Document on Circular Dichroism and Optical Rotation

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 representation 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 molecular 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 perturbation 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 perturbation theory states that the coefficients are governed by the differential equation

Thus, H = nRE. R only operates on the polarization vector 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 -. i, and r represents the coordinates (including the spin) of all the electrons and nuclei of the molecule and 'i: is the interaction Hamiltonian of the per- turbing field and the molecule. In this paper the interaction Hamiltonian consists simply of the interaction of the for right and left cpl, respectively. Substituting these ex- electric (p) and magnetic (rn) moments of the molecule pressions in (4) we have with E and H of the radiation field. Circular Dichroism and Optical Rotation Chemical Reviews, 1975, Vol. 75, No. 3 325

u= -p.E - m.H (8) This is the standard formula for a periodic perturbation. It is preferable to leave u unspecified for the moment The 6 function guarantees conformity to the Bohr fre- except for its time dependence which will be assumed to quency condition w = wmo. its presence demands an be periodic with circular frequency w. eventual integration over frequency which will be sup- plied physically by the fact that neither the incoming radiation nor the energy states are infinitely sharp.15 It should also be remarked that the conditions assumed for V is the time-independent part of 'u and represents the the above derivation (the growth of a population of mole- amplitude of Z, in its time variation. Only the time dependence is required for the integration of eq 7. In normal cules in state m) does not accord with the conditions of spectroscopy virtually all molecules are in the ground normal spectroscopic measurements which are per- electronic state so that for a selected excited state m formed under conditions of a steady state. A more care- ful analysis shows that the rate of absorption predicted for steady-state conditions and for the boundary condi-

\ tions given above are identical, provided the fraction of molecules in the excited state remains very small. This is usually the case. where Vmo = S$,*V$,dT. Following the usual course of time-dependent perturbation theory, the expression for V. Broad Electronic Bands of Molecules da,/dt is integrated with boundary conditions a, = 1 The factors which dictate the shape of electronic ab- and a, = 0 at t = 0. The first term in the above expres- sorption bands of molecules containing more than two sion leads to a term with (wmo + w) in the denominator. atoms are extremely complex,'6 particularly in solution. This leads to negligible contributions unless (am0 + w) Transitions from the ground state must be summed over r 0 in which case there is spontaneous emission to a all vibrational levels with Boltzmann weighting factors. A lower energy state. Such states are precluded by our as- molecule sufficiently large to be optically active has sumptions. With the deletion of the first term and with the many atoms and, as a consequence, many low-energy given boundary conditions degrees of freedom which are thermally excitable. The excited state is still more complicated. A typical band width of 2000 cm-1 may encompass thousands of different combinations of vibrational excitations. The spectrum The probability that a molecule has absorbed a photon of a moderately simple molecule in the vapor phase con- to jump to state m is given by sists of a large number of lines that are not resolvable with an ordinary spectrophotometer. In solution, the solvent cage provides an external and varying force field which alters the electronic energy of both the ground and excited states. Further, rotational and translational degrees of freedom are quenched into librational motions by interaction with solvent molecules. The result is a where the manipulations have followed the course: (err highly degenerate continuum of energy levels, which may - l)(e-Lz - 1) = 2(1 - cos x) = 4 sin2 (x/2). We note or may not show bumps arising from the varied excitation that (Vom(2 = V,,*V,, = VomVom* = VomVmo, etc., of large vibrational quanta. because V is Hermitian. The time dependence- of la, I It is desirable to bypass consideration of band shape in is contained in the factor in brackets. The detailed beha- a general discussion of electronic absorption and circular vior\of this function with time and frequency is discussed dichroism. This can be done by introducing a weighting in texts on quantum mechanics (see especially function p(w,,) for the probability of a transition within Kauzmannl' and Schiff13). For our purposes, its most in- the manifold of the electronic transition o - m where teresting property is its relationship with the d function. a,, is considered as a variable within the limits of the We introduce the abbreviation A = (amo- w)/2 and electronic band. Thus instead of the expression 1 V,,,(* rewrite the expression as follows for the electronic transition at a single frequency, we substitute p(w,,) 1 Vom(2 which describes the probability of the electronic transition taking place at the frequency am,. It is clear that p has the shape of the absorption The expression in brackets has all the properties of the band. It may be regarded either as the subject for eventual theoretical consideration or as an empirical function delta function:14 at A = 0 it is given by f/H - m as f -, which determines band shape. It will also be assumed to 03. For A # 0, it is of the order of l/t -+ 0 as t -* m. The factor of H was introduced to normalize the integra- be normalized to unity so that tion. This permits the writing of (10) as

