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Least-Squares Is Not the Only Yardstick for Estimating the Absorption Limit of an Infinitely Long Conjugated Chain from Spectra of Oligomers
F
It will be well to begin with a somber word of counsel: “Advice to persons about to use eq 2 or eq 3Don’t.” The three-parameter equation
ew topics have exerted a greater influence, or commanded a more enduring interest, than the correlation between the length of a conjugated molecule and its electronic properties. The initial interest, ignited by the urge to puzzle out the relation between chemical constitution and color,1−6 is sustained at present by the drive to fabricate molecular wires and other tailor-made components for nanodevices7 and by the necessity to interpret the performance of conjugated polymerssystems that are ill-defined and polydispersein terms of the behavior of well-characterized long oligomers. A plethora of formulas have been proposed in an effort to relate λN (a suitably chosen wavelength in the most prominent absorption band of a conjugated oligomer) to N, the number of repeat units,1−6,8,9 and three excellent reviews have appeared during the past decade.10−12 The authors of these contributions have focused primarily on comparing experimental observations with the predictions of various equations in their “raw” forms, not allowing themselves to be distracted by (or benefit from) the fact that, more often than not, an algebraic formula can be manipulated so as to bring one or another of its features into prominence. As a result, expressions that are closely related have been treated as dissimilar rivals, like has been compared with unlike, the role of constraints, whether purposely enforced or inadvertently incorporated, has been overlooked, and equations which should have long been discarded, or never adopted in the first place, have continued to enjoy undeserved popularity. Theoretically, λ[00] (the wavelength representing the first N peak in the vibrational progression of the most prominent band) is the quantity of prime importance, but the use of λ[max] N (the wavelength corresponding to the peak absorbance) is more practical when one is working with molecules whose spectra are devoid of vibrational structure, provided that one is certain that the longest wavelength band arises from a single electronic transition. Except when confusion is likely to arise or when the contrary is stated, I will use the simpler symbol λN instead of λ[00] or λ[max] . The necessity to distinguish the N N measured value λN from the corresponding predicted value will be met by using the symbol λN for the latter quantity. The number of π electrons will be denoted by Nπ. In the formulas given below, X will invariably stand for the parameter whose optimized value is to be interpreted as the best prediction for λ∞ provided by a given fitting function, and Y ≡ X−1 will represent the corresponding estimate for the limiting energy; wavelengths will always be measured in nanometers, and the units for energy E = C0/λ are inverse centimeters (i.e., C0 = 107). The predicted value for the energy will be denoted by E N; the measured value, by EN. We will also need the symbols Z = N −1 , and θN =
π N+1 © 2016 American Chemical Society
E̲ N = A 0 + A1Z + A 2 Z2
(2)
is often used in the literature on polyenes,
13
⎡ ⎤ 1 U ⎥ E̲ N = Y ⎢1 − ( N 1) + + π Nπ (Nπ + l)2 ⎣ ⎦
and
(3)
8
is another three-parameter relation, obtained by extending the formula of H. Kuhn,4 who chose l = 1. The inadequacy of eq 2 has already been demonstrated by others,9,10,12,14,15 and that of eq 3 by the present author.16 We will be interested, for the most part, in fitting functions that can be expressed as λ̲ N = X[1 − K Ξ(k , N )]q
(4)
or as λ̲ N = X[1 − A Θ(a , N )]q
(5)
where K and k (and A and a) are adjustable parameters, and 1 q = 1 or 2 . The functions Ξ (exponential kernel) and Θ (trigonometric kernel) are defined below as Ξ = exp( −kN ) Θ=
(6)
1 − cos θN 1 − acos θN
(7)
A concrete versions of eq 4 will be called X3pU or X3pH 1 according as q = 1 or q = 2 ; likewise, the two concrete versions of eq 5 will be called T3pU and T3pH. The first character identifies the kernel (X for exponential, T for trigonometric), the last character identifies the value of q (unity or half), and the middle two specify the number of free parameters. 1 If one sets Ξ = exp(−kN) and q = 2 in eq 4, it becomes equivalent to the relation λ̲ N = (α − βγ N )1/2 , (γ < 1)
(8)
6
