Dynamic perturbation effects upon the circular dichroism intensity induced in an aggregate of dye chromophores bound to biopolymers

Chemical Physics 52 (1980) 337-351 © North-Holland Publishing Company

DYNAMIC PERTURBATION EFFECTS UPON THE CIRCULAR DICHROISM INTENSITY INDUCED IN AN AGGREGATE OF DYE CHROMOPHORES BOUND TO BIOPOLYMERS Mamoru KAMIYA Laboratory of Physical Chemistry, Shizuoka College of Pharmacy, 2-2-1, Oshika, Shizuoka-Shi, Japan 422 Received 16 January 1980

The dynamic perturbation effects of polarizable monomer perturbers upon the circular dichroism intensity arising from absorption transitions of an arbitrary aggregate of dye chromophores bound to a large host polymer are formulated using the linear response theory in the decorrelation approximation, where the interchromophoric retardation phase factors are eliminated by a first-order Taylor expansion which is compatible with the use of the retarded helix selection rules in the long-wavelength approximation. A space-averaged and closed-form formulation of the non-conservative circular dichroism intensity which is perturbed by intensity sharing with the outside perturber transitions is derived in the limit of the weak dynamic perturbation where perturber-perturber interactions are negligible. The relevant formulation is applied in order to investigate the intercalation model dependence of the non-conservative circular dichroism intensity induced at the visible absorption band of proflavine molecules intercalated in either poly(A-T) or poly(G-C).

1. Introduction Along with recent progress of theoretical treatment o f polymer optical activity, circular dichroism has been used as a powerful information source for conformational changes in solutions of large polymers. In the present work we are concerned with some essential problems contained in theoretical formulations of the circular dichroism induced in absorption transitions o f an aggregate of dye chromophores like the carcinogenic acridines bound to a large host biopolymer with random and local lattice disruptions [ 1 - 2 ] . The induced circular dichroism, which should be non-conservative as a consequence o f the host polymer perturbation fields, was found to be very sensitive to bound dye concentration, temperature and ionic strength o f solvent each of which makes considerable contributions to changes in the configurational effects upon d y e - m o n o m e r perturbation interactions [ 3 - 5 ] . Since the relevant circular dichroism induced in the bound dye aggregate provides valuable information not only on dye-binding mechanism but also on host polymer structures, it is important to obtain a theoretical formulation of the non-conservative circular dichroism induced in an arbitrary dye aggregate bound to a large polymer and to apply it to real systems o f dye-biopolYmer complexes. The optical activity induced in d y e - p o l y m e r complexes was formulated b y Philpott [6] b y applying the exciton theory coupled with the primary trap approximation to a model system where a single dye chromophore is bound to an infinite single-stranded helical polymer. In this treatment the coupling o f different electronic states of the bound dye by the host polymer was not considered, and optical activity arising from the one-electron mechanism with the host polymer providing a static or dynamic perturbation fields was left as an interesting possibility to be studied separately. The present work treats the dynamic perturbation effects o f monomer perturbers upon the circular dichroism arising from the Frenkel exciton states of an arbitrary bound dye aggregate. The dynamic perturbation interactions will be considered in terms o f the linear response theory for instantaneous c0ulombic interactions between single-photon transitions on different chromophores. A space-averaged and closed-form expression of the non-conservative circular dichroism intensity which is perturbed b y intensity sharing with the outside perturber transitions will be formulated on the basis of the linear relations between the

3 38

M. Kamiya / Dynamic perturbation effects

diagonal elements of the full resolvent operators defined for a set of perturber exciton states which are approximately valid in the limit of the weak dynamic perturbation interactions. The resulting formulation will be applied in order to investigate the intercalation model dependence of the non-conservative circular dichroism intensity induced at the first 7r ~ 7r* transition of proflavine molecules intercalated in polynucleotides.

2. Theoretical formulation of the dynamic perturbation interactions Let us consider an arbitrary aggregate of tight-binding dye chromophores bound to a large host polymer where the excited eigenstates of the bound dye aggregate are perturbed by the dynamic interactions with a group of neighbouring polarizable monomers of the host polymer. For simplicity it is assumed that the excitation energy levels of the monomer perturbers lie far outside the exciton state energy levels of the unperturbed dye aggregate and that no perturber-perturber interactions exist before the dynamic interactions with the light-absorbing dye aggregate are switched on. A familiar method to treat the mechanism of dynamic interactions is the application of polarizability theories for the dispersive and absorptive response of coupled chromophores. These theories, which are typified by the early papers of Kirkwood and co-workers [7,8] and have been utilized in many distinct forms [9-12] in order to formulate the optical activity of an assembly of symmetric chromophore units, may be classified into the following three categories; classical treatment assuming coupled oscillators, static quantum mechanical treatment based on the first-order stationary perturbation theory, and dynamic quantum mechanical treatment based on the firstorder time-dependent perturbation theory. A more general method to formulate the mechanism of dynamic interactions is the use of a field theoretical technique based on the linear response functions for a coupled chromophoric system. The linear response theory has been applied by Rhodes and co-workers [ 13-16] in order to formulate the polymer optical activity under two basic approximations involving partial decorrelation of charge motion and spatial separability of coupled chromophores. In the present work we apply an extended linear response model coupled with the two basic approximations used by Rhodes and co-workers in order to formulate the internal dynamic perturbation interactions between aggregated dyes and monomer perturbers which are essentially related to coulombic interactions between transition moments on different chromophores. By expanding the internal electric field acting on a given chromophore in infinite orders about its center point, the dynamic perturbation operator may be expressed in the form: Hint(t)=- ~

