HüCKEL, ERICH ARMAND ARTHUR

(b. Charlottenburg, Germany, 9 August 1896; d. Marburg, Germany, 16 February 1980),

physical chemistry, quantum chemistry, chemical bond, molecular orbitals, aromaticity.

Hückel was born in a suburb of Berlin on 9 August 1896, the second of three sons of Marie and Armand Hückel. The intellectual development of the young Hückel was strongly influenced by his father, who was a physician with interests in the natural sciences. In the cellar of their family home in Göttingen, Hückel’s father had built a chemical laboratory. There he introduced his sons to the world of science, and together they conducted experiments in the laboratory while studying books on physics and chemistry, such as Wilhelm Ostwald’s Die Schule der Chemie (Hückel, 1975). It was thus at home that the Hückel brothers experienced their first contacts with chemistry. Whereas the one-year-older Walter decided to study chemistry in 1914, Erich entered Göttingen University shortly before the outbreak of World War I to study physics and mathematics. There he came under the active influence of the mathematician David Hilbert.

Assistant to David Hilbert, Max Born, and Peter Debye . Hückel attended several of Hilbert’s lecture courses. He was particularly influenced by the course “Denkmethoden der exakten Wissenschaften,” (Methods of thinking in the exact sciences), which Hilbert gave during the winter semester 1919–1920 and which offered a rich blend of mathematical and physical ideas. Hückel’s notes from this course can be found in his papers preserved at the Staatsbibliothek zu Berlin—Preussischer Kulturbesitz, Hand-schrifttensammlung (see also Bernays, 1992).

After the course, Hückel served for one year as Hilbert’s assistant. Hückel’s main activity consisted in helping him as interlocutor in preparing Hilbert’s lecture course on the theory of special and general relativity. Hilbert emphasized in his series of one-hour lectures that Albert Einstein’s theory of relativity demanded a special intellectual effort because it set forth how the conceptions of space and time from classical mechanics had to be replaced by a far subtler physical conception that could only be grasped by thinking abstractly or by means of analogy. Thus Hückel learned from Hilbert’s lecture course that thought structures more abstract than the classical conceptions of space and time were necessary if one wanted to arrive at a deeper insight into the laws of nature. Hilbert also presented the importance of relativity theory in other sciences beyond physics. A repercussion of Hilbert’s message was that Hückel was sensitized to approaching critically the foundations of other sciences as well. In an indirect way, Hilbert’s axiomatic thinking and methodological approach were to influence Hückel’s theoretical ideas, including possibly his quantum mechanical treatment of unsaturated and aromatic compounds during the early thirties.

In 1921 Hückel presented his thesis, written under the supervision of Peter Debye. This was an experimental study of the scattering of x-rays in substances then thought of as liquid crystals. One year later he became assistant to Max Born. Born provided him with his first theoretical task, namely, to assist him in developing a quantum theory of the molecular spectra of molecules consisting of two or more atoms. In 1923 Hückel went to Zürich to work as Debye’s assistant until 1927. In this period Hückel made his first important contribution to theoretical chemistry in the form of the theory of strong electrolytes, known in the early twenty-first century as the Debye-Hückel theory. During these years, Hückel learned to use mathematics and physical intuition in the form of models, from which he tried to understand physical and chemical phenomena. This process was accompanied by a comparison between the results of his calculations and experimental data.

Quantum-Mechanical Interpretation of Double Bonding . With this training and methodological background, Hückel spent the following two years, from 1928 to 1930, on a Rockefeller Foundation scholarship that took him to Frederick Donnan in London, Paul Dirac in Cambridge, and Niels Bohr in Copenhagen. Under Bohr’s guidance, he began to learn the new quantum mechanics. In his autobiography, Hückel implies that it was the stimulus given to him by Bohr that led him to his work on unsaturated molecules in the summer of 1929. The following year, supported by a scholarship from the Deutsche Notgemeinschaft, Hückel was in Leipzig working with Werner Heisenberg and Friedrich Hund. There he finished his first landmark paper in the new field of quantum chemistry, “Zur Quantentheorie der Doppelbindung” (1930).

