Calculations of molecular structure

Type of physical science: Chemistry

Field of study: Chemistry of molecules: general theory of molecular structure

The structure of a molecule determines its chemical reactivity. Calculations of molecular structure utilizing the techniques of quantum mechanics are extremely valuable for understanding chemical reactivity and the nature of chemical bonds.

Overview

Calculations of molecular structure based on quantum mechanics yield great insight into the nature of chemical reactivity. Such computations are performed so that the structures and energies of all the participants in a chemical reaction can be determined.

For example, in the reaction between hydrogen gas and oxygen gas to produce water, the structures of the reactant gases, the structure of the product water molecule, and the relative energies of all these structures can be calculated with quantum mechanical methods. In addition, quantum mechanics allows for the calculation of the structure and energy of the hydrogen-oxygen complex throughout the course of the reaction. This procedure yields the activation energy of the reaction, the highest energy required to reach the transition state structure of the hydrogen-oxygen complex. Once the activation energy is supplied to the reactants and they have formed the high-energy transition state structure, then products are created.

All quantum mechanical calculations are based on solving Schrodinger's equation.

Solving Schrodinger's equation for the motion of electrons about the nuclei in a molecule is simplified by using the Born-Oppenheimer approximation. The Born-Oppenheimer approximation states that since electrons are much smaller than nuclei, they move much faster than nuclei. Therefore, for any particular position of nuclei in a molecule, electrons will instantaneously rearrange to new positions. Thus, the nuclei can be considered fixed in one position and the Schrodinger equation can be solved. The Born-Oppenheimer approximation allows for the calculation of the molecular structure of reacting molecules; at any particular nuclear configuration, Schrodinger's equation is solved for the movement of electrons.

In principle, using the Born-Oppenheimer approximation allows for the electronic solution of the Schrodinger equation. Unfortunately, the Schrodinger equation can be solved exactly only for the hydrogen molecular ion, two hydrogen atoms sharing one electron, having an overall molecular charge of plus one. Further addition of nuclei or electrons creates a molecule for which an exact solution of the Schrodinger equation is impossible.

Fortunately, there are methods that allow the Schrodinger equation to be solved after simplifications are made to the molecular system. The solutions based on initial approximations of the molecular system always have an energy higher than the true energy for the molecular system. This last statement is known as the variation principle. It may seem that such solutions are not very valuable. In fact, however, solutions based on approximations to the Schrodinger equation are very close to the true value, so that the calculation can replace experimentation. This is particularly true for small molecular systems.

Two different ways to obtain useful solutions to the Schrodinger equation are ab initio and semiempirical methods. With ab initio methods, simplifications are made to the actual molecular system, resulting in a model that describes how the electrons interact in the molecule of interest. Then a calculation is performed to solve the Schrodinger equation exactly, without further approximations or input from experiment. Thus, in ab initio calculations, the approximation lies in the choice of the model. The term ab initio does not necessarily imply that the calculations are accurate, as the choice of the model affects the validity of the results. An inappropriate choice of model will yield a result that is inaccurate and chemically misleading. The calculation of chemically accurate ab initio molecular structure requires large amounts of supercomputing time, and even then only small molecular systems can be studied thoroughly. For this reason, much effort has been devoted to devising computationally simpler methods for calculating molecular structures and energies. Many of these methods incorporate experimental information and are called semiempirical methods.

The semiempirical methods were developed in the knowledge that it is extremely difficult to obtain molecular structures and energies that are chemically accurate. Therefore, to achieve a high level of accuracy, actual experimental data are incorporated into the method.

Semiempirical methods are developed by taking the exact equations of ab initio methods and making approximations so that the computation time is greatly reduced. Accuracy is achieved by making use of experimental data to parameterize the method. It is the extensive parameterization that assures chemical accuracy, while the approximations to the original ab initio equations allow for efficient calculations.

Both ab initio and semiempirical methods are based on the molecular orbital method. The molecular solutions to the Schrodinger equation are based on the Hartree-Fock Self-Consistent-Field (HF-SCF) method. The HF-SCF method was developed for atoms and molecules with many electrons. In a molecule with many electrons, the electrons are strongly attracted to the nuclei but repel one another. This electron-electron repulsion is the reason that the Schrodinger equation cannot be solved exactly unless a simplified model is introduced. In the HF-SCF method, the electron-electron repulsion terms are ignored and each electron is considered to interact with the average electrical field resulting from all the electrons. The Schrodinger equation can then be solved exactly (using the variation method), resulting in an improved average electrical charge distribution for the electrons. This process is repeated until the lowest energy reached no longer changes, that is, until the solution becomes self-consistent.