Fm +

Fx

exp(r?t-iwt)+c.c.=Hint

exp(r~t-iwt)+c.c.,

(1)

where Ptm is a generalized electric transition moment operator referred to the center point of one of the aggregated dye chromophores labeled with the summation suffix m, written as a column matrix arranged in (Dm) ~, (O.m)a#, .... i.e., electric-dipole, electric-quadrupole ..... p,, is the transpose o f p , , . F m is the amplitude of a generalized internal perturbation field acting on the center of the chromophore m, written as a column matrix arranged in (Era)a, l ( V E m ) 1 3 a . . . . , i.e., electric field, electric-field gradient, .... The Greek suffices denote the vector and tensor components referred to a cartesian coordinate system. The same definitions of the symbolic notations are also valid for the monomer perturbers of the host polymer labeled with the summation suffix x. co is the frequency at which the internal dynamic perturbation field oscillates. If the dynamic perturbation field is regarded as selfconsistent, ~o must be correlated with the eigenfrequency of a stationary excited state of the whole chromophoric system composed of dye and monomer chromophores, r/is a positive infinitesimal which has been introduced to guarantee the adiabatic switching on of the internal dynamic perturbation field from the infinite past, and which is understood to take the limit 77~ +0 after integration. By applying the original linear response formula [ t 7,1 8] to a generalized internal dynamic perturbation in the present representation, the statistical average of the transition moment lan on a given chromophore n which

M. Kamiya / Dynamic perturbation effects

339

is induced by the internal dynamic perturbation is expressed in the form:

(~tn(t))

exp(r/t - icot) + c.c.,

= ((It n ; H i n t ) ) c o

(2)

where the symbolic notation of ((lan ; Hint))oo stands for the Fourier transform of the two-time retarded Green function of the operator p n and Hin t

] dr O(r) Tr{,oo [Pn(r), H i n d ) exp(ioar - r/r).

((~ ; Hint))~o = - ( i / h )

(3)

Here P0 denotes the density matrix representing the initial thermal equilibrium distribution, and Pn(r) denotes the interaction representation of the transition moment operator I1n in terms of the unperturbed hamiltonian. If all chromophoric transitions are assumed to originate or return to the ground level, the trace in eq. (3) may be replaced with an average over the ground state. With this assumption and with the single-photon transition approximation, eqs. (2) and (3) are combined to yield the following equation for the monomer perturbers

(Px (t)) = -(1/h) ~ ((0r I Gg-) (z)l 0r> (0~ IHint 10)(01p~ 10r) r

r +) r r + (OxIG~ (z)l(~x)(Ox[llxlO)(OlHintlO~x)) exp(r/t -- icot) + c.c.,

where z = co + it/, and

(4)

G~+-)(z)denotes the resolvent operator defined for the unperturbed hamiltonian Ho 1

Gg+-)(z) =

(5)

(Ho/h - Eo/h ) ± z '

E0 being the ground (exciton vacuum) state of the whole chromophoric system. IOr) is the single-exciton basis belonging to the perturber subsystem, x and r being the position and excitation suffix respectively. For the aggregated dye chromophores, the corresponding equation is expressed as

(lain(t)) = - ( I / h ) ~

((c)Sm[Gg-)(z)lc)~,)(OS/n'lgintlO)(OtPm[OSm)

~ S

m',s'

+
(6)

where l0 s ) is the single-exciton basis belonging to the dye subsystem. In the present treatment Go (z) is assumed to be diagonal with respect to 10r), but such is not the case with 10s). By use of the hermitean property of the operator lax one obtains the relations (OilaxlO~)(O~lHintlO)=

1 r ~ * Fx - ~Px(lax)

=

-

(7)

and 1 ~ , "pxFx, ; ~(lttx)

(8)

where I1r stands for <0haxl0~>, and the superscript asterisk means the operation of takinn.g complex conjugate (c.c.). r r , Assuming that the eigenfunctions of each chromophore are real so that the matrix Px(Px) is real and symmetric, eqs. (4), (7) and (8) are combined to give

s s exp(r/t - iwt) + c;c., (i,x(t))=_(1/2h)~la~(qfxlGo(z)l~r)(~,(O r IV IOx') / ax' r' + ~(OrxlVlOm)am) r

where

\x

,r

m,s

s denote the amplitude of the transition moment pr', and Go(z) = G~-)(z) + G~÷)(z), and where a r,' and am

(9)

M. Kamiya / Dynamic perturbation effects

340

s ram, respectively. The V matrix refers to coulombic exciton interactions between different chromophores. Note that the V matrix elements for a pair of single-exciton basis both labeled with the excited states of an identical chromophore unit vanish because of the orthogonality of eigenfunctions belonging to any isolated chromophore unit. One obtains from eq. (9) the following linear equations with respect to the transition moment amplitudes

a~' + (1/2h)(~xlao(z)lO~x)

(~xlrlc)x,)ax, " + \xt,r

'

s rH,S

s

!

=0

(10)

Following the same procedure as used in deriving eq. (10), one obtains from eq. (6) the following linear equations a s + (1/2h) ~

~

((osm Ia~-)(z)l~b~ ,) + (os/n, lG~+)(z)[c)s))(~s/n,lVlOrx)arx = 0 .

