OPTICAL ROTATORY DISPERSION DATAAND
THE
DRUDEEQUATION
2663
Optical Rotatory Dispersion Data and the Drude Equation.
A Parametric Curve Fitting Problem’
by The0 Burer2and Leonard I. Katzin Chemistry Division, Argonne National Laboratory, Argonne, Illinois
(Received September $0, 1966)
The relationships between the parameters of the Drude equation best fitting optical rotatory dispersion data and the parameters of the circular dichroism spectra from which the optical rotation is derived are mapped out through computations on model ORD “data.” These are obtained by transforming gaussian circular dichroism spectra into optical rotatory dispersion data through the Kuhn-Braun function. The effects of circular dichroism band width, interband separation, experimental error, and typical approximations made in the Drude equation upon the correspondence between circular dichroism parameters and Drude parameters are shown.
Introduction Early in the history of optical rotation studies, the variation of rotation with wavelength was discovered, and the pattern of this was recognized in Biot’s law: Theoretical work by ORD(X) = C/(Xz - io2). Drudea gave an equation ORD(h) =
cck/(X2 - AX2) k
(1)
explicitly relating the rotatory dispersion to absorption centers, X k , of the observed entity. Drude’s equation was meant to describe the wing of the dispersion curve, i.e., the shape in the wavelength region removed from the absorption centers. It was found, especially by L ~ w r y to , ~ provide satisfactory fit with data accumulated on rotatory dispersion of organic molecules. The most exacting test of the application of the Drude equation to dispersion data has been made with crystalline quart^.^^^ A striking feature of such fitting of optical rotatory dispersion data with the Drude function is that, in the hands of different investigators, data on the same material have been described with quite different sets of parameters (e.g., quartz516). The conclusion has been drawn by some that Drude-fit parameters cannot be related reliably to the physically meaningful parameters of absorption wavelength and rotational strength or that at least a careful, independent test is indispensable in each The direct measurement of
optical rotation or circular dichroism within the region of optically active absorption is the preferred method of these critics. Although recent progress in instrument development has expanded the accessible wavelength range and increased the sensitivity of the spectropolarimetry, the Drude-fit technique is still the only practical approach for the investigation of many systems for which the absorption region is still insufficiently accessible. With this technique, we take advantage of the fact that the optical rotatory dispersion (ORD) is an expanded function, in the sense that it decreases only slowly outside the region of primary optically active absorption (circular dichroism). This allows measurements of high precision to be made in the wing region even when it is impossible to measure through the center of the function at X I , either because the center is outside the range of the instrumentation or because the light ab(1) Based on work performed under the auspices of the U. S. Atomic Energy Commission. (2) On leave from the Federal Institute of Technology. Zurich, Switzerland. (3) P. Drude, “Lehrbuch der Optik,” 1st ed, Hirzel, Leipsig, 1900. (4) T. M.Lowry, “Optical Rotatory Power,” Longmans, Green and Co., London, 1935. (5) L. I. Katzin, J. Phys. Chem., 68, 2367 (1964). (6) S. Chandrasekhar, Proc. Rov. SOC.(London), A259, 531 (1961). (7) C. Djerassi, “Optical Rotatory Dispersion,” McGraw-Hill Book Co., Inc., New York, N. Y., 1960. (8) W. Heller, J. Phys. Chem., 6 2 , 1569 (1958).
Volume 70, Number 8
August 1966
TIIEOBURERAND LEONARD I. KATZIN
2664
sorption in that region reduces the instrumental sensitivity too farug Basically the problem is that of fitting a smooth and largely featureless curve with a function which, even when it contains only two Drude terms, involves four adjustable parameters and which further attempts to describe the inaccessible wavelength region by extrapolation from the wing. It is obvious that this places very exacting requirements on the data to be used, on the mathematical fitting procedure, and on the criteria of “fit.” The empirical application of a versatile least-squares computer program to data of relatively high precision has been described in earlier papers from this labore t0ry.~~’O,~1 We wish now to develop some general guidelines for effective use of the technique and to establish the necessary correlations between Drudefit parameters and the physical parameters of absorption Wavelength and rotational strength. To demonstrate these relations we use model calculations on “data” synthesized from accepted equations which describe optical rotation in the absorption region. The Drude Equation According t o a general concept of relationships between absorption and dispersion, each circular dichroism component12CDk(v)corresponds to an optical rotatory dispersion component, ORDk(v), through the relationship ORDk(v) = {CDk(v’)f(v,v’)dv’ (see Appendix). The full spectrum is the summation over the components. Several mathematical expressions have been proposed to describe the characteristic shape of the individual rotatory dispersion component in the absorption region, usually called the Cotton effect. Most of them involve a Gaussian function for CDk(v). It has been demonstratedla that the Drude function is an asymptotic approximation to the wings of the corresponding ORD,(v) (see Appendix). The nature of this asymptotic relation is shown in Figure 1, with the relative deviations in a sample case listed in Table I. The circular dichroism component, CDk(v),is characterized by its intensity (CDko), its location (vJ, and its width (Avk). The Drude function asymptotic to ORDk(v) is characterized by intensity and location, but in effect assumes an infinitesimally narrow width. (It is most simply defined as the Kramers-Kronig transformate of a 6 function for cDk(v>.) From the mathematical standpoint, the Drude expression is a two-parameter approximation to a threeparameter function. Therefore, the precision of the approximation The Journal of Physical Chemistry
Figure 1. Diagrammatic representation of the relations between a circular dichroism peak, CDk, centered a t wavelength Ak, and its optical rotatory dispersion, ORDk, and the asymptotic Drude relation, DR,,. T h e Drude term “best fit” to the wing of ORDk, DRfit, and the corresponding critical wavelength, Xfit, are also indicated.
