<\/p>\n
Michelle Francl dusts off Pauling\u2019s notes on bonding to explore the illusory link between electron promotion and hybridization.<\/h6>\n<\/div>\n<\/div>\n\n <\/p>\n
<\/p>\n
Pauling haunts my classes. Metaphorically, and sometimes I suspect, literally.<\/p>\n<\/div>\n
\nIn the fall of 1957, Linus Pauling paid an impromptu visit to Bryn Mawr College to hear my colleague Frank Mallory, a new faculty member and CalTech alumnus, speak1<\/a><\/sup>. Pauling sat in the front row, four feet from the lecturer\u2019s table; four feet from where I now teach first-year chemistry. Each time I turn to the periodic table on the wall and encourage students to think about atomic valences or electronegativity, I catch a glimpse of Pauling in his prime, still stretched out in my front row, iconic black beret on his head, holding forth in his Oregon twang.<\/p>\n <\/p>\n
\nFig. 1: A page from Linus Pauling\u2019s notes.<\/b><\/figcaption>\n![\"Fig.](\"https:\/\/media.springernature.com\/m685\/springer-static\/image\/art%3A10.1038%2Fs41557-018-0099-3\/MediaObjects\/41557_2018_99_Fig1_HTML.jpg\")
<\/a><\/div>\n\nWritten on a single evening in December 1930, Pauling\u2019s derivation of a tetrahedral set of hybrid orbitals using\u00a0s<\/i>,\u00a0p<\/i>\u00a0and\u00a0d<\/i>\u00a0functions. Photograph courtesy of Ava Helen and Linus Pauling Papers, Oregon State University Libraries.<\/p>\n<\/div>\n<\/div>\nFull size image<\/a><\/div>\n<\/figure>\n<\/div>\n<\/div>\n\n <\/p>\n
<\/p>\n
Even when I\u2019m not teaching in that particular classroom, Pauling\u2019s shade insinuates itself into my syllabus, driving what I teach \u2014 and sometimes what I\u2019m trying to unteach. Pauling\u2019s hybrid orbitals are a de facto language for organic chemistry. They are one of our most beloved and powerful loci of argument, invoked to explain phenomena ranging from conformational preferences to bond lengths to acidity. But imbued with such power, orbitals have taken on a reality they do not actually possess. The claims of orbital tomographers are to the contrary2<\/a><\/sup>: there are no such observables as orbitals \u2014 hybrid, atomic or molecular. My introductory quantum-chemistry students, after a year of mentally tagging every carbon atom they encounter with its hybridization and seeing pi clouds floating high above phenyl rings, enticing targets for electrophiles, are loath to believe me when I insist orbitals are nothing more than a mathematical convenience, figments of quantum mechanics\u2019 imagination. Ironically, the power of Pauling\u2019s approach is one reason quantum mechanics is a routine part of a chemist\u2019s training.<\/p>\n<\/div>\n\nChemists\u2019 contentions as to the material reality of orbitals \u2014 atomic or molecular \u2014 aren\u2019t what concern me here. Buu Pham and Mark Gordon, among others, have cogently laid out the reasons why orbitals cannot be observed3<\/a>,4<\/a>,5<\/a><\/sup>. I\u2019ve nothing to add to this, other than to say Pauling didn\u2019t believe orbitals were observable either6<\/a><\/sup>. Instead, I\u2019d like to take a step back and examine the narratives (or perhaps I should say \u2018myths\u2019) we use to explain to new students how hybrid orbitals emerge from the canonical set of atomic orbitals. At the very moment we introduce them to a quantum-mechanical view of chemistry, we unnecessarily ignore the quantum-mechanical underpinnings and send our students down a rabbit hole that can be hard to get them out of later.<\/p>\n<\/div>\n\nIn many, if not most, general chemistry texts, the introduction of hybrid orbitals begins by noting that the ground-state electron configuration for carbon has a filled 2s<\/i>\u00a0subshell, which makes those electrons unavailable for forming a two-electron bond and limits carbon to the formation of, at most, two bonds. In order to account for the known tetravalence of carbon, textbook authors often suggest that an electron from the 2s<\/i>\u00a0subshell is first promoted to the 2p<\/i>\u00a0level, the cost of this promotion being recouped from the extra bonds the atom can now form. As a result of this promotion, the narrative implies that not only do you have the ability to form four bonds, you can make new hybrid orbitals. Authors generally note that you can\u2019t make the same argument for either oxygen or nitrogen.