(11)

The simultaneous linear equations of eqs. (10) and (11), whose order is equal to the total number of single-photon chromophoric transitions under consideration, are regarded as self-consistent with respect to coulombic exciton interactions between chromophores which have been treated hereupon on the basis of the decorrelation approximation which neglects interchromophoric permanent moment interactions and transitions between chromophore excited states [ 14]. In the present treatment the decorrelation approximation was introduced under neglect of the internal static perturbation field so that the linear response tensor for coupled chromophores was formally expressible in terms of electric multipolar polarizability in infinite orders. The present treatment of the dynamic perturbation interactions is not entirely complete in that it does not take into account magnetic interactions, but it is adequate for the present purpose of utilizing the self-consistent solution of the simultaneous linear equations in order to describe the perturbed eigenstates of the whole chromophoric system which have counterparts to the excited eigenstates of the unperturbed dye aggregate and which cannot be formulated appropriately on the basis of the exciton limit approximation. If the mixing coefficient of the perturber single-exciton basis l0 r) which takes part in the perturbed eigenstate I ~v) of the whole chromophoric system corresponding to the excited state of the unperturbed dye aggregate is chosen so as to satisfy relation (10), it follows that ($rlfv)=

_

~

~ ( l + A ) x r , - - 1 x,r, Ax,r,,ms(OSi$v ),

(12)

x ' r ~ rn, S

where (I + A) - l stands for the inverse of the interaction matrix (I + A)n~.n'.' = 8n, n'8~,,u' + ( 1 / 2 h ) ( ~ IGo (z~)l ~ ) ( ~ l V I ~ " ) ,

(13)

and where Zv = coy + iT. Note that the frequency co has been substituted with the eigenfrequency coy of I ~ ) since the internal dynamic perturbation field is regarded as self-consistent to generate the perturbed eigenstate l~v). Eq. (12) will be used to express the dynamic perturbation effects upon the circular dichroism intensity arising from absorption transitions of the bound dye aggregate.

3. Theoretical formulation of the dynamic perturbation effects upon the non-conservative circular dichroism intensity induced in the bound dye aggregate Let us consider left and right circularly polarized light with a radiation wavevector (co~c) ea referred to a right-handed laboratory coordinate system (el, e2, e3). The circular dichroism arising from single-photon light absorptions by any chromophoric system can be formulated using the current operator matrix method for interaction of the chromophoric system with a radiation field in transverse gauge [19-20]. Then, the circular dichroism distribution of the chromophoric system, which is represented by the difference between molar extinction

M. Kamiya / Dynamic perturbation effects

341

coefficientof left and right circularly polarized light, is expressed as ('4~r2NACC°~ ~ P v ( ~ ) , f Ae(co) = \h1031 n 10 ]

dco(COv/COa)6(co-cov) oo

× [l
(14)

whereeL and eR are the polarization unit vectors for left and right (circularly) polarized light respectively, I Co) and [@) are the ground and excited eigenfunction, respectively, to the total hamiltonian, N a is Avogadro's number, pv(co) is the phenomenological normalized lineshape function for the 0 -+ v transition, 6(co - coy) is the Dirac delta function for a discrete lineshape, and J(-co) denotes the current operator defined as ei • J ( - w ) = ( . . ~ 1 ~ \ mec / m

~exp[-i(co/c)(e3-rmy)](e /

i "Pray).

(15)

Here,rm] and Pm] are the position and linear momentum operator, respectively, of the electron / in the chromophore m. It should be noted that thermal effects and the solvent index of refraction, which were formally retained in the formulation of the referenced papers [19,20], have been neglected for simplicity in the present work. The current operator of eq. (15) is transformed by the first-order radiation expansion into a multipolar form truncated up to quadrupolar order ei • J ( - w ) = ~

e x p [ - i ( w / c ) ( e 3 • rm) ]

m

× ((i/ch)[H, (e i • Din) ] + i(6d/c) e i • (e 3 × Mm) + (6d/2c2h)[n, (e i • Qm " e3)]},

(16)

whererm denotes the center point of the chromophore m, H denotes the total hamiltonian, and Din, Mm and Qm are the electric-dipole, magnetic-dipole and electric-quadrupole transition operator, respectively, referred to the chrornophore center. The transformation procedure is equivalent to a particular canonical transformation of the radiation perturbation hamiltonian by use of a generating function defined in terms of a first-order Taylor expansion about chromophore centers of the radiation vector potential [21 ]. By applying the transformed current operator of eq. (16) to the excited eigenstates of the whole chromophoric system expanded in a complete set of singleexciton basis IX), the circular dichroism distribution formulated by eq. (14) reduces to the form:

( 16 3NA

Ae(co) = ~ch -~5 ~ - { 0 } ~

~

~

rt,/.t n ' , ~ '

Pv(C~) Im(exp[-i(c°v/c)(e3 " rnn')] (~l~v><~vl~;)

× [½(D~ X D ~ : ) . e3 + [D~ • (M~;)*] - (D~ • e3)[(21~;)* • e3] + i(wu/2c)((D~n × e3)" Q~: • e3)]},

(17)

wherernn, is the vector distance pointing from the center of chromophore n to that of chromophore n'(rnn' = rn, - rn). It should be noted that the summation suffix v refers only to the excited eigenstates of the whole &rornophoric system which have counterparts to the excited states of the unperturbed dye aggregate. In deriving eq. (17) the ground state is taken to be exciton vacuum state, and DUn and Q~n are assumed to be real and M~n is assumed to be pure imaginary since all chromophoric eigenfunctions have been regarded as real. ~v is the excitation eigenfrequency of the excited state of the whole chromophoric system. To eliminate the phenomenological bandshape contributions, we consider a chirality index, to be called R hereafter, which corresponds to the imaginary (Ira) part of eq. (17) and which represents the sum of the circular

342

M. K a m i y a / D y n a m i c perturbation effects

dichroism intensity under consideration. The chirality index R can be measured experimentally by integrating over frequency the circular dichroism curve ~ch 103 in 10 1 R -= t- 16~-N~ ] ;

Ae(co)dCO ' w

(18)

o

If the retardation phase factors in eq. (17), exp [ - i ( w v / c ) ( e 3 • rnn')], are negligible because the size of the d y e polymer complexes is small enough compared to the wavelength of incident light, the chirality index R reduces to the familiar rotatory strength terms in the dipole approximation. To proceed further, we substitute eq. (12) into the term (q~[~v) of eq. (17). Then, the chiral coupling amplitudes between transition moments on the perturbers and those on the aggregated dyes, which should make substantial contributions to the dynamic perturbation effects upon the chirality index R, are expressible in the form: exp[-i(cov/c)(e3 " rxm)] (q~l~v)(qJvl¢~,)

Jxr, ms =- ~ = __

~ p

xt,r

~

exp[--i(cov/c)(e

' m~,s

3

.

