Knowledge Bank

“In principle, molecular force constants for a given electronic state are determined by the electronic charge distribution $\Psi \ast \Psi$, , where $\Psi$ is the solution of the electronic wave equation. Expressions for the force constants in terms of the electronic charge distribution are derived which differ depending upon whether (1) $\Psi$ is an exact wave function which must necessarily satisfy both the variational principle and the virial theorem, (2) $\Psi$ is an approximate wave function which satisfies the variational principle but \emph{not} the virial theorem. For a diatomic molecule, the force constant (for the case of exact wave functions) is given by, [ K = -\Sigma_{i} \frac{\partial \bar{F}}{\partial a_{i}} \frac{\partial a_{i}}{\partial R} ] where $F¯$ is the quantum mechanical force, or -$F¯$ the expectation value over the electronic wave function of the gradient of the potential (Hellmann-Feynman theorem), $\alpha i$ are the variational parameters of the electronic wave function, R is the internuclear distance, and $R0$ is the equilibrium distance. A side result of the procedure employed is to emphasize the logical relations between the variational, virial, and Hellmann-Feynman theorems. Some numerical results are presented, and some problems suggested by these calculations are discussed. For example, the scheme permits the investigation of the role of electron correlation in molecular vibrations as well as the relative contribution of $\sigma$ - and $\Pi$-electrons to the value of the force constants in the case of polyatomic molecules.”