Theory and theoretical
input data
The energy levels of hydrogenlike atoms are determined mainly by the Dirac
eigenvalues, quantum electrodynamic (QED) effects such as self energy and
vacuum polarization, and nuclear size and motion effects. The theory and
theoretical data needed for the evaluation of the energy levels in hydrogen
and deuterium are summarized on this page.


The energy levels and transition frequencies are calculated using the
results of the 2002 CODATA leastsquares adjustment of the fundamental physical
constants. The calculation is based on the formulation described by
U.D. Jentschura, S. Kotochigova, E.O. Le Bigot,
P.J. Mohr, and B.N. Taylor in
Precise
calculation of transition frequencies of hydrogen and deuterium based on a
leastsquares analysis, Phys. Rev. Lett. 95, 163003 (2005). 


Matrices referred to in the above paper can be accessed from the fundamental
constants Web site physics.nist.gov/constants.
Although those complete matrices from the 2002 CODATA leastsquares adjustment
of the fundamental physical constants can provide the basis for the calculation
of the energy levels, in this Web database, we use matrices from a leastsquares
adjustment that includes only data relevant to the hydrogen and deuterium
energy levels. The results for the energy levels are essentially the same in
either case. 


The contributions to the theoretical values of the individual energy levels are
summarized by P.J. Mohr and B.N. Taylor in CODATA recommended
values of the fundamental physical constants: 2002, Rev. Mod. Phys.
77, 1 (2005), Appendix A, which is
reprinted here. 


The largest QED contribution is the onephoton selfenergy correction. This
correction includes the nonrelativistic Bethe logarithm which requires a
separate calculation for each state. For states with principle quantum number
n ≤ 20, we employ the values given by G.W.F. Drake and
R.A. Swainson, Phys. Rev. A 41, 1243 (1990). For states with
21 ≤ n ≤ 200, we use the results given in the
attached table of Bethe logarithms by U.D. Jentschura and P.J. Mohr,
NIST Technical Note 1467. The calculation of
the values in the latter table is described by U.D. Jentschura and
P.J. Mohr in Calculation of hydrogenic Bethe
Logarithms for Rydberg states, Phys. Rev. A 72, 012110
(2005). Higherorder contributions to the self energy are summarized by
U.J. Jentschura, S. Kotochigova, E.O. Le Bigot, and
P.J. Mohr in Precise theory of levels of hydrogen and deuterium: The
onephoton self energy correction, NIST Technical
Note 1469. 


The secondlargest QED contribution is the onephoton vacuum polarization. The
calculated data needed to evaluate this contribution is given by
S. Kotochigova and P. Mohr, Precise theory of levels of hydrogen
and deuterium: The Coulomb vacuum polarization correction, NIST Technical Note 1468. 


The twophoton corrections are somewhat smaller, but still very important at
the level of accuracy considered here. A summary of the status of the theory
and the data necessary for the present work is given by U.D. Jentschura
and P.J. Mohr, Precise theory of levels of hydrogen and deuterium: The
twophoton radiative corrections, NIST Internal
Report 7217. 