of the Drude function to ORDk(v) depends on the extent to which the individual band shape of CDk(v), epitomized by Av/vk, can be neglected. Analysis of Experimental Data The two interrelated facets of the analysis of experimental data are (a) the empirical reproduction of the data within their statistical accuracy by a fit-function with the fewest adjustable parameters and (b) the physical significance of those parameters. With respect to empirical fitting, to a first approximation any so-called “plain” ORD curve7 (Le., one showing no change in sign of rotation and no extremum within the wavelength range of the data) can be approximated with a one-term Drude function ORD(X) = C,/(X2
- Xi2)
(2)
in which C1 and XI are adjustable parameters. However, as the number of data points, wavelength range of the data, and data precision are increased, the systematic deviation from the one-term description increases, and consideration of additional adjustable parameters becomes unavoidable. Introduction of a second Drude term, with its two additional parameters, (9) T.Burer, “Advances in Molecular Spectroscopy,” Vol. 111, A . Mangini, Ed., Pergamon Press, New York, N. Y., 1962,p 1268. (10)L. I. Katain and E. Gulyas, J. Am. Chem. Xoc., 86, 1655 (1964). (11) L. I. Katairi and E. Gulyas, J. Phys. Chem., 66, 494 (1962). (12) TO avoid multiplying symbols, C D ( u ) will be used as an abbreviation for “circular dichroism component” even when the context of the discussion involves wavelength rather than frequency units. (13) A. Moscowitz in “Optical Rotatory Dispersion,” C. Djerassi, Ed., McGraw-Hill Book Co., Inc., New York, N.Y.,1960,Chapter 12.
OPTICALROTATORY DISPERSION DATAAND
THE
2665
DRUDEEQUATION
Table I: Deviation of the Rotatory Dispersion Function ORD(v) from the Drude Function DRaaymp(v) Distance from center
-
J%)/LiVo
1
1.5
2
4
8
16
32
Relative deviation, yo (ORD(v) - DR(v))/ORD(V)
7.2
22.6
17.5
3.6
0.90
0.25
0.072
Z = (Y
into the fit-function increases its flexibility, and will therefore reduce the deviations in the fit. However, whereas the one-term function can readily be fitted graphically, there is no comparably simple way of treating the two-term equation with four adjustable parameters. Many approximation procedures have been used, including assumptions of fixed values for one or more of the pararneters.l4-” These in turn have led to uncertainties in the values of the parameters which, to a certain extent, have been used as arguments to discredit the Drude equation itself. With the current availability of fast and efficient computers, it is unnecessary to settle for less than direct numerical fit according to accepted statistical criteria (ie., the “least-squares” criterion) and simultaneous adjustment of all of the parameters. Modern instrumentation has also made it possible to extend greatly the wavelength range and number of data points, and, importantly, their precision. The accurate fitting of ORD data of this sort has been demonstrated in several publications from this l a b ~ r a t o r y ~using ~ ’ ~ the ~ ~ ~technique outlined in the Appendix. In this procedure, the absolute deviation (rather than the relative) was minimized since, from the origin of the data, the experimental error is expected to be relatively constant in magnitude, independent of the absolute value of the rotation at the given wavelength, in contrast to some other types of measurements in which percentage of accuracy is more likely. The “fit” is considered satisfactory when the computation has converged to give an essentially random deviation pattern, with a root-mean-square deviation comparable to the expected reliability of the data points.
Analysis of Synthetic Data Accepting that the Drude function is a good approximation to the wing of the ORD function expected from theoretical considerations and that it is suitable for fitting experimental data, three fundamental questions remain. (1) How do computed Drude parameters for a ~ to the parameters for the limiting Drude single J J relate term of the generating function, especially as a function of width of the generating CDk(v)?
( 2 ) Can one distinguish between data sets generated by a single CD(v) and those generated by two or more CD(v), and what are the limiting conditions to such discrimination and identification? (3) With two generating CD(v), what are the characteristics of the Drude-fit parameters for the data sets, in terms of the relative positions, the intensities and breadths of the CD(v), and the approach to the absorption region? To attempt to answer such questions, we have chosen to produce synthetic rotatory dispersion “data” from equations for the ORD functions and then to analyze such data by the computer program for the Drudefunction fitting, mentioned above. From the variety we chose to test the of proposed ORD Kuhn function19 and the Moscowitz functioml8 Since the numerical results relevant to our problem from the different functions turned out to be essentially the same, we are reporting in detail only the findings using the Kuhn function (see Appendix). ORD,(~) = B(exp(-r2)Sdi-exp(~l)di exp( -x+’)S’*exp( 0 P)dt
}
To restrict the investigation to a useful area, we have simulated laboratory data of the type we have analyzed previously. We have also presented them in conventional laboratory wavelength units rather than frequencies. Therefore, the wavelength range used has been from 0.65 to 0.2 p, with 0.005, (50-A) intervals between “data points.” I n one series, X k was taken as 0.2 p (approximating the carboxyl chromophore), and, in another, X k was 0.3 p (about the absorption of the carbonyl chromophore), Computation series included single CD(v), pairs of the same sign, and pairs of the opposite sign. I n all cases, the effects of varying the (14) T. M. Lowry and J. 0. Cutter, J . Chem. SOC.,121, 532 (1922). (15) E. Iiauka and J. T. Yang, Biochemistry, 3, 1519 (1964). (16) M. K. Hargreaves and P. J. Richardson, J . Chem. SOC.,2260 (1957). (17) E. Schechter and E. Blout, Proc. Natl. Acad. Sci. U . S., 51, 695, 794 (1964). (18) A. Moscowitz, Thesis, Harvard University, 1957. (19) W. Kuhn and E. Braun, Z. Physik. Chem.. 88, 281 (1930).
