J . Phys. Chem. 1986, 90, 2560-2566
2560
at higher energy. In this correlation the basic variable seems to be primarily the distance between traps and not cluster size.35 Closely trapped pairs emit at higher energy because of the ground-state Coulomb attraction which exists between traps of opposite charge. They also tunnel faster due to a greater exchange (wave function overlap) matrix element. The excited-state decay properties seem to be a determined by pairs of trapped carriers and not by isolated trap (i.e., defect) electronic states. In this sense they bear some resemblance to molecular states, specifically to the biradical states often observed in organic chemistry.37
T I M E RESOLVED EMISSION SPECTRA
S H O R T DELAY LONG DELAY
I
360
440
520
600
680
7
I
EMISSION WAVELENGTH ( n m )
Figure 7. Time-resolved emission spectra of luminescence from N 22-A CdS clusters in a frozen organic glass at N 10 K, adapted from ref 34. Short delay refers to the period 0-1.5 X 10” s, and long delay to the period (17-34) X IO” s. There is no resolvable vibronic structure at 1-8, resolution under the present conditions.
-
which should have a purely radiative lifetime of E s. The broad emission spectrum in Figure 7 and the temperature dependence of the lifetimes indicates that the electrons are strongly coupled to lattice phonons.35 The long lifetimes indicate that the hole and electron are separately trapped at different locations in the crystallite. Decay at low temperature occurs by a combustion of radiative and nonradiative tunneling of the electron back to the hole. In the luminescence spectra, there is a correlation between the emission wavelength and the rate of decay. Faster emission occurs
Final Remarks Metal and semiconductor clusters represent entire new classes of large molecules. Their structural and dynamical properties contain novel aspects and should be of interest to the molecular spectroscopy and chemical physics communities. Our understanding of their chemistry and physics is growing rapidly, yet remains primitive. I believe that the potential for additional interesting discoveries is present. It is already clear that semiconductor clusters must grow to a very large physical size before bulk electronic and optical properties are present. The clusters themselves may find useful chemical and optical applications. Acknowledgment. It is a great pleasure to acknowledge the essential contributors of my collaborators-J. M. Gibson, R. Hull, and S. Nakahara in electron microscopy; and R. Rossetti, N. Chestnoy, and T. D. Harris in synthesis and luminescence spectroscopy. K. Ragavachari, T. D. Harris, and C. J. Sandroff offered valuable suggestions on an earlier version of this manuscript. (37) Turro, N. J. Modern Molecular Photochemistry; Benjamin/Cummings: Menlo Park, CA, 1978; Section 7.8.
SPECTROSCOPY AND STRUCTURE a-Helical Polypeptide Circular Dichroism Component Band Analysis David A. Rabenold? and William Rhodes** Institute of Molecular Biophysics and Department of Chemistry, The Florida State University, Tallahassee, Florida 32306 (Received: June 24, 1985; In Final Form: January 27, 1986)
Equations are presented for calculating circular dichroism component bands for short helical polymers. Also presented are equations for calculating the effect of high-frequency transitions, which are approximated by transition moment dyadics, on the low-frequency region of the spectra. For a rigid model of an a-helix calculations show the following: (1) The isotropic part of the transition moment dyadic for high-energy transitions located on the amide is responsible for large induced long-wavelength circular dichroism contributions. (2) The net long-wavelength circular dichroism helix band is skewed with the negative high-frequency lobe being significantly reduced in intensity in comparison to the unperturbed helix band. (3) The chain length dependence of the circular dichroism is largely due to that of the helix band.
I. Introduction Calculations of the circular dichroism (CD) of helical polypeptides have been successful to various degrees, e.g., for the a-helix’” and for poly-L-proline (lI).%’ Theoretical formulations of the CD for infinitely long helices with two transitions per chromophore, the m r * and the n**, predict a conservative spectrum with five or six bands, namely one helix band, two radical Institute of Molecular Biophysics. ‘Institute of Molecular Biophysics and Department of Chemistry
0022-3654/86/2090-2S60$01.50/0
bands, and the n r * band which is balanced in intensity by a contribution(s) at the position of one or both radial bandssa (1) (a) R. W. Woody, J. Chem. Phys., 49,4797 (1968); (b) R. W. Woody and I. Tinoco, Jr., J . Chem. Phys., 46, 4927 (1967). (2) F. M. Loxsom, L.Tterlikkis, and W. Rhodes, Biopofymers, 10, 2405 (1971). (3) J. Applequist, J . Chem. Phys., 71, 4332 (1979). ( 4 ) E. S . Pysh, J . Chem. Phys., 52, 4723 (1970). (5) E. W. Ronish and S. Krimm, Biopolymers, 13, 1635 (1974). ( 6 ) J. Applequist, Biopolymers, 20, 2311 (1981).
