HIGHLY ACCURATE CALCULATIONS OF SPIN-DEPENDENT RELATIVISTIC CORRECTIONS IN SMALL ATOMS WITH ONE AND TWO P-ELECTRONS

Abstract

I present an algorithm to calculate spin-dependent relativistic corrections using variationally optimized non-relativistic wave functions expanded in terms of explicitly correlated Gaussian basis functions. All matrix elements required for the calculations were derived analytically, with detailed derivations provided. The algorithm can be applied to systems with one and two p-electrons, or a single d-electron. Using the newly developed algorithm, I studied how the relativistic effects affect the stability of positronic beryllium in the ground singlet S and excited triplet S and P states. I found that the inclusion of relativistic corrections changes the binding energies of the considered states only by 2.2% at most. Interestingly, this small change persists even for triplet P states, for which the spin–orbit and spin–spin contributions are not canceled out when the binding energy is computed. In the second application, I investigated the fine structure of the carbon atom's ground and first excited triplet ³Pᵉ states as well as the lowest ³Dᵉ state. I accounted for the leading-order (∝ α²), the electron anomalous magnetic moment (∝ α³), and the dominant part of the second-order perturbation theory (∝ α⁴) contributions. To my knowledge, these are the first high-precision calculations at the α⁴ level of theory performed for a system of this size. The computed values of the fine-structure splittings represent the most accurate calculations of the carbon atom reported to date and are in agreement with experiment at the level of 0.0001–0.01 cm⁻¹. In addition, I report the isotopic shifts in the fine-structure levels of ¹³C, ¹⁴C, and ∞C relative to ¹²C.