Relativistic correlation Wenjian Liu Peking University QED QC

Relativistic correlation Wenjian Liu (Peking University)

QED | QC: a “ 3+1”-d problem A 2 C No-pair DEQ Q 4 C X 2 C | ? ? ? Explicit correlation? SEQ Is there a consistent theory between no-pair and QED? How to interface RQC with QED?

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

No-pair Hamiltonians: 4 c vs. 2 c No-pair Dirac-Coulomb-Breit Hamiltonian SC E U RI-R A 2 C 1950~ 200 9 1950 KB Visscher 09: Mol. Mean-Field “from atoms to molecule H” X 2 C 2010 2006 Q 4 C DKH: IOTC: X 2 C = ∞: 2: 1 2007 RK B 2005 U C S E For a review see Liu, Mol. Phys. 108, 1679 (2010)

Spectroscopic constants: E 1172 Re (Å) 3. 172 ωe (cm-1) 121. 7 De (e. V) 1. 96 SR DKS 3. 075 3. 52074 118. 1 68. 5803 1. 84 0. 72329 p. DKS Q 4 C NESC 0. 00001 0. 00003 0. 00024 -0. 0009 -0. 0035 0. 0086 0. 00007 -0. 00005 0. 00026 SESC -0. 00046 0. 0142 -0. 00011 NR For a review see Liu, Mol. Phys. 108, 1679 (2010)

No-pair Hamiltonians: 4 c vs. 2 c 4 C = Q 4 C = X 2 C All know-how correlation methods under the orbital approximation For a review see Liu, Mol. Phys. 108, 1679 (2010)

Local correlation/excitation: “from fragments to molecule for wave function” link atom cap 3 link atom cap 2 buffer link atom buffer fragment buffer cap 2 Wu, Liu, et al. JCTC, 2011, ASAP cap 3

Fragmentation of C 20 H 22 (Divide) Fragment Buffer (primitive fragment LMO, p. FLMO)

Global SCF (Conquer)

Localization of CMO in the p. FLMO 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 X 2 C! (from atoms/fragments to molecular Hamiltonian/wave function)

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

Locality of FLMO pairs a. Ii. J Post-SCF should be linear scaling, >10(-η) and may even C ben. Hcheaper than SCF! n+2

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

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

Configuration space: empty Dirac picture 1. 2. 3. 4. 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 + + BR disease (1951) Configuration + Isn’t it a mathematical failure?

Configuration space: empty Dirac picture + + BR disease (1951) Configuration + FCI: -------minimization---- maximization (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

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);

1 st order wave functions of CS & FS

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)

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

2 nd order energies of CS, FS, QED 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 -) (2+) (1 -) R 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 NES is 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 + retardation + recoil) applied only up to 3 e systems NR QED: applicable to molecules of light atoms Rel. QED: DCB-{CC}++ + LS (? ? ? ) DCB-{ (CC)++ (val. ) + [(MP 2)++ – (MP 2)--](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 (50 th anniversary issue)

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

Future plans Relativistic WFT: Conventional WF methods l relativistic explicit correlation: extended no-pair Hamiltonian l Potential independent npp correlation (QED for NES) l O(N) correlation with FLMO l 4 c/X 2 C-MB-GIAO-based correlation for NMR l Relativistic theories for NSR l

Acknowledgments Ø Dr. Yunlong Xiao (4 c-NMR) Dr. Lan Cheng (4 c-NMR, MB-GIAO) Drs. Daoling Peng, Yong Zhang (X 2 C) Dr. Fangqin Wu (FLMO-TD-DFT) Mr. Qiming Sun (X 2 C-NMR) Mr. Zhendong Li (open-shell TDDFT) Ø ¥NSFC for RMB Ø Ø Ø