Recently, a staged progress in few-body bound-state quantum electrodynamic (QED) theory has been achieved from the theoretical atomic and molecular physics group at Wuhan Institute of Physics and Mathematics (WIPM), Chinese Academy of Sciences, by deriving the complete spin-averaged m(alpha)**6 order effective Hamiltonian for hydrogen molecular ions.

The hydrogen molecular ion H2+ (and its isotopes D2+, HD+ etc.) is the simplest molecular ion in nature, which allows its ro-vibrational transition frequencies to be precisely measured experimentally and calculated theoretically using bound-state QED theory. Therefore, we can test bound-state QED theory and determine fundamental physical constants, in particular the proton-to-electron mass ratio, by studying precision spectroscopy of hydrogen molecular ions. Currently, several experiments are underway, including the two-photon transition measurements in H2+ by L. Hilico and J.-Ph. Karr of LKB (France), the ro-vibrational transition measurements in HD+ by J. Koelemeij of VU (Netherlands) and by S. Schiller of Heinrich-Heine University (Germany), and the (v=0->v=6) transition measurements in HD+ by X. Tong of WIPM (China). The targeted precision of these experiments 10**(-10) is a great challenge to theoretical calculations that need high-order relativistic and QED corrections up to m(alpha)**7 or even higher.

The current status of theory is as follows. First, the theory group at WIPM has obtained the ultra-high precision results for the non-relativistic energy levels, as well as the leading-order relativistic and QED corrections. Second, a series of works on higher-order relativistic and QED corrections were published by V. I. Korobov of Dubna and co-workers. In these works, the leading-order relativistic and QED corrections were calculated in a three-body Coulomb scheme without any approximation; however higher-order relativistic and QED corrections, including orders m(alpha)**6, m(alpha)**7 and m(alpha)**8, were treated by taking the external-field approximation and the Born-Oppenheimer approximation. Meanwhile, the two-body bound state results were also used to deal with the nuclear recoil correction of order (m/M) m(alpha)**6. In order to reduce theoretical uncertainties in transition frequencies due to the use of these approximations, a rigorous theory at orders of m(alpha)**6 and (m/M) m(alpha)**6 must be developed.

In order to derive the order m(alpha)**6 and (m/M)m(alpha)**6 effective Hamiltonian, significant efforts were made by the theory group at WIPM to extend successfully the so-called non-relativistic quantum electrodynamic (NRQED) theory from atomic systems to one-electron molecular systems, without assuming the Born-Oppenheimer approximation. Renormalization procedure was also used to completely remove theoretical singularities among various operators. As a nontrivial test of the correctness of their derivation, the obtained theory was applied to the atomic hydrogen and all the results of the Dirac equation for hydrogen were reproduced. This progress signifies the first successful application of NRQED theory to molecular systems. The obtained effective Hamiltonian can now precisely describe the m(alpha)**6 order relativistic and QED corrections, including the nuclear recoil effects. Furthermore, their theory can also be applied to other one-electron two-center problems, such as the anti-protonic helium. The next step of the theory group is to calculate those effective operators numerically, which is as challenge as the derivation of the effective Hamiltonian.

This work has been supported by the National Natural Science Foundation of China and the Strategic Priority Research Program of the Chinese Academy of Sciences.