J . Phys. Chem. 1988, 92, 4863-4868 spin-rotation coupling constant for 13C in singly labeled benzene may also be compared to theoretical and experimental chemical shift data. Calculated principal axis components of the paramagnetic shielding tensor, up(xx) = -446.57 ppm, .,by) = -351.56 ppm, and u (zz) = -307.27 ppm, evaluated at a carbon nucleus in b e n ~ e n e ~ ~consistent ~ a r e with a C eof~ 1712 Hz, which is close to our experimental value. The estimated Cenfor 13C can be used to estimate the density at which Tl is a minimum for I3C nuclei in benzene by using the relationship
A = (./4r2)(1
/Ced)(WI/Prnin)
(6)
where A is the slope in the extreme narrowing region and the other parameters are defined above.22 At 381 K, the calculated pmin is 1.32 mol/m3 (24.08 Torr). Although data were obtained at densities as low as 1.0 mol/m3 in the present study, the data were not of sufficient quantity or quality to detect a departure from linear density dependence for the lower density data points. The lack of appreciable dipole-dipole spin-lattice relaxation for the I3C nuclei is an interesting result of this study. In the extreme narrowing region, Tl,dd is given by (Tl.dd)-' = CddTO
(7)
where Cdd is the dipole-dipole coupling constant and T~ is the molecular reorientation correlation time. Tlgrcan be calculated from eq 4. The dipole-dipole coupling constant, Cdd, for a 13C nucleus and attached proton is 2.00 X 1Olo s-*, assuming an effective bond length of 1.101 A2.14 The quantity (4r2/a)Cef:
4863
is 1.38 X 10" s - ~by using the spin-rotation coupling constant estimated above. A N O E enhancement factor, 7, of 0.14 is expected at densities well above the T1minimum, by assuming that the geometric and angular momentum reorientation correlation times, TO and 75,are identical. Experimentally is substantially longer than Tl,srand does not contribute significantly to the relaxation process. The uncertainty in the experimentally determined NOE factor, 7,of 2% allows an estimate of the lower limit for Tl,dd to be made. For 7 of 0.02, the corresponding lower limit T1,dd is 3 s at 10.8 mo1/m3 and 20 s at 80.7 moi/m3. The estimated lower limit Tl,dd values are consistent with molecular framework reorientation correlation times of 1.7 X lo-" s and 2.5 X lo-'* s at 10.8 and 80.7 mol/m3, respectively. The angular momentum reorientation correlation times determined above for the benzene protons are 2.68 X 10-'0 and 3.41 X lo-" s, respectively, at 10.8 and 80.7 mol/m3.
Conclusion Spin-rotation interactions provide the dominant spin-lattice relaxation mechanism for both protons and I3C nuclei of gaseous benzene. 13Crelaxation is facilitated by the larger spin-rotation coupling constant, and (13C)benzene TI values in the gas phase are ca. 1000 times shorter than in the liquid phase. Acknowledgment. We are pleased to acknowledge support from the National Science Foundation (CHE-83-511698-PYI and CHE-85-03074) and the Alfred P. Sloan Foundation for support of this research. Registry No. Benzene, 71-43-2.
Circular Dichroism Band Shapes for Helical Polymers David A. Rabenold Biochemistry and Biophysics Department, Iowa State University, Ames, Iowa 5001 1 (Received: October 26, 1987; In Final Form: March 14, 1988)
Three approaches are discussed for handling band shapes in circular dichroism (CD) and absorption spectra calculations for helical polymers. The effect of interaction between chromophores upon polymer transition band shapes is explored by employing Gaussian band shapes for isolated chromophore transitions in the classical coupled oscillator scheme. The results are compared to those obtained from using Lorentzian band shapes and to those obtained by using 6 function line shapes that are replaced by Gaussians after interactions are taken into account. The three approaches yield very different calculated AT* absorption and circular dichroism spectra for an a-helix. Skewing of band shapes derived from input Gaussians is analyzed by employing input bands that are composite bands of several Lorentzians and mimic the input Gaussians. Skewing is shown to result from coupling of different vibronic components. Distortion of the CD helix band is discussed by using an approximate formulation based on periodic boundary conditions that includes end effect corrections. The classical scheme is also related to the so-called matrix method.