0 1986 American Chemical Society
Circular Dichroism Analysis of a-Helical Polypeptides Inclusion of high-energy transitions predicts a possible change in intensity of the radial and n r * bands and a change in shape and intensity of the helix band.8a Such work on very long he lice^"^ has provided a guide for resolving experimental spectra into component bands. I6v1' Recently, Applequist has presented results of calculated C D He spectra for a variety of helical polypeptide applied the dipole coupling approximation, a coupled oscillator t h e ~ r y , which ~ ~ , ~is~the same as the time-dependent Hartree (TDH) ~ c h e m e , ~and l - ~a~dispersive-nondispersive normal-mode partitioning technique. The results of the C D calculations are presented as whole spectra with no decomposition into helix and radial bands. Our purpose in this paper is to show how such a decomposition can be made for helices much shorter than the wavelength of the light. This should help facilitate comparison of calculated spectra with experimental spectra that have been resolved into component bands and also help lend more understanding of how the low-energy bands are perturbed by the high-energy transitions located on the amide, backbone, and side chain. Our approach is to employ the structure of the helix to obtain expressions for the rotational strengths that are more explicit than those usually used. Our work incorporates exciton-exciton system coupling and a partitioning technique and yields expressions for each of the various bands that are related to the.bands predicted for infinitely long helices. By this approach some of the details may be found as to which high-frequency transitions and their polarizations or which polarizabilities and their anisotropies are important in influencing the low-frequency spectrum. Experimental C D spectra of a-helices do not display the negative high-frequency lobe of the m*helix band.25 The helix band extracted from an experimental spectrum was found to be onefourth as intense as that predicted by theory for infinitely long helices.16 Also, we have recently shown for infinitely long helixes that long-range solvent effects tend to increase the intensity of the helix band.26 An nu* transition, assumed to be weakly electric dipole allowed and strongly magnetic dipole allowed and assumed to contribute positive C D in the region of the negative lobe of the helix band, has been suggested as a candidate that might overshadow the negative However, Applequist's work3 suggests that electric dipole allowed high-frequency transitions located on the amide-backbone and side chain contribute enough to the CD through coupling of the low- and high-energy systems to overshadow the negative lobe. Our effort here is also to make a contribution that may help resolve this puzzle.
(7) V. Madison and J. Schellman, Biopolymers, 11, 1041 (1972) (8) (a) D. A. Rabenold, J . Chem. Phys., 74, 941 (1981); (b) J . Chem. Phys., 74, 5988 (1981); (c) Chem. Phys. Lett., 79, 86 (1981). (9) W. Moffitt, D. Fitts, and J. G . Kirkwood, Proc. Nutl. Acad. Sci. U.S.A., 43, 723 (1957). (10) I. Tinoco, J . Am. Chem. SOC.,86, 297 (1964). (11) T. Ando, Prog. Theor. Phys., 40, 471 (1968). (12) F. M. Loxsom, J. Chem. Phys., 51,4899 (1969); Phys. Rev.B: Solid Srure, 1, 858 (1970); Int. J . Quantum Chem., Symp., 3, 147 (1969). (13) C. W. Deutsche, J. Chem. Phys., 52, 3703 (1970). (14) W. Rhodes, J . Chem. Phys., 53, 3650 (1970). (15) M. R. Philpott, J . Chem. Phys., 56,683 (1972). (16) R. Mandel and G.Holzwarth, J . Chem. Phys., 57, 3469 (1972). (17) R. Mandel and G . Holzwarth, Biopolymers, 12, 655 (1973). (18) J. Applequist, Biopolymers, 20, 387 (1981); 21, 779 (1982). (19) J. Applequist, J . Chem. Phys., 58, 4251 (1973); J. Applequist, K. R. Sundberg, M. L. Olson, and L. Weiss, J. Chem. Phys., 70, 1240 (1979); J. Applequist, J. Chem. Phys., 71, 1983, 4324 (1979). (20) H. DeVoe, J . Chem. Phys., 41, 393 (1964); 43, 3199 (1965). (21) A. D. McLachlan and M. A. Ball, Mol. Phys., 8, 581 (1964). (22) R. A. Harris, J . Chem. Phys., 43, 959 (1965). (23) W. Rhodes and M. Chase, Rev.Mod. Phys., 39, 348 (1967). (24) D. A. Rabenold, J . Chem. Phys. 62, 376 (1975). (25) W. C. Johnson, Jr., and I. Tinoco, Jr., J. Am. Chem. SOC.,94,4389 (1972). (26) D. A. Rabenold and W. Rhodes, J. Chem. Phys., 80, 3866, 3873 (1984).