Volume 70, Sztmber 8 August 1966
THEOBURERAND LEONARD I. KATZIN
2666
Table 11: Fit Results for One-Term Test Functions with Varying Widthasb Test function width
t iF---
Fit range,
function, Cl/(X2 - Xiz)--10aRMSN XI
z
PT
ROT
C1
From 0,6500 To 0,3500 0.3000 0.2700 0.2500
34.6 21.4 16.7 13.0 10.0
1 61 71 77 81
26.2 121.3 200.2 304.3 445.3
10.0019 10.0016 10.0010 9.9998
0.20007 0,20008 0.20009 0.20011
0.0416 0.188 0.627 1.83
10.0016 10,0009 9.9997 9.9976
0.20007 0.20009 0.20010 0.20012
0.00113 0.00297 0.00694 0.01544
0,0157 0.0313 0.355 1.15
From 0.6500 To 0.3500 0.3000 0.2700 0.2500
17.3 10.7 8.3 6.5 5.0
1 61 71 77 81
26.2 121.5 200.6 305.5 448.1
10.0077 10.0064 10.0039 9.9987
0.20027 0,20032 0,20037 0.20045
0.169 0.767 2.58 7.61
10.0062 10,0035 9.9987 9.9899
0.20030 0.20035 0.20041 0,20050
0.00456 0.01209 0.02844 0.06400
0.0638 0,373 1.46 4.83
From 0,6500 To 0,3500 0.3000 0.2700 0.2500
8.65 5.35 4.14 3.25 2.50
1 61 71 77 81
26.3 122.3 202.7 310.4 460.3
10.0308 10.0253 10.0138 9.9890
0.20111 0.20129 0.20153 0.20189
0.713 3.32 11.6 36.3
10.0244 10.0128 9.9911 9.9481
0.20122 0.20143 0,20171 0.20211
0.01927 0.05198 0,12620 0,30114
X,
P
0.02
0.04
0.08
Cl/(h2
-
X12)
+ cz/x-
-C1/(X*
-
X12)
+ C2/(X2
0.242 1.63 6.67 23.9
- 0.02)-
c1
A1
c2
lOJRMSN
c 1
XI
c 2
10JRMSN
9.9960 9.9937 9.9902 9.9846
0.20010 0.20012 0.20014 0.20016
0.00626 0.00866 0.01247 0.01880
0.00319 0.0238 0.113 0.432
9.9847 9.9809 9.9754 9.9669
0.20013 0.20015 0.20017 0.20019
0.01751 0.02136 0.02706 0.03595
0,00179 0.0140 0.0693 0.278
9.9837 9.9744 9.9597 9.9367
0,20042 0.20048 0.20056 0.20066
0.02522 0.03504 0.05072 0.07721
0.0130 0.0980 0.469 1.84
9.9387 9.9234 9.9007 9.8656
0.20054 0.20060 0.20068 0.20078
0.07007 0.08566 0.10903 0.14601
0.00738 0.0580 0.291 1.19
9.9318 9.8913 9.8262 9.7116
0.20173 0.20198 0,20232 0.20282
0.10410 0.14676 0.21796 0.34916
0.0571 0.443 2.24 9.73
9.7543 9.6893 9,5890 9.4198
0,20220 0.20246 0.20281 0.20335
0.28103 0.34725 0,45103 0.63040
0.0329 0.268 1.43 6.60
Width = A v k / v k , relative half-intensity band width of CDk(v). 1: = ( v l - v ) / A v l , relative DR,.,,, = 1O.O/(X2 - (0,2)2). distance from the center of CDI(v) (ie., from the closer CD function). PT, number of points. ROT, rotation of the test function in degrees a t given wavelength. RMSN = RMS X 10/ROT (0.5 p ) , root-mean-square deviation, normalized for a test function with calculated parameters for the Drude function asymptotic to the wings of the test function (for each ROT = 10" a t X 0.5 p. DRasymp, ORDk(v)individually). Note that these characterize the test functions used to generate the data.
band width and the wavelength range fitted were tested, and with pairs of CD(v), the influences of the relative intensities and of the spacing between their centers were also investigated. I n Tables 11-V, we present typical results for a selection of cases. For convenient comparison, the examples listed have equal numerical magnitudes of the rotational strength R, for the individual circular dichroism components, and the longest wavelength component is centered at 0.2000 p. I n discussing the results, we make use of the parameters Ck and XI, of The Journal of Physical Chemistry
the Drude function asymptotic to the wings of the dispersion curve, DRa8Ymp, and the relative half-intensity band width, Avr/v,, of the generating circular dichroism band($) (numerically equal to AX/Xk). The fundamental equations fitted are eq 2 and
but fittings were also done, where appropriate to the purpose, to the approximate, abbreviated forms of the two-term Drude equation
OPTICALROTATORY DISPERSIONDATAAND THE DRUDEEQUATION
2667
Table I11 : Fit Results for Test Functions with Two Terms of Same Sign, Varying Width& ,Test functionWidth SepC
-Fit
Fit range, X,
0.02 1.0
x
P
From 0.6500 34.6
To
0.3500 21.4 0.3000 16.7 0.2700 13.0 0.2500 10.0
function, CI/(X~ XI (72
XIS)
+ Ca/(X2 - hz9-
-
----CI/(X~
X+I-
PT
ROT
c1
1 61 71 77 81
51.2 235.7 386.7 583.6 845.7
9.2225 9.2287 9.1353 8.9439
0.20086 0.20086 0.20088 0.20094
10.3934 0.19569 10.3872 0.19569 10.4806 0.19571 10.6721 0.19576
0.0299 0.0403 0.0985 0.580
19.615 19.614 19.611 19.606
0.19818 0.19820 0.19823 0.19827
0.092 0.410 1.33 3.79
1 61 71 77 81
51.2 236.1 387.7 585.8 850.6
8.5540 8.4602 8.1498 7.7217
0.20268 0.20272 0.20285 0.20305
11.0747 11.1685 11.4789 11.9070
0.0357 0.0877 0.728 4.82
19.6263 0.19838 19.6232 0.19843 19.6170 0.19850 19.6051 0.19860
0.212 0.949 3.12 8.99
lO6RMSN
Xz
Xi
c.1
lO*RMSN
0.04 0.5 From 0.6500 17.3
To
0.3500 0.3000 0.2700 0.2500
10.7 8.3 6.5 5.0
0.19483 0.19487 0.19498 0.19514
0.08 0.25 From 0,6500 8.65 1 51.4 To 0.3500 5.35 61 237.6 7.4988 0.20773 12.1812 0.19326
0.0798 0.3000 4.14 71 391.4 7.0207 0.20815 12.6592 0.19357 1.21 0.2700 3.25 77 594.7 6.3409 0.20880 13.3389 0.19402 12.0 0.2500 2.50 81 872.2 5.3263 0.20985 14.3531 0.19470 102
Cl/(X2
c1
- X12)
Xi
+ -c Cz
I
c1
lOaRMSN
Ci/(Xn XI
-
112)
f CX z/Cz
--Cl/(h2
lOaRMSN
CI
19.6719 0.19921 0.874 19.6610 0.19938 3.32 19.6388 0.19963 11.3 19.5926 0,19999 34.8
-
+ Cd(X2 - 0.02)-
XI
Cz
lOJRMSN
19.613 19,610 19,605 19.597
0,19819 0,19822 0.19825 0,19830
0.00491 0.0123 0.0294 0,0639
0.0346 0.198 0.75 2.38
19.588 19.578 19.563 19.539
0.19826 0,19830 0,19834 0,19839
0.0284 0.0389 0.0551 0.0814
0.064 0.469 2.17 8.14
19.534 19.517 19.493 19.457
0.19833 0,19837 0,19841 0,19846
0.082 0.099 0.124 0,162
0.035 0.270 1.31 5.09
19.6225 19.6160 19,6044 19.5837
0.19841 0.19848 0,19855 0,19866
0.01131 0.02960 0,06847 0,15065
0.0798 0.459 1.76 5.66
19.5643 19.5411 19.5060 19.4505
0.19858 0.19865 0.19874 0.19887
0.06495 0.08931 0.12744 0.19017
0.0160 0.118 0.553 2,lO
19.4417 19.4030 19.3469 19.2620
0.19874 0.19881 0.19891 0.19904
0.18730 0.22658 0.28430 0.37332
0.00888 0.0658 0.336 1.34
19,6591 19,6361 19.5940 19.5133
0.19932 0.19953 0.19981 0,20022
0,03865 0,10291 0.24528 0,56922
19.4649 19,3836 19.2552 19.0362
0.19986 0.20012 0.20046 0.20096
0.21710 0.30270 0.44248 0.69160
0,0566 0.432 2.13 8.89
19.0734 18.9409 18.7404 18.4122
0.20039 0.20065 0.20101 0.20154
0.60759 0.74249 0.94938 1.29547
0.0320 0,256 1.33 5.91
' DR,,,,
=
0.275 1.62 6.48 22.4
+
10.O/(Xz- (0.2)2) 9.6177/(X2- (0.19608)2). * For meaning of notations, see footnote b of Table 11.