Introduction Calculations of the circular dichroism (CD) and absorption spectra of helical polypeptides have been based on a variety of formulations. Use of the so-called matrix method yields polymer rotational and oscillator strengths and polymer transition frequencies to which band shapes are assigned.'+ On the other hand, a classical coupled oscillator scheme, with Lorentzian band shapes, all of the same half-peak width, for the oscillators yields the same Lorentzian band shapes for the normal modess-' In another
method,8 a time-dependent Hartree schemeg-" is used that is equivalent to the decorrelation approximation,12each normal mode of which obtains a &function line shape that to do calculations is replaced, e.g., by a Gaussian. The latter two methodss,8 can yield identical polymer rotational and dipole strengths and polymer transition frequencies. All of the above methods yield polymer transition frequencies that are shifted from unperturbed values
(1) Woody,R. W.J . Chem. Phys. '1968, 49, 4797. (2) Pysh, E.S.J. Chem. Phys. 1970, 52, 4723. (3) Ronish, E.W.;Krimm, S.Biopolymers 1974, 13, 1635. (4) Madison, V.;Schellman, J. Biopolymers 1972, 1 1 , 1041. (5) Applcquist, J.; Sundberg, K. R.; Olson, M. L.; Weiss, L. C. J. Chem. Phys. 1979, 70, 1240; 1979, 71, 2330.
I70 ._.
0022-3654/88/2092-4863$01.50/0
(6) Applequist, J. J. Chem. Phys. 1979, 71, 1983;1979, 71, 4324;1979, 71,4332;1980, 73, 3521. (7) Applequist, J. Biopolymers 1981, 20, 387;1981, 20, 2311; 1982, 21,
.
(8) Rabenold, D. A,; Rhodes, W. J. Chem. Phys. 1986, 90,2561 (9) McLachlan, A. D.;Ball, M. A. Mol. Phys. 1964, 8, 581. (10)Harris, R. A. J . Chem. Phys. 1965, 43,959. (11)Rabenold, D.A. J. Chem. Phys. 1982, 77,4265. (12)Rhodes, W.;Chase, M. Reu. Mod. Phys. 1967, 39, 348.
0 1988 American Chemical Society
4864
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988
by amounts determined by interactions present in the system. In these approaches interactions do not affect polymer transition band shapes. Except for the above mentioned classical scheme these methods are considered to be strong coupling theories.13 They are considered to be applicable to systems for which the largest interaction between subunits is greater than the half-peak width of the absorption band involved in the intera~tion.’~ Weak coupling theories, considered to be applicable to systems for which the largest interaction between subunits is less than the above mentioned half-peak width, allow for interactions to affect polymer band shapes by using band shapes of absorption spectra of the isolated subunits as input factors in the calculation of the C D and absorption spectra. 13,14 The purpose of this paper is to study the effect of interactions on the component aa* CD and absorption bands of an a-helix. We initially employ what is essentially DeVoe’s classical normal mode polarizability f o r m ~ l a t i o n . The ~ ~ use of input bands that resemble chromophore absorption bands, e.g., Gaussians, yields polymer bands that are skewed in shape.I6-I9 Our objective is to study interaction effects on the shapes of aa* C D helix and radial bands and on the shapes of the aa* absorption bands for perpendicular and parallel polarized transitions and also to analyze the mechanisms that yield skewed band shapes. The analysis is carried out by employing a composite input absorption band, made up of several vibronic components, that equals the Gaussian-shaped absorption band in a least-squares sense. To discuss the skewing of the CD helix band, we use an approximate formulation based on periodic boundary conditions that also includes end effect corrections.20a
