Quantum Mechanics Of Molecules

• Type of physical science: Chemistry
• Field of study: Chemistry of molecules: general theory of molecular structure

The methods of quantum mechanics can provide a definitive account of the structure of molecules, including the nature of chemical bonding, molecular geometry, and spectroscopic properties.

Overview

A molecule is a combination of atoms bonded together to form a single chemical entity.

Whereas an atom is the smallest unit of an element, a molecule is the smallest unit of a compound. The atoms of the noble gases, such as helium, neon, and argon, are stable in their uncombined form and are usually designated as monatomic molecules. Molecules containing two or three atoms are termed "diatomic" and "triatomic," respectively. Those with three or more atoms can be described as polyatomic. The chemical formula for a molecule is a composite of the element symbols, with subscripts indicating more than one atom of an element. For example, water is the triatomic molecule H₂O, indicating two atoms of hydrogen combined with one atom of oxygen. Gaseous oxygen and nitrogen in the atmosphere occur in the form of the diatomic molecules O₂ and N₂. The molecular formula for ethyl alcohol is C₂H₆O, where C is carbon. An alternative way of writing this formula that represents more closely the actual grouping of atoms is CH₃CH₂OH. In fact, the compound dimethyl ether contains the same atoms connected in a different order, namely, CH₃OCH₃.

From another perspective, a molecule can be regarded as a collection of nuclei and electrons. This viewpoint is particularly useful for theoretical purposes. A proton, the nucleus of a hydrogen atom, has a mass about 1,836 times greater than that of an electron. Other nuclei will outweigh the electron by even greater multiples of this ratio. This mass disparity between nuclei and electrons has profound consequences for the structure of matter. According to quantum mechanics, the electrons in an atom or molecule can be viewed as a cloud of negative charge, in which their individual identities are subsumed. If nuclei had masses closer to those of electrons, a molecule would correspondingly consist of what can be described as a "soup" of nuclei and electrons, with none of the rigidity implied by chemical structures. Life would be impossible, since, at the very least, DNA (deoxyribonucleic acid) molecules could not carry a genetic code based on a stable sequence of units. (Incidentally, if protons were much heavier, they would not be able to take part in the hydrogen bonds needed for transcription of the genetic code.)

Because they are so much lighter, the electrons in a molecule can move at much greater speeds than the nuclei. This is the basis of the Born-Oppenheimer approximation, proposed by Max Born and J. Robert Oppenheimer in 1927. From the viewpoint of the electrons, the nuclei appear to be standing still, like the tortoise to the hare. From the viewpoint of each nucleus, the cloud of whirling electrons also appears to be stationary, like the spinning blades of an electric fan. As a consequence of the Born-Oppenheimer approximation, the quantum mechanical problem of molecular structure can be reduced to two somewhat more tractable problems, one treating the motion of the electrons and the other the motion of the nuclei. The electron problem proceeds from a specified geometric arrangement of the nuclei. In principle, the electronic calculations can be repeated for different molecular geometries to determine the most stable one.

Typically, distances between neighboring nuclei at equilibrium are between 1 and 2 angstrom units and provide a convenient "yardstick" for distances on the atomic scale. The most stable (lowest-energy) electronic state of a molecule is known as the "ground state." Higher-energy states, called "excited states," are also possible, each with its own equilibrium molecular geometry. Transitions among electronic states of molecules typically give rise to spectra in the visible and ultraviolet regions.

The motion of the nuclei within the Born-Oppenheimer approximation is chiefly of two types. First, nuclei execute small vibrations around their equilibrium positions, with amplitudes in the range of 1 or 2 percent of the internuclear distances. Infrared and Raman spectra of molecules are produced by transitions among vibrational energy levels. Second, the molecule as a whole can rotate, more or less rigidly. If the molecules are in a gas, these rotations often can be detected by microwave spectroscopy.

