Density Functional Theory. Martin Lüders Daresbury Laboratory

Transcription

1 Density Functional Theory Martin Lüders Daresbury Laboratory

2 Ab initio Calculations Hamiltonian: (without external fields, non-relativistic) impossible to solve exactly!! Electrons Nuclei Electron-Nuclei interaction In principle, the system is fully determined by specifying the number, mass and charge of all its constituents. We have to deal with mutually interacting particles!!!

3 Born Oppenheimer Approximation Nuclei are 3 orders of magnitude heavier than electrons Their motion will be much slower Electrons can adjust instantaneously to nuclear positions. When calculating the electronic motion, nuclear coordinates can be treated as parameters. Formally we set the nuclear mass to infinity: the kinetic energy of the nuclei drops out. We are still left with the full interacting electronic problem!

4 The Electronic Problem T W V lattice potential: Coulomb potential of the nuclei Experiments show: electrons (often) behave as if they were independent particles This suggests an effective single-particle approach photo-emission

5 Mean field approximation (Hartree) Replace e-e interaction by interaction with the average potential of the other electrons (Hartree approximation) Independent electrons in effective field Equations to be solved: Potential depends on the density: Self-consistency!!

6 Mean field approximation (Hartree) Total Energy:,, double counting term Very crude approximation Quantum many-body effects completely neglected Self-Interaction error

7 Mean field approximation (Hartree-Fock) Hartree-Fock equations: - Total Energy: Coulomb and exchange energy double counting term

8 Beyond Hartree-Fock: Many-body perturbation theory: Finite systems: Atoms and Molecules consider higher order terms (finite systems): configuration interaction (CI), coupled clusters, etc. very expensive!! Extended systems: Solids finite order diverges infinite order resummation possible (RPA) (See GW) Alternative: Density Functional Theory

9 Hohenberg-Kohn Theorem One to one mapping between density and external potential: Therefore the many-body wave function is a functional of the density: Observables are functionals of the density n(r)

10 Hohenberg-Kohn Theorem In particular: Ground state energy is a functional of n(r) Ground state energy for given external potential: Functional F[n] is universal: It does not depend on the external potential It is the same for all electronic systems The exact form is unknown. It has to be approximated!

11 Variational principle: Hohenberg-Kohn Theorem Ground state density and energy can, in principle, be found by minimizing the energy functional: Euler-Lagrange equation: But: Free minimization is difficult!

12 Summary: 1. One-to-one mapping: Hohenberg-Kohn Theorem 2. Variational principle: 3. Universal functional: Extensions to degenerate ground states.

13 Kohn-Sham Approach The Hohenberg-Kohn (HK) theorem: independent of the interaction W(r,r ) Apply HK twice: HK HK Defines effective potential of a non-interacting system (Kohn- Sham system) which reproduces the exact ground state density of the full system.

14 Consider non-interacting system: Two ways to calculate the density: 1. Schrödinger equation: Kohn-Sham Approach 2. Euler-Lagrange equation: with

15 Rewrite universal functional F[n]: Kohn-Sham Approach Kinetic energy Classical Coulomb energy exchange energy Exchange-correlation energy: contains all many body effects, but is unknown.

16 Kohn-Sham Approach Insert this in Euler equation This corresponds to with Solve this with method 1. (Schrödinger equation)

17 Kohn-Sham Approach Kohn-Sham equations: (to be solved self-consistently) Total Energy: (electronic contribution) single particle energy + double counting terms

18 Input crystal structure; Practical realizations of DFT Lattice; basis vectors+atomic numbers. Implementation-specific input, eg Radius of augmentation spheres Which states are core, valence? (Ga 3d) Construct pseudopotential 0. Trial density n in (r) (e.g. Mattheis) 1. Generate V in (r) from n in (r). *Potentially rate-limiting step 2. Solve KS equations. *Usually rate-limiting step 3. Generate output density n out (r). *Usually rate-limiting step 4. If not self-consistent (n out n in )? make new n in and iterate

19 Functionals and the Variational Principle In practice: E KS = T s [n]+ E pot [n], E pot [n] = drv ext (r) n(r) + E H [n]+ E xc [n] No explicit functional for T KS. Use non-interacting ansatz: ψ i V eff ψ i = ψ i ε i ψ i = ε i We get T KS = T KS = eff 1 δ E 2 eff eff pot[ n( r)] KS ψi = ψi + () rψi = εψ i i, ( r) = H V V 2 i ε i i ψ i ψ i δ n drv eff (r) ψ * i (r)ψ i (r) = E i bs drv in (r)n out (r) E KS [n] = E bs drv[n(r)] n(r) + E pot [n]

20 The Harris-Foulkes Functional The Kohn-Sham functional is then a functional of n out : E KS [V in ] = E bs drv in (r)n out (r) + E pot [n out ] An alternative functional for which n in is the independent variable: E HF [n in ] = E bs drv[n in (r)]n in (r) + E pot [n in ] Original functional by Harris, but found independently and better explained by Foulkes and Haydock (PRB 39, ). At self-consistency, the two functionals give the same energy, but they differ for non-converged densities. E KS [V in ] E HF [n in ] = drv in (r)(n out (r) n in (r)) + E pot [n out ] E pot [n in ]

21 The Harris-Foulkes Functional II Write difference in terms of n = n out n in : E KS [V in ] E HF [n in ] = drv in (r)δn(r) + E pot [n out ] E pot [n in ] Expand as Taylor series in n 2 δ Epot δ Epot 2 Epot[ n] = Epo t[ nin] + Δn + ( Δn) δ n δ n δ Epot = ( Vext + VH + Vxc) in = V n= n δ n Linear term in E pot cancels V in n term. Quadratic term remains. 2 δ E in in pot[ n] EKS[ V ] EHF[ n ] = drr d ʹ Δn( r) Δn( rʹ ) δn() δn( ) r rʹ in

22 The Harris-Foulkes Functional II Establishes that E HF is stationary at self-consistency just as E KS is. Key points: E KS is variational (E KS [V in ] > E KS [V KS ]) while E HF is only stationary : it can be a maximum, a minimum, or a saddle point. Usually E HF approaches a maximum so E HF and E KS bound the selfconsistent energy. Typically E HF is better: less sensitive to n.