Theory and Calculation Three Types of Band Shape Inputs. We consider an isotropic solution of identical a-helical polypeptides with each polymer containing N identical repeating units. Each isolated repeating unit is characterized by a complex polarizability. For example, for the j t h repeating unit
Rabenold
+ it)2. The lime.+, is taken after, e.g., the polymer rotatory parameter is obtained to yield &function polymer line shapes. The CD is given by 5 , 1 1 ~ 1 5 At@) = ( 2 a / c ) C ( ~Im ) @(o)
(6)
where C(8) = 4 N o ( 2 a c ~ ) 2 / ( 2 3 0 3 cin) , which No is Avogadro’s number and where the rotatory parameter @ ( s ) isz2
P(B) = (?,/3hcN)CDgi
X
D%*Rjk{[B(v)+ ( 2 / h c ) ~ , W ] - ’ ) j k
ik
(7) in which Rjk = Rj - Rk is the vector distance between repeating units j and k. Wjk,an element of W , is the Coulombic coupling of aa* transition charge densities on repeating unitsj and k and Bjj($ = Bo($ is an element of B(v). The normal model problem [ B ( B ) (2/hc)s,W] is N X N in dimension and is diagonalized with its eigenvector matrix Y as follows:
+
cyjn[BO(3)6jk + (2/hC)3ryklYkm = (BO(v) + 2vrEnn)6nm (8) jk
which gives
([B(n) + (2/hc)i7,WI-’1jk = CI;nYkn/[Bde) + 2vnEnnI n
to yield @(3)
= ( 8 , / 3 h ~ N ) C D g iX D$$RjkCqnYkn/[Bo(B) ik
+ 2B,E,]
n
(10)
The strength factor D& X D$-,)-Rjk in eq 10 is decomposed into helix and radial parts by employing the structure of the helix for which Bo and d respectively are the angle of rotation about the helix axis and displacement along the helix axis between sequentially repeating units.* This gives
+ R h + R?k
(11)
H,$ = -Zj&,*Dd
Sin (SoZjk)
(12)
Rj’k = -2b&:&
COS
(SoZjk)
(13)
DFi
X
D$j*Rjk = H h
where
&(B) = (2/hc)o,D&D:d[g’(~) + ig”(n)] = ( 2 / h c ) ~ , D &D 9 Bo(8)-’ ( 1 )
RJk = 2b&,D5 where h is Planck’s constant and c is the speed of light and where, for a Lorentzian dispersion,21 B0(3) = 8, - v2 - ivr, in which r is the half-peak width. D f i is the unperturbed air*electric dipole transition moment in repeating unit j . For a Gaussian dispersion
g”(p) = [ ~ ~ / ~ / ( 2[exp(-(v, ~ , 7 ) ] - B ) ~ / Y ~-) exp(-(v,
+ ~ ) ~ / - y (~2 ) ] (3)
(9)
(14)
in which so = Bo/d and z j k = Zj - z k = d(j - k ) . bo is the distance from the helix axis to the location of the m*transition moment and D&-Dd = (gr)2 (D&)2. Substituting eq 1 1 into eq 10 and summing over j and k yields
+
= (BW/3hcN)C[H,I, + R,I, + Rinl/[B0(8)+ 2PJnnI
P(v)
n
(15) where, e.g.
H,I, = CH1iI;nYkn jk
where a principal value integral is indicated, which gives
g’(n)/[g’(v)2+ g”(s)2]
(4)
B$(B) = g”(v)/[g’(s)2+ g”(v)2]
(5)
Bo’(8) =
and where H a n d R refer to helix and radial band strengths and where the symbols 1and I/ are used because the net bands tend to be located at the positions of the perpendicular and parallel polarized absorption bands. The as* absorption spectrum is given by
For the method in which &function line shapes are replaced by Gaussians, B is replaced by B it to give B0(n i t ) = i j r 2 - ( 8
+
+
(13) Tinoco, I., Jr.; Bustamante, C.; Maestre, M. F. Annu. Reu. Biophys. Bioeng. 1980, 9, 107. (14) Cech, C. L.; Hug, W.; Tinoco, I., Jr. Biopolymers 1976, 15, 131. (15) DeVoe, H. J . Chem. Phys. 1964, 41, 393; 1965, 43, 3199. (16) Briggs, J. S.; Herzenberg, A. Mol. Phys. 1971, 21, 865. (17) Ziv, A. R.; Rhodes, W. J . Chem. Phys. 1972,57, 5354. (18) Redmann, S. M., Jr.; Rhodes, W. Biopolymers 1979, 18, 393. (19) Ito, H.; Eri, T.; I’Haya, Y . J. Chem. Phys. 1975, 8, 68. (20) (a) Rabenold, D. A,; Rhodes, W., to be submitted for publication. (b) Rabenold, D. A,; Rhodes, W. Biopolymers 1987, 26, 109. (21) In ref 17-19 this dispersion is called a Kettelar Helmholtz dispersion.
