Optical Properties of Molecular Aggregates. I. Classical Model of ElectronicAbsorption and RefractionHoward DeVoe Citation: J. Chem. Phys. 41, 393 (1964); doi: 10.1063/1.1725879 View online: http://dx.doi.org/10.1063/1.1725879 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v41/i2 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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ESR LINE SHAPE 393

correlation of the peaks and other characteristic details to stationary resonance fields. However, there may arise some complications due to the fact that parallel SRF's could appear in perpendicular spectra and vice versa.

(b) Calculations based on the use of an isotropic g tensor and a constant linewidth make it possible to fit reasonably well a number of observed spectra. Recently dynamical effects have been shown to occur in some cases.6 It is therefore of interest to be able to decide whether a given spectrum can or cannot be reproduced from the usual Hamiltonian with appropriate zero­field splittings and linewidth parameters.

(c) The fact that the electron spins are not quantized in a simple way may complicate considerably the task of relating the quantities which express the electron­nucleus magnetic hyperfine couplings to the linewidth or the hyperfine splittings. However, the canonical orientations play a privileged role, as for these one should be able in most cases to interpret the data in the usual manner.

THE JOURNAL OF CHEMICAL PHYSICS

Note added in proof: Dr. Wasserman has kindly in­formed us that as a result of further investigations on nitrenes: (a) The lower of the two resonance fields observed for one of them at 1620 G is not due to the nitrenej it may be interpreted as the Hmin of pairs of weakly interacting radicals. (b) The peak at 6720 G has the shape characteristic of X = Y, and Z is near 0.66 em-I. The tensor T6 of Table II obtained from an attribution of the two peaks to the same species agrees with these values. This coincidence is due to the fact that when X - Y is small the field H / occurs near 5j2g{3, as do also the Hmin of species with a very small spin-spin coupling tensor.

ACKNOWLEDGMENTS

The authors thank Dr. van der Waals and Dr. de Groot for several helpful discussions and for communi­cating their spectra before publication. Part of the computing time used in the present work was made available through a NATO grant.

VOLUME 41, NUMBER 2 15 JULY 1964

Optical Properties of Molecular Aggregates. I. Classical Model of Electronic Absorption and Refraction

HOWARD DEVOE

Section on Physical Chemistry, NIMH, National Institutes of Health, Bethesda, Maryland

(Received 20 January 1964)

A theoretical classical model is developed to predict the absorption and refraction of an aggregate of monomer units (a molecular aggregate, molecular crystal, or polymer) at any frequency. The monomers are treated as having complex electronic polarizabilities whose frequency dependence is determined by the absorption bands of the isolated monomers. Polarizations in the aggregate induced by incident light are modified by Coulombic interactions between the monomers. No first-order approximation is involved as in exciton theory. The molar extinction coefficient and molar refraction are obtained from normal mode polarizabilities found by solving an eigenvalue problem. The predicted absorption spectra agree (to first order in interaction energy) with exciton theory in the limit of weak coupling, with the hypochromism theory of Tinoco and Rhodes, and (for a classical oscillator model) with exciton theory for strong coupling. The oscillator strength sum rule is obeyed. The predicted spectrum of a pair of dyelike monomers is illus­trated for the cases of weak, intermediate, and strong coupling.

I. INTRODUCTION

THIS paper derives the absorption and refraction properties of an aggregate by a classical model in

which the electronic polarizations of the constituent monomer units, induced by the incident light, are modi­fied by Coulombic interactions between the monomers. The theory predicts the dependence of these properties on the frequency and polarization direction of the inci­dent light. The aggregate can be a molecular aggregate, molecular crystal, macromolecular polymer, or other system which is built up from chromophore monomer units.

The classical model and the commonly used quan­tum mechanical exciton model (Frenkel model) are shown to predict similar aggregate absorption spec­tra. The paper which introduced the exciton concepti pointed out its close similarity to a coupled oscillator model, which is the simplest form of the present model. Other coupled oscillator treatments similar to the pres­ent model have recently been suggested as alternatives to the quantum theory of hypochromism.2,a

1 J. Frenkel, Phys. Rev. 37,17 (1931). 2 H. C. Bolton and J. J. Weiss, Nature 195, 666 (1962). 3 H. DeVoe, Nature 197, 1295 (1963).

