Atomic Structure Calculations of Helium with Correlated Exponential Functions

• Main Authors: Vladimir A. Yerokhin, Vojtěch Patkóš, Krzysztof Pachucki
• Format: Article
• Language: English
• Published: MDPI AG 2021-07-01
• Series: Symmetry
• Subjects: helium atom; Lamb shift; nonrelativistic QED
• Online Access: https://www.mdpi.com/2073-8994/13/7/1246
• ISSN: 2073-8994

The technique of quantum electrodynamics (QED) calculations of energy levels in the helium atom is reviewed. The calculations start with the solution of the Schrödinger equation and account for relativistic and QED effects by perturbation expansion in the fine structure constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. The nonrelativistic wave function is represented as a linear combination of basis functions depending on all three interparticle radial distances, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>r</mi><mn>1</mn></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>r</mi><mn>2</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>r</mi><mtext> </mtext><mo>=</mo><mtext> </mtext><mo>|</mo></mrow><msub><mover accent="true"><mi>r</mi><mo>→</mo></mover><mn>1</mn></msub><mo>−</mo><msub><mover accent="true"><mi>r</mi><mo>→</mo></mover><mn>2</mn></msub><mrow><mo>|</mo></mrow></mrow></semantics></math></inline-formula>. The choice of the exponential basis functions of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">exp</mo><mo>(</mo><mo>−</mo><mi>α</mi><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><mi>β</mi><msub><mi>r</mi><mn>2</mn></msub><mo>−</mo><mi>γ</mi><mi>r</mi><mo>)</mo></mrow></semantics></math></inline-formula> allows us to construct an accurate and compact representation of the nonrelativistic wave function and to efficiently compute matrix elements of numerous singular operators representing relativistic and QED effects. Calculations of the leading QED effects of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>α</mi><mn>5</mn></msup><mi>m</mi></mrow></semantics></math></inline-formula> (where <i>m</i> is the electron mass) are complemented with the systematic treatment of higher-order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>α</mi><mn>6</mn></msup><mi>m</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>α</mi><mn>7</mn></msup><mi>m</mi></mrow></semantics></math></inline-formula> QED effects.