This text originates from:
     Kostas Gavroglu and Ana Simões :
     The Americans, the Germans,
     and the beginnings of quantum chemistry :
     The confluence of diverging traditions.
Historical Studies in the Physical Sciences,
25(1), 47-110 (1994).
Photographic materials added by this website.

QUANTUM CHEMISTRY is usually regarded as a success story of quantum mechanics. The community of physicists, in any case, came under the spell of Dirac's reductionist program, expressed as a theoretically correct, but practically meaningless dictum (1) :



P.A.M. Dirac 1902-1984 Photo from Nobel Foundation.

The general theory of quantum mechanics is now almost complete, the imperfections that still remain being in connection with the exact fitting of the theory with relativity ideas ... The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It appeared theoretical chemistry amounted to no more than quantum mechanical calculation. We argue that the development of quantum chemistry involved issues that transcended the application of quantum mechanics to chemical problems. Quantum chemistry developed an autonomous language. What appeared to be disputes over computational details were discussions about the collective decision of the chemical community on methodological priorities and ontological commitments. The outstanding issue turned out to be the character of theory for chemistry and, therefore, a reappraisal of the praxis of chemists.

Nearly all books and research papers consider two approaches to atomic bonding: the Heitler-London-Pauling-Slater (H-L-P-S) method of valence bonds and the Hund-Mulliken (H-M) method of molecular orbitals. We argue for a different conception : Heitler-London versus Mulliken-Pauling {set in italic by this website}. We regard as the dominant criterion the views of the protagonists about theory building and the role of theory for chemistry. Heitler and London insisted on an approach close to Dirac's reductionist pronouncement of 1929. Pauling and Mulliken had a strong inclination toward semi-empirical methods. This characterization does not imply that either Heitler and London, or Mulliken and Pauling, agreed on everything. Heitler proved in the end to be comfortable with a much more reductionist approach than London. The disagreement between Mulliken and Pauling over valence theory concerned questions for more fundamental than computational details.

The differences between the two approaches (H-L or M-P) can be understood in terms of two different cultures. To some Germans (at least to Heitler and London), American pragmatism appeared flippant. To some Americans (at least to Mulliken, Pauling, Slater, and Van Vleck), the German mania to do everything from first principles appeared as unnecessarily cumbersome and tortuous. At times there appeared to be a confluence of different styles of research and at other times an uneasy feeling that not all differences could be reconciled.


The wonderful matchmaker: G.N. Lewis and his pairing of electrons

In 1916, Gilbert Newton Lewis proposed that chemical bonding - both the ionic and the homopolar type - could be explained in terms of shared electron pairs. He supposed that each atom in a molecule in most cases had eight electrons in outer orbits, and that in homopolar bonds the "rule of eight" could be realized by the sharing of pairs of electrons between atoms.


"In my paper on ' The Atom and the Molecule ', I laid much stress upon the phenomenon of pairing of electrons. I have since become convinced that this phenomenon is of even greater significance than I then supposed... .There is nothing in the known laws of electric force, nor is there anything in the quantum theory of atomic structure, as far as it has yet been developed, to account for such a pairing." (2)



J.J. Thomson 1856-1940 Photo originating from here.

Before 1916, the possibility of explaining homopolar bonding by "sharing" or "exchanging" one or two electrons had occurred to several people. In 1914, J.J. Thomson proposed a model in which the stability of an atom could be guaranteed if the electrical forces between the nucleus and the orbiting electrons were confined to narrow tubes. (3) Extending these ideas, he made polar and non-polar bonds two distinct types. Polar bonds were realized through the transfer of electrons, non-polar bonds via a tube of force connecting an electron of one atom with the nucleus of another. Also in 1914, William Arsem proposed that the non-polar bond consisted in the sharing of one electron. Hence the electrically neutral units of matter were the molecules and the electronic charge should have been twice the value used for the ionic mechanism.

(4) In 1915 Alfred Parson published a paper exploring the possibilities provided by magnetism for the formation of molecules. (5) He proposed a model of the atom with magnetons arranged at the corners of a cube. Each magneton was a circular band of electricity in rapid revolution; chemical affinity arose from the magnetic moments generated by the motions. In 1916 Walther Kossel put forth a number of proposals about valence, insisting on the "rule of eight." (6) He did not commit himself on the crucial point of the shared electron pair; the mechanism he proposed for chemical bonding depended on electron transfer. He made one exception: a special non-polar bond created by rings of two to five electrons with orbits perpendicular to the bond axis. Lewis thought that Bohr's theory provided the theoretical basis of his ideas for the chemical bond. But he regarded the orbit as a whole, rather than as a succession of positions of an electron within the orbit, as the essential aspect of Bohr's theory. "If these orbits are in fixed positions and orientations they may be used as the building stones of an atom which has an essentially static character." (7)

Walther Kossel (1888-1956) Photo from here.

Assigning an independent orbit to each electron, considering it as having a fixed position in space, and identifying the fixed position of the electrons in the static atom with the average position of the electron in the orbit, reconciled the physicists' atom with the chemists' atom. Furthermore, this interpretation of Bohr provided a theoretical justification for the notion of shared pairs and, by making the notion less dependent on the "cubic" model, it anchored Lewis's theory of the chemical bond more securely than the octet rule.

Lewis's reconciliation of the two views rested not only on his ontological commitments ("it is the same atom and the same molecule that is being studied by the organic chemist, inorganic chemist and physicist)," (8) but also on the hope that the sharing of the electron pairs could unify the two disparate theories of the chemical bond. "The suggestion of two entirely distinct kinds of chemical union, one for polar and the other for non-polar compounds was repugnant to the chemical instinct which leads so irresistibly to the belief that all types of chemical union are essentially one and the same." (9) But such an attitude implied the sacrifice of the notion of the discreteness of the electric charge. He wrote to A.A. Noyes that he could no longer "subscribe to [your] fundamental idea that an atom must possess an integral muItiple of the elementary charge." (10)



This was Lewis's view in 1916. The purpose of his work on valence was to show how it would be possible to obtain a complete continuity between extremely polar and extremely non-polar compounds by giving up the idea of discreteness. As he told Noyes, he intended to substitute for it the "idea of shared electrons which could range all the way between complete possession by one atom, which corresponds to what you still call the polar bond, and the most equitable division between two atoms which would correspond with your idea that the charge of an electron pair is shared equally by the two bonded atoms." In 1923, soon after the publication of Lewis' Valence and the structure of atoms and molecules, the Faraday Society organized a meeting at the University of Cambridge. The theme of the meeting was the "Electronic theory of valency." In the opening address, J.J. Thomson said that "The bond dominates the field of chemistry, which finds its most suggestive mode of expression in terms of electrons. Admitting the presence of electrons, their repulsion involves important chemical consequences." (11) Lewis gave the introductory talk. He declared that the two views of the atom derived from the study of chemistry and physics were "completely reconcilable." The "cardinal phenomenon of all chemistry" was the formation of electron pairs, actual physical pairings; eventually quantum theory would explain the pairing. Lewis insisted that the two kinds of bonds, polar and non-polar, could be interpreted in terms of the relative position of the electron pair with respect to the nuclei of the molecule and, hence, that there was fundamentally but a single mechanism for chemical bonding.

G.N. Lewis 1875–1946 Photo from chemheritage.org

While discussing the problem of multiple bonds, Lewis expressed opinions that would be incorporated in the resonance approach introduced by Pauling in the early 1930s. Where more than one electron pairs exist, they cannot be on the line joining the two atomic centers. There arose a very serious problem of lack of definiteness, "which is to be ascribed, not to our modes of expression, but to the nature of the actual structure of the molecule." How far can an electron pair be removed from the line joining the atomic centers and still be regarded as belonging to both atoms? Lewis had no second thoughts about the implicit arbitrariness of an answer to such a question: one should consider all the possible arrangements and the actual state of affairs lies "somewhere between" these possible structures.

By 1927, most chemists had become aware of the amazing explanatory power of the new quantum mechanics, but they did not see how to assimilate it. Moreover assimilation might bring lasting changes to their disciplinary cuIture. It was, nevertheless, a risk worth taking. N.V. Sidgwick, in his influential book The electronic theory of valency, would have no inhibitions about letting the new quantum mechanics invade the realm of chemistry. Faced with the full development of the new mechanics by Heisenberg and Schrödinger, but not with an application of the theory to a chemical problem, Sidgwick attempted to clarify the methodological stumbling block in the way of his fellow chemists : (12)


In developing a theory of valency there are two courses open to the chemist. He may use symbols with no definite physical connotation to express the reactivity of the atoms in a molecule, and may leave it to the subsequent progress of science to discover what realities these symbols represent: or he may adopt the concepts of atomic physics - electrons, nuclei, and orbits - and try to explain the chemical facts in terms of these. But if he takes the latter course, as is done in this book, he must accept the physical conclusions in full, and must not assign to these entities properties which the physicists have found them not to possess: he must not use the terminology of physics unless he is prepared to recognize its laws. I have endeavored to conform to this principle, and not to lay myself open to the reproach of an eminent physicist, that "when chemists talk about electrons they use a different language from the physicists." I have been careful to avoid as far as possible the introduction of any physical hypotheses which are not already sanctioned by those who are best qualified to judge of them.



