We derive the effective Hamiltonian of the Nonrelativistic Quantum Electrodynamic up to mα 8 by using scattering matching approach. At mα 6 order, these results are coincide with Pachucki's, which is obtained by applying Foldy-Wouthuysen transformation. And by using the NRQED Hamiltonian, we derive the photon-exchange interaction in non-retarded approximation and the retardation correction up to mα 8 . The energy shift of the photon-exchange interaction is obtained by studying the pole of the total Green function.a Haoxue Qiao; electronic mail: qhx@whu.edu.cn where the subscript (n) means the mα n . The leading terms E (2) , which is calculated by solving the Schrödinger equation, is at mα 2 order. The high-order energy shift, relativistic, radiation and recoil corrections, are the means value of the high-order Hamiltonian. The precision of the energy is restricted by the accuracy of E (2) , the non-relativistic wave-functions and the order of relativistic, radiation and recoil corrections obtained. As the non-relativistic problem has been solved by using Rayleigh-Ritz variational method in Hylleraas coordinates [4][5][6][7]. The core issue of the high-precision calculating the energy-level is deriving these high-order corrections.The most effective method to do this work is the non-relativistic quantum electrodynamic(NRQED) [8][9][10], which is an effective field theory describing the non-relativistic electromagnetic systems. The key point is the contributions of the virtual photon are separated into the high-energy region ω > mα and low-energy region ω ∽ mα 2 . In the low-energy region, the energy shift can be calculated by using the low-energy effective Hamiltonian. In the High-energy region, the contributions can be treated in QED approach with scattering approximation.The NRQED has achieved a great success in the light Hydrogenlike atoms. Their energy-level has been calculated up to mα 7 [11] In the three-body systems, Helium and Hydrogen molecule ions, the effective Hamiltonian of the energy up to mα 6 order has been complete obtained [12] And at mα 7 order, only the fine-structure corrections are derived [13]. The studying about the mα 8 order corrections are seldom reported.The fine-structure splitting in the 2 3 P J level of Helium is recognized as the best atomic system for the determination of the fine-structure constant α. Comparing with calculations of the mα 7 order energy, the measured result can be used to determine the fine-structure constant with a precision of about 2 × 10 −9 [2] It is necessary to derive the mα 8 order Hamiltonian to reduce the uncertain 2.3 × 10 −10 in CODATA [14] At present, the low-energy effective Hamiltonian of NRQED obtained by using Foldy-Wouthuysen transformation [15] is at mα 6 order. The mα 8 order result hasn't been obtained. In this work, we

The Foldy-Wouthuysen Hamiltonian of a light atomic system that has an 𝑚𝛼 8 contribution to energy levels is calculated. The case of a Dirac-Coulomb field is discussed. The results can be used for relativistic and radiative corrections to energy levels in the low-energy part. A divergent operator 𝛿 2 (𝑟) emerges. This is probably due to the nature of the point-like charge source. The effective method of radiation calculation may be re-checked.

The one-loop correction and two-loop contribution to black-body radiation (BBR) shift are restudied. The S-matrix approach and nonrelativistic quantum electrodynamics (NRQED) are adopted in finite temperature case. The relativistic correction to one-loop BBR-shift has a (Zα) 2 αT 2 /m-order contribution. In the two-loop case, the pure thermal (real) photon part is too tiny to be detected; while the corrections induced by the thermal and virtual mixing diagram are at (Zα) 2 α 2 T 2 /m order. We calculate the relativistic correction to one-loop BBR-shift in the ground state of hydrogen and ionized helium, which is larger than the leading term. As the leading term is proportional to T 4 /Z 4 . We estimate these higher-order corrections may be larger than the leading term, when the system is a highly ionized (large Z) or a cold (small T ) one.

The Hylleraas-B-splines basis set is introduced in this paper, which can be used to obtain the eigenvalues and eigenstates of helium-like system's Hamiltonian. Comparing with traditional Bsplines basis, the rate of convergence of our results has been significantly improved. Through combine this method and pseudo-states sum over scheme, we obtained the high precision values of static dipole porlarizabilities of the 1 1 S−5 1 S, 2 3 S−6 3 S states of helium in length and velocity gauges respectively, and the results get good agreements. The final extrapolate results of porlarizabilities in different quantum states arrived eight significant digits at least, which fully illustrates the advantage and convenience of this method in the problems involving continuous states.

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