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Relativistic correlation Wenjian Liu (Peking University)

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QC: a “3+1”-d problem No-pair | | SEQ A2C X2C Q4C DEQ QED ??? Is there a consistent theory between no-pair and QED? How to interface RQC with QED?

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E np is always potential dependent ! Outline With virtual pairs (charge-conserving): correlation from NES: interfacing RQC with QED Without virtual pairs (particle-conserving): orbital approximation: 4c = 2c explicit correlation: 4c with extended npp

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RI-RKB ESC U X2C ESC RKB U No-pair Dirac-Coulomb-Breit Hamiltonian For a review see Liu, Mol. Phys. 108, 1679 (2010) 1950 2005 2009 A2C 1950~ DKH:IOTC:X2C = ∞:2:1 2010 Q4C 2006 “from atoms to molecule H” 2007 Visscher 09: Mol. Mean-Field No-pair Hamiltonians: 4c vs. 2c

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Spectroscopic constants: E117 2 R e (Å)ω e (cm -1 )D e (eV) NR 3.172121.7 1.96 SR 3.075118.1 1.84 DKS3.5207468.5803 0.72329 pDKS0.00001-0.0009 0.00007 Q4C0.00003-0.0035 -0.00005 NESC0.00024 0.0086 0.00026 SESC -0.00046 0.0142-0.00011 For a review see Liu, Mol. Phys. 108, 1679 (2010)

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No-pair Hamiltonians: 4c vs. 2c 4C = Q4C = X2C All know-how correlation methods under the orbital approximation

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fragment buffer link atom cap fragmentbuffer link atom cap2 fragment buffer link atom cap3 Local correlation/excitation: “from fragments to molecule for wave function” Wu, Liu, et al. JCTC, 2011, ASAP

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Fragmentation of C 20 H 22 (Divide) BufferFragment (primitive fragment LMO, pFLMO)

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Global SCF (Conquer)

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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! (from atoms/fragments to molecular Hamiltonian/wave function) Localization of CMO in the pFLMO basis

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Locality of FLMO The global FLMO still localized on the parent fragments of pFLMO!

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Locality of FLMO pairs a I i J C n H n+2 >10 (-η) Post-SCF should be linear scaling, and may even be cheaper than SCF!

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Caveats with the no-pair Hamiltonian 1.Incompatible with explicitly correlated methods! 2.Potential dependent (even “FCI”)! PHPPHP Extended no-pair projection (All algebraic 2c Hamiltonians do not fit!) (dual-basis projector) f 12

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How to go beyond no-pair ? + mc 2 NES - mc 2 (part of the basis in a L 2 discretization) O(c 0 ) for odd operators! No relativistic diamagnetism! (for a recent review see Xiao, Sun, Liu, TCA 2011)

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Configuration space: empty Dirac picture 1.Normal ordered w.r.t. |0> 2.Number of electrons is conserved 3.NES are regarded as virtual orbitals 4.Just like the Schrödinger equation

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Configuration space: empty Dirac picture ++ + BR disease (1951) Configuration Isn’t it a mathematical failure?

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Configuration space: empty Dirac picture ++ + BR disease (1951) Configuration FCI: (1) Bunge (1997): bona fide bound states bounded from 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 H 1 (R 3 ) 4 (3) Sapirstein (1999): mathematically correct, physically wrong -------minimization-------- maximization

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No-photon Fock space: filled Dirac picture p-h normal charge-conserving only PES; VP particle-conserving both PES and NES Chaix, Iracane (1989); Saue, Visscher (2003); Eliav, Kaldor (2010); Kutzelnigg (2011);

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1 st order wave functions of CS & FS

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2 nd 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)

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FS vs. QED p-h normal (time ordering)

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2 nd order energies of CS, FS, QED correlating anti-correlating

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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

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Why do FS and QED differ?

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(1+) (2-) (2+) (1-) I R configuration space

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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)

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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.

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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 NES is responsible for resolving the (Zα) 3 uncertainty in the eigenvalues of the DC/DCB equation

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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

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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 (50 th anniversary issue)

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Open questions CBS = PES + NES Liu, perspectives of RQC, PCCP (in press) Time-independent treatment of NES? ‘No-photon’ QED beyond BDF? Correlation of NES to, e.g., NMR? QED: time-dependent!

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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

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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

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