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394 HOWARD DEVOE

In Sec. II expressions are derived for the aggregate's refraction and absorption in terms of the monomer properties. The predicted absorption spectra are com­pared with exciton theory in Sec. III. In Sec. IV, the model is discussed and predicted spectra are illustrated for a pair of dyelike monomers.

II. CLASSICAL MODEL

The aggregate is an assembly of N monomers, not necessarily identical. The over-all dimensions are small relative to the wavelength of the incident light which is used to measure the optical properties; thus retarda­tion effects can be neglected and the electric field of the incident light can be assumed to be uniform throughout the aggregate. The monomers are treated as rigidly oriented with respect to one another within the aggregate. If they are actually mobile, the aggre­gate properties must be averaged over their orienta­tions.

Monomers

We start with the basic assumption that the optical properties of each monomer can be described by one or more one-dimensional complex electronic pol ariza­bilities with differing polarization directions and fre­quency dependence. The real part of each polarizability describes its polarization in-phase with the incident electric field of frequency v, causing optical refraction. The imaginary part describes the out-of-phase polari­zation, causing energy dissipation (i.e., optical absorp­tion) .4

The aggregate contains a total of M different po­larizabilities belonging to the N monomers (M?:..N). These polarizabilities are denoted by complex quanti­ties aP(v), and their respective polarization directions by unit vectors ei, where the index i has values 1 to M.

The polarizabilities belonging to a given monomer are each associated in quantum theory with a dipole­allowed transition from the ground state to an elec­tronically excited state of the monomer. Consider a particular aPe v) and the transition associated with it. The unit vector ei has the same direction as the elec­tronic transition moment t'iO. If the spectrum given by the molar extinction coefficient ~P(v) of the electronic band (broadened by vibrational structure) can be measured for the isolated, randomly oriented monomer, then the real and imaginary parts of aiO(v) are given by the equations

ReaNv) =Ri(V) = 2Cllco~Nx)dx, (1) 7r ° X

2-V

2

(2)

where C1 =3X2303c/87r2No, c is the velocity of light, and No is Avogadro's number. Equation (1) comes

• W. Kauzmann, Quantum Chemistry (Academic Press Inc., New York, 1957), Chap. 15.

from one of the Kronig-Kramers relations5 between the real and imaginary parts of a complex polarizabil­ity; the bar across the integral sign denotes the princi­pal value of the integral.

Aggregate

We now investigate the behavior of the entire inter­acting aggregate of N monomers when placed in plane­polarized incident light of frequency v. The amplitude and direction of the electric field of the light is given by the vector E. Each monomer undergoes polariza­tions depending only on the local field at the monomer; this field is the incident field plus the field from the polarizations of the other monomers. Let mi(v) be the complex amplitude of the induced moment of polariza­bilityap(v) belonging to a particular monomer. It can be written

mi(v) =aiO(v) X (amplitude of the local field component along ei)

M

=aiO(V)[ei·E- .L:Gi,mi(v)], i=!

(3)

w here the coefficients G ij are zero if ap (v) and al (v) belong to the same monomer.

Equation (3) expresses the induced moment of aiO(v) as a linear function of the induced moments of all the other monomers. Strictly speaking, the equation is valid only if each monomer is well separated from the others so that the local field is uniform throughout the monomer. Each interaction coefficient Gij can then be interpreted simply as the contribution from a point unit dipole along ej, located at one monomer, to the component of the local field of another monomer along ei. Equivalently, G ij is the Coulombic interaction en­ergy between two point dipoles with unit dipole mo­ments which are oriented along ei and ej and located at the respective monomers. In this point-dipole as­sumption, Gij is given by

where eij is a unit vector along the line between the two monomers and rij is the distance between them.

In aggregates with closely spaced monomers where the point-dipole assumption is poor, Eq. (3) should be approximately valid when a transition density expres­sion is used

(5)

where Vij is the Coulombic interaction energy between the transition densities6 or transition monopoles7 treated

6 J. H. Van Vleck, in Propagation of Short Radio Waves MIT Radiation Laboratory Series, edited by D. E. Kerr (McGra~-Hill Book Company, Inc., New York, 1951), No. 13, p. 641.