Sep =
(P,
- v ~ ) / A vrelative ~, separation of two CDR(Y).
ORD(X) = Ci/(Xz - Xi') ORD(X) = Ci/(X2
- Xi2)
+ C2 + Cz/X2
(4) (5)
Although the fit results are given in terms of C, and XI, values, the formulation of the parameters in the
actual computation was often different, to speed convergence (see .Appendix), or for convenience of programming (e.g., use of X k 2 as a parameter instead of A t ) . It is easy to demonstrate that with convergent calculations such convenience alterations do not affect the final fit results. An additional quantity important in discussion of the fit results is the root-mean-square deviation. We designate this quantity by RMSN, for computed ORD data normalized to a rotation value of 10" a t 0.5000 p , so that more direct comparisons can be made
between different series of computations. Other quantities appearing in our discussions of results are the fit range (wavelength span, in microns, from 0.65 p to shorter wavelengths, of the data being fitted), the relative distance (distance from the center of the CD band to the wavelength in question in multiples of the band width), and the number of data points (starting from 0.65 p, with a point every 0.005 p ) . This last has some significance as an indicator of the statistical base on which the computation rests, as well as a measure of wavelength span of the data. We first show the results for a single CD(v) centered at 0.2 p . Table I1 gives the computed Drude parameters and other characteristics for three cases, with relative half-intensity band widths Avk/vk of the CD( Y ) equal, respectively, to 0.02, 0.04, and 0.08. In Volume 70,Number 8 August 1966
THEOBURERAND LEONARD I. KATZIN
Table IV: Fit Results for Test Function with Two Terms of Opposite Sign, Varying Width and Separation4 Part A -Test functionWidth SepC
0.02
Fit range, X, P
0.3000 0.2700 0.2500
Cl
ROT
c2
XI
To
0.3500 0.30OO 0.2700 0.2500
34.6 21.4 16.7 13.0 10.0
1 61 71 77 81
1.12 6.86 13.58 25.04 45.04
5.6508 5.6465 5.6331 5.6140
0.20159 0.20160 0.20160 0.20161
17.3 10.7 8.3 6.5 5.0
1 61 71 77 81
1.12 6.88 13.64 25.24 45.66
3.2835 3.2690 3.2454 3.2063
0.20438 0.20441 0.20445 0.20452
8.65 5.35 4.14 3.25 2.50
1 61 71 77 81
1.12 6.96 13.90 26.09 48.51
1.7444 1.7127 1.6583 1.5587
0.21092 0.21111 0.21144 0.21203
-5.2624
0.19454 0.19454 0.19453 0.19451
0.0318 0.0366 0.0945 0,630
-2.8567 -2.8177
0,19191 0.19187 0.19181 0.19171
0.0199 0.0970 0.964 7.76
-1.3547 -1.3231 -1.2687 -1.1693
0.18583 0.18547 0.18474 0.18307
0.0996 1.68 21.1 233
- 5.2581 -5.2447 -5.2256
-2.8949
-2.8804
0.25 From 0.6500
To
0.3500 0.3000 0.2700 0.2500
Part Bd Fit range,
,Test functionSepc Width
0.02
P
To
0.3500 0.3000 0.2700 0.2500
c1
ROT
PT
F i t function, Ci/(X* - Xi’) X1 CP
+ Cz/(X* - X
P ~
XP
1OsRM SN
34.6 21.4 16.7 13.0 10.0
1 61
17.3 10.7 8.3 6.5 5.0
1 61
8.65 5.35 4.14 3.25 2.50
2.16 13.11 25.65 46.66 82.47
8.0468 8.0370 8.0183 7.9892
0.20101 0.20101 0.20102 0.20104
-7.2922 -7.2824 -7.2637 -7.2346
0,19135 0.19135 0.19133 0.19131
0.0149 0.0252 0.159 1.15
77 81
2.16 13.14 25.77 47.01 83.52
5.6453 5.6193 5.5761 5.5066
0,20330 0.20333 0.20338 0.20345
-4.8902 -4.8642 -4.8210 - 4.7515
0.18921 0.18917 0.18910 0.18897
0.0126 0.125 1.20 9.13
1 61 71 77 81
2.17 13.29 26.24 48.51 88.34
3.2748 3,2154 3.1167 2.9408
0.20940 0.20959 0.20991 0.21049
-2.5177 -2.4584 -2.3597 -2.1842
0.18356 0.18317 0.18245 0.18082
0.0964 1.69 19.6 201
71 77 81
1 .o From 0.6500
To
0.08
2
2.0 From 0.6500
0.04
>
lOsRMSN
X2
0.5 From 0.6500
0.08
PT
1.0 From 0.6500 To 0.3500
0.04
Fit function, Ci/(X* - Xis) C C z / ( X P - X z 2 )
r 2
0.3500 0.3000 0.2700 0.2500
71
0.50 From 0.6500 To 0.3500
0.3OOO 0.2700 0.2500
*
-
- 9.6117/(X2- (0.19608)2).