The interactions that occur within molecules are exclusively of the electromagnetic type, which was anticipated as long ago as 1838 by Michael Faraday. To a good approximation, one may consider only electrostatic forces among the nuclei and electrons. Recall that electrons, having like negative charges, repel one another, as do nuclei, with their positive charges; however, nuclei and electrons, having opposite charges, attract one another. A theorem derived independently by Hans Hellmann in 1937 and Richard P. Feynman in 1939 shows that nuclei respond only to the average electrostatic forces produced by the electron cloud (in addition to the other nuclei). The Hellmann-Feynman theorem implies that, in an equilibrium molecular geometry, the net force on each nucleus will be exactly zero.

From the middle of the nineteenth century, a number of empirical generalizations were developed that brought some coherence to the vast body of experimental knowledge in chemistry. Notable among these was the periodic table of the elements formulated by Dmitry Ivanovich Mendeleyev, which uncovered recurring patterns in the chemical behavior of the elements. Another was the concept of linkage, or bonding, between atoms, which came to be represented by dashes. Later, it was recognized that molecules actually had a three-dimensional geometry; for example, the four atoms bonded to a carbon atom lie at the corners of a tetrahedron. Gilbert Newton Lewis proposed in 1916 that each bond in a molecule consists of a pair of electrons shared by two atoms. Moreover, the number of valence electrons associated with an atom, both shared and unshared, totals eight; this is known as the octet rule. For hydrogen and helium, the total is two, and for some atoms later in the periodic table, the total may exceed eight.

These and other chemical principles were explained by quantum mechanics in the few years following the introduction of the Schrodinger equation in 1926. The periodic structure of the elements is accounted for by the following facts. Each electron in an atom occupies an atomic orbital, which determines its energy and how its charge cloud is distributed in space. An electron possesses, in addition, an internal attribute known as its spin, with two possible orientations. By the Pauli exclusion principle, enunciated in 1925 by Wolfgang Pauli, no two electrons can occupy the same orbital with the same spin. Thus, as one goes down the periodic table, adding electrons as the atomic number is increased, a sequence of atomic orbitals will successively fill up; no orbital occupied by two electrons of opposite spin can accept any more electrons. Filled groups of orbitals form closed shells. The integers 2 and 8 encountered above are, in fact, the contents of the first two closed shells. Electrons outside of closed shells are called valence electrons; these determine the chemical behavior of the atom. For example, sodium (Na) and potassium (K) contain, respectively, eleven and nineteen electrons; however, each atom has only one valence electron, which accounts for the chemical similarity of the two elements.

Walter Heitler and Fritz London gave a quantum mechanical account of chemical bonding in 1927, specifically for the hydrogen molecule, H₂. They showed that when two hydrogen atoms come together, the two electrons lose their individual identities and can, with equal probability, be associated with the other atom. This is known as "exchange" and accounts for a major fraction of the binding energy of the molecule. This phenomenon has no analogue in classical physics. The fact that exchange requires two electrons very neatly explains the concept of the electron pair bond. The Heitler-London approach can be extended to describe more complex molecules, in what is known as valence bond theory. Linus Pauling has been its leading proponent and has applied the valence bond method with great success in many areas of structural chemistry.

An independent approach to the electronic structure of molecules is molecular orbital theory. Conceptually, molecular orbital theory perceives molecules as being made of nuclei and electrons. In contrast, valence bond theory sees molecules as being made out of atoms. In molecular orbital theory, the electronic structure of molecules is built up in the same way that it is for atoms. Nevertheless, instead of a single nucleus, one starts with the entire nuclear "skeleton" of the molecule. Electrons are then added successively, as in the atomic case, and one finds analogues of closed shells and valence electrons. The prototype system for the molecular orbital method is the hydrogen molecule ion H₂⁺. This species, which can be generated by an electric discharge through hydrogen gas, consists of two protons but only one electronhence the net positive charge of one unit. (Recall that ions are atoms or molecules with a net electric charge.) The Schrodinger equation for H₂⁺ was solved by Ø. Burrau in 1927. The two protons were assumed stationary, in accordance with the Born-Oppenheimer approximation. The fact that this problem could be solved was important, for it showed the validity of quantum mechanics for molecules as well as for atoms.