This involves an approximation which is discussed in the Appendix. Each monochromatic transition at wm0 given and for the energy absorbed per molecule per second by eq 12 is weighted by the "transition density" p(Wm0) to give

q%+d(wrno - w) (12)

In the form at the right, A has been eliminated in favor of This equation applies for radiation at the frequency w. If the circular frequency utilizing the formula 6(ax) = eq 14 is now integrated over wmo in keeping with the (1/a)b(x). continuous nature of the absorption, we find 326 Chemical Reviews, 1975, Vol. 75, NQ.3 John A. Schellman

TABLE II. Formulas Relating Extinction Coefficients and Strengths of Transitions"

~~ where the 6 function has permitted the substitution of w, Absorption Circular dichroism the frequency of the radiation, for wmo. The subscript on 4**PNXP(X)Dom 16*z~NXu(X)R,, E= Ac = p keeps track of the fact that p is the shape factor for the 3(2303)kc 3(2303)Fic o - m transition. This extremely simple formula is the cornerstone of molecular spectroscopy. The nature of the perturbation V has not yet been specified except for its periodicity in time. Selection of V appropriate to the problem at hand In Numerical Formb leads to formulas for ordinary absorption, linear and circular dichroism, etc. Superposition of a constant magnet- D~, = 9.18 x 10-3 ic field leads to the basic formulas for magnetic CD and nuclear and electronic spin spectroscopy. The equation (de byez) (Debye-Bohr magnetons) has the form of the "golden rule" for the transition to the aTo convert to wave numbers, replace X by F. To convert to continuous states of free particles. l3 The interpretation of ellipticities, replace AC by Ma/3300, where (4500 In 10)/~= 3300," p(w) is different, however, since in the free particle case p = (na + 2)/3. bThe unit for dipole strength is (debyes)' = it represents a state density (including spatial degenera- 10-36 cgs unit. The unit for rotatory strengths is the (Debye- cy) whereas in (15) it is a transition density. For each Bohr magnetons) = 0.9273 X 10-38 cgs unit. Very frequently p p(w) there is a large variety of initial states, final states, and pa/n are equated to unity and X is replaced by A,,, in the and selection rules at work. A formula for p(w) is derived integral. in the Appendix. of light of half the intensity, polarized in the x and y di- VI. Absorption Spectra and Linear Dichroism rections. If the molecular system is in a crystal, oriented film, It is easier to recognize the unique terms which are re- electric or hydrodynamic field, etc., an assembly of mole- if ordinary absorption is sponsible for circular dichroism, cules will produce linear dichroism. This is measured by treated as a previous case. This will be done in this sec- comparing the extinction coefficient for two different po- tion. larization directions, x and y. Then, from (17) The perturbation producing ordinary electronic spectra is the interaction of the electric field of the radiation with the electric moment of the molecule. In this case, V = -1.1 Eo where 1.1 = e2rL,with e the electronic charge and ri the position vector of the ith electron of the molecule, and Eo is the amplitude vector of the electric field. The where 8 is the angle the transition dipole makes with the time dependence of E has been removed by integration; direction of propagation of light, 4 is the standard meridi- the space dependence does not contribute to ordinary nal angle relative to the x axis, and the averages are over absorption, i.e., electric dipole absorption. Thus Eo = the distribution function appropriate to the method of ori- veEo and Vom = Eove . porn. Substituting this relation into entation. (15) we have The above formulas contain no correction for the local field on the chromophore in condensed media. It is com- JEo*porn12Pm(w)- raJve.pomI2Pm(w)Eo2 'W(W) = - mon practice either to ignore the local field problem or to 2h 2h apply the Lorentz correction. The latter is easily introduced. According to the Lorentz theory, the field on the chromophore is given by F = @E with @ = (n2 + 2)/3, rather than the macroscopic field itself. This introduces the factor [(n2 + 2)/3]2€02 into eq 16a instead of Eo2 so where in (16b) Eo2 has been eliminated in favor of the that, instead of (18), we have intensity by eq 6. w is proportional to the intensity as de- manded by the Beer-Lambert law. Using this formula for wineql Finally, circular frequency is an inconvenient unit for experimental work where wavelength in nanometers or frequency in cm-' (ij) is usually preferred. Conversion to If impinging light is linearly polarized in the x direction, these other units follows directly from the requirement the orientational part of (17) becomes Iv - PomI' = that the shape function be normalized in its own units; I (px)oml2. For randomly oriented molecules, the average that is, lp(w)dw = .fp(ij)dij = lp(X)dX = 1, from which over all orientations gives we deduce that up(@) = ijp(ij) = Xp(X). Thus the change in spectral units introduces no changes in the constants of the absorption formula. The results are summarized in Table I I.

