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Definition and Calculation of Electron Densities in Relativistic Quantum GRC 2010 Definition and Calculation of Electron Densities in Relativistic Quantum Theory Markus Reiher Laboratorium f¨ur Physikalische Chemie, ETH Zurich, Switzerland http://www.reiher.ethz.ch Gordon Research Conference on Electron Distribution and Chemical Bonding Mount Holyoke College, 15 July 2010 Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction Electron density Powerful quantity for structure and reactivity studies Topological analyses provide deeper understanding of bonding In contrast to wave function, it is available in experiment Fully determines a given system (→ Hohenberg–Kohn theorems, DFT) Basis for conceptual DFT Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction But ... why relativistic quantum theory? This is the most fundamental many-electron theory ! (Just an observation: If you make a strong statement about your favourite analysis tool, make sure that it fits in this framework.) Practical reason: It is mandatory for heavy-element chemistry (i.e., beyond organic chemistry) Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction World of relativistic quantum theory Quantum Electrodynamics Local QED Model Potentials first−quantized, N−electron Dirac−Fock−Coulomb (DFC) Hamiltonian 2−component methods 4−comp. Dirac−Fock & post−Dirac−Fock methods transformation techniques elimination techniques numerical methods basis set methods (atoms) (molecules) DKH transformation (Pauli approximation) regular approximations clumsy, accurate, but less efficient basis sets: X−operator techniques (BSS,IOTC) GRASP (Oxford) ZORA, FORA MOLFDIR (FW transformation) Desclaux/Indelicato DIRAC NESC (Dyall) Johnson BERTHA ADRIEN (M.R.) UTChem DFT (e.g., BDF) Scalar−relativistic 1−component formulations by . neglect of spin−dependent terms or use of ECPs Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction For all details on the theory see: Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction How to define the electron distribution? Born interpretation: abs. square of wave function (1 electron only) ρ(r)= |Ψ(r)|2 Many-electron systems: integrate out all but one electronic coordinate +∞ +∞ ! 3 3 2 ρ(r) = N d r2 · · · d rN |Ψ({ri},t)| Z−∞ Z−∞ ... yields sum of squared occupied (spin) orbitals for Hartree–Fock or DFT wave function occ 2 ρ(r) → |ψi(r)| Xi=1 → Let us look at a simple example ... Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction Squared orbital of Hg79+ 1e+08 9e+07 Dirac,fn,-3529.440 8e+07 7e+07 6e+07 5e+07 4e+07 squared radial function 3e+07 2e+07 1e+07 finite nucleus Dirac 0 0 0.0002 0.0004 0.0006 0.0008 0.001 r / bohr Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction Squared orbital of Hg79+ 1e+08 9e+07 Dirac, fn, -3529.440 a.u. Dirac, pn, -3532.192 a.u. 8e+07 7e+07 6e+07 5e+07 4e+07 squared radial function 3e+07 point charge nucleus Dirac 2e+07 1e+07 finite nucleus Dirac 0 0 0.0002 0.0004 0.0006 0.0008 0.001 r / bohr Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction Squared orbital of Hg79+ 1e+08 9e+07 DKH(5,0), fn, -3530.397 a.u. Dirac, fn, -3529.440 a.u. 8e+07 Dirac, pn, -3532.192 a.u. 7e+07 6e+07 finite nucleus DKH(n,0) 5e+07 4e+07 squared radial function 3e+07 point charge nucleus Dirac 2e+07 1e+07 finite nucleus Dirac 0 0 0.0002 0.0004 0.0006 0.0008 0.001 r / bohr Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction Squared orbital of Hg79+ 1e+08 9e+07 DKH(5,0), fn, -3530.397 a.u. Dirac, fn, -3529.440 a.u. 8e+07 basis set artifact Dirac, pn, -3532.192 a.u. DKH(5,0), pn, -3532.367 a.u. 7e+07 point charge nucleus DKH(n,0) 6e+07 finite nucleus DKH(n,0) 5e+07 4e+07 squared radial function 3e+07 point charge nucleus Dirac 2e+07 DKH(n,0) 1e+07 finite nucleus Dirac 0 0 0.0002 0.0004 0.0006 0.0008 0.001 r / bohr Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction Squared orbital of Hg79+ 1e+08 9e+07 DKH(5,0), fn, -3530.397 a.u. Dirac, fn, -3529.440 a.u. 8e+07 basis set artifact Dirac, pn, -3532.192 a.u. DKH(5,0), pn, -3532.367 a.u. 7e+07 point charge nucleus DKH(n,0) DKH(5,0), pn, lbs, -3532.454 a.u. DKH(5,0), fn, lbs, -3530.397 a.u. 6e+07 finite nucleus DKH(n,0) 5e+07 4e+07 squared radial function 3e+07 point charge nucleus Dirac 2e+07 DKH(n,0) 1e+07 finite nucleus Dirac 0 0 0.0002 0.0004 0.0006 0.0008 0.001 r / bohr Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Introduction Squared orbital of Hg79+ 1e+08 9e+07 DKH(5,0), fn, -3530.397 a.u. Dirac, fn, -3529.440 a.u. 8e+07 basis set artifact Dirac, pn, -3532.192 a.u. DKH(5,0), pn, -3532.367 a.u. 7e+07 point charge nucleus