A Resonating-Valence-Bond Theory of Metals and Intermetallic Compounds, 1949

Linus Pauling, 1949

As we have written elsewhere, Linus Pauling developed and championed the theory of resonance early on in his career. But for almost 20 years, the ways that metals might conform to the theory remained elusive. Even though he had a hunch that they too adhered to the tenets of resonance, he was not able to prove it definitively at the outset. Not until 1949, with the publication of “A Resonating-Valence-Bond Theory of Metals and Intermetallic Compounds” was he able to demonstrate that metals do resonate. 

Pauling’s interest in metals dated at least as far back as his undergraduate years at Oregon Agriculture College and continued to flourish during his graduate training at the California Institute of Technology. Later in his career, when asked to reflect on his contributions to the field of chemistry, he often spoke of his early work with metals as being important. This was especially so with his work on metals and resonance.


“A Resonating-Valence-Bond Theory of Metals and Intermetallic Compounds,” which was published in the Proceedings of the Royal Society London, serves as an addendum of sorts to Pauling’s previous papers on the nature of the chemical bond. Even though Pauling’s theory of resonance had been well-received for several years, confusion still existed amongst chemists (including Pauling) about how the theory might apply to metals; particularly the iron-group transition metals. Pauling believed that, like other elements, resonance must be used to explain the way that these metals bond, but the specifics proved difficult to pin down.

Pauling had always aligned himself with the notion that the properties of an element were connected to the configuration of its valence electrons. For example, Pauling knew that the arrangement of valence electrons gave Carbon its stable tetrahedral properties and salts their ionic properties. Metals, however, were harder to define due to the fact that they “showed great ranges of values of their properties, such as melting and boiling point, hardness and strength, and magnetic properties.”

Pauling initially focused on magnetism, and from this work, it was determined that metals had a high ligancy of either 8 or 12. Ligancy, or the number of compounds that each metal could bond to, (often referred to as Coordination Number) is similar but still distinct from valency, and for the metals that Pauling was studying, a ligancy of 8 or 12 was higher than their valence. This discrepancy seemed to indicate that resonance could not be applied to explain how metals bonded. But Pauling still believed that resonance was the answer, and set about trying to confirm this belief.


In his quest to understand if or how metals resonated, Pauling first needed to clarify whether or not previous thinking about metals was correct. Prior to the publication of his 1949 paper, it was believed that the d orbitals did not participate in bonding with the iron-group transition metals, such as Manganese, Iron, Cobalt, Nickel, and Copper. Using these assumptions, Pauling modeled the predicted properties of such metals and found that they would have low melting points and weak bonds. The lab work indicated otherwise however, leading Pauling to conclude that the current understanding was incorrect, and that something else had to be going on with the bonds in order to account for their physical properties.

For several years, Pauling had been unable to devise a competing theory that would explain the unique properties of metals, but he felt certain that the answer would be revealed through the application of quantum mechanics. By specifically using the wave theory of quantum mechanics, Pauling calculated that metals used 8.28 orbitals instead of the expected 9. While he was sure that the math was correct, he struggled to understand what was accounting for the missing 0.72 orbitals, and for almost nine years he worked to reconcile the discrepancy. Ultimately though, he realized that 0.72 orbitals was, in fact, the answer he was looking for; they were what gave metals their unique properties.

In its essence, Pauling breakthrough was that the extra 0.72 orbitals was not a mathematical anomaly, but instead an extra orbital, one that he called a “metallic orbital.” The metallic orbital was, according to Pauling, “required” to give metals their “characteristic properties, especially that of electronic conductivity of electricity.” In Pauling’s view, substances lacking the metallic orbital that still maintained some metallic properties, such as electric conductivity, should be classified as metalloids.

Pauling’s conceptualization of the metallic orbital allowed him to subsequently reframe his understanding of resonance. When Pauling developed the theory, it was more specifically known as synchronized resonance; a state in which all bonds resonate in the same way and valence is undisturbed by the process. Pauling realized that, in order for metals to resonate, a different kind of resonance – unsynchronized resonance – would be needed. In unsynchronized resonance, instead of all bonds resonating simultaneously, a single bond could resonate on its own. In the case of metals this happened in the metallic orbital, and it was found that unsynchronized resonating metals conferred unusual stability and high ligancy.


Once Pauling made these connections, he was ready to publish his paper, which, first and foremost, sought to prove the existence of a metallic orbital. To do this, Pauling had to show that the existing understanding of valency was incorrect, and then to demonstrate that resonance – specifically unsynchronized resonance – could account for both the predicted and the observed properties of metals.