VII. Circular Dichroism In (18) pornz has been replaced by the dipole strength To obtain a theory of circular dichroism, it is necessary which is defined as Do, = Jpom12. to add the small magnetic interaction of radiation with This equation applies equally to linearly polarized light electrons to the dominant electric interaction. Thus in eq in the, y direction and to nonpolarized light, which can be 15, V = -p.E - m-H, where rn is the magnetic moment regarded as the superposition of two independent beams (operator) of the molecule given by Circular Dichroism and Optical Rotation Chemical Reviews, 1975, Voi. 75, No. 3 327

where Li is the orbital angular momentum and Si the spin angular momentum of the ith electron. Spin magnetic moments usually make negligible contributions to circular Substituting in eq 22 and separating the equation into its dichroism and will be dropped from the discussion which follows. With known or assumed wave functions, magnet- real and imaginary parts ic transition moments are easy to calculate. What hap- (V&)om = -[(px)omEo f (gx)ornHo)l - pens in the interaction of cpl with molecules can be made much more transparent if circukarly polarized com- i[(ht(py)omEo+ (gy)omHo)l ponents of the transition moments are introduced. Thus This is now in the form V = a + ib with a and b real so in place of wX and py, we utilize the electric moments that I VI2 = a2 + b2 so that

and I.L = ccxi + clyi + Pzk (py)om(gy) rnolEoHo (23) becomes Averaging over all orientations of the molecules utilizing M = p-e+ + p+e- + pzk the relation (axbx)= (a,b,) = (a-b)/3, we have where e+ and e- are the unit polarization vectors for the 2 2 2 (V*),,2 = *E02 *H,2 f ~(~m.gom)EoHo electric field of right and left circularly polarized light. + The introduction of p+ and p- involves the decomposi- The field quantities can be eliminated by means of eq 6. tion of linear moments into circular components. This is a We also introduce the quantities: familiar device in the discussion of magnetic resonance interactions. The magnetic moment m can be similarly Do, = (I.L,,)~ = /~,~~p,,electric dipole strength (24a) decomposed, but in so doing it will be desirable to main- Go, = (g)om2 = mom*mmo magnetic dipole strength (24b) tain the same kind of phase relationships between m and - p as exist between H and E. In accordance with Table I R,, - -Cc,,.gom = Im(fio,.mmo) rotatory strength (24c) we define Do,, Go,, and Rom are molecular measures of electric -imx m im, + m dipole absorption, magnetic dipole absorption, and circu- m+=Tx+ m-=+ lar dichroism, respectively. In the second part of eq 24c, R,, has been converted to its customary form by substi- mx=.w my=* tuting -go, = +im,, = +im,,* = -immo. But this equals Im(mm0) in the sense that if z = x 4- iy, -iz = In these terms the magnetic moment operator is Im[z] = y, when x = 0. m = m-h+ + m+h- + m,k Making the substitutions from the three forms of eq 6 and the three terms of eq 23 In terms of the circularly polarized moments the interaction takes a very simple form. For right and left circularly polarized light we have respectively E+ = e+€,, H+ = h+H,, and E- = e-€,, H- = h-Ho. Utilizing the relations e+ .e+ = h+ .h+ = 0 and e, .e- = h+ .h- = 1, we We may now utilize eq 15 and 1 to obtain expressions for find the extinction. In so doing it is necessary to recall that the transition density function depends not only on the (!'