The young Mulliken

In the year that Lewis's Valence appeared, E.C. Kemble and R.T. Birge, together with W.F. Colby, F.W. Loomis, and L. Page, started preparing a comprehensive report under the auspices of the National Research Council on the spectra of molecular diatomic gases. (13) The molecular spectroscopists assumed three different contributions: rotational, vibrational, and electronic. An increasingly sophisticated model of the rotational and vibrational nuclear motions of diatomic molecules guided them through the maze of band spectra to a detailed knowledge of molecular structure. (14)

In 1921, R.S. Mulliken completed his Ph.D. at the University of Chicago working on the separation of isotopes with the physical chemist W.D. Harkens. He stayed in Chicago one more year as a National Research Council postdoctoral fellow and then moved to Harvard. There he worked on molecular spectroscopy with F.A. Saunders and Kemble and later helped prepare the report to the National Research Council on the spectra of diatomic molecules.



Isotope effects in the spectra of diatomic molecules aroused Mulliken's interest in the electronic distribution in molecules. (15) By 1925 several electronic levels had been identified in very simple molecules and molecular fragments: CO (five electronic levels), N2 and NO (four levels), and BO, CN, CO+, and 02 (three levels). (16) As their number increased, so did the need for classification. Spectroscopists searched for analogies in the spectroscopic behavior of different compounds. Following earlier suggestions on the similarities between certain molecular and atomic spectra and on the physical similarities of isosteric molecules (compounds with the same number of elements and the same total number of electrons), Mulliken looked for similarities in the spectra of isosteric molecules. (17) He found that the spectroscopic analogy between isosteric molecules could be extended to the chemical element with the same number of electrons. The parallels between molecular and atomic spectra served as the basis for the classification of diatomic molecules into different families and suggested that similar electronic distributions were responsible for corresponding systems of energy levels.

R.S. Mulliken, 1929 1896-1986 Photo from R.S. Mulliken, Life of a Scientist, Springer 1989.

Although "pretty speculative", (18) the analogy had been used by several 'scientists, including R. Mecke and H. Sponer in Germany and Birge in the United States. (19) It became a recurring theme in the extensive correspondence between Birge and Mulliken. (20) Together with evidence that the electronic levels of CO, NFR₂, and H₂ approximately fit the formulas for line spectra (such as the Rydberg or Ritz formulas), these analogies led Birge to postulate that "the energy levels associated with the valence electrons of molecules agree in all essential aspects with those associated with the valence electrons of atoms." (21)



Hertha Sponer 1895-1968 Austin 1955

With Birge's proposal one could classify electronic states in diatomic molecules by means of the same nomenclature (Russel-Saunders notation) used for atomic states (2S, 2P, 3S, 1S, 1P). Mulliken immediately looked for corroborative evidence. (22) Going one step further, he introduced three postulates that accounted for known band spectra and predicted yet unanalyzed band spectra. (23) Mulliken soon addressed the question of molecule formation and for the first time hinted at what he would later call "electron promotion", a concept essential to his theory of chemical binding: in the formation of molecules a radical rearrangement of some electrons may take place, corresponding to their "promotion" to orbits with a higher n quantum number. (24)

With Birge's proposal one could classify electronic states in diatomic molecules by means of the same nomenclature (Russel-Saunders notation) used for atomic states (²S, ²P, ³S, ¹S, ¹P). Mulliken immediately looked for corroborative evidence. (22) Going one step further, he introduced three postulates that accounted for known band spectra and predicted yet unanalyzed band spectra. (23) Mulliken soon addressed the question of molecule formation and for the first time hinted at what he would later call "electron promotion", a concept essential to his theory of chemical binding: in the formation of molecules a radical rearrangement of some electrons may take place, corresponding to their "promotion" to orbits with a higher n quantum number. (24)



F. Hund 1929 1896-1997 Photo taken from here.

When Mulliken read Friedrich Hund's theoretical discussion of the nature of the electronic states, (25) which introduced electron spin into band structure, he immediately recognized its importance and excitedly confided to Birge: (26) Hund really has everything in his paper. It's most remarkable. Nature of electronic states and fine structure both although the experimental evidence at the time was rather small (and he didn't discuss it very carefully). But almost all of my conclusions seem to agree with his theory. Of course it may not all be right. Dr. Sommer (of Göttingen) says Hund is going right into molecule spectra now. Mulliken published a summary of Hund's theory and provided an extensive discussion of the empirical evidence, which relied heavily on his own previous work. (27) Hund's quantum mechanical approach to molecules was largely confirmed by Mulliken's systemization of band spectra, and Mulliken's phenomenological theory found a legitimizing framework.

In the summer of 1927 Mulliken went to Europe. He visited Göttingen, Zürich, and Geneva, and ended the summer with a hiking trip to the Black Forest with Hund and some friends. He wanted to discuss with Hund his new contributions to a quantum mechanical theory of molecular structure. Contrary to Hund, Mulliken's attitude toward the new quantum mechanics was pragmatic; "[I] was satisfied with a general knowledge of [quantum mechanical] methods and principles sufficient to help me understand particular molecules or types of molecules and their properties; especially their spectra. In short, I was more interested in getting better acquainted with molecules than with abstract theory about them. (28)

Mulliken soon succeeded in assigning quantum numbers to electrons in molecules. He presented his preliminary findings at the meeting of the American Physical Society in February 1928, (29) and circulated the draft of the paper, certain that his colleagues would find mistakes. (30)

Mulliken first presented the aims of his program and then explained his methods. He formulated a set of working rules for assigning quantum numbers to electron states in molecules and gave examples of their application. He cautiously noted that the method had so far been applied exclusively to diatomic molecules made of atoms in the first row of the periodic table. Only a few of the molecular states discussed were stable. With an eye to future chemical applications Mulliken remarked that, besides their purely theoretical importance, a knowledge of the numerous excited states and chemically unstable molecules was indispensable in deducing the electron configurations for stable molecules and the intermediate steps in chemical reactions. The "essential ideas and methods" came from Hund; Mulliken's achievement lay "in the attempt to assign individual electronic quantum numbers" and obtain a knowledge of the energies of individual electrons in molecules, similar to the existing knowledge of atoms. (31) In a way, Mulliken attempted to assign electronic configurations to experimentally observed molecular orbitals.

Hund had conceived of an adiabatic transition from the separated atoms to the diatomic molecule and then to the united atom, thus supporting Mulliken's earlier hypothesis that electronic quantum numbers could change drastically in the process of molecule formation. (32) Mulliken thought more along the lines of the old quantum theory rather than the new quantum mechanics. He used neither Schrödinger's equation nor the language of quantum mechanics. Mulliken's highly visual spectroscopic approach seemed to be consistent with the existence of orbits. The only paragraph where Mulliken addressed "the meaning of quantum states of electrons in the new mechanics" was merely a cosmetic appendage to his largely "pre- quantum mechanical" language.