In this paper, Hückel undertook to solve a deep problem of classical organic chemistry: the restricted rotation of double bonds and the stability of cis-, trans-, syn-, and anti-isomers. The persistence of stereochemical configurations about C=C or C=N bonds was well known to organic chemists. J. H. van ‘t Hoff had provided a classical explanation fifty-six years earlier: He imagined the carbon-carbon double bond to be formed by joining two nearby edges of two tetrahedrally disposed valences of each atom. The four remaining valences were conceived as lying in the same plane as the carbon atoms. Hückel attempted to give a quantum theoretical explanation for the hypothesis that valency forces in an unsaturated molecule have a definite direction. He argued that this phenomenon could not arise from forces of a classical nature between the substituted groups; rather it must depend on features of the structure of the double bond that were amenable to treatment only by quantum theory. In his paper Hückel emphasized that the explanation of the chemical facts—in agreement with the conception of the chemists—can only be found in the nature of the double bond (1930, pp. 432–433). According to Hückel’s model, in ethylene the two C=C bonds (σ and π) were nonequivalent. In his view, the pair of electrons that corresponds to the second valency bond (π bond) has a positional eigenfunction whose nodal plane coincides with that of the molecule and which is symmetrical in relation to the two carbon atoms. The corresponding statistical charge distribution stabilizes the planar arrangement of the two carbon and four hydrogen atoms. At the same time, the eigenfunction of the remaining electron pair (σ bond) associated with the double bond is nearly axially symmetrical about the C-C axis and has no stabilizing effect for the planarity of the molecule.

Hückel wanted to show that the quantum mechanical treatment of the restricted rotation of double bonds led to a different interpretation from the classical model. This treatment led him to conclude that the direction of the four valency lines in space has no real meaning. According to Hückel, quantum theory can assign a meaning only to the plane in which van ‘t Hoff’s four C-H valency lines lie. Hückel showed that, contrary to van ‘t Hoff’s assumption, two planes must be taken into consideration to account for the restricted rotation of double bonds. The first is the plane with the four C-H valency lines and the C-C bond. The second, perpendicular to the first, is the plane with the charge distribution of the πbond. Hückel remarked that the reason for the restricted rotation had to do with the electrostatic energy between the charge distributions in the two planes. The formation of this charge distribution, according to Hückel, is a quantum mechanical effect without any classical analogues (1930, pp. 454–455).

Thus Hückel opened the way for a critical revision of the classical system of valency and overcame its inability to account for the case of unsaturated molecules and especially for the length of the double bond. His critique of the conventional interpretation of van ‘t Hoff represented a decisive step away from a naive picture of directed valence lines and toward a new abstract understanding of chemical properties based on quantum mechanics.

Quantum Theory of Aromaticity . In the meantime, through Debye’s intervention and with the support of Paul Ewald and Erich Regener, Hückel obtained a position at the Technische Hochschule in Stuttgart teaching “Chemische Physik” from 1930 to 1937. This period was the most fruitful for his scientific production. In the 1931–1932 period, Hückel published three long papers on aromatic and unsaturated molecules, the first providing a quantum theoretical description of benzene (1931, 1932). This was his habilitation thesis leading to the venia legendi (right to teach) in theoretical physics at the Technische Hochschule Stuttgart. In this paper Hückel gave two descriptions of benzene: his first method, which eventually came to be known as the valence bond method, and a second, which involved the application of molecular orbital methods. Hückel believed that the experimental data gave him good reasons for preferring the second approach, which was utilized by John Lennard-Jones, Hund, and Robert Mulliken. This second approach is widely known as the HMO method (Hückel’s molecular orbital method).

