Module 6 1. Density functional theory

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Module 6 1. Density functional theory Updated May 12, 2016

B A DDFT C K A bird s-eye view of density-functional theory Authors: Klaus Capelle G http://arxiv.org/abs/cond-mat/0211443 R https://trac.cc.jyu.fi/projects/toolbox/wiki/dft O P. Hohenberg and W. Kohn. Phys. Rev., 136:864B, 1964. U W. Kohn and L.J. Sham. Phys. Rev., 140:1133A, 1965. N

Beyond Hartree Fock theory Wavefunction methods: Post-HF Add determinants + + + + +... Configuration interaction: excite from reference Ψ Optimize weight of each determinant Density functional methods: DFT Introduces correlation in H via the exchange correlation functional Correlation functional derived from systems with dynamical correlation. Static correlation is still lacking. Still one determinant with integer occupation! Static correlation (due to near degeneracy) can be introduced with fractional occupation. Free energy functionals (metals) or Multi-configurational DFT

Density functional theory, DFT A formally exact and practically approximate but cheap way to access the correlation energy = E exact E Hartree Fock Static correlation: Near degeneracy one determinant not sufficient Dissociation limits AND certain systems F 2(g), NiO(s) Dynamical correlation: Electrons avoid each other instantaneously Fe(CO) 5 Dispersion forces Metals Optimized geometry Fe C axial Fe C equatorial (Å) Exp. 1.807 1.827 Hartree Fock 2.071 1.884 HF+static correlation 1.884 1.879 HF+static+dynamical 1.810 1.803 DFT BLYP 1.832 1.832 DFT PBE 1.809 1.807 Static correlation problem remains in practical implementations of DFT!

Bring order in the chaos of density functionals Slater Xα B3LYP PW VWM B3PW91 PBE0 HSE06 Becke88 BPW91 BLYP HCTH PBE P86 BHandH MO8-HX TPSS CAM-B3LYP B3LYP-D BMK Becke95 B2PLYP

Bring order in the chaos of density functionals Slater Xα LDA PW VWM Becke88 BPW91 Pure DFT (LDA and GGA) BLYP HCTH PBE P86 B3LYP B3PW91 PBE0 HSE06 Hybrid DFT (DFT+HF) BHandH CAM-B3LYP B3LYP-D MO8-HX BMK Meta hybrid (tau-gga) TPSS Becke95 Double hybrid (KS+MP2) B2PLYP

Useful link www.nist.gov Computational Chemistry Comparison and Benchmark Database

Bring order in the chaos of density functionals Reduced density matrix DFT Hohenberg Kohn theorems Orbital-free DFT only electron density Kohn Sham formulation of DFT 1-electron orbitals as in HF Various density functional approximations: Local density approximation Generalized gradient approximation Hybrid functionals exact HF exchange Meta hybrid functionals Double hybrid functionals D-functionals dispersion term added local semi-local non-local

Density functional theory, DFT Wavefunction methods search for: Exact H complicated Ψ Ψ(r 1, r 2,..., r i,..., r n) wavefunction, n is the number of electrons DFT methods search for ρ(r) electron probability density ρ(r) is a much simpler entity than Ψ(r 1, r 2,..., r i,..., r n) The main advantages with DFT: Much faster than Hartree Fock, plus correlation (wavefunction), particularly for large systems Easier to use (and faster) than multireference methods Modern DFT can often be as accurate as correlated wavefunction methods

Density functional theory Brief history In 1927, Thomas and Fermi made the first attempt to treat the many-body problem of the electronic structure of an atom in a statistical model to approximate the electron distribution. DFT owes it credibility and applicability to the work of Walter Kohn, Nobel laureate 1998 through the Hohenberg Kohn theorem 1964 and the Kohn Sham formulation 1965 of DFT Kohn at the 2012 Lindau Nobel Laureate Meeting, Vien, Austria. CC BY-SA 3.0, via Wikimedia Commons DFT has been very popular for calculations in solid state physics since the 1970s. However, DFT was not considered accurate enough for calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better model the exchange and correlation interactions. Solid state physics meets molecular physics! E.g. basis set limitations plane waves versus local Gaussian functions

