Book explains how Schrödinger equation might be solved

Quantum mechanics theorists Paul Dirac (left) and Richard Feynmann at the International Conference on Relativistic Theories of Gravitation, Warsaw, Poland, July 1962. The photograph was taken by A. John Coleman, one of more than 20 contributors to a new book, Reduced-Density-Matrix Mechanics with Application to Many-electron Atoms and Molecules, edited by University of Chicago chemist David Mazziotti. Coleman published an early landmark paper on the book’s subject in Reviews of Modern Physics in 1963. The photograph appeared on the cover of Physics Today following the conference and will be reprinted in Mazziotti’s book.

David Mazziotti, Assistant Professor in Chemistry and the College, has edited a new book that addresses a major challenge in computational chemistry and physics.

The book develops a new paradigm for calculating the distribution of electrons in many-electron molecules and molecular processes.

For decades, chemists have sought to better understand the behavior of electrons in atoms and molecules. Understanding this behavior is a key ingredient in predicting chemical properties and reactions that govern everything from the efficiency of combustion to the design of new medicines.

A significant hurdle to describing the electrons in atoms and molecules is their sheer number—tens to hundreds or even thousands—in a given molecule. This is far too many electrons for their distribution in the molecule to be determined exactly, even with modern supercomputers, for the computational cost of describing the electrons grows exponentially with their number.

Mazziotti’s book, Reduced-Density-Matrix Mechanics with Application to Many-Electron Atoms and Molecules, details recent advances that are realizing a 50-year dream of using only two electrons to represent all the electrons in a molecular system.

“That quest has been a ‘holy grail’ of theoretical chemistry for more than 50 years,” said Nobel laureate Dudley Herschbach, the Frank Baird Jr. professor of science at Harvard University and Mazzioti’s graduate mentor.

The book will be published this month by John Wiley and Sons as Volume 134 in its Advances in Chemical Physics Series. The Editor-in-Chief of the series is Stuart Rice, the Frank P. Hixon Distinguished Service Professor Emeritus in Chemistry and the College. Mazziotti invited an international group of more than twenty scientists to contribute chapters to the book, which summarizes advances, both historical and recent.

The book includes a discussion of a major theoretical advance that Mazziotti published in the Oct. 6, 2006, issue of Physical Review Letters. Herschbach described the work as “an innovative approach that brings him much closer to the holy grail than anyone has managed to get before.”

The work solves for the energies and properties of many-electron atoms and molecules with only two electrons.

If it were applicable to football, Herschbach explained, Mazziotti’s highly complex work would mean that the Chicago Bears offense could operate with just a quarterback and one running back/pass receiver, with auxiliary help from just one lineman. Plotting the roles of just three players would automatically determine the actions of the entire team. “His method requires dealing with just pairs and trios of electrons,” Herschbach said.

Mazziotti’s work has its roots in the Schrödinger equation. Formulated in 1926 by Erwin Schrödinger, it is the primary equation of quantum mechanics. “While the equation determines the behavior of all the electrons in an atom or molecule, the computational cost of its solution increases exponentially with the number of electrons,” Mazziotti explained.

In 1976, scientists produced a contracted Schrödinger equation that depended upon only four of a molecule’s electrons. A solution of the contracted Schrödinger equation remained elusive until 1993, when Carmela Valdemoro at the Consejo Superior de Investigaciones Cientificas in Spain recognized that the behavior of the remaining four electrons could be expressed in terms of just two. “The first successful calculations were her calculations,” Mazziotti said.

As a graduate student in the late 1990s, Mazziotti verified and extended Valdemoro’s work, as did Kyoto University’s Hiroshi Nakatsuji. Accuracy, however, remained an issue until Mazziotti’s current advance. Results that formerly ranged from 71 to 96 percent accurate have jumped to 95 to 100 percent. The contracted Schrödinger equation may soon become solvable with a package of computer software, according to Mazziotti.

Theoretical and computational development of the two-electron approach to many-electron molecules during the last eight years has been particularly rapid when compared with one of the most popular methods, known as density functional theory. While density function theory was proposed by Pierre Hohenberg and Walter Kohn in Physical Review in 1964, it took thirty to forty years for practical implementations to be realized.

Potential applications of the two-electron approach include the study of chemical reactions that govern a wide variety of systems and processes from semiconductors and solar energy to the design of new medicines.