By analogy to what Bohr had done in his "grand synthesis," Mulliken pictured the molecules as being formed by feeding electrons into orbits which encircled the nuclei. (33) As he recalled: "Bohr's Aufbauprinzip for atoms made a very great impression on me and so I thought something similar for molecules would be nice. If you translate orbits into orbitals for atoms, then for molecules it is molecular orbitals; it is something that goes around all the atoms or however many atoms there are and the Aufbauprinzip transferred to molecules simply means molecular orbitals." (34)

To apply the Aufbauprinzip to molecules, two sorts of questions had to be clarified. The first concerned the quantum numbers for electrons in molecules and the nature of closed shells, molecular states, and multiplets. The second concerned binding energies in the molecule and the energy relations resulting from the union of the two atoms. To reply to the first set of questions, theorists emphasized the relation between a molecule and a molecule-as-united-atoms. To reply to the second, they emphasized the relation between a molecule and the separated atoms. Mulliken suggested that the possible quantum numbers for each electron could be obtained from those of the associated united atoms by placing the atoms in a strong axially symmetric electric field, to fix the two nuclei in the molecule. He justified this simplification on the ground that "we are not directly interested here in the effects of nuclear rotation and vibration." Several coupling schemes could be applied and, contrary to the atomic case, "in molecules, no one limiting case is ordinarily approached, ..... . the actual condition usually lies more or less in the midst of a region between several limiting cases." (35)

The relation to the separated atoms enabled Mulliken to discuss the energy conditions favorable to the formation of the molecule. In order to obey the Pauli principle for a molecule-as-united-atoms, some electrons could have their value of n increased in the process of molecule formation. Mulliken dubbed these "promoted electrons" and the associated increase in energy, "energy of promotion." To analyze the energy necessary for the formation of the molecule, Mulliken divided the total energy into two components: the positive potential energy of nuclear repulsion and the negative binding energies of each electron in the field of the nuclei and the other electrons. To form a molecule, the electronic binding energy must increase more rapidly than the nuclear repulsion energy for r>r₀ (where r₀ is the internuclear equilibrium distance) For r = r₀, the two types of energy increased at the same rate and the total energy of the molecule attained a minimum. For r<r₀, the nuclear repulsion energy had to increase faster than the electronic binding energy. To form a reasonably stable molecule, "the binding energy has to increase considerably faster than the nuclear energy over a considerable range of r values, as r decreases towards r₀". (36)

As the nuclear distance diminishes, the binding energy of the unpromoted electrons should increase steadily, since the electrons are influenced by two nuclei, reaching a maximum when the molecule forms. In the case of promoted electrons the binding energy may increase or decrease, because the increase in effective nuclear charge is at least partially offset by the increase in energy associated with the increase of the quantum number n. This qualitative analysis threatened some of the most cherished concepts in chemistry. Although, following Lewis, electrons were usually classified as either "bonding" (the paired electrons that hold the molecule together) or "non-bonding," Mulliken concluded that various degrees of "bonding power" could be assigned for various orbit types. Electrons could have positive bonding power if their presence in a molecule made the dissociation energy large, or the equilibrium internuclear distance small. (37) The converse was also true. Two possible definitions of bonding power resulted: energy-bonding power or distance-bonding power, arising either by the application of the energy criterion or the distance criterion, and a set of rules for the analysis of spectroscopic data and for the assignment of quantum numbers to electron states of actual molecules. (38) As will appear, these considerations were at the heart of Mulliken's criticism of the valence theory of Heitler and London.

Mulliken completed this phase of his work in 1928, as he moved to the University of Chicago as an Associate Professor in the Physics Department. In recognition of his outstanding contributions at New York University, where he had moved after Harvard, Mulliken was offered several jobs, all of them related to research programs on molecular structure: to replace Loomis as head of the Physics Department at NYU and continue developing the program he helped to start; to accept an offer from R.W. Wood of Johns Hopkins University and start a program there on the study of molecules; to go to Harvard to help Kemble and Slater develop a molecular research program; or to accept the offer of A.H. Compton and the Department of Physics of the University of Chicago.

Mulliken opted for Chicago. Besides sentimental reasons - according to Slater, "he liked everything about the great city, even its gangsters" (39) - the physicists at Chicago, especially Compton and the spectroscopist H.A. Gale (head of the Physics Department), were the most persuasive. Chicago already possessed a good spectroscopic laboratory, Eckart Hall (designed by the spectroscopist G.S. Monk), and Gale had promised Mulliken a new high-resolution grating. Their program could expand with the transformation of the Ryerson Laboratory into a molecular research center. Endowments would fund new apparatus and the university had always been willing to hire research assistants and pay visiting professors. Hund, Heisenberg, and Dirac were to spend the summer of 1929 in Chicago. {underlined by this website}.

click to enlarge

Slater was being pressed to join the department. Mulliken wanted to develop a molecular program along theoretical as well as experimental lines. He did not consider himself a theoretical physicist, but a sort of middleman between experiment and theory; interaction with theoretical physicists was crucial for "stimulus and cooperation." Mulliken was asked to give an advanced undergraduate course and a graduate course that boiled down to the supervision of three or four graduate students. Chicago offered exactly what Mulliken was looking for: a minimum teaching load, just for "stimulus," and expert colleagues in associated fields. (40)

Due to a delay in getting the high-resolution spectrograph, Mulliken shifted into more theoretical matters and wrote his review articles on "The interpretation of band spectra," a series originally intended for a book. (41) The correlation diagram for homonuclear diatomic molecules appeared for the first time in these articles. These diagrams provided a comparison of the electronic energy levels of diatomic molecules (relative to the energy levels of the separated atoms and corresponding united atom) as a function of the internuclear separation. They completely described the electronic structure of diatomic molecules made up of identical atoms. In their influential review article of 1935, Van Vleck and Sherman recommended that the correlation diagram be "placed on the walls of chemistry buildings, being almost worthy to occupy a position beside the Mendeleeff periodic table so frequently found thereon. Just as the latter affords an understanding of the structure of atoms, so does the former afford an understanding of the structure of molecules, with which the chemist is often concerned." (42)

The review articles on the interpretation of band spectra and the notation for diatomic molecules marked the end of Mulliken's work on the spectra and structure of diatomic molecules. (43) He then shifted to the study of polyatomic molecules and to problems related to valence. He also wanted to disseminate his work on band spectra, his preliminary ideas on the chemical bond, and his criticism of Heitler and London's suggestions. In the meantime, another promising young American physical chemist started to address the question of the chemical bond.



The young Pauling

Linus Pauling attended the California Institute of Technology from 1922 to 1925, studying for his Ph.D. He worked under R.G. Dickinson on the determination of the structure of crystals by means of X-ray diffraction. He stayed on at Pasadena one more year as a National Research Council postdoctoral fellow. Pauling made his first contributions to the subject of the chemical bond in this period. (44)



L. Pauling 1901-1994 Photo from C.J. Ballhausen: Quantum Mechanics and Chemical Bonding in Inorganic Complexes, J. Chem. Ed. 56, II: 294-297 (1979).

Although these contributions came within the framework of the old quantum theory, they also contained the interplay of theory and experiment that characterized Pauling's successful explanation of the chemical bond. In one paper, Pauling used information on crystal structure together with Lewis's shared-electron bond to suggest a new way to analyze the relative stability of groups of molecules composed of the same atoms and having the same total number of electrons. (45) In another, he represented Lewis's shared electrons by means of binuclear orbits and, together with experimental evidence from his work on crystal structure, suggested dynamic models for the ammonium ion (NH4 +), benzene (C6H6), and other aromatic molecules. (46) By the end of 1925, the two papers on the chemical bond bad been sent for publication. Following the advice of his mentor A.A. Noyes, Pauling applied for a Guggenheim Foundation fellowship. He planned to stay in Munich at Sommerfeld's Institute for Theoretical Physics for a year and to visit Bohr in Copenhagen. He also planned to pay brief visits to other centers where work on crystal structure, either theoretical or experimental, was being carried out, including Born's Institute in Göttingen and the Braggs's laboratory in Manchester. Pauling also arranged for a stay with Schrödinger in Zürich.

Pauling adapted smoothly to Munich's scientific lifestyle. He attended Sommerfeld's lectures on the new wave mechanics, which then had just been formulated by Schrödinger. He realized how much he missed the "invigorating discussions" of seminars devoted to "formal presentations" of new papers, and not to presentations of students' research problems. Pauling also realized that, in his first meeting with Sommerfeld, he had undiplomatically suggested a research topic he would like to work on, instead of waiting for Sommerfeld's advice. However, he feIt that he would probably be assigned a problem that met his interest since electronic motions in binuclear orbits were a topic of research at the Institute. (47)



Sommerfeld suggested that Pauling work on the spinning electron, (48) but Pauling mentioned his earlier paper on the effect of electric and magnetic fields on the dielectric constant of hydrogen chloride, and Sommerfeld allowed him to extend that work to fields of arbitrary strength. (49) Pauling compared the results obtained with the old quantum theory to those obtained with the new mechanics, and proved that the new gave values of the dielectric constants in good agreement with experiment. This result, more than anything else, convinced Pauling that quantum mechanics was necessary for the solution of chemical problems. He wrote to Noyes: "I am now working on the new quantum mechanics, for I think that atomic and molecular chemistry will require it. I am hoping to learn something regarding the distribution of electron-orbits in atoms and molecules." (50) While Pauling was in Munich a new line of research emerged as a result of a paper published by Gregor Wentzel, a former doctoral student of Sommerfeld and a Privatdozent at the Institute. Taking into account the electron spin and the new quantum mechanics, Wentzel calculated the screening constants for electrons in complex atoms, but there was still poor agreement with experimental results.