6 H. C. Longuet-Higgins, Proc. Roy. Soc. (London) A235, 537 (1956) .

7 I. Tinoco, Jr., J. Chern. Phys. 33, 1332 (1960); 34, 1067 (1961).

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OPTICAL PROPERTIES OF MOLECULAR AGGREGATES. I 395

as charge densities on the two monomers, and PiG and pl are the monomer transition moments.

There are M simultaneous equations like Eq. (3). They may be written in matrix form

[mi(V) J= [Gii+Oij/a.o{v) J-l[eio EJ, (6)

where [mi(v)] and [eioEJ are column matrices with M rows, [GiiJ is a square symmetric matrix, and Oij is the Kronecker delta. (In the matrix equations, i is the row index and j is the column index.)

The total amplitude and direction mt(v) of the com­plex moment of the aggregate is given by

mt(v) = Leimi(V) = [ei]'[Gii+Oij/aiO(v) jl[e;· EJ,

(7)

where [eiJ is a column matrix and [ei]' is its trans­pose.

One way to obtain the optical properties of the aggregate is to solve Eq. (7) to obtain explicit expres­sions for Remt(v) and Irnmt(v). For instance, the molar extinction coefficient, ~(v), of a randomly ori­ented aggregate depends on the average component of Immt(v) in the direction of E

~(v)=-3v<Immt(v)·E>Av/Cll E 12. (8)

For a dimer with one polarizability per monomer (N=M=2), one can solve Eq. (7) for Immt(v), in­sert in Eq. (8), and perform the averaging over ran­dom directions of E relative to the aggregate using

« ei' E) (ej" E) )Av= H ei' ej I E 12). (9)

The result for the dimer is

E(V) LLE;O(V) i J~i

1-2G12Rj el' e2+ G122 (/rI2+ Rl) (10) x [1-G122(R1R2-Id2) J2+[G122(R2I 1+ R1I 2) J2 ' where the subscripts i and j take the values 1 and 2.

Normal Modes

The optical properties of the aggregate are most easily obtained in general from normal mode polariza­bilities. Two cases are considered: first, aggregates with identical monomer polarizabilities, which are sim­plest to treat; and second, aggregates with nonidentical monomer polarizabilities.

I dentical Monomer Polarizabilities

This is the case of an aggregate of identical monomers whose optical properties are sufficiently well described by one polarizability per monomer (or three identical mutually perpendicular polarizabilities for isotropic monomers). The M monomer polarizabilities all have the same frequency dependence. They can be sepa-

rated into the real and imaginary parts

(i= 1,2, .. ·M). (11)

The aggregate can be described by M complex nor­mal mode polarizabilities denoted by ak(v) , expressible in the form

(12)

where the coefficients ak(v) are complex and Vk is a vector in the polarization direction of Mode k. The amplitude and direction of the aggregate's total mo­ment is then given by

M

mt(v) = Lak(V)Vk(Vk·E). (13) k~l

To find the normal mode vectors Vk, let them be written as linear combinations of the monomer polari­zation direction vectors

(14)

Then Eqs. (7), (13), and (14) lead to the matrix equation

Since [GiiJ is real and symmetric and the right-hand side of Eq. (15) is a diagonal matrix, Eq. (15) is an eigenvalue problem. The eigenvectors of the [GijJ ma­trix are the columns of the real orthogonal [CikJ matrix, yielding the required vectors Vk through Eq. (14). The orthogonality of [C;kJ gives

M. (16)

Let the real eigenvalues of the [GijJ matrix be de­noted by gk (k= 1,2, "', M); they obey

(17)

By solving Eq. (17) for the real and imaginary parts of ak(v) and inserting in Eq. (12), one finds the rela­tions

(18)

(19)

These two equations relate the normal mode polariza­bilities of the aggregate to the monomer polarizabilities and the eigenvalues and eigenvectors of the [GijJ ma­trix.

The real and imaginary parts of each ak (v) as given by Eqs. (18) and (19) are related to one another by the Kronig-Kramers relations, one of which, analogous

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396 HOWARD DEVOE

to Eq. (1), is

Reak(v) = _~l°Ox Imak(x)dx . (20) 1r ° X

2-V

2

This follows because the preceding equations show that if aO(v) satisfies the conditions for a physically realistic complex polarizability a(v), so does each ak(v). These conditions, which are sufficient for the Kronig­Kramers relations to hold, are5 : (1) a(v) analytic in the lower half of the complex v planeS; (2) Rea(v) = Rea(-v); (3) Ima(v)=-Ima(-v); (4)a(oo)=0.

