solute Y in terms of the known solvation tcf 2R(t) for the f’ + S’ transition of a reference solute R, both in the same solvent. Here P and S (P’ and S’) are. respectively, the ground and excited states of the solute Y (R). The new approximate power law for 2(t) is quite consistent with the results predicted by the more complete SH-RDT theory. II. SH-RDT
Theory of the Solvation Time Correlation Function
The photoexcitation of the solute in the time-dependent fluorescent Stokes shift experiment carries the molecule from the precursor (P) or ground electronic state to the successor (S) or excited electronic state. The solvation time correlation function 2(t) [cf. Eq (1. l)] that monitors the subsequent, solvent relaxation has the statistical mechanical formulationg-ll (ti; t) - (ti; 3(j) (2.1) 2(t) = (ti; 0) - $4; cc) ’ where G is the solvation vertical energy gap dynamical variable: It represents, for a given configuration of the system, the difference in solvation energy between the two electronic states of the solute. We consider systems in which lf has the Coulombic form
where AQx = Qf - Qf; is the difference between the partial charges Qf and Qf at the interaction site X of the solute in states P and S respectively, and qj is the partial charge on interaction site j of a solvent molecule. Furthermore, rx+j E Ir~,~jl = lxaj - XXI is the distance between the solute site X (located at xx relative to the laboratory reference frame) and the site j of the solvent molecule a (located at xaJ). In Eq. (X.1) (U; t) is t,he
422
expectation value of the vertical energy gap in a non-equilibrium ensemble at time t after the mstantaneous P + S transition caused by photoexcitation at t = 0. In the SH t,heory of solvation dynamics the system is described by a time-dependent surrogate (subscript C) Hamiltonian Hz(t) of the same form as the bask Hamiltonian 3.1(t).g-12 The surrogate Hamiltonian BE(~), however, is formulated in terms of renormalized solute-solvent interactions &z(t). so that a simple linear response treatment [using the Hamiltonian of the pure solvent Hw as the reference Hamiltonian and the solute-solvent renormalized potential energy of interaction 6z(t) as the perturbation], is sufficient to describe the equilibrium solvent structure around the solute in the electronic states P (at t = 0) and S (at t = 30).~-l~ One of the implementations of the SH theory of solvation is the Renormalized Dielectric theory (SH-RDT). In the SH-RDT theory the renormalized solute-solvent potential energy of interaction has the form
where B(t) is the unit step function
and (2.4)
is the surrogate analog of the vertical energy gap c of the basic description of the system. Both Eqs. (2.3) and (2.4) are expressed in terms of the renormalized solute solvent interactions !&$,, (where the superscript D refers to the solute electronic state, P or S). These potential energies are of the formg-15
D=P,S.
(2.5)
where (2.6) is the Fourier component of the solvent polarization charge density at a point r relative D k) is the Fourier transform of a renormalized electrostatic to the solute site A, while ‘P~,~\( potential state.
cpg A(r) associated
The SH-RDT
with the interaction
theory gives an expression
site X of the solute in the D-electronic
for (pgx( k ) in terms of the site-site
solute-
state solvent RISM direct correlation functions 21 when the solute is in the D-electronic and certain response functions of the pure solvent. We refer the reader to Refs. 9-15 for more details. The renormalized linear response formulation of the SH-RDT theory gives the following estimate of the solvation tcf 2(t) (see Refs. 9-12 for a detailed derivation)
(2.7)
423
where, from Eqs. (2.4) and (2.5), the SH-RDT
renormalized
vertical
energy gap is given
(2.8) where Acpc,x(k)
= &,Jk)
(2.9)
- &Jk)
is the change in the Fourier transform of the renormalized electrostatic potential due to the solute charge redistribution in the P + S transition. It is important to noticegel that the tcf &or(t) in Eq (2.7) is governed by the Hamiltonian H,,, of the homogeneous solvent, i.e. both the equilibrium average operation (, .)w and the evolution of E,(t) are determined by Hw. From our earlier work,‘-15 it may be shown that Eq. (2.7) can be rearranged to the form &Yr(t)
=
dk@,(k,t) W,,,(k) , JO” 0
@,(k,t) =
(&(k t) l$-k) )w (i%(k),4(-k) )w
(2.10) (2.11)
is the tcf of the Fourier component
of the fluctuating solvent polarization charge density &(r). We recall that @/‘(k, t) is the tcf that determines the frequency- and wavevector-dependent longitudinal dielectric, function cL(k.u) of the pure solvent. 14,22 Furthermore, the weight function W,u,T(k) may be interpreted Here
=
C(k) I’m(k)
as a normalized
C(k) = 1 -
-!--. ‘L(k)
/
irn C(k)ram(k) dk
distribution = -47rP k2
of solute-solvent
+
(Mk)
&A-k)
(2.13)
interaction
)w }
M
coupling. (2.14)
is the wave-vector dependent Pekar factor.15922123 Here /3 = (kgT)-’ is the inverse of the temperature T in energy units (kB is the Boltzmann constant), while V is the volume of the system. The symbol {. . .}m indicates the thermodynamic limit of the quantity inside the braces. Notice that we take too = 1 because in this study we ignore the electronic polarizability of the solvent. Finally, (2.15)
is the SH-RDT
solute-solvent
coupling
factor.15 It is defined in terms oP,14~15 (2.16)
424
the Fourier coefficient of the change in the effective partial charge density at solute site X, while wxx~(lc) is the Fourier transform of the solute intramolecular site-site correlation function2* According to Eq. (2. lo), it is the dielectric dynamics of the homogeneous solvent, as expressed in 9,(/c, t), that is the source of the time dependence of the SH-RDT estimate of 2(t). In this approximation the effect of the solute-solvent interactions is carried by the static weight function W,,,(k). This “factorization” (to a function of the homogeneous solvent dynamics times a function of the static solute-solvent structure) is an Important approximation and is a distinctive feature of the SH theory of solvation. The renormalized character of WRDT(k) allows us to bypass the two-time many-point correlation functions that would necessarily appear in a more complete dynamical theory that addresses the inhomogeneity of the solvent in the field of the solute particle. We distinguish the SH-RDT from alternative versions of the surrogate Hamiltonian theory of solvation. 12,15 In particular it should be distinguished from the Renormalized Site Density theory (SH-RST), in which the primary dynamical variables are the microscopic solvent interaction-site densities. 12,15,24 III. Generalized Langevin Equations. We discuss now two approximate methods for calculating time correlation functions of collective dynamical variables. In Sets. IV and V they are applied to calculate the surrogate estimate 2,,,(t) of the solvation tcf. A. GLEr and the Reference Memory finction Approximation. The reference memory function approximation22*25y26 (RMFA) is a method for calculating, using the memory function equation27,28
d@y(t)/dt the normalized
time correlation
= -
Jot
dt’ Ky(t - t’) @jy(t’) ,
(3.1)
function
of the desired primary dynamical variable Y. In this section the symbol (. ’.) denotes a general equilibrium ensemble average; a particular choice would be the ensemble average Hw of the pure solvent. (...)uI with the Hamiltonian The memory function equation (3.1) is a consequence of the generalized Lsngevin equation (GLE1) derived by Mori. 27128 A distinctive feature of both the GLEr and the memory function equation is the convolution operation of the function of interest with the memory function KY(t). Rather than attempting the calculation of KY(t), the RMFA method relies on the information contained in the tcf
@x(t) =
(x(t) (xx*)
x*)
(3.3)
425 of a related reference dynamical variable X. We assume that @x(t) is already known, say from experiment, computer simulation, or by an independent theoretical approximation. The RMFA is based on the assumption that the normalized first memory functions of the primary and reference dynamical variables are equal at all times:22T25,26 Ky(t)lKy
= Kx(t)lxx.
