Chemical bond angles and lengths

Type of physical science: Chemistry

Field of study: Chemistry of molecules: structure

The geometry of chemical molecules is dictated by the bond lengths and bond angles of the atoms that compose the molecules. These lengths and angles are in turn dictated by the mathematical descriptions of positions allowed for electrons about the atoms, as given by quantum mechanics.

Overview

Covalently bonded chemical molecules have definite shapes--or structures, or geometry, or architecture, depending on which term is preferred. A depiction of the DNA (deoxyribonucleic acid) molecule is familiar to most, either as a drawing or possibly as an actual three-dimensional model. The regularity of structure of the intertwined helices of the sugar-phosphate chains is striking, as is the orientation of the organic bases, turned to the interior of the helices where they can interact with each other. What is even more striking, on close examination, is that this larger structure is the product of an absolute regularity of geometry--that is, of bond angles and lengths--around each of the carbon, oxygen, phosphorus, and nitrogen atoms that form the structure of the molecule. There is no bond angle around the carbons, oxygens, and phosphoruses of the main chain that deviates appreciably from 109.5 degrees, nor any in the carbons and nitrogens of the bases that is not 120 degrees or 108 degrees, with little deviation.

What is true of the very large DNA molecule is equally true of small molecules.

Carbon dioxide, O=C=O, is always linear, with a rigidly maintained 180 degree bond angle (O-C-O) around the carbon atom. Water, H-O-H, on the other hand, is angular, with the bond angle around the oxygen atom approximating 109.5 degrees. Ammonia, NH₃, which one might imagine as a kind of three-bladed propeller with nitrogen at the hub, is in fact a little three-cornered pyramid, with nitrogen at the apex and with bond angles around the nitrogen of very nearly 109.5 degrees.

Quantum mechanics, a branch of physics, explains why molecules form with rigid geometric shapes; why, for example, the carbon atom in carbon dioxide exerts all of its bonding capacity in two bonds 180 degrees apart from each other and none anywhere in between; and why a hydrogen atom forms one bond and stops. Quantum mechanics is basically a mathematical discipline, and the mathematics is formidable, but a nonmathematical picture can be extracted that explains chemical bonding.

First, one abandons the notion of electrons as billiard balls, or as planets rotating in orbits around a sun that is the nucleus of the atom. The electron is retained only as a quantized negative charge (that is, one can have one electron's worth of charge, or two electron's worth, but nothing in between). The orbits are not retained, but the "orbital" is used to describe the mathematically defined volume in which an electron may exist. Also, other concepts are retained from the simple picture of the atom, notably the picture of "shells" of electrons around the nucleus, such as the layers of an onion, composed of inner and outer shells of electrons, even though the quantum mechanical picture is at variance with this.

For nonbonded atoms, four kinds of orbitals are defined and are given the designations s, p, d, and f. These are diagrammed in figure 1. Figure 1a shows the simple spherical s orbital. It is tempting to relate this to the planetary orbit picture, but in fact the electron fills this orbital completely, as a kind of cloud of negative charge. The only relation to the orbit picture is that the charge density is greatest in a spherical shell at a distance from the nucleus that corresponds to the diameter of the former orbit. Each electron shell has a single s orbital, which may hold zero, one, or two electrons but no more. The s orbitals increase in diameter with increasing shells.

When one turns to the p orbitals, (figures 1b-1d), one can see that the planetary picture must be given up altogether. There are three p orbitals in each shell from the second on. They take the x, y, and z directions of the Cartesian coordinates, and they are each two-lobed. Each orbital may hold up to two electrons. Clearly, one can no longer consider the electron as a solid spherical particle. Only the charge-cloud picture makes sense here.

Figures 1e and 1f show only a representative d and f orbital. There are five four-lobed d orbitals in each shell from the third on, and seven eight-lobed f orbitals in each shell from the fourth on. Each separate orbital may hold up to two electrons. When one tries to imagine an atom of uranium, for example, element 92, with forty-six assorted orbitals, each with two electrons, all occupying more or less the same volume, the mind boggles, and must accept the word of the quantum mechanicians that one's understanding is mathematical, not pictorial. For this reason, only a few orbitals are considered when constructing pictures of atoms that form bonds in molecules.