Selection Rules

For large aggregates with translational or helical symmetry, simplifying selection rules can be taken from exciton theory for degenerate transitions. Equa­tion (5) shows that the [GijJ matrix is proportional to the quantum-mechanical [ViiJ matrix, so that both matrices have the same eigenvector components Ck. In exciton theory, the zero-order transition moment t'k for the transition to Exciton Level k is given by

M

t'k= LCikt'P. (21) i~1

Comparison of Eqs. (14) and (21) shows

Vk= t'k/ I t'iO I, (22)

so that the polarizability of the normal mode k is zero unless the exciton transition to Level k is allowed.

A molecular crystal with re-entrant boundary con­ditions, zero wave vector (no retardation), and one polarizability per monomer has (1 allowed exciton tran­sitions with indices k= 1, 2, "', (1, where u is the number of monomers in a unit cell.9 The monomer index i is replaced by the double index na, where n labels the unit cell and a labels the site within the unit cell. Then the eigenvector components for the exciton transitions are

(23)

where N' is the number of unit cells in the aggregate and Bel is an element of the orthogonal matrix which diagonalizes Gna •n (3 (or Vna .n(3) for the monomers within one unit cell. Values of Bak are 1 if u= 1 and ±2-i if u= 2. Thus the values of gk and Vk to be used in Eqs. (18) and (19) to evaluate the normal mode polariza­bili ties are

M

gk= LLCkCjkGij i-1#i

N' u

= (N')-1L L L BakBiGna •m(3 (24) ~1 a=1 mfjr"na

(k=1,2,"·,u). (25) na

8 Equations (12) and (17) showthatak(v) is a rational function of aO(v); therefore ak(v) is analytic in the same domain as aO(v).

9 A. S. Davydov, Theory of Molecular Excitons, translated by M. Kasha and M. Oppenheimer, Jr. (McGraw-Hill Book Com­pany, Inc., New York, 1962).

If the aggregate is an essentially infinite single­or double-stranded helix, equations given by Tinoco, Woody, and BradleylO provide values of the eigen­values Ek of the [ViiJ matrix and values of the dipole strengths I t'k 12 for the three allowed exciton transi­tions. These values divided by I t'iO 12 give the gk and I Vk 12 values needed in Eqs. (18) and (19).

Nonidentical Monomer Polarizabilities

For this general case the aggregate has M real nor­mal mode polarizabilities akR(v) and M imaginary normal mode polarizabilities iakI(V). The directions of the real polarizabilities may not coincide with those of the imaginary ones, and the directions may depend on v. These normal mode polarizabilities can be ex­pressed in the form

akR(v) =akR(v) I VkR(V) 12,

akI(v) =akI(v) I VkI(V) 12,

(26)

(27)

w here the coefficients ak R (v) and ak I (v) are real and the vectors VkR(V) and VkI(V) are in the polarization directions of the respective polarizabilities. The com­plex amplitude and direction mt(v) of the aggregate's total moment is given by

M

Remt(v) = LakR (v) VkR (v) [VkR(V) ·EJ (28) k=1

M

Immt(v) = LakI(V)VkI(V) [VkI(V) ·EJ. (29) k=l

The normal mode vectors are written as frequency­dependent linear combinations of the monomer po­larization direction vectors

VkR(V) = LCikR(v) ei, i

VkI(V) = LCkI(v)ei. i

(30)

(31)

Equations (7) and (28)-(31) lead to the matrix equations

[Cit (v) J' Re{[Gij+oij/aNv) J-11 [ct(v) J = [oijaf(v)J, (32)

[Ci/(v) J' 1m {[Gi;+Oii!aiO(V) J-11[C/(v) J =[oija/(v)]. (33)

Equations (32) and (33) are eigenvalue problems which show that the columns of [CikR(V)J and the values of akR (v) equal respectively the eigenvectors and the eigenvalues of the real part of the symmetric matrix [Gij+Oii!aiO(V)]-1. Likewise [CkI(V)] and akI(v) are given by the eigenvectors and eigenvalues of the imaginary part of the same matrix.