Using Lithium as his model element, Pauling explained that the current understanding posited a valence structure that consisted of 2s orbitals, and that this was the commonplace belief because, “the [proposed] molecular orbitals correspond to electron energies.” By using the wave function however, Pauling found that the assumed orbital structure did not correspond to observed electron energies. Specifically, he calculated that if Lithium had 2s orbitals, the predicted heat of formation would be much different than was the observed heat of formation; a calculated difference of 32.4 kcal/g. In short, the current understanding was incorrect.

Pauling’s next step was to use resonance to explain Lithium’s metallic properties, and to devise an arrangement of valence electrons that would match predicted energies with observed energies. Using the idea of unsynchronized resonance, Pauling suggested that if, instead of 2s orbitals, Lithium’s valence electrons were actually in resonance, this new arrangement could be “responsible for the difference in energy.” In other words, without resonance the electrons existed in a fixed energy state. But if the electrons were in resonance, their energy state would instead be a hybrid of all possible energy states. This circumstance, Pauling postulated, would confer lower energies to the molecule and bring predicted and observed energies into alignment. In the case of Lithium specifically, Pauling found that the valence electrons were in 2s orbitals, but also 2p orbitals. Pauling worked through different metals throughout the remainder of the paper to further support his thinking.

The theoretical work that Pauling put into this paper completed the framework for resonance. And interestingly, even though the ideas that he presented were accepted and widely used for decades, some of the math in the paper was not fully validated at the time. In fact, it wasn’t until 1984, when Pauling revisited the 0.72 orbitals, that the math was completed. By then Pauling had acknowledged that the calculations he used to get to 0.72 orbitals had been crude, and based on in part on educated guesses. But by the 1980s, many of the unknowns were known, which allowed Pauling to revisit the math with more precision and compare it to his work from the late 1940s. Using clean, contemporary data, Pauling confirmed that the new calculation was “in exact agreement with the observed value of 0.72.”

The Nature of Interatomic Forces in Metals, 1938

Linus Pauling, ca. 1930s

“In recent years I have formed, on the basis of mainly empirical arguments, a conception of the nature of the interatomic forces in metals which has some novel features.”

­-Linus Pauling, 1938

Prior to the publication of this article, which appeared in the December 1938 issue of Physical Review, much about the interatomic forces operating in metals was either unknown, or theoretical predictions did not align properly with observed data. In publishing this paper, Linus Pauling first sought to align the incongruencies between theory and data for the transition metals, such as iron, cobalt, nickel, copper, palladium, and platinum. He was then able to correctly predict properties including “interatomic distance, characteristic temperature, hardness, compressibility, and coefficient of thermal expansion” by discarding previously held assumptions and inserting new – and correct – assumptions about transitions metals.

The most significant idea that Pauling introduced with this paper was the notion that the valence shell electrons – those in the outer shell – play a part in bonding. Previously, scientists believed that these electrons made “no significant contribution” to bond formation. Pauling was able to establish otherwise, and used this breakthrough to both align observable data with theoretical data, and make other predictions about transition metals.


The 1938 paper was written in the wake of a revolution within the world of chemistry. A raft of new theories brought about by a widening understanding of quantum mechanics was generating intense excitement for scientists world-wide, and the tools that quantum mechanics provided for helping to “correct” previous understandings of the chemical bond were of paramount interest to many. Pauling, of course, was a leader in this area, his body of work ultimately garnering the 1954 Nobel Chemistry Prize for “research into the nature of the chemical bond and its application to the elucidation of the structure of complex substances.”

Within this area of focus, many scientists were especially interested in exploring the ways that metals bonded because, as noted, the observed data did not match up with theory. Pauling sought to mend this gap by using quantum mechanics to look at interatomic forces in a novel way. Prior to the Physical Review paper, chemists believed that when metals bonded, their valance shell electrons played only a small role in the resulting structures. Pauling argued otherwise, and put forth an important new theory that the valence shell electrons contributed to the process through resonance, a theory that he had developed earlier in the decade and continued to champion.


Because the crux of Pauling’s scientific intervention was to prove that valence shell electrons are involved in bonding, most of the paper is devoted to supporting this claim. The primary tool that Pauling uses to craft his argument is an analysis of temperature predictions. According to the reigning theory regarding metals and valence shell bonds, when bonding occurred, the electrons would bond in a manner that would create a moment of ferromagnetism. Specifically, it was theorized that these ferromagnetic moments would be temperature dependent, meaning that as the temperature of the metal changed, its degree of magnetism would also change in a predictable way. Experiments had shown however, that when metals bonded, their ferromagnetism remained independent of temperature.

