A Misapplication of the Virial Theorem

Frank Rioux
Department of Chemistry
Saint John's University
College of Saint Benedict


When asked what motivated the creation of his model of the atom Bohr replied "the stability of matter, a pure miracle when considered from the standpoint of classical physics." Kinetic energy is not, as Carlton claims, redundant at the atomic level. It is essential for an understanding of atomic stability and electronic structure. The virial theorem does not say kinetic energy is redundant anymore than it says that potential energy is redundant.

For example, the misapplication of the virial theorem has stood in the way of a clear understanding of the nature of the chemical bond. More than 35 years ago Ruedenberg demonstrated "the virial theorem notwithstanding" (1) that kinetic energy played an essential role in chemical bond formation. In other words chemical bonding is not simply an electrostatic phenomenon as many have been led to believe on the basis of the virial theorem. Twenty-five years ago in a review of Ruedenberg's model of the chemical bond, Kutzelnigg (2) commented incisively on the misuse of the virial theorem in interpreting the chemical bond (see following letter by DeKock).

Furthermore, the jurisdiction of the virial theorem is not such as to prohibit the partitioning of a process (bond formation or ionization, for example) into a sequence of steps for interpretive purposes. Two recent publications (3,4) show that using Ruedenberg's approach, chemical bond formation can be interpreted simply as a two step process consisting of charge delocalization followed by orbital contraction. Our quantum mechanical analysis of the H/He ionization energy trend (5) runs parallel to this analysis of the chemical bond. In our approach we factor ionization into two steps: frozen ionization followed by relaxation. Our wavefunction and our calculation satisfy the virial theorem. Just as Ruedenberg's goal was to develop a mechanism for chemical bond formation, our goal was to develop a quantum mechanical mechanism that would reveal the essential features of the ionzation process.

In summary, the uncritical adherence to the virial theorem has prevented generations of chemists from recognizing the importance of kinetic energy at the atomic and molecular level. It is because of this potential to mislead, rather than inform, that we chose not to explicitly invoke the virial theorem in our analysis. The virial theorem serves as a criterion for acceptable wavefunctions and calculations, but one has to go beyond it to interpret phenomena such as chemical bond formation and atomic ionization.

Literature cited:

  1. Ruedenberg, K. Rev. Mod. Phys. 1962, 34, 326.
  2. Kutzelnigg, W. Angew. Chem. Int. Ed. Eng. 1973, 12, 546.
  3. Rioux, F. The Chemical Educator, 1997, Volume 2, Number 6.
  4. Backsay, G.B.; Reimers, J.R.; Nordholm, S.J. Chem. Educ. 1997, 74, 1494.
  5. Rioux, F.; DeKock, R.J. Chem. Educ. 1998, 75, 537.

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