Roger E. Gerkin Ohio State University Columbus
A Molecular Spectral Corroboration of Elementary Operator Quantum Mechanics
The central, integrating concepts of the first quarter in the freshman honors chemistry sequence at the Ohio State University are structure (lecture) and physical measurements (laboratory). Since our students come to us with a substantial qualitative introduction to the structure of atoms and molecules but have not always mastered this material, it has seemed essential to redirect their attention to these very fundamental topics. In an attempt to make their understanding more quantitative, we have introduced a discussion of operator calculus which culminates in a detailed mathematical description of hydrogen-like oneelectron atomic orbitals as eigenfuuctions of a particular (Hamiltonian) operator, in the introduction of linear combination functions, and in the development of hybridized atomic orbitals and molecular orbitals. JIathematics through differential calculus (which some students take concurrently) is necessary for this development. The experiment described here, entitled "Molecular Structure and Spectra: the Absorption Spectra of a Series of Conjugated Dyes; a Simple Quantum Mechanical Derivat~onof Electronic Spectral Properties," is particularly valuable--apart from its obvious connection with the central course content-because it provides a simple chemical (physical) application of conclusions that the students can themselves verify using their operator calculus. It thus imparts a sense of reality to what is otherwise admittedly a rather abstract mathematical development. The molecules studied in this experiment belong to the class whose ground and low-lying electronic states can be well described by the one-dimensional free-electron model, which has been extensively developed in
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this connection by Kuhn.' In its simplest form this model is sufficiently sophisticated to be challenging to the students yet is sufficiently straightforward to be comprehended by them. The assumptions made iuelude: So long as a. free electron is "on" the molecule it has a finite constant potential energy. If the free electron were "off" the molecule it would have an infinite polentid energy.
Under these assumptions the problem of finding these electronic energy levels becomes identical with the problem of finding the allowed energies of a particle confined in a one-dimensional box whose length equals the "length" of the molecule. The resulting energies are
where n = 1,2,3. . ., m = electron mass, and I = "length" of the molecule. The free electrons are assigned to the lowest available of these levels by pairs (consistent with the molecular Pauli principle) until all are accommodated. This yields the ground (free) electronic state. The lowest-lying exclted (free) electronic state is realized when one electron from the highest filled level is promoted to the lowest unJilled level; the upward transition between these two eigenstates is effectedby absorption of a photon of frequency v such that hv = A E , which is thus the least energetic photon capable of causing a (free) electronic transition. (It is KUHN,H., J . Chem. Phys., 17, 1198 (1949).
this least energet,ic elect,ronic absorption which is observed in this experiment.)
Experimental
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The Experiment
The material given to the student comprises 10 single-spaced typed pages, in five parts2: 1. Theoretical background. (1) General remarks concerning absorption of light in the visible region. (2) The electronic structure of conjugated organic molecules. (3) The free-electron model of the polymethine dyes (derivation of the wavelength of the lowest electronic transition). 2. Experimental background. (1) Absorption spect,rophotometry(a functional description of the hasic element6 of an absorption spectrophotomet,er). (2) The quantitative description of absorption. 3. Experimental directions. 4. Data and calculat,ians. 5. Discussion and questions.
We are aware of the description of a similar experiment3; other presentations for physical chemistry laboratory may also exist. However, the present experiment has been prepared especially for first year students and assumes no prior familiarity with either theory or practice of spectrophotometry. It was intended, too, that the material given to the students should be sufficiently detailed to provide, together with the lectures on operator calculus, an adequate background for the experiment. The l,l'diethyl4,4'-cyanine iodide series of polymethine dyes was chosen partly because of the ready availability of four members [l,lr-diethyl4,4'-(cyanine, carbocyaninc, dicarbocyanine, and tricarbocyanine) iodide]. Our source is Koch-Light Laboratories Ltd. in England.& I n addition the corresponding four members of the 1,l'-diethyl-2,2'-cyanine iodide series and of the 3,3'-diethyl-thiacyanine iodide series are available. Although the prices of these substances range from -86-$110 per gram, thelong term expenditure is modest since we find that solutions need to be no more concentrated than M to give easily observed maxima. Using methanol as solvent, solutions of appropriate concentrations are made just p ~ i o rto the laboratory meeting. This saves class time for actual spectral measurements, thus permitting each student to obtain his own spectra. Also it makes more efficient use of these rather expensive reagents. Typically, the sample cell is filled once and left in place until each student has obtained a satisfactory spectrum so that 25 ml of 10-5 M solution suffices. However, we find that the tricarbocyanine is sufficiently unstable in solution to permit only one spectral scan; this compound is perhaps not satisfactory for direct measurement by each student. We find also that this procedure keeps attention focused on the spectroscopic nature of the experiment. The students are instructed as a group concern-
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' Copies will be provided upon request.
See SHOEMAKER, D. P., AND GARLAND, C. W.. "Experiments in Physical Chemistry," McGraw-Hill Book Co., Inc., New York, 1963, p. 298ff. All orders for Koeh-Light compounds should now be placed directly with Pierce Chemical Company, P. 0.Box 117, Rockford. Illinois. J
ing operation of the spect,rometer by means of a lecture demonstration. Student res1,onse has been very satisfactory. The absorption maxima of the first four 1,l'-diethy!4,4'-(c anine)iodides occur at approximately 5920 A, 7080 8130 A. and 9290 A res~ectivelv. - . so that all four cah be studied on a single \Gde-range instrument. The students compare their expcriment,al measurements of the wavelengths of the absorption maxima with the values calculated from theory. One datum is used to establish the value of t,he one adjustable parameter, and the remaining maxima are predicted. However, introduction of this parameter contributes in a rather minor way to the fit and could be omitted. The theoretical expression given for the wavelength of the first absorption maximum (lowest electronic transition) is:
where b = number of carbon atoms in polymethine chain and a = adjustable parameter, constant for a given series of molecules. Theoretical and experimental results for the l,lf-diethyl4,4'-cyanine iodides are summarized in the table above. Since the experimental values for the first and second members lie above the theoret,icalvalues calculated with a = 0 whereas the experimental values for the third and fourth lie below the theoretical values calculated with a = 0, it is clear that no single value of a will produce a good fit for all four members. Even so, the introduction of an adjustable parameter is common enough in approximate theory to warrant its introduction here. It should also be pointed out, as Kuhn did, that more sophisticated approaches lead to predictions that the carhocyanine species absorption should lie in the ultraviolet. The free-electron model predicts absorption in the visible; thus although there are discrepancies -200400 A between free-electron theory and experiment, these must be regarded as remarkably small, taking the crudeness of the model into account. It should be emphasized t,hat the experimental procedures are sufficiently straightforward that the student's concern remains centered on what was intended to be the crux of this experiment: the application of a. rather abstract mathematical technique (t,he general importance of which has already been established) to the practical problem of predicting part of the absorption spectrum of a real molecule. The agreement between theory and experiment is decidedly impressive. The writer wishes to acknowledge the interest and encouragement of Professor W. T. Lippincott, t,he cooperation of Professor D. W. Meek in obtaining the tricarbocyanine spectrum, and the aid of Mr. Dennis Foreman in presenting this experiment to the students. - -
BROOKER, L.G. S., J. Am. Chem. Soc., 64, 199 (1942). Volume 42, Number 9, September 1965
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