Preliminaries for DFT The electronic Hamiltonian within the Born Oppenheimer approximation E = T e + U ee + V en + V NN H el = 1 2 + 1 1 + V 2 r 2 ext(r) i 2 r ij i What kind of operator is the kinetic energy operator 1 2 r? 2 i 0-electron operator 1-electron operator 2-electron operator i,j 2

Preliminaries for DFT The electronic Hamiltonian within the Born Oppenheimer approximation E = T e + U ee + V en + V NN H el = 1 2 + 1 1 + V 2 r 2 ext(r) i 2 r ij i What kind of operator describes electron electron repulsion 1 2 0-electron operator 1-electron operator 2-electron operator i,j i,j 1 r ij?

Preliminaries for DFT The electronic Hamiltonian within the Born Oppenheimer approximation E = T e + U ee + V en + V NN H el = 1 2 + 1 1 + V 2 r 2 ext(r) i 2 r ij What kind of operator describes V en? 0-electron operator 1-electron operator 2-electron operator i i,j

Preliminaries for DFT The electronic Hamiltonian within the Born Oppenheimer approximation E = T e + U ee + V en + V NN H el = 1 2 + 1 1 + V 2 r 2 ext(r) i 2 r ij What kind of operator describes V NN? 0-electron operator 1-electron operator 2-electron operator i i,j

Preliminaries for DFT The electronic Hamiltonian within the Born Oppenheimer approximation E = T e + U ee + V en + V NN One-electron operators H el = 1 2 + 1 1 + V 2 r 2 ext(r) i 2 r ij Two-electron operators V ext = V en = A kinetic {}}{ T + H el = i i Coulomb {}}{ J + ZA r ia exchange {}}{ K + i,j The external potential and N elec determine Ψ and E tot electrons, nuclei {}}{ V How can we use the ground state electron density? E tot = F + V = T + U + V[ρ(r)] U = J[ρ(r)] + K + E correlation J Hartree[ρ] and V[ρ] are known. V[ρ] = ZAρ(r) R A r dr A Self-interaction! J Hartree[ρ] = 1 ρ(r)ρ(r ) 2 r r dr Self-interaction in J and K cancels in HF! All include correlation

Preliminaries for DFT Reduced Density Matrix Functionals In statistical physics, the density matrix is used to describe the statistical mixture of pure states. Probability density: Ψ (r 1, r 2,..., r N 1, r N)Ψ(r 1, r 2,..., r N 1, r N) Density matrix: Ψ (r 1, r 2,..., r N 1, r N)Ψ(r 1, r 2,..., r N 1, r N) Density matrix operator: e βh Tr(e βh ) Resolution of identity 1 = i Ψ i Ψ i (based on complete set of eigenstates) i Resolution in eigenfunctions e βe i Ψ i Ψ i i e βe i

Preliminaries for DFT Reduced Density Matrix Functionals In statistical physics, the density matrix is used to describe the statistical mixture of pure states. Probability density: Ψ (r 1, r 2,..., r N 1, r N)Ψ(r 1, r 2,..., r N 1, r N) Density matrix: Ψ (r 1, r 2,..., r N 1, r N)Ψ(r 1, r 2,..., r N 1, r N) Reduced density matrices Electron density: ρ(r 1) = N... Ψ (r 1, r 2,..., r N)Ψ(r 1, r 2,..., r N)dr 2dr 3,... dr N

Preliminaries for DFT Reduced Density Matrix Functionals In statistical physics, the density matrix is used to describe the statistical mixture of pure states. Probability density: Ψ (r 1, r 2,..., r N 1, r N)Ψ(r 1, r 2,..., r N 1, r N) Density matrix: Ψ (r 1, r 2,..., r N 1, r N)Ψ(r 1, r 2,..., r N 1, r N) Reduced density matrices Electron density: ρ(r 1) = N... Ψ (r 1, r 2,..., r N)Ψ(r 1, r 2,..., r N)dr 2dr 3,... dr N 1st order DM ρ(r 1) = γ 1(r 1, r 1) : γ(r 1, r 1) = N... Ψ (r 1, r 2,..., r N)Ψ(r 1, r 2,..., r N)dr 2dr 3,... dr N