+)om = -(p+)omEo -(m+)omHo upper and lower states, but on the matrix elements con- (22) (V-)orn = -(/l-)ornEo -(m-)omHo necting them. There are three combinations of matrix elements in (24) and each has its own transition density where V+ and V- refer respectively to the interaction function. We will call these p(w), ~(w),and a(w). Utiliz- with right and left cpl. ing the equations mentioned above Evaluation of IVo,12 can be cumbersome because of the three possible sources of imaginary quantities contained in (22). These are the wave functions themselves, the complex expression for circular polarization, and the (264 magnetic moment operator. The latter is a pure imaginary since, in a coordinate representation, L in eq 21 is If we take into account the possibility that the local field, given by L = (h/i)Zjr, X Vi, where the summation is F, differs from E, Le., F = @E, then this equation takes over electrons. The work proceeds more simply if we the form take the following steps: (1) Assume all wave functions are real. This is in accord with a theorem of Van Vleck17 4x'wN which states that the wave functions of a molecule in a '* = 3(2303)hc nondegenerate state are necessarily real in the absence of an external field. A complex phase factor may be introduced, but it is devoid of physical significance. (2) Temporarily define a real operator g = -im. In terms of The three terms in (26) represent electric dipole ab- this operator sorption, magnetic dipole absorption, and circular di- 328 Chemical Reviews, 1975, Vol. 75, No. 3 John A. Bchellman chroism. Magnetic dipole absorption has only been de- field corrections but with the fact that the wavelength in tected in a few simple molecules in the gaseous state. It the medium is shortened by a factor of l/n. In the ener- is extremely weak since the ratio of magnetic intensity to gy balance this compression of the wave plus the electric electric intensity is of the order of the square of the ratio polarization requires that H = nE. If the Lorentz formula of the Bohr magneton to the debye unit, i.e., about is applicable n/P = 3n/(n2 + 2). This is close to unity The term in Go, will be dropped in subsequent formulas. for most realistic values for n. It is unity at n = 1, reach- If the incident light is linearly polarized or unpolarized, es a maximum of 1.06 at n = d2, goes through unity at the radiation may be treated as a mixture of rcpl and lcpl n = 2, and attains a value of 0.91 at n = 2.5. We see so that the CD term cancels. Thus then that the validity of the formula g = 4Rom/Do, depends not on the absence of an internal field but in the cancellation of the intrinsic effect of refractive index by the internal field dependence. This cancellation is very where circular frequency has been replaced by wave- effective for a Lorentz internal field, and one may at least length. This permits the evaluation of Do, from experi- assume this to be so for the more complex fields which mental da’ta. Recalling that arise in situations which are not cubic in symmetry as required for the Lorentz expression. SPom(A)dX = 1 Frequently circular dichroism spectra are reported as molar ellipticities. This usage is related to an older tech- nique of measuring CD by measuring the ellipticity of light produced by optically active substances. It is not This can be replaced with a moderately good approxima- difficult to show’s that the molar ellipticity is related to tion by considering n, p, and Y constant over the width of Ac by the relation the absorption; thus Me = 4.5(2303)A At 3300Ac

