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## Relativistic correlation

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**Relativistic correlation**Wenjian Liu (Peking University)**QC: a “3+1”-d problem**QED ??? DEQ Q4C X2C No-pair | | Explicit correlation? A2C SEQ Is there a consistent theory between no-pair and QED? How to interface RQC with QED?**Outline**Enpis always potential dependent! • With virtual pairs (charge-conserving): • correlation fromNES: interfacing RQC with QED • Without virtual pairs (particle-conserving): • orbital approximation: 4c = 2c • explicit correlation: 4c with extended npp**No-pair Hamiltonians: 4c vs. 2c**No-pair Dirac-Coulomb-Breit Hamiltonian ESC RI-RKB A2C 1950~ U 2009 1950 DKH:IOTC:X2C = ∞:2:1 Visscher 09: Mol. Mean-Field X2C Q4C 2006 2010 “from atoms to molecule H” 2007 U RKB ESC 2005 For a review see Liu, Mol. Phys. 108, 1679 (2010)**Spectroscopic constants: E1172**For a review see Liu, Mol. Phys. 108, 1679 (2010)**No-pair Hamiltonians: 4c vs. 2c**4C = Q4C = X2C All know-how correlation methods under the orbital approximation For a review see Liu, Mol. Phys. 108, 1679 (2010)**link atom**link atom cap2 buffer fragment cap2 buffer link atom link atom link atom link atom fragment cap3 buffer buffer cap3 cap buffer buffer cap fragment Local correlation/excitation:“from fragments to molecule for wave function” Wu, Liu, et al. JCTC, 2011, ASAP**Fragment**Buffer Fragmentation of C20H22 (Divide) (primitive fragment LMO, pFLMO)**Localization of CMO in the pFLMO basis**Physics: transferability Mathematics: block-diagonalization Globally monotonic, locally cubic convergence (or non-iteratively) Least change in the diagonal blocks The same trick as from Dirac to X2C! (fromatoms/fragments tomolecular Hamiltonian/wave function)**Locality of FLMO**The global FLMO still localized on the parent fragments of pFLMO!**Locality of FLMO pairs aIiJ**Post-SCF should be linear scaling, and may even be cheaper than SCF! >10(-η) CnHn+2**Caveats with the no-pair Hamiltonian**• Incompatible with explicitly correlated methods! • Potential dependent (even “FCI”)! (dual-basis projector) Extended no-pair projection PHP f12 (All algebraic 2c Hamiltonians do not fit!)**-mc2**NES How to go beyond no-pair ? +mc2 (part of the basis in a L2 discretization) O(c0) for odd operators! No relativistic diamagnetism! (for a recent review see Xiao, Sun, Liu, TCA 2011)**Configuration space: empty Dirac picture**Normal ordered w.r.t. |0> Number of electrons is conserved NES are regarded as virtual orbitals Just like the Schrödinger equation**Configuration space: empty Dirac picture**BRdisease (1951) Configuration + + + Isn’t it a mathematical failure?**Configuration space: empty Dirac picture**BRdisease (1951) Configuration + + + -------minimization-------- maximization FCI: (1) Bunge (1997): bona fide bound states boundedfrom below by the no-pair states; NES anti-correlating (2)Pestka, Karwowski, Tatewaki (2006-2011): resonances only The DC Hamiltonian is NOT self-adjoint, although the Dirac operator is self-adjoint on H1(R3)4 (3) Sapirstein (1999): mathematically correct, physically wrong**No-photon Fock space: filled Dirac picture**charge-conserving only PES; VP p-h normal particle-conserving both PES and NES Chaix, Iracane (1989); Saue, Visscher (2003); Eliav, Kaldor (2010); Kutzelnigg (2011);**2nd order energies of CS & FS**X Configuration space with filled Dirac picture No contractions among the NES, viz., No effective potential from the NES (a weird feature of the filled Dirac picture)**FS vs. QED**p-h normal (time ordering)**2nd order energies of CS, FS, QED**correlating anti-correlating**Why do FS and QED differ?**Quantized Dirac fields in the CBS of PES+NES; charge-conserving Positive energy electrons propagate forward in time Negative energy electrons propagate backward in time Positive energy positrons propagate forward in time NES are taken as the basis (image) describing virtual positrons**Why do FS and QED differ?**Time ordering is an essential ingredient In relativistic QM, we must allow time to go backwards**Why do FS and QED differ?**I (1+) (2-) R (2+) (1-) configuration space**Why do FS and QED differ?**The QED and CS electron propagators: (Both PES and NES are particles in CS due to improper time flow)**Why do FS and QED differ?**Under the no-pair approximation, the system of electrons is held on by the projection and is hence closed and stationary. So both time dependent and independent approaches work. However, when the projection is lifted, the number of electrons is no longer conserved. The system of electrons becomes an open and non-stationary subsystem entangled with the NES, just like the Schrödinger cat entangled with the environment. So only time dependent treatment works: PES and NES propagate in opposite directions in space and time.**Configuration space, Fock space & QED**Agree on one-electron and non-interacting electrons correlation within the PES manifold Disagree on correlation involving the NES,even in the one-body terms CS: mathematically correct, physically wrong FS: mathematically correct, physically plausible QED: mathematically correct (yet nasty), physically correct The contribution of NESis responsible for resolving the (Zα)3 uncertainty in the eigenvalues of the DC/DCB equation**Configuration space, Fock space & QED**Agree on one-electron and non-interacting electrons correlation within the PES manifold Disagree on correlation involving the NES,even in the one-body terms Full QED: (non-radiation + radiation + retardation + recoil) applied only up to 3e systems NR QED: applicable to molecules of light atoms Rel. QED: DCB-{CC}++ + LS (???) DCB-{ (CC)++ (val.) + [(MP2)++– (MP2)--](core’) } + LS**Two classes of properties**1. Even (diagonal): electric field (scalar potential) 2. Odd (off-diagonal): magnetic field (vector potential) For a review, see Sun, Xiao, Liu, TCA (50th anniversary issue)**Open questions**CBS = PES + NES QED: time-dependent! • Time-independent treatment of NES? • ‘No-photon’ QED beyond BDF? • Correlation of NES to, e.g., NMR? Liu, perspectives of RQC, PCCP (in press)**Future plans**Relativistic WFT: • Conventional WF methods • relativistic explicit correlation: extended no-pair Hamiltonian • Potential independent npp correlation (QED for NES) • O(N) correlation with FLMO • 4c/X2C-MB-GIAO-based correlation for NMR • Relativistic theories for NSR**Acknowledgments**• Dr. Yunlong Xiao (4c-NMR) • Dr. Lan Cheng (4c-NMR, MB-GIAO) • Drs. Daoling Peng, Yong Zhang (X2C) • Dr. Fangqin Wu (FLMO-TD-DFT) • Mr. Qiming Sun (X2C-NMR) • Mr. Zhendong Li (open-shell TDDFT) • ¥NSFC for RMB