Ab initio and semiempirical calculations are used to determine the geometries of the equilibrium structures of molecules and molecular ions. These quantum mechanical methods can also be used to follow the course of a chemical reaction and to determine the transition state structure for a given chemical reaction. The structures and energies calculated by high-level ab initio and semiempirical methods help describe the chemistry of the molecular system.

Indeed, the dissociation energy, that is, the energy required to break a molecule into its component atoms, is easy to determine using these methods. High-speed supercomputers are essential for accurate, high-level ab initio computations. The best semiempirical methods are thousands of times faster than the corresponding ab initio methods, and these programs can be run on microcomputers. Large biological molecules are beyond the scope of semiempirical calculations, even on supercomputers, but this will change as computer technology improves.

Applications

The hydrogen molecule, two hydrogen atoms joined together sharing two electrons, is historically significant: It was the first molecule for which the dissociation energy was correctly predicted by quantum mechanical calculations. In 1968, two Polish physicists working at the University of Chicago, W. Kolos and L. Wolniewicz, reported their work in the JOURNAL OF CHEMICAL PHYSICS (1968). Kolos and Wolniewicz were working in the Chicago facilities of Professor Robert S. Mulliken and Professor C. C. J. Roothaan when they did these calculations.

By invoking the Born-Oppenheimer approximation, the two scientists were able to calculate the energy for the hydrogen molecule system in a systematic way. Kolos and Wolniewicz followed the procedure of fixing the hydrogen-hydrogen bond distance and then solving the Schrodinger equation using ab initio methods. They continued this procedure at greater and greater bond lengths. By plotting the solved energy of the hydrogen molecule system versus increasing bond distance, Kolos and Wolniewicz generated a potential energy curve for the dissociation of the hydrogen molecule. Potential energy curves provide a wealth of information, including the energy at which the hydrogen molecule dissociates into two individual hydrogen atoms. This energy, the dissociation energy for the hydrogen molecule, was calculated to be 432.072 kilojoules per mole. This energy was 0.045 kilojoule per mole lower than the best experimental energy in the literature, in violation of the variation principle, which states that the calculated energy must always be greater than the true value. Since the true value is determined directly by experiment, a discrepancy exists between the calculations of Kolos and Wolniewicz and the experimental energy measurement. The work of the Polish scientists in the famous laboratories of Mulliken and Roothaan stimulated the remeasurement of the experimental dissociation energy of the hydrogen molecule.

In 1969, the great experimentalist Gerhard Herzberg, working in Canada, remeasured the dissociation energy of the hydrogen molecule with a more sensitive experiment. This new high-resolution work showed that the dissociation energy for hydrogen was greater than 432.059 kilojoules per mole and lower than 432.083 kilojoules per mole, in excellent agreement with the ab initio calculations of Kolos and Wolniewicz.

One of the goals of theoretical quantum chemistry is to provide predictions of molecular properties that can be tested by experiment. The example of the dissociation energy of hydrogen discussed above is a classic case of theory guiding experimentation. Another interesting example is the case of the methylene radical, which is a highly reactive molecule containing two hydrogen atoms bound to one carbon atom. The methylene radical, while not present in appreciable amounts in the lower atmosphere on earth, is an important constituent in combustion of hydrocarbon fuels in the upper atmosphere, in stars, and in interstellar space.

The fundamental question concerning the methylene radical is whether it is a linear molecule, with the hydrogen-carbon-hydrogen bond angle equal to 180 degrees, or a bent molecule, with the hydrogen-carbon-hydrogen bond angle less than 180 degrees. In 1960, J. M. Foster and S. F. Boys reported ab initio calculations for the methylene radical. Their results predicted a hydrogen-carbon-hydrogen bond angle of 129 degrees. This prediction was followed by an experimental determination by Herzberg, who came to the conclusion that the methylene radical is linear, in contrast to the results of Foster and Boys. In 1970, C. F. Bender and Henry F. Schaefer III used a more rigorous model in their ab initio calculations to predict a bent molecule with a bond angle of 135.1 degrees. This research stimulated a new type of experiment, which showed that the methylene radical was indeed bent, with a bond angle of 137.7 degrees. In 1972, D. R. McLaughlin, Bender, and Schaefer performed a higher level ab initio calculation which predicted a bond angle of 134.0 degrees. This result is in excellent agreement with the latest experimental value of 133.8 degrees obtained by laser spectroscopy.