Substituting from eq 28a where n and ,d are evaluated at A = Amax. If circular dichroism is measured, t- - t+ is determined. From eq 25b this is Both (27a) and (29) are useful since the former permits direct comparison of circular dichroism with absorption spectra and the latter with optical rotation. In eq 19, we had a relation for the absorption of linear- Rearranging and integrating over the band ly polarized light of an oriented system. It may appear that a similar possibility of development for oriented CD is contained in eq 23. This is not so. In an oriented system, quadrupole terms which we have neglected contrib- or ute to circular dichroism. Their contribution vanishes when the system is optically isotropic. Thus the derivation given in this paper is valid for circular dichroism only with the same approximations as before. This provides if uniform spatial averaging is employed. Theories which the experimental method of evaluating rotatory strengths. include quadrupole terms for oriented systems have been In discussing the shapes of circular dichroism bands, a developed by a number of ~riters.~~~~~~-~~ parameter of considerable importance is the anisotropy factor g, which is the ratio of CD to ordinary absorption g VIII. Optical Rotatory Dispersion = A€/€. This is in general a function of frequency. From As mentioned in the introduction, the theory of optical eq 26a and 27a it is given by rotation is usually derived directly from Maxwellian sus- ceptibility theory or as a coherent scattering process. Here we adopt the approach of considering optical rotation as a dispersive property determined by the form of g is seen to depend on the ratio of the shape factors the absorptive property, CD, rather than the converse. for CD and absorption as well as a weak variation with Going either way the correlation is rigorously dictated by n/P. Moscowitz18 has shown that in many strong bands the Kronig-Kramers transform relations between the dis- the CD signal and the absorption signal have the same persive and absorptive properties of material media.’ s723 shape. In this case a(v)/p(v)= 1 and If the wavelength is chosen as the independent variable, the form of the Kronig-Kramers transform which is appropriate for the transformation of the circular dichroic spectrum into the optical rotation spectrum is given byl9 The conditions for this approximation are discussed in the Appendix. This formula is usually given as g = 4Ro,/Do,, which is probably a good approximation. On the other hand, it is where Ma is the molar rotation. The symbol fmeans the usually understood that the latter formula represents “ig- principal value is to be taken: noring the internal field correction.” This is not the case. If the internal field is ignored, p = 1 and, for a/p = 1, g = 4n(RO,/Dom). The refractive index, n, can differ appreciably from unity, especially in the ultraviolet part of the spectrum where CD is usually measured. The appear- Referring back to eq 28a, it is seen that the form of the ance of n in this equation has nothing to do with internal rotatory dispersion is governed by the form of the shape Clrcuiar Dichrolsm and Optical Rotation Chemical Reviews, 1975, Vol. 75, No. 3 329 function a(h). In the absence of a general theoretical The Kronig-Kramers transform can now be used to ob- form for the shape of circular dichroism bands, we shall tain the appropriate expression for Lorentzian optical utilize the three most commonly considered empirical rotation. Defining x = (A - Am)/?,, eq 39 can be put forms: infinitely sharp lines, Lorentzian or forced oscilla- into the form const[l/(x2 + l)].One of the simplest tor response, and Gaussian bands. transforms is --x1 A. Line Spectra x2+1 x2+1 If the spectrum may be considered as a series of lines This gives for the molar rotation of negligible breadth, a(A) may be represented as a sum of 6 functions a(X) = Za, = Z,6(h - A,), where the Am are the wavelengths of the lines. Substituting eq 31 into eq 32 and utilizing the properties of the 6 function, we obtain Rosenfelds’ equation Equations 37 and 38 give a good representation of the shape of the circular dichroism and optical rotation within (33) the absorbing region of a Lorentzian band but are less satisfactory outside the band. In particular when (h - Note that the rotation is undefined at A = A, because in Am) >> Ym, eq 38 gives a term in 1/(A - A,) rather this case the 6 function is located at the singularity of the than a Drude term. To obtain the correct form for disper- integral. Equation 33 may be also expressed sion far from the absorbing region, eq 35 must be used. In this case

(34) with Ai = 48PN/hc. In this form it is called the Drude equation since it was first derived by Drude on the basis This has essentially the same shape as (40) within the of a helical oscillator without damping. It has been shown band but reduces to a Rosenfeld-Drude equation when by Mosc~witz~~and Kuhn and Braun3 that the above ex- (A2 - Am2) >> hm2(eq 33 and 34). Since (41) and (37) pression holds with high accuracy for broad bands pro- represent the in-phase and out-of-phase response of a vided that the separation between the wavelength of forced oscillator, they are obtained simultaneously in the measurement and the band center is sufficiently greater solution of the problem so that it is not necessary to per- than the band width. form the Kronig-Kramers transform. A pair of equations of the form of (37) and (41) were first proposed by Na- B. Lorentzian Bands tan~on~~on the basis of an outmoded classical model. In any event, the Lorentzian shape has had less impor- It is possible to develop classical forced oscillator tance in the interpretation of electronic CD spectra than theories of optical activity with typical damping terms. in other branches of spectroscopy. The width of an elec- This approach was first developed by Natan~on~~and tronic band of a molecule complicated enough to show was later elaborated upon considerably by K~hn~~with circular dichroism arises from the wide range of energies his coupled oscillator model. As would be anticipated, of the initial and excited states and not from the intrinsic the result is a Lorentzian dependence on wavelength or width of absorption. As a result, a Lorentzian which is frequency, so that the shape factor takes the form broad enough to represent the shape of the band close to constant the maximum tails off much too slowly to represent band a(h) = (35) (x2 - xm2p + r,w shape far from the center. where Am is the wavelength of maximum CD and rmis C. Gaussian Bands the damping parameter. Since a is normalized, we must have Sam(h)dA = 1, which gives for the constant the This case has been developed by Mo~cowitz~~and approximate value 2hm2rm/r. Thus Kuhn3 and has found the most extensive application in the interpretation of CD band shapes. In this case the shape factor has the normalized form