--1 r x m ) ] (I + A ) x r , x , r , A x , r , , m , s ' ( 4 )

s' , m [qjv)@vl4~m).

(19)

'

To eliminate the retardation phase factors in eq. (19), these are expanded in a first-order Taylor series exp [-i(cov/c)(e3 • rxm)] ~ 1 - i(wv/c)(e3 • r x m ) .

(20)

With this and with a first-order expansion of ( I + A ) -1 ~ I - A one obtains from eq. ( 1 9 ) Jxr, ms = (1/2h) ~

~

I)

+(1/2h) ~ ~ V

m

r Igl Cm')(CPm' s' s' I¢~><¢~1G1 (COx)[ r ¢~><¢~Iq~m s > (Ox

,s

~. . . .~ X ,r

~ r' ~' ,' ~' ((~xlVl(~x,)(~x,iVlq~m,)(C~m,l~v)(~vlGl(Jx) r' )l ~v)(OJui gPrn) s G1 (COx'

~'l , S

-- (i/2hc)(e3 " rxm) ~

~, , ((9 rx IV] ~)m')(~) s' s'm' I ~ v ) ( t ~ v IG2 (¢Ox)[ r ~ v ) ( ~ v I(Orn s )

V

rn ,s

-- (i/4h2c)(e3 • rxm) ~

~, , ~, , ( O ~x l V l O x r', ) ( O x ,r' l V l ~ ms', ) ( ( ~ ms', l t ) v ) ( ~ v l G a ( c O x ,r C O,'x , ) l ~ v ) ( ~ v l O ms) ,

l~

X ,r

(21)

rH , S ,

where the operators GI(CO~), r G2(eOx) r and G3(cox, r c@) are expressible in terms of the resolvent operator defined for the total hamiltonian H of the whole chromophoric system including the perturber-dye interactions: G(_+)(c4) =

1 r + it/' ( W h - Fo/h) ± ~ x

(22)

GI(CO r ) = G(-)(co r ) - G(+)(cor),

(23)

a~(~,)=

(24)

a 3 ( ¢ J xr,

r (-) (Wx) r + G(+)(c°r)] , COx[G c o rx' ' ) = 1

, r G2(c@) + Gl(COx') r' G2(coxl] r [Gl(cOx) .

(25)

Here co~ denotes the excitation eigenfrequency of an isolated perturber. The full resolvent operator G(-+)(cor) is introduced through the effective equality lim

1 r

n-*+O 09 + i77 ± 69x

- P

1 r

CO ± COx

irrS(w + w r ) ,

(26)

M. Kamiya /Dynamic perturbation effects

343

where the symbols P and 6 represent the principal value and the Dirac delta function, respectively. Note that the symbol P is omitted from the G(-+)(cor~)operator of eq. (22) which is now used in conjunction with a summation over the discrete eigenstates [ff~). Since the subset of {l~v}} corresponding to the excited states of the bound dye aggregate is not complete for the total hamiltonian H, Jug,ms of eq. (21) is not transformable directly into a closed-form expression. To proceed further, we now assume

1~°,))@(0)1~) ~ Iff(°)),

~,

i.e. <0~ [~,) ~ (0~ I~(°)).

(27)

v

It follows then from eq. (27) that

<~la(_+)(~)l~> = + 23 r r' r' , , (~v(o) [G (+_)(~Ox)lOx,>(Ox,14,~> x,r

+ Z) . <~.lof,><~',lo(-+)(~5,)lv,(y)> . . . . . . . + 23 X ,r

X ,r

23 (~lOx')(Ox'lG r, r' (~) (~x)lOx")(~Zl~v), r /, r"

(28)

X ,r

where the summation suffices x', r' and x", r" refer to various single-exciton states of the perturber system. If the third summation terms of eq. (28) due to interactions between the perturber single-exciton states are omitted from consideration in the limit of the weak perturbation interactions, substitution of eqs. (12), (13) and (27) into eq. (28) leads to the following linear equations with respect to the diagonal matrix elements of the operators G(+-)and Gt-+) both defined for a set of the perturber slngle-exclton • . states

(~v[Gl(f.D~')lt])v}(((~' [Wll~(yo))(l~/(v°)lG(+-)(cor)l(~',)

(~[G(~)(w~)iffv) + (1/2h) ,~, X,E

(29)

Note that eq. (29) has been derived using the identity I

r I

S

S

(30)

(0%,lrl q;~°)>.= 23(Ox, lrl()m)(Om[~(°)). ;T/, S

It follows from eq. (29) that

Z;i (i

t x,r

+ B1)xr, x'r' (~vlG,(~')l~v)=(~(vO)lGl((-Dr)[

l~(v0 ) )

,

(31)

and thus (~v[Gl(~rx)l~v) -- ~

(I +

r' (o) Ba)~x'r'(~(°)lGl(Wx')lt~v }

X,r

, x,r

,

(32)

where (o) (o) [GI( ~ rx)l~:,,)+ r' (,~',IG,(~)I@(0))@(0)IVIq)~',)) (B0x~,~,/ = (1/2h)((O~,lVl~,,)@. It follows from eqs. (29) and (32) that

(33)

344

M. Kamiya / Dynamic perturbation effects

<~,vlO:(cor)le~> = <~,(.O)lo:(fo[<)l~,(~o)> - ~,

~,,

X ~#" X

(B:)x.,x,.,(l

+

B,);,~,,x,,/,

,1"

(~,(O)lG2(fo~)l ~(o)) - ~, , (B2)xr, x ' / ( $ v (o) IGl(fox')l~v / (o) ) ,

(34)

X,r

where

(II2)x.,x, r, = (1/2h)((O/, l Vl ~°)><$~°)1G2(foDI ¢>~',>+ <~{, t G2(for)i¢,~°)><¢,(,,°)I Vl ~/,)).