(3.4)
Ky
= Ky(t=O)
= (Ii’(2)/(jY12),
(3.5a)
Kx
F Kx(t=O)
= (lji12)/(IX12),
(35b)
Here27,28
are, respectively, the initial values of the first memory functions of the dynamical variables Y and X. With Eq. (3.4) and the memory function equations [cf. Eq. (3.1)] for @y(t) and @)x(t) it is possible to show22,25,26 that the corresponding Fourier-Laplace transforms @Y(W) =
om dteetwt .I
and @X(W) [defined as in Eq. (3.6)] are connected @y(u)
= {Ryx
+ iw@x(w)[l
where the renormalization factor ‘Ryx Icy of the respective memory functions Ryx
(3.6)
@Y(t) by the equation
- R&j-*
@x(w) I
is defined in terms of the initial Ky(t) and Kx(t): ??
(3.7) values Kx
KyfKx.
and
(3.8)
B. GLE+ and the Reference fieqnency Modulation Approximation. The memory function equation (Eq. (3. l)] of the GLEl is not the only possible effective equation of motion for the primary tcf @y(t). An alternative is the frequency modulation equation” $ln@y(t)
= -qJy(t)
.
that relates t,he normalized tcf @y(t) [Eq (3.2)] to the frequency modulation @y(t). This equation is derived from the convolutionless generalized Langevin
(3.9) function equation
(GLE2) first derived by Tokuyama and Mori.17 In a previous reportl’ we showed that an approximate relation analogous to Eq. (3.4) could be invoked for the frequency modulation functions of the primary dynamical variable 1’ and an appropriate reference dynamical variable X. Thus, in the RFMA we assume that (3.10) @y(t) llii~(O) = tix(t)l?j'x(O)~, at, every t. It is straightforward to show that this assumpbion, together with the frequency modulation equation (3.9) for @y(t) and @x(t), leads to the useful relation @y(f)
= @X(t)-
(3.11)
426
between the primary and reference time correlation functions. The renormalization factor RyX in Eq. (3.11) is also given by Eq. (3.8) because of the relations KY = $y(O) and Kx = TJx(O).“~‘~ IV. SH-RDT
Solvation tcf Z&(t)
with the R.MFA Approximation
To calculate 2,,,(t) according to Eq. (2.10) we need to evaluate the solvent polarization charge density tcf @,(lc, t) and the weight function W,nr(lc). In previous reports25>26 we showed that @,(/c, t) can be evaluated reliably with the RMFA method discussed in Sec. 111.A. We review the method in this section. From Eq. (2.11), it is clear that the primary dynamical variable is Y = /3,(k). For the reference dynamical variable X we choose the total dipole polarization
(4.1)
(PCltis the z-Cartesian component of the dipole moment pa of the solvent molecule a in the laboratory reference frame). In this case @y(w) and @x(w) in Eq. (3.7) correspond, respectively, to @,(lc, w) and (4.2) where (4.3) is the normalized tcf of fin,, that determines the frequency-dependent dielectric constant c,.26 The idea of using of Eq. (4.1) as the reference dynamical variable X is due to Fried and Mukamel.2g For the present choices of X and Y, the renormalization factor RyX [which we now call R,(/c)] becomes 22,25,26 [cf. Eqs. (3.5) and (3.8)]
Rp(k)=
Rick) _ ( I bJk)I2)w/ ( I kc(k)I2)w x;(k) (Id(k)12),/(l~~(k)12)w
(4.4)
The second equality defines, respectively, the kinetic 7$(k) and structural R;(k) parts of the renormalization factor. Their calculation has been discussed at length in Refs. 22, 25, and 26. With these choices of primary and reference dynamical variables, Eq. (3.7) becomes @,(k,w)
= {R,,(k)
+ ~w@M,L(w) [l -
R,Vdl I-’ @M,L&J)
(4.5)
As in most of our previous reports, in the present work we shall use the “sim” implementation of Eq. (4.5), in which the reference tcf GM,L(~) is obtained from computer simulation of the ISM model solvent.
421
V. SH-RDT
Solvation tcf 2,,,(t)
with the RFMA
Approximation
The RFMA result, Eq. (3.1 l), suggests a simple procedure for estimating the solvation tcf 2y(t) for the P + S transition of a solute Y in a given solvent in terms of the (presumed) known solvation tcf 2~(t) for the P’ + S’ transition of a reference solute R in the same solvent. Thus, since the SH-RDT estimate of 2(t) [Eq. (2.7)] is a normalized tcf, we identify the primary tcf @y(t) of Eq. (3.11) with 2y(t) and, correspondingly, the reference tcf @x(t) with ZR(t). In this way Eq. (3.11) becomes zy(t)
(5.1)
= 2R(tp
From the derivation of the RFMA in Section 1ll.B it follows that Eq. (5.1) corresponds to the choice of the SH-RDT vertical energy gaps ($)y and (&)R [cf. Eq. (2.8)], respectively, as the primary and reference dynamical variables Y and X. Therefore the renormalization factor %?,YRin Eq. (5.1) is given by the ratio of the initial values of the memory functions (Kc,)y and (!$)R of 2y(t) and ZR(t), respectively:
R
(Ql
YR=
)Y (K&p)x =
( il~,‘I”)w/(lwu)y ((l~d’)w/(l~p12)w)R
(5.2)
On the other hand, by using the formula9~27~28 WC,,),
=
4d22Y
wdt2)t=o
(5.3)
[and a similar expression for (KC,,),] together with Eq. (2.10) for the SH-RDT of the solvation time correlation functions we calculate
where
Waft
is tl le solute-solvent
similar interpretation value of the memory
coupling
weight function
for the solute
estimate
Y [with a
for kVcR) RDT(k)] while K,(k) = -(d2@,(k,t)/dt2&o is the initial function of the polarization charge density tcf @,(k, t).