<\/p>\n<\/div>\n\nBut how does this promotion of an electron connect to the formation of hybrid orbitals? The texts are silent on this. As they then immediately apply those hybrid orbitals to both nitrogen and oxygen compounds, my alert students are left rightfully confused about the connection between promotion and hybridization. Despite what the text says, I assure them, notions of electron promotion have no bearing on Pauling\u2019s creation of\u00a0sp<\/i>n<\/i><\/sup>\u00a0hybrids.<\/p>\n<\/div>\n\nI\u2019ve wondered for years how this misplaced notion found its way into general chemistry textbooks. Pauling never mentions the promotion of an electron in the foundational papers on the nature of the chemical bond6<\/a>,7<\/a><\/sup>\u00a0\u2014 or in the notes he made on a December night in 1930 when his thinking about hybrids crystallized (Fig.\u00a01<\/a>)8<\/a><\/sup>. Neither is such an explanation found in Pauling\u2019s own general chemistry textbook9<\/a><\/sup>\u00a0nor in the lectures he recorded on the nature of the chemical bond10<\/a><\/sup>. Examples can be found in general chemistry, organic chemistry and physical chemistry texts for at least the past 40 years. Since a substantial collection of Pauling\u2019s own notes on the theory of bonding have been digitized by Oregon State University archives, I dug into them to see if I could find the seed of the promotion myth. I suggest there are two potential sources for this confounding of promotion and hybridization.<\/p>\n<\/div>\n\nPauling\u2019s hybrid orbitals emerge from his attempts to show that the tetrahedral nature of carbon, first proposed by van \u2018t Hoff in 1874, could be derived\u00a0a priori<\/i>\u00a0from the principles of quantum mechanics. Pauling\u2019s first foray involved reducing a bond to the interaction of two atoms, A and B, each of which was described by a collection of one-electron eigenfunctions7<\/a><\/sup>. He treated the approach of the two atoms as a perturbation and noted that in some cases the quantization of the\u00a0s<\/i>\u00a0and\u00a0p<\/i>\u00a0orbitals was \u2018destroyed\u2019 by a strong interaction. In contemporary terms, the set of unperturbed functions on the two atoms that gave the most stable interaction was not the canonical\u00a0s<\/i>\u00a0and\u00a0p<\/i>\u00a0subshells, but a degenerate set of four linear combinations of those orbitals with a different energy than either the\u00a0s<\/i>\u00a0or\u00a0p<\/i>\u00a0levels.<\/p>\n<\/div>\n\nPauling later asserts that comparing the interaction energy and the energy separation between the\u00a0s<\/i>\u00a0and\u00a0p<\/i>\u00a0subshells provides a rough criterion for whether this \u2018change in quantization\u2019 was important: if the bond energy was greater than the\u00a0s<\/i>\u2013p<\/i>\u00a0gap, use the linear combinations6<\/a><\/sup>. I wonder if this is the source of the current explanation that the cost of promotion is balanced by the bond energy, but I note that Pauling wasn\u2019t constructing his perturbation diagram from an excited atomic state, but from a new state, with a fourfold degenerate shell at a new energy, a subtlety lost in translation over the years.<\/p>\n<\/div>\n\nThe second potential source stems from Pauling\u2019s 1931 paper: \u201cThe nature of the chemical bond,\u201d in which he brought variational theory to bear on the problem, again treating a bond as the interaction between two atomic wavefunctions built from single-electron functions. A single such function on each atom would overlap to create a two-electron bond. In the days before the available computational machinery could tackle the calculation of the integrals for finding the variational energy, Pauling had to make reasonable guesses as to the most important terms contributing to the variational energy and what they depended on. He expressed6<\/a><\/sup>\u00a0the energy of the interaction as:<\/p>\n <\/p>\n
E<\/span>=<\/span>W<\/span><\/span>A<\/span><\/span><\/span>+<\/span>W<\/span><\/span>B<\/span><\/span><\/span>+<\/span>J<\/span><\/span>E<\/span><\/span><\/span>+<\/span>J<\/span><\/span>X<\/span><\/span><\/span>\u2212<\/span>\u2211<\/span><\/span>Y<\/span><\/span><\/span>J<\/span><\/span>Y<\/span><\/span><\/span>\u2212<\/span>2<\/span>\u2211<\/span><\/span>Z<\/span><\/span><\/span>J<\/span><\/span>Z<\/span><\/span><\/span><\/span><\/span>E=WA+WB+JE+JX-\u2211YJY-2\u2211ZJZ<\/span><\/span><\/div>\n<\/div>\n\n <\/p>\n