(35)

By substituting eqs. (27), (32) and (34) into eq. (21), Jxr ms is reduced to a form expressible by the overall • . + r • summation over the unperturbed aggregate elgenstates I$(°)).' Furthermore, by expanding G(-)(fox) in terms of the practical resolvent operators defined for the independent-chromophore system and by truncating up to second order the resultant interchromophoric interactions in Jxr, ms, we obtain finally a simple closed-form expression for Jxr, ms based on the enclosure property of I$ (°)) for the unperturbed aggregate hamiltonian: Jxr

fox

'

ms =

r

(l/h) • s i-(fom)

( l / h ) 2 m~s

[(cos) 2

'

r s', s' s ~¢'°x(fOm ~+ COrn') <~xlVl¢~)((~m'lVlOm) - ( f o 2 l [(fom') -- (foS<)~]

[(co~)2- -

+ (l/h) 2

s

r 2 (~xlVl(Pm)

-- (fOx)

(

fo~fox' , $7,~-,

(COX)

for (.Ds

] [(fom)

2

<¢~1 -- (foSc")2]

/

~'

VlOx')<~x'lVl~m)

]

- (i/hc)(e3 • rxm){~ s ~ X - - ~ r.21 <¢srlVl¢~> ( tfom) - tfox)

~ x [~m fore' + (rex) 1

+ (i/hac)(e3 " rxm)

s i - - 7--~r~- - - ~ - - [(fOnt)

-- (i/h2c)(e3 • rxm) x

-- (fOx)

] [(fOm')

r z

-- (fOx)

]

fOxfOx'fOm

7 ~ , 2 ,], 7[(tOm) ~2 [(foSm)2_ (fOx)

__ (fo~,)2]

(qsr[vi ~m'>(0m'l s' s' Vl~>

<~rl Vl~x')<~x'lVlqJm) "' "' s

= / o 2 ~ s _ ( ~ . ,x~) #)!m~ •

(36)

Since the coulombic exciton interactions between chromophores fall off rapidly with the increase of interchrombphoric separations, the consequence of eq. (36) indicates that the first-order Taylor expansion of the retardation phase factors can be used with impunity. Note that there is no singularity due to the denominators contained in eq. (36) since the excitation energy levels of the perturbers have been assumed to lie far outside those of the aggregated dye chromophores. After substitution of eq. (36) into the leading contribution terms of the perturbed part of the chirality index (R'), ~ I m [ J x r . m s ( - ~ (1 D x rX D S z )

R'=~

X,r m,s

* m s ( ~, ( D ms • e3 } + Jxr,

X

D r ) • e3}

+Jxr, ms([Drx " (M~)*] - (Dr " e3)[(MSm) * " e3]}+Jxr, * m s ( [ D ms "(Mr) *] _ (D sm . e3) . [(Mr) * . e31) 1

1)

r

r

+ ~J(xr, ms(l(Dx X e3)" Qsm • e3] + [(DSm X e3)" Qx " e3l)],

one obtains

(37)

345

M. Kamiya / D y n a m i c perturbation effects

R'= ~ ~Jflx°,.!ms Im{[D~ • (MS) *] + [DSm • (M~)*] - (D~" e3)[(MSm)*" e31 - (DSm" e3) [(M~)* • e31} X,Y ;'gl,S

_ ~ ~_Jlm[J(xlr!ms](rxm • e3)[(D~: X D S ) • e3] X,Y m , s

+51 ~

lm[fi~)ms] {[(D~,

X e3)" Q~m " e3] + [(D s X e3)" Q~: • e3]}

(38)

X,/" m , s

The space average of R' over random orientations is readily derived using the averaging formula of Av[(ei • A ) ( e j • B)l = ~(A • B) ~ 0 ,

(R'):~ ~ f l ° ) r , , -s,

lm{[D r

.

and

A v [ ( e , • A ) ( e 2 • B ) ( e 3 • c)] = 4 • (B X

(MS,n) *] + [DS~ . (h~x)* ]}

1

c)/6"

5 ~ i m [ f l x r . m s l ( D,)x X D S mr) . r x m

X,P gn,S

"

(39)

X,r m,s

The D-D contribution terms in (R') correspond to the important helical terms of the optical activity of an infinite linear helix which can be incorporated only by the correct helix selection rules [22,23]. Note that the D-Q contribution terms are averaged to zero by the symmetry properties of electric-quadrupole moments. Similar results are obtained for the unperturbed part of the chirality index: R (°) = ~

I m { [ / ~ • (MS) *] - (D~m • e3)[(MSm)* • e3]}

-- ( 1 / c h )

• • (Om[V[d~m'}(rmrn' "

e3)[(OSrn X O S , )

•

e3]

gtl ,S I~lr,S ~

+ (1/2ch)

2 2, (~nlVt~,}((DSmX e3)" QSn~,"e3),

(40)