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986 2561 In section I1 we use the T D H scheme and a partitioning technique to obtain the electric dipole contribution to the C D of a system of helical polymers. Some discussion concerning anisotropic systems is also given. In section I11 we present a recipe for transforming vectors and tensors from the amide coordinate system to that of the helix, and in section IV we discuss the results of a simple application to a rigid a-helix. 11. Theory By neglecting contributions from intrinsic magnetic dipole transition moments, the CD of an isotropic system may be written as
e(@) = (i/3)[eH(O)'
+ &(o)' + eR(w)"]
(1)
where the symbols H and R are used to represent the helix and radial bands, respectively, and where the perpendicular and parallel notation is used because the bands tend to be centered at the transition frequencies of the perpendicular and parallel polarized absorption bands. In the TDH scheme, in which solvent effects are neglected, eq 1 takes the form2' e(w) =
( 2 ~ / c )Im p(w)
(2)
where
P ( 0 ) = b ( ~ ) ( 1 / 3 h ) C C ( w , w s ) " ~ Y , ~ Y -, ~z2)-'[Do, (Q~ X us
y
D~o*R,,- (4/h)CwoDool x P B ~ . R . ~ / -( ~Q,2)l O ~ (3) Y
where we have used a partitioning technique to handle the lowand high-frequency transition systems and their couplings and where the operator b(z) = (No/NcV lim, z2 in which No is the number of polymers each with N repeating units in a volume V and z = w k. Do,is an electric dipole transition moment for transition (Y with frequency w,. R,, is the vector distance between the positions of transitions a and p and is constructed according to the scheme used to label transitions. Y is the eigenvector matrix used to diagonalize the level-shifted low-frequency normal-mode equation, whereby
+
+
(2/h)w'/2WLLw'/2 (4/h2)o'/2WLHWHLW1/2wo/(wo2 - U;)]Y
Y-'[w2 - Iz2
=
n2- I Z 2
(4)
where represents the Coulombic coupling of transition densities for transitions a and on different unperturbed chromophores. WLHand its transpose WHLaccount for the coupling of the low-frequency system with the high-frequency system. Note that we have neglected high-frequency intrasystem coupling and we have assumed that all high-energy transition frequencies equal wg. w, is the unperturbed m*transition frequency. In the second half of eq 3 P,, has the form
p,, = D,o*T(R,7).Dopro
(5)
where D,DY0 is a transition moment dyad& for high-frequency transition y and T(RBY)= (R,,I-3[I - 3R,,R,,] is the dipole coupling tensor. Also Im lim z 2 / ( 0 : - z2) = (rw/2)[6(Q2,- w ) f-0
+ 6(Q, + w ) ]
(6)
where the 6 functions are replaced by Gaussians for calculations. Equations 2 and 3 give the C D in radians per unit length per repeating unit. For calculations we convert this to Ac units. W e consider the possibility of several chromophores per repeating unit, each with several transitions, and we sequentially label all low-frequency transitions according to increasing energy on chromophore 1 in repeating unit 1 followed by a similar labeling for chromophore 2 in repeating unit 1, etc., for the remaining chromophores in repeating unit 1, and then in the same manner we label the low-frequency transitions on the rest of the polymer. Next, the high-frequency transitions are labeled in the same manner. (27) D. A. Rabenold, J . Chem. Phys., 77, 4265 (1982).
2562 The Journal of Physical Chemistry, Vol. 90, No. 12, 1986
Rabenold and Rhodes For the absorption spectrum the dipole strengths become
+dt
= CY,,Yp,(Do,.D,
DBdZ sin
COS(^,^)
(rap)- (4/h)C[Do,.Pp, Y P , , ) ~sin
DllY
+ (Do,
-
cos
X
(rap)+ (Do, x
Q , ~ ) I ( W , W ~ ) ~ (24) /~Q~Y-~
+ dll = Y
C Y a Y Y p v ( G , ~-o(4/h)CW'(?,oo/(wo2 a@
- QY~)I(w,wo)"~QY-~
Y
(25) where the dot products in eq 24 omit the Z components. Equation 1 for the CD may now be written as O(w) =
(1/3)
[&(W)'
+
X 8{(W)'
+ 8 $ ( W ) ' l + AH(w)' + AR(w)' + A R ( W ) " ] (26)
where the superscript zero indicates the low-frequency system contribution to the CD. The A terms, stemming from Do, X PpY-R,,in eq 3 and handled according to eq 15, are corrections to this due to the coupling of the low- and high-frequency systems. For a mr* system O ~ ( w ) ' tends to have a derivative bandshape as the length of the chain becomes large. However, the perturbation AH(w) tends to have an ordinary Gaussian-like bandshape as the length of the chain becomes large.sa For light propagating parallel to the helix axis it is well-known that only OH(w)' results. (This, of course, is true only within the model which omits intrinsic magnetic dipole and electric quadrupole contributions to the CD.) When working with extrinsic magnetic dipole and electric quadrupole contributions to the CD, for light propagating parallel to the helix axis, the extrinsic electric quadrupole contribution yields ' / 2 0 H ( ~ ) i- '/20R(w)L while the extrinsic magnetic dipole contribution yields '/28H(W) + 1/28R( w ) with ~ the net result being OH(w)I. This had been shown before for short28and very long helices,8band Snir and Schellman have shown how the decomposition of the rotational strength tensor is artificial and leads to this cancellation.28 Use of the full rotational strength tensor yields just b $ + ( ~ ) ~ . For light propagating along the direction of orientation of a partially oriented system with a degree of orientationf, defined by Mandel and H~lzwarth,'~*'' the C D Op0(w) is 8''(W)
= (1
-f)$(U)
(27)
+feH(W)'
and thus
AO(W) = $'O(W) - $(LO) = ( f / 3 ) [ 2 8 H ( ~ ) ' - $R(W)'
- 8R(w)II]
(28) which is the result employed by Mandel and H~lzwarth'~*'' and later derived for infinitely long helicessb For completeness we point out that for light propagating perpendicular to the helix axis CD is not obtained because of the anisotropy of the polymer absorption spectrum. A nonlinear combination of linear and circular dichroism and birefringence results.sb,C For this case the induced transverse current has the strength factors, e.g., -'IZH& '/2R& derived from extrinsic electric quadrupole terms and '12H& + ' / z R $ + RLgderived from extrinsic magnetic dipole terms. The net result, R,$ R!@,is derived from the full strength tensor.