Hückel’s second method was essentially adopted from a theory developed by Felix Bloch in a 1928 article to explain phenomena such as the electric conductivity of a metal lattice. Ignoring the mutual interactions, Bloch treated the motion of the electrons in a crystal lattice not as free but rather as influenced by a field of force of the same periodicity as the crystal’s lattice structure. Hückel retraced Bloch’s considerations in his analysis of the quantum states of a single p_h electron in a field of force of the same periodicity as the structure of the cyclic compound. This field of force is caused by the structure and the p _h electrons other than the one under consideration. One could thus imagine Hückel’s benzene lattice as composed of regular hexagons of p_h electrons of carbon. The procedure Hückel used was to set up such a lattice for benzene, calculate the eigenvalues and eigenfunctions in terms of two parameters by solving the Schröndiger equation.

For the calculation of the eigenvalues and eigenfunctions, Hückel employed an approximative procedure indicated by Bloch in which the overlapping fields of potential of the individual atoms are treated as perturbations. Bloch’s method yielded the following result for eigenvalue W:

where α and β are two parameters resulting from integrals over the values contributed by the individual atoms. The quantity α represents the potential energy of the unperturbed charge distribution of the [p]_h electron located with the individual atom in the fields of the neighboring atoms. Since α describes an electron bound to an atom, α is negative. The quantity β describes the quantum-mechanical “resonance interaction” between the two neighboring atomic eigenfunctions (ψ^o_f and ψ^o_f+1). This is the stabilizing energy of an electron relative to α, when two such entities are interacting within a certain distance. Hückel showed that β is likewise negative for p_h electrons. The eigenfunction for eigenvalue W^k is:

The eigenvalues and eigenfunctions are characterized by the following values for k: k = 0, ±1, ±2 ± n/2, where n is even. Such “quantum numbers” characterize the electron states in their energetic sequence. Hückel then plugged in the electron states, taking resonance effects and Wolfgang Pauli’s exclusion principle into account, in a manner similar to the molecular orbital (MO) method employed by Hund and Mulliken for diatomic molecules. Drastic negligence of the “interactions between the spins and electronic motions,” the energy exchanges between electrons in k states, and skillful use of molecular symmetry led Hückel to the following general finding:

There consequently results for an n ring the electronic states characterized by the k “quantum numbers.” k = 0 yields (without spin) one state, k = 1 is doubly degenerate and yields (without spin) two states, k = 2 likewise, etc. [...] Because, according to the Pauli principle, each state can only be occupied doubly, one obtains a first complete electron shell for 2 electrons, a second for 4 additional electrons (2 + 4 = 6) and another for 4 more electrons (2 + 4 + 4 = 10). (1931, p. 255)

These results, derived from Schrödinger’s differential equation, led Hückel to a quantum-theoretical interpretation of aromaticity: The numbers 2, 6, and 10 describe “complete electron shells,” which endow monocyclic aromatic molecules with particular stability. The occurrence of the numbers 2, 6, and 10 for “complete electron shells” in a quantum-mechanical analysis of cyclic compounds is characteristic of such compounds. They resulted only from the second method, not from the first. This is based on the fact that Hückel’s second method (HMO) yields from k = 0 the state of lowest energy, which is not degenerate, whereas for k = 1 the corresponding state is doubly degenerate—that is, it appears as a pair of equal energy. Hückel’s first method yielded fewer states than the second method. Chemists in the twenty-first century still employ the “Hückel rule” of the form: “4n + 2,” where n = 0, 1, 2, 3, etc., to determine whether a given organic cyclic compound with 4n + 2 π-electrons is classifiable among the aromatic compounds.

According to Hückel’s description, the electron states of benzene are occupied as shown in Figure 1. Hückel emphasized that it was not the number of atoms forming the ring but the number of electrons forming a “complete electron shell” that determined aromaticity. Such “complete electron shells” explained the chemical stability of aromatic cyclic systems—in particular, their strong resistance to chemical addition. Among the aromatic species, that is those with “complete electron shells,” only the lowest energy states are occupied. That is why they are particularly low in energy and as a consequence stable and especially chemically resilient. In Hückel’s mind, the aromatic molecule constituted the model example of a chemically stable species. Hückel appealed to the analogy between the atomic stability of the noble gases and aromatic molecules. The unusual stability of both species is caused by the possession of “complete electron shells.” In the case of the noble gases, the valence electron octet performs the key role. He extended his theory for benzene to other aromatic species and was able to predict the existence of other forms.

