Concepts and Math Problems in Electronic Structure Calculations - PowerPoint PPT Presentation

Concepts and Math Problems in Electronic Structure Calculations Lin-Wang Wang Scientific Computing Group • Many-body Schrodinger’s equations • Density functional theory and single particle equation • Selfconsistent calculation/nonlinear equation/optimization • Optical properties • Basis functions for wavefunctions • Pseudopotentials • Technical points in planewave calculations

Many body Schrodinger’s equation Schrodinger’s equation (1930’s): the great result of reductionism ! ∂ 1 1 Z ∑ ∑ ∑ − ∇ + + Ψ = Ψ 2 { } ( r ,.. r , t ) i ( r ,.. r , t ) i − − 1 N ∂ 1 N 2 | r r | | r R | t i i , j i , R i j i All the material science and chemistry is included in this equation ! The challenge: to solve this equation for complex real systems. Ψ = − ω Ψ i t ( r ,.. r , t ) e ( r ,.. r ) For stationary solution: 1 N 1 N 1 1 Z ∑ ∑ ∑ − ∇ + + Ψ = Ψ 2 { } ( r ,.. r ) E ( r ,.. r ) i 1 N 1 N − − 2 | r r | | r R | i i , j i , R i j i The famous Einstein formula: E= ħω Ground state: the lowest E state; Excited state: higher E state.

Many body wavefunctions Electrons are elementary particles, two electrons are indistinguishable Ψ = α Ψ ( r ,.. r ... r ... r ) ( r ,.. r ... r ... r ) 1 i j N 1 j i N 2 = α 1 α = 1 , Boson: phonon, photon, W-boson, Higgs-boson, …. (usually particles which transmit forces) α = − 1 , Fermion: electron, proton, neutron, quark,muon, …. (usually particles which constitute the matter) For our case: electron Ψ = − Ψ ( r ,.. r ... r ... r ) ( r ,.. r ... r ... r ) 1 i j N 1 j i N

Many body wavefunctions Ψ = − Ψ ( r ,.. r ... r ... r ) ( r ,.. r ... r ... r ) antisymmetric 1 i j N 1 j i N ∫∫∫ Ψ = 2 | ( r ,... r ) | dr ... dr N normalized 1 N 1 N One example of the antisymmetric wavefunction: Slater determinate ……… Φ 1 (r 1 ) Φ N (r 1 ) Ψ = ( 1 r .... r ) ……………………… N Φ 1 (r N ) ……… Φ N (r N ) This is the exact solution for: 1 1 Z ∑ ∑ ∑ − ∇ + + Ψ = Ψ 2 { } ( r ,.. r ) E ( r ,.. r ) i 1 N 1 N − − 2 | r r | | r R | i i , j i , R i j i The partial differential equation becomes separable

Another way to look at it: variational methods 1 1 Z ∑ ∑ ∑ ∫∫∫ = Ψ − ∇ + + Ψ 2 E ( r ,.. r ){ } ( r ,.. r ) dr .. dr 1 N i − − 1 N 1 N 2 | r r | | r R | i i , j i , R i j i The ground state corresponds to the optimized state Ψ which is antisymmetric and normalized. So, we can try variational Ψ for whatever expressions we like variational linear eigen value nonlinear problem problem on simplified functions approximation Plug in the Slater determinate for Ψ , we have (Hartree-Fock equation): ϕ ρ ( r ) 1 Z ( r ' ) ∑ ∑∫ ∫ − ∇ + + ϕ + j ϕ ϕ = ϕ 2 { dr ' } ( r ) ( r ' ) ( r ' ) dr ' E ( r ) i j i i i − − − 2 | r R | | r r ' | | r r ' | R j

Some concepts and terminologies ……… Φ 1 (r 1 ) Φ N (r 1 ) Ψ = ( 1 r .... r ) ……………………… N N+2 N+1 Φ 1 (r N ) ……… Φ N (r N ) E i Φ j (r) : single particle orbital N One orbital can only have one electron (2 include spin) ------- Pauli exclusion principle Φ 1 Φ 2 ,… Φ N , the N occupied single particle orbitals 2 1 We also have: Φ N+1 Φ N+2 , ,… the unoccupied orbitals Using one of Φ N+1 Φ N+2 , ,… to replace one of Φ 1 Φ 2 ,… Φ N , the resulting Slater determinant will correspond to one excited state − ≈ − (band gap) E E E E For the lowest excited state: + 1 excited ground N N

Energy breakup ρ ρ 1 Z 1 ( r ' ) ( r ) ∑ ∑ ∫ ∫ ∫ HF = − ϕ ∇ 2 ϕ + ρ + E ( r ) ( r ) dr ( r ) dr dr ' dr tot i i − − 2 | r R | 2 | r r ' | i R kinetic Electron-ion Electron Coulomb ϕ ϕ ϕ ϕ ( r ) ( r ) ( r ' ) ( r ' ) ∑∫ + j i j i dr ' dr − | r r ' | i , j Exchange energy E corr =E exact -E HF Whatever left from HF Kinetic: ~ 40 eV/atom Coulomb: ~ 40 eV/atom Exchange: ~ 20eV/atom Correlation: ~ 4 eV/atom Typical chemical bond: ~ 2 eV Every term is important For chemical accuracy, we need: ~ 0.05 eV/atom

Different configurations: CI electron j,c … … Φ 1 (r 1 ) Φ j,c (r 1 ) Φ N (r 1 ) N+2 …………………………… SD conf (r 1 ,..r N )= N+1 … Φ 1 (r N ) … Φ j,c (r N ) Φ N (r N ) N CI: configuration interaction hole i,v ∑ Ψ = ( r ,... r ) C ( config ) SD ( r ,..., r ) 1 N config 1 N 2 config 1 The number of configuration is exponential, only feasible for a few atom systems. Judicious selection of configurations: � MP2, coupled-cluster, etc Traditional quantum chemistry approaches