A. Sommerfeld, 1928 Photo from M. Eckert, Die Atomphysiker, Vieweg, Braunschweig, 1993.

Sommerfeld had introduced the concept of screening of the outer electrons by inner ones to explain the fine-structure of X-ray levels. Pauling examined this question systematically. He noticed that some of the previous approximations were not correct and, when properly refined, gave good agreement with experiment. (51) Although Noyes advised Pauling to keep publishing in American journals while in Europe, this paper, being a reply to Wentzel, came out in the Zeitschrift für Physik. (52) Buoyed by this success, Pauling continued to work on screening constants, using quantum mechanics to study the electronic structure and physical properties of complex atoms and ions. (53) Encouraged by Sommerfeld, Pauling published his results in the Proceedings of the Royal Society of London. (54) Pauling called this paper the "start of quantum mechanics of polyelectronic atoms." (55)



The Heitler-London paper of 1927

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W. Heitler 1904-1981 Photo from here : Theor.Physik,Univ.Frankfurt,Germany

In April of 1927, Walter Heitler and Fritz London, both on Rockefeller fellowships, went to Zürich hoping to work with Schrödinger. Heitler had initially intended to work with Nils Bjerrum in Copenhagen on problems of solutions. He convinced the authorities to allow him to go to Zürich, since he wanted to work in quantum mechanics. London had already been working in quantum mechanics while assisting Ewald at Stuttgart, contributing to what came to be known as transformation theory. He felt that Heisenberg's matrix mecbanics, would be a useful guide for his research. What they expected did not come to pass, since Schrödinger did not like collaborations.

F. London 1900-1954 Photo from here

About a month after arriving at Zürich, Heitler and London decided to calculate the van der Waals forces between two hydrogen atoms as "just a small 'by the way' problem." Nothing indicates that Schrödinger gave them the problem of the hydrogen molecule or that they talked with him about it. Schrödinger knew about their work, however, and told Mulliken that there were two people working in his Institute who had some results "which he thought would interest me very much; he then introduced me to Heitler and London whose paper on the chemical bond in hydrogen was published not long after." (56)

Heitler and London initially calculated the interaction of the charges of two atoms "without even thinking of the exchange." They found that the "Coulomb integral" could not account for the van der Waals forces: "So we were really stuck and we were stuck for quite a while; we did not know what it meant and did not know what to do with it." Heisenberg's work on quantum mechanical resonance did not help them, since the exchange was part of the resonance of two electrons, one in the ground state, the other excited, and both in the same atom. Then came some hot air: (57)


Then one day was a very disagreeable day in Zürich; [there was the] Föhn. It's a very hot south wind, and it takes people different ways. Some are very cross... and some people just fall asleep... I had slept till very late in the morning, found I couldn't do any work at all... went to sleep again in the afternoon. When I woke up at five o'clock I had clearly - I still remember it as if it were yesterday - the picture before me of the two wave functions of two hydrogen molecules joined together with a plus and minus and with the exchange in it. So I was very excited, and I got up and thought it out. As soon as I was clear that the exchange did play a role, I called London up; and he came as quickly as possible. Meanwhile I had already started developing a sort of perturbation theory. We worked together until rather late at night, and by that time most of the paper was clear... Well, I am not quite sure if we knew it in the same evening, but at least it was not later than the following day that we knew we had the formation of the hydrogen molecule in our hands. And we also knew that there was a second mode of interaction which meant repulsion between two hydrogen atoms - also new at the time - new to the chemists too. Well the rest was then rather quick work and very easy, except, of course, that we had to struggle with the proper formulation of the Pauli principle, which was not at that time available, and also the connection with spin... There was a great deal of discussion about the Pauli principle and how it could be interpreted.


Heitler and London started by considering the two hydrogen atoms slowly approaching each other. They assumed electron 1 to belong to atom a and electron 2 to atom b, or vice versa. Because the electrons were identical, the total wave function of the system was the linear combination of the wave functions for both cases:
The problem was to calculate the coefficients c₁ and c₂. They minimized the energy,


and found two values for it:

The integrals C (Coulomb integral) and A (exchange integral) had negative values, A being larger than C. E₁ implied c₁/c₂ = 1 and E₂ implied that c₁/c₂ = -1. Hence the wave function of the system could now be written as




They had not yet taken the spin of the electrons into account. The symmetry required by the exclusion principle was satisfied only by Ψ1, when the electrons had antiparallel spins; but Ψ1 corresponded to E1. E1 was less than 2E0, the sum of the energies of the two separate hydrogen atoms, and hence, it signified attraction. Ψ II, which denoted parallel electron spins, corresponded to E2. But E2 was greater than 2E0, implying repulsion. Bonding between two neutral hydrogen atoms could take place only when the spins of the electrons were antiparallel. (This, according to Heitler and London, was the justification for the electron pairing that Kossel - but not Lewis - had talked about). An electron pair required not only energetically available electrons, but also properly oriented ones. The homopolar bonding could be understood as a pure quantum effect, since its explanation depended wholly on the electron spin, which had no classical analog. Heitler and London found the energy to be 72.3 kcals and the internuclear distance 0.86 Angstroms (compared with the experimental values of 109.4 and 0.74). (58)

Reactions were generally positive. London later told W.M. Fairbank that Schrödinger had been very favorably surprised since he did not expect his equation to describe the whole of chemistry. (59) Born and Franck also were very enthusiastic about the paper. Sommerfeld initially responded coolly, but he warmed up once Heitler explained certain points directly to him.

The exchange force remained a mystery. Heitler and London had not expected to find one, because they began with van der Waals forces in mind. (60) The proposed exchange mechanism confronted them with a fundamentally new phenomenon. Experimental physicists and chemists posed questions about what was exchanged and the frequency of exchange: (61)


I believe that for a long time I thought it was a major ununderstood problem of quantum mechanics to explain what this exchange really means. We were pressed by the questions of other people and one always likes to understand the unfamiliar lines on the basis of something that is familiar with one. But this was something quite new ... it became gradually clear to me that it has to be taken as a fundamentally new phenomenon that has no proper analogy in older physics. And that is, I think, the only proper way to describe it. One can define a frequency of exchange in a certain manner - one has to be very careful about it - by a frequency of exchange of spin directions; that is a possibility. But this does not occur in the finished molecule; that would be a non-stationary state where the spin directions exchange at the frequency that is identical to the value of the exchange integral. There are many misunderstandings and I dare say that even today not all chemists, and not even all physicists arc really clear about it. But I think the only honest answer today is that the exchange is something typical for quantum mechanics, and should not be interpreted - or one should not try to interpret it - in terms of classical physics.


Both London and Heitler in all their early writings repeatedly stressed the "non-visualizability" of the exchange energy; commentators have consistently misinterpreted this.

Although the treatment of the homopolar bond of the hydrogen molecule appeared to be an "extension" of the methods successfully used for the hydrogen molecule ion, a radical difference existed between the two cases: the role of the Pauli principle. John L. Heilbron, in his penetrating study of the origins of the exclusion principle, called it "one of the oddest of the instruments of microphysics," and stated that Pauli's first enunciation in December 1924 had the form not of a dynamical principle but of the Ten Commandments. (62) Hermann Weyl talked of Pauli's principle as something which revealed a "general mysterious property of the electron." (63)

Pauli's paper on spin appeared while London and Heitler were in Zürich. (64) They judged it to be an unsatisfactory "hybrid between a wave equation and some matrix mechanics superposed on it. It was, so to speak, glued together, but not naturally combined together." (65) Now the problem of the hydrogen molecule ion could be solved by application of the Schrödinger equation where only electromagnetic forces determine the potential. A similar approach to the hydrogen molecule led to a physically meaningless solution that did not account for the attractive forces. An additional constraint was required. Heitler and London applied this constraint not in the form of further assumptions about the forces involved, but as an invocation of the Pauli exclusion principle, leading to a quite amazing metamorphosis of the physical content of the mathematical solutions. The terms previously giving strongly repulsive forces gave strongly attractive forces under the new constraint. These terms not only became physically meaningful, but their interpretation in terms of the Pauli principle led to a realization of the new possibilities provided by the electromagnetic interaction.

London proceeded to a formulation of the Pauli principle for cases with more than two electrons, which proved convenient for his later work in group theory: the wave function can contain arguments symmetric in pairs, and electron pairs on which the wave function depends symmetrically have antiparallel spin. He considered spin to be the constitutive characteristic of quantum chemistry. Since two electrons with antiparallel spin are not identical, the Pauli principle did not apply to them. The symmetric solution therefore could be chosen: (66)


The possibility of a homopolar chemistry rests quite essentially on the fact that the electrons can still differ from each other in one respect, which we have completely ignored until now: namely in the orientation of the rotation axis of their eigen-rotational movement. There are so to speak two different kinds of electrons, depending on the direction of their spin vectors (parallel or antiparallel to a magnetically preferred direction), and as the Pauli principle is only a statement concerning absolutely identical particles, it implies no restriction at all when the spin vectors point in opposite directions.