23 Extensions Spin-DFT for magnetic systems in principle, the magnetization is a functional of n in practice, this functional is unknown (and would have to be complicated) Therefore: include magnetization m(r) as additional density. include conjugate field B(r) which couples to m(r) xc functional: E xc [n,m] effective magnetic field effective field can maintain finite magnetization even without external field

24 Extensions of DFT Spin-DFT for magnetic systems collinear magnetism: magnetization in z-direction Kohn-Sham equation: effective potential:

25 Current-DFT include current density describes orbital magnetism Extensions of DFT Multi-Component DFT goes beyond Born-Oppenheimer approximation includes nuclei as quantum mechanical particles DFT for Superconductors based on multi-component DFT for electrons and nuclei includes superconducting order parameter as additional density ab initio description of superconducting properties

26 Kohn-Sham Approach Remarks: Kohn-Sham quasi-particles KS system is a auxiliary system, defined to reproduce the ground state density KS quasi particles do not represent the physical quasi particles: Infinite lifetime In general wrong excitation energies DFT (as discussed) is a ground state theory! In many cases KS quasi-particles are good approximations to the physical quasi particles, but they should be used with care.

27 Exchange-correlation (xc) Functionals Local Density Approximation (LDA): Approximate the system locally as homogeneous e-gas density is constant: n(r)=n xc energy per particle is a function of n Ansatz for E xc [n]: e xc (n) calculated from Quantum Monte Carlo (Ceperley, Alder) and then parameterized.

28 xc Functionals Local Spin-Density Approximation (LSDA): Approximate the system locally as homogeneous magnetic e-gas spin-densities are constant xc energy per particle is a function of (n,n ) Ansatz for E xc [n,n ]: e xc (n,n ) is parametrisation of Quantum Monte Carlo data

29 xc Functionals Generalized Gradient Approximations (GGA) L(S)DA is first term in an expansion in gradients of the density Inclusion of higher terms worsens results Reason: exact relations, fulfilled by L(S)DA! GGA s are constructed to incorporate some of these relations

30 Orbital functionals: Kinetic energy: xc Functionals KS already treats kinetic energy as orbital functional: Exact exchange (Fock term): unknown as density functional, but exact representation in terms of orbitals. implicit functional of the density:

31 xc Functionals Orbital functionals: optimized effective potential: apply chain rule for functional derivative: inverse response function OEP integral equation So far: no good orbital functional for correlations (adding LDA/GGA correlations removes some error cancellation)

32 Implementations of the DFT-KS The DFT Kohn-Sham approach (as well as other methods) is a formal theory, with no reference to how it is implemented. Many different approaches have been employed, especially for the LDA. Single-particle orbitals ψ i expanded in some basis set. Similarly n and V must be represented somehow. Classification #1: plane waves, or atom-centered real-space orbitals. Advantage of PW: can be made arbitrarily complete. Advantage of RS: efficient. Classification #2: treatment of the core electrons: Valence electrons must be orthogonalized to the inert core states. Options: (1) approximate effect of the core by an effective pseudopotential. The valence functions are smooth and nodeless. Advantage: simplicity and efficiency. (2) Augment the wave functions in spheres around each nucleus with numerical solutions of the radial Schrodinger equation. Advantage: accuracy.

33 Add some artificial, or pseudo-potential V PS to V KS replace the true wave function ψ with smooth analog ψ PS, with the same eigenvalues. No core! ψ PS is not unique! Pseudopotentials The envelope defines the basis, Shown here as green. Blue, red show augmentation region: region where PP is nonzero. Can be Plane Waves, or Atom-centered functions

34 Augmentation All-electron basis sets have envelope functions like PP methods: Far from nuclei ψ is smooth and described by smooth envelope functions (green). Choice of Plane Waves or local functions Valence ψ must be orthogonal to core states many wiggles that envelope functions cannot well represent. Solution: augment spheres around nucleus with numerical solutions of the Schrodinger equation (red, blue) At boundary envelope and augmented functions must match continuously and differentiably. Both value and slope must match quantization condition (see later).

35 4 types depicted in figure. Envelope functions are green. Four basic variants Blue, red show augmentation region (or region where PP is nonzero) PP methods have smooth w.f; All electron methods make ψ core. Atom-centered basis sets are largest at (blue) head site where they are centered; they decay ~exponentially (or faster) and reach a few shells of neighbors (red). All-electron basis sets have a smooth envelope function, augmented (i.e. replaced) inside spheres at each nucleus that match smoothly onto envelopes

36 ψ = ψ(e,r). What about E dependence? Classification #3: the basis can be (1) Fixed (PP methods) -- simplest (2) potential-and-energy dependent (KKR,APW). Accurate but expensive. (Nonlinear eigenvalue problem; see later) (3) potential dependent. LAPW = linearized APW LMTO = linearized KKR Energy dependence Linearization turns nonlinear problem into linear algebraic eigenvalue problem like fixed-basis case (Later). Modest loss of accuracy with large efficiency gains. Caveat: generalizations do not always apply