Pauling considered the terms\u00a0W<\/i>A<\/sub>,\u00a0W<\/i>B<\/sub>\u00a0and\u00a0J<\/i>E<\/sub>\u00a0to be independent of any interaction between the atoms. Though he was unable to quantitatively calculate the exchange terms,\u00a0J<\/i>X<\/sub>,\u00a0J<\/i>Y<\/sub>\u00a0and\u00a0J<\/i>Z<\/sub>, he assumed them to be negative based on the value of similar known integrals. The best variational energy would thus be obtained when\u00a0J<\/i>X<\/sub>, the exchange term between the bonding orbitals, was maximized. As the exchange terms were directly proportional to the overlap of the orbitals involved, the best basis functions for the molecule would have their maxima directed along the bond axis.<\/p>\n<\/div>\n\nTo find the best functions under these constraints, Pauling took linear combinations of the canonical\u00a0s<\/i>\u00a0and\u00a0p<\/i>\u00a0functions. Drawing on his perturbation theory analysis, he assumed that the quantization of the subshells was destroyed, which would make the radial parts of\u00a0s<\/i>\u00a0and\u00a0p<\/i>identical and radically simplify his calculations. This assumption is another possible source of the promotion myth, as it is tantamount to moving the energy of the\u00a0s<\/i>\u00a0orbital to match that of the\u00a0p<\/i>\u00a0orbital.<\/p>\n<\/div>\n\nIt was now straightforward to show that an\u00a0sp<\/i>\u00a0hybrid had a larger value along the bond axis than either a pure\u00a0s<\/i>\u00a0or\u00a0p<\/i>\u00a0orbital8<\/a><\/sup>. Creating a function orthogonal to this hybrid, its maximum is found to be at the tetrahedral angle. Thus, Pauling brilliantly used a quantum-mechanical framework to show that quadrivalent carbon had to be tetrahedral.<\/p>\n<\/div>\n\nMy reading of Pauling\u2019s work is that the formal promotion of electrons has nothing to do with his hybridization theory, and we should summarily exorcise it from the textbooks (along with a number of other myths)11<\/a><\/sup>. Pauling\u2019s own qualitative reasoning is accessible to undergraduates, the only math you need is trigonometry. It has the advantage of showing them the route from quantum mechanics to chemical structure, as well as setting the stage for those who will go on in chemistry to understand orbitals as mathematical constructs, not real objects. But perhaps most importantly, we miss an opportunity to help students see equations not as machines that spit out numbers, but as another way to describe the natural world. They can and should appreciate the power of qualitatively \u2018reading\u2019 the mathematical expressions that were at the core of the groundbreaking connection Pauling forged between quantum mechanics and chemical structure. If Pauling\u2019s ghost is going to haunt our classrooms, we should take a moment to listen to what he has to say.<\/p>\n<\/div>\n<\/div>\n <\/p>\n
<\/p>\n
<\/p>\n
<\/p>\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
<\/p>\n
<\/p>\n
Pauling haunts my classes. Metaphorically, and sometimes I suspect, literally.<\/p>\n<\/div>\n
In the fall of 1957, Linus Pauling paid an impromptu visit to Bryn Mawr College to hear my colleague Frank Mallory, a new faculty member and CalTech alumnus, speak1<\/a><\/sup>. Pauling sat in the front row, four feet from the lecturer\u2019s table; four feet from where I now teach first-year chemistry. Each time I turn to the periodic table on the wall and encourage students to think about atomic valences or electronegativity, I catch a glimpse of Pauling in his prime, still stretched out in my front row, iconic black beret on his head, holding forth in his Oregon twang.<\/p>\n <\/p>\n Written on a single evening in December 1930, Pauling\u2019s derivation of a tetrahedral set of hybrid orbitals using\u00a0s<\/i>,\u00a0p<\/i>\u00a0and\u00a0d<\/i>\u00a0functions. Photograph courtesy of Ava Helen and Linus Pauling Papers, Oregon State University Libraries.<\/p>\n<\/div>\n<\/div>\n <\/p>\n <\/p>\n Even when I\u2019m not teaching in that particular classroom, Pauling\u2019s shade insinuates itself into my syllabus, driving what I teach \u2014 and sometimes what I\u2019m trying to unteach. Pauling\u2019s hybrid orbitals are a de facto language for organic chemistry. They are one of our most beloved and powerful loci of argument, invoked to explain phenomena ranging from conformational preferences to bond lengths to acidity. But imbued with such power, orbitals have taken on a reality they do not actually possess. The claims of orbital tomographers are to the contrary2<\/a><\/sup>: there are no such observables as orbitals \u2014 hybrid, atomic or molecular. My introductory quantum-chemistry students, after a year of mentally tagging every carbon atom they encounter with its hybridization and seeing pi clouds floating high above phenyl rings, enticing targets for electrophiles, are loath to believe me when I insist orbitals are nothing more than a mathematical convenience, figments of quantum mechanics\u2019 imagination. Ironically, the power of Pauling\u2019s approach is one reason quantum mechanics is a routine part of a chemist\u2019s training.