With the Pauli principle it became possible to comprehend "valence" saturation: whenever two electrons of different atoms combine to form a symmetric Schrödinger vibration, a bond will result. Spin became one of the most significant indicators of valence behavior. As Van Vleck put it, spin lay "at the heart of chemistry." (67)


Polyelectronic molecules and group theory

By September 1927 Heitler had gone to Göttingen as Born's assistant and London to Berlin as assistant to Schrödinger (who had succeeded Planck). Heitler found physics at Göttingen exciting, especially Born's course in quantum mechanics, which started with the matrix forinulation and then derived "God knows how' Schrödinger's equation." (68) Heitler felt that group theory provided the only solution to the many-body problem, and he outlined his program to London in two long letters.

Heitler began by clarifying the meaning of the line chemists drew between two atoms in diagrams. He held that every bond line represented the exchange of two electrons of opposite spin between two atoms. He examined the case of the nitrogen molecule and, in analogy with hydrogen, the term containing the outermost three electrons of each atom. Their spins and ) signified attraction: (69)


The general proof for something like this cannot be given, except group theoretically. In all probability it is about the theory of reducible representations, which as representations of the totality of permutations, e.g. of 6 elements, is not possible to be further reduced. But now it should be considered as a representation of a subgroup with 3 elements, and as such they are not at all impossible to reduce. It appears to be quite difficult and in any case it is necessary to dip into group theory.

Let us assume for the moment that the two atomic systems and are always attracted in a homopolar manner. We can, then, eat Chemistry with a spoon.


Heitler's program to explain all of chemistry got him into trouble more than once. Wigner used to tease him. He would ask: " ' what chemical compounds would you predict between nitrogen and hydrogen?' And of course, since he did not know any chemistry he couldn't tell me." (70) Heitler confessed as much: "The general program was to continue on the lines of the joint paper with London, and the problem was to understand chemistry. This is perhaps a bit too much to ask, but it was to understand what the chemists mean when they say an atom has a valence of two or three or four ... Both London and I believed that all this must be now within the reach of quantum mechanics." (71)

Heitler then worked out in detail the methane molecule CH₄. C is in the state (C has to be excited from its ground state in order to be in which agrees with experience). Four different "cells" in the L- shell exist for four electrons combined antisymmetrically. The four hydrogen atoms accordingly would be . Methane could therefore be reduced to a simple formula: the four hydrogen atoms are attracted in a homopolar manner to the carbon atom, without, however, any repulsion among them. The tetrahedron resulted. It was all very promising: "The long story for the foundational aspects of this matter, could give us, I believe, a series of very simple and equally interesting observations, if it were possible to approximate better the whole damn thing." (72)

London agreed with Heitler that group theory could help generalize the results derived by perturbation methods. They hoped to prove that, among all the possible combinations of spins between atoms, only one term provides the necessary attraction for molecule formation. Nevertheless, unlike Heitler in the company of Wigner and Weyl at Göttingen, London was not carried away by the spell of the new techniques. London "did not join in [Heitler's] studies of group theory. He thought it was too complicated and wanted to get on in his own more intuitive way." (73)

Wigner's papers implied that group theory could be used for classifying the energy values in a many-body problem as well as for calculating perturbation energies. The theory of the irreducible representations of the permutation group provided the possibility of dealing mathematically with the problems of chemical valence and avoided the difficulties involved in many-body problems. This lack of reliable methods for tackling many- body problems haunted London all his hife. Many years later, however, this difficulty helped him articulate the concepts related to macroscopic quantum phenomena.

Not everyone welcomed the use of group theory in physics. Hartree wrote to London: (74)


I am afraid that having studied physics, not mathematics, I find group theory very unfamiliar, and do not feel I understand properly what people are doing when they use it. (In England, "Physics" usually means "Experimental Physics," until the last few years "Theoretical Physics" has hardly been recognized like it is here [at the time Hartree was in Copenhagen], and, I understand, in your country. In Cambridge particularly the bias has been very much towards the experimental side, and most people now doing research in theoretical physics studied mathematics, not physics). I have been waiting to see if the applications of group theory are going to remain of importance, or whether they will be superseded, before trying to learn some of the theory, as I do not want to find it is going to be of no value as soon as I begin to understand something about it! 15 it real!y going to be necessary for the physicist and chemist of the future to know group theory? I am beginning to think it may be.


After moving to Göttingen in late 1927, Heitler published papers dealing with valence by group theory. In a significant paper with Rumer, he studied the valence structures of polyatomic molecules and determined the closest possible analog in quantum mechanics to the chemical formula which represented the molecule simply by fixed bonds between two adjoining atoms. (75) They found that the bonds were not strictly localized. Nevertheless, in general the dominant structure corresponded to the chemical formula. But other structures turned up that also were useful in understanding chemical reactions: (76)


And it was really in this paper that we then could see what the chemists mean by a chemical formula, when they write down a chemical formula .... There were a few new things for the chemists; one is that to each chemical formula there corresponds a wave function, but the wave function is not such that it corresponds to one chemical formula alone, but is in general a combination of several. There are some structures which are quite important for the calculation of energy contribution which the chemists never think of. London was the first [a long time before the Heitler-Rumer paper] who showed that the activation energies in the treatment of the three hydrogen atoms could be understood in quantum mechanics, and this method gave us then a general understanding for it. Later Pauling called this a resonance between several structures, which is a name perhaps not quite in agreement with the use of the word resonance by physicists ... A further point which was violently objected to by the chemists was that both London and I stated that the carbon atom with its 4 valences must be in an excited state ... all this was later accepted by the chemists, but at that time I don't think the chemists did find this of much use for them.


Convinced that it was impossible to continue his work in chemical valence by more analytic methods, London also turned to group theory, for three reasons. Firstly, the correlation between qualitative assessments of a theoretical calculation and the "known chemical facts" confirmed the methodology chosen by expressing the observed regularities as rules. Secondly, since analytic calculations were hopelessly complicated, group theory was the only convenient way to deal with the valence numbers of polyelectronic atoms. Thirdly, the interpretation of the chemical facts was compatible with the conceptual framework of quantum mechanics. Using group theory it might be possible "to discover in the quantum mechanical description conceptual facts which in chemistry have proven themselves in complicated cases as a guide through the diversity of possible combinations, and see them in their connection with the structure of atoms." (77) London attempted to interpret the valence numbers of the homopolar combinations based "on the conceptual representations" of wave mechanics, similar to the interpretation of the polar valence numbers by Kossel and Lewis.

By the middle of 1928, London had drawn up a three-pronged program to tackle "the most urgent and attractive problem of atomic theory: the mysterious order of clear lawfulness, which is the basis for the immense factual knowledge of chemistry and which has been expressed symbolically in the language of chemical formulas." (78) He intended to study the mutual force interactions between the atoms; he wished to examine the semi-empirical rules that chemists had found and to place them on a sound theoretical basis; and he would attempt to determine the limits of these rules and, if possible, treat them quantitatively.

But he doubted that the principles considered so far in atomic theory would enable him to realize such a program. The characteristic interaction of the chemical forces deviated completely from other familiar forces: chemical forces seemed to "awaken" after a previous "activation," and they suddenly vanished after the "exhaustion" of the available "valences." Elementary symmetry considerations showed that the homopolar valence forces could be mapped onto the symmetry properties of the Schrödinger eigenfunction of the atoms and could be interpreted as quantum-mechanical resonance effects. This interpretation was formally equivalent to its chemical model; it produced the same valence numbers and it satisfied the same formal combination rules, as expressed in the symbolic representation of the structural formulas of chemistry. These followed from group theory as an immediate consequence of the Pauli principle; the valences "saturated" because the Pauli principle restricted the occupation of equivalent states. Group theory demonstrated that the "uniqueness of the chemical symbolism is actually a consequence of the most fundamental theorems of the theory of the representations of the symmetric group." (79)

London's "spin theory of valence" dealt mainly with cases in which each electron in a pair comes from a different atom. He examined the condition that electrons from different atoms can combine with each other into a spin-zero pair. An electron already paired with another electron in the same atom did not come into consideration. But a paired electron could become available for bonding with an electron from another atom if it could be unpaired without the expenditure of much energy. London claimed that an electron could be unpaired provided that its total quantum number n did not change. He considered the unpairing an intennediate step in the formation of a compound. (80)



