Positron binding to neutral atoms, which to this day have not been detected experimentally, is thought to be rather weak. According to a few reliable nonrelativistic calculations reported previously, the positron affinity for small atoms, even if it happens to be positive, is predicted to be very small in magnitude, of the order of just few millihartrees $(<0.1\mathrm{eV})$. In this work, we used a highly accurate variational expansion in terms of explicitly correlated Gaussians to investigate how relativistic effects may affect the stability of bound states of positrons with atoms. We performed calculations of positronic beryllium, $e^{+}\left[\mathrm{Be}\right]$, in its ground singlet $S$ and excited triplet $S$ and $P$ states, all of which are bound. According to our findings, neither the inclusion of scalar relativistic nor spin-dependent corrections alters the predictions regarding the existence of the bound states. When leading-order relativistic effects are taken into account, positron affinities change only by 2.2% or less. Notably, this is so even for the triplet $P$ state of positronic beryllium, where the spin-orbit correction is not at all canceled out when the energy difference with the parent system is computed.

In this work we consider an efficient algorithm for variational calculations of quantum few-particle systems in $S, P$, and $D$ states of the even parity using all-particle explicitly correlated Gaussian (ECG) basis sets. We primarily focus on the description of states where the dominant configuration contains either two particles in $p$ states or a single particle in a $d$ state (all other particles are in $s$ states). The basis functions we consider are products of spherically symmetric ECGs and bipolar harmonics. We introduced a scheme for deriving expressions for matrix elements of the overlap, kinetic and potential energy, as well as their derivatives with respect to the nonlinear parameters of the Gaussians. This allowed us to improve the efficiency of numerical calculations of the matrix elements (which is the most critical part of any code that uses ECGs) by one to two orders of magnitude compared to previous implementations. We provide a complete set of formulas for all basic matrix elements using the formalism of matrix differential calculus and discuss some technical details relevant to their efficient implementation. Lastly, we report a few example calculations of the ${}^2{D}^e$ states of the Li atom, ${}^4{P}^e$ and ${}^2{D}^e$ states of the B atom, and ${}^3{P}^e$ states of the C atom.

We present benchmark theoretical values of the fine-structure splittings in the ground 3 P e and low-lying excited 3 P e and 3 D e states of the C 12 , C 13 , and C 14 isotopes of the neutral carbon atom. We handle the correlations in the dynamics of all particles (six electrons and a nucleus) in all considered states in the practically exact manner by employing very large expansions of the wave functions in terms of all-particle explicitly correlated Gaussian basis sets, whose nonlinear parameters were extensively optimized. The resulting wave functions were used to compute the fine-structure splittings for the three isotopes by including the leading-order spin-orbit and electron spin-spin corrections ( ∝ α 2 ) , contribution from the electron anomalous magnetic moment ( ∝ α 3 ) , and by taking into account the most significant couplings between the targeted and neighboring electronic states ( ∝ α 4 ) . Our results show a notable improvement in accuracy compared to any previous theoretical calculations, are in good agreement with the best available experimental data, and demonstrate that the calculations of six-electron systems are now sufficiently advanced for precision measurement applications.