+
+
111. Geometry
Here we show how to transform from the amide frame to the helix frame. This derivation, in contrast to other^,^^-^' is based on three invariances to change of coordinate systems, distance (28) J. Snir and J. Schellman, J. Phys. Chem., 77, 1653 (1973). (29) T. Shimanouchi and S. Mizushima, J . Chem. Phys., 23,707 (1955). (30) H. Sugeta and T. Miyazawa, Biopolymers, 5, 673 (1967); 6, 1387 (1968). (31) H. Kijima, T. Sato, M. Tsuboi, and A. Wada, Bull. Chem. Soc. Jpn., 40, 2544 (1967).
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986 2563
Circular Dichroism Analysis of a-Helical Polypeptides between chromophore centers and dipole and rotational strength. Bond lengths and angles32 and the m*transition moment magnitude and direction5 are taken from literature sources. Well-known transformation procedures3*for transforming vectors from one frame to another along the polypeptide backbone provide the matrix that gives the m*transition moment Pi? on amide 2 in frame 1 as Pi: = WP& where W(4,Ji) is a function of the Ramachandran angles 4 and Ji. We put the center of the amide chromophore on the carbonyl carbon. The above-mentioned transformation technique also provides the means to obtain the position of the center of amide chromophore j referred to frame 1, which we indicate as Sj(4,Ji). We require the transformation matrix 9 that takes a vector from the amide to the helix frame as Do, = SPOT.We also need the helix structure parameters bo,Bo, and d. Matrix and vector algebra techniques used to obtain these are described in what follows. Noting that Di? = RDi; = OWO-'Di; gives R = 9W9-I so that T r R = T r W = 2 cos 0, 1 which yields the absolute value of eo. Since the square of the absolute value of the vector distance between two amide centers is independent of the coordinate system, the two equations
+
+ dz SI3.Sl3 = RI3.R13 = 2b,2(1 - COS 280) + 4 8 S12-S12 = Rl2'Rl2 = 2b02(1- cos 0,)
(29) (30)
provide bo and d. In order to determine the sign of Bo, we first use (dropping Or subscripts)
P(I)-P(~) = D(~).D(Z) = (oX2 + 0:) P(l).P0)= D(1).D(3) = (Dx2
+ D:)
COS
eo + ~~2
cos 2e0
+ Dz2
(31) (32)
to obtain Ox2 + Dy2 and Dz2. Then the two equations p(l) x P(2).Sl2= D(1) X D(2).RI2= 2b$yD~(l - cos 8,) - d(Dx2 02) sin Bo (33)
+
p(l)X
p(3).S13= D(I) X D(3).R13= 2bflYDz(l - cos 2B0) - 2d(Dx2 + 0:)
sin 200 (34)
provide DyDz and sin Bo, which with cos Bo specifies the sign of 0,. Next, to obtain 9 we set up three sets of three equations from R12 = QSI2, R13= ETl3, and RI4 = 9 S l l which give
which determines 9. This procedure, for 4 = -48' and Ji = - 5 7 O , ives D& = 0.986 D, Oyo, = -2.142 D, lu,, = 1.938 D, bo = 1.646 d = 1.509 A, and 8, = 98.7' and
1,
-0.389 0.853 -0.349
-0.137 0.321 0.937
0.911 0.412 -0.008
1
(36)
Repeating unit transitions that are not centered on the amid group have position vectors S,. These transformed from the amide frame to the helix frame as b:
COS
8, -bo (37)
which gives
z, = r',
[(e+ boy + ( r y ) 2 ] 1 / 2 = t a d [ r { / ( e + bo)]
bo, =
(38)
8,
(39)
Transition moments located at S, are transformed to the helix frame as
9Po, = D:, = 3?(0,)Dou
(40)
(32) P. J. Flory, Statistical Mechanics of Chain Molecules, Wiley, New
York, 1969, Chapter VII.
which gives
Dr,, = DXZ,
(41)
and the two equations
e,cos (e,)
(e,) = e, sin (e,) + ot;, cos (e,)
0:: =
- ot;, sin
(42)
D: (43) provide D& and WOu. Transition moment dyadics located at S, are transformed to the helix frame as 9PouP,,&'-1 = @,DL = R(B,)Do,D,oR(B,)-'
(44)
which gives
&&,,
= DP$Z
(45)
and three equations which provide D&D& 4 O , Y d ,and G p d and two equations which yield qpdand Dy,pd.