proposed by Hirayama, who used a different notation. Let us pause to observe that eqs 4 and 5 can each be reduced to a two-parameter equation by the imposition of a constraint; in labeling the reduced version of a three-parameter equation, the numerical character will be changed from 3 to 2. The process of reduction will be illustrated by considering a special case (q = 1) of eq 4, namely
(1)
Published: February 18, 2016 676
DOI: 10.1021/acs.jpclett.6b00060 J. Phys. Chem. Lett. 2016, 7, 676−679
Viewpoint
The Journal of Physical Chemistry Letters λ̲ N = X[1 − K e−kN ]
To understand the small-Z behavior of curves generated by functions with exponential kernels, it is helpful to recall the definition of a flat f unction.17 A function f(x) is said to be flat at x0 if it is infinitely differentiable at x0 and all its derivatives at this point vanish. This implies that a formal Taylor expansion of f(x) at x = x0 does not exist. The four functions that use an exponential kernel, being flat functions, appear (on the scale of ordinary plots, such as those shown here) to become parallel to the horizontal axis well before meeting the vertical axis. In contrast, the small-Z behavior of functions with a trigonometric kernel can be expressed as
(9)
Setting N = s in eq 9, where s is a suitably chosen constant, and solving for K, one gets ⎛ λ̲ ⎞ K = eks⎜1 − s ⎟ ⎝ X⎠
(10)
If we impose the demand λ s = λs, introduce a new symbol Ks through the relation
⎛ λ⎞ eks⎜1 − s ⎟ = e ksK s ⎝ X⎠
(11)
λ̲ N = X − αZ2
and replace K by Kse on the right-hand side of eq 9 (X3pU), we arrive at X2pU, the two-parameter variant of X3pU,
2
ks
λ̲ N = X[1 − K se−k(N − s)]
E̲ N = Y + βZ
which can also be expressed as (13)
Equation 13 subsumes the equation proposed by Meier and coauthors (M&Co), who confined themselves to the special case s = 1; they did not use different symbols for the measured and calculated wavelengths and overlooked, or failed to point out, that their equation was a three-parameter equation in disguise. M&Co demonstrated that their equation provided excellent fits to many data sets.9,10,14,15 Now, M&Co, who proposed eq 13 (with s = 1), also recommended, as an independent recipe for computing the energy, the relation E̲ N = Y − (Y − E1)e−ϵ(N − 1)
(17b)
where α and β are constants (>0) satisfying the relation α/X = β/Y. To find out which of the small-Z behaviors mentioned above fits better with experimental observations, one needs a data set that is sufficiently dense and precise. Fortunately, a trove of spectroscopic data satisfying these stringent requirements has been made available by Izumi and coauthors,18 who synthesized a long series of N-mers of cyclopentathiophenes (Chart 1; N =
(12)
λ̲ N = λ̲ ∞ − ( λ̲ ∞ − λs)e−k(N − s)
(17a)
Chart 1. Molecular Structure of the Oligocyclopentathiophenes Synthesized by Izumi et al.18
(14)
which will henceforth be called Meier’s equation for energy (MfE). Evidently, MfE and the equation (cf. eqs 12 and 13) E̲ N = [ λ̲ ∞ − ( λ̲ ∞ − λ1)e−k(N − 1)]−1
E̲ N = Y [1 − (1 − Y /E1)e
−k(N − 1) −1
]
(15a)
2, 4, 6, 12, 18, 24, 36, 48, 72, 96). They have listed both λ[max] N for their (unstructured) absorption spectra and λ[00] for their N (structured) fluorescence emission spectra; because the precision of their absorption measurements is stated to be 0.2 nm for absorption and 0.4 nm for emission spectra, I have chosen their absorption data for further inspection. Plots resulting from some two-parameter fits are displayed in Figure 1; in order to be able to focus attention on the large-N
(15b)
cannot both provide equally good fits to the same data set. When MfE and eq 15, an algebraically incongruent pair, are applied to the same spectral data, one must find that the relations E ∞ = C0 /λ∞ and ϵ = k are not obeyed; M&Co did perform such a comparison9,14 but did not comment on the discrepancies. So far as the experience of the present author goes, eq 15 provides a closer fit than MfE, which means that the latter should be discarded. Two fitting formulas from the past (both using two parameters) will now be recalled and stated, in keeping with the general theme of this Viewpoint, in a single relation λ̲ N = W (1 − A cosθN )−q
(16)
1
If we put q = 2 in eq 16, we recover W. Kuhn’s formula,3 whereas the other option (q = 1) gives Davydov’s formula,2 which was derived by using a simplified quantum mechanical description analogous to that used by W. Kuhn. It is easily verified that eq 5 reduces to eq 16 when one sets a = A, and uses the relation X(1 − A)q = W, which follows from eq 16. As previous works have paid scant attention to the small-Z (large-N) behavior of functions (other than those based on polynomials in Z), it is necessary to fill this gap here.