VI. Applications In this section we test the ability of the RFMA procedure [Eqs. (5.1) and (5.4)] to reproduce the results for 2,,,(t) calculated with the more familiar RMFA methodg-‘2 [Eqs. (2.10). (2.13), and (4.5)]. With this aim in mind, we examine the dependence of the solvation tcf on the features of the solute charge distribution. The solvents considered are acetonitrile (MeCN) and methanol (MeOH). A. Models and Methods for the Solution Structure. A distinctive feature of the SH theory of solvation is the representation of both the solute and solvent, molecules by interaction site models. The site-site interaction potential energies used in this work have the 12-6-l form uij(?.) = 4Eij [ (Ly”_ (y-1 + !!!$ (6.1)
428 for sites i and j on different molecules. ISM models for acetonitrile an,rsrethanol, are taken from the literature. -
whether solvent-solvent or solute-solvent. The each with a united atom for the methyl group,
To calculate 2,,,(t) with either RMFA or RFMA we need a host of equilibrium solute-solvent and solvent-solvent structure functions. As in Refs. 9-15, we calculate the wavevector-dependent Pekar factor C(k) and the solute-solvent coupling function &r(k) with the RISM-HNC integral equation method (XRISM) of Rossky et a1.32 These determine the coupling weight function W RD~(k) [cf. Eq. (2.13)]. We also apply the XRISM Integral equation method to calculate the renormalization factors R,(k) and RyR. For detailed expositions of our calculations we refer the reader to our earlier reports.g-15,25)26 We consider the solvation dynamics of a family of four “benzene-like” model solutes in the solvents MeCN and MeOH. These are the solute models already studied by Kumar and Maroncelli by molecular dynamics computer simulations;4 a study of a related set of solute models was reported elsewhere.12 The solutes have twelve interaction sites arranged in the same geometry as the atomic sites of a benzene molecule (cf. Figure 1). For convenience. we refer to the ring interaction sites as ?he carbon atoms” (C) and to the remaining interaction sites as “the hydrogen atoms” (H). The relevant intramolecular distances of the models are foe = 1.41 A; &n = 1.09 A. and the Lennard-Jones potential parameters are crcc = 3.50 A, &Cc/kg = 40.3 K; unn = 2.50 A; and EHH/kB= 25.18 K. M
D
+ -c;* 0
??
Figure 1: Benzene-like solutes with twelve interaction site. The labels M, D. Q, and 0 correspond to the leading electric multipole moment of the change Ano in the charge distribution in the P + S transition. See the text for details. The partial charges of all the solutes in the electronic state P are: Qg = -0.135e for all the carbon sites and Q[ = 0.135 e for all the hydrogen sites. The charges are expressed in terms of the protonic charge e. The change in the charge distribution of each of the
solutes is represented in Figure 1: a + (or -) sign attached to the solute interaction site X indicates that in the electronic state S the partial charge at the site is Q: = Qf; + e (or Qf = (a,’ - e). The partial charges of the interaction sites not marked with a + or a - sign in the figure do not change in the P + S transition (i.e., Qf = Qr). Figure 1 shows that the solute species are labelled after the leading electric multipole moment of the change A7~(x) E n:(x) - n,‘(x) in the charge distribution in the P - 5’ transition:
monopole
(M), dipole (D), quadrupole
(Q), and octopole
(0).
Here n:(x)
=
CA Qf: 6(x - xx) is the bare charge distribution of a solute in the electronic statt II. Notice that because the molecular charge distributions have finite spatial extension, the higher order multipoles of An,(x) are nonzero
1.0
1--
z(t)
-T---I
-l
.8 .? .6 .5 .4 .3 .2 .1 0
Figure 2: Solvation tcf’s 2,,,(t) p: Calculated with the RMFA 2,,,(t) for the monopole solute RFMA theory using 2,,,(t) for B. Results.
We now turn
for the benzene-like solutes of Figure 1 in acetonitrile. theory. - -: Calculated with the RFMA theory using as the reference in Eq. (5.1). - - - -: Calculated with the the octopole solute as the reference. 1 is in units of ps.
to the results
for the solvation
tcf’s of the M, D, Q and 0
430
solutes in MeCN and MeOH, as calculated with the RMFA and RFMA methods; the results are shown in Figures 2 and 3. The solid lines in the figures correspond to Z,,,(t) calculated with the RFMA approximation [Eqs. (2.10), (2.13), and (4.5)). The simulation data for the reference tcf SM,L(~) used in the calculations are taken from Refs. 30 and 33 for MeCN and MeOH, respectively.
1.0
20)
MeOH
.8
]
.? .6 .5 .4 .3 .2 .1 0
--L.__1.__1_..~_-c.1___~l_._i._~
0
.1
.3
.5
.?
t
Figure 3. As in Figure 2 but for the solutes in methanol. It is evident from the figures that the dependence of the solvation tcf on the order, L, of the leading multipole moment of A%(x) is quite pronounced in either solvent. This feature was already reported by Kumar and Maroncelli4 and in our previous report.12 It should be noted that the solid curves (Z,,,(t) with RMFA) are in very good agreement with the simulation results of Kumar and Maroncelli.4 This comparison is important, since both studies use the same ISM models for the solute and solvent molecules. The two other sets of curves in Figures 2 and 3 (long and short dashed lines) correspond to calculations of &o,(t) with the new RFMA power-law formula [Eqs. (5.1) and (5.4)] using two different reference solutes R. For the reference solvation tcf [2~(t) in Eq. (5.1)) we take ZRDT(t) for one of the benzene-like solutes calculated with the RMFA method
431
(i.e., one of the solid curves in Figures 2 and 3). Thus the long-dashed curves in the figures correspond to the solvation tcf’s 2y(t) for the benzene-like solutes Y = D, Q, 0 taking 2.&~(t) as the reference tcf (i. e., the reference solute is the monopole R = hf and 2M((t) is calculated with the RMFA). The short-dash curves are the solvation tcf’s 2y(t) for the benzene-like solutes Y = A4, II,& taking 20(t) as the reference tcf (i. e., the reference solut,r is the ortopole R = 0). The 2y(t) results with the RFMA should be compared with the corresponding solid lines in Figures 2 and 3. Pigure 2 shows that the new RFMA power-law approximation gives very reasonable results in acetonitrile. Figure 3 shows that in methanol the agreement between the RMFA and RFMA estimates is poorer. but still qualitatively correct. It is clear from Eq. (5.1) that the distinctive feat.ures of the reference tcf 2~(t) will be reproduced in the RFMA estimate 2y(t). Therefore, it is not surprising that 2~(t), 2,(t), and 20(t) calculated with the RM FA reference 2~ (t) will show pronounced oscillatory features at intermediate times. In c’ontrast. the oscillatory c,omponent is missing in the RFMA estimates 2~(t), Zu(l). and zQ(t) calculated with the RMFA tcf 20(t) as the reference. Th~l figures illustrate, that the RFMA tcf’s 2y(t) based on either of the alternative RMFA reference functions 2~ (1) or 20(t) are remarkably consistent with each other MKI with the corresponding RMFA results for times less that 300 fs. We recall” that the generic, RFMA power-law formula, Eq. (3.1 I), is exact at very short times, in the sense that t,he initial curvature (d2@y (t)/dt2),,,o of the primary tcf calculated with Eq. (3.1 1) has the correct value. It IS also noteworthy that the correct ordering of the tcf’s calculated with the RFMA pow<‘r-law IS always maintained at long times. although the deviations with the corresponding RMFA curves are larger when methanol is the solvent. At long times we expect” tll