Hückel thus offered a chemical explanation, not a physical one, for a molecule’s energetic stability. Complete electron shells imply, speaking quantum-theoretically, that the total angular momentum of the electrons equals zero, and that all states are occupied by two electrons of antiparallel spins. Hückel explicates these facts in his paper:

As already mentioned, the energy content furthermore does not by any means alone govern the stability of a compound in a chemical sense. In this the reactivity of a compound is also decisive. This reactivity depends, among other things, on how the energy responds to a disturbance in the atomic arrangement (changes in the separating distances), how easily the molecule is excited, how easily it takes on electrons, and so on. In general, much experimental data has been gathered about the correspondence between the constitution and reactivity of organic molecules. Yet only a very modest number of satisfactory theoretical conceptions about it exist. We now believe we can contribute a new aspect in the case of cyclic compounds considered here. The introduction alluded to the importance ascribed to the number 6 for “double-bond electrons” in chemistry and stated that in a certain sense, this number 6 corresponded to a complete electron shell. (1931, pp. 254–255; emphasis added)

Hückel extended his theory for benzene to other aromatic species and was able to predict the existence of other forms. In fact, for n = 6 members to a ring corresponding to benzene, Hückel obtained 6 electron states in the ground state with a complete electron shell formed of 6 electrons (Figure 1).

For n = 4 and n = 8 members of a ring, such as for cyclobutadiene (at that time yet to be synthesized) or cycloalkene, Hückel did not obtain complete electron shells from his second method, because their electron states are occupied as shown in Figure 2.

According to Hückel’s interpretation, this signified that these cyclic systems are not aromatic in character and are more reactive than benzene, as R. Willstätter and E. Waster had already demonstrated in their contribution “Über Cyclo-octatetraen” published 1911 in Berichte der Deutschen Chemischen Gesellschaft. This, Hückel emphasized, was only valid under the condition that the nonplanar arrangement of the ring was unable to have a major influence on its stability and chemical properties. Thus Hückel managed to solve the enigma for classical structural theory in organic chemistry posed by the differing chemical behaviors of benzene, cyclobutadiene and cycloalkene. Moreover, Hückel’s quantum-mechanical approach to aromaticity was able to predict the existence of a 10-membered ring (10-annulene) of low reactivity from its closed electron shell. His explicit prediction inspired its eventual synthesis: “It would therefore be interesting to try to produce this compound, and if it

worked, to watch whether, unlike the 8-membered ring, it manifests a more aromatic character” (1931, pp. 255–256). Experimental confirmation of Hückel’s prediction in fact had to await the end of World War II, after which 10-annulene and other cyclic polyenes (annulenes) of the same aromatic character were synthesized. Hückel’s preliminary results on the stability of then-unknown aromatic compounds triggered research specifically focusing on the synthesis of new substances of that class (Garratt, 1986).

Symmetry plays as fundamental a role as does the principle of complete electron shells in Hückel’s theory of aromaticity. Andrew Streitwieser, one of Hückel’s postwar advocates in America, emphasizes in his autobiographical notes: “The real value of Hückel Theory is that it gives the correct nodal properties of MOs, and these alone can lead to important predictions and understanding of chemistry” (Streitwieser, 1997).

Hückel regarded Friedrich August Kekulé’s valence line diagram as a symbolic representation that failed to do adequate justice to the quantum-mechanical forces and resonance interactions between the electrons and carbon atoms. For this reason, a full understanding of the valence line diagram was not possible without appealing to the more fundamental conceptions of quantum theory in which abstract configurational space afforded a better description of the experimentally confirmed equivalence of the six carbon atoms.