For more complicated molecules, the molecular orbitals are more generally obtained as a linear combination of atomic orbitals, or LCAO, approximation. For example, in an H₂⁺ molecule, in the vicinity of one of the protons, relatively far away from the other proton, the environment for an electron closely resembles that of a hydrogen atom. It stands to reason that the molecular orbital the electron occupies in the vicinity of this proton (and in the vicinity of the other proton) should closely resemble a hydrogen atomic orbital. A simple way to construct a molecular orbital with these properties is to take the sum of the two atomic orbitals, each centered on its proton. For molecular orbitals involving two nonidentical atomsfor example, hydrogen and chlorine (Cl)one would likewise add together the corresponding atomic orbitals, but in this case in unequal proportions. Such a sum is, in fact, denoted as a linear combination.

Generally, the more electronegative atom, in the above case the Cl, makes a larger contribution to the molecular orbital, and the electron density is accordingly greater in its vicinity.

Actually, the interaction of two atomic orbitals produces two molecular orbitals: a bonding molecular orbital and an antibonding molecular orbital. The bonding molecular orbital has a lower energy than either of the atomic orbitals and contributes to the stability of a chemical bond between the two atoms. The antibonding molecular orbital has a higher energy than either atomic orbital and favors the separation of the two atoms. When one electron is contributed by each atomic orbital, both can occupy the bonding molecular orbital, leaving the antibonding molecular orbital empty. In other cases, antibonding molecular orbitals can be occupied, but their effect is canceled by a greater number of occupied bonding molecular orbitals.

The molecular orbitals obtained by the LCAO procedure are distributed in the vicinity of only two nuclei. Other molecular orbitals are predominantly localized around a single nucleus and differ little from atomic orbitals; these correspond to either inner shell or lone pair electrons.

Inner shell electrons are from atomic closed shells and do not participate in chemical bonding.

Neither do lone pair electrons, even though they are from the valence shell; these correspond to the unshared electrons in Lewis' terminology. Another important type of molecular orbital is delocalized over more than two nuclei. The classic case is the benzene molecule, C₆H₆. The carbon atoms are known to form a six-membered ring in the shape of a regular hexagon, with each C atom bonded to one H atom.

All but six of the forty-two electrons in benzene occupy molecular orbitals of the one- and two-center types. Yet, the remaining six electrons occupy molecular orbitals completely delocalized over the benzene ring. Such delocalization imparts a special chemical stability to benzene and related organic compounds classed as aromatic.

The quantum mechanical calculations described are all based on approximation methods. The only problems that are exactly solvable are the hydrogen atom and the H2+ molecule (and then only within the Born-Oppenheimer approximation). Both of these are one-electron systems. Systems involving two electrons (and at least one nucleus) belong in the category of three-body problems, which cannot be solved exactly in either classical or quantum mechanics. Clearly, exact methods are out of the question, given the number of electrons in chemically interesting systems.

The variational theorem governs approximate solutions to quantum mechanical problems. Given an approximate wave function for the ground state of a system, one can calculate the corresponding approximation of the energy. The variational theorem states that this energy must be greater than the exact ground-state energy, which might be an experimentally determinable quantity. Suppose another scientist, using another approximate wave function, calculates a lower energy. Since the result must still be greater than the exact value, it must be closer to the exact value. Thus, this wave function can be rated as superior to the original one. A useful technique is to employ wave functions containing variable parameters. The parameters can subsequently be optimized so that they minimize the energy, thus giving the "best possible" wave functions of the form considered.

With the powerful computers available to quantum chemists in the twenty-first century, sophisticated computations are possible even for molecules approaching the size of proteins. When no empirical data are used, such computations are termed ab initio. For larger molecules, such methods might become too costly in computer time and programming effort. Semiempirical methods have been developed for such cases. Often, these are needed to answer questions of a chemical nature. In the early 2020s, Princeton University physicists managed to entangle individual molecules, meaning they chemically linked molecules in such a way that the state of one directly affects the state of another, no matter how far apart they are. This breakthrough could lead to advancements in quantum computing, simulations, and sensors. Advances in other technology have allowed other important discoveries, including a new class of quantum particles called fractional excitons.