Conclusions
Wc have explored 11simplified illlpleli~entatioil nf the SH-RDT formula for the solvation t cf. The new power-law formula is based on the convoln tionless generalized Langevin equation of Tokuyama and Mori. l7 The resulting simplified procedure gives 2(t) in reasonablr agreement with the more complete in~pletl~eiltatio~~ of the SH-RDT theory under t 11txRMFA approximation. We note that we have not conditioned the application of Kq. (5.1) to reference, (R) and target (Y) solute pairs that have the same symmetry. the same pconirtry. or even the same number of interaction sites. Iii another communication 34 \vo sllon tllat t,he INV procedure ac.c,urately predicts the solvation tcf of coumarin 153 (36 interaction sites) in acet.onitrile whrn 2 R[jT(t) of a benzene-like solute is used as the rcfc~reiicr. ‘I’hc simpler RI:MA approach discussed in this paper should be useful in cases where the more robust’-‘j RMFA (Se<.. IV) proc4ure is very difficult to implement. This situation arises ht‘ll the required dielectric information of the solvent is not available,
432 especially with regards to the far-infrared contribution to the frequency-dependent dielectric constant tW. In such cases, if the solvation tcf of another solute R has already been measured, the RFMA method can be very helpful. A case in point is the very interesting paper by Barbara. Fonseca, and coworkers, 35 where the required 2(t) of bianthryl was assumed to be equal to the solvation tcf of coumarin probes. We believe that Eq. (5.1) will improve on this type of approximation. Acknowledgements It is a pleasure to dedicate this article to Professor Josef Barthel, who has contributed enormously to our current knowledge of the dielectric behavior of liquids and solutions. We are very grateful to Professor Mark Maroncelli for useful discussions. F.O.R. gratefully acknowledges a fellowship from the Consejo Nacidnal de Investigaciones Cientificas y TCcnicas (CONICET) de la Reptiblica Argentina. This work was made possible by the support provided by the National Science Foundation of the United States under Grant No. CHE-932 1963 References 1. (a) P. F. Barbara and W. Jarzeba, Adv. Photochem. 15, (1990) 1; (b) R. Jimenez, G. R. Fleming, P. V. Kumar, and M. Maroncelli, Nature 369, (1994) 471; (c) S. J. Rosenthal, R. Jimenez, G. R. Fleming, P. V. Kumar, and M. Maroncelli, J. Mol. Liqs. 60, (1994) 25: (d) W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, Chem. Phys. Lett. 247. (1995) 264; (e) M. L. Horng, J. A. Gardecki, A. Papazyan, and M. Maroncelli, J. Phys. Chem. 99, (1995) 17311. 2. For a recent. general discussion see:R. M. Stratt and M. Maroncelli, J. Phys. Chem. 100, (1996) 12981. 3. J. T. Hynes. J. Phys. Chem. 90, (1986) 3701. 4. P. V. Kumar, and M. Maroncelli, J. Chem. Phys. 103, (1995) 3038. 5. (a) R. M. Levy, D. B. Kitchen, J. T. Blair, and K. Krogh-Jespersen. J. Phys. Chem. 94, (1990) 4470; (b) M. Belhadj, D. B. Kitchen, K. Krogh-Jespersen, and R. M. Levy, J. Phys. Chem. 95, (1991) 1082; (c) P. L. Muino and P. Callis, J. Chem. Phys. 100, (1994) 4093; (d) R. Olender and A. Nitzan. J. Chem. Phys. 102, (1995) 7180; (e) M. S. Skaf and B. M. Ladanyi, .I. Phys. Chem. 100, (1996) 18258. 6. B. M. Ladanyi and R. M. Stratt, J. Phys. Chem. 100, (1996) 1266. 7. For recent references along these lines see: (a) R. Biswas and B. Bagchi, J. Phys. Chem. 100, (1996) 1238; and (b) A. Ch an d ra. D. Wei, and G. N. Patey, J. Chem. Phys. 99, (1993) 4926. and the references to previous work quoted therein. 8. X. Song, D. Chandler, and R. A. Marcus. J. Phys. Chem. 100, (1996) 11954. 9. F. 0. Raineri. H. Resat. B.-C. Perng, F. Hirata, and H. L Friedman, J. Chem. Phys. 100, (1994) 1477. 10. F. 0. Raineri, B.-C. Perng, and H. L. Friedman, Chem. Phys. 183, (1994) 187. 11. H. L. Friedman, B.-C. Perng, and F. 0. Raineri, J. Phys. Condensed Matter 6, (1994), A131. 12. H. L. Friedman, F. 0. Raineri, F. Hirata, and B.-C. Perng, J. Statist. Phys. 78, (1995) 239.
Journal of Molecular Liquids 73,74 (1997) 419432
A SIMPLE POWER-LAW APPROXIMATION FOR THE SOLVATION TIME CORRELATION FUNCTION Fernando 0. Raineri, Harold L. Friedman, and Baw-Ching Perng Department of Chemistry, State University of New York at Stony Brook, Stony Brook, New York 11794-3400, USA Abstract In previous work we developed a new class of molecular theories of solvation, the “surrogate Hamiltonian” (SH) th eories, to elucidate the structural, energetic, and dynamical aspects of the solvation process relevant to ultrafast time-domain spectroscopy and charge transfer reactions in solution. In these theories we represent both the solute and the solvent molecules by interaction site models of the sort already used in many computer simulation studies of solutions. Of special interest to the characterization of the nonequilibrium solvation process is the solvation time correlation function 2(t), that can be obtained in time-dependent fluorescence Stokes shift experiments. The experimental and theoretical studies of 2(t) in polar solvents have revealed an important dependence of the solvation time correlation function on the nature of the solute, especially with respect to the multipolar order of the charge jump. Here we investigate a simplified version of our theory of solvation dynamics, which is based on the straightforward application of a method previously developed by our group! the reference frequency modulation approximation (RFMA). In the earlier paper we showed that when applied to the solvation tcf, the RFMA leads to a simple interpretation of the power law equation for 2(t) found empirically by Maroncelli and coworkers. Here we propose a new power-law approximation to the solvation time correlation function that retains the dependence of 2(t) on the features of the solute. More specifically, the new application of t&e RFMA discussed here enables us to exoress the solvation tcf Zv(tj of solute Y in terms of the solvation tcf 2R(t) of another solute R in the same solvent. The solutes R and Y may differ in size, shape, and charge distribution. The consistency of 2(t) calculated with more accurate methods and the simpler RFMA theory is quite satisfactory. I
1
\
I
I. Introduction 111the last few years a vast experimental’ and theoretical2 effort has been directed to uncover the molecular details of the dynamical solvent response triggered by a “jump” in the charge distribution of a solute molecule. Typically the charge jump (by photoexcitation) carries the solute from its ground (or precursor P) state to its first electronic excited (or successor S) state. The solvent, which is initially equilibrated under the field of the P-state solute, is abruptly destabilized by the field of the newly created S-state solute charge distribution. In suitable cases the solute fluoresces, and the ensuing process of nonequilibrium solvation is monitored by the evolution of the solvation time correlation 0167-7322/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PI1 SO167-7322(97)00085-8
420
function
(tcf)1>2
4t) - u(m)
2(t)= u(0)
- u(m)
’
(1.1)