~'/,S tY/ ,S

where the summation suffices m, s and m', s' extend over the single-exciton basis belonging to the bound dye aggregate. After the space average of R (°) over random orientations, the D - M contribution terms in eq. (40) vanish by the conservation rule for rotatory strengths of individual dye chromophores, and the D - Q contribution terms are averaged to zero. If all of the bound dye chromophores are assumed to be inherently achiral so that the center origins of them can be chosen in such a way as dropping the magnetic-dipole transition moments referred to the chromophore centers, the D - D contribution terms in the space-averaged (R (°)) reduce to a closed-form sum over rotatory strengths of the aggregate transitions and thus vanish again by the conservation rule. This is readily known by the commutation relation between the total linear momefitum and electric-dipole operators and by the permutation property of the summation suffices m, s and m', s'. Since the magnetic-dipole transition moments on chromophores are usually much less than the corresponding electric-dipole transition moments, the conservative.property of the unperturbed circular dichroism intensity is valid approximately even without the assumption of inherently achiral dye chromophores. 4. Application of the theoretical formulation to the non-conservative circular dichroism intensity induced in proflavine dyes bound to double-stranded polynucleotides In this section formula (39) in space-averaged form will be applied in order to investigate the non-conservative circular dichroism intensity induced by the dynamic perturbation effects at the first ~r-+ o* transition of proflavine (3,6-diamino-acridine) dyes bound to either poly(A-T) or poly(G-C). The circular dichroism induced at the visible absorption band of bound proflavines, which is assigned to the first rr -+ 7r* transition polarized along the molecular long axis, has proved to be a useful information source for the dye-binding mechanism and the host

346

M. Kamiya / Dynamic perturbation effects

polymer helicity as well [ 3 - 5 , 2 4 - 2 6 ] . The helical angle and stacking distance for both poly(A-T) and poly(G-C) were set to 36 ° and 3.4 A, respectively, by analogy with the double-stranded DNA helix. The origin of the base-pair chromophore units, which is now substituted with the position of the helix axis, was set to the middle point between the N(1) atom of the pyrimidine base and the N(9) atom of the purine base each of which forms the glycoside linkage. Geometrical structures of the A - T and G - C base-pairs were taken from the Watson-Crick model. The origin of the proflavine chromophore unit was set to the center point of the N-containing hexagonal ring. In order to avoid the origin dependence of the calculated rotatory strengths, the electric-dipole transition moments as well as the magnetic-dipole transition moments were calculated using the dipole velocity operator formalism [27]. The electronic excited wavefunctions of the chromophore units were determined using the Pariser-ParrPople method [28,29] including singly excited configuration interactions, where the methyl group in thymine was treated referring to a hyperconjugation model [30]. Coulombic interactions between excited chromophore units were calculated using the dipole expansion approximation, where the interactions of a given chromophore unit with the remaining ones were restricted within a center-to-center separation equal to the single-turn height along the helix axis since the interactions were found to converge rapidly with further increase of interchromophoric separations. Dye-binding mechanisms adopted in the present calculation were set up on the basis of the intercalation models which have been regarded as an important source for the non-conservative induced circular dichroism. On the contrary, the conservative property of the circular dichroism for an isolated helical array of acridine dyes externally stacking outside the DNA helix core was treated by Tinoco et al. by exciton-splitting theory [31 ]. According to the internal intercalation model proposed by Lerman et al. [1,32,33], proflavine is inserted into the polynucleotide helix core with its molecular plane parallel to the stacking plane of the base-pairs and possibly with its molecular long axis along a radial direction so that the stacking interactions with adjacent base-pairs might be maximized. According to the modified intercalation model firstly proposed by Pritchard et al. [34], proflavine is half-inserted between the successive nucleotide bases attached to the same sugar-phosphate chain, where the positively charged acridine-N atom is retained in electrostatic interactions with the neighbouring negatively charged phosphate group. In model A for the internal intercalation, the proflavine long axis was set along the helical radius which bisects a helical angle of the successively stacking base-pairs between which the relevant proflavine is intercalated. In model B for the modified intercalation, the long axis of proflavine half-inserted with its amino-groups positioned externally was set parallel to a tangential direction perpendicular to the above-considered helical radius. The distance between the helix axis and the proflavine origin, to be called the intercalation distance hereafter, was varied in a range that does not exceed the observed helical radius of the double-stranded DNA of about 10 A. In both models proflavine intercalation was assumed to take place at successive intercalation sites located between the stacking base-pairs. The chirality index representing the dynamic perturbation effects upon the non-conservative circular dichroism intensity induced at intercalated proflavines is plotted against the intercalation distance in figs. 1-4. In the case that proflavines are intercalated in poly(A-T) or poly(G-C) by the model A mechanism, the curves obtained for various intercalation numbers have positive sign at intercalation distances shorter than about 1 )~, and they take negative minimum at an intercalation distance of about 2 A. On the contrary, the corresponding curves for the model B type intercalation have negative sign and fall deeply as the intercalation distance becomes shorter. Since the. molecular plane of proflavine intercalated by models A or B is taken to be perpendicular to the helix axis, the triple scalar product of rxm (Drx × DSm) contained in the D - D contribution terms which are now restricted within interactions between ~ ~ ~r* transitions on the chromophore units does not actually depend on changes in the intercalation distance. Consequently, the distinct intercalation model dependence of the curves may be regarded as reflecting the configurational effects of the intercalation model upon the coulombic interactions of the ~ -~ 7r* transition dipoles of intercalated proflavines with those of the neighbouring base-pairs. It is worth noting that the positive sign of the curve as associated with the model A type intercalation of a "

M. Kamiya / Dynamic perturbation effects (

-'" t0 c.g.s, u n i t

347

)

5 '> (

'" IO c.g.s, u n i t )

5 4

3

2

1

0

N=I N=I

-1

-2 1~

I

[

I

I

I

I

0

i

2

3

4

5

6

r(~)

Fig. 1. The
0

I

I

I

I

~

2

3

z,

r(,&

[

I

5

6

Fig. 2. The (R'> values (in c.g.s, units) plotted against distance r (in A) from the helix axis to the origin of proflavine intercalated with poly(G-C) by model A. N denotes the number of intercalated proflavines.