<\/p>\n<\/div>\n Chemists\u2019 contentions as to the material reality of orbitals \u2014 atomic or molecular \u2014 aren\u2019t what concern me here. Buu Pham and Mark Gordon, among others, have cogently laid out the reasons why orbitals cannot be observed3<\/a>,4<\/a>,5<\/a><\/sup>. I\u2019ve nothing to add to this, other than to say Pauling didn\u2019t believe orbitals were observable either6<\/a><\/sup>. Instead, I\u2019d like to take a step back and examine the narratives (or perhaps I should say \u2018myths\u2019) we use to explain to new students how hybrid orbitals emerge from the canonical set of atomic orbitals. At the very moment we introduce them to a quantum-mechanical view of chemistry, we unnecessarily ignore the quantum-mechanical underpinnings and send our students down a rabbit hole that can be hard to get them out of later.<\/p>\n<\/div>\n In many, if not most, general chemistry texts, the introduction of hybrid orbitals begins by noting that the ground-state electron configuration for carbon has a filled 2s<\/i>\u00a0subshell, which makes those electrons unavailable for forming a two-electron bond and limits carbon to the formation of, at most, two bonds. In order to account for the known tetravalence of carbon, textbook authors often suggest that an electron from the 2s<\/i>\u00a0subshell is first promoted to the 2p<\/i>\u00a0level, the cost of this promotion being recouped from the extra bonds the atom can now form. As a result of this promotion, the narrative implies that not only do you have the ability to form four bonds, you can make new hybrid orbitals. Authors generally note that you can\u2019t make the same argument for either oxygen or nitrogen.<\/p>\n<\/div>\n But how does this promotion of an electron connect to the formation of hybrid orbitals? The texts are silent on this. As they then immediately apply those hybrid orbitals to both nitrogen and oxygen compounds, my alert students are left rightfully confused about the connection between promotion and hybridization. Despite what the text says, I assure them, notions of electron promotion have no bearing on Pauling\u2019s creation of\u00a0sp<\/i>n<\/i><\/sup>\u00a0hybrids.<\/p>\n<\/div>\n I\u2019ve wondered for years how this misplaced notion found its way into general chemistry textbooks. Pauling never mentions the promotion of an electron in the foundational papers on the nature of the chemical bond6<\/a>,7<\/a><\/sup>\u00a0\u2014 or in the notes he made on a December night in 1930 when his thinking about hybrids crystallized (Fig.\u00a01<\/a>)8<\/a><\/sup>. Neither is such an explanation found in Pauling\u2019s own general chemistry textbook9<\/a><\/sup>\u00a0nor in the lectures he recorded on the nature of the chemical bond10<\/a><\/sup>. Examples can be found in general chemistry, organic chemistry and physical chemistry texts for at least the past 40 years. Since a substantial collection of Pauling\u2019s own notes on the theory of bonding have been digitized by Oregon State University archives, I dug into them to see if I could find the seed of the promotion myth. I suggest there are two potential sources for this confounding of promotion and hybridization.<\/p>\n<\/div>\n Pauling\u2019s hybrid orbitals emerge from his attempts to show that the tetrahedral nature of carbon, first proposed by van \u2018t Hoff in 1874, could be derived\u00a0a priori<\/i>\u00a0from the principles of quantum mechanics. Pauling\u2019s first foray involved reducing a bond to the interaction of two atoms, A and B, each of which was described by a collection of one-electron eigenfunctions7<\/a><\/sup>. He treated the approach of the two atoms as a perturbation and noted that in some cases the quantization of the\u00a0s<\/i>\u00a0and\u00a0p<\/i>\u00a0orbitals was \u2018destroyed\u2019 by a strong interaction. In contemporary terms, the set of unperturbed functions on the two atoms that gave the most stable interaction was not the canonical\u00a0s<\/i>\u00a0and\u00a0p<\/i>\u00a0subshells, but a degenerate set of four linear combinations of those orbitals with a different energy than either the\u00a0s<\/i>\u00a0or\u00a0p<\/i>\u00a0levels.