where v(t) is the frequency of the solute fluorescence at time t after the charge jump. This description corresponds to the time-dependent fluorescence Stokes shift experiments. * Establishing the detailed molecular mechanism leading to 2(t) is crucially important for understanding the role of nonequilibrium solvation in the rate constants of electron and charge transfer reactions in solution.3 It has been found 1,2 that the solvation response in polar solvents is intimately related to the dielectric properties of the solvent, an area in which Josef Barthel’s contributions are essential. A useful discussion of the wide variety of theoretical methods that have been applied to analyze the solvent dynamics responsible for the features of 2(t) has recently appeared.2 Because of the amount of detailed information that they provide, molecular dynamics computer simulation studies of solvation dynamics have,been important aids in the development of additional approximate theories. More recently, however, the simulation effort has been directed toward the study of systems using realistic representations of the solute and solvent molecules4y5 Among the more recent approximate treatments are the instantaneous normal mode analysis6 (which is also capable of detailed representation of solute and solvent molecules); semianalytical methods that focus on the solvent response and regard the solute as the source of an external potential7 and a novel continuum dielectric formulation of the solvent that satisfies the dielectric boundary conditions for arbitrary shapes of the solute.* In the last few years our group has developed a molecular theory of 2(t) (and other equilibrium and nonequilibrium solvation processes) that applies to model solutions in which both the solute and solvent molecules are represented by rigid non-polarizable Interaction Site Models (ISM’S).~-‘~ We recall that the ISM are models in which the potential energy of interaction between two molecules is a sum of pairwise additive site-site terms including, in addition to Lennard-Jones or similar short range interactions,Coulombic interactions between partial charges located at the molecular sites. The theory is based on a simple renormalized linear response development that incorporates non-linear aspects of equilibrium solvation. The renormalization is carried over the solute-solvent potential energy of interaction; the outcome is a statistical mechanical theory based on a surrogate (renormalized) Hamiltonian. We refer to this approach as the surrogate Hamiltonian (SH) theory of solvation. In passing over less detailed models, the SH theory recognizes that typical solutes and solvents of chemical interest are comprised of polyatomic molecules that cannot be adequately represented by simple models. Like the computer simulation studies, the SH theory consistently accounts for the “size”, the “shape”, and the charge distribution of the actual molecules. There are several implementations of the SH theory that have been discussed in our previous reportsg-l5 In the case of solvation dynamics in polar solvents, the best results for the solvation tcf 2(t) are obtained with one of the versions of the SH theory, the Renormalized Dielectric Theory (SH-RDT). There has also been some effort in obtaining friendlier expressions for 2(t) that would be convenient to implement. For example. we reported” a simple new method for estimating time correlation functions of collective dynamical variables, the Reference Fre-
421
quency Modulation Approximation (RFMA). It is based on the convolutionless generalized Langevin equation reported by Tokuyama and Mori.17 By applying the RFMA approximation to a simplified version of the expression for the solvation tcf given by the SH-RDT. we recovered the “power-law” relation 2(t)
=
@l(V
,
(1.2)
found empirically by Maroncelli and coworkers.18~1g Both the pure-solvent single-dipole tcf @r(t) = (p(t) . /.J)/(P. P) and the exponent o on the’right hand side of this equation depend only on the solvent. Unfortunately, the approximations that lead to Eq. (1.2) are so drastic that any dependence of 2(t) on the solute features is lost. In contrast, experimental resultsa as well as detailed studies by computer simulation4 and various approximate theories al2 have revealed an important dependence of the solvation tcf on the nature of the solute, especially with respect to the multipolar order of the charge jump. In this paper we present a new and straightforward application of the RFMA to derive an approximate power-law form for the solvation time correlation function that depends realistically on the nature of the solute molecule. More specifically, we investigate the possibility of estimating the solvation tcf 2y(t) for the P -+ 5’transition of a solute Y in terms of the known solvation tcf 2R(t) for the f’ + S’ transition of a reference solute R, both in the same solvent. Here P and S (P’ and S’) are. respectively, the ground and excited states of the solute Y (R). The new approximate power law for 2(t) is quite consistent with the results predicted by the more complete SH-RDT theory. II. SH-RDT
Theory of the Solvation Time Correlation Function
The photoexcitation of the solute in the time-dependent fluorescent Stokes shift experiment carries the molecule from the precursor (P) or ground electronic state to the successor (S) or excited electronic state. The solvation time correlation function 2(t) [cf. Eq (1. l)] that monitors the subsequent, solvent relaxation has the statistical mechanical formulationg-ll (ti; t) - (ti; 3(j) (2.1) 2(t) = (ti; 0) - $4; cc) ’ where G is the solvation vertical energy gap dynamical variable: It represents, for a given configuration of the system, the difference in solvation energy between the two electronic states of the solute. We consider systems in which lf has the Coulombic form
where AQx = Qf - Qf; is the difference between the partial charges Qf and Qf at the interaction site X of the solute in states P and S respectively, and qj is the partial charge on interaction site j of a solvent molecule. Furthermore, rx+j E Ir~,~jl = lxaj - XXI is the distance between the solute site X (located at xx relative to the laboratory reference frame) and the site j of the solvent molecule a (located at xaJ). In Eq. (X.1) (U; t) is t,he
422
expectation value of the vertical energy gap in a non-equilibrium ensemble at time t after the mstantaneous P + S transition caused by photoexcitation at t = 0. In the SH t,heory of solvation dynamics the system is described by a time-dependent surrogate (subscript C) Hamiltonian Hz(t) of the same form as the bask Hamiltonian 3.1(t).g-12 The surrogate Hamiltonian BE(~), however, is formulated in terms of renormalized solute-solvent interactions &z(t). so that a simple linear response treatment [using the Hamiltonian of the pure solvent Hw as the reference Hamiltonian and the solute-solvent renormalized potential energy of interaction 6z(t) as the perturbation], is sufficient to describe the equilibrium solvent structure around the solute in the electronic states P (at t = 0) and S (at t = 30).~-l~ One of the implementations of the SH theory of solvation is the Renormalized Dielectric theory (SH-RDT). In the SH-RDT theory the renormalized solute-solvent potential energy of interaction has the form
where B(t) is the unit step function
and (2.4)
is the surrogate analog of the vertical energy gap c of the basic description of the system. Both Eqs. (2.3) and (2.4) are expressed in terms of the renormalized solute solvent interactions !&$,, (where the superscript D refers to the solute electronic state, P or S). These potential energies are of the formg-15
D=P,S.
(2.5)
where (2.6) is the Fourier component of the solvent polarization charge density at a point r relative D k) is the Fourier transform of a renormalized electrostatic to the solute site A, while ‘P~,~\( potential state.