Reactions to the Heitler-London paper

The Heitler-London paper opened a new era in chemistry. Quantum mechanics led to the significant conclusion that two hydrogen atoms form a molecule and two helium atoms do not. Such a "distinctjon is characteristically chemical and its clarification marks the genesis of the science of subatomic theoretical chemistry," remarked Pauling. (81) Van Vleck wrote: "Is it too optimistic to hazard the opinion that this is perhaps the begin- nings of a science of ' mathematical chemistry ' in which chemical heats of reaction are calculated by quantum mechanics just as are the spectroscopic frequencies of the physicist?" (82)

Both de Broglie and von Laue regarded the paper as a classic. (83) In their book on quantum mechanics for chemists, Pauling and E. Bright Wilson hailed Heitler and London's work as the "greatest single contribution to the clar;fication of the chemists' conception of valence" since Lewis's suggestion of the electron pair. (84) Heisenberg considered the theory of valence of Heitler and London to "have the great advantage of leading exactly to the concept of valence which is used by the chemist." (85) Buckingham quoted McCrea, who recalled his own attempts to solve the problem of the hydrogen molecule bond: (86)


This was the most important problem I considered these days, but I got nowhere. Then one day in 1927, I was able to tell Fowler that a paper by Walter Heitler and Fritz London apparently solved the problem in terms of a new concept: a quantum mechanicai exchange force. He grasped this idea at once, and bade me to expound it in the next colloquium - which is how quantum chemistry came to Britain.


Although "impressed with the skill" of London's papers, Wigner thought that "the quantum mechanics of chemistry is not a resounding success ... the Heitler-London paper is very beautiful, but much of what came afterwards is only ' all right '." (87) London's papers on group theory provoked positive comments by Weyl in his book Group theory and quantum mechanics. (88) Born thought related the group theoretical treatment so "difficult and involved" that few people could definitely state what it had achieved, and stressed the importance of Weyl's generalization of it.

In 1928 two review articles appeared that exerted a deep influence on chemists. Published in the Chemical Reviews, both had the explicit aim of "educating" the chemists in the ways of the new mechanics. (89) Pauling presented the details of Burrau's calculation of the electron charge density distribution of the hydrogen molecule ion. (90) Pauling considered the Heitler-London treatment of the structure of the hydrogen molecule "most satisfactory" and repeatedly stated that spin and resonance would explain chemical valence. Concerning the recent work of Heitler, London, and himself, Pauling commented that their results agreed with the qualitative conclusions of Lewis: (91)


When I examined the papers pubiished by Heitler and London, I was unable to formulate a good relationship between their calculations and my knowledge of the properties of atoms and molecules. I realized that the chemists' ideas about the sharing of electron pairs fit more closely with the theory of quantum mechanics if the assumption of Russell-Saunders coupling were to be abandoned for atoms in molecules. In 1928 I published a short paper in the U.S.A. Proceedings of the National Academy of Sciences, in which I discussed briefly a number of results obtained in this way, including the statement that the resonance phenomenon in quantum mechanics gives rise to the tetrahedral arrangement of the four single bonds formed by a quadrivalent carbon atom in the molecules. I mentioned that a detailed account of the calculations wöuld be published iater. In fact, the calculations that I made were so complicated that I felt there was some uncertainty about convincing other people of their correctness, and it was not until three years later that I was able to formulate a remarkable method of simplifying the quantum mechanical treatment of the chemical bonds and to publish a long paper on application of quantum mechanics to the problem of the nature of the chemical bond.



John Van Vleck 1899–1980 Photo taken from J. Chem. Ed. 56, 296 (1979).

Van Vleck's review of quantum mechanics concentrated on explaining the principles and the internal logic of the new theory. (92) He gave full credit to the work of Heitler and London, something found in most of Van Vleck's papers through 1935 (93) before going over to use the more "practical" methods of Pauling and Mulliken. He summarized the achievements of quantum mechanics in physics in ten points and titled the section about chemistry, "What the quantum mechanics promises to do for the chemist." He emphasized the importance of spin for chemistry and showed that the Pauli exclusion principle provided a remarkably coherent explanation of the periodic table. He stressed its extreme importance elsewhere as well: "The Pauli exclusion principle is the cornerstone of the entire science of chemistry." (94) Van Vleck shared Dirac's attitude that the laws for the "whole of chemistry are thus completely known" and thought that the dynamics that successfully explained atomic energy levels for the physicist should support calculations of molecular energy levels for the chemist.

The calculations, though formidable, were unavoidable: the mathematical problem confronting the chemist was "to investigate whether there are stable solutions of the Schrödinger wave equation corresponding to the interaction between two (or more) atoms, using only the wave functions which have the type of symmetry compatible with Pauli's exclusion principle." Van Vleck drew attention to the crucial feature of London and Heitler's approach, lest chemists "get the wrong idea." The absence of certain compounds was not because the calculations yielded energetically unstable combinations, but because the corresponding solutions of the Schrödinger equation did not satisfy the symmetry requirements of the Pauli principle: "Thus considerations of symmetry (group theory) are often quite as vital as those of energetics. The failure of the chemist to find a compound does not necessarily mean that the corresponding molecule is ' energetically unstable, but may mean rather that it would demand electronic groupings contrary to the exclusion principle." (95) Few other papers at that time did so much to advocate quantum mechanics for chemists.

Among the meetings that discussed chemical bonding, one in particular expressed the radical changes occurring in the chemists' culture. The chemists in the 1928 meeting of the American Chemical Society appeared sufficiently fluent in the new physics, as appears from G.L. Clark's open- ing remarks: (96)


It may be asserted, in spite of discrepancies and disagreements, that in the first quarter of the twentieth century, the actual existence of an atomic world became an established fact. Part of the difficulties in details may be ascribed to failure of chemists to test their well-founded conceptions with the facts of physical experimentation, and far too few physicists inquired critically into the facts of chemical combination. So, firmly entrenched each in his own domain, a certain long-range firing of static cubical atoms against infinitesimal solar atoms has ensued, with few casualties and few peace conferences. The position of the Bohr conception has seemed so convincing that perhaps the majority of thinking chemists were coming to accept the dynamic atom, which is fully capable of visualization.


Clark was not alone in specifying the problematic relationship between physicists and chemists. Worth Rodebush, one of the first to receive a doctorate in 1917 from the newly established Department of Chemistry at Berkeley under the chairmanship of Lewis, went a step further than Clark. The divergent paths of physicists and chemists were drawn together after the advent of quantum theory and especiaily after Bohr's original papers. But in this process, "the physicist seems to have yielded more ground than the chemist. The physicist appears to have learned more from the chemist than the chemist from the physicist. The physicist now tells the chemist that his way of looking at things are really quite right because the new theories of the atom justify that interpretation, but, of course, the chemist has known all the time that his theories had at least the justification of correspondence with a great number and variety of experimental facts." He gracefully remarked that the physicist enabled him to calculate the energy of formation of the hydrogen molecule by using the Schrödinger equation. But the difficulty in a theory of valence was not to account for the forces that bind the atoms into molecules; the outstanding tasks were to predict the existence and absence of various compounds, and the unitary nature of valence as expressed by the law of multiple proportions. The "brilliant theories" of Lewis accounted for the features of valence "in a remarkably satisfactory manner, at least from the chemist's point of view." (97) Rodebush considered London's group theoretical treatment of valence to be important even though it did not answer all the queries of the chemist.

Perhaps the most cogent manifestation of the characteristic approach of American chemists was Harry Fry's articulation of what he called the pragmatic outlook. He posed a single question for organic chemists: what modifications to the structural formulas would conform with the current concepts of electronic valence? No change should obscure the fundamental purpose of a structural formula, which was to present the number, the kind, and the arrangement of atoms in a molecule as weh as to correlate the manifold chemical reactions displayed by the molecule: (98)


It should here be noted that no theory in any science has been so marvelously fruitful as the structure theory of organic chemistry.... When we are considering methods of modifying this structure theory of organic chemistry, by imposing upon its structural formulas an electronic valence symbolism, are we not, as practical chemists, obligated to see to it that such a system be one that is calculated to elucidate our formulas rather than render them obscure through the application of metaphysically involved implications on atomic structure which are extraneous to the real chemical significance of the structural formulas, per se .... the opinion is now growing that the structural formula of the organic chemist is not the canvas on which the cubist artist should impose his drawings which he alone can interpret .... Many chemists believe that the employment of a simple plus and minus polar valence notation is all that is necessary, at the present stage of our knowledge, to effect the further elucidation of structural formulas. On the grounds that practical results are the sole test of truth, such a simple system of electronic valence notation may be termed "pragmatic."