single proflavine close to the helix axis of p o l y ( A - T ) as well as p o l y ( G - C ) is in agreement with the positive nonconservative circular dichroism which is observed at the visible absorption band of proflavine isolatedly intercalated in the double-stranded DNA helix core in the limit of high phosphate-per-proflavine ratio [3], whereas the negative sign of the corresponding (R') curves for the model B type intercalation fails to reproduce the observed non-conservative circular dichroism intensity. Though definitive conclusions have to be avoided in the present calculation which does not include chirality sources other than the dynamic perturbation field and the helix deformation of host polynucleotides concurrent with the proflavine intercalation, the distinct intercalation model dependence of the (R'> curves as predicted for isolatedly intercalated proflavine suggest that the preferable orientation of the long axis of the deeply intercalated proflavine should be radial rather than tangential at least when the repulsive forces between the intercalated proflavines are not operative. Recent circular dichroism anisotropy studies [26] indicate that the molecular plane of proflavine intercalated in the DINA helix is inclined slightly from the plane perpendicular to the helix axis. Then, instead of models A and Bboth assuming completely planar intercalation, we consider a modified intercalation model, to be called the model C, where the long axis of the intercalated proflavine is directed radially like in model A but is not perpendicular to the helix axis. For the model C type intercalation of a single proflavine into the helix core of p o l y ( A - T ) or poly(G-C), the curves w.ere calculated by changing angle formed by the helix axis and the proflavine long axis. The relevant consequences shown in figs. 5 and 6 indicate that the intercalation distance dependence of the (R') sign characteristic in the model A type intercalation tends to be counterbalanced by increase of non-planarity between the stacking plane of the base-pairs and the molecular plane of the intercalated proflavine.

<~>(

( I0 c.g.s, u n i t ) 0

-,7 io c.g.s, u n i t )

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O

-1

-3


-2

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-4

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-5

-4

-6

-5

-7

./

0

I

I

[

I

I

l

1

2

3

4

5

6

r(Z) Fig. 4. T h e values (in c.g.s, units) p l o t t e d against distance

Fig. 3. The values (in c.g.s,units) plotted against distance r (in A) from the helix axis to the origin of proflavine intercalated with p o l y ( A - T ) by model B. N denotes the number of intercalated proflavines.

r (in A) from the helix axis to the origin o f proflavines intercalated with p o l y ( G - C ) by model B. N denotes the number of intercalated proflavines. (RX c.g.s, u n i t ) [

× i0 -J|

7 ~_ ( Io- " c.g.s, u n i t )

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3

t 4

I

J

5

6

]

o

I

i

~

3

I

4(~)

I

I

s

6

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Fig. 5, The values (in c.g.s, units) plotted against distance r (in A) from the helix axis to the origin of proflavine singly intercalated with p o l y ( A - T ) by model C. 0 denoted angle formed by the helix axis and the proflavine long axis.

Fig. 6. The values (in c.g.s, units) plotted against distance r (in A) from the helix axis to the origin of proflavine singly intercalated with p o l y ( G - C ) by model C. 0 denotes angle formed by the helix axis and the proflavine long axis.

M. Kamiya / Dynamic perturbation effects

349

5. Supplementary remarks For the particular case that the unperturbed excited states of the aggregate are expressible by non-overlapping exciton bands each of which is described by a correct degenerate set of single-exciton basis, the present formulation is readily applicable to treating the dynamic perturbation effects of the remaining exciton states upon the partial circular dichroism intensity due to absorption transitions where the final states are limited to energy sublevels belonging to one of the separate exciton bands. If the exciton band forms a quasi-continuous spectrum so that the perturbed eigenstates depend on the effects of intraband exciton scattering as well, these must be determined in correlation with excited scattering eigenstates for the exciton band which will be derived by solving the integral equation of scattering theory in the Born approximation. The first-order Taylor expansion of the retardation phase factors can be used with impunity only when adopting the Frenkel exciton model which guarantees a rapid decrease of coulombic exciton interactions in a large separation between chromophores. In this context it should be mentioned that the non-newtonian property of coulombic exciton interactions does not fulfill some of the general higher-order sum rules in closed-form for rotatory strengths because this property does not allow the commutation of the total hamiltonian with the total dipole-velocity or angular momentum operator [35]. The phase variation due to the retardation between oscillators belonging to different chromophores was neglected in the present treatment since it is much smaller in linear systems compared to the phase variation caused by the optical wavelength. It is worth noting that, when applied to an infinite linear helix system, the first-order Taylor expansion of the phase variation factors leads to a consequence which is compatible with the use of the correct helix selection'rules [22,23] in the long-wavelength approximation. If the excited eigenstates of a chromophore aggregate are represented by the helical Bloch function t~a(k) and band energy Ea(k) both labelled with band state a and exciton wavevector k, and if the helix axis is taken to be parallel to the propagation direction of incident circularly p0Iarized light, one obtains the relation of

(~s [~a(k)/~k ) = i(e3 • rm)(~ s ]~ba(k)). Substitution of this relation into the closed-form sum over the D-D contribution terms typified by the one appearing in eq. (40) leads to the following transformation

- ~

~ ((pSm]V](Ps/n')(e3"rmm')[(DSm × DSm') " e31

m,s mf,s '

=~

~

~ ~Im[i(¢Sml~a(k))Ea(k)(q2a(k)lOS/n')I(rm -rm')" e3]

m,s m',s'

x ((o%

•

a

k • e_)

-

(D m •

• e+))]

a L~-k(Ea(k)lDa(k)

(41)

which is completed by use of the relations (DSm × D ~ , ) " e3 = i((DSm " e+)(DSm' ' e_) - (DSm " e_)(DSm ' ' e+)}