For meaning of notations, see footnote b of Table 11. DE,,, = 10.O/(Xa (0.2)2) DR,,,,, = 10.0/(X2- (0.2)2) 9.2456/(X2 (0.19231)e). note e of Table 111.
-
all cases the asymptotic Drude expression for the “data” corresponds to I)Rssymp(X)= 10.0/(X2 - 0.04)
(6) Table I1 shows that C1 is computed slightly too high, and X1 slightly too long, relative to the asymptotic values. This effect becomes more pronounced for AI as the data extend more toward 0.2 p and as the The Journal of Physical Chemistry
-
See foot-
band width becomes larger. As A1 increases, C1 decreases in compensation, and RMSN becomes larger. This is the expected nature of the discrepancy, according to Figure 1. The test function approaches the asymptotic Drude function from above, so that toward shorter wavelengths (up to about two half-widths from the center) its curvature is always greater. The fitted Drude parameters respond
OPTICALROTATORY DISPERSION DATAAND
THE
DRUDEEQUATION
2669
Table V : Fit Results for Test Functions with Two Terms of Opposite Sign and Varying Widtha7b ,Test function-. Width Sepc
0.02
0.04
0.08
-----Fit
Fit range, A, c
X
PT
ROT
c1
From 0.6500 To 0,3500 0.3000 0.2700 0.2500
34.6 21.4 16.7 13.0 10.0
1 61 71 77 81
1.12 6.86 13.58 25.04 45.04
From 0.6500 To 0.3500 0.3000 0.2700 0.2500
17.3 10.7 8.3 6.5 5.0
1 61 71 77 81
From 0,6500
8.65 5.35 4.14 3.25 2.50
1 61 71 77 81
-
AIS)
+ Cl/(A* - 0.02)-
A1
c1
IO’RMSN
0.79075 0.83793 0.90013 0.98328
0.22202 0.21955 0.21700 0 . 2 1442
- 0.40329 - 0.45225 - 0.51850 - 0,61036
1.28 8.96 38.9 133
1.12 6.88 13.64 25.24 45,66
0.78756 0,83373 0.89441 0,97519
0.22245 0.22002 0.21751 0 . 2 1500
- 0,39985 - 0.44778 - 0.51249 - 0,60195
1.27 8.91 38.8 133
1.12 6.96 13.90 26.09 48.51
0.77462 0.81632 0.86976 0.93715
0.22422 0.22197 0.21971 0,21758
- 0,38586 - 0.42926 - 0,48655 - 0,56201
1.23 8.61 37.2 124
1.0
0.5
0.25
To
0.3500 0.3000 0,2700 0.2500
Ci/(A*
c1
O.O/(X2
108RMSN
-10.9105
0.19636 0.19640 0.19646 0.19655
10.7379 10.8594 11.0402 11.2991
0,0108 0.0826 0.435 1.88
- 14,4550 - 14.3273 - 15,8284 - 18.5516
0,19739 0.19737 0.19763 0.19799
14.8437 14,7160 16.2173 18,9410
0.246 0.339 1.81 7.95
25,9581 25.7575 15.6063 9.0576
0.20145 0.20146 0,20238 0,20398
- 25.3673 - 15,2154
-10.3495
DRmyrnp
- A i 9 4- Cz/(Xl - 0.04) c2
hi
- 10.4710 - 10,6517
a
function, Cl/(hl
- (0.2)*) - 9.6117/(X2
- (0.19608)2).
- 25.5680
-8.6641
1.13 2.50 8.54 42.5
’For meaning of notations, see footnote b c
Table 11.
See ootnote
c of Table 111.
to this curvature and give back an absorption wavelength longer than that of the asymptotic Drude function. One may attempt to compensate for the band width effect by introducing a third adjustable parameter in various arbitrary ways, as by eq 4 or eq 5 , or by using eq 3 with A2 set, at, say, 0.2 p . One does get improvement of RMSN with such successive changes but at the expense of even further decrease in C1and further increase in A 1 over the asymptotic values. The deviation pattern is still systematic, showing as many nodes as there are adjustable parameters used in the fitting function. Significantly, when the data are analyzed by the full two-term Drude function, eq 3, with four adjustable
parameters, the wavelengths for the two Drude-fit terms become essentially identical, indicating that the computer “sees” the solution effectively as a one-term solution. It, therefore, seems that one will not obtain a true least-squares solution containing more Drude terms than are appropriate to the original data. In consequence of Kuhn’s sum rule20
CCk/Xi?
=
0
k
(sum over all absorptions), which is based on fundamental symmetry considerations, every optically active (20) W.Kuhn, Z . Physik. Chem., B4, 14 (1929); W.Kuhn and K. Freudenberg, “Hand und Jahrbuch der chemischen Physik,” Vol. 8, Part 3,Akademische Verlagsgesellschaft, Leipzig, 1932, p 47;E.U. Condon, Rev. Mod. Phys., 9, 432 (1937).
