Fine-structure splittings in low-lying triplet states of the carbon atom

We present benchmark theoretical values of the fine-structure splittings in the ground <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msup> <a:mrow/> <a:mn>3</a:mn> </a:msup> <a:msup> <a:mi>P</a:mi> <a:mi>e</a:mi> </a:msup> </a:mrow> </a:math> and low-lying excited <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:msup> <b:mrow/> <b:mn>3</b:mn> </b:msup> <b:msup> <b:mi>P</b:mi> <b:mi>e</b:mi> </b:msup> </b:mrow> </b:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:msup> <c:mrow/> <c:mn>3</c:mn> </c:msup> <c:msup> <c:mi>D</c:mi> <c:mi>e</c:mi> </c:msup> </c:mrow> </c:math> states of the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mmultiscripts> <d:mi mathvariant="normal">C</d:mi> <d:mprescripts/> <d:none/> <d:mn>12</d:mn> </d:mmultiscripts> </d:math> , <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mmultiscripts> <f:mi mathvariant="normal">C</f:mi> <f:mprescripts/> <f:none/> <f:mn>13</f:mn> </f:mmultiscripts> </f:math> , and <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mmultiscripts> <h:mi mathvariant="normal">C</h:mi> <h:mprescripts/> <h:none/> <h:mn>14</h:mn> </h:mmultiscripts> </h:math> 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 <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:mo>(</j:mo> <j:mo>∝</j:mo> <j:msup> <j:mi>α</j:mi> <j:mn>2</j:mn> </j:msup> <j:mrow> <j:mo>)</j:mo> </j:mrow> </j:mrow> </j:math> , contribution from the electron anomalous magnetic moment <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mo>(</k:mo> <k:mo>∝</k:mo> <k:msup> <k:mi>α</k:mi> <k:mn>3</k:mn> </k:msup> <k:mo>)</k:mo> </k:mrow> </k:math> , and by taking into account the most significant couplings between the targeted and neighboring electronic states <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:mo>(</l:mo> <l:mo>∝</l:mo> <l:msup> <l:mi>α</l:mi> <l:mn>4</l:mn> </l:msup> <l:mo>)</l:mo> </l:mrow> </l:math> . 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.