C(B) = CA(B) Im X(B)
where
CA(3) =
~ ( 8 )=
(16)
(8a2N0/2303)Sand where
(2s,/3hcN)E(D,I, + D!n)/[&(v) + 2eJnnI n
(17)
in which the dipole strengths D,& and Din are given by (22) @(s) in ref 5 is half of @(P)in eq 7. The prefactor for the CD in eq 6 is ( 4 ~ / c ) C ( s )in ref 5. Reference 5 is based on the work in ref 22a while the @(s) that we employ here is derived in ref 22b via a multipole expansion of the transverse current linearly induced in the system by the light. (a) Appliquist, J. J. Chem. Phys. 1973, 58, 4251. (b) Rabenold, D. A. J . Chem. Phys. 1975, 62, 376.
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988 4865
C D Band Shapes for Helical Polymers
D,&
+ Din = C(D&*Dd COS (SoZjk)
D&D"&)qnYkn
""
(18)
jk
For Lorentzian dispersions the imaginary part of the inverse [Bo(v) 2 ~ ~ E ~in~ eq 1 - 15 l and 17 is
IA
+
+
Im [BO(e) 23,Enn]-1 = Im :D[
+
- ij2 - i e r 2er E nnI-' = 3r/[(p: - 1 2 ) ~ ~ 2 r 2 1(19)
+
+
where :e = vT2 23,Enn. For the case of replacing 6 functions by Gaussians we have Im lim
[v:
- (8
f-0
+ ic)z + 2v,Enn]-'
=
+
(*/28,)[6(8, - 2 ) - 6(Dn e)] (~~/~/2v,y)[exp(-(v, - ~ ) ~ / -y exp(-(e, ~ )
-+
~ ) ~ / y (20) ~)]
For a Gaussian dispersion
+
Im [Bo(s) 2i~,E,,,,]-l = B'$(V)/[(B'o(V)
+ 28,E32 + B'b(e)2]
(21)
For calculations we put the m*transition at 1900 A. We use the parameters of Ronish and Krimm3 for the *a* transition moment. We also use their split monopoles in computing elements of W. The AB* transition moment is arbitrarily located on the / ~ y = 3300 carbonyl carbon. The band widths are r = 2 y / ~ ' with cm-'. Bond lengths and angles for the repeating unit are those used by Applequist.6 All calculations are for a-helices with 6 = -47.1O and = -56.0°, which yields Bo = 99.963', b - 1.616 A, d = 1.552 A, @, = 1.043 D, D& = -2.192 D, a n d d , ; 1.850 D. Figure 1, parts A and B, respectively, display the C D helix and radial bands for N = 18 obtained for the three cases of input band. For the a-helix the perpendicular and parallel polarized radial bands respectively appear as high- and low-energy bands. The CD helix bands have first-derivative-like band shapes. Figure 1C displays the net TT* CD for the three cases. The more intense bands result from the Gaussian input band. Figure 2, parts A and B, respectively display the component and net AT* absorption bands for the three cases. Again, the more intense and/or skewed bands derive from the Gaussian input band. Composite Band Approach: Coupling of Vibronic Components. To obtain understanding of the nature of the skewing of the absorption and radial C D bands, we consider in eq 1 to be a sum of vibronic components, namely
-80
I
I
I
I
I
I
I
I
1
I
B
6ot 40
+
B ~ ( V ) =- ~C(s,/e,)f,/[e,2 - e2 - ier,]
-
20
A€ 0 -20
-40 I
-60
C 6o
t
40
(22)
0
where e, = ij, - nut and Ed, = 1. The components 8, are symmetrically distributed about e,. The ~,,f,, and roare chosen in a least-squares fit so that
