NANO266 - 04 - Introduction to Density Functional Theory

Transcript

1. Introduction to Density Functional Theory Shyue Ping Ong
2. Two broad approaches to solving the Schrödinger equation Variational Approach Expand wave function as a linear combination of basis functions Results in matrix eigenvalue problem Clear path to more accurate answers (increase # of basis functions, number of clusters / configurations) Favored by quantum chemists Density Functional Theory In principle exact In practice, many approximate schemes Computational cost comparatively low Favored by solid-state community NANO266 2
3. NANO266 3 The birth place of DFT > 20,000 citations each! (Web of Science, March 2015)
4. NANO266 4 http://www.nature.com/news/the-top-100-papers-1.16224
5. The Hohenberg-Kohn Theorems The Hohenberg-Kohn existence theorem •  For any system of interacting particles in an external potential Vext (r), the density is uniquely determined (in other words, the external potential is a unique functional of the density). The Hohenberg-Kohn variational theorem •  A universal functional for the energy E[n] can be defined in terms of the density. The exact ground state is the global minimum value of this functional. NANO266 5 Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas, Phys. Rev., 1964, 136, B864, doi:10.1103/ PhysRev.136.B864.
6. Proof of H-K Theorem 1 NANO266 6 Assume there are two different external potentials, V a and V b , (with corresponding Hamiltonians H a and H b ) consistent with the same ground state density, ρ0 . Let the ground state wave function and energy for each Hamiltonian be ψ0 and E 0 . From the variational theorem: E 0,a < ψ0,b H a ψ0,b E 0,a < ψ0,b H a − H b ψ0,b + ψ0,b H b ψ0,b E 0,a < ψ0,b V a −V b ψ0,b + E 0,b E 0,a < (V a −V b )ρ0 (r)dr ∫ + E 0,b Similarly, E 0,b < (V b −V a )ρ0 (r)dr ∫ + E 0,a Summing the two, we have E 0,a + E 0,b < E 0,b + E 0,a
7. Consequence of H-K theorems Schrodinger equation in 3N electronic coordinates reduced to solving for electron density in 3 spatial coordinates! In theory, H-K theorems are exact. Unfortunately •  No recipe for what the functional is •  In other words, beautiful theory, but practically useless (until one year later…) NANO266 7
8. A switch of notation Electronic Hamiltonian (atomic units) From H-K theorem, energy (and everything) is a functional of the density. Therefore, NANO266 8 H = − 1 2 ∇i 2 i ∑ 2 − Z k r ik k ∑ i ∑ + 1 r ij j ∑ i ∑ H = T +V ne +V ee E[ρ(r)]= T[ρ(r)]+V ne [ρ(r)]+V ee [ρ(r)]
9. The Kohn-Sham Ansatz Fictitious system of electrons that do not interact and live in an external potential (Kohn-Sham potential) such that ground-state charge density is identical to charge density of interacting system NANO266 9 E[ρ(r)]= T ni [ρ(r)]+V ne [ρ(r)]+V ee [ρ(r)]+ ΔT[ρ(r)]+ ΔV ee [ρ(r)] Non-interacting KE Corrections to non- interacting KE and Vee
10. The Kohn-Sham Equations NANO266 10 E[ρ(r)]= − 1 2 ψi *(r)∇2ψi (r)dr ∫ − ψi *(r) Z k r ik ψi (r)dr ∫ k ∑ % & ' ( ) * i ∑ + 1 2 ρ(r') r i − r' ψi (r) 2 dr ∫ ∫ 'dr i ∑ +E xc [ρ(r)] h i KS = − 1 2 ∇2 − Z k r ik + k ∑ 1 2 ρ(r') r i − r' dr ∫ ∫ '+V xc [ρ(r)] KS one-electron operator h i KSψi (r) =εi ψi (r) ρ(r) = ψi (r) i ∑ 2
11. Solution of KS equations Follows broadly the general concepts of HF SCF approach, i.e., construct guess KS orbitals within a basis set, solve secular equation to obtain new orbitals (and density matrix) and iterate until convergence Key differences between HF and DFT •  HF is approximate, but can be solved exactly •  DFT is formally exact, but solutions require approximations (Vxc ) NANO266 11
12. Flowchart for KS solution NANO266 12
13. Limits of KS Theory Eigenvalues are not the energies to add / subtract electrons, except the highest eigenvalue in a finite system is the negative of the ionization energy. NANO266 13 Silicon bandstructure from www.materialsproject.org Exp bandgap: 1.1eV But KS orbitals and energies can be used as inputs for other many- body approaches such as quantum Monte Carlo.
14. Exchange-Correlation Thus far, we have constructed an elegant system, but we have convenient swept all unknowns into the mysterious Vxc . Unfortunately, the H-K theorems provide no guidance on the form of this Vxc . With approximate Vxc , DFT can be non-variational. What’s the simplest possible assumption we can make? NANO266 14 E[ρ(r)]= T ni [ρ(r)]+V ne [ρ(r)]+V ee [ρ(r)]+V xc [ρ(r)]
15. Local Density Approximation (LDA) Independent particle kinetic energies and long-range Hartree contributions have been separated out => Remaining xc term can be reasonably approximated (to some degree) as a local or nearly local functional of density LDA: XC energy is given by the XC energy of a homogenous electron gas with the same density at each coordinate With spin (LSDA): NANO266 15 E xc LDA[ρ]= ρ(r) εx hom (ρ)+εc hom (ρ) ! " # $ ∫ dr E xc LSDA[ρ↑,ρ↓]= ρ(r) εx hom (ρ↑,ρ↓)+εc hom (ρ↑,ρ↓) # $ % & ∫ dr
16. LDA, contd For a homogenous electron gas (HEG), the exchange energy can be analytically derived as: Correlation energy for HEG has been accurately calculated using quantum Monte Carlo methods NANO266 16 E x σ [ρ]= − 3 4 6 π ρσ " # $ % & ' 1/3
17. Does LDA work? NANO266 17 Haas, P.; Tran, F.; Blaha, P. Calculation of the lattice constant of solids with semilocal functionals, Phys. Rev. B - Condens. Matter Mater. Phys., 2009, 79, 1–10, doi:10.1103/PhysRevB.79.085104. Over-binding evident Radial density of the Ne atom, both exact and from an LDA calculation
18. Error in LDA xc energy density of Si Exchange energies are too low and correlation energies that are too high => Cancellation of errors! NANO266 18 Hood, R.; Chou, M.; Williamson, a.; Rajagopal, G.; Needs, R. Exchange and correlation in silicon, Phys. Rev. B, 1998, 57, 8972–8982, doi:10.1103/PhysRevB.57.8972.
19. Phases of Si from LDA – an early success story NANO266 19 Yin, M. T.; Cohen, M. L. Theory of static structural properties, crystal stability, and phase transformations: Application to Si and Ge, Phys. Rev. B, 1982, 26, 5668–5687, doi:10.1103/ PhysRevB.26.5668.