Binding of muonated hydrogen molecules and Born–Oppenheimer approximation revisited

The stability of four fermionic particles with unit charge, of which two are positively and two negatively charged, is discussed. Except for using the simplest approximation of a single Gaussian orbital per particle, the problem is exactly solved variationally by varying the masses to simulate molecular di-hydrogen, mono-muonated di-hydrogen, and di-muonated di-hydrogen. We illustrate the celebrated Born–Oppenheimer approximation 2 years after the occasion of its 90th anniversary. It is suggested that this method is valid only for di-hydrogen.

Three-center molecular integrals and derivatives using solid harmonic Gaussian orbital and Kohn–Sham potential basis sets

Three-center integrals over Gaussian orbital and Kohn–Sham (KS) basis sets are reviewed. An orbital basis function carries angular momentum about its atomic center. That angular momentum is created by solid harmonic differentiation with respect to the center of an s-type basis function. That differentiation can be brought outside any purely s-type integral, even nonlocal pseudopotential integrals. Thus the angular factors associated with angular momentum and differentiation with respect to atom position can be pulled outside loops over orbital and KS Gaussian exponents.