When one atom forms a covalent bond with another, the atoms approach to bonding distance and their atomic orbitals overlap to become molecular orbitals--that is, orbitals (one or more) associated with two nuclei, defining a volume that can hold the negative charge of two electrons to glue together two positive nuclei. Figure 2 shows this process for two hydrogen atoms with their s (atomic) orbitals overlapping to form a molecular orbital, which is given the designation σ.

Hydrogen is, for practical purposes, the only element that forms chemical bonds using a purely atomic (s) orbital. All other elements go through a process of combining two or more atomic orbitals (as many as eight, in some cases) to create clusters of "hybrid orbitals," with new shapes, but with directivities just as rigid as those of the atomic orbitals from which they are made. Figure 3 shows the more common hybrid orbitals: their shapes, their designations (derived simply from the names of the atomic orbitals appropriated to make the hybrid), and their bond angles.

(Note that a "single" lobe of each hybrid orbital is being viewed, not the double lobes of the p orbitals. Each of the lobes shown can overlap with another orbital and form a bond. There is, in fact, a tiny second lobe for each of these hybrids, as befits their partial p character, but it does not bond and, to simplify the diagram, it is not depicted.)

Hybrids are principally for the use of a number of quite different atomic orbitals, whose differences are smoothed out into a set of energy-equivalent hybrids with clear-cut directivities that overlap with orbitals of other atoms to make molecules of definite geometric shape. Making the hybrids requires energy; hybrid orbitals have energies higher than any of the simple orbitals from which they are made. When bonds are made, however (usually, more bonds than could be made with the simple atomic orbitals), energy is returned in greater quantity than it was laid out, and stable compounds are formed.

Another gain of hybrids is that of definite and equal bond lengths for the cluster of hybrids around an atomic nucleus. Close inspection of figure 3 will show that different types of hybrid orbitals have different lengths--they "reach out" to a greater or lesser degree. The mathematical statements that describe a molecular orbital include a distance factor that gives a bond length of optimum stability: maximum orbital overlap without pushing two nuclei so close together that they repel each other excessively.

Applications

Detailed application of the geometrical ideas addressed (in greatly simplified form) in this article can occupy a chemist for many years. A few examples are presented, which may serve to indicate the range of problems that can be addressed by this approach.

Consider the simplest carbon compounds such as methane, CH₄, and carbon tetra-fluoride, CF₄. In these molecules, the carbon is at the center and is sp³ hybridized. In methane, each hydrogen overlaps its unhybridized s orbital with one of carbon's sp³'s. Four molecular orbitals (sigma bonds) form to hold the electron pairs that glue together the carbon nucleus with the four hydrogen nuclei. The resulting compound is tetrahedral in shape; that is, if lines were drawn connecting the hydrogens, the resulting three-dimensional figure is a tetrahedron, a pyramid with an equilateral triangle as the base. The bond angles around the tetrahedral carbon atom are 109.5 degrees. Because of the discrete nature of the sp³ hybrids, and because the resulting molecular orbitals are the only places (by the mathematics of quantum mechanics) where electrons may exist, methane has no tendency to take on more hydrogen atoms. Species such as CH₅, CH₆, and so on, do not exist, because there are no more orbitals around the carbon atom to make bonds.

The carbon tetrafluoride molecule is made in the same way as methane, except that the orbitals of fluorine that overlap with carbon's sp³ orbitals are also sp³. The σ bonds that form are the same as those in methane, and the tetrahedral shape, with 109.5 degree bond angles, is the same.

If the sp³ hybridized carbon atom is allowed to bond to other similar carbon atoms, the extended structures of organic chemistry are obtained: "straight chain" hydrocarbons, which are not straight, but zigzag, with 109.5 degree bond angles; branched-chain and cyclic structures; and the ultimate three-dimensional structure, diamond. Trials with three-dimensional models show that cyclic structures of five or more carbons can be constructed without straining the 109.5 degree bond angle. It follows that three- and four-carbon ring structures must be highly strained. This makes them relatively unstable and very reactive, opening the ring readily to release the strain.