IV. Results and Discussion In doing the calculations, we employed one m*transition per amide a t 1900 8, plus three effective high-frequency transitions at 1000 8,. We used the standard parameters employed by Ronish ~ also used their and Krimm for the m*transition m ~ m e n t .We split monopoles in computing the Coulombic coupling of r?r* transition densities. All transitions were arbitrarily located on the carbonyl carbon, and all bandwidths were chosen to be 3300 cm-'. All CD calculations were for a-helices with 4 = -48O and Ji = - 5 7 O . We have employed two types of high-frequency transition moment dyadics. First, we used an isotropic dyadic obtained by averaging Applequist's bond-centered core p ~ l a r i z a b i l i t ywhich ~*~~ gives all(0)= L U ~ ~ (=O q) 3 ( 0 ) = 1.79 A3 for the components of the static polarizability. Second, we employed Applequist's ana l l ( 0 )= isotro ic bond-centered core p o l a r i ~ a b i l i t yfor ~ , ~which ~ 1.94 a ~ ~ (=0 0.76 ) A3,and C Y ~ ~ ( O=) 2.58 A3. For the isotropic dyadic, Figure la shows the unperturbed C D and its components, the helix and radial bands. The net unperturbed conservative C D has extrema of -10.9, +20.8, and -7.8 at 176, 191, and 208 nm, respectively. The extrema of the unperturbed helix band are -30.1 and 27.5 at 180 and 196 nm, repsectively, with the latter intensity being about 82% of that calculated for an infinitely long helix.1aJ6 The intensity of the perpendicular radial band at 187 nm is 43.8 while that of the parallel radial band at 196 nm, is -42.0, which are close to the calculated values for an infinitely long helix of 45.0 and -42.7, respectfully. Figure lb displays the C D perturbation and its components. The radial CD perturbations have Gaussian-like bandshapes with extrema of -9.5 and -3.5 at 186 and 197 nm, respectively. This amounts to intensity changes of (-9.5/43.8) X 100 N -22% and [-3.5/(-42.0)] X 100 = -8%. The perturbation of the CD helix band has a highly skewed derivative bandshape with extrema of 20.7 and -3.6 at 182 and 201 nm, respectively, which added to the unperturbed helix band in Figure l a yields the skewed helix band in Figure IC with extrema of -1 1.5 and 26.2 at 178 and 195 nm, respectively. The net CD band with extreme of -2.0, 23.4, and -8.8 at 173, 189, and 206 nm, respectively, is in line with what is experimentally observed. The high-frequency negative lobe of the unperturbed helix band is essentially cancelled by the result of the coupling of high- and low-energy transition systems, which is in agreement with the work of Applequist.3 However, an no* transition is probably present near 175 nm, which produces the shoulder in the observed spectrum.25 An important result of these calculations is that an isotropic high-energy transition moment dyadic is effective in producing the intensity changes of the radial bands and in skewing the helix band. The helical structure of the polymer provides the asymmetric environment that produces the induced C D components,
i3,
A.,-
(33) We have approximated Applequist's polarizability components a as a,,(O) where w5893 is the frequency corresponding to 589.3
(w5893)
2564
The Journal of Physical Chemistry, Vol. 90, No. 12, I986
Rabenold and Rhodes
TABLE I: Circular Dichroism Intensities (Ac Units) and Positions (nm) . . for a-Helices with N = 10., 14., 18., and 20 Amides" AE units (positions) N = 10 N = 14 N = 18 N = 26 (1/3)v (1 / 3 ) k L (1 /3)@ 80
(1/3)Ak ( 1/ 3111; (1/3)ab A (1/3YA (1/3)6( (1/3)8i7 B
-24.4 (180), 22.2 (198) 42.9 (188) -42.1 (195) -6.7 (177), 12.2 (191), -4.3 (208) 19.4 (183), -1.5 (204) -9.7 (186) -2.1 (197) 10.5 (181), -5.7 (199) -7.7 (178), 24.1 (195) 33.4 (188) -44.2 (195) -0.6 (171), 14.5 (188), -7.8 (205)
-27.7 (180), 25.2 (197) 43.4 (187) -42.0 (196) -8.9 (177), 17.7 (191). -6.9 (207) 22.9 (183), -2.4 (203) -10.2 (186) -3.2 (197) 13.8 (181), -8.4 (198) -7.9 (177), 26.5 (195) 33.3 (188) -45.1 (196) -0.8 (171), 19.9 (188), -12.0 (205)
-29.5 (180), 26.9 (197) 43.8 (187) -41.9 (197) -10.2 (176), 21.1 (191), -8.6 (207) 25.1 (182), -3.1 (202) -10.5 (187) -3.8 (197) 15.9 (181), -10.1 (1518) -7.9 (177), 27.8 (194) 33.3 (187) -45.7 (197) -0.9 (171), 23.1 (187), -14.6 (205)
-31.5 (180), 28.8 (196) 44.2 (187) -41.9 (197) -11.7 (176), 24.8 (191), -10.5 (207) 27.2 (182), -3.8 (201) -10.9 (187) -4.6 (198) 18.4 (180), -12.1 (198) -7.8 (177), 29.3 (194) 33.3 ,(187) -46.4 (197) -0.9 (171), 26.8 (187), -17.5 (205)
@EmployedApplequist's anisotropic static polarizability (ref 3) which are collective in nature requiring at least three different amides. This can be seen from the second part of eq 3 which contains
TABLE II: Absorption Spectral Intensities
E: EA
(0