E(v,/v,)f~ro/[(n:- 32)2 + 82roz] N
20
A€ 0
u
( ~ ~ / ~ / 2 y eexp(-(8, ,)
- 8)z/y2) (23)
Our objective here is to show that by employing eq 22 in P(e) there is an interaction between individual vibronic components and that the effect of this interaction yields skewed bands. There are at least two routes to take, both of which are identical but which appear to be quite different on the surface. For the first route we substitute eq 22 into eq 7 and expand the inverse to obtain P(V)
= (1/3hcN)CCDg? X D$j*Rjk(Jddv8T)'/ZX jk
(6jk6,,r/(3,2
07
- 8* - i3ro) - (2/hC)(8ddTe,)'/'Wjk/ [(e2 - e2 - ier,)(e,2 - e2 - i~r,,)] + ...I (24)
To handle eq 24, we first relabel indices so that the pairsjo and kr respectively refer to a and P. This enlarges the dimensions of the problem. We then define Tap = D& X D~~*Rjk(8$dr3r)1/2
sa, = (ad&T)"2wjk Oa6,,
= Qj,6,,
-20
-60
160
I
I
I
I
I
170
180
190
2W
210
WAVELENGTH (nm) Figure 1. (A) Helix bands, B) radial bands, and (C) net *I* CD respectively for Lorentzian (-), 6 function replaced by Gaussian (+), and Gaussian dispersions (0). Then by putting the geometric series in eq 24 into closed form, we obtain
@(e) = ( ~ / ~ ~ C N ) ~ -TI(r2 ~ ~+ (i8Fo) [ O +~ (2/hc)S]-'JaB US
(25)
For M vibronic components per repeating unit the order of the normal mode problem is N M . The form of eq 25 displays the coupling of individual vibronic components. For large M the
4866
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988
6
Rabenold
" he
-a0
WAVELENGTH (nm)
V,[Bo(V)
+ 2V,E,,]-'
+
= B,Bo(V)-' - P,BO(V)-' 2VffE,$0(V)-' ... = XF,,{[v2 - I($ + iVr0) 2Ennl;l-1)0T(26)
+
UT
where Fur = ( v J J p , ) l / z . This approach yields N normal mode problems each of order M , which is the number of vibronic components. We diagonalize the normal mode problem in eq 26 for each n with the eigenvector matrix X(n) as follows: EX,,(n)[(n,Z - v2 - ivro)6uT+ 2E,,For]X,,(n) = r,
( ~ ~ (- nV~ )-~inI'o)6sl (27)
Employing eq 27 in eq 15 gives
I
I
I
B
-d IEQ
Figure 2. (A) Component and (B) net AT* absorption bands respectively for Lorentzian (-), 8 function replaced by Gaussian (+), and Gaussian dispersions (A).
elements of S are small because of the small weighting factors (fJT)1/2. But there are many components contributing to make a significant effect, which is the skewing of the bands. This skewing, however, is hidden in the formalism. Extracting it requires numerical calculation. The drawback of this approach is that it yields a normal mode problem that has an impractically large order. Our second approach is to substitute eq 22 into v,[B,(v) + 2ij&,J1 in eq 15, which is the result of diagonalizing the normal mode problem in eq 7. Making this substitution, expanding the inverse, and then closing the geometric series yield
i
I
I
I
I
I
I
170
iao
190
200
210
220
WAVELENGTH (nm) Figure 3. (A) Component and (B) net AT* CD bands respectively for Gaussian (-) and composite dispersion (+).