X - Am and from eq 31 a,(A) = -Am16exp [-(TT] (42’ where Am is the exponential half-width. Again utilizing eq 31 In performing the integration it has been assumed that h h, except in the term (Az - Am2)2. The exact expression for the constant gives a more complicated formula. It is customary to simplify the Lorentzian band by mak- The Kronig-Kramers transform3.19 is ing the same approximation in the shape function itself. Putting A N A,, we find Me., =

where y = r/2 is the half-width of the band. Utilizing this form for a(h)in eq 31 where x = (X - A,)/Am. Moscowitz has shown that this equation goes over to Rosenfeld’s equation when x > 4. The integral in (44) is known as a Dawson integral (39) and is tabulated in a number of places (see especially ref 330 Chemical Reviews, 1975, Vol. 75, No. 3 John A. Schellman

TABLE 111. Optical Activity for Three Band Shapes"

48PNR Om X Lines mS(X - A,) RC

Lorentzia n (eq 39 and 40)

Lorentzian (eq 37 and 41)

Gaussian

0 MO@,, = 3300A~,,, is the maximum in molar ellipticity. F(X,A,,A,) = { exp[-x2] J' exp[t2]df+ [Am/2(X+ A,) 1 with x = (A - X,)/A,. The frequently occurring constant (48N/fic) = 9.145 X 1041. If one wishes to go directly from molar ellipticities or rotations to rotatory strengths in Debye-Bohr magnetons, this constant is to be replaced by 8.480 X IO3, the reciprocal of which is 1.179 X Other sym- bols are defined in the text and in Table II,

27). Carver, Shechter, and Blout28 have devised a rapidly tion so that the assumptions made in a number of cur- converging procedure for evaluating the integral which is rently used formulas can be clearly stated. utilized in most contemporary curve fitting programs for evaluating and resolving circular dichroism curves. X. Appendix Chemical absorption bands are frequently represented as a sum of Gaussian-shaped bands. Unless this repre- The transition densities p and CT are extremely compli- sentation has physical significance in the realm of spec- cated functions for solvated, polyatomic molecules, but it troscopic assignments, i.e., each term represents a 'dis- is possible to obtain formal expressions for them. We ex- tinct transition with a Gaussian shape, the rotatory pand the notation so that the ground and excited states strengths associated with each Gaussian lose their signif- are designated by ocy and mp, respectively. o and m are icance and only the sum of the absorption envelope has labels for the electronic state as defined for an isolated significance. With CD bands obviously separated into vi- molecule in its lowest state of nuclear motion, and cy and bronic components, the rotatory strength associated with are descriptive labels for the entire set of nuclear coor- a given electronic transition can be partitioned into vi- dinates and their states of motion. These include not only bronic components. the internal vibrations and librations of the absorbing mol- Table I I I summarizes the results of this section. ecules but also the solvent cage. An energy is associated with each label cy and p, but the situation is highly degen- IX. Discussion erate in that for a given energy there will be a large number of combinations of internal vibrations, librations, and In general the theory of optical activity may be divided solvent perturbations which are possible. The probability into three parts: (1) the relationship between experimen- that a transition ocy -+ mp takes place is governed by eq tal observations of circular dichroism and optical rotation 15 with IVom12 replaced by JVo,,mS12. On the other and the quantum mechanical quantity-rotatory strength; hand, spectroscopic observations combine all transitions (2) the development of mechanisms for the rotatory of the same Bohr frequency so that the transition density strength in terms of the electronic structure of the mole- for a given frequency must be summed over all initial and cule under consideration (here the one-electron mecha- final states separated by the same frequency. The proba- nism of Condon, Altar, and E~ring,~~the coupling mecha- bility of the initial state cy is governed by nism of Kirkw~od,~~and the exciton mechanism of Mof- fitt3' come prominently to mind): and (3) the relationship between the rotatory strength and the three-dimensional structure of the molecule. This paper has been con- Grouping transition probabilities with regard to frequen- cerned entirely with the first problem. In particular, the cies instead of states, the probability of a transition at en- formulas developed apply to single conformations. If a ergy wmo will be proportional to molecule possesses conformational freedom, the formulas must be averaged over conformation to provide a rep- CPnC'IVon,mA* (A2) resentation of experimental results. The method of this paper consists in utilizing semiclas- where the prime on the second sum indicates that only sical perturbation theory retaining only electrical and those states (I are included for which Eg - E, = hwmo. In magnetic dipole interactions with radiation. Band shape the text the probability of this transition was represented questions are handled by means of a golden-rule formu- by p(~,,) I Vom12 which is a definition of p(o,,). Thus lation with transition density functions. Optical rotation is approached as the transform of circular dichroism. The P(wmo) = IVon,m312/ I vornl2 (A3) familiarity of the mathematical apparatus makes this ap- flB proach especially simple. Moreover, the relationship of If ~(w,,) is defined also as being normalized over the CD with absorption as well as its distinctive features are band o -+ m, then I VOml2 is simply proportional to the brought out with considerable clarity. Questions con- integrated absorption intensity. One would like I Vo,I2 to cerning the internal field correction and its relationship be the transition probability for an isolated molecule in its with absorption mechanism (eq 26b) are automatically lowest state of nuclear motion since this is frequently answered. Finally, the problem of bandshape and its vari- calculable from molecular quantum mechanics. This is ation with mechanism is obtrusively present in the deriva- not necessarily so, though it is often a good approxima- Circular Dichroism and Optlcal Rotation Chemical Reviews, 1975, Vol. 75, No. 3 331