Development of ab initio computer packages that can be used by chemists interested in utilizing quantum mechanics to understand the chemistry of molecules was a major focus of Professor John A. Pople and his coworkers at Carnegie Mellon University in Pittsburgh, Pennsylvania. Pople, the John C. Warner University Professor of Natural Science at Carnegie Mellon University, was a pioneer in work on molecular orbital theory. He is one of the most frequently cited theoretical chemists in the world (the other being Linus Pauling), and he has made many significant contributions to both ab initio and semiempirical methods. His group in Pittsburgh had a goal of developing a program that could be applied to molecular systems of all sizes, with the ab initio model selected according to the type of computer available. This flexibility is necessary since a chemist at one institution may have access to a supercomputer while a chemist at another institution may have access only to a microcomputer.

The name of their ab initio computer package is GAUSSIAN. To use GAUSSIAN, a chemist selects an ab initio model that can perform the calculation subject to the limits imposed by the available computer. The results of the model can then be compared with experimental data. If the agreement is good, the model can then be used to make predictions for other systems. A major problem with GAUSSIAN is that it is easy to perform calculations with a low-level model which then lead to poor results. It is the responsibility of users of the program to ensure that the model they have selected is adequate to describe the molecular system. Ab initio calculations are often worthless because the user of a computer software package like GAUSSIAN fails to justify the ab initio model on scientific grounds, simply picking a model based on the size of the available computer. If the model is not evaluated against experimental data, the calculations are useless. Thus, it is important to justify the model used in an ab initio calculation for a particular molecular system. When performed correctly, as in the examples of the hydrogen molecule and the methylene radical, ab initio calculations guide and interpret experimental results.

The other quantum mechanics-based approach to calculations uses semiempirical methods. These procedures are much faster than ab initio methods because a low-level ab initio model is selected and then parameterized using experimental results. Michael J. S. Dewar and his coworkers at the University of Texas, Austin, pioneered the use of highly accurate semiempirical methods. They developed three models that have been widely used by the chemical community: MINDO/3, MNDO, and AM1. In 1989, a colleague of Dewar, James J. P. Stewart (Air Force Academy), developed the PM3 model. Stewart wrote a general molecular orbital package for the chemistry community known as MOPAC. MOPAC contains the MINDO/3, MNDO, AM1, and PM3 semiempirical methods.

Because of the high speed of the procedures available in MOPAC, they have been used to provide guidance for a variety of experimental projects. For example, chemists have long been interested in forming ringlike structures of carbon compounds that contain double and triple bonds. Richard O. Angus, Jr., and Richard P. Johnson, working in the Department of Chemistry at Iowa State University, used MNDO calculations in 1983 in order to explore the possibility of incorporating a 1,2,3-butatriene moiety in various-sized cyclic carbon rings. The 1,2,3-butatriene moiety consists of four carbon atoms bonded together with three double bonds and a hydrogen attached to the first and last carbon. A ringlike structure of carbon atoms containing 1,2,3-butatriene would have a variable number of carbon atoms connecting the first carbon in 1,2,3-butatriene with the fourth carbon, thus completing the ring. Angus and Johnson used the MNDO semiempirical method to calculate the geometries for cyclobutatrienes with five to nine carbons in the ring. The calculations suggested that the 1,2,3-butatriene moiety, which was expected to be constrained to a linear geometry by the three double bonds, is readily bent and should remain intact in essentially any ring size. Armed with this knowledge, Angus and Johnson then completed the first synthesis of the nine-member ring 1,2,3-cyclononatriene. They reported their work in the JOURNAL OF ORGANIC CHEMISTRY (1984).

Context

Calculations of molecular structure have been made possible by the development of quantum mechanics and the advent of high-speed computers. Toward the close of the nineteenth century, many scientists thought that physics was completely understood. Yet experiments were setting the stage for the most profound revolution in the history of physics. This revolution in physics reinvigorated the discipline of physical chemistry.

The spectral shape of black body radiation, the photoelectric effect, the Compton effect, line spectra of atoms, and the wave-particle duality properties of light were all naturally occurring phenomena that could not be explained by classical physics. These experiments led to the conclusion that the energies of microscopic particles such as electrons were quantized, with only certain discrete energy levels allowed for a given microscopic system. In 1905, Albert Einstein, then a low-paid patent clerk in Switzerland, postulated that light itself is quantized. His model was proved correct by the American physicist Robert Millikan in 1916. In 1924, the French physicist Louis de Broglie suggested that just as light has particle-like properties as shown by Einstein and Millikan, so too could microscopic particles such as electrons and molecules have wavelike particles. Experimental verification of de Broglie's hypothesis was first provided in 1927 by the American physicists Clinton J. Davisson and Lester H. Gerner at Bell Laboratories.