which is zero for j = I or for j = k . Thus, the nonzero induced CD contributions are at least quadratic in the coupling of chromophores. Nonzero contributions that are linear in the chromophore coupling can arise only if DorDd is anisotropic. But, as we show below for our present model of an a-helix, the former terms are by far the largest for reasonable anisotropies. The induced CD contributions that arise from (1/3) Tr (DO,D,,J have sometimes been negle~ted.~ However, they are implicitly contained in Applequist's ~ o r k . ~ * ~ , ' ~ The skewing of the helix band can be understood from formulations of the CD of infinitely long helices for which sharp selection rules e ~ i s t . ~ For * ~ light ~ J ~propagating along the helix axis the CD helix band results from the difference of absorption bands centered a t the closely spaced frequencies Q,(q, so) and Q,(q, - so) where qr is the light's wavevector. For a single RT* transition per chromophore the helix band is a derivative-shaped band because the oscillator strengths f;(q2 + so) andf;'(q, - $0) are equal and hypolhyperchromism does not result. But for a system with several transitions per chromophore exciton-exciton system, coupling occurs yielding oscillator strengthsf;L(q, so) and f,'(q2 - so) that for each v are not equal. A borrowing and lending of intensity occurs. This produces a skewed derivativeshaped band. Of course, over the whole spectrum the CD is conservative due to the sum rule for oscillator strengths, which demands that the sum of oscillator strengths for the coupled system equal that of the uncoupled system. For the anisotropic dyadic, parts a, b, and c of Figure 2 display the unperturbed CD's, the CD perturbations, and the net CD's for a-helices, respectfully, of 10, 14, 18, and 26 chromophores. As can be seen from Table I, the chain length dependence in each case is largely due to that of the helix band component. The unperturbed CD's for N = 18 with isotropic and anisotropic high-energy transition moment dyadics are essentially the same. Small differences result due to the different level shift functions in the normal-mode equations, which produces slightly different transition frequencies and eigenvectors. The CD perturbations are quite different, the major difference occurring for the CD helix band perturbations with intensities of 20.7 and -3.6 at 182 and 201 nm, respectively, for the isotropic case (Figure IC) and 25.1 and -3.1 at 182 and 202 nm, respectively, for the anisotropic case (Figure 2c). The isotropic dyadic produces significant intensity, and its omission would lead to large error. We have also performed CD calculations for an a-helix with 18 chromophores for which we used a high-energy transition moment dyadic that is isotropic in the amide plane, Le., the one employed by Ronish and Krimm.5 The results are qualitatively the same as those calculated from Applequist's bond-centered core
+
+
(X
and Wavelengths
(nm)'
cL E'
e
TDH 5.32 3.61 7.78 3.04 3.43 5.57
(187) (197) (191) (187) (197) (191)
matrix
experimentalb
5.38 (188) 3.48 (197) 7.76 (191) 3.61 (188) 3.42 (197) 6.10 (192)
4.06 (187.8) 1.23 (205.4) -4.1 (189)
'Employed Applequist's anisotropic static polarizability (ref 3). bData from Mandel and Holzwarth (ref 16). polarizability, curves for N = 18 in Figure 2, a, b, and c. For the net CD the large negative high-frequency lobe of the helix band is absent, in contrast to the work of Ronish and Krimm, in which the CD contribution derived from the isotropic part of the dyadic is omitted. We have performed similar calculations with the matrix method'a,34-37which yields slightly different level shifts and hypo/ hyperchroism of the various CD However, the small differences in the results for the two schemes are more easily seen from calculations of the absorption spectrum for an a-helix with 18 chromophores. The peaks and wavelengths are displayed in Table 11. The unperturbed spectra are quite similar whereas the hypochromism of the perpendicular polarized absorption band is underestimated by the matrix method. The overall hypochromism predictions are not in agreement with experimental observation, nor is the exciton splitting.16 This could be due to our omission of certain transitions on the amide-backbone and side chain. However, long-range solvent effects are expected to increase the intensity of the perpendicular polarized band and, by twice that amount, decrease the intensity of the parallel polarized band. For a continuum model for water as the solvent, a 24% increase and a 48% decrease are p r e d i ~ t e d . ~This ~ ? ~would ~ tend to create agreement with experimental results. In summary, for our rigid model of an a-helix calculations show the following: (1) The isotropic part of the high-energy transition dyadic located on the amide is responsible for large induced long-wavelength CD contributions. (2) Due to the coupling of low- and high-energy transition systems, the net long-wavelength CD helix band is skewed with the negative high-frequency lobe being significantly reduced in intensity. (3) The chain length dependence of the CD is largely due to that of the helix band. Before we can suggest a method for extracting the helix band from experimental CD spectra, we feel that knowledge of the effect of chain flexibility on the helix band is required. But, at present ~