<\/p>\n<\/div>\n Pauling later asserts that comparing the interaction energy and the energy separation between the\u00a0s<\/i>\u00a0and\u00a0p<\/i>\u00a0subshells provides a rough criterion for whether this \u2018change in quantization\u2019 was important: if the bond energy was greater than the\u00a0s<\/i>\u2013p<\/i>\u00a0gap, use the linear combinations6<\/a><\/sup>. I wonder if this is the source of the current explanation that the cost of promotion is balanced by the bond energy, but I note that Pauling wasn\u2019t constructing his perturbation diagram from an excited atomic state, but from a new state, with a fourfold degenerate shell at a new energy, a subtlety lost in translation over the years.<\/p>\n<\/div>\n The second potential source stems from Pauling\u2019s 1931 paper: \u201cThe nature of the chemical bond,\u201d in which he brought variational theory to bear on the problem, again treating a bond as the interaction between two atomic wavefunctions built from single-electron functions. A single such function on each atom would overlap to create a two-electron bond. In the days before the available computational machinery could tackle the calculation of the integrals for finding the variational energy, Pauling had to make reasonable guesses as to the most important terms contributing to the variational energy and what they depended on. He expressed6<\/a><\/sup>\u00a0the energy of the interaction as:<\/p>\n <\/p>\n <\/p>\n Pauling considered the terms\u00a0W<\/i>A<\/sub>,\u00a0W<\/i>B<\/sub>\u00a0and\u00a0J<\/i>E<\/sub>\u00a0to be independent of any interaction between the atoms. Though he was unable to quantitatively calculate the exchange terms,\u00a0J<\/i>X<\/sub>,\u00a0J<\/i>Y<\/sub>\u00a0and\u00a0J<\/i>Z<\/sub>, he assumed them to be negative based on the value of similar known integrals. The best variational energy would thus be obtained when\u00a0J<\/i>X<\/sub>, the exchange term between the bonding orbitals, was maximized. As the exchange terms were directly proportional to the overlap of the orbitals involved, the best basis functions for the molecule would have their maxima directed along the bond axis.<\/p>\n<\/div>\n To find the best functions under these constraints, Pauling took linear combinations of the canonical\u00a0s<\/i>\u00a0and\u00a0p<\/i>\u00a0functions. Drawing on his perturbation theory analysis, he assumed that the quantization of the subshells was destroyed, which would make the radial parts of\u00a0s<\/i>\u00a0and\u00a0p<\/i>identical and radically simplify his calculations. This assumption is another possible source of the promotion myth, as it is tantamount to moving the energy of the\u00a0s<\/i>\u00a0orbital to match that of the\u00a0p<\/i>\u00a0orbital.<\/p>\n<\/div>\n It was now straightforward to show that an\u00a0sp<\/i>\u00a0hybrid had a larger value along the bond axis than either a pure\u00a0s<\/i>\u00a0or\u00a0p<\/i>\u00a0orbital8<\/a><\/sup>. Creating a function orthogonal to this hybrid, its maximum is found to be at the tetrahedral angle. Thus, Pauling brilliantly used a quantum-mechanical framework to show that quadrivalent carbon had to be tetrahedral.<\/p>\n<\/div>\n My reading of Pauling\u2019s work is that the formal promotion of electrons has nothing to do with his hybridization theory, and we should summarily exorcise it from the textbooks (along with a number of other myths)11<\/a><\/sup>. Pauling\u2019s own qualitative reasoning is accessible to undergraduates, the only math you need is trigonometry. It has the advantage of showing them the route from quantum mechanics to chemical structure, as well as setting the stage for those who will go on in chemistry to understand orbitals as mathematical constructs, not real objects. But perhaps most importantly, we miss an opportunity to help students see equations not as machines that spit out numbers, but as another way to describe the natural world. They can and should appreciate the power of qualitatively \u2018reading\u2019 the mathematical expressions that were at the core of the groundbreaking connection Pauling forged between quantum mechanics and chemical structure. If Pauling\u2019s ghost is going to haunt our classrooms, we should take a moment to listen to what he has to say.<\/p>\n<\/div>\n<\/div>\n <\/p>\n <\/p>\n <\/p>\n <\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":"<\/a><\/div>\n