Accurate fine structure of low-lying triplet states of the carbon atom

We report benchmark theoretical values of the fine structure splittings in the ground ³P^e and low-lying excited ³P^e, ³D^e, and ³P^o states of the of the neutral carbon atom. First, nearly exact nonrelativistic calculations of the ¹²C, ¹³C, and ¹⁴C isotopes are performed in the framework of the variational method. The wave functions of the targeted states are expanded in terms of large sets of all-particle explicitly correlated Gaussians. Next, we use these wave functions to calculate the fine structure of the three isotopes perturbatively. Apart from the leading-order of the spin-orbit and electron spin-spin corrections (α²), we also calculate the contribution due to the electron anomalous magnetic moment (α³), and take into account the most significant couplings between the targeted and neighboring electronic states (α⁴). As our calculations do not assume frozen nucleus, we can extract the isotopic shifts to the fine structure splittings. Our fine-structure results show a notable improvement in accuracy compared to any previous theoretical calculation, are in good agreement with the best available experimental data, and demonstrate that the calculations of six-electron systems are now becoming sufficiently advanced for precision measurement applications.