~~
(34) P. M. Bayley, E. B. Nielsen, and J. A. Schellman, J . Phys. Chem., 73, 228 (1969). (35) E. S . Pysh, J . Chem. Phys., 52, 4723 (1970). (36) W. C. Johnson, Jr., and I. Tinoco,Jr., Biopolymers, 7, 715 (1969). (37) D. A. Rabenold, J . Chem. Phys., 65, 4850 (1976); 73, 5942 (1980). (38) D. A. Rabenold, J . Chem. Phys., 80, 1326 (1984). (39) H DeVoe and I. Tinoco, Jr., J . Mol. Biol., 4, 518 (1962).
Circular Dichroism Analysis of a-Helical Polypeptides I
a
UNPERTURBED CD
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Figure 1. (a) Unperturbed circular dichroism and components as a function of wavelength (nm) for an a-helix with 18 amides with isotropic dyadics. R", is the unperturbed perpendicular radial band. $ is the unperturbed parallel radial band. is the unperturbed helix band. The unlabeled curve is the net unperturbed circular dichroism. (b) Circular dichroism perturbations and components as a function of wavelength (nm) for an a-helix with 18 amides with isotropic dyadics. AR, is the prependicular radial band perturbation. MII is the parallel radial band perturbation. A H L is the helix band perturbation. The unlabeled curve is the net circular dichroism perturbation. (c) Perturbed circular dichroism and components as a function of wavelength (nm) for an a-helix with 18 amides with isotropic dyadics. R, is the perturbed perpendicular radial band. Rllis the perturbed parallel radial band. H, is the perturbed helix band. The unlabeled curve is the net perturbed circular dichroism.
i
-20 L
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100
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Figure 2. (a) Unperturbed circular dichroism as function of wavelength (nm) for a-helices with 10, 14, 18, and 26 amides with anisotropic dyadics. Intensity increases with increasing N. (b) Circular dichroism perturbations as a function of wavelength (nm) for a-helices with 10, 14, 18, and 26 amide with anisotropic dyadics. Intensity increases with increasing N. (c) Perturbed circular dichroisms as a function of wavelength (nm) for a-helices with 10, 14, 18, and 26 amides with anisotropic dyadics. Intensity increases with increasing N.
2566
J . Phys. Chem. 1986, 90, 2566-2569
it appears that in a resolution scheme two Gaussians of competing and unequal intensity should be employed instead of a derivative band. Also, the present formulation is essentially an exciton theory in which transition change densities for electronic transitions are Coulombically coupled. The calculated results, especially for the CD helix band, could be different if individual vibronic components of electronic transitions are coupled. Both of the above aspects
of the problem will be dealt with in future papers,
Acknowledgment. This work was supported by Contract No. DE-AS05-78EV05784 between the Division of Biomedical and Environmental Research of the Department of Energy and Florida State University and by Grant No. 5 R 0 1 GM23942 from the National Institute of General Medical Sciences.
Infrared Matrix Isolation Studies of the 1:l Molecular Complexes of HCI and HBr with Substituted Cyclopropanes Candace E. Truscott and Bruce S. Ault* Department of Chemistry, University of Cincinnati, Cincinnati, Ohio 45221 (Received: August 12, 1985; In Final Form: February 3, 1986)
Infrared matrix isolation spectroscopy has been used to isolate and characterize the complexes formed between either HC1 or HBr with cyclopropanes which contain an electron-withdrawing substituent, such as -Br, -CN, -CH2Br, -COHCH,, or -COCH,. In each case, two distinct product bands were observed: a broad, intense absorption that is attributed to the stretch of the hydrogen halide in the complex, and a perturbed mode of the substituent. No perturbed deformation modes of the cyclopropane ring were noted, which stands in contrast to the results that were obtained for the cyclopropane-hydrogen halide complexes. These observations suggest that the structure of the complex is one in which the hydrogen halide is hydrogen bonded to the substituent, rather than to the cyclopropane ring.
