Quantum Chemistry for Atoms, Molecules and Solids: Virial Theorem for Diatomic Molecules

equilibrium separation

(Å)

exponential parameter a

-8

-6

-4

-2

R (Å)

energy E(R) (eV)

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For both classical and quantum systems with any number of particles interacting by inverse-square forces ( potentials)—electrostatic or gravitational—the average potential and kinetic energies obey the virial theorem. This states that the average of the potential energy equals the negative of half the average of the kinetic energy: . Since the total energy is given by , , while In classical mechanics, and pertain to the time averages of these quantities per period of the motion, while in quantum mechanics they mean the expectation values of the potential and kinetic energy operators. For example, in the ground state of the hydrogen atom, with eV, we have eV and eV.