Volume 70, Number 8 August 1966
THEOBURERAND LEONARD I. KATZIN
2670
compound must have more than one CD(v). In fact, the simplest case to meet this requirement will be two CD(v), of equal rotational strength and opposite sign. One may, therefore, concentrate on the twoterm Drudc case, without any more detailed discussion of the one-term computations. Looking now to the cases in which two CD(v) contribute to the ORD, we may consider equal rotational strengths (magnitude and sign) and band width and a separation of, say, one band width. Algebraic analysis of the two-term Drude equationlOpllsuggests strongly that this will be very difficult to distinguish from a single contribution, with Drude parameters Co = C1 Cz and Xo2 = 1/z(X12 Xz2). If a test of the “data” with the one-term equation is made, such a match is indeed found, with an RMSN of the order of to depending on the specific band widths, separation, and fit range. However, in contrast to the one-term case, the computation converges reasonably well, with a favorable choice of the analytical expression for eq 3, for the full two-term Drudefit function. The RSISS is improved over that for the one-term analysis by a factor of lo4,and the resolution can be effected even for peak separations which are only 0.25 of the band width. The intensity of the more remote term is overestimated somewhat, andsince the sum (C1 Cz) of eq 3 is roughly constantthe intensity of the nearer contribution is underestimated correspondingly (Table 111). Other properties of the computation will be mentioned together with those for the further cases to be discussed. When the two CD components are of opposite sign, the fitting of the resultant ORD “data” with eq 3 is relatively uncomplicated and straightforward. The fit value for X1, that for the nearer contribution, is longer than the asymptotic value, as in the case for the single CD(v). The fit value for Xz, the second component, however, is smaller than its asymptotic value, so that the wavelength spacing is definitely greater than that of the original components. (This is true also when the two components are of the same sign.) I n consequence (Table IV), the intensity parameters are decreased from the asymptotic values, approximately according to the relation
+
+
+
tC1 - CZ)(Xl2- Xz2) = constant
If the fit range is extended toward the absorption region, if the bnnd widths of the CD(v) are increased, or if the spacing between the CD(v) is decreased (both of the latter increasing their overlap) , the spacing computed relative to the input value is increased (Table IV). If, on the other hand, the separation of the two CD components is increased considerably, keeping their A,
The Journal of Physical Chemistry
intensity and band width constant, the fit parameters for the nearer component are improved, whereas those of the more remote component deteriorate with increasing separation from the wavelength region of the “data.” I n many cases in the literature, fitting procedures have been used with experimental systems, in which a three-parameter form of eq 3 has been used. This has been generally either in the form either of eq 5 or of eq 3 where XZ is fixed at some “known” value, e.g., the wavelength of an absorption maximum. If we repeat this procedure, by setting Xz in eq 3 to 0.2 k , the results of Table V are obtained. It is seen that values of the parameters are sensitively dependent on the band width and overlap of CD(v) generating the “data.” The numerical values of C1 and Cz may be quite different from the asymptotic values, and even the sign will depend on whether the computation interprets the fixed wavelength parameter as relating to the positive or the negative term. If the computation is set so that not only the wavelength parameter hz but also the sign of Cz is fixed, then the fitting computation could give quite misleading results. In either case, the position of X1 will be unpredictably related to that of the generating function-it may be at longer or shorter wavelength. The RIISX values, of course, are much poorer than for the free four-parameter fitting. For a poor guess at Xz ( e . g . , 0.1414), the solution takes on the character of the one-term solution with three parameters (Table V), as discussed below. To this point we have been using data precise to the fifth or sixth decimal place. We must now consider the effects of finite experimental error in the data on the fitting results. To this end, we have introduced random Gaussian errors into the data, corresponding to a preset absolute RnlSN value. In one case this was times the rotation value at 5000 A (corresponding to =kO.OOl”, for the data normalized to 10.0” a t 5000 A), and in a second case it was taken as ten times as large. The important effect of the superimposed error, judged from the results, was to flatten the least-squares surface so that efficient convergence was interfered with. With an error of ~ 0 . 0 0 1 ”it, was still possible to obtain essentially the same values of the parameters as without the superimposed errors, though convergence was considerably slower. The RJISN obtained essentially matched the input error, being 1 or 2’3, on the low side. With error of *O.0lo, however, the surface was flattened so much that, with the normal four-parameter computation, convergence was not found. Presumably, however, if one of the parameters had been assigned, convergence to a three-parameter solution could still have been obtained. The action
OPTICALROTATORY DISPERSION DATAAND
THE
DRUDEEQUATION
of experimentd error is, therefore, a loss of precision in the fitting, with a diminution in the number of parameters which can be resolved. The sensitiveness of the effect is perhaps greater than intuition might have anticipated.
Discussion As has been pointed out elsewhere,’l from the twoterm Drude equation in the polynomial form
(CI
+ Cz)(Xz - L ) + (Ci - Cz)A
(7) - L)’ - A* Xzz) and 2 A = ( X l 2 - X Z ~ ) ,one can (Az
+
where 2L = (Xl2 make a number of qualitative predictions. Thus, at very long wavelengths (X >> 11, XZ) and CI = Cz, the rotations will approximate to (Ci Cz)/(Xz - L ) . If C1 >> Cz, the rotations will look like Cl/(X2 - X12). As the absorption region is approached, the relative importance of (Cl - Cz)A and A2, respectively, will increase, and resolution of the two Drude terms should increase. In the case that C1 and Cz are of opposite sign but comparable magnitude, there may actually appear a change in sign of the rotation in the wavelength span of measurement, so that the fit function cannot at all s:ttisfactorily be approximated by a single Drude term. The computations reported in this paper illustrate that with proper fitting of sufficiently precise and extensive data, it may even be possible to demonstrate when two terms of the same sign are present. In the practical case, however, Kuhn’s rule indicates that these would be accompanied by one or more terms of the opposite sign, leading to a t least the complicated three-term, six-parameter situation. This in turn would probably be approximated by the soluble twoterm, four-parameter equation, in which the wavelength parameters would represent some sort of average of the values for the terms of the same sign (see below). As has been seen, the analysis of “data” from the twoterm test function with the two-term Drude function leads to quite good determination of the parameters in many cases. The wavelength separation of the fit terms, however, is larger than that of the generating functions. This is in qualitative agreement with the analysis of the corresponding circular dichroism pattern: The superposition of two Gaussian circular dichroism bands with comparable magnitude but opposite sign gives an apparent separation of the two bands not smaller than their half-intensity band width even when the actual separation is far below this magnitude.z1~2zFrom our model calculations,
+
2671
we conclude that the Drude-fit analysis of the corresponding ORD data gives an apparent separation not smaller than about 1.5 times the half-intensity band width. As has been pointed out, imprecise measurements and also numbers of points insufficient to give good statistical evaluation of parameters serve, in effect, to reduce the number of parameters which can be resolved and defined meaningfully. Perhaps even less recognized have been the effects of fixing one or more of the parameters. As has been shown above, fixing one of the wavelength parameters at a “known” value will result in unpredictable variations of the remaining three parameters. If the fixed value chosen is a poor one, the resulting fit parameters will bear no useful relationship at all to the expected parameters, and one deals, in effect, with the one-term approximation to the two-term situation. If one fixes both wavelength parameters, as has been done in the l i t e r a t ~ r e , ~ ~ , ~ ’ one reduces the problem essentially to one of a single parameter. This can be seen through eq 7 . Fixing the wavelengths determines both L and A. In addition, the range for the values C1 and Cz (opposite sign assumed) is narrowly fixed through (C1 - Cz)A, so that for practical purposes one is determining only the difference of the C’s through (C1 Cz)(Xz - L ) . This could be done equally well through measurements at a single wavelength. Still other approximations6jz3 reduce the computation to two parameters, (Cl CZ) and L (cf. ref 5 ) . Molecules in general have a number of energy levels to which transitions may occur, and when these molecules are optically active, each one can lead to a CD(v) (see also ref 24). Real molecules therefore can be expected generally to have more than two CD(v) and a corresponding number of Drude terms for the rotational components. The two-term Drude analysis will therefore always be an approximation. Operating in its favor are the tendency of closely spaced terms of the same sign to appear as a single tern1 and the general tendency for distant spacing of levels of comparable intensity. The greater the spectral distance between transitions, the more precisely does the Drude analysis give the characteristics of the closer transition. The more remote terms get lumped together, with a net effect that is rather moderate, through the operation
+
+
(21) T. Burer, “Habilitationsschrift,” Eidgenossische Technische Hochschule, Zarich, 1963; T. Burer, H e h . Chim. Acta, 46, 2388 (1963). (22) K. M. Wellman, R. H. A. Laur, W. S. Briggs, A . Moscowits, and C. Djerassi, J. Am. Chem. SOC.,87, 66 (1965). (23) V. Sasisekharan, 2. Physik. Chem. (Frankfurt), 29, 219 (1961). (24) W. Kauzmann, F. B. Clough, and I. Tobias, Tetrahedron, 13, 57 (1961).