where F,(n) = C,J,,&(n) X,(n). Thus, for each normal mode n the factors Fss(n) differ from the originalfp, because of the coupling of individual vibronic components. For example, we expand [Bo(v) 2~$,,,]-l in terms of the diagonal elements of the inverse in eq 26 to obtain the approximation
+
vJB,(v)
+ ~v,E,,]-~= ES,,/[v,Z -
ij2
- inr,
+ 2E,,Fu,] (30)
where Suu
=
Fuo
- 4Enn C F n / [ v , Z rfo
V?+ 2Enn(Fn - Fw)I
(31)
For u = p , corresponding to a high-frequency component, the tendencies Spp> Fpp
for E,, positive
< Spp
for E,, negative
S,
are obtained. Each normal mode band shape, Le., the imaginary part of G,(B) = ZFss(n)/(v,2- v2 - iijr,)
(32)
S
R!,)zF,,CX,,(n) UT
Summing over P(5) =
CJ
and
s 7
X & ) / ( V , ( ~ )-~v2 - ivr',) (28)
in eq 28 gives
is skewed toward high (low) energy for positive (negative) E,,, in agreement with the work of Briggs and Herzenberg.16 Figure 3 A displays the component CD bands obtained from a composite input band of nine components for which ro= 1836.9 cm-' and a Gaussian input band for which y = 3300 cm-'. Figure 3B displays the net m* CD bands for these cases. The differences between the curves in these spectra are due to a lack of a perfect
C D Band Shapes for Helical Polymers TABLE I: Values of f,, 8, F,,(n)/ij,, and
0.100 -2125
fc
8,
- 8,
8 , - 8, -1511 ( n = -142 ( n = 660 ( n = 1376 ( n = 1817 ( n = 1874 ( n =
F,,(n),
vI.
and
8,
- D,
for Which There Are Significant Dipole or Rotational Strengths'
0.114 -1062
0.107 -1593 s = l 0.514 0.1 13 0.048 0.026 0.0 19 0.019
5) 9) 11) 14) 15) 16)
ij,
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988 4867
s = 2 0.114 0.116 0.060 0.035 0.026 0.025
0.123 -531
s = 3 0.089 0.120 0.072 0.043 0.032 0.03 1
0.131
0.121 531
0
s = 4 0.074 0.126 0.088 0.053 0.039 0.038
s = 5 0.061 0.130 0.110 0.069 0.051 0.049
0.110 1062
s = 6 0.048 0.116 0.123 0.085 0.063 0.060
0.101 1593
s = 7 0.040 0.104 0.135 0.105 0.077 0.074
0.092 2125
s=8
s = 9
0.033 0.093 0.153 0.137 0.098 0.093
0.025 0.082 0.210 0.446 0.594 0.61 1
are in cm-'.
least-squares fit of the composite band to the Gaussian in eq 23. The near agreement of the curves shows that assigning to each repeating unit one transition with a Gaussian dispersion is essentially the same as assigning to each repeating unit one transition with many vibronic components that have Lorentzian dispersions. Table I displays the vu andf, used for the composite band and the F s s ( n ) / 8 ,for which there are significant dipole or rotational strengths. In Table I energy increases with increasing values of n and s. For example, for a particular n for which E,, > 0, 8, - 8, is positive and the value of F,(n)/8, increases with increasing energy as s increases. For this case the band shape is Im G,(P) in eq 32, which is a sum of Lorentzians with greater weigths given to the higher energy bands. That is, the distribution of F,(n)/s,, which is normalized for each n within the limits of the least-squares fit, shows the origins of the skewing of the absorption and radial C D bands. Periodic Boundary Conditions with End Effect Corrections. To analyze the skewing of the C D helix band, we consider ideal a-helices with 3.6 repeatingunits per turn and with N = 18, and we extend a previously developed approximate formulation based on periodic boundary conditions. For this case we approximate Bo N looo. The formulation yields the rotatory parameters asZh P(8)
= PH(3)
+ &(a)
(33)
where
PH(D) = (8,/2hc)D,I,.D,o( [Bo(F)+ p * W ~ o )- q,~,W~o)l-'[Bo(i~) + 8,Wso) + qP,Wso)I-')(l/q) (34) in which qr = q cos 0, where q = 2 a / h is the light's wave vector and where 0 is the angle between the helix axis and the direction of light propagation. The angular brackets indicate an average over orientations, namely c f ( q ) / q ) = (1/2q)Jxd0
sin 0 cos Of(q cos 0)
(35) -60'
I
W
I
I
I
I
1
N- I
V ( s o )= 2
(1 - m / N ) cos (somd) W ( m d ) ( 2 / h c )
(36)
WAVELENGTH (nm)
m>O
N- 1
( m - m 2 / N ) sin (somd) W ( m d ) ( 2 / h c )
LIs(so)= 2d
(37)
m>O
in which the W ( m d ) are elements of the first row of W in eq 7. Expanding the inverses in eq 34 and averaging yield
= (1 /~~C)V,~D&*D&.~US(SO) [Bo(J)+ P,UC(SO)]-~
&(J)
For
(38)
&(a) we have
p ~ ( p )=
Figure 4. (A) Component and (B) net m r * CD bands respectively for Gaussian dispersions calculated exactly (A)and calculated with the use of periodic boundary conditions (-).