Hückel did not limit the application of his second method to cyclic systems in which “the number of electrons is not assignable to simple pairs of bonds” agreeing with the number of atoms on the ring. He also applied it to charged monocyclic polyenes (monocyclic ions). He was able to explain, for example, why cyclopentadiene (C₅ H₆)—unlike cycloheptatriene (C₇ H₈)— forms a stable potassium salt thereby demonstrating that the C₅ H₆ – ion had to be quite stable. Thus one finds a 5-membered ring with 6 electrons occupied, like the 6-membered ring of benzene. Hückel interpreted this “as a tendency toward completion of the closed 6-membered electron shell” (1931, pp. 257–258). By contrast, the C₇ H₇ – ion would be occupied in the same way as cyclooctatetraene, which lacks a complete outer electron shell. Hückel again expected cyclononatetraene (C₉ H₁₀) to form a stable C₉ H₉ – ion because it could form a complete electron group of 10 electrons.

Hückel’s theory thus clarified many long-familiar peculiarities of aromatic compounds observed in organic chemistry, even explaining the anomalous properties of annulenes and monocyclic charged polyenes with 4n + 2 [p]_h-electrons. Cyclobutadienes, benzenes, and cyclooctatetraenes (cycloalkenes) were regarded after 1945 as members of a homologous series of conjugate, monocyclic carbon atoms of the general form (C₂ H₂)_n, where the index n denotes the size of the ring. Such systems were called “annulenes.” According to this general nomenclature, benzene is referred to as a 6-annulene and cyclooctatetraene as an 8-annulene. Hückel’s theoretical considerations about the molecular stability of as yet unknown compounds also served as a starting point for research concentrating on the synthesis of new substances. Until the end of World War II, the main advances in the field of aromaticity were of a rather theoretical nature. It was only afterward that various research teams synthesized the monocyclic aromatic compounds predicted by Hückel’s theory.

The team of researchers led by Franz Sondheimer is a prime example. They started working on the synthesis of a series of aromatic annulenes in 1956. As Jerome Berson pointed out, the most persuasive proof of acknowledgment of Hückel’s ideas about aromaticity appears to be the carbocations tropeoline and cyclopropenyl, synthesized in the 1950s and 1960s (1999, p. 53). After much effort and using great synthetic resourcefulness, in 1962 Ronald Breslow was able to confirm the Hückel rule for n= 0 by producing the cyclopropenyl cation (Figure 3). This is a three-membered monocyclic ring with a cyclic

bond between the three centers and with one delocalized positive charge on the three carbon atoms.

Aside from a few rare exceptions, the overwhelming majority of chemists apparently did not understand Hückel’s theoretical considerations. Hückel’s efforts to promote his theory among chemists and chemical physicists in various talks and survey articles could not change this state of affairs. The reason for this failure lay in the experimental and heuristic mentality predominating among chemists in Germany. Traditional training impeded the chemist’s ability to adjust conceptually to an unintuitive and purely mathematical treatment of problems in their field. Moreover, Hückel’s clumsy communication skills, especially when set against Linus Pauling’s persuasive charm, had negative consequences for the general acceptance of his pioneering results. Besides traditionalist tendencies among German chemists, there were other institutional and ideological factors at work during the National Socialist regime that had negative consequences—not just on Hückel’s theories but also on the further development of quantum chemistry as a whole inside Germany. In 1965, during the centennial celebration of Kekulé’s formula for benzene, Hückel was awarded the Otto Hahn Prize for chemistry and physics for his theory on aromatic compounds. One year later, the Stuttgart polytechnic conferred upon him the honorary degree of Doctor of Natural Sciences—“probably as compensation for the seven years of shame,” was Hückel’s characteristically wry comment. Hückel was awarded a number of other distinctions for his accomplishments in the field of quantum chemistry before his death on 16 February 1980.