Volume 70,Number 8 August 1968
THEOBURERAND LEONARD I. KATZIN
2672
of the Kuhn sum relation. Thus, if one has a single strong rotat,ion in, say, the visible region, Ck/Xkz,and nothing at longer wavelengths, the sum of the effects at shorter wavelengths will be -C,/XkZ. If the nearest strong member of this group is in the far-ultraviolet, the rotational effect a t wavelengths in the vicinity of X, will be approximated by a Drude term Ceff/(X2
-
Xeff')
= (-C,/X,z)Xefi2/(X2
- Xeft')
where Xeff is some wavelength in the far-ultraviolet region which is appropriate to the approximation. It will be seen that the greater the separation between X k and the ultraviolet transitions, the less is the apparent contribution of the latter, since the apparent Drude rotatory constant, C e t f , is the C, value diminished by Xe:f2/Xk2. An experimental example of some of these points is furnished by the ORD of camphor,26 with a positive strong Cotton effect at ca. 290 mp, consisting of a number of vibronic components of the same sign26r27 and a negative contribution from the farultraviolet region, below 200 mp. The components at 290 mp act effectively as a single unit in the Drude analysis, and the rotation(s) below 200 mp act as a second unit, so distant spectrally from the measurement region as to be poorly characterized. An illustration of a different set of relations is given by ephedrine hydrochloride in water.28 Here the strong rotational effect is due to transition(s) in the far-ultraviolet region, while in the near-ultraviolet region a complex of weak positive and negative effects is associated with the phenyl absorption. The self-cancellation effects of this complex further weaken its net influence at longer wavelengths. I n consequence, analysis of Ihe rotation shows a satisfactory match to a far-ultraviolet, one-term Drude description for data down to 330 mp,29with the first Cotton extremum of the phenyl absorption as close as about 268 mp. Attempts to obtain a normal two-term fit give essentially the same rmults, the second term being a miniscule correction (C, < for a slight irregularity in the data in the 600-mp region. Attempts to force a twoterm solution with one term giving a close wavelength in the 250-270-mp region are unsuccessful, unless one extends the data range down to 305 mp. A leastsquares fit of reasonable stability is now obtained showing a relatively weak term with a X, a t about 253 mp. The strong systematic deviations at the short wavelengths, however, indicate one is too close to the absorption region. The question naturally arises, as to the circumstances under which an apparent single Drude term relation fits experimental data. The ephedrine case seems to be typical of a number of amines in the literature, and The JozLrnal of Physical Chemistry
we would attribute it to a combination of remoteness of the first strong dichroism and a significant spectral distance to the first strong term of the opposite sign. The camphor discussion above may be used as an analogy. With precise data, and terms of comparable strength, even remoteness from the measurement region will not prevent recognition of two close terms of opposite sign. The demonstration of this is quartqs with its first absorption bands in the vicinity of 100 mp. I n fitting experimental data on several systems containing the carboxyl chromophore, we have repeatedly and reproducibly found very narrow spacings between two Drude terms of opposite sign, e.g., 1-2-mp spacing for wavelengths approximating 200 mp.lo~ll If this result is assumed to conform to the fitting characteristics pointed out above, it would require very sharp and intense circular dichroism and absorption bands. Such bands do not correspond to those seen in the absorption spectra of these systems. To attempt to obtain fitting solutions of the above separation characteristics from two CD(v) components of normal width, we have tried many expedients. We have tested the effects of varying relative intensities of the two generating functions, the possible effects of broad band width of the shorter wavelength band, of solvent refraction corrections (cf. ref 25), and of contributions of a third CD(v), remote in wavelength. Although the apparent separation of the two Drude terms might be slightly reduced by these influences, none gave a result comparable with that sought. It has been found possible, with very careful setting of the initiating parameters for the fitting computation, to start on a plateau of the four-dimensional parameter surface which is so flat that the fitting does not fall off in a reasonable time and which corresponds, after a fashion, to the type of solution sought. The RMSN for this region, holw ever, is larger by a factor of lo2 than for the normasolution, and it is only by very careful planning of the initial parameters for a given test function that one can arrive a t this metastable solution. The computations for the laboratory data that give the narrow separations, in contrast, have the characteristics, both in convergence and in RMS, of a normal least-squares solution. Simulated experimental errors have also ~
(25) A. Singh and L. I. Katzin, J. Phys. Chem., 69, 3708 (1965). (26) A. Moscowita, K. M. Wellman, and C. Djerassi, Proc. Nail. Acad. Sci. U.S.,50, 799 (1963). (27) T. Burer, A. Singh, and L. I. Katzin, Symposium on Molecular Structure and Spectroscopy, Ohio State University, Columbus, Ohio, June 14-18, 1965. (28) I. P. Dirkx and Th. J. De Boer, Rec. Trau. Chim., 8 3 , 535 (1964). (29) L. I. Katzin and E. Gulyas, unpublished data.