Figure 1. Observed and predicted values of transition energy (E[max] N and E [max] , respectively) for oligomers whose structure is shown in N Chart 1. 677
DOI: 10.1021/acs.jpclett.6b00060 J. Phys. Chem. Lett. 2016, 7, 676−679
Viewpoint
The Journal of Physical Chemistry Letters Table 2. Values of λ∞a and Δb for Oligocyclopentathiophenes of Izumi et al.18,c
behavior, the data for the shorter molecules (N = 2, 4, 6) have not been included in these plots. On the whole (that is to say, when the entire set is considered), excellent fits ensue from X2pU/H, T2pU/H, and W. Kuhn’s function, but T2pU and T2pH (which are indistinguishable on the scale of the plot) stand out in particular. The reason for the inferior performance of flat functions is easy to understand if one divides the oligomers into two subsets, one for which N ≤ 24 and another for which N ≥ 36, and notes that δN is positive (negative) for one set and negative (positive) for the other. Curves generated by flat functions (or exponential kernels), where saturation sets in abruptly (between N = 24 and N = 36), minimize the sum of squared residuals (Δ = ∑Nδ2N) by changing the sign of δN≡ λN − λN in this region, which produces systematic deviations on either side of the divide. The situation of the greatest practical interest is that when one wishes to find λ∞ after one has acquired spectral data for short and moderately long oligomers. To explore whether extrapolation is a shot in the dark or a reliable predictive tool with a narrow margin of uncertainty, a long string of precise spectral data is needed. One can use an initial segment of such a string for fitting purposes, and the remainder for checking how well a given fitting equation accounts for the (longer) oligomers not included in the fitting; the outcome of one such enquiry, which was based on quantum chemical calculations of the HOMO−LUMO gaps of oligothiophenes, has been described elsewhere.16 Another route, taken here, is to return to the data set for oligocyclopentathiophenes, which has a preponderance of long oligomers, and to use interpolation for producing data for shorter, “virtual” oligomers. One set, produced by fitting a second-degree polynomial in N to the spectral data for N = 2, 4, 6 and a second-degree polynomial in Z to the data for N = 6, 12, 18, is shown in Table 1; it meets the requirement that λN
2 313.0 8 484.7 18 517.0
3 366.6 9 492.7 24 521.5
4 408.5 10 498.6 36 524.4
5 438.8 11 503.0 48 525.6
6 457.5 12 506.5 72 526.8
N = 2−12
λ∞
Δ
λ∞
Δ
λ∞
Δ
T3pU T3pH T2pU T2pH X3pU X3pH X2pU X2pH W. Kuhn
528.1 528.2 528.1 528.2 523.7 524.4 523.7 524.4 529.0
5.9 8.8 6.0 8.9 91.6 48.4 93.7 48.8 38.6
534.2 535.8 534.4 536.2 513.8 522.0 514.1 522.9 538.2
6.0 8.4 6.0 9.0 5.5 17.1 5.8 19.2 13.0
532.7 533.6 532.7 533.9 514.1 519.6 514.1 520.2 536.3
8.9 13.0 9.1 14.1 5.9 19.9 6.1 23.0 25.2
a
In nanometers. bIn square nanometers. cFor the structure of the oligomers, see Chart 1.
notices that the X-equations (rows 5−8) underestimate λ∞ and lead to disconcertingly large values of Δ. As for the merits of the T-equations vis-à-vis W. Kuhn’s formula and for U-versus-H forms of the X-equations, the following remarks may be made on the basis of Figure 1 and Table 2: the T-equations (rows 1−4) outperform W. Kuhn’s formula (in dealing both with the full data set for “real” oligomers and with the data for “virtual” oligomers); X(3/2)pH should be preferred over X(3/2)pU. For more details on the issues discussed in this Viewpoint, recourse may be had to a preprint.16 Of the four X-equations presented above, only X3pU and X2pH are new, even X2pU has more than a morsel of originality, and all four T-equations are unprecedented.
K. Razi Naqvi
Table 1. Values of λN Oligocyclopentathiophenes; the Italicized Entries Belong to “Virtual” Oligomers N λN N λN N λN
N = 2−10
original set equation
■ ■
7 473.6 13 509.2 96 527.7
Department of Physics, Norwegian University of Science and Technology, 7491 Trondheim, Norway
AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS I thank Hans-Richard Sliwka for keeping alive my interest in the topic through discussions and advice stretching over more than a decade.
■
must coincide with the original data for N ≤ 18. A different procedure for interpolation is not expected to yield data differing by more than the uncertainty in the original data. Table 2 shows the results found by analyzing the original set of Izumi and co-workers,18 and two segments (N = 2−10 and N = 2−12) from the data for “virtual” oligocyclopentathiophenes. The data in the last four columns will be discussed first. If one declares the best fit to be that with the smallest value of Δ, one must conclude that X3pU and X2pU outperform all the other fits. However, one can see from Table 1 that λ∞ must be very close to 528 nm; because the very purpose of making a fit is to deduce the value of λ∞ from a data set that does not contain sufficiently long oligomers, one must conclude that X3pU and X2pU are, in fact, particularly poor fits, because they predict a value for λ∞ that is close to 514 nm. The inadequacy of flat functions to portray the N-dependence of λN becomes obvious when one examines the values of λ∞ and Δ which result from the application of various fitting formulas to the f ull data set of “real” oligomers (columns 2 and 3 in Table 2), and
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DOI: 10.1021/acs.jpclett.6b00060 J. Phys. Chem. Lett. 2016, 7, 676−679