101. Tinoco" Jr., R. W. Woody, and D. F. Bradley, J. Chem. Phys.38, 1311 (1963).

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OPTICAL PROPERTIES OF MOLECULAR AGGREGATES. I 397

Aggregate Optical Properties

The molar refraction and molar extinction coefficient of the aggregate can now be written down in terms of the normal mode polarizabilities which have been found. For generality the expressions will be written for aggregates with nonidentical monomer polarizabili­ties. If the monomer polarizabilities are identical, VkR

and VkI should be replaced by Vk, ail (lI) replaced by Reak(II), and ak l (lI) replaced by Imak(lI) in the fol­lowing expressions.

Refraction

The molar refraction of the aggregate for light po­larized along a principal axis, which determines the aggregate's contribution to the refractive index of a macroscopic sample, can be written

4 IDt(II)·E M (ekR·E)2 [R]=s7l'No Re \ E \2 tn-No~akR(II) \ E \2 '

(34)

where ekR is a unit vector along VkR(II). The difference in the refractive index for light polarized along differ­ent principal axes of a sample of oriented aggregates determines the sample's birefringence. Isotropic or randomly oriented aggregates have an average molar refraction given by

M

[R]=t[47l'NoL:akR (II) ]. (35) k-l

Extinction Coefficient

The molar extinction coefficient of the aggregate in a vacuum, for incident light polarized along E, is

( __ 311 IDt(lI) ·E _ 311 ~ I() (ekI ·E)2

f 11) - C1lm \ E \2 C1 f;:tak

11 \ E \2

(36)

where ek l is a unit vector along VkI(II). The difference in absorption of a sample of oriented aggregates for different directions of E determines the sample's di­chroism. If the aggregate is isotropic or randomly ori­ented, its molar extinction coefficient for light polarized in any direction is

f(lI) = -11 Eakl(lI) . (37)

k-l C1

Normally one would wish to divide Eqs. (36) and (37) by the number of monomers in the aggregate which are considered to be responsible for the absorp­tion, giving the extinction coefficient per mole of ab­sorbing monomers.

III. COMPARISON WITH EXCITON THEORY

Coupling Strength

The quantum-mechanical exciton theory of aggre­gate absorption makes a fundamental distinction be-

tween the limits of strong coupling and weak coupling. Strong coupling treats the electronic and vibrational parts of the wavefunction as Born-Oppenheimer sepa­rable for the aggregate, while weak coupling treats them as separable for the individual monomers within the aggregate. Simpson and Petersonll derived criteria for these limits which may be stated in general terms for aggregates with identical monomer absorption bands: If the largest interactions between monomers given by Vii (in frequency units) are large or small compared to the monomer band half-widths, the cou­pling should be treated as strong or weak, respectively. Between these limits the coupling is intermediate.

For strong coupling, exciton theory treats the mon­omer transitions as electronic lines. The vibrational structure which is neglected broadens the exciton lines unpredictably. However, the spectra of large aggre­gates in which each monomer is excited only a small fraction of the time should consist of the nonbroadened exciton lines.ll

In the usual strong coupling treatment, exciton tran­sition energies are calculated to first order and the selection rules are found from the zero-order transition moments. The Tinoco-Rhodes theory of hypochrom­ism7.12.13 calculates the first-order transition moments and computes the sum of the oscillator strengths to give the aggregate absorption intensity. This theory obeys the Kuhn-Thomas sum rule for the conservation of oscillator strength,4 so that an intensity decrease in one region of the spectrum (hypochromism) is accom­panied by intensity increases in other regions.

For weak coupling, exciton theory can take into account the empirical vibrational structure of the mon­omer absorption bands. 14.15 The aggregate spectrum for weak coupling has been derived from the first-order vibronic transition moments. 14 This treatment gives the same expression for hypochromism as the Tinoco­Rhodes theory, and likewise obeys the oscillator strength sum rule.

We turn now to the classical model. The use of the empirical monomer absorption band shapes clearly corresponds to the exciton treatment of weak coupling. However, an electronic line in quantum theory has a classical analog in the narrow Lorentz absorption peak of a damped harmonic oscillator.' By assuming that the monomers are classical oscillators without vibra­tional structure, we should have an aggregate corre­sponding to the exciton model of strong coupling.