"Chemical pragmatism" resisted the attempts to embody ' in the structural formulas what Fry considered to be metaphysical hypotheses: questions related to the constitution of the atom and the disposition of the valence electrons. The chemical behavior of molecules was the primary concern of the pragmatic chemist, rather than the imposition of an electronic notation complicated by metaphysical speculations. Fry had to admit that as chemists learned more about the constitution of the atom, they would be able to exphain chemical properties more fully.

At the meeting of the Faraday Society in 1929, London (and to a lesser extent Heitler) appeared to agree with the molecular orbital approach as a way of providing a quantitative dimension to the valence group theory. This was not the sole reason for the formulation of the molecular orbital approach, an obvious point needing emphasis nonetheless since the widely held view that regards the molecular orbital approach as diametrically opposed to the valence bond method, while methodologically justified, appears to be historically untenable. Hund addressed a weakness of the group theoretical approach: chemical binding could not be understood in terms of energetics; the saturation of the valences was explainabhe only in terms of spin, (99) and he suggested mechanisms for molecule formation to account for the characteristic differences between H₂ and HeH. Lennard- Jones proposed a set of rules for the assignment of electrons to molecules consistent with the group theoretical considerations of both London and Heitler - especially with London's suggestion of two kinds of unsaturated valences that could give rise to molecular binding: one owing to an incomplete group of electron spins, the other to an incomplete group of angular momentum spins. (100)



Pauling introduces resonance

After staying with Sommerfeld in Munich for approximately thirteen months, Pauling went to Copenhagen, where he stayed less than a month. There he worked with Goudsmidt on the hyperfine structure of bismuth without much success. He then moved to Zürich, where he stayed for about two months. On the whole, he "rather regretted" the time spent in Zürich. He saw Schrödinger only at the weekly seminars, never found out what he was working on, and did not manage to interest him in his own work. "I offered to make any calculation interesting to him since he was not interested in my work; but without success." (101) He met Heitler and London, who, shortly before his arrival, had finished writing their paper on the formation of the hydrogen molecule, but they did not invite his collaboration. Thus, he continued alone to muse over the nature of the chemical bond.

Pauling's notebooks display some of his early thoughts about the nature of the chemical bond. (102) The first set of notes are from 1926, when he was still in Munich. Before his trip to Europe, Pauling had thought about the electron-pair bond in terms of electrons in binuclear orbits, (103) and in his application for the Guggenheim Fellowship he proposed to apply quantum theory to their study. Burrau was the first successfully to integrate the wave equation for the simplest molecule - the hydrogen-molecule ion. (104) He found a numerical expression for the electronic wave function in the field of the two nuclei - that is, he obtained the first numerical expression of a moledular (binuclear) orbital, together with values for the equilibrium internuclear distance, total energy, and vibrational energy of the lowest state.

In order to study diatomic molecules of identical atoms, Pauling tried to determine the form of molecular (binuclear) orbitals by approximating the internuclear potential by two identical one-dimensional square-well potentials. He then integrated Schrödinger's equation for the electron in the different regions of this simplified potential. He used the boundary conditions to determine some of the constants of integration and analyzed the extreme situations corresponding to the separated atoms (infinite distance between the atoms) and the united atom (zero distance between the atoms). The eigenfunctions obtained were either symmetrical or antisymmetrical in the position coordinate, implying symmetrical or antisymmetrical electron orbits. Then, by assuming that shared electrons were necessarily symmetrical in two-center orbitals (whereas unshared electrons were half symmetrical and half antisymmetrical), Pauling wrote the Lewis formulas for F₂, 0₂ and N₂. Where more than one formula could be written, he used the available empirical information together with his theoretical results to devise rules to decide among them. He noted that "the spinning electron accounts for electron pairs." He also referred to resonance energy: "There is, of course, a continual interchange of energy among the various electrons, so that one electron cannot be assigned to a given state; and there is the corresponding resonance effect on the energy of the system." (105) Finally, he tried to extend his square-well potential model to the case of a diatomic molecule composed of two different atoms, and of polyatomic molecules composed of two large and two small atoms.

Pauling was thinking about the electron-pair bond in terms of a Burrau- like two-nuclei orbital, but he could not derive any results he considered worth publishing. If the electrons had spin, then the Pauli principle implied that they should have opposite spins. In his notes Pauling never stated this consequence explicitly, but he later recalled having discussions with several people "about the explanation of chemical bonding in terms of the Burrau paper on the hydrogen molecule ion and the Pauli principle." (106) He might have been thinking already of the electron pair bond in terms of two electrons with opposite spins in a two-center molecular orbital.

While still in Munich, Pauling learned about Condon's "empirical method" for the hydrogen molecule, (107) and was immediately struck by "Condon's ingenuity." Condon assumed that the interaction of the two electrons in the hydrogen molecule was small in comparison with the electron-nuclei attraction, and represented each electron by an H2 ⁺ eigenfunction. To estimate the value for the perturbation energy, Condon assumed the same relation between interelectronic energy and total energy for the hydrogen molecule and for its united atom (the helium atom), and used the known value for the latter to estimate the value for the former. In this way Condon "got results as good as Heitler and London got later." (108)

Although Pauling kept on thinking about the chemical bond while in Copenhagen, his next set of important notes were written in Zürich. (109) There, Pauling discussed the chemical bond with Heitler and London. He did not agree entirely with their results, an understandable reaction since for more than three years he had been thinking about the electron pair bond as a pair of electrons in molecular (binuclear) orbits. (110) In Munich he had made the first calculations using a very rough approximation to the molecular potential. Calculations with molecular orbitals were not easy. Burrau and Condon used molecular orbitals for the hydrogen molecule ion and the hydrogen molecule, respectively, but the integrations had not been straightforward. In the case of the hydrogen molecule, Condon's ingenious "empirical method" enabled him to circumvent some mathematical obstacles and arrive at a good estimate for the interaction energy. Burrau and Condon's success undoubtedly reassured Pauling about the potential for a method alternative to that of Heitler and London. In Zürich he labored to develop such an alternative computational method. In doing so he used both the idea of resonance and the Pauli principle, which he later classified as the two fundamental factors influencing chemical valence." (111)

Pauling took two hydrogen atoms separated by a distance d and represented each electron by an H₂ ⁺ wave function, as Condon had; according to earlier calculations, (112) the function could be either symmetric (Ψ) or antisymmetric (Φ) relative to the nuclei. By also neglecting at first the repulsion between the two electrons, Pauling approximated the wave function for the hydrogen molecule by the product of the two electronic wave functions; that is, Ψ(H₂) = Ψ₁Φ_;2 (or Ψ₂Φ_;1).However, the indistinguishability of the electrons produces a resonance or exchange phenomenon, so that two different complete eigenfunctions can be formed for the hydrogen molecule:
Taking the Coulomb interaction between the two electrons as the perturbation force, represented by f = e² /r₁₂ , Pauling used perturbation theory to express the perturbation energy due to the electrons:
Representing the two-center wave functions in terms of one-center wave functions Ψ and Φ
and substituting in the expression for the perturbation energy, Pauling obtained
The first term represents the electrostatic repulsion between the electrons and the second term the resonance between them.

Pauling next considered the electrons in the two hydrogen atoms as spinning particles. He distinguished six different cases: one with both electrons represented by the symmetric H₂ ⁺ eigenfunctions; another with both electrons represented by the antisymmetric H₂ ⁺ eigenfunctions; and four with one symmetric and one antisymmetric H₂ ⁺ eigenfunction. Pauling concluded that the latter four situations corresponded to the formation of the hydrogen molecule from neutral atoms, whereas the other two corresponded to the formation of the hydrogen molecule from the ions H⁺ and H^-. By representing the two-center wave functions (molecular orbitals) in terms of one-center wave functions (atomic orbitals), Pauling could relate his six cases (with two-center wave functions) with the four cases considered by Heitler and London (with one-center wave functions). As we know, Heitler and London obtained three cases where the electrons attracted each other and one in which they repelled; they did not consider the formation of the hydrogen molecule from two ions, so they had two solutions less than Paul- ing. Pauling proved that by starting with two separated like nuclei (one- center wave functions) and two electrons, and taking into account the degenerations owing to similar nuclei and indistinguishable electrons, one would arrive at the six cases he obtained with two-center wave functions. He analyzed his conclusions in his notebooks: (113)


London and Heitler have a different method of treatment, starting with separate atoms (or ions). Their eigenfunctions are not those obtained by starting from H₂ ⁺ eigenfunctions. I suppose, though, that they are roughly correct for large distances, and that in some way there is transition to the H₂ ⁺ eigenfunctions, probably through degeneration. Thus if it could be shown that their H⁺ + H^- and H + H eigenvalues approach each other; it would be satisfactory. There would have to be triple degeneracy - both H⁺ + H^- and one H + H coming together.

