The Valence Stroke and the Valence Sphere One- and Three-Dimensional Models of Localized Molecular Orbitals

This book is about old wine (valence stroke diagrams) in new bottles (valence sphere models).

Strokes and spheres represent localized molecular orbitals, in the sense that, divorced from their diagrams and models they have no significance. The essence of organic stereochemistry lies in locations of substituents off single, double, and triple bonds.

Valence Sphere Models generate those locations almost magically. Illustrated is an observation made many years ago, in the 19th century, by Thomson and Tait, in the preface to their classic Elements of Natural Philosophy. "It is particularly interesting to note how many theorems, even among those not ordinarily attacked without the help of the Differential Calculus [and, today, in valence theory, with quantum mechanics and computers], have here been found to yield easily to geometrical methods of the most elementary character" [emphasis added].

Both representations above — stroke and sphere — place a tetrahedral arrangement of electron pairs about sites of atomic cores (not shown in the valence sphere models), whether the cores are involved in single, double, or triple bonds; or with lone pairs.

One arrangement (the tetrahedral arrangement) fits all (atomic cores that obey the Octet Rule: usually B+3, C+4, N+5, O+6, and F+7). The arrangement may be generated in two ways: occupancy of alternate corners of a cube, with the atomic core in question at the cube's center; and by the centers of four close-packed spheres, with the atomic core in question in the spheres' interstice, called in packing of, e.g., oxide ions about smaller cations in crystalline oxides a "tetrahedral interstice". The most important figure in chemistry, it's been said, is the tetrahedron. And, we would add, the sphere.

The Tetrahedron, the Cube, and the Tetrahedral Angle

A tetrahedron's corners occupy alternate corners of a circumscribed cube.

The tetrahedron's faces face the cube's other corners.

A face lies opposite a tetrahedron's corner.

(Those facts, we shall see, have enormous consequences for reaction mechanisms.)

The symbol "C" stands for the atomic core C+4. In attack by reagents that have protruding electron domains, the C+4 core moves outward, thereby lengthening its bond that lies opposite the attacked face.

Chemistry's most famous bond angle is "the tetrahedral angle": 109.47º

Bond angles HCH, HNH, and HOH of CH4, NH3, and H2O are 109.47º, 107.1º, and 104.5º.

PERSONAL NOTE: The author's interest in molecular structure was initially ignited on reading about those cited bond angles in an article by R. S. Mulliken, in 1952.

The Tetrahedron's Back Story

Chemistry has but one noteworthy theory and but one set of hypothetical ideas, the theory of the combination of atoms into molecules with its fundamental idea of valence. It is a most beautiful theory, surpassed by none other in the intellectual satisfaction it affords. NORMAN CAMPBELL

The tetrahedron's back story in chemistry, presented here for newcomers to chemical thought, illustrates the nature of the evolution of thought in an inductive science. The story begins with two familiar facts and ends with one of the leading inductions in the history of chemistry (along with John Dalton's Atomic Hypothesis and G. N. Lewis's conjecture regarding electron pairs), by van't Hoff, chemistry's first Nobel Laureate, in 1901.

TWO FAMILIAR FACTS. Molecules of water and carbon dioxide have the chemical formulas H2O and CO2.

A FACT ABOUT HYDROGEN ATOMS. No molecules have chemical formulas of the type HXn for n > 1. (HN3 — "hydrazoic acid" — is an exception.)

AN INFERENCE ABOUT HYDROGEN ATOMS. Hydrogen atoms are never attached by chemical bonds to more than one atom. (HN3's order of atomic attachments is HNNN.)

AN INDUCTION REGARDING H2O. H2O's order of atomic attachments is HOH (not HHO).

A LINGUISTIC CONVENTION REGARDING H. Chemists assign hydrogen atoms a "combining capacity" or "valence" of 1. Hydrogen, they say, is monovalent.

A DEDUCTION REGARDING OXYGEN ATOMS. Chemists deduce — from their linguistic convention for hydrogen; from the reason for that convention; and from the formula for water molecules — that oxygen atoms are divalent.

FACTS ABOUT COMPOUNDS OF CARBON AND OXYGEN. COO designates a peroxide, not thermodynamically stable carbon dioxide.

A CONCLUSION REGARDING CARBON DIOXIDE. Both atoms of oxygen of a molecule of carbon dioxide are attached to the carbon atom.

A DEDUCTION REGARDING CARBON ATOMS. Carbon atoms are tetravalent; or quadrivalent.

GRAPHIC EXPRESSIONS OF VALENCE ASSIGNMENTS. Chemists represent valence assignments by "valence strokes": 1 for H, 2 for O, 4 for C.