The fact that microscopic particles display wavelike behavior suggests the existence of a wave equation. Such a wave equation was first proposed by the Austrian physicist Erwin Schrodinger in 1926. With the Schrodinger equation, physicists and chemists could, in principle, calculate the allowed energy levels for any microscopic particle imaginable, whether it be an electron, an atom, or a molecule. Chemists needed to wait for the development of computers before calculations of molecular structure became feasible.

Rapid developments in both computer technology and experimentation in the 1960's, 1970's, and 1980's were highly beneficial for molecular structure calculations. High-resolution spectroscopy experiments have provided very good data that chemists try to match with ab initio calculations. The accuracy of semiempirical methods is directly dependent on the degree of reliability of the experimental values used in the parameterization of these methods.

The future is promising for calculations of molecular structure. Continued improvements in the design of computers and in quantum mechanical software will allow larger molecules to be investigated with high-level ab initio methods. Improvements in the accuracy of experimental values will result in better semiempirical methods. Calculations of small biological molecules will be made possible by these advances.

Principal terms

AB INITIO METHOD: a computational procedure utilizing an approximate model instead of a complete description of the molecular system; the Schrodinger equation is then solved without further approximations

BORN-OPPENHEIMER APPROXIMATION: a principle stating that the nuclei in a molecule are fixed so that the Schrodinger equation can be solved for the motion of the electrons about the nuclei; used to calculate potential energy curves

DISSOCIATION ENERGY: the energy required to break a molecule into smaller molecular fragments or atoms

ELECTRON: a small negative charge that is attracted to positively charged protons in a nucleus; electrons are responsible for chemical bonds

MOLECULE: the smallest particle of a substance that retains the properties of the substance and is composed of atoms

NUCLEI: composed of positively charged protons and neutral neutrons; nuclei are much heavier and slower than electrons and attract electrons to their positive charge

POTENTIAL ENERGY CURVE: a diagram illustrating how the energy of a molecule changes as the distance between two atoms in a molecule changes

QUANTUM MECHANICS: a general mathematical theory dealing with the interactions of matter and radiation

SCHRODINGER EQUATION: a mathematical construct formulated by Erwin Schrodinger as the basis of quantum mechanics; solutions to the Schrodinger equation reveal all the possible information about a molecular system

SEMIEMPIRICAL METHOD: a computational procedure in which approximations are introduced to the exact equations of ab initio methods; experimental parameters are added to ensure high accuracy; this method is much faster than ab initio methods

TRANSITION STATE: a unique point on the potential energy curve; addition of a very small amount of energy pushes the transition state structure to yield products

Bibliography

Condon, E. U. "Sixty Years of Quantum Physics." PHYSICS T 17 (October, 1962): 37. An interesting historical account of the development of quantum mechanics. Written by one of the pioneering contributors to quantum theory.

Feynman, Richard P. QED. Princeton, N.J.: Princeton University Press, 1985. Explains in a nonmathematical way the interaction of light and electrons. QED stands for quantum electrodynamics, and Feynman has written this book specifically for a nontechnical audience. Masterfully written.

Offenhatz, Peter. ATOMIC AND MOLECULAR THEORY. New York: McGraw-Hill, 1970. Although dated, it is a good, nontechnical account of molecular theory for college undergraduates.

Pagels, Heinz R. THE COSMIC CODE. New York: Simon & Schuster, 1982. An entertaining book written for a nontechnical audience covering the history of quantum mechanics from a physicist's viewpoint. The first part of the book, describing the development of the quantum theory of the atom, is especially recommended.

Polkinghorne, J. C. THE QUANTUM WORLD. Princeton, N.J.: Princeton University Press, 1984. A short book written by a mathematical physicist. This book contains a few equations and is slightly technical. Well worth the effort.

Rae, Alastair I. M. QUANTUM PHYSICS: ILLUSION OR REALITY? New York: Cambridge University Press, 1986. This short, paperback book provides an explanation of the conceptual problems of quantum physics without requiring the reader to be trained in math or physics. Many diagrams and a few equations. Similar to Polkinghorne's book in level of approach.

Zukav, Gary. THE DANCING WU LI MASTERS. New York: William Morrow, 1979. This book, written by a layperson, contains the whole evolution of quantum mechanics from its beginning through the late 1970's. A very philosophical approach. Winner of the American Book Award in 1980.

Chemical Bond Angles and Lengths

Quantum Mechanics of Chemical Bonding

Chemical Reactions and Collisions

Quantum Mechanics of Molecules

Photon Interactions with Molecules