I suppose that this is what happens, and that neither my treatment nor London's and Heitler's is correct, but that the correct treatment of the secular problem would give London and Heitler's results for large distances, and mine for small, with intermediate ones in between, and with degeneracy at points, so that all the transitions indicated are possible adiabatically (switchings being called adiabatic). Thus both H⁺ + H^- and H + H could go adiabatically to H₂.


Besides working on the interaction between two hydrogen atoms, Pauling spent most of his time in Zürich on the interaction between two helium atoms, but he ran into some integrals for which he could not find good approximate values. But he persevered, convinced that "if I worked in this field I probably would find something, make some discovery, and that the probability was high enough to justify my working in the field. Of course, it led to hybridization and all of this stuff." (114)



1931: One valence theory

Pauling returned to the California Institute of Technology in the fall of 1927 as Assistant Professor of Theoretical Chemistry. By 1931 he was a Professor and the first recipient of the A.C. Langmuir Prize of the American Chemical Society. During that time, he avidly followed the papers on group theory of Heitler and London; in contrast to a number of physicists he was at ease with group theory, although it did not appeal very much to him." (115) Among his extensive notes on their papers, and in his comments on one of London's papers, is the first mention of the equivalence between the spin theories of valence and Lewis's empirical ideas of the electron-pair bond, the first thoughts on "changed quantization," and the first reference to how Pauling intended to clarify the nature of the chemical bond." (116)

In his important paper, "The shared-electron chemical bond," (117) Pauling pointed out that chemical valence was the result of two factors: the Pauli exclusion principle and the quantum mechanical resonance phenomenon of Heisenberg and Dirac. Pauling recalled that resonance had been an essential ingredient in Heitler and London's explanation of the formation of the H₂ molecule and emphasized the relation to Lewis's theory: Heitler and London's theory often was "entirely equivalent" to Lewis's phenomenological theory of the shared pair bond. The quantum mechanical explanation specified the meaning of a shared electron bond - two electrons belonging to different atoms, in identical states, and with opposite spins. But Pauling already had realized that the quantum mechanical explanation of valence provided a "more detailed" and "more powerful" picture than the old description. If the extension of London's theory had already produced results, more were still to come.



John C. Slater 1900-1976 Photo © American Institute of Physics

Pauling further predicted that quantum mechanics alone could not solve the problems posed by valence theory. "It is to be especially emphasized that problems relating to choice among alternative structures are usually not solved directly by the application of the rules resulting from the mechanics; nevertheless, the interpretation of valence in terms of quantities derived from the consideration of simpler phenomena and susceptible to accurate mathematical investigation by known methods now makes it possible to attack them with a fair assurance of success in many cases." (118) He hinted at what he later called hybridization (which he then called "changed quantization") and promised a detailed account of his claims on "changed quantization." It took him three years to deliver his promise. The paper, which Pauling classified as his single most important contribution to the understanding of the chemical bond, (119) came out almost simultaneously with one by the physicist John Slater containing many of the same suggestions. At the same time, Mulliken was proposing an alternative approach to the problem of the chemical bond.

In 1931, Pauling went to Berkeley to deliver a series of lectures on "The nature of the chemical bond," which spanned almost a month, from March 23 to April 20. Birge enthusiastically wrote to Mulliken: (120)


Pauling has just finished a series of 12 lectures here, which I think are the most exciting lectures I ever attended. The main subject matter of the lectures is contained in his article in the April J.A.C.S. [ref. (124) ] I think this is an article of the very highest importance, and I hope you will read it carefully, if you have not already done so. He has gotten many results since this article was sent in for publication... He is interested in the actual character of the binding near equilibrium, and I think he is really on the right track. He is of course duplicating some of Slater's work, but according to him, this is his idea, and he told Slater about it, some time ago. (See his paper in P.N.A.S. in 1928, I think [ref. (111) ].) Certainly we are on the threshold of a very large new development.


Confident of the importance of the results that had eluded him for nearly three years, Pauling wrote a letter to A.B. Lamb, the editor of the Journal of the American Chemical Society, urging him to publish the paper as soon as possible. In his letter to Lamb, Pauling wrote: (121)


The paper could be shortened a little by omitting references to previous work and by condensing the explanatory discussions of new results; but I feel that this would render it more difficult for chemists to understand what has been done.

The paper seems to me to be primarily of chemical interest, and I hope that it may be published in the Journal of our American Chemical Society; but if prompt publication might be prevented through time-consuming consideration by referees, I must ask you to return the manuscript for I shall then feel obliged to publish it in the Physical Review or in an European journal where quick publication can be secured.


Pauling also sent a letter to the Physical Review in which he called the attention of physicists to the article in the Chemical Society's Journal, outlined its main conclusions, and emphasized the differences between his paper and an earlier note by Slater. (122)

Pauling made evident his debt to Lewis. "[I considered myself] not a stranger bringing something from outside, but rather a member of the group here carrying on the work begun by Professor Lewis in 1916; for ever since I first learned of the electron-pair bond, in 1920, I have devoted my efforts to attempting to understand the properties of substances from this viewpoint, so that even though I never matriculated in the University of California, I like to consider myself as to some extent a student of Professor Lewis's." (123)

In the opening paragraph of the first paper in the series in the Journal of the American Chemical Society, Pauling assessed the work on the chemical bond and outlined his own method: (124)


During the last four years the problem of the nature of the chemical bond has been attacked by theoretical physicists, especially Heitler and London, by the application of quantum mechanics. This work has led to an approximate theoretical calculation of the energy of formation and of other properties of simple molecules... and has also provided a formal justification of the rules set up in 1916 by G.N. Lewis for his electron bond. In [this] paper it will be shown that many more results of chemical significance can be obtained from the quantum mechanical equations, permitting the formulation of an extensive and powerful set of rules for the electron-pair bond supplementing those of Lewis. These rules provide information regarding the relative strengths of bonds formed by different atoms, the angles between bonds, free rotation or lack of free rotation about bond axes, the relation between the quantum numbers of bonding electrons and the number and spatial arrangement of the bonds.


Pauling considered his own work as an extension of the program of the theoretical physicists who had applied quantum mechanics to chemistry. He provided "many more" results and rules, which he sketched out using approximations and arbitrary assumptions which might have inhibited a theoretical physicist. He postponed the quantum mechanical justification of the rules to the end of the lectures, but never gave them because of lack of time. Their justification was not important; their usefulness in describing the different bonds in chemical substances alone mattered. (125)

Pauling implicitly assumed that the presence of other atoms in the molecule did not much influence the conditions for bond formation. He stated that bonds resulted from the overlapping of two atomic eigenfunctions, the larger the overlap the stronger the bond. The bond formed in the direction of the highest concentration of individual bond eigenfunctions. The strength and directional character of bonds was thus a consequence of the overlapping of individual bond eigenfunctions, which itself reflected a greater density of charge concentrated along that particular direction. The "remarkable" simplification Pauling had been searching for consisted of working just with the angular component of the atomic wave functions. For compounds of hydrogen and elements from the first row of the periodic system, only s and p eigenfunctions figured in the formation of stable bonds.

Pauling also assumed implicitly that s and p eigenfunctions did not change during bonding and thus explained the bond angle of the water molecule. But, according to Pauling, sometimes change in quantization occurred, producing novel results. He gave a loose criterion for the change of quantization: when the bond energy exceeded the s - p separation, then s - p quantization broke. Carbon satisfied the criterion. Pauling proved that the four best bond eigenfunctions that could be formed had a maximum value along lines making tetrahedral angles (109º) with each other.

By applying the rules for the electron-pair bond Pauling removed the apparent incompatibility between chemistry and quantum theory. One more step had been taken in the reconciliation between the physicist's and the chemist's conceptions of the atom. This time the physicist's picture was modified, by the new idea that quantization could be broken under certain conditions. Furthermore, Pauling's approach provided valuable insights for the case of polyatomic molecules too complex to be treated on a rigorous quantum mechanical basis. The fluidity of Pauling's style and the clarity of his explanations made it relatively easy for interested chemists to take the first steps into a territory largely unknown to them. (126)


1931: Another valence theory

Mulliken's papers rarely shared the clarity of presentation of Pauling's. Seldom able to restrain himself from discussing the whole story, he often did not distinguish between crucial points and minor details. Until 1929, he published in journals read mainly by physicists and it took him some years to realize the importance of writing for a chemical audie