^{1,a)}and Ingvar Lindgren

^{2}

### Abstract

The current field of relativistic quantum chemistry (RQC) has been built upon the no-pair and no-retardation approximations. While retardation effects must be treated in a time-dependent manner through quantum electrodynamics (QED) and are hence outside RQC, the no-pair approximation (NPA) has to be removed from RQC for it has some fundamental defects. Both configuration space and Fock space formulations have been proposed in the literature to do this. However, the former is simply wrong, whereas the latter is still incomplete. To resolve the old problems pertinent to the NPA itself and new problems beyond the NPA, we propose here an effective many-body (EMB) QED approach that is in full accordance with standard methodologies of electronic structure. As a first application, the full second order energy E 2 of a closed-shell many-electron system subject to the instantaneous Coulomb-Breit interaction is derived, both algebraically and diagrammatically. It is shown that the same E 2 can be obtained by means of 3 Goldstone-like diagrams through the standard many-body perturbation theory or 28 Feynman diagrams through the S-matrix technique. The NPA arises naturally by retaining only the terms involving the positive energy states. The potential dependence of the NPA can be removed by adding in the QED one-body counter terms involving the negative energy states, thereby leading to a “potential-independent no-pair approximation” (PI-NPA). The NPA, PI-NPA, EMB-QED, and full QED then span a continuous spectrum of relativistic molecular quantum mechanics.

The authors are grateful to Professor W. Kutzelnigg and Professor D. Muhkerjee for stimulating discussions. The research of this work was supported by the National Natural Science Foundation of China (Project Nos. 21033001, 21273011, and 21290192), the Swedish Research Council, Vetenskapsrådet, as well as the Swedish National Infrastructure for Computing (SNIC).

I. INTRODUCTION

II. EFFECTIVE MANY-BODY QED APPROACH

A. The non-retarded QED Hamiltonian

B. The second order energy

1. Treating the occupied PES as holes

2. Treating the occupied PES as particles

3. The S-matrix approach

III. CONCLUSIONS AND OUTLOOK

## Figures

Diagrammatical representation of the second order energy (65) . (a) two-body direct; (b) two-body exchange; (c) one-body. The horizontal dashed line represents the instantaneous Coulomb/Breit interaction. For the state, the particles (upgoing lines) and holes (downgoing lines) are {a, b} and , respectively. The one-body potential represented by the square is V 1 (57) . For the state, the particles and holes are {a, b, i, j} and , respectively. The one-body potential is (59) . A global negative sign should be inserted to the terms of .

Diagrammatical representation of the second order energy (65) . (a) two-body direct; (b) two-body exchange; (c) one-body. The horizontal dashed line represents the instantaneous Coulomb/Breit interaction. For the state, the particles (upgoing lines) and holes (downgoing lines) are {a, b} and , respectively. The one-body potential represented by the square is V 1 (57) . For the state, the particles and holes are {a, b, i, j} and , respectively. The one-body potential is (59) . A global negative sign should be inserted to the terms of .

Diagrammatical representation of the potential (77) . Free orbital lines directed upwards and downwards represent PES and NES, respectively. The internal orbital lines for the electron vacuum polarization (6) and (7) and self-energy (8) and (9) can be both PES and NES, whether occupied or not. Both the dashed and wavy lines represent the instantaneous Coulomb/Breit interaction. The cross indicates the counter potential −U.

Diagrammatical representation of the potential (77) . Free orbital lines directed upwards and downwards represent PES and NES, respectively. The internal orbital lines for the electron vacuum polarization (6) and (7) and self-energy (8) and (9) can be both PES and NES, whether occupied or not. Both the dashed and wavy lines represent the instantaneous Coulomb/Breit interaction. The cross indicates the counter potential −U.

Diagrammatical representation of the first order wave operator Ω(1) (72) . Free orbital lines directed upwards and downwards represent PES and NES, respectively. The internal orbital lines for the electron vacuum polarization (6) and (7) and self-energy (8) and (9) can be both PES and NES, whether occupied or not. Both the dashed and wavy lines represent the instantaneous Coulomb/Breit interaction. The cross indicates the counter potential −U.

Diagrammatical representation of the first order wave operator Ω(1) (72) . Free orbital lines directed upwards and downwards represent PES and NES, respectively. The internal orbital lines for the electron vacuum polarization (6) and (7) and self-energy (8) and (9) can be both PES and NES, whether occupied or not. Both the dashed and wavy lines represent the instantaneous Coulomb/Breit interaction. The cross indicates the counter potential −U.

Time-ordered Feynman diagrams for the S-operators (79)–(106) with instantaneous interactions.

Time-ordered Feynman diagrams for the S-operators (79)–(106) with instantaneous interactions.

Non-retarded Feynman diagrams for the second order energy. The numbers in brackets refer to the diagrams in Fig. 4 . The retarded diagrams are obtained by replacing one or two Coulomb photons with transverse photons.

Non-retarded Feynman diagrams for the second order energy. The numbers in brackets refer to the diagrams in Fig. 4 . The retarded diagrams are obtained by replacing one or two Coulomb photons with transverse photons.

Non-retarded Feynman diagrams for the first order energy. The retarded diagrams are obtained by replacing the Coulomb photon with a transverse photon.

Non-retarded Feynman diagrams for the first order energy. The retarded diagrams are obtained by replacing the Coulomb photon with a transverse photon.

## Tables

The spectrum of relativistic Hamiltonians. SC: Schrödinger-Coulomb; A1C: spin-free part of A2C; A2C: approximate two-component; X1C: spin-free part of X2C; X2C: exact two-component; Q4C: quasi-four-component; DCB: no-pair Dirac-Coulomb-Breit; PI-DCB: potential-independent DCB; eQED: effective (non-retarded) QED.

The spectrum of relativistic Hamiltonians. SC: Schrödinger-Coulomb; A1C: spin-free part of A2C; A2C: approximate two-component; X1C: spin-free part of X2C; X2C: exact two-component; Q4C: quasi-four-component; DCB: no-pair Dirac-Coulomb-Breit; PI-DCB: potential-independent DCB; eQED: effective (non-retarded) QED.

Degeneracy (n d ) of low-order Feynman diagrams. n F2: number of electron-field contractions between two different vertices enumerated in an ascending order; n F1: number of electron-field contractions within the same vertex; n P : number of possible assignments of all the photon interactions; n d = max(1, n F1 + 2n F2) × max(1, n P ).

Degeneracy (n d ) of low-order Feynman diagrams. n F2: number of electron-field contractions between two different vertices enumerated in an ascending order; n F1: number of electron-field contractions within the same vertex; n P : number of possible assignments of all the photon interactions; n d = max(1, n F1 + 2n F2) × max(1, n P ).

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