Applications

The substantial progress in quantum mechanical computations in the late twentieth century has had a significant impact on practical aspects of chemistry, biology, biochemistry, pharmacology, and materials science. In the best possible case, the properties of a substance or the course of a chemical reaction might be predicted. At the very least, correlations can be made from related substances or reactions, on the basis of quantum mechanical principles, to forecast the likely outcome of an experiment. One of the most beneficial areas has been molecules of biochemical and pharmacological significance. Quantum mechanical computations are becoming a powerful tool for molecular design of new biologically active compounds. For example, from the computed electron distribution in pharmacologically active molecules, the active centers can sometimes be identified. This can suggest alternative structures of drugs with the same benefits but without undesirable side effects. Another active area of research concerns the dynamics of protein conformation using computer simulations of interatomic interactions extrapolated from the known behavior of smaller molecules. The goal of much of this work is to engineer new proteins with desired properties. Studies of the atomic geometry of enzyme-substrate and drug-receptor interactions continue adding to the understanding of the biochemical processes of life.

The carcinogenic (cancer-causing) nature of certain compounds was also suggested by theoretical analysis of their molecular structure.

Life entails a complex network of molecular processes. Quantum theory has helped scientists to understand some of the fundamental aspects of biology and genetics: for example, how the genetic code is carried by the base pairs in DNA molecules, how this code is passed from one strand of DNA to another, and how the code is eventually expressed in the synthesis of proteins and enzymes. Molecular theory has made vital contributions to understanding these questions.

Catalysts are substances that increase the speed of a chemical reaction without being used up themselves; more precisely, they do take part in the reaction but are regenerated at the end. Many catalysts contain transition metal atoms, atoms near the middle of the periodic table, including iron, manganese, nickel, and palladium. In the course of a catalyzed reaction, these metal atoms form temporary compounds with the main reactant and product species. In some cases, the structure of these reactant-catalyst intermediates can be understood, which may suggest a related catalyst that works faster or more cleanly. Probably half the products of the chemical industry depend at some stage on the action of catalysts, so this is a matter of immense economic significance. Increasing the efficiency of a process by 1 percent might result in an annual savings of millions of dollars. Two very important processes that depend on catalysis are the synthesis of ammonia and the cracking of petroleum. Ammonia, used to make fertilizer and many other industrial chemicals, is manufactured by the Haber process, by reacting nitrogen with hydrogen at high pressure over a metallic catalyst. Cracking converts crude oil into combustible fuels by breaking down large hydrocarbon molecules into smaller, more volatile ones. A related process, known as reforming, causes these molecules to rearrange into more highly branched chains, to improve antiknock properties of fuels. Again, an understanding of these catalytic processes on a molecular level has contributed to making these reactions more efficient.

"Designer materials" are substances that are synthesized specifically to have certain desired propertiesfor example, great mechanical strength, resistance to high temperature or corrosive environments, or high flexibility. Basic knowledge of the structure of atoms, molecules, and solids has now given materials scientists the capability of creating a growing number of such materials. These have made important contributions to the space program, to automotive technology, and even to the construction of light but rugged tennis rackets.

Quantum mechanics of molecules continues driving new technology in computing, healthcare, drug development, and space science. High-precision spectroscopy using quantum mechanics helps scientists study molecules in extreme conditions, like space exploration or fundamental physics experiments. Quantum sensors using entangled molecules can detect tiny changes in magnetic and electric fields, which is useful in medical technology like brain imaging as well as navigation.

Context

The germinal ideas of modern chemical theory can be traced back to John Dalton at the beginning of the nineteenth century. Dalton proposed that all matter was made up of atoms and that chemical reactions involved rearrangements of atoms. The term molecule was first introduced by Amedeo Avogadro, who also deduced that some of the common gases occur in nature as diatomic molecules, for example, O₂, N₂, H₂, and Cl₂. The work of Friedrich August Kekule and Archibald Scott Couper established that each carbon atom could form four bonds, including bonds to other carbon atoms to build up chains or rings. The concept of valence emerged, the valence of an atom being equal to the number of bonds that atom could form. For example, C, H, O, and N could be assigned the valences 4, 1, 2, and 3, respectively. Alexandr Mikhaylovich Butlerov introduced the concept of a structural formula to make explicit the bonds between the atoms in a molecule. This accounted for the difference between dimethyl ether and ethyl alcohol. In 1874, Jacobus Henricus van't Hoff and Joseph-Achille Le Bel extended the concept of structure to include the three-dimensional shape of a molecule. In particular, they proposed that a carbon atom can form four single bonds directed toward the corners of a tetrahedron.