More on variational many-body wavefunctions e One electron at r will repulse other Correlation effects: e e e electrons near r due to Coulomb inter. e ……… Φ 1 (r 1 ) Φ N (r 1 ) ∑ ∑ Ψ = − χ − − ( 1 r .... r ) exp[ ( r ) u (| r r |)] ……………………… N i i j i ij Jastrow factor Φ 1 (r N ) ……… Φ N (r N ) Unfortunately, cannot break down the following integration. 1 1 Z ∑ ∑ ∑ ∫∫∫ = Ψ − ∇ + + Ψ 2 E ( r ,.. r ){ } ( r ,.. r ) dr .. dr 1 N i − − 1 N 1 N 2 | r r | | r R | i i , j i , R i j i Using Monte-Carlo method to do the integration: variational quantum MC.

Diffusion quantum Monte-Carlo approach ∂ 1 1 Z ∑ ∑ ∑ − ∇ + + Ψ = Ψ 2 { } ( r ,.. r , t ) i ( r ,.. r , t ) i − − 1 N ∂ 1 N 2 | r r | | r R | t i i , j i , R i j i This looks like a classical diffusion equation with finite temperature ∂ { } r r r ∇ + − µ = 2 S(r,t) � particle density D V ( r ) S ( r , t ) S ( r , t ) ∂ t Using classical Monte-Carlo to simulate the random movements of particles in a 3N dimension space. Problem: S is always positive, but ψ has both positive and negative due to antisymmetry � the famous sign problem ! ……… Φ 1 (r 1 ) Φ N (r 1 ) to divide the 3N space into positive and negative Fix nodal approx: use ……………………… compartments, move articles within. Φ 1 (r N ) ……… Φ N (r N )

Another approach: the density matrix method 1 1 Z ∑ ∑ ∑ ∫∫∫ = Ψ − ∇ + + Ψ 2 E ( r ,.. r ){ } ( r ,.. r ) dr .. dr 1 N i − − 1 N 1 N 2 | r r | | r R | i i , j i , R i j i 1 Z ∑ ∫∫∫ = δ − δ − −∇ + + ρ 2 E ( r r ' ) ( r r ' ){ } ( r , r ' ; r , r ' ) dr dr ' dr dr ' 1 1 2 2 1 1 1 2 2 1 1 2 2 − − | r r | | r R | R 1 2 1 = ∫∫ ρ Ψ Ψ ( r , r ' ; r , r ' ) ( r , r , r ,... r ) ( r ' , r ' , r ,... r ) dr ... dr 1 1 2 2 1 2 3 N 1 2 3 N 3 N Great, reduce the N variable function into a 4 variable function !! Problem: ρ might not be N-representable ! ( r , r ' ; r , r ' ) 1 1 2 2 • Many necessary conditions to make ρ N-representable • The ρ is within some hyperdimension convex cone. • Linear programming optimization approach • Recent work: Z. Zhao, et.al, it can be very accurate, but it is still very expensive (a few atoms). • No known sufficient condition

The density functional theory = ∫∫ ρ Ψ Ψ ( r ) ( r , r , r ,... r ) ( r , r , r ,... r ) dr ... dr 1 1 2 3 N 1 2 3 N 2 N Any single particle ρ (r) is N-representable. Can we use ρ as one basic variable to determine all other things ? 1 1 ∑ ∑ ∑ ∫∫∫ = Ψ − ∇ + + Ψ 2 E ( r ,.. r ){ V ( r )} ( r ,.. r ) dr .. dr 1 N i i 1 N 1 N − 2 | r r | i i , j i i j V(r) is one basic variable which determines everything. So V � ρ , Now, can ρ � V ? ( ρ uniquely determine V) We need to prove: we cannot have V 1 � ρ , and V 2 � ρ .

Density functional theory (continued) 1 1 ∑ ∑ ∑ ∫∫∫ = Ψ − ∇ + + Ψ 2 E ( r ,.. r ){ V ( r )} ( r ,.. r ) dr .. dr 1 N i i 1 N 1 N − 2 | r r | i i , j i i j We need to prove: we cannot have V 1 � ρ , and V 2 � ρ . Suppose this happens, then V 1 � Ψ 1 � ρ and V 2 � Ψ 2 � ρ Ψ < Ψ E ( V , ) E ( V , ) • Since Ψ 1 is the variational minimum of V 1 , so: 1 1 1 2 ∫ ∫ Ψ + Ψ + ρ < Ψ + Ψ + ρ E [ ] E [ ] V ( r ) ( r ) dr E [ ] E [ ] V ( r ) ( r ) dr K 1 Coul 1 1 K 2 Coul 2 1 Eq(1) Ψ + Ψ < Ψ + Ψ E [ ] E [ ] E [ ] E [ ] K 1 Coul 1 K 2 Coul 2 Ψ < Ψ • Since Ψ 2 is the variational minimum of V 2 , so: E ( V , ) E ( V , ) 2 2 2 1 Ψ + Ψ < Ψ + Ψ E [ ] E [ ] E [ ] E [ ] Eq(2) K 2 Coul 2 K 1 Coul 1 Eq(1),(2) contradict with each other, so we cannot have V 1 � ρ , and V 2 � ρ We can also prove, smooth ρ is V-representable (i.e, can find a V � ρ ) In summary, V is a functional of ρ , thus everything is a functional of ρ