It is shown next that the classical model predicts absorption band shapes for weak coupling, or line posi­tions and intensities for strong coupling, which are the same to first order in the interaction energy as pre-

11 W. T. Simpson and D. L. Peterson, J. Chern. Phys. 26, 588 (1957).

121. Tinoco, Jr., J. Am. Chern. Soc. 82, 4785 (1960); 83, 5047 (1961) .

13 W. Rhodes, J. Am. Chern. Soc. 83, 3609 (1961). 14 H. DeVoe, J. Chern. Phys. 37, 1534 (1962). IIi E. G. McRae, Australian J. Chem. 16, 295, 315 (1963).

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398 HOWARD DEVOE

dieted by the exciton treatments. The oscillator strength sum rule is shown to be obeyed.

Weak Coupling

The absorption of an aggregate having identical monomer polarizabilities is given by the classical model from Eqs. (2), (19), and (37)

For a dimer, one has

g±=±G12

1 v± 12=1±el,e2,

(39)

(40)

respectively,

R(v) = C2(V02_V2) JO/[(V02-V2)2+Z2V2],

l(v) = -C2ZvJO/[(V02-V2)2+Z2V2],

(44)

(45)

where C2=3e2/41T2m (e and m are the electronic charge and mass). The aggregate absorption spectrum is found by inserting the above equations in Eq. (19), rearrang­ing the expression for Imak(v) to a Lorentzian form analogous to Eq. (45), and inserting in Eq. (37)

1 N C2Zv2jo I Vk 12 €(v) = - C

1 £;(VOL v2+C2JOgk)2+Z2V2 (46)

The spectrum is a sum of Lorentz peaks, one for each normal mode. The peak for Mode k has a frequency Vk given by

where + and - represent the two values of k. Ex- Vk2=V02+C2/0gk. panding Eq. (38) in powers of G12 and retaining only

( 47)

the zero- and first-order terms, one obtains the first- The oscillator strength of this peak is

order absorption expression fk = I Vk 12fo. ( 48)

which is equivalent to the expression previously de­rived from exciton theory16 for a dimer with weak coupling.

The simplest way to derive the first-order absorption expression predicted by the classical model for any aggregate is to start with Eq. (3), making the ap­proximation

(42)

This allows mi(v) to be expressed as a linear function of the Gds. By averaging over random directions of E using Eq. (9) one obtains

M

€(v)=-3v(ImLmi(v)ei,E)Av/C11 E 12 i=l

M M

= L€iO(V) [1- 2LRj(v) Gijei' ej], (43) i~l i~l

which again is equivalent to the expression from weak coupling exciton theory.16 The quantity in brackets makes the aggregate spectrum differ from the sum of the monomer spectra, changing the band shapes and positions and altering band intensities14 in agreement with the Tinoco-Rhodes hypochromism theory.

Strong Coupling

The classical and exciton models for strong coupling are compared here for the case of an aggregate of N identical monomers. For the classical model we assume at present that the aggregate contains N classical damped oscillators, each with resonance frequency Vo,

line half-width z, and oscillator strength fO, The real and imaginary parts of the oscillator polarizability are,

16 Reference 14, Eq. (19).

The same normal mode Vk andfk values are predicted by the classical model when the empirical monomer spectrum is used, provided the Gij interaction coeffi­cients are large enough to shift the aggregate absorp­tion to regions where the monomer absorption is small.J7 To show this, we assume the absorption peaks of each monomer and of Normal Mode k are concentrated at Vo and Vk, respectively. The corresponding oscillator strengths are given by the integrated intensities

( 49)

2303mcCljImak(v)dv. 1Te2No v

(SO)

The value of Vk must be such that -Imak(vk) is large while -l(vk) is small. From Eq. (19), the condition for this is

(51)

Equations (1) and (49) can be used to approximate R(v) away from Vo by

R(v):=:::;CdO/(voL V2) , (52)

which together with Eq. (51) leads to Eq. (47) for Vk. Equations (20) and (SO) can be used to approximate Reak(v) away from Vk by

(53)

Then Eqs. (18), (52), and (53) evaluated at v= 0 (where 1=0) combined with Eq. (47) result in Eq. (48) for fk. Since the height of each normal mode peak is proportional to -l/l(v) whereas fk is constant, as the peak shifts it approaches a line in shape.

