Molecular structure calculations: A unified quantum mechanical description of electrons and nuclei using explicitly correlated Gaussian functions and the global vector representation

Abstract

We elaborate on the theory for the variational solution of the Schrödinger equation of small atomic and molecular systems without relying on the Born-Oppenheimer paradigm. The all-particle Schrödinger equation is solved in a numerical procedure using the variational principle, Cartesian coordinates, parameterized explicitly correlated Gaussian functions with polynomial prefactors, and the global vector representation. As a result, non-relativistic energy levels and wave functions of few-particle systems can be obtained for various angular momentum, parity, and spin quantum numbers. A stochastic variational optimization of the basis function parameters facilitates the calculation of accurate energies and wave functions for the ground and some excited rotational-(vibrational-)electronic states of H(2) (+) and H(2), three bound states of the positronium molecule, Ps(2), and the ground and two excited states of the (7)Li atom.

Experimental progress in positronium laser physics

“…Shortly after Wheeler's paper was published [16] the stability of Ps 2 was demonstrated by Hylleraas and Ore, who obtained a binding energy of 0.11 eV using variational methods [507]. Since then there have been many calculations of the Ps 2 binding energy [565,[574][575][576][577]. Recent calculations, which include relativistic corrections, yield 0.4341373 eV [508].…”

“…Just as with Ps ± ions, the Born-Oppenheimer approximation [515] is completely inapplicable to the Ps 2 system [565], perhaps maximally so. A wide variety of theoretical techniques have been used to deal with this system (e.g.…”

Relativistic kinetic-balance condition for explicitly correlated basis functions

“…Essentially, we normalize each component of each basis function individually to ensure numerical stability when solving the eigenvalue problem. This procedure can be understood as the relativistic counterpart of the quasi-normalization in pre-Born-Oppenheimer theory [50]. Hence, explicit normalization of a trial wave function has to be taken into account when the energy is calculated.…”

“…Trial values for α kl,i were generated from a normal distribution as in Ref. [50] (see also references therein). The optimized parameter values of A i are deposited in the supplementary information.…”

Including nuclear quantum effects into highly correlated electronic structure calculations of weakly bound systems

“…17,19 The total wavefunction is then obtained by minimizing the energy functional through the variational principle. Following these ideas, Mátyus and Reiher have very recently developed a new numerical approach, 20 obtaining rotational-vibrationalelectronic energy levels with spectroscopic accuracy.…”