BOND FORMATION. To generate models of molecules, chemist indicate the "mutual saturation of chemical affinities" by pointing at each other valence strokes of two different moieties. Generated with hydrogen atoms are the figures -

Molecular hydrogen is, indeed, diatomic, molecular formula H2. Illustrated is -

THE RULE OF NO DANGLING VALENCE STROKES. Molecular hydrogen, for instance, is not H—.

Produced by the Rule are correct atomic linkages and molecular formulas for molecules of dihydrogen (H—H), water (H—O—H), and hydrogen peroxide (H—O—O—H). Not accounted for, however, by the drawing —O—, is -

A FACT ABOUT OXYGEN MOLECULES. They're diatomic!

AN INFERENCE REGARDING OXYGEN ATOMS. As for adjacent valence strokes of polyvalent carbon atoms, the pair of directed valence strokes of an oxygen atom are not collinear.

In an oxygen atom's "valence shell", together with its two valence strokes that represent shared electron pairs (discussed later; and illustrated on p1) are two pairs of unshared electrons.

For molecules of dioxygen and water, valence assignments together with "the mutual saturation of chemical affinities" procedure, the Rule regarding dangling affinities, and the angular inference regarding oxygen's affinities yield these valence stroke diagrams:

[FORMULA NOT REPRODUCIBLE IN ASCII]

H2O is, indeed, a bent molecule (

(OO bonds of O2, whose valence strokes don't point at each other, are called "bent bonds".)

A FACT ABOUT HYDROGEN CHLORIDE. Its molecular formula is HCl.

AN INFERENCE REGARDING CHLORINE ATOMS. They're monovalent, in HCl.

A DEDUCTION REGARDING COMPOUNDS OF C, H, AND Cl THAT CONTAIN ONE CARBON ATOM. Their molecular formulas are CH4, CH3Cl, CH2Cl2, CHCl3, and CCl4.

A FACT ABOUT CH2Cl2. Only one substance with the molecular formula CH2Cl2 exists!

CONCLUSION: Carbon's four valence strokes, suggested van't Hoff, are directed toward the corners of a tetrahedron.

For CH3Cl all three C—H bonds are (therefore, by supposition) equivalent to each other. In passing to CH2Cl2 from CH3Cl it makes no difference which H atom of CH3Cl is replaced by a Cl atom.

Mutual saturation by two carbon atoms of 1, 2, and 3 of their combining capacities yields carbon-carbon single, double, and triple bonds (p1). The remainder of organic stereochemistry is in large measure different combinations of those possibilities. A quadruple bond between two carbon atoms, according to the tetrahedral model, is impossible — and is, indeed, unknown.

The tetrahedral arrangement of valence strokes (left and middle figures above) introduced into chemistry the third dimension. That advance was continued (right figure), over half a century later, by replacement of the one-dimensional valence strokes of valence stroke diagrams (in which the strokes never cross each other) by three-dimensional domains (for wave-like electrons, only two of which — postulated Lewis — can be at the same place at the same time.

The Nature of the Double Bond

This story of the double bond begins with organic chemistry's leading idea.

The Tetrahedral Tetravalent Carbon Atom

The idea accounts for the formula and structure of CH4, H monovalent. To save the idea for C2H6, Kekule introduced the idea of a carbon-carbon single bond. For C2H4 he postulated "some denser arrangement", later called a double bond.

Bent and Banana Bond Representations of a Carbon-Carbon Double Bond

Lewis's identification of a valence-stroke as a pair of electrons led to the idea of describing bond directions in molecules in terms of atomic orbitals for electrons.

For carbon's valence-shell the available orbitals are (from atomic spectroscopy) 2s and 2p+1, 2p0, and 2p-1 orbitals. They haven't, however, the desired directional character. The s-orbital is spherically symmetric. The p±1 orbitals are doughnut-shaped. Linear combinations of them yield dumbbell-shaped orbitals 2px and 2py that, with 2pz, point along the x, y, and z directions. It's more directionality than none at all, but not the directionality of tetrahedrally directed affinities. Slater and Pauling continued with the idea of linear combinations of atomic orbitals. The vector sum, so to speak, of px, py, and pz points toward the upper right rear corner of the cube above, the sum –px – py + pz to the upper left front corner, &c.

Mixing in the s-orbital, in order to achieve four linearly independent orbitals that point in the tetrahedral directions, yields the tetrahedral sp3 "hybrid orbitals", te. They're descriptions, not explanations, of carbon's directional affinities.

te1 s + px + py + pz

te2 s - px - py + pz

te3 s + px - py - pz

te4 s - px + py - pz

To account for the HCH bond angles of ethylene, thought, initially (and incorrectly) to be 120º, and linear acetylene's 180º HCC bond angles, investigators introduced the hybrid orbitals sp2 and sp.

s/p Hybrid Orbitals, their Fractional s-Character, and Inter-Orbital Angles

Two descriptions of double bonds between small-core atoms emerged: a Molecular Orbital "sigma-pi" description (below, on the left) designed, initially, for use with double-bonded species in electronically excited states, and, following in the footsteps of organic chemistry's classical structural theory of molecules in their ground electronic states, a Valence Bond "bent bond" or "banana bond" equivalent orbital description.