17 In this case the criterion of Simpson and Peterson (Ref. 11) for strong coupling is satisfied.

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OPTICAL PROPERTIES OF MOLECULAR AGGREGATES. I 399

The exciton model for strong coupling9,lO,18 predicts that the monomer line at Vo splits (Davydov splitting) into N exciton lines in the aggregate. Exciton Line k has first-order frequency

(54)

is polarized along the exciton transition moment tlk, and has dipole strength 1 tlk 12• Here Ek is the kth eigenvalue of the Vij matrix, and t'k is given in zero order by the eigenvectors C;k of this matrix

N

tlk= L:CiktiP. (55) i=1

The magnitude of the monomer transition moment tiP is related to 10 by

(56)

From Eq. (5) one can see that the eigenvalues (gk and Ek ) of the Gij and V ij matrices are proportional and the eigenvectors are the same. With the aid of Eqs. (5) and (14), the above expressions lead to the following relations between the classical (normal mode) quantities and the exciton quantities

Vk=VkE+O(gk2) (57)

Vk = [(VkE) L (C2jOgk/2vo) 2J! (58)

~=~I~I (~

jk/fO= 1 t'k 12/ 1 tIP 1

2• (60)

Equations (57) -( 60) show, respectively: (1) the frequency Vk of the peak of a normal mode is the same to first order in interaction energy as the frequency VkE of the corresponding exciton line; (2) compared to the classical model, exciton theory overestimates blue shifts and underestimates red shifts; (3) the polariza­tion directions of a normal mode and the corresponding exciton line are the same; (4) the oscillator strengths of a normal mode peak and the monomer peak are in the same ratio as the zero-order dipole strengths of the corresponding exciton line and monomer line.

In exciton theory there is, in addition to Davydov splitting, a line-shift term which represents a change in the interaction of the static dipole moments when one of the monomers becomes excited.I8 The effects of this term can be reproduced in the classical model by in­cluding diagonal elements Gii~O in the Gij matrix.

Sum Rule and Hypochromism

The theoretical spectra of various aggregates (with differing numbers of monomers and various absorp­tion band shapes) were generated from the equations of Sec. II using an electronic computer. In all cases the integrated intensities of the complete spectra of the aggregate and of the monomers were equal (within

18 D. S. McClure, Solid State Phys. 8, 1 (1959).

6

/ !

4 i

'" ! . i

'5? / 2

0 ./ /

19 20 21 22 23 Wavenumber (kK)

FIG. 1. Dimer spectra predicted by exciton theory. The two monomers are identical and parallel. Dashed curve: assumed monomer spectrum, generated as the sum of two Gaussian peaks with maxima at 20.0 kK and 21.2 kK (1 kK = 103 em-I) and with heights equal to 50X 103 and 25X 103 and half-widths equal to 0.6 kK and 1.0 kK, respectively. Solid curve: dimer spectrum (extinction coefficient per mole of monomer) for weak coupling (see Ref. 14), calculated by an electronic computer program from Eq. (41). Dashed vertical line: position of the monomer line assumed for strong coupling, located at center of gravity of the monomer band. Solid vertical lines: calculated positions of the allowed exciton lines for strong coupling. Assumed values of the interaction parameter GI2 (10-3 A-3) are indicated.

the error of the numerical integration). A general mathematical proof of this oscillator strength sum rule is not given here, but probably could be found.

The sum rule is easily proved for an aggregate with N identical classical oscillators, corresponding to strong coupling. Equations (16) and (48) give

(61)

which equates the total oscillator strength of the ag­gregate (left-hand side) with the sum of the monomer oscillator strengths (right-hand side).

Aggregates with more than one absorption region can have hypochromism and hyperchromism. In the computed spectra the lowest frequency band was some­what less hypochromic than the first-order hypochrom­ism predicted by the Tinoco-Rhodes theory or by the first-order spectrum of Eq. (43).

IV. DISCUSSION

The basic equations of the classical model are Eqs. (1) and (2), which determine the complex monomer polarizabilities from the monomer band spectra; Eqs. (14), (18), and (19) for identical monomer polariza­bilities or Eqs. (26), (27), (30), and (31) for non­identical ones, which provide the directions and mag­nitudes of the complex normal mode polarizabilities of the aggregate; and Eqs. (34)-(37), which give the molar refraction and molar extinction coefficient of the aggregate at any frequency.