Introduction There has been much interest in the weakly bound molecular complexes which are formed between Lewis acids and bases. Many of these complexes may be of mechanistic importance in catalyzed reactions, such as the complex formed between cyclopropane and the hydrogen halidesz4 The structure of this complex, as deduced from gas-phase microwave s p e ~ t r o s c o p y ~ - ~ and supported by matrix isolation studies,*" is of C , symmetry with the hydrogen halide hydrogen bonding to the midpoint of one carbon-carbon bond. The formation of this complex may represent the first step in the acid-catalyzed isomerization of cyclopropane. Recently the matrix isolation technique has been used to characterize the initial complex formed between several methyl-substituted cyclopropanes and the hydrogen halides." A hydrogen-bonded complex was formed in each case, with the site of coordination being to the carbon-arbon bond adjacent to the methyl group. The manner in which a substituent affects the cyclopropane ring has been examined in detai1.lz-l4 When the substituent is (1) Jensen, W. B. The Lewis Arid-Base Concepts; Wiley-Interscience: New York, 1980. (2) Ross, R. A.; Stimson, V. R. J. Chem. SOC.1962, 1602. ( 3 ) Dorris, D. B.; Sowa, F. J. J . Am. Chem. SOC.1938, 60, 358. (4) Lee, C. C.; Hahn, B.; Wan, K.; Woodcock, D. J. J. Org. Chem. 1969, 34. 3210. (5) Aldrich, P. D.; Kukolich, S. G.; Campbell, E. J.; Read, W. G.J. Am. Chem. SOC.1983, 105, 5569. (6) Legon, A. C.; Aldrich, P. D.; Flygare, W. H. J . Am. Chem. Soc. 1982, 104, 1486. (7) Buxton, L. W.; Aldrich, P. D.; Shea, J. A,; Legon, A. C.; Flygare, W. H. J. Chem. Phys. 1981, 75, 2681. (8) Legon, A. C. J . Phys. Chem. 1983,87, 2064. (9) Truscott, C. E.; Ault, B. S. J . Phys. Chem. 1984, 88, 2323. (10) Barnes, A. J. J . Mol. Struct. 1983, ZOO, 259. (11) Truscott, C. E.; Auk, B. S. J . Phys. Chem. 1985, 89, 1741. (12) Penn, R. E.; Boggs, J. E. J. Chem. SOC.,Chem. Commun. 1972,666.
0022-3654/86/2090-2566$01.50/0
electron withdrawing in character, the distal bond is shortened and the two vicinal bonds are lengthened, while the electron acceptor substituent acts in the opposite manner. Ion cyclotron resonance studied5 have shown that for cyclopropanes that contain the electron-withdrawing groups -C1, -Br, -NH2, -OH, and -OCH3 as the substituents, the chemistry observed is that of the substituent and not the cyclopropane ring. However, ICR studies of protonated cyclopropyl cyanide have shown that the parent ion undergoes "cyclopropane-like" reactions, quite possibly due to appreciable charge localization on the ring.I5 This differs from solution studiesI6 of cyclopropyl cyanide, where hydrogen bond formation has been shown to occur between the nitrogen of the nitrile group and Bronsted acids. In addition, recent solution studies of the interaction of metal ions (e.g. Pt2+)with substituted cyclopropanes have shown that ring opening occurs to form a metalocyclopropane when the substituents are alkyl groups, but that no reaction occurs when the substituents are electron withdrawing in nature.17 However, reactions of cyclopropane carbinol with HC1 carried out in solution give cyclobutanol as the major product, suggestive of protonation on the cyclopropane ring.'* The isolation and characterization of the complexes formed between cyclopropanes containing electron-withdrawing groups over a range of basicities (-Br, -CN, -COCH,, -COHCH3, -CHzBr) and HCI and HBr will be useful in furthering understanding of the chemistry of these molecules. Whether the hy(13) Allen, F. H. Acta Crystallogr.,, Sect. B 1981, B37, 890. (14) Allen, F. H. Acta. Crystallogr., Sect. 8 1980, 836, 81. (15) Luippold, D. A.; Beauchamp, J. L. J . Phys. Chem. 1976, 80, 795. (16) Allerhand, A.; Schleyer, P. v. R. J . Am. Chem. Soc. 1963, 85, 866. (17) Ekeland, R. A.; Jennings, P. W.; Presented at the 189th National Meeting of the American Chemical Society, Miami Beach, FL, April 30, 1985, Inorganic Division. (18) (a) Lee, C. C.; Cessna, A. J. Can. J . Chem. 1980, 58, 1075. (b) Tunemoto, D.; Kondo, K. J. Synth. Org. Chem., Jpn. 1977, 35, 1070. (c) Mazur, R. H.; White, W. N.; Semenow, D. A,; Lee, C. C.; Silver, M. S.; Roberts, J. D. J. Am. Chem. SOC.1959, 81, 4390.
0 1986 American Chemical Society
