Calculation of the relativistic Bethe logarithm in the two-center problem

Abstract

We present a variational approach to evaluate relativistic corrections of order α^2 to the Bethe logarithm for the ground electronic state of the Coulomb two center problem. That allows to estimate the radiative contribution at mα^7 order in molecular-like three-body systems such as hydrogen molecular ions H_2^+ and HD^+, or antiprotonic helium atoms. While we get 10 significant digits for the nonrelativistic Bethe logarithm, calculation of the relativistic corrections is much more involved especially for small values of bond length R. We were able to achieve a level of 3-4 significant digits starting from R=0.2 bohr, that will allow to reach 10^{-10} relative uncertainty on transition frequencies.

19 pages, 5 tables, 7 figures

References in corpus (5)

• Nonrelativistic QED approach to the Lamb shift
• Relativistic corrections of mα^6 order to the ro-vibrational spectrum of H_2^+ and HD^+ molecular ions
• Calculation of the One- and Two-Loop Lamb Shift for Arbitrary Excited Hydrogenic States
• Relativistic corrections of mα^6(m/M) order to the hyperfine structure of the H_2^+ molecular ion
• Relativistic corrections of order mα^6 to the two-center problem

Cited by in corpus (29)

• Fundamental transitions and ionization energies of the hydrogen molecular ions at the few ppt level
• Theoretical transition frequencies beyond 0.1 ppb accuracy in H${}_2^{+}$, HD${}^{+}$, and antiprotonic helium
• The Born-Oppenheimer approximation in an effective field theory language
• Bounds on fifth forces at the sub-Angstrom length scale
• Ab initio potential energy curve for the ground state of beryllium dimer
• Proposed method for laser spectroscopy of pionic helium atoms to determine the charged-pion mass
• Two-photon spectroscopy of trapped HD${}^{+}$ ions in the Lamb-Dicke regime
• Non-adiabatic, Relativistic, and Leading-order QED Corrections for Rovibrational Intervals of ${}^4$He${}_2^{+}$ ($X{}^2{\Sigma }_{\mathrm{u}}^{+}$)
• Relativistic corrections to the Bethe logarithm for the $2^{3}S$ and $2^{3}P$ states of He
• All-order relativistic computations for atoms and molecules using an explicitly correlated Gaussian basis
• Ro-vibrational states of H${}_2^{+}$. Variational calculations
• Precise computation of rovibronic resonances of molecular hydrogen: $EF{}^1{\Sigma }_{\mathrm{g}}^{+}$ inner-well rotational states
• Higher-order corrections to spin-spin scalar interactions in HD${}^{+}$ and H${}_2^{+}$
• Nonradiative ${\alpha }^{7}m$ QED effects in Lamb shift of helium triplet states
• The Bethe Logarithm for resonant states. Antiprotonic helium
• Spectroscopic tests for short-range modifications of Newtonian and post-Newtonian potentials
• Evaluation of the Bethe logarithm: from atom to chemical reaction
• Explicitly correlated Gaussian functions with shifted-center and projection techniques in pre-Born-Oppenheimer calculations
• Calculation of Araki-Sucher correction for many-electron systems
• QED corrections to the correlated relativistic energy: one-photon processes
• Relativistic Bethe logarithm for triplet states of helium-like ions
• One-loop vacuum polarization at m$\alpha$7 and higher orders for three-body molecular systems
• One-particle operator representation over two-particle basis sets for relativistic QED computations
• One-loop vacuum polarization at $m{\alpha }^7$ order for the two center problem
• Rovibrational computations for He${}_2^{+}$ X ${\Sigma }_{\mathrm{u}}^{+}$ including non-adiabatic, relativistic and QED corrections
• Ro-vibrational spin-averaged transitions in the hydrogen molecular ions
• Double-pair Coulomb and Breit photon correction to the correlated relativistic energy
• QED corrections of orders $m{\alpha }^6$ and $m{\alpha }^6(m/M)$ for HD${}^{+}$ rovibrational transitions beyond Born-Oppenheimer approximation
• Rovibrational computations for the He${}_2$ a ${}^3{\Sigma }_{\mathrm{u}}^{+}$ state including non-adiabatic, relativistic, and QED corrections