Preliminaries for DFT Reduced Density Matrix Functionals In statistical physics, the density matrix is used to describe the statistical mixture of pure states. Probability density: Ψ (r 1, r 2,..., r N 1, r N)Ψ(r 1, r 2,..., r N 1, r N) Density matrix: Ψ (r 1, r 2,..., r N 1, r N)Ψ(r 1, r 2,..., r N 1, r N) Reduced density matrices Electron density: ρ(r 1) = N... Ψ (r 1, r 2,..., r N)Ψ(r 1, r 2,..., r N)dr 2dr 3,... dr N 1st order DM ρ(r 1) = γ 1(r 1, r 1) : γ(r 1, r 1) = N... Ψ (r 1, r 2,..., r N)Ψ(r 1, r 2,..., r N)dr 2dr 3,... dr N 2nd order DM: γ(r 1, r 2, r 1, r 2) = N(N 1)... Ψ (r 1, r 2,..., r N)Ψ(r 1, r 2,..., r N)dr 3,... dr N

Preliminaries for DFT Reduced Density Matrix Functionals E tot = T + U + V Recall that the kinetic energy calculated for an n-electron wavefunction Ψ is T = Ψ T Ψ = Ψ 1 2 2 r Ψ. 2 i i Which of the following reduced density matrices is needed for evaluation of the functional T[γ(?)]? γ 1(r 1, r 1) γ 1(r 1, r 1) γ 2(r 1, r 2, r 1, r 2) γ 2(r 1, r 2, r 1, r 2)? Reduced density matrices Electron density: ρ(r 1) = N... Ψ (r 1, r 2,..., r N)Ψ(r 1, r 2,..., r N)dr 2dr 3,... dr N 1st order DM ρ(r 1) = γ 1(r 1, r 1) : γ(r 1, r 1) = N... Ψ (r 1, r 2,..., r N)Ψ(r 1, r 2,..., r N)dr 2dr 3,... dr N 2nd order DM: γ(r 1, r 2, r 1, r 2) = N(N 1)... Ψ (r 1, r 2,..., r N)Ψ(r 1, r 2,..., r N)dr 3,... dr N

Preliminaries for DFT Reduced Density Matrix Functionals E tot = T + U + V T[γ 1(r 1, r 1)] = 1 2 γ 1(r 1, r 1) 2 dr 1 r 1 =r 1 γ2(r 1, r 2, r 1, r 2) U[γ 2(r 1, r 2, r 1, r 2)] = 1 dr 1dr 2 2 r 1 r 2 V[γ 1(r 1, r 1)] = V[ρ(r 1)] = a,nuclei Za(R a)ρ(r 1) dr 1 R a r 1 Exact! Local! Reduced density matrices Electron density: ρ(r 1) = N... Ψ (r 1, r 2,..., r N)Ψ(r 1, r 2,..., r N)dr 2dr 3,... dr N 1st order DM ρ(r 1) = γ 1(r 1, r 1) : γ(r 1, r 1) = N... Ψ (r 1, r 2,..., r N)Ψ(r 1, r 2,..., r N)dr 2dr 3,... dr N 2nd order DM: γ(r 1, r 2, r 1, r 2) = N(N 1)... Ψ (r 1, r 2,..., r N)Ψ(r 1, r 2,..., r N)dr 3,... dr N

Density matrix natural orbitals = eigenvectors of the density matrix Exact T[ρ exact] = n i φ i 1 2 2 φ i N = ρ exact = i=1 n i φ i 2 i=1 n i, 0 < n i < 1 i=1 If we have discrete occupations of natural orbitals: T[ρ] = N electrons i=1 φ i 1 2 2 φ i occupied orbitals (n i = 1)

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