Circular Dichroism and Optical Rotation? JOHN A. SCHELLMAN Department of Chemistry, University of Oregon, Eugene, Oregon 9 7403 Received April 29, 7974 (Revised Manuscript ReceivedJune 7 7, 7974) Contents method is capable of producing a quite accurate repre- sentation of the theory of optical activity. As pointed out I. Introduction 323 by Dirac,lo the use of the quantum theory of radiation I I. Extinction Coefficient 323 adds nothing to the discussion for an absorptive process II I. Classical Description of Radiation 324 at low densities of radiation. The electric and magnetic IV. Molecules in Periodic Fields 324 dipole terms are all that are required for isotropic sys- V. Broad Electronic Bands of Molecules 325 VI. Absorption Spectra and Linear Dichroism tems containing molecules small compared to the wave- 326 length of light. By developing the absorptive property, VI I. Circular Dichroism 326 CD, rather than the dispersive property, optical rotation, VI I I. Optical Rotatory Dispersion 328 the accent is placed on the spectroscopic character of A. Line Spectra 329 B. Lorentzian Bands optical activity in agreement with current experiment and 329 interpretation. C. Gaussian Bands 329 In this paper, the theory of optical activity will be de- IX. Discussion 33 0 veloped following the program outlined above. Since the X. Appendix 33 0 primary concern is the relationship of experimental data XI. References 33 1 with quantum mechanical matrix elements, it will be nec- 1. Introduction essary to consider real absorption bands with complex shapes. To do this, an empirical distribution of absorptive For well over a hundred years optical activity was re- intensity will be used which is a generalization of a dis- garded primarily as a refractive property of transparent cussion by Kauzmann.’ Optical rotation and, in particu- media which was strongly, but obscurely, dependent on lar, Rosenfeld’s equation for optical rotation are generat- the configuration and conformation of the constituent ed from the CD expressions via the Kronig-Kramers molecules.’ More recently in roughly three stages, the transform. study of optical activity has entered into the realm of mo- lecular spectroscopy. These stages were the establish- 11. Extinction Coefficient ment of the behavior of optical activity within absorption bands by Cotton2 and later by K~hn,~the realization that In differential form the Beer-Lambert law is given by the isolated optical activity of a single band is much sim- 6//1 = -2.303tC62, where t is the extinction coefficient, pler to interpret than optical activity at a single wave- C the concentration, 6z the thickness of the thin layer of length,4.5 and, finally, the availability of commercial sample on which the light is impinging normally, I the in- equipment which converts the study of optical activity tensity of the radiation in ergs cm-* sec-’, and 61 the into a direct and accurate measurement of differential intensity lost in passage through the layer of thickness 6z absorption of right and left circularly polarized light. by the absorption of the molecules therein. 61 is equal to This essentially inverted development of the field has the number of molecules in the layer, 6N, times w, the led to a peculiar structure to the theory as it exists in the average energy absorbed per molecule per second. 61 = literature. An analogous situation would be if the theory w6N. If the unit of concentration is moles per liter, 6N = of the electronic absorption of atoms and molecules had (NC/1000)6z because I is defined in terms of unit cross- been approached from the point of view of the refractive sectional area and index. The result is that experimental workers in the field t= --Nw at the present time place most weight on the measure- 2303 I ment and interpretation of circular dichroism (CD) but, on seeking theoretical guidance, find that the literature is where N is Avogadro’s number. Note that if w is not pro- mostly concerned with the optical rotatory dispersion portional to I, the extinction coefficient will depend on the (ORD) of line spectra. These remarks apply only to theo- intensity of radiation. From an examination of the units it retical discussions at the elementary level. From a more can be seen that w/I is the cross section in cm2 per mol- advanced point of view, the topic has been covered in a ecule for the absorption process. very satisfactory manner by a number of writers (for ex- Linear dichroism and circular dichroism are defined in ample, ref 6-9). terms of differences in extinction coefficients as follows: What seems to have been bypassed in the literature is linear dichroism a discussion of circular dichroism at the minimum level of complexity, i.e., straightforward, time-dependent per- turbation theory