tion. Only the product p(omo)I Vmo l2 is defined so that, if Assuming the Born-Oppenheimer approximation and (VmoI2 refers to the electronic states of the isolated mol- equilibrium nuclear coordinates in electronic integrals ecule, p(wmo) is not necessarily normalized since changes in intensity produced by solvent interactions or vibrations must be reflected in its behavior. On the other aB hand, if p(wmo) is defined as normalized as in the text, This is the same formula as for p(wmo) so that, under then IVmo12 in general must contain the effects of sol- these circumstances, circular dichroism and absorption vent and vibrations. have the same shape and eq 29b is applicable for the It turns out that the crux of the matter is whether or anisotropy factor of the band.6 not the Born-Oppenheimer approximation can be ap- There are a number of instances in which one or the plied. If it can, the wave functions for oa and mp can be other of the two approximations do not apply. Circular di- written in the form chroism is frequently observed in exciton systems for which the Born-Oppenheimer approximation is not applicable.33 It often arises with weak absorption bands which are essentially electric-dipole forbidden, but magnetic-dipole allowed. In these circumstances, it is not permissi- The electronic wave functions are of the form $m(q>Q) ble to use equilibrium nuclear coordinates in evaluating and the nuclear wave functions are xa(Q), where the q porn. It is, however, usually possible to equate the areas are electronic coordinates and the Q are nuclear coordi- under absorption and CD curves with the dipole strengths nates including solvent nuclei as parameters. and rotatory strengths of electronic transitions, but the Looking first at the dipole strength, we need to evalu- shapes of the curves for absorption and CD can differ ate J/.Lu,a,rna)2. In the Born-Oppenheimer approximation markedly. In particular, contributions to D and R can we have arise in mutually exclusive vibrational progression^.^,^^