cpg A(r) associated
The SH-RDT
with the interaction
theory gives an expression
site X of the solute in the D-electronic
for (pgx( k ) in terms of the site-site
solute-
state solvent RISM direct correlation functions 21 when the solute is in the D-electronic and certain response functions of the pure solvent. We refer the reader to Refs. 9-15 for more details. The renormalized linear response formulation of the SH-RDT theory gives the following estimate of the solvation tcf 2(t) (see Refs. 9-12 for a detailed derivation)
(2.7)
423
where, from Eqs. (2.4) and (2.5), the SH-RDT
renormalized
vertical
energy gap is given
(2.8) where Acpc,x(k)
= &,Jk)
(2.9)
- &Jk)
is the change in the Fourier transform of the renormalized electrostatic potential due to the solute charge redistribution in the P + S transition. It is important to noticegel that the tcf &or(t) in Eq (2.7) is governed by the Hamiltonian H,,, of the homogeneous solvent, i.e. both the equilibrium average operation (, .)w and the evolution of E,(t) are determined by Hw. From our earlier work,‘-15 it may be shown that Eq. (2.7) can be rearranged to the form &Yr(t)
=
dk@,(k,t) W,,,(k) , JO” 0
@,(k,t) =
(&(k t) l$-k) )w (i%(k),4(-k) )w
(2.10) (2.11)
is the tcf of the Fourier component
of the fluctuating solvent polarization charge density &(r). We recall that @/‘(k, t) is the tcf that determines the frequency- and wavevector-dependent longitudinal dielectric, function cL(k.u) of the pure solvent. 14,22 Furthermore, the weight function W,u,T(k) may be interpreted Here
=
C(k) I’m(k)
as a normalized
C(k) = 1 -
-!--. ‘L(k)
/
irn C(k)ram(k) dk
distribution = -47rP k2
of solute-solvent
+
(Mk)
&A-k)
(2.13)
interaction
)w }
M
coupling. (2.14)
is the wave-vector dependent Pekar factor.15922123 Here /3 = (kgT)-’ is the inverse of the temperature T in energy units (kB is the Boltzmann constant), while V is the volume of the system. The symbol {. . .}m indicates the thermodynamic limit of the quantity inside the braces. Notice that we take too = 1 because in this study we ignore the electronic polarizability of the solvent. Finally, (2.15)
is the SH-RDT
solute-solvent
coupling
factor.15 It is defined in terms oP,14~15 (2.16)
424
the Fourier coefficient of the change in the effective partial charge density at solute site X, while wxx~(lc) is the Fourier transform of the solute intramolecular site-site correlation function2* According to Eq. (2. lo), it is the dielectric dynamics of the homogeneous solvent, as expressed in 9,(/c, t), that is the source of the time dependence of the SH-RDT estimate of 2(t). In this approximation the effect of the solute-solvent interactions is carried by the static weight function W,,,(k). This “factorization” (to a function of the homogeneous solvent dynamics times a function of the static solute-solvent structure) is an Important approximation and is a distinctive feature of the SH theory of solvation. The renormalized character of WRDT(k) allows us to bypass the two-time many-point correlation functions that would necessarily appear in a more complete dynamical theory that addresses the inhomogeneity of the solvent in the field of the solute particle. We distinguish the SH-RDT from alternative versions of the surrogate Hamiltonian theory of solvation. 12,15 In particular it should be distinguished from the Renormalized Site Density theory (SH-RST), in which the primary dynamical variables are the microscopic solvent interaction-site densities. 12,15,24 III. Generalized Langevin Equations. We discuss now two approximate methods for calculating time correlation functions of collective dynamical variables. In Sets. IV and V they are applied to calculate the surrogate estimate 2,,,(t) of the solvation tcf. A. GLEr and the Reference Memory finction Approximation. The reference memory function approximation22*25y26 (RMFA) is a method for calculating, using the memory function equation27,28
d@y(t)/dt the normalized
time correlation
= -
Jot
dt’ Ky(t - t’) @jy(t’) ,
(3.1)
function
of the desired primary dynamical variable Y. In this section the symbol (. ’.) denotes a general equilibrium ensemble average; a particular choice would be the ensemble average Hw of the pure solvent. (...)uI with the Hamiltonian The memory function equation (3.1) is a consequence of the generalized Lsngevin equation (GLE1) derived by Mori. 27128 A distinctive feature of both the GLEr and the memory function equation is the convolution operation of the function of interest with the memory function KY(t). Rather than attempting the calculation of KY(t), the RMFA method relies on the information contained in the tcf
@x(t) =
(x(t) (xx*)
x*)
(3.3)
425 of a related reference dynamical variable X. We assume that @x(t) is already known, say from experiment, computer simulation, or by an independent theoretical approximation. The RMFA is based on the assumption that the normalized first memory functions of the primary and reference dynamical variables are equal at all times:22T25,26 Ky(t)lKy
= Kx(t)lxx.
(3.4)
Ky
= Ky(t=O)
= (Ii’(2)/(jY12),
(3.5a)
Kx
F Kx(t=O)
= (lji12)/(IX12),
(35b)
Here27,28
are, respectively, the initial values of the first memory functions of the dynamical variables Y and X. With Eq. (3.4) and the memory function equations [cf. Eq. (3.1)] for @y(t) and @)x(t) it is possible to show22,25,26 that the corresponding Fourier-Laplace transforms @Y(W) =
om dteetwt .I
and @X(W) [defined as in Eq. (3.6)] are connected @y(u)
= {Ryx
+ iw@x(w)[l
where the renormalization factor ‘Ryx Icy of the respective memory functions Ryx
(3.6)
@Y(t) by the equation
- R&j-*
@x(w) I
is defined in terms of the initial Ky(t) and Kx(t): ??
(3.7) values Kx
KyfKx.
and
(3.8)
B. GLE+ and the Reference fieqnency Modulation Approximation. The memory function equation (Eq. (3. l)] of the GLEl is not the only possible effective equation of motion for the primary tcf @y(t). An alternative is the frequency modulation equation” $ln@y(t)
= -qJy(t)
.
that relates t,he normalized tcf @y(t) [Eq (3.2)] to the frequency modulation @y(t). This equation is derived from the convolutionless generalized Langevin
(3.9) function equation
(GLE2) first derived by Tokuyama and Mori.17 In a previous reportl’ we showed that an approximate relation analogous to Eq. (3.4) could be invoked for the frequency modulation functions of the primary dynamical variable 1’ and an appropriate reference dynamical variable X. Thus, in the RFMA we assume that (3.10) @y(t) llii~(O) = tix(t)l?j'x(O)~, at, every t. It is straightforward to show that this assumpbion, together with the frequency modulation equation (3.9) for @y(t) and @x(t), leads to the useful relation @y(f)
= @X(t)-
(3.11)
426
between the primary and reference time correlation functions. The renormalization factor RyX in Eq. (3.11) is also given by Eq. (3.8) because of the relations KY = $y(O) and Kx = TJx(O).“~‘~ IV. SH-RDT
Solvation tcf Z&(t)
with the R.MFA Approximation
To calculate 2,,,(t) according to Eq. (2.10) we need to evaluate the solvent polarization charge density tcf @,(lc, t) and the weight function W,nr(lc). In previous reports25>26 we showed that @,(/c, t) can be evaluated reliably with the RMFA method discussed in Sec. 111.A. We review the method in this section. From Eq. (2.11), it is clear that the primary dynamical variable is Y = /3,(k). For the reference dynamical variable X we choose the total dipole polarization
(4.1)
(PCltis the z-Cartesian component of the dipole moment pa of the solvent molecule a in the laboratory reference frame). In this case @y(w) and @x(w) in Eq. (3.7) correspond, respectively, to @,(lc, w) and (4.2) where (4.3) is the normalized tcf of fin,, that determines the frequency-dependent dielectric constant c,.26 The idea of using of Eq. (4.1) as the reference dynamical variable X is due to Fried and Mukamel.2g For the present choices of X and Y, the renormalization factor RyX [which we now call R,(/c)] becomes 22,25,26 [cf. Eqs. (3.5) and (3.8)]
Rp(k)=
Rick) _ ( I bJk)I2)w/ ( I kc(k)I2)w x;(k) (Id(k)12),/(l~~(k)12)w
(4.4)
The second equality defines, respectively, the kinetic 7$(k) and structural R;(k) parts of the renormalization factor. Their calculation has been discussed at length in Refs. 22, 25, and 26. With these choices of primary and reference dynamical variables, Eq. (3.7) becomes @,(k,w)
= {R,,(k)
+ ~w@M,L(w) [l -
R,Vdl I-’ @M,L&J)
(4.5)
As in most of our previous reports, in the present work we shall use the “sim” implementation of Eq. (4.5), in which the reference tcf GM,L(~) is obtained from computer simulation of the ISM model solvent.