When sp² hybridized carbon atoms bond with each other, the result is the rigid structure of the double bond C=C. When an sp² hybrid on one carbon has overlapped with the sp² hybrid on the other carbon to form a σ bond between the carbons, each carbon atom has a leftover p orbital that was not used in making the hybrids. These two-lobed p orbitals are oriented perpendicular to the plane defined by the three hybrid orbitals, like the shaft of a propeller, and they can overlap above and below the σ bond to form a new kind of two-lobed bond called a π bond. This bond is a molecular orbital, associated with both carbon atoms, containing two electrons, and because it is outside the axis of the σ bond, it prevents rotation around the σ bond. Thus, the two carbon atoms and whatever atoms are bonded by the remaining four sp to the power of 2 hybrids are held in a rigid plane. When this system is part of a long carbon chain (-C-C=C-C-), the rigid geometry causes different molecular shapes, depending on whether the carbons that continue the chain are on the same side of the double-bond carbons (cis) or on opposite corners (trans). This has important consequences in natural product and physiological chemistry.

The compounds nitrogen trichloride (NCl₃) and phosphorus trichloride (PCl₃) are known. Both have sp³ orbitals around the central atom (N or P), with three Cl atoms connected by σ bonds, and the fourth orbital containing a nonbonding electron pair. Thus, they have the same triangular pyramid structure for ammonia, NH₃.

Phosphorus also forms the pentachloride, PCl₅, but there is no corresponding NCl₅, because the five bonds of the pentachloride call for a dsp³ hybrid, which is possible for the third-shell element phosphorus, but not for the second-shell element nitrogen, because the second shell contains no d orbitals.

Context

The principles discussed have been extended since the 1950's to explain more and more complex systems. In 1951, Linus Pauling and Robert Corey announced their determination of the α-helix structure for protein molecules. This was the culmination of nearly two decades of work that began with determination by X-ray crystallography of the bond lengths and bond angles in the amino acid units that compose the protein chain. Using these data, Pauling and Corey carefully constructed three-dimensional models of the chain and discovered that if it was coiled into a helix with 3.6 amino acid units per turn, the resulting structure held its shape because of hydrogen bonding between the carboxyl oxygen of one amino acid and the amide hydrogen of the amino acid four units ahead. They also constructed another hydrogen-bonded structure, the β-pleated sheet. For the first time, it was possible to understand how protein chains could assume and hold their (already known) shapes--the shapes that allow them to function as enzymes, structural molecules, transport proteins, and so forth.

In 1953, Francis Crick and James D. Watson established the structure of DNA by essentially the same method--X-ray crystallographic data translated into three-dimensional models. This time the helices, built of sugar-phosphate units, were 10 amino acid units per turn for each chain. The result that grew from this finding was that there was room inside the helices only for paired organic bases, provided a one-ring (pyrimidine) base always paired with a two-ring (purine) base. Further structural details dictated that a particular pyrimidine always paired (by hydrogen bonding) with a particular purine. Thus, the genetic code was revealed: The genetic information is contained in the sequence of bases, and it can be passed from an existing DNA molecule to a new molecule that forms using the old molecule as a template, because the structure of the double helix leaves room for only certain bases to pair. The new molecule therefore has a base sequence that is dictated by that of the old one, and the genetic information is preserved.

Synthetic polymers can also be understood through structural analysis. High-density polyethylene is composed exclusively of linear polyethylene molecules, which can be shown through models to associate in a crystalline structure that is rigid and resistant to deformation.

Branches on the polymer chain hold the molecules apart, reducing crystallinity and making a softer product. A similar effect is found in polypropylene, in which regular structures are found to be more crystalline and rigid than random structures.

At the time when the structures of proteins, DNA molecules, and many of the synthetic polymers were first deduced, the only method available was the painstaking construction of molecular models to see how the parts of the molecules related to one another. Since then, however, computer software has been developed that places structural analysis within the grasp of many researchers. As there are many such molecules available in the living world, there is confidence that structural analysis based on the principles of quantum mechanics will be providing answers to important questions about molecules for many years in the future.

