Quantum-mechanical derivation of angular and torsional forces in well-bonded systems

A quantum-mechanically based method of deriving real-space interatomic potential functions for covalently bonded systems, without broken bonds, is developed. The method focuses on the one-electron energy, which is modeled via a tight-binding Hamiltonian. The potentials are derived via a general formalism based on perturbation theory, using a starting state in which the electrons reside in bond orbitals. The perturbing terms correspond to overlap and Hamiltonian couplings between the bond orbitals and with other occupied and unoccupied states. The interactions are given in terms of simple trigonometric functions, and the parameters of the quantum-mechanical Hamiltonian. A major contribution to the angular forces comes from the overlap between occupied bonding orbitals. Examples are given for model Hamiltonians relevant to phosphorus, carbon, sulfur, and the ethane molecule. The functional forms of the derived potentials are generally similar in form to those assumed in simulations. However, the actual appearance of the potentials is sometimes quite different from that obtained by an empirical fitting to molecular properties. In addition, it is found that the “improper” torsion terms that are often included in polymer simulations can be replaced by angular terms that are much more physically transparent.