421
V. SH-RDT
Solvation tcf 2,,,(t)
with the RFMA
Approximation
The RFMA result, Eq. (3.1 l), suggests a simple procedure for estimating the solvation tcf 2y(t) for the P + S transition of a solute Y in a given solvent in terms of the (presumed) known solvation tcf 2~(t) for the P’ + S’ transition of a reference solute R in the same solvent. Thus, since the SH-RDT estimate of 2(t) [Eq. (2.7)] is a normalized tcf, we identify the primary tcf @y(t) of Eq. (3.11) with 2y(t) and, correspondingly, the reference tcf @x(t) with ZR(t). In this way Eq. (3.11) becomes zy(t)
(5.1)
= 2R(tp
From the derivation of the RFMA in Section 1ll.B it follows that Eq. (5.1) corresponds to the choice of the SH-RDT vertical energy gaps ($)y and (&)R [cf. Eq. (2.8)], respectively, as the primary and reference dynamical variables Y and X. Therefore the renormalization factor %?,YRin Eq. (5.1) is given by the ratio of the initial values of the memory functions (Kc,)y and (!$)R of 2y(t) and ZR(t), respectively:
R
(Ql
YR=
)Y (K&p)x =
( il~,‘I”)w/(lwu)y ((l~d’)w/(l~p12)w)R
(5.2)
On the other hand, by using the formula9~27~28 WC,,),
=
4d22Y
wdt2)t=o
(5.3)
[and a similar expression for (KC,,),] together with Eq. (2.10) for the SH-RDT of the solvation time correlation functions we calculate
where
Waft
is tl le solute-solvent
similar interpretation value of the memory
coupling
weight function
for the solute
estimate
Y [with a
for kVcR) RDT(k)] while K,(k) = -(d2@,(k,t)/dt2&o is the initial function of the polarization charge density tcf @,(k, t).
VI. Applications In this section we test the ability of the RFMA procedure [Eqs. (5.1) and (5.4)] to reproduce the results for 2,,,(t) calculated with the more familiar RMFA methodg-‘2 [Eqs. (2.10). (2.13), and (4.5)]. With this aim in mind, we examine the dependence of the solvation tcf on the features of the solute charge distribution. The solvents considered are acetonitrile (MeCN) and methanol (MeOH). A. Models and Methods for the Solution Structure. A distinctive feature of the SH theory of solvation is the representation of both the solute and solvent, molecules by interaction site models. The site-site interaction potential energies used in this work have the 12-6-l form uij(?.) = 4Eij [ (Ly”_ (y-1 + !!!$ (6.1)
428 for sites i and j on different molecules. ISM models for acetonitrile an,rsrethanol, are taken from the literature. -
whether solvent-solvent or solute-solvent. The each with a united atom for the methyl group,
To calculate 2,,,(t) with either RMFA or RFMA we need a host of equilibrium solute-solvent and solvent-solvent structure functions. As in Refs. 9-15, we calculate the wavevector-dependent Pekar factor C(k) and the solute-solvent coupling function &r(k) with the RISM-HNC integral equation method (XRISM) of Rossky et a1.32 These determine the coupling weight function W RD~(k) [cf. Eq. (2.13)]. We also apply the XRISM Integral equation method to calculate the renormalization factors R,(k) and RyR. For detailed expositions of our calculations we refer the reader to our earlier reports.g-15,25)26 We consider the solvation dynamics of a family of four “benzene-like” model solutes in the solvents MeCN and MeOH. These are the solute models already studied by Kumar and Maroncelli by molecular dynamics computer simulations;4 a study of a related set of solute models was reported elsewhere.12 The solutes have twelve interaction sites arranged in the same geometry as the atomic sites of a benzene molecule (cf. Figure 1). For convenience. we refer to the ring interaction sites as ?he carbon atoms” (C) and to the remaining interaction sites as “the hydrogen atoms” (H). The relevant intramolecular distances of the models are foe = 1.41 A; &n = 1.09 A. and the Lennard-Jones potential parameters are crcc = 3.50 A, &Cc/kg = 40.3 K; unn = 2.50 A; and EHH/kB= 25.18 K. M
D
+ -c;* 0
??
Figure 1: Benzene-like solutes with twelve interaction site. The labels M, D. Q, and 0 correspond to the leading electric multipole moment of the change Ano in the charge distribution in the P + S transition. See the text for details. The partial charges of all the solutes in the electronic state P are: Qg = -0.135e for all the carbon sites and Q[ = 0.135 e for all the hydrogen sites. The charges are expressed in terms of the protonic charge e. The change in the charge distribution of each of the
solutes is represented in Figure 1: a + (or -) sign attached to the solute interaction site X indicates that in the electronic state S the partial charge at the site is Q: = Qf; + e (or Qf = (a,’ - e). The partial charges of the interaction sites not marked with a + or a - sign in the figure do not change in the P + S transition (i.e., Qf = Qr). Figure 1 shows that the solute species are labelled after the leading electric multipole moment of the change A7~(x) E n:(x) - n,‘(x) in the charge distribution in the P - 5’ transition:
monopole
(M), dipole (D), quadrupole
(Q), and octopole
(0).
Here n:(x)
=
CA Qf: 6(x - xx) is the bare charge distribution of a solute in the electronic statt II. Notice that because the molecular charge distributions have finite spatial extension, the higher order multipoles of An,(x) are nonzero
1.0
1--
z(t)
-T---I
-l
.8 .? .6 .5 .4 .3 .2 .1 0
Figure 2: Solvation tcf’s 2,,,(t) p: Calculated with the RMFA 2,,,(t) for the monopole solute RFMA theory using 2,,,(t) for B. Results.
We now turn
for the benzene-like solutes of Figure 1 in acetonitrile. theory. - -: Calculated with the RFMA theory using as the reference in Eq. (5.1). - - - -: Calculated with the the octopole solute as the reference. 1 is in units of ps.
to the results
for the solvation
tcf’s of the M, D, Q and 0
430
solutes in MeCN and MeOH, as calculated with the RMFA and RFMA methods; the results are shown in Figures 2 and 3. The solid lines in the figures correspond to Z,,,(t) calculated with the RFMA approximation [Eqs. (2.10), (2.13), and (4.5)). The simulation data for the reference tcf SM,L(~) used in the calculations are taken from Refs. 30 and 33 for MeCN and MeOH, respectively.