During the nineteenth century, chemistry developed, for the most part, independently of physics. With the evolution of quantum theory in the twentieth century, however, all of chemistry became explicable, at least in principle, as applications of physical theory. In 1916, Lewis proposed the electron pair model of the chemical bond. This work was further elaborated in later years by Walther Kossel, Irving Langmuir, and Nevil Vincent Sidgwick into a comprehensive electronic theory of valence that retains its descriptive validity even today. The Schrodinger equation, has become the basis for all subsequent theoretical work on molecules. Heitler and London gave the first quantum mechanical account of the chemical bond. The quantum theory of valence was further developed in the 1930s by Linus Pauling, Henry Eyring, Erich Huckel, and others and applied to a wide variety of chemical problems. At the same time, Friedrich Hund, Robert S. Mulliken, J. E. Lennard-Jones, and others were developing the alternative approach to molecular electronic structure known as molecular orbital theory.

Mulliken first introduced the term "orbital" and exploited the molecular orbital method for the elucidation of both ground and excited states of molecules.

Beginning in the 1960s, electronic computers began to play an increasing role in molecular theory. Ab initio computations have become feasible for larger and larger molecules, up to significant fragments of proteins and nucleic acids. Using semiempirical methods, even the most complex biochemical processes are now within the scope of quantum chemistry.

Principal terms

ATOM: the fundamental unit of a chemical element

CHEMICAL BOND: a link holding two atoms in a molecule together

ELECTRON: a negatively charged elementary particle; electrons play a fundamental role in the structure of atoms and molecules

MOLECULE: a group of atoms bonded together to form a single chemical species

NUCLEUS: the central core of an atom, containing its positive charge and almost all its mass

ORBITAL: the spatial distribution of electrons in an atom or molecule

PERIODIC TABLE: systematic tabulation of the elements according to atomic number, showing regular patterns in chemical and physical properties

PROTON: the nucleus of the hydrogen atom, with one unit of positive charge

QUANTUM MECHANICS: the fundamental laws governing matter on the atomic scale

SCHRODINGER'S EQUATION: the fundamental equation of quantum mechanics for the wave function of a particle

SPECTRUM: a measured pattern of absorption or emission of radiation by a substance; yields information on the energy levels of atoms and molecules

Bibliography

Companion, Audrey L. Chemical Bonding. 2nd ed., McGraw-Hill, 1979.

Coulson, Charles A. The Shape and Structure of Molecules. Oxford UP, 1973.

Coulson, Charles A. Valence. 2nd ed., Oxford UP, 1961.

Garlinghouse, Tom. "Physicists ‘Entangle’ Individual Molecules for the First Time, Bringing About a New Platform for Quantum Science." Princeton University, 8 Dec. 2023, www.princeton.edu/news/2023/12/08/physicists-entangle-individual-molecules-first-time-hastening-possibilities-quantum. Accessed 13 Feb. 2025.

Gray, Harry B. Electrons and Chemical Bonding. Benjamin, 1964.

Levine, Ira N. Quantum Chemistry. 2nd ed., Allyn & Bacon, 1974.

Pauling, Linus. The Nature of the Chemical Bond. 3rd ed., Cornell UP, 1960.

Slater, John C. Quantum Theory of Molecules and Solids, vol. 1, McGraw-Hill, 1963.

Weidman, Jared D., et al. “Quantum Computing and Chemistry.” Cell Reports Physical Science, vol. 5, no. 9, 2024, doi:10.1016/j.xcrp.2024.102105. Accessed 13 Feb. 2025.