Which description is most useful? For whom? Students? Or computational chemists? And for what purpose? Description of molecules' ground states? Or molecules' electronically excited states?

It's like asking: Which mathematical coordinates are most useful: rectangular coordinates or polar coordinates? It depends on the situation. Whatever the situation, however, when viewed properly, in the light of the indistinguishability of electrons and the consequent antisymmetric character of electronic wave functions, the two descriptions of a double bond are seen to be mathematically equivalent to each other.

Consider a bent-bond description of a double bond between two carbon atoms A and B. Four hybrid orbitals are involved: two from atom A, two from atom B. (In the expressions below the bond axis lies along the x coordinate, with the z-axis in the plane of the page, also.)

The two bent-bond components of the double bond are represented by the sums of the two top hybrids and the two bottom hybrids. Addition of those two partial sums yields, on division by 2, the following expression for the sigma component of a sigma-pi description of the double bond (as the overlap of two s/p hybrids of the form s + apx).

(sA + apxA) + (sB - apxB)

Subtraction of the lower hybrid partial sum from the upper one yields, on division by 2b, the following expression for the pi component of a sigma-pi description of the double bond (as the overlap of two pz orbitals on adjacent atoms).

pZA + pZB

Since the system's full wave function may be expressed, in the orbital approximation, as a determinant whose columns' or rows' terms represent electron occupancy of individual orbitals, and since addition or subtraction of a determinant's rows or columns to or from each other does not change its numerical value, it's a matter of indifference from a purely mathematical point of view which orbitals one uses for expression of the wave function of double bonds: those of MO Theory or those of VB Theory.

On the other hand, if one thinks of a double bond solely in terms of its individual orbitals, rather than in terms of its full wave function, and if one then estimates, from the shapes of the individual orbitals, what the most likely distribution of electrons in a double bond is, then the difference between the MO and VB descriptions of the bond is like the difference between "night and day".

Frontier Orbitals

A Localized Electron Domain Perspective

The two traditional theories of the chemical bond, Valence Bond Theory and Molecular Orbital Theory, are jointly, in a way, paradoxical, from the point of view of electron-pair donor-acceptor interactions. What VB Theory lacks MO Theory has, for chemical purposes, a surfeit of: namely, orbital lobes for the description of electrophilic sites. Neither imperfection seems to have attracted much, if any, attention.

Description of valence-strokes of valence-stroke diagrams in terms of hybrid atomic orbitals provides Valence Bond Theory with a localized orbital account of its occupied electron domains. Symmetry adapted linear combinations of those localized orbitals generate the delocalized orbitals of Molecular Orbital Theory and, accordingly, provide an orbital account of systems' highest occupied molecular orbitals. Missing from valence stroke diagrams, however, is an account of molecules' lowest unoccupied orbitals. It is true, on the other hand, that the "pockets", "hollows", or "dimples" of valence sphere models of valence-stroke diagrams correspond to empty electrophilic sites, indicated in one instance below by the short arrow.

Conventional valence bond theory provides for such sites, however, no description in terms of hybrid atomic orbitals. Molecular orbital theory, on the other hand, provides, in addition to a lobe for a sigma bond A—B, from the sum of overlapping hybrid orbitals of A and B, two lobes from their algebraic difference: one off one end of the bond, the other off the other end. In "attack" on a molecule by a photon (in, e.g., a lonepair/sigma* electronic transition), both lobes are occupied, simultaneously. In backside attack by nucleophilic reagents, however, only one lobe of a sigma antibonding molecular orbital is occupied. The other lobe at the other end of the bond is superfluous.

Valence Stick Theory, in summary, generates no lobes and Molecular Orbital Theory generates two lobes where only one lobe is needed in orbital descriptions of inter- and intra-molecular electron-pair donor-acceptor interactions. Description in orbital language of the pockets, hollows, or dimples of the "bumps and hollows" of valence-sphere models of molecules' valence-stroke diagrams requires directional, localized orbitals that for Octet-Rule-satisfying atomic cores point, as indicated in the following figure by the dashed arrows (of antibonding te* orbitals), in the opposite directions of the cores' conventional tetrahedral orbitals (te).

In the case of double bonds, MO theory's pi antibonding orbitals have four lobes. Chemical attack by a nucleophilic reagent occurs, however, as indicated below by the arrow on the right, at the site of a single electrophilic site opposite one of the four hybrid atomic orbitals that comprise a double bond in its bent- or banana-bond formulation (at the left).