OPTICAL ROTATORY DISPERSION DATAAND
THE
DRUDEEQUATION
been tried, iz the effort to explain the narrow spacings, but, as indicated, these only affect the ease of convergence in the fitting. Experimental investigation of the absorption region of carboxyl compounds, by rotatory dispersion and circular dichroism techniques, is in progress in this laboratory to obtain further clues to the answer to this conundrum of the wavelength relations. In conclusion, one can say that Drude-equation fitting, with two terms, is capable of giving very good results, when two CD(v) are involved, if the necessary precautions with respect to precision and statistical number of data points are observed. The relations of the derived parameters and those of the CD(v) on which the optical rotatory dispersion curve is based have been outlined. Many systems exist for which this is the only practical mode of investigation, and there are many more in which the information from this mode is a necessary adjunct to such observations as can be made in the absorption (Cotton effect) region. With more than two CD(v), relations are less predictable, but Kuhn’s sum rule operates to keep the apparent complexity down.
Appendix Dispersion-Absorption Relation. Moffitt and Moscowite3O formulate a Kramers-Kronig type of relation between circular dichroism, CD(v), and optical rotatory dispersion, ORD(v), for measurements in dilute solutions 2 ORD(v) = -S T
[
m v”v‘ CD(v’) [ b ( v ’ ) / b ( v ) ] y ’ ~dv’ ~ (Al)
*
- v
2673
band width A& by Axk = 1.665Ak.1a,27Further approximation steps (for x+, x-)
finally give the Drude function, asymptotic to the wing region (large x)
Kuhn Function. Similarly, but not in as straightforward fashion, a Gaussian, symmetrical in frequency CDk(v) = CDkoe-”-z leads to the function given by Kuhn and Braunlg 2 ORDk(v) = -(S)CDko-
4;
vk
exp(-x-2)
LX
exp(t2) dt - e x p ( - x + 2 ) ~ + exp(t2)dt}
(A4)
with XT = (vk ‘f .)/Ak and Avk = 1.665Ak, which in turn can also be reduced to t,he Drude function by the same approximation steps. Line Spectrum. The integral transformation (Al) of a 6 functiona2 C D ~ ( V=) Rk6(V - ~ k ) leads immediately to the Drude function ORD~(V)
2
2
V2
-TR k - V k 2
Xk2
-T Rx t X rk 2 (A5)
- v2
Computation of ORD Functions. The integral I =
where b(v) is the refraction correction for the solvent. We shall neglect here its dispersion, setting b(v‘) = bb). For ORD in molar rotation and CD in molar extinction the factor S becomes S = (180/~)(100)(ln 10)/4. 139 31 Moscowitz Function. The integral transformation (Al) of a Gaussian CD function, symmetrical in wavelength CD,(X) = CDkoe-”-’
e - ” ~ e “ d t was approximated for the range 7
< 1x1 by
the semiconvergent series 1
1
using up to ten terms, or, for the range 0 < 1x1 < 7 by
{
2 3
I =e-”x+-+-+ 1!3
26
2!5
,..
~
using up to 200 terms (minimum 1Olzj terms). was performed to a precision of , The~computation ~ better than (contribution of the last term), carrying ten significant digits.
(A2)
(30) W. Moffitt and A. Moscowitz, J. Chem. Phys., 30, 648 (1959). (31) F. Woldbye in “Technique of Inorganic Chemistry,” H. B.
where XT = (A =F &)/Ak; Ak is the half-band-width at l / e intensity and is related to the half-intensity
Jonassen, Ed., Interscience Publishers, Inc., New York, N. Y., in press. (32) E. Madelung, “Die Mathematischen Hilfsmittel des Physikers,” Springer-Verlag, Berlin, 1957, p 20.
leads within excellent approximation t
~
~
2 ORDk(x) = --(s)cDko{ exp( -x-z)
6
exp( -z+2)r+exp(12) 0 dt}
Volume 70, NumbeT 8
August 1966
THEOBURERAND LEONARDI. KATZIN
2674
Least-Squares Fit. Given a series of n points (experimental data), FMt, i = 1, . . ., n, to fit with a mathenlatical expression, FC, = fct{2,; B1, B2, . . . , B k ) with weighing factors p,, minimizing the sum of the squared deviations 5 ( B ) 2
W )= tCpd? =1
The fit result can be characterized-in addition to the values of B-by the sum of the squared deviations 5, by the standard deviation STD = dqj,f = n m 1 (with: f, degrees of freedom; rn, number of adjustable parameters), or simply by the root-meansquare deviation RMS = d5T. For the two-term Drude-fit function
-
where B = B1,B2, B S , . . . , Bk are the adjustable parameters and d,(B) = FM, - FC, are the deviations. The Taylor expansion of the function 5 to first order a t the point determined by the estimated initial values Bo = BlO,B20, . . , , Bko becomes d,(B) = &(Bo)
+
k
x G l l A B , , where G,, = bdl/bB,. A linearized set of 3=1
normal equations is obtained with the known solution AB =
- (G'PG)-'G'Pd"
where P is a diagonal matrix of the weighting factors p,, and do is the vector of the residuals belonging to BO. A new set of values B = Bo AB is now considered as a new initial solution, and the process is repeated until the sum 5 ( B ) is stable within a predetermined limit. The process converges, if the guessed values for Bo are shch that the nonlinearity of the function 5 is small. The actual computations were performed using ten significant digits, all weighting factors pr = 1, and a termination criterion
+
where q and q
the fastest convergence was obtained with the following choice of the adjustable parameters
For very small values of (Xi2 - h2), however, it was found advantageous to choose: B1 = C1/h2, B2 = hi2, B3 = C Z / X B4 ~ ~=, X22. Some improvement for the convergence for "illconditioned" systems was possible, using a damping technique similar to the one described by L e ~ e n b e r g , ~ ~ but with continuous adjustment of the damping factors. However, this procedure has not been tested systematically.
~ * + l /6/ v10-4
+ 1 are successive iterations.
The Journal of Physical Chmiatry
(33)
K.Levenberg, Quart. A p p l . Math.,
2, 164 (1944).