BIBLIOGRAPHY

WORKS BY HÜCKEL

With Peter Debye. “Zur Theorie der Elektrolyte: I. Gefrierpunkterniedrigung und verwandte Erscheinungen; II. Das Grenzgesetz für die elektrische Leitfähigkeit.” Physikalische Zeitschrift 24 (1923): 185–206, 305–325.

“Zur Theorie der Elektrolyte.” Ergebnisse der exakten Naturwissenschaften 3 (1924): 199–276.

“Zur Theorie konzentrierter wässeriger Lösungen starker Elektrolyten.” Physikalische Zeitschrift 26 (1925): 93–147.

“Zur Quantentheorie der Doppelbindung.” Zeitschrift für Physik 60 (1930): 423–456.

“Quantentheoretische Beiträge zum Benzolproblem. I. Die Elektronenkonfiguration des Benzols und verwandter Verbindungen.” Zeitschrift für Physik 70 (1931): 204–286.

“Quantentheoretische Beiträge zum Benzolproblem. II. Quantentheorie der induzierten Polaritäten.” Zeitschrift für Physik 72 (1931): 310–337.

“Quantentheoretische Beiträge zum Problem der aromatischen und ungesättigten Verbindungen, III.” Zeitschrift für Physik 76 (1932): 628–654.

“Die freien Radikale der organischen Chemie. Quatentheoretische Beiträge zum Problem der aromatischen und ungesättigten Verbindungen, IV.” Zeitschrift für Physik 83 (1933): 632–668.

“Die Bedeutung der neuen Quantentheorie für die Chemie.” Zeitschrift für Elektrochemie und angewandte physikalische Chemie 42 (1936): 657–662.

“Grundzüge der Theorie ungesättigter und aromatischer Verbindungen.” Chemie 43 (1937): 752–788, 827–849.

“Zur modernen Theorie ungesättigter und aromatischer Verbindungen.” Chemie 61 (1957): 866–890.

Ein Gelehrtenleben: Ernst und Satire. Weinheim, Germany: Verlag Chemie, 1975.

OTHER SOURCES

Bernays, Paul. Natur und mathematisches Erkennen: Vorlesungen, gehalten 1919–1920 in Göttingen. Edited by D.E. Rowe. Basel: Birkhäuser, 1992. Transcript of David Hilbert’s lectures for the course “Denkmethoden der exacten Wissenschaften.”

Berson, Jerome A. “Erich Hückel, Pioneer of Organic Quantum Chemistry: Reflections on Theory and Experiment.” Angewandte Chemie International Edition in English 108 (1996): 2922–2937.

———. Chemical Creativity: Ideas from the Work of Woodward, Hückel, Meerwein, and Others. New York: Wiley-VCH, 1999.

Bloch, Felix. “Über die Quantenmechanik der Elektronen in Kristallgittern.” Zeitschrift für Physik 52 (1929).

Garratt, Peter J. Aromaticity. New York: Wiley, 1986.

Heilbronner, Edgar, and Hans Bock. Das HMO-Modell und seine Anwendung. Weinheim, Germany: Verlag Chemie, GmbH, 1968.

Karachalios, Andreas. “On the Making of Quantum Chemistry in Germany.” Studies in the History and Philosophy of Modern Physics, 31B (2000): 493–510.

———. “Erich Hückel (1896–1980): Von der Physik zur Quantenchemie.” Diss., Johannes Gutenberg-Universität, Mainz, Germany, 2003.

Kragh, Helge. “Before Quantum Chemistry: Erich Hückel and the Physics-Chemistry Interface.” Centaurus 43 (2001): 1–16.

Streitwieser, Andrew. A Lifetime of Synergy with Theory and Experiment. Washington, DC: American Chemical Society, 1996.

Woodward, Robert B., and Roald Hoffmann. Die Erhaltung der Orbitalsymmetrie. Weinheim: Verlag Chemie, 1970.

Yates, Keith. Hückel Molecular Orbital Theory. New York: Academic Press, 1978.

Andreas Karachalios