Principal terms

ATOMIC ORBITAL: a volume or space, not necessarily spherical, around the nucleus of an atom, in which up to two electrons may be accommodated; the shape of the space is defined and described mathematically by the equations of quantum mechanics

BOND ANGLE: in a three-atom group, A-B-C, B may be considered as the vertex of an angle formed by the bonds to A and C; this is the bond angle around B

BOND LENGTH: the center-to-center distance between the nuclei held together by a covalent bond

BONDING DISTANCE: the center-to-center distance to which two atoms must be drawn together to obtain bonding by overlap of their orbitals, without bringing them so close together that powerful nuclear repulsion will destabilize the bond

COVALENT BOND: a bond between two atoms in which the atoms are held together by a shared electron pair

MOLECULAR ORBITAL: a mathematically defined volume, associated with two atoms, in which up to two electrons may exist to form a bond between the atoms

ORBITAL OVERLAP: as two atoms are brought to bonding distance, orbitals from each atom interpenetrate each other's volumes, or overlap; this overlap volume becomes the molecular orbital that contains the bonding electrons

PI BOND: a two-lobed bond formed by overlap of both lobes of two p orbitals on adjacent atoms; the bond exists entirely outside of the axis between the atoms

SHELL: properly, a set of atomic orbitals defined by a single principal quantum number; loosely, a set of orbitals at about the same distance from the nucleus; the outermost shell of an atom contains the orbitals that enter into chemical bonds

SIGMA BOND: a bond formed by overlap of orbitals along the axis between two atomic nuclei; the orbitals that enter into a σ-bond may be of any type, s or hybrid

Bibliography

Brown, Theodore L., H. Eugene LeMay, Jr., and Bruce E. Bursten. CHEMISTRY: THE CENTRAL SCIENCE. 5th ed. Englewood Cliffs, N.J.: Prentice-Hall, 1991. Chapter 9 on molecular geometry contains a thorough discussion; chapter 6 on electronic structure of atoms is less so, but adequate.

Fessenden, Ralph J., and Joan S. Fessenden. ORGANIC CHEMISTRY. 4th ed. Pacific Grove, Calif.: Brooks/Cole, 1990. A discussion of the hybrid orbitals needed to understand organic chemistry is presented in the first two chapters.

Morrison, R. T., and R. N. Boyd. ORGANIC CHEMISTRY. 3d ed. Boston: Allyn & Bacon, 1973. A standard text in the field. The material on hybrid orbitals is presented in the chapters where it applies: sp³ with methane and other alkanes; sp2 with alkenes; sp with alkynes. Provides many immediate examples of the working out of orbital principles.

Peters, Edward I. INTRODUCTION TO CHEMICAL PRINCIPLES. 5th ed. Philadelphia: Saunders, 1990. Good diagrams; sound explanations.

Radel, Stanley R., and Marjorie H. Navidi. CHEMISTRY. St. Paul, Minn.: West, 1990. Chapters 7 and 8 contain a good discussion of atomic electron structure, and chapter 10 has an excellent discussion on molecular geometry.

Ryschkewitsch, George E. CHEMICAL BONDING AND THE GEOMETRY OF MOLECULES. New York: Reinhold, 1963. From a series, "Selected Topics in Modern Chemistry." Although dated, the series consists of short volumes addressing limited topics in chemistry, very thoroughly, but at a level demanding no more than a high school chemistry background. Contains a complete discussion of molecular geometry.

Stryer, Lubert. BIOCHEMISTRY. 3d ed. New York: W. H. Freeman, 1988. Although this book has a limited discussion on orbitals, superb detailed explanations are given of structure of proteins in chapter 2 and of DNA/RNA in chapter 4.

Zumdahl, Steven S. CHEMISTRY. Lexington, Mass.: D. C. Heath, 1986. A thorough general chemistry text. Relevant material is found in chapters 7 to 9.

Atomic orbitals

Interaction of two hydrogen atoms

Common hybrid (molecular) orbitals

Quantum Mechanics of Chemical Bonding

Calculations of Molecular Structure

Quantum Mechanics of Molecules

X-Ray Determination of Molecular Structure