XI. References If the electric transition moment is large, it is not greatly (1) T. M. Lowry. "Optical Rotatory Power," Longmans Green 8 Co.. affected by nuclear motions so that $(q,Q) may be re- London, 1935; Dover Publications, New York, N.Y., 1964. placed by $(q,O) where the nuclei are placed in their (2) A. Cotton, Ann. Chim. Phys., 8, 347 (1896). (3) W. Kuhn and E. Braun, Z. Phys. Chem. 6,8, 445 (1930). equilibrium positions. Then (A5) may be factored and we (4) P. A. Levene and A. Rothen. "Organic Chemistry," Vol. 2, H. Gil- have Ipoa,mal' = pornz where the (alp) are man, Ed., Wiley, New York, N.Y., 1938, Chapter 21. Frank-Condon nuclear overlap integrals. l6 As is well (5) C. Djerassi, "Optical Rotatory Dispersion," McGraw-Hill, New York, N.Y., 1960. known, these partition the intensity of electronic bands (6) W. Moffitt and A. Moscowitz, J. Chem. Phys., 30, 648 (1959). among the vibronic levels of the excited state. Substitut- (7) A. D. Buckingham and H. C. Longuet-Higgins, Mol. Phys., 14, 63 (1968). ing in (A3), we have (8) C. W. Deutsche, J. Chem. Phys., 52, 3703 (1970). (9) F. M. Loxsam, J. Chem. Phys., 51,4899 (1969). ~crC'c("m2(4B)a (10) P. A. M. Dirac, "Quantum Mechanics," Oxford University Press, "0 New York, N.Y., 1947, pp 245-6. P(Wmo) = = >,C'(aIP, (A6) (11) W. Kauzmann, "Quantum Chemistry," Academic Press, New York. Porn' B N.Y., 1957. (12) W. A. Shurcliffe, "Polarized Light," Harvard University Press, Cam- Thus p(umo) contains the combined effect of tempera- bridoe. Mass.. 1962. (13) L. I: Schiff. "Quantum Mechanics," 2nd ed, McGraw-Hill, New ture (in p,) and the Frank-Condon principle (in (alp)). York, N.Y.. 1955. Integrating over the whole band is equivalent to eliminat- (14) L. D. Landau and E. M. Lifschitz, "Quantum Mechanics," Pergamon ing the stricture on the summation over p since the sum Press, London, 1958. (15) A. Somerfeld, "Optics," Academic Press, New York, N.Y., 1954. is then over all frequencies (16) G. H. Herzberg, "Electronic Spectra of Polyatomic Molecules," Van Nostrand, New York, N.Y., 1966. (17) J. H. Van Vleck, "Theory of Electric and Magnetic Susceptibilities," Oxford University Press, New York, N.Y., 1932, pp 272-3. (18) A. Moscowitz, Thesis, Harvard University, 1957; see also ref 6. If and are members of complete orthonormal (19) A. Moscowitz, in ref 5. la) (p) (20) I. Tinoco and W. G. Hamerle, J. Phys. Chem., 60, 1619 (1956). sets, it can be shown that I;~(alp)~= Since Zp, = (21) T. Ando, Progr. Theor. Phys. (Kyoto), 40, 471 (1968). 1 by definition, p(wm0) is normalized to unity. The condi- (22) L. D. Barron, Mol. Phys., 21, 241 (1971). (23) L. D. Landau and E. M. Lifschitz, "Electrodynamics of Continuous tions required are the applicability of the Born-Oppen- Media," Pergamon Press, New York, N.Y., 1963. heimer principle and the assumption that the transition is (24) L. Natanson, J. Phys. Radium, (4) 8, 321 (1909). (25) W. Kuhn, "Stereochimie," K. Freudenberg, Ed., Deuticke, Leipzig, sufficiently strongly allowed that small nuclear displace- 1933. ments do not appreciably affect the transition so that the (26) J. Irving and N. Mullineux, "Mathematics in Physics and Engineer- Q can be equated to their equilibrium values in the elec- ing," Academic Press, New York, N.Y., 1959. (27) W. L. Miller and A. R. Gordon, J. Phys. Chem., 35, 2785 (1931). tronic integrals. (28) J. P. Carver, E. Shechter, and E. R. Blout, J. Am. Chem. SOC.,88, Applying the same analysis for the rotatory strength, 2550 (1966). (29) E. V. Condon, W. Altar, and H. J. Eyring, J. Chem. Phys., 5, 753 we find (1937). (30) J. G. Kirkwood, J. Chem. Phys., 5, 479 (1937). (31) W. Moffitt, J. Chem. Phys., 25, 467 (1956). (32) M. Tinkham, "Group Theory and Quantum Mechanics," McGraw- Hill, New York, N.Y.. 1964. (33) W. T. Simpson and D. L. Peterson, J. Chem. Phys., 26, 588 (1957). The symbol Im is not necessary since the factors of i (34) 0. Weigang, J. Chem. Phys., 42, 2244 (1965). cancel. (35) Reference 26, p 637.