DKH(n,0) DKH(5,0), pn, lbs, -3532.454 a.u. DKH(5,0), fn, lbs, -3530.397 a.u. 6e+07 finite nucleus DKH(n,0) DKH(2,0), pn, lbs, -3523.321 a.u. DKH(2,0), fn, lbs, -3521.878 a.u. 5e+07 4e+07 squared radial function 3e+07 point charge nucleus Dirac 2e+07 DKH(n,0) 1e+07 finite nucleus Dirac 0 0 0.0002 0.0004 0.0006 0.0008 0.001 r / bohr Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Outline Electron density in quantum theory Calculation of electron density distributions Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Outline Electron density in quantum theory Definition of electron density Approximate Hamiltonian operators Electron density distributions for various Hamiltonian operators Calculation of electron density distributions Effects from approximate Hamiltonian operators A difficult case: contact densities Electron density in conceptual theories Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Definition of electron density in quantum theory Electron density is an observable → Operator can be assigned Density operator defined as N (3) ρˆr = δ (r − ri) Xi=1 (compare to classical expression for set of point charges) Expectation value yields particle density ! ρ(r) = hΨ({ri},t)|ρˆr|Ψ({ri},t)i +∞ +∞ +∞ 3 3 3 ∗ = d r1 d r2 · · · d rN Ψ ({ri},t) ρr Ψ({ri},t) Z−∞ Z−∞ Z−∞ Charge density: ρc(r)= −e ρ(r) Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Definition of electron density in quantum theory Continuity equation The continuity equation is the fundamental equation relating electron and current density ∂ρ + ∇ · j = 0 ∂t → Can be used to define these quantities ! Deduction of continuity equation from expectation value of density operator uniquely defines electron distribution of an N-electron system in any QM (model) theory. Note: the continuity equation also follows from Heisenberg equation of motion for the density operator Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Definition of electron density in quantum theory Consider change of electron density with time Apply the Ehrenfest theorem dhΨ|Oˆ|Ψi i ∂Oˆ = hΨ|[H,ˆ Oˆ]|Ψi + Ψ Ψ dt ~ ∂t Insert density operator d Ψ({ri},t) ρr Ψ({ri},t) i = Ψ({ri},t) [H,ˆ ρˆr] Ψ({ri},t) dt ~ ∂ρ(r,t) −∇ · j ∂t | {z } | {z } (2nd term on r.h.s. vanishes as density operator does not depend on t) Choice of Hamiltonian Hˆ and wave function Ψ determine the current density j Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Outline Electron density in quantum theory Definition of electron density Approximate Hamiltonian operators Electron density distributions for various Hamiltonian operators Calculation of electron density distributions Effects from approximate Hamiltonian operators A difficult case: contact densities Electron density in conceptual theories Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Approximate Hamiltonian operators Dirac–Coulomb–Breit (DCB) operator Fundamental Hamiltonian for M nuclei and N electrons: M 2 N M M 2 N N pI ZI ZJ e HˆDCB = + hˆD(i)+ + g(i, j) 2mI |RI − RJ | XI=1 Xi=1 XI=1 J=XI+1 Xi=1 j=Xi+1 ‘Fully relativistic’ single-electron Hamiltonian M 2 ˆ e 2 ZI e hD(i)= cαi · pˆi + A + (βi − 1)mec − eφ − c |rˆi − RI | XI=1 Dirac matrices αi =(αx, αy, αz) and β 10 0 0 0 0 σ 0 0 σ 0 0 σ 0 0 x 0 0 y 0 0 z 01 0 0 αx = , αy = , αz = , β = − σ 0 0 σ 0 0 σ 0 0 0 0 1 0 „ x 0 0 « „ y 0 0 « „ z 0 0 « „ 0 0 0 −1 « which are built from the Pauli spin matrices σ =(σx,σy,σz) 0 1 0 i 1 0 σx = 1 0 , σy = −i 0 , σz = 0 −1 ` ´ ` ´ ` ´ Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Approximate Hamiltonian operators Electron–electron interaction Is actually the reason why DCB is approximate; first-quantized Coulomb–Breit operator (Gaussian units) e2 α · α (α · (r − r ))(α · (r − r )) g(i, j)= 1 − i j − i i j j i j |r − r | 2 2|r − r |2 i j i j → Includes Coulomb and magnetic interactions → Derives from classical Darwin energy → Considers retardation effects on transmission of interaction Gaunt interaction is first approximation to Coulomb–Breit operator e2 g(i, j)= [1 − αi · αj] |ri − rj| (neglects retardation; includes instantaneous magnetic interaction) Electron Density in (Relativistic) Quantum Theory Markus Reiher GRC 2010 Approximate Hamiltonian operators Consequences of the Dirac formalism Since 4×4 matrices appear in the Dirac Hamiltonian, orbitals have four components (4-spinors) 1 2 3 4 T L S T ψi = (ψi , ψi , ψi , ψi ) = (ψi , ψi ) L S [ψi ≡large component (2-spinor) ψi ≡small component (2-spinor)] Dirac equation yields positive- (electronic) and negative-energy (positronic) eigenstates Negative-energy states are source of pathologies: → Hamiltonian is no longer bound
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