with simple electric and magnetic dipole interaction with the radiation. Though lacking the power circular dichroism and generality of other approaches to this problem, this N w- - w+ 7 This research was supported by the National Institutes of Health bt t- (GM-20195)and the National Science Foundation (GB-41459). - - t+=2303 I (3) 323 324 Chemical Reviews, 1975, Vol. 75, No. 3 John A. Schellman TABLE 1. Polarization Vectors for Circularly Polarized Light E .E+ = le+E,ei$j = -$$(i cos $ - j sin $) Electric Magnetic (5a) 1 E- = {eE,e[+/ = €0 (i cos $ + j sin +) rcpl e+ = -- (I) h + --- i2 (Yi) 77 42 i That these represent right and left cpl can be seen by lcpl e-=-(')1 putting t = 0 and observing that the vectors trace out 42 --i h-=z1 right and left spirals in space on substituting $ = -~Kz. In forming the magnetic polarization vectors for cpl, The subscripts x and y designate extinctions with linearly care must be taken that these have the proper phase polarized light parallel to arbitrarily selected orthogonal relations with the electric vectors. This is accomplished axes. The subscripts - and + refer to extinctions ob- with the rotation matrix R defined above. h+ = Re+ and tained with left and right circularly polarized light. In eq 2 h- = Re-. The results are and 3, it is assumed that the pairs of extinctions were measured at the same intensity I. H+ = {h+H,e'+l with h+ = - 41 111. Classical Description of Radiation In a transparent, isotropic medium of refractive index n, electromagnetic radiation may be described in terms The results and notation for the polarization vectors are of perpendicular electric and magnetic fields E and H. E, summarized in Table I. H, and the direction of propagation of the radiation k are The instantaneous intensity of the radiation is given by mutually orthogonal and have the same relation to one the Poynting vector S = (c/4r)E X H, where the real another as the x, y, and z axes of a right-handed coordi- values of E and H must be used in forming the product. nate system. In the Gaussian system, E and H have the For linearly polarized light ve X Vh = k and the phase de- same dimension and are related by I H 1 = n I E 1. pendence is cos2 $. Averaging the latter over a cycle If the z axis is defined as the direction of propagation yields a factor of l/2. The mean scalar intensity is then in the medium, the electric and magnetic fields may be represented by two-dimensional vectors in the xy plane (Jones vectors12). The electric field may then be written The three forms of the intensity expression on the right E = IveEoeLJ') (4) will lead to different refractive corrections for different modes of interaction. where ve is a normalized unit vector describing the state Though (6) was derived for the linearly polarized case, of polarization of the electric field, and E, and $ are the it may be shown that it is valid for any polarization, pro- amplitude and phase of the wave, respectively. $ = ut - vided v is normalized such that v* - v = 1. Further, the nKz = 2r(u - z/v), where w is the circular frequency above discussion assumed a single frequency for the ra- 2ru, K is 27r/X, and v is the velocity c/n. It is to be un- diation. Any real radiation is polychromatic, and the oper- derstood that the electric field is given by the real part of ative electric field, magnetic field, and the intensity must the complex expression. Braces will be used to mean the be obtained by a summation or integration over the fre- real part of any vector. Thus if v is real, E = veEo cos $ quency. Absorption and circular dichroism spectroscopy = {veEoe4$1. For linearly polarized light, ve may be writ- belong to the domain of linear optical effects where this ten as a column vector is a valid procedure. (:E) IV. Molecules in Periodic Fields where CY is the azimuth of polarization relative to the x The electronic state of a molecule may be described axis. by the formula \k' = zai'€',,where the 9, $,e-Lbrt'h The magnetic vector may be obtained from the electric are the time-dependent wave functions of the eigenstates vector by multiplication by n and a rotation of 90" ac- with energy E,. The probability that a molecule is in state cording to the right-hand rule stated above. This rotation i is [ail2 = a,*a,, which is a function of time in the pres- is accomplished with the rotation matrix ence of an external perturbation. Time-dependent pertur- bation theory states that the coefficients are governed by the differential equation Thus, H = nRE. R only operates on the polarization vec- tor v. For the linearly polarized case where (7) For circularly polarized light (cpl) the electric polariza- Here Wij is the circular Bohr frequency (EL - Ej)/n for tion is most conveniently represented by the vectors the transition j -.