displays the net aa* C D bands. The more intense bands are derived from eq 38 and 39. Our objective now is to analyze the CD helix band with the use of eq 34, which with eq 22 we rewrite as PH(8)
(2J,boo~,o~oo/3hC)([Bo(P) + Prp(0)]-I J,~(So)l-'l
=
-
[&(8)
P W +Prim
+ (39)
=
(1 /2h~q)D&.D,~( CFu,([8* - I(p2 .J7
q,US(so)F]-l-
[v2 - I($
Diagonalizing [s2 - I($ N- I m>O
+ i8r0) + V(sO)F+ ~,US(S~)F]-~),,,) (41)
where V ( 0 )= 2 C (1 - m / N ) W ( m d ) ( 2 / h c )
+ i8ro) + V(so)F-
(40)
For Gaussian input bands Figure 4A displays the CD component bands obtained from eq 15, 3 8 , and 39, while Figure 4B
+ i8r0)+ UC(so)F]as follows
CXu,(so)[(v,Z - v2 - iijro)fju,+ Wso)FurlX,,(so) = UT
(P,(S~)~ - i j 2 - i8r0)6,, (42)
yields
4868 {[p2 -
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988
I(v2
+ ier,) + UC(so)F f qzC'S(so)F]-l}u,= + iero)
XAso){[i4fo)2 -
CXUS(S0) UT
* q z W s 0 ) Fb0)IHSf (43)
where F(s0) = X(S~)-'FX(s0)
(44)
Substituting eq 43 into eq 41 and expanding the inverses in terms of their diagonal elements yield PH(J) = (1/2hcq)D,I,'D,(C[F,,(sO) + Hm(qz + S
- v2 - iero - q z W s 0 ) F,,(So)l - C[F,s(so) +
so11
S
HJq,
+ SO)]/
-
e2 - iero + q z W s o ) F J s o ) l ) (45)
where
*
H d q z $0) = f2qzWfO) CF,,(so)2/ [ef(so)2 -
qz(F,,(so) - Fs,(so))l
tfs
(46)
For each component s in eq 45 a CD helix band results from the difference of two closely spaced bands of unequal intensity. These two bands are unequal in intensity due to the chiral hypo/hyperchromism displayed in eq 46. This yields a skewed CD helix band for each component and a skewed net CD helix band, which remains conservative. The above intraband effect is similar to the interband effect discussed elsewhere.20b For practical purposes of calculation, expanding the denominators in eq 43, keeping terms that are linear in qz,and averaging over orientations give P H ( ~= ) (1 /3h~)~(so)D~.D,oC(F,,(s,)~/ [ e A ~ o-) ~e2 S
ier0l2 + 2 C ( ~ , , ( s ~ ) ~ / [ e -~ ~( ,s (~S)~~) ~ ] ) / [ J ,-( Se2~-) ~iero]} t#S
(47) which gives the CD helix band as a sum of bands that are skewed in shape. Substituting eq 22 into eq 39 yields the rotatory parameter for the radial bands as
&(v)
= (2~,b,D&D$/3hc)
X
{CF,,(O)/[J,(O)~ - ez - iero] - CF,s(so)/[e,(so)2 - ez - iero]] I
S
(48) in which Fss(0)and ~ ~ ( are 0 )obtained ~ respectively with eq 44 and eq 42 with so set equal to zero. Discussion Within the following approximation for positive frequency absorption Im JIde :8(
- e - ieI'-I
N
( r / 4 e , ) S m-md e [(e, - e)* + r2/41-' (49) the three types of input bands for an isolated m*transition (Lorentzian, 6 function replaced by a Gaussian, and Gaussian) all have the same integrated intensity. Since we choose r = 2y/a1l2, they also have the same peak intensity at 8 = 8, for the isolated chromophore. However, in the normal mode formulation they yield significantly different m*absorption and C D spectra for the polymer. The skewing of the input Gaussian can be understood in terms of the coupling of different components of the composite band, although using more components would be more realistic and allow roto be smaller. Since the periodic boundary conditions results