Pauling exploited this piece of information and used it to support his theory. According to Pauling, if metals bonded through resonance, they would create ferromagnetic moments that were temperature independent, a hypothesis that correctly aligned with the observed data.

To develop his argument, Pauling made specific use of the element Vanadium, which has an electron configuration of 3d34s2. Under the old model, Vanadium’s valence electrons could only interact weakly in bonding if, at most, two of the 4s2 electrons were involved in a bond. This, according to Pauling, would create ferromagnetism which would decrease with increasing temperature, meaning that it was temperature dependent. On the contrary, the experimental evidence showed that Vanadium’s magnetism was temperature independent. This meant, therefore, that weak valence interaction during bonding was not possible.

Pauling’s alternative suggestion was that all of Vanadium’s valence electrons were involved in bonding through resonance; not just the two 4s2 electrons, as previously believed. Further, if the valence electrons bonded through resonance, the ferromagnetism of their structure would be temperature independent, a prediction that aligned with the observed data.


Once Pauling was able to prove that the valence electrons in Vanadium bonded through resonance, he then began to apply the concept to all transition metals. As with the previous example, Pauling continued to support the concept by comparing predicted outcomes with empirical data. And once again, when viewing the bonding through the prism of resonance, predicted outcomes of magnetic moments began to align with the empirical data.

Pauling then took it another step by repeating the exercise with interatomic distances. As demonstrated in his paper, a resonant structure would correctly predict the interatomic distances that had been observed for many bonds. Pauling also claimed that other properties, such as the “compressibility, coefficient of thermal expansion, characteristic temperature, melting point, and hardness” would likewise correctly align with experimental evidence, once resonance was used to explain valence shell bonding.

Though clearly a significant breakthrough, the assertions that Pauling made in his paper were grounded in work done by others — notably the quantum mechanical theory of ferromagnetism developed by Heisenberg, Frenkel, Bloch, Slater, et al., and Wolfgang Pauli’s theory of the temperature-independent paramagnetism of the alkali metals. And while the 1938 article acknowledges these debts, it also attempts to improve upon them.

This was especially so with the quantum mechanical theory of ferromagnetism. As we have seen, Pauling successfully applied the idea of temperature independence and ferromagnetism to support his claims, but he also found one aspect of the theory to be needlessly bothersome. As Pauling noted, in order for much of the theory to work on a mathematical level, scientists were compelled to assign positive numbers to all unpaired valence electrons. Pauling recognized that this was only necessary if it was assumed that valence electrons did not play a large role in bonding for metals. Under a resonant scenario, Pauling was able to show that the math could still work if the valence electrons were negative and that, once again, “this conclusion agrees with the observation.”

New Insights into Metals and More

Linus and Peter Pauling at Warwick Castle, England. 1948.

[The Paulings in England: Part 3 of 5]

In his lab, a five minute walk from his office at Balliol College (where he was once caught boiling an egg on his electric space heater), Linus Pauling’s research took a turn from the contents of his lectures – intermolecular forces and biological specificity – and he found himself devoting his research time to metal theory. Pauling had planned to revise the index for his newly published freshman text, General Chemistry, during his Eastman Professorship, but couldn’t seem to get metals off his mind.  As he wrote in a letter to his Caltech colleague J. Holmes Sturdivant, “I thought that I would be doing work in connection with my freshman text while in England, but it has turned out that I have devoted all of my time, and presumably shall continue to do so, to work on the theory of metals and intermetallic compounds.”

He was aided in his lab by three other researchers – David Shoemaker, Hans Kuhn, and a young man from Holland, Dr. F. C. Romeyn. Pauling’s circumstances were proving to be highly productive, and in a March letter to Robert Corey, Pauling wrote of the impact that the change of setting was having in stimulating his thoughts:

I have been having wonderful success in my development of a theory of metals. I think that it has really been very much worthwhile for me to get away for this period of time, under circumstances favorable to my thinking over questions and trying to find their solution. The problem of metals has been on my mind for a number of years, and I haven’t been able to leave it alone, so it is a good thing that I have now managed to get it solved.

This new theory of metals was an extension of Pauling’s valence-bond approach to determining the structure of molecules, as initially developed in the late 1920s. Pauling was first exposed to quantum mechanics as an undergraduate at Oregon State University (then known as Oregon Agricultural College) and retained that interest as he transitioned to graduate studies and faculty employment at the California Institute of Technology.