(42)

and

where e+_ = 2 -~/2 (el -+ ie2), and S is the wavenumber corresponding to a wavelength equal to the helix pitch. In the long-wavelength approximation the consequence of eq. (41) corresponds to the first-order Taylor expansion

350

M. Kamiya / Dynamic perturbation effects

term o f the difference in oscillator strengths for left and right circularly polarized light propagating down the helix axis which can be formulated by use o f the correct helix selection rules [22,23]. The space-averaged and closed-form formulation o f the perturbed part o f the chirality index R contains the dipole and rotatory strength terms which are obtainable b y the stationary first-order perturbation theory. In principle the present formulation should be more elaborate compared to the perturbation treatment since the former starts from regarding as self-consistent perturbation the linear response interactions between chromophore transition moments. The truncated formula of eq. (36) is adaptable for a practical calculation o f the non-conservative circular dichroism intensity which is induced in absorption transitions of any aggregate o f dye chromophores bound to a large host biopolymer and which changes depending on intensity sharing with the perturber transitions lying outside the aggregate transitions. It is worth noting that the van der Waals correction terms involving difference between excited d y e - g r o u n d perturber interactions and ground d y e - g r o u n d perturber interactions were neglected in deriving eq. (36) since these are much smaller compared to the corresponding exciton interaction terms. This approximation is compatible with the neglect o f the permanent moment difference between the ground and excited state of chromophores under consideration. Since the intrinsic optical properties o f isolated chromophores can be measured from the absorption spectrum and optical activity o f a dilute solution of chromophores, the present formulation may be convenient for calculating the induced non-conservative circular dichroisrn intensity as a function of the geometrical factors which determine interchromophoric exciton interactions, and for searching the reasonable dye-polymer configuration interactions which reproduce the observed circular dichroism intensity.

Acknowledgement The author acknowledges fundamental supports by Professor Yukio Akahori to practical use of the formulation which will be published elsewhere. Thanks are also due to Professor Masamichi Tsuboi for reading this manuscript. The author's interest in the circular dichroism of biopolymers owes much to the early guidance of Professor W. Curtis Johnson Jr. o f Biophysics Laboratory, Oregon State University, where the author stayed partly in support from Public Health Service Grant. References [1] L.S. Lerman, J. Mol. Biol. 3 (1961) 18. [2] D.M. Neville Jr. and D.R. Davis, J. Mol. Biol. 17 (1966) 57. [3] H.J. Li and D.M. Crothers, Biopolymers 8 (1969) 217. [4] K. Jackson and S.F. Mason, Trans. Faraday Soc. 67 (1971) 966. [5 ] D.G. Dalgleish, E. Dingsoeyr and A.R. Peacocke, Biopolymers 12 ( 1973) 445. [6] M.R. Philpott, J. Chem. Phys. 55 (1971) 4005. [7] J.G. Kirkwood, J. Chem. Phys. 5 (1937) 479. [8] W.W. Wood, W. Fickett and J.G. Kirkwood, J. Chem. Phys. 20 (1952) 561. [9] H. DeVoe, J. Chem. Phys. 41 (1964) 393. [10] A.D. McLachlan and M.A. Ball, Mol. Phys. 8 (1964) 581. [11] H. DeVoe, J. Chem. Phys. 43 (1965) 3199. [12] E.G. Hohn and O.E. Weigang Jr., J, Chem. Phys. 48 (1968) 1127. [ 13 ] D.G. Barnes and W. Rhodes¢ J. Chem. Phys. 48 (1968) 817. [14] W. Rhodes, J. Chem. Phys. 53 (1970) 3650. [15] A.R. Ziv and W. Rhodes, J. Chem. Phys. 57 (1972) 5354. [16] W. Rhodes and S.M. Redmann, Chem. Phys. 22 (1977) 215. [17] R. Kubo and K. Tomita, J. Phys. Soc. Japan 9 (1954) 888. [18] R. Kubo, J. Phys. Soc. Japan 12 (1957) 570. [19] P.J. Stephens, J. Chem. Phys. 52 (1970) 3489. [20] S.H. Lin, J. Chem. Phys. 55 (1971) 3546.

M. Kamiya / Dynamic perturbation effects [21] Aa.E. Hansen and E.N. Svendsen, Moh Phys. 28 (1974) 1061. [22] F.M. Loxsom, Intern. J. Quantum Chem. $3 (1969) 147. [23] C.W. Deutsche, J. Chem. Phys. 52 (1970) 3703. [24] A.R. Peacocke, Chemistry of heterocyclic compounds, Vol. 9, ed. R.M. Aeheson (Wiley, New York, 1973). [25] V.L. Makarov, A.I. Poletaev and M.V. Volkenshtein, Mol. Biol. 11 (1977) 228. [26] A.I. Poletaev, P.G. Sveshnikov, V.L. Makarov and M.V. Volkenshtein, Stud. Biophys. 67 (1978) 55. [27] R.R. Gould and R. Hoffmann, J. Am. Chem. Soc. 92 (1970) 1813. [28] R. Pariser and R.G. Parr, J. Chem. Phys. 21 (1953) 466. [29] J.A. Pople, Trans. Faraday Soc. 49 (1953) 1375. [30[ Y. Morita, Bull. Chem. Soc. Japan 33 (1960) 1486. [31] I. Tinoco Jr., R.W. Woody and D.F. Bradley, J. Chem. Phys. 38 (1963) 1317. [32] V. Luzzati, F. Masson and L.S. Lerman, J. Moh Biol. 3 (1961) 634. [33] L.S. Lerman, Proc. Natl. Acad. Sci, US 49 (1963) 94. [34] N.J. Pritchard, A. Blake and A.R. Peacocke, Nature 212 (1966) 1360. [35] Aa.E. Hansen, Mol. Phys. 33 (1977) 483.

351