Electron Correlation

Electron Correlation

Levels of QM Theory HΨ=EΨ Born-Oppenheimer approximation Nuclear equation: H n Ψ n =E n Ψ n Electronic equation: H e Ψ e =E e Ψ e Single determinant SCF Semi-empirical methods Correlation approaches: 1.WF based. Multi-determinantial methods - variational (MCSCF, CI) - non-variational (MBPT, CC) 2. DFT: - orbital free (ρ based) - Single determinantial (Kohn-Sham)

Why correlation? The Hartree Product Φ HP (x 1 ; x 2 ; ; x n ) = φ 1 (x 1 ) φ 2 (x 2 ) φ n (x n ) is completely uncorrelated, in the sense that the probability of simultaneously finding electron 1 at x 1, electron 2 at x 2, etc., is given by : Φ HP (x 1 ; x 2 ; ; x n ) 2 dx 1 dx 2 dx n = φ 1 (x 1 ) 2 φ 2 (x 2 ) 2 φ n (x n ) 2 dx 1 dx 2 dx n which is the probability of finding electron 1 at x 1 times the probability of finding electron 2 at x 2, etc... the product of the probabilities. This makes the Hartree Product an independent particle model. Electrons move independently; their motion is uncorrelated

The Hartree-Fock approximation The n-electronic wave function ψ in the case of Hartree-Fock (HF) approximation: HF 1(1) 2(1)... n(1) (1,2,... n) det 1(2) 2(2)... n(2)......... 1( n) 2( n)... n ( n) One-electron density: 2 HF x, y, z n j j j n j is the occupation number (n j = 0,1,2)

Hartree-Fock Energy n n n 1 E h ( J K ) i ij ij i1 2 i1 j1 Sums run over occupied orbitals. h i is the one-electron integral, J ij is the Coulomb integral, and K ij is the exchange integral. 1 Z e 1 Z e h dr ( r)[ ] ( r) ( dr [ ( r) ( r) ( r)] 2 2 * I * I i i i i i i 2 I R I e 2 RI e 1 1 J dr dr ( r ) ( r ) ( r ) ( r ) dr dr ( r ) ( r ) * * ij 1 2 i 1 i 1 j 2 j 2 1 2 i 1 j 2 r12 r12 1 K dr dr ( r ) ( r ) ( r ) ( r ) * * ij 1 2 i 1 j 1 j 2 i 2 r12

WF approaches beyond the SCF. Correlated Methods (CM) Include more explicit interaction of electrons than HF : E corr = E - E HF, where E = H Most CMs begin with HF wavefunction, then incorporate varying amounts of electron-electron interaction by mixing in excited state determinants with ground state HF determinant The limit of infinite basis set & complete electron correlation is the exact solution of Schrödinger equation (which is still an approximation)

The N-electron Basis A collection of atom-centered Gaussian functions can be used as a basis set for expanding one-electron functions (molecular orbitals). b c c c... c j sj s 1 j 1 2 j 2 bj b s1 We need to solve the electronic Schrodinger equation to get Ψ e (r 1 ; r 2 ; ; r n ), a function of n electrons. What can we use as a basis for expanding Ψ e? Slater determinants are proper n-electron basis functions: they are functions which can be used to expand any antisymmetric) n-electron function. In the limit of an infinite number of Slater determinants, any n- electron function can be expanded exactly.

Configuration Interaction (CI) and other correlation methods CI method for many-electron WF N HF e r1 r2 rn c0 0 r1 r2 rn ci I r1 r2 rn I 1 (,,..., ) (,,..., ) (,,..., ) where N is the number of excited determinants Ways of Coefficients c I definition: 1. Variations - CI 2. Perturbation expansions MBPT 3. SCF-like iterations Coupled Cluster

Classification of CI methods The Hartree-Fock reference" determinant Φ 0 should be the leading term. Expect the importance of other configurations to drop off rapidly as they substitute more orbitals. ab... c Let denote a determinant ij... k which differs from Φ 0 by replacing occupied orbitals ij k with virtual orbitals ab c. HF a a ab ab abc abc c c c c... 0 0 i i ij ij ijk ijk Reference Singles Doubles Triples CIS CISD CISDT

Example of CI calculation: H 2 molecule

WF methodology: infinite basis set Schrodinger 6-311++G** 6-311G** 6-311G* 3-21G* 3-21G STO-3G increasing size of basis set increasing accuracy, increasing cpu time STO increasing level of theory full e - HF CI QCISD QCISDT MP2 MP3 MP4 correlation

Density Functional Theory 1998 Nobel Prize in Chemistry (Kohn and Pople) recognized work in this area. Main idea: Use the density instead of complicated many electron wavefunctions. Basic approach: minimize the energy with respect to the density. Relationship of energy to density is the functional E[ρ] (true form of this functional is unknown: use approx.)

One-electronic density The probability density ρ of finding an electron (ANY!!!) in the neighborhood of point (x,y,z) is ( x, y, z) n... ( x, y, z, x,..., z, m,..., m ) dx... dz all m s In most cases - knowing the ρ is knowing the system! Aˆ A( x, y, z) ( x, y, z) dxdydz Z e( x, y, z) dxdydz en m s Main question How does ρ look like? 2 n s s 2 n 1 n 2

The energy functional = density functional (W. Kohn) Model Hamiltonian: H λ = T+(1-λ)V ext (λ)+λv ee 0 λ 1 ; ρ λ=1 = ρ λ=0 = ρ H λ=1 = H H λ=0 = T+V ext (0) single-e-h Kohn - Sham : E=E[Ψ]= Ψ*ĤΨdV =E[ρ]=? HF: E HF [ρ]=t[ρ(φ)]+e ne [ρ]+(j[ρ]+k[ρ(φ)] DFT: E[ρ]=T[ρ]+E ne [ρ]+(j[ρ]+k[ρ]+e cor [ρ])- single-electron theory including correlation! Alternatives: x y z,, n j j j 2 I II

Orbital-free DFT E[ ] T[ ] E [ ] J[ ] K[ ] E [ ] ne In the HF method E corr [ρ] is missed corr General case: Homogenious e-gas (Tomas-Fermi-Dirac) simplest case: E [ ] T [ ] E [ ] J[ ] K [ ] TFD TF ne D

Kinetic and exchange functionals corrections Kinetic functional: Exchange functional:

Correlation functionals An example Lee, Parr, Yang (LYP) E corr [ρ]) :

Kohn-Sham theory (DFT with orbitals) Exact WF: x, y, z n j j n j is the generalized occupation number (n j 0 or 1); φ j natural orbitals j=1,, Kohn-Sham (KS) WF: 1(1) 2(1)... n(1) KS (1,2,... n) det 1(2) 2(2)... n(2)......... 1( n) 2( n)... n ( n) Exact exchange + KS kinetic functional - very precise j 2

Jacob s Ladder