There are both similarities and differences between the classical model and quantum exciton theory. The classical model assumes that the aggregate spectrum depends on the shapes of the monomer absorption bands, regardless of the coupling strength which is determined by the values of the interaction param-

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400 HOWARD DEVOE

w .. (,

Wavenumber (kK)

FIG. 2. Dimer spectra predicted by the classical model. Dashed curve: same assumed monomer spectrum as in Fig. 1. Solid curves: dimer spectra (extinction coefficient per mole of monomer) calcu­lated by an electronic computer program from Eqs. (38)-(40). Assumed values of the interaction parameter G12 (1o-a A-3) are indicated.

eters Gij. Exciton theory takes the vibrational structure of the monomer bands into account only in the limit of weak coupling, and indeed the spectra predicted by the classical model are the same to first order as pre­dicted by exciton theory for weak coupling. For aggre­gates with strong coupling, the classical model may give band spectra whereas exciton theory calculates line spectra. The line spectra are reproduced by re­stricting the classical model to a classical oscillator model. However, in the case of an aggregate whose absorption is shifted well away from the monomer bands, the classical model predicts lines which agree to first order with exciton theory for strong coupling.

The advantages of the classical model are that it allows detailed aggregate spectra to be calculated for any coupling strength (intermediate as well as weak or strong), and it is free of the first-order approxima­tion of exciton theory.

The classical model obeys the oscillator strength sum rule, which means that hypochromism in an absorption band requires bands in other frequency regions to bor­row intensity. Recent classical oscillator2 and quantum dispersion theory19,2o treatments of dimers reached the opposite conclusion that hypochromism is possible as­suming only one absorption band. They reached this incorrect conclusion by calculating the dimer absorp­tion only at the frequency of the monomer band, thereby neglecting the intensity which is shifted to other frequencies. In the case of two parallel monomers, the equations of both of these treatments can be shown to actually predict the same spectrum, obeying the sum rule, as the present model would predict.

19 R. K. Nesbet, Mol. Phys. 7, 211 (1964). 20 R. K. Nesbet, Biopolymers Symposia No.1, 129 (1964).

DyeIike Dimers

To illustrate spectra predicted by the classical model for various coupling strengths, we take the example of a pair of identical parallel dyelike monomers. The re­sults are relevant to the study of aggregated cationic dye molecules in aqueous solution.

The assumed spectrum of both monomers (dashed curves of Figs. 1 and 2) is an asymmetric band in the visible region with a shoulder located about 1.2 kK above the absorption maximum. This is the typical shape of the lowest 1r-1r* band of many planar aro­matic dyes. Since higher bands are neglected there is no hypochromism.

Figure 1 indicates the dimer spectra which are pre­dicted by exciton theory assuming the limits of strong coupling (solid vertical lines) and weak coupling (solid curve). For strong coupling, the allowed exciton line is shifted from the monomer line (dashed vertical line) by a frequency V12/h. By comparing the shifts with the monomer band width, one can see that the three indicated values of the interaction parameter G12 cor­respond in order of increasing magnitude to weak, intermediate and strong coupling.ll

The dimer spectra predicted by the present classical model for the same values of G12 are shown in Fig. 2 (solid curves). The band shape for weak coupling (G12=0.OO2 A-8) differs considerably from the corre­sponding first-order exciton curve of Fig. 1 (solid curve), which moreover has negative values of E in one region. The spectrum for strong coupling (G12= 0.010 X-8) in Fig. 2 is almost a line, as stated in Sec. III.

The dimer band shapes predicted by the classical model (Fig. 2) show a smooth continuity on going from zero coupling (monomer spectrum) to progres­sively stronger coupling strengths. However, the posi­tions of the absorption maximum are not continuous. It is as if the main peak of the monomer band pro­gressively lends its intensity to the shoulder, which qualitatively is what is observed in the experimental spectra of dye aggregates.21 In fact, the spectrum shown for intermediate coupling (G12 =0.004 X-8) has a great similarity in shape and position of the maximum to typical dye aggregate spectra. The classical model thus appears to be useful in interpreting the spectral changes of dyes when they aggregate.

ACKNOWLEDGMENT

I wish to thank Dr. Dan F. Bradley for helpful discussions.

21 S. E. Sheppard and A. L. Geddes, J. Am. Chern. Soc. 66, 1995 (1944).

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