1.0
20)
MeOH
.8
]
.? .6 .5 .4 .3 .2 .1 0
--L.__1.__1_..~_-c.1___~l_._i._~
0
.1
.3
.5
.?
t
Figure 3. As in Figure 2 but for the solutes in methanol. It is evident from the figures that the dependence of the solvation tcf on the order, L, of the leading multipole moment of A%(x) is quite pronounced in either solvent. This feature was already reported by Kumar and Maroncelli4 and in our previous report.12 It should be noted that the solid curves (Z,,,(t) with RMFA) are in very good agreement with the simulation results of Kumar and Maroncelli.4 This comparison is important, since both studies use the same ISM models for the solute and solvent molecules. The two other sets of curves in Figures 2 and 3 (long and short dashed lines) correspond to calculations of &o,(t) with the new RFMA power-law formula [Eqs. (5.1) and (5.4)] using two different reference solutes R. For the reference solvation tcf [2~(t) in Eq. (5.1)) we take ZRDT(t) for one of the benzene-like solutes calculated with the RMFA method
431
(i.e., one of the solid curves in Figures 2 and 3). Thus the long-dashed curves in the figures correspond to the solvation tcf’s 2y(t) for the benzene-like solutes Y = D, Q, 0 taking 2.&~(t) as the reference tcf (i. e., the reference solute is the monopole R = hf and 2M((t) is calculated with the RMFA). The short-dash curves are the solvation tcf’s 2y(t) for the benzene-like solutes Y = A4, II,& taking 20(t) as the reference tcf (i. e., the reference solut,r is the ortopole R = 0). The 2y(t) results with the RFMA should be compared with the corresponding solid lines in Figures 2 and 3. Pigure 2 shows that the new RFMA power-law approximation gives very reasonable results in acetonitrile. Figure 3 shows that in methanol the agreement between the RMFA and RFMA estimates is poorer. but still qualitatively correct. It is clear from Eq. (5.1) that the distinctive feat.ures of the reference tcf 2~(t) will be reproduced in the RFMA estimate 2y(t). Therefore, it is not surprising that 2~(t), 2,(t), and 20(t) calculated with the RM FA reference 2~ (t) will show pronounced oscillatory features at intermediate times. In c’ontrast. the oscillatory c,omponent is missing in the RFMA estimates 2~(t), Zu(l). and zQ(t) calculated with the RMFA tcf 20(t) as the reference. Th~l figures illustrate, that the RFMA tcf’s 2y(t) based on either of the alternative RMFA reference functions 2~ (1) or 20(t) are remarkably consistent with each other MKI with the corresponding RMFA results for times less that 300 fs. We recall” that the generic, RFMA power-law formula, Eq. (3.1 I), is exact at very short times, in the sense that t,he initial curvature (d2@y (t)/dt2),,,o of the primary tcf calculated with Eq. (3.1 1) has the correct value. It IS also noteworthy that the correct ordering of the tcf’s calculated with the RFMA pow<‘r-law IS always maintained at long times. although the deviations with the corresponding RMFA curves are larger when methanol is the solvent. At long times we expect” tll
Conclusions
Wc have explored 11simplified illlpleli~entatioil nf the SH-RDT formula for the solvation t cf. The new power-law formula is based on the convoln tionless generalized Langevin equation of Tokuyama and Mori. l7 The resulting simplified procedure gives 2(t) in reasonablr agreement with the more complete in~pletl~eiltatio~~ of the SH-RDT theory under t 11txRMFA approximation. We note that we have not conditioned the application of Kq. (5.1) to reference, (R) and target (Y) solute pairs that have the same symmetry. the same pconirtry. or even the same number of interaction sites. Iii another communication 34 \vo sllon tllat t,he INV procedure ac.c,urately predicts the solvation tcf of coumarin 153 (36 interaction sites) in acet.onitrile whrn 2 R[jT(t) of a benzene-like solute is used as the rcfc~reiicr. ‘I’hc simpler RI:MA approach discussed in this paper should be useful in cases where the more robust’-‘j RMFA (Se<.. IV) proc4ure is very difficult to implement. This situation arises ht‘ll the required dielectric information of the solvent is not available,
432 especially with regards to the far-infrared contribution to the frequency-dependent dielectric constant tW. In such cases, if the solvation tcf of another solute R has already been measured, the RFMA method can be very helpful. A case in point is the very interesting paper by Barbara. Fonseca, and coworkers, 35 where the required 2(t) of bianthryl was assumed to be equal to the solvation tcf of coumarin probes. We believe that Eq. (5.1) will improve on this type of approximation. Acknowledgements It is a pleasure to dedicate this article to Professor Josef Barthel, who has contributed enormously to our current knowledge of the dielectric behavior of liquids and solutions. We are very grateful to Professor Mark Maroncelli for useful discussions. F.O.R. gratefully acknowledges a fellowship from the Consejo Nacidnal de Investigaciones Cientificas y TCcnicas (CONICET) de la Reptiblica Argentina. This work was made possible by the support provided by the National Science Foundation of the United States under Grant No. CHE-932 1963 References 1. (a) P. F. Barbara and W. Jarzeba, Adv. Photochem. 15, (1990) 1; (b) R. Jimenez, G. R. Fleming, P. V. Kumar, and M. Maroncelli, Nature 369, (1994) 471; (c) S. J. Rosenthal, R. Jimenez, G. R. Fleming, P. V. Kumar, and M. Maroncelli, J. Mol. Liqs. 60, (1994) 25: (d) W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, Chem. Phys. Lett. 247. (1995) 264; (e) M. L. Horng, J. A. Gardecki, A. Papazyan, and M. Maroncelli, J. Phys. Chem. 99, (1995) 17311. 2. For a recent. general discussion see:R. M. Stratt and M. Maroncelli, J. Phys. Chem. 100, (1996) 12981. 3. J. T. Hynes. J. Phys. Chem. 90, (1986) 3701. 4. P. V. Kumar, and M. Maroncelli, J. Chem. Phys. 103, (1995) 3038. 5. (a) R. M. Levy, D. B. Kitchen, J. T. Blair, and K. Krogh-Jespersen. J. Phys. Chem. 94, (1990) 4470; (b) M. Belhadj, D. B. Kitchen, K. Krogh-Jespersen, and R. M. Levy, J. Phys. Chem. 95, (1991) 1082; (c) P. L. Muino and P. Callis, J. Chem. Phys. 100, (1994) 4093; (d) R. Olender and A. Nitzan. J. Chem. Phys. 102, (1995) 7180; (e) M. S. Skaf and B. M. Ladanyi, .I. Phys. Chem. 100, (1996) 18258. 6. B. M. Ladanyi and R. M. Stratt, J. Phys. Chem. 100, (1996) 1266. 7. For recent references along these lines see: (a) R. Biswas and B. Bagchi, J. Phys. Chem. 100, (1996) 1238; and (b) A. Ch an d ra. D. Wei, and G. N. Patey, J. Chem. Phys. 99, (1993) 4926. and the references to previous work quoted therein. 8. X. Song, D. Chandler, and R. A. Marcus. J. Phys. Chem. 100, (1996) 11954. 9. F. 0. Raineri. H. Resat. B.-C. Perng, F. Hirata, and H. L Friedman, J. Chem. Phys. 100, (1994) 1477. 10. F. 0. Raineri, B.-C. Perng, and H. L. Friedman, Chem. Phys. 183, (1994) 187. 11. H. L. Friedman, B.-C. Perng, and F. 0. Raineri, J. Phys. Condensed Matter 6, (1994), A131. 12. H. L. Friedman, F. 0. Raineri, F. Hirata, and B.-C. Perng, J. Statist. Phys. 78, (1995) 239.