Ra benold yield spectra that are close to those obtained by the more exact approach, they also help to explain the distortion of the various bands. This approachzoayields three normal modes, two of which are essentially degenerate. Equation 48 for the radial rotatory parameters simply contains two sets of bands. The CD helix band characterized by eq 34 and 38 is seen to be a difference of two skewed bands. The net CD helix band is skewed, conservative, and a sum of skewed first-derivative-like bands. The interactions present in the system are less than the bandwidths. For example, eq 36 and 40 respectively yield V(s0)/2 = 1101 cm-' and V(0)/2 = -1408 cm-'. Therefore, on the surface it appears that the proper approach is to employ input band shapes obtained from absorption spectra of isolated repeating units, Le., the weak coupling method. This method also yields splitting in the absorption spectrum that is more in line with experiment. Of course, more transitions would have to be included to obtain agreement with experiment. In closing, we relate our work to that in the literature in the hope of stimulating future work. By approximating Bo(2)-' in eq 22 as b --I= o(u) (1/2%)%/(SU - e - iF0/2) (50) 0
the matrix m e t h o d ' ~ ~ . ~ J form, e.g., for the polymer polarizability is obtained as 1,23v24
aM(v)= (1 /3h~iv)CD&.D$)([b(e)+ W/hc]-]],k
(51)
Jk
in which an element of b(v) is bo(^) given by eq 50. & e()' in eq 51 is an excellent approximation for x(e) in eq 17 because only one electronic transition per repeating unit is considered. For example, for a single Lorentzian dispersion for each repeating unit the resonance frequencies for aM(e) are V, = 2, E,,, which are close to those for x(e), which are vn = v,[ 1 + ~ E , , / D , ] ~ / ~ . With a Green operator method and their coherent exciton scattering approximation Briggs and HerzenbergI6 have derived what is essentially eq 5 1, which is the weak coupling scheme if b o ( ~ ) -is] given, e.g., by eq 50. Hemenger, also using a Green operator technique, has derived the degenerate ground-state approximation, which has been shown to be superior to the weak coupling scheme for a wide range of interactions.2s28 This method does not yield severe distortion of the band shapes. But, for realistic model aggregates it is not readily amenable to numerical calculations. However, for helical systems by employing periodic boundary conditions the calculational complexity is reduced,28and, as shown here, by including end effect corrections the method should yield informative results. In applications of the weak coupling method the interactions are assumed to be screened, by as much as 50%, which reduces the intensities of the CD bands and brings them in line with e ~ p e r i m e n t . ' ~ ~This ~ ~ -local * ~ dielectric effect is one handling of the need for the screening. An alternative explanation might be deviation from the fixed nuclei approximation or a failure of the coherent exciton scattering approximation. Screening the interaction array W by 50% essentially yields, for our model a-helix, a T P * CD that is reduced in intensity by 50%. The input Gaussian method with screening then yields a CD that is closer to the CD derived from the method of replacing &function line shapes after the interactions are accounted for by Gaussians. The effect of polymer band distortion is thus reduced by this screening method.
+
Acknowledgment. I am grateful to Professor Jon Applequist for support through Grant No. 5 R01 G M 13684-20 from the National Institute of General Medical Sciences. (23) Bayley, P. M.; Nielsen, E. B.; Schellmann, J. A. J . Phys. Chem. 1969, 73, 228. (24) Rabenold, D. A. J . Chem. Phys. 1976, 65, 4850; 1980, 73, 5942. (25) Hemenger, R. P. J . Chem. Phys. 1977, 66, 1795. (26) Hemenger, R. P. J . Chem. Phys. 1977,67, 262. (27) Hemenger, R. P. J . Chem. Phys. 1978, 68, 1722. (28) Hemenger, R. P.; Kaplan, T.; Gray, L. J. J . Chem. Phys. 1979, 70, 3324. (29) Rizzo, V.; Schellmann, J. A. Biopolymers 1984, 23, 435