In 1926 Pauling traveled on a Guggenheim Fellowship to study the developing field of quantum mechanics with physicists in Europe, and especially Germany. He brought these new ideas back to Caltech in the form of quantum chemistry, which he used to compute the electronic structures of molecules. This intuitive valence-bond approach was quickly judged a success and had been popular since the 1930s as a simple model for studying the electron dispersal in the bonds between molecules.

But all the while another chemist, Robert Mulliken (recipient of the 1966 Nobel Prize for Chemistry) had been steadily fostering a rival approach: the molecular orbital theory. While the Pauling family enjoyed springtime in Paris at the beginning of April, Pauling and Mulliken met head to head at a conference on Isotopic Exchange and Molecular Structure. There an entire day was devoted to the comparison of the two theories before a group of quantum chemists. Pauling had written earlier that molecular orbitals were confusing to students, but he learned at this meeting that one always has to stay one’s toes: with more mathematics under their belts, advanced chemistry students were increasingly hungry for the more quantitative approach that Mulliken’s theory offered.


Sometimes ideas come upon the great thinker at surprising times, and Pauling experienced just such a eureka moment during one of his twice-weekly Oxford lectures in February.  As he wrote to Holmes Sturdivant,

I have just had a great stroke of luck. While giving my lecture on Tuesday I suddenly realized that a calculation about resonance energy of metals that I had just made and was reporting contained the key to the strange valence numbers and numbers of atomic orbitals and unused orbitals that have turned up in my theory of valency of metals.

Notes on intermetallic compounds by Linus Pauling, March 1948.

Pauling worked out his ideas on electron theory and the structure of metals and intermetallic compounds through pages and pages of careful handwritten calculations. In looking at each manuscript now, Pauling presents a hypothesis about some aspect of metal theory and then proceeds to calculate, revise, and recalculate until the theory and the experimental x-ray diffraction data line up. For instance, on one day in March, Pauling was exploring intermetallic compounds from several different angles.  He writes “I shall now treat intermetallic compounds, with my new ideas – resonance of bonds when an extra orbital is available, importance of n=1/2, 1/4 etc., concentration of bonding electrons into strong bonds (Zn-Zn, etc as compared with Na-Na) , transfer of electrons with increase in valence.” Hybrid orbitals, bond lengths, and the overall stability of structures were other items on Pauling’s research agenda.

Of course, not every idea is a winner and a few theories led Pauling down the wrong path; in one manuscript Pauling set out to, as he wrote, “consider sp hybridization – how can we set up a secular equation to give the results given by my bond-strength postulate?”  In the end Pauling found that “the ratio does not come out as desired. It is evident that my assumption that the energies can be taken proportional to ‘bond strengths’ is not right.”  Missteps such as these didn’t deter Pauling from pressing on with his research, for as he often said, “The way to have good ideas is to have lots of ideas, and throw away the bad ones.”


Chemistry boasts its own special language, or nomenclature, and chemists like Pauling are to thank for the terms that make chemical jargon unique. As research advances, sometimes an entire new word is needed to describe an innovative concept. While tackling the nuances of metal theory at Oxford, Pauling wrote to Sturdivant about this very problem.

By the way, I think that we should do something toward improving the nomenclature. For example, coordination number is an awkward and unwieldy expression – we need one short, precise word for this concept. Perhaps ligancy could be used. It would fit in well with ligand and the verb to ligate. We also need some general words to express the bonds between one atom and the surrounding atoms – we now use the word bond to refer both to the electron pair bond that is resonating around among alternative positions and to the fraction of an electron pair bond that is a portion to a particular position. I have also felt troubled about using the word position in this way – to mean the region between two atoms. If we do introduce any change in nomenclature, it must be very well thought out, and must not involve too great a strain on the memory, or too great a departure from the past.

New fields also call for innovations in instrument development and research programs. Pauling was in constant communication with his colleagues back home about new tools that might be constructed to aid the researchers. He admired the Cavendish’s vast x-ray crystallography laboratory and also gained new insights from reading British journals devoted to scientific instrumentation. He would frequently send word back as to how Caltech workers could improve on a complex apparatus such as the specialized cameras for x-ray diffraction of metallic crystals.

Pauling was likewise intrigued by the English system of graduate education, wherein graduate students would take class work completely during the first year and then spend practically 100% of their time on research during the other two years. Pauling was always looking to improve upon existing programs, but as appealing as the English system was, he acknowledged that in implementing it one would run the risk of not knowing whether a student was an apt researcher for their entire first year!