Simple molecular orbital treatment of diatomic force constants

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Abstract
From the molecular orbital theory for diatomic molecules, the virial theorem, and the Heisenberg equation of motion, expressions are derived for off‐diagonal matrix elements of the one‐electron Hamiltonian for a limited basis set of atomic orbitals that display exact zero differential diatomic overlap. The resulting expressions are combined to yield analytic equations for the force constants ke and higher derivatives of the diatomic potential energy surface. These expressions depend only on the density matrix of a single calculation near enough to the equilibrium geometry. Under the approximation that all nonvanishing Coulomb interactions are given by classical electrostatics, a simple formula for the force constant is obtained which has mean square error of 3% for a number of first‐ and second‐row diatomic molecules. If reasonable approximations are made for the density matrix and the equilibrium separation Re is estimated by a ’’tangent sphere’’ model, the following simple equations are obtained: keR3e=η/2(1+nAζB+nBζA) (two atoms with an s,p basis), keR3e=η/2[1+nbζa +1/2ζb(na+ζa)] (hydrides), keR3e=η/2[1+1/2(nAζB+nbζA)] (hydrogen molecule). Here, η is the number of bonds, n the principal quantum number, and ζ the orbital exponent of the valence electrons as, for example, given by Slater’s rules. These equations yield predictions generally within ±10% of the experimental values.