First-principles modeling: The evolution of the field from Walter Kohn s seminal work to today s computer-aided materials design

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1 First-principles modeling: The evolution of the field from Walter Kohn s seminal work to today s computer-aided materials design Peter Kratzer 5/2/2018 Peter Kratzer Abeokuta School 5/2/ / 34

2 Outline of this talk I. From the many-particle problem to the Kohn-Sham functional II. How to perform a total energy calculation III. High-throughput computations: crystal structure prediction and big data analytics IV. From the total energy to materials science Peter Kratzer Abeokuta School 5/2/ / 34

3 General condensed-matter Hamiltonian ( ˆT e + ˆT ion + ˆV e e + ˆV e ion + ˆV ion ion )Ψ = EΨ wavefunction Ψ(r 1, r N ; R I, R M ) electronic coordinates r k, k = 1, N ionic coordinates R I, I = 1, M ˆT e = N k=1 p 2 k 2m M ˆT ion P 2 I = 2M I =1 I ˆV e e = 1 N 1,N e2 2 4πɛ 0 r k k k r k ˆV ion ion = πɛ 0 ˆV e ion (r k, R I ) = M,M Z I Z I R I I I R I N M vi ion ( R I r k ) k=1 I=1 Peter Kratzer Abeokuta School 5/2/ / 34

4 Born-Oppenheimer approximation convenient, frequently made approximation (but not compulsary) separation of variables parametric dependence on set of coord. {R I } Ψ(r 1, r N ; R 1, R M ) = ν Λ ν ({R I })Φ ν,{ri }(r k ) electronic Schrödinger equation H e {R I } Φ ν,{r I }(r k ) = E e ν,{r I } Φ ν,{r I }(r k ) H e = T e + V e e + V e ion frequently made approximations: neglect of non-adiabatic couplings (terms of order m/m I ) only one Λ ν 0 (in a solid, this means neglect of electron-phonon couplings) electronic and nuclear degrees of freedom decouple! Peter Kratzer Abeokuta School 5/2/ / 34

5 Limitations of Born-Oppenheimer doesn t account for correlated dynamics of ionic and electronic coordinates Example: suprafluid 3 He, polaron-induced superconductivity breakdown of the restriction to a single ground-state Born-Oppenheimer surface Example: chemoluminescence breakdown of the adiabatic approximation Example: excitation of surface plasmons during scattering of an ion from a metal surface time-dependent density functional theory Peter Kratzer Abeokuta School 5/2/ / 34

6 electronic many-particle Hamiltonian N k=1 W (r, r ) = v (0) (r) = 2 N k 2m + v,n (0) 1 (r k ) + k k e 2 4πɛ 0 r r M I =1 vi ion ( R I r ) 2 W (r k, r k ) Φ(r 1, r N ) = EΦ(r 1, r N ) still many (for a typical solid: ) degrees of freedom The many-particle problem can be solved only for small systems (atoms, molecules and clusters). wavefunction-based methods Configuration-interaction, Coupled Cluster method Quantum Monte Carlo method Peter Kratzer Abeokuta School 5/2/ / 34

7 Density Functional Theory Hohenberg-Kohn theorem (1965): For any given external potential v (0), the wavefunctions can be considered as functionals in the space of ground state densities, n: E v (0)[n] = Φ[n] ˆT e + 1 2Ŵ + ˆv (0) Φ[n] The energy functional is stationary at the ground state energy, and the true ground state density n 0 coincides with n at the stationary point. Therefore a universal functional F exists with the property E v (0)[n] = F[n] + dr v (0) (r)n(r) Proof: Φ[n] n : trivial n Φ[n] : Note that v ext n Raleigh Ritz Φ[n] Also: v ext Φ[n]. Thus, E v (0)[n] is uniquely defined. Peter Kratzer Abeokuta School 5/2/ / 34

8 Kohn-Sham theorem idea: decompose F[n] into its major contributions F[n] = T 0 [n] + 1 dr dr n(r)w (r, r )n(r ) + E XC [n] 2 T 0 : kinetic energy of a system of non-interacting particles E XC [n] exchange-correlation energy defined as the rest approximations V XC [n](r) := δe XC [n] : exchange-correlation potential δn(r) popular approximations for E XC [n]: local-density approximation (LDA) generalized gradient approximations (GGAs) exact exchange formalism (EXX). Peter Kratzer Abeokuta School 5/2/ / 34

9 Kohn-Sham Hamiltonian To find the stationary point, we do variations at fixed N = dr n(r), which leads to δe v (0) δn(r) = µ (Lagrange parameter) If we write the density as a sum over single-particle functions, n(r) = N j=1 k BZ ϕ j,k (r) 2, the variational principle δe v (0)[ϕ ]/δϕ (r) = 0 leads to the Kohn-Sham equations ) ( 2 2m + V eff [n](r) ϕ j,k (r) = ɛ j,k ϕ j,k (r) with the effective potential V eff [n](r) = v (0) R (r) + dr e 2 n(r ) I 4πɛ 0 r r + δe XC [n] (r). δn Peter Kratzer Abeokuta School 5/2/ / 34

10 Nobel prize Nobel prize for chemistry awarded to Walter Kohn in Peter Kratzer Abeokuta School 5/2/ / 34

11 How can we specify E XC? At this point, we need to make approximations to get further. Example: local density approximation (LDA) E XC [n(r)] = dr e XC [n(r)] n(r) dr [ex hom (n(r)) + ehom C (n(r))] n(r) ex hom (n) = (81/64π)1/3 n 1/3 (r) ( r s r s) 1 if r s 1, ec hom (n) = lnr s r s r s lnr s if r s < 1. r s := (4πn(r)/3) 1/3 Wigner-Seitz radius [see, e.g. J. Perdew & A. Zunger, Phys. Rev. B (1981)] Peter Kratzer Abeokuta School 5/2/ / 34

12 improved functionals Jacob s ladder (as seen by John Perdew) DFT heaven AC-FDT hybrid meta-gga (τ[n(r)]) GGA ( n(r)) LDA Peter Kratzer Abeokuta School 5/2/ / 34

13 The total energy (for static ions) two equivalent definitions: E tot [n] = T 0 [n] + dr v (0) R (r)n(r) + 1 dr dr e2 n(r)n(r ) i 4πɛ 0 2 r r +E XC [n] + V ion ion R I E tot [n] = N ε j,k + E e e [n] + E XC [n] + V ion ion R I j =1 k BZ E e e [n] = 1 dr dr e2 n(r)n(r ) 4πɛ 0 2 r r = E e e [n] E XC [n] = E XC [n] dr V XC [n](r)n(r) E tot is stationary with resp. to variations of n around n 0, but the individual terms are not! Peter Kratzer Abeokuta School 5/2/ / 34

14 Basic steps in an electronic structure calculation 1 guess a starting charge density (e.g. superposition of atomic densities) 2 set up the Hamiltonian for this charge density (usually done in a small, preliminary basis set) 3 diagonalize this approximative Hamiltonian 4 use the eigenvalues and wavefunctions to set up a new charge density 5 try to improve the wavefunctions using the variational principle for E tot, thereby simultaneously approaching self-consistency n (i 1) V (i 1) eff ε (i 1) j,k, ϕ(i 1) j,k n (i) V (i) eff ε(i) j,k, ϕ(i) j,k, Peter Kratzer Abeokuta School 5/2/ / 34

15 Flowchart move ions construct new charge density and potential {Ψ (i-1) } {Ψ (i) } Ψ (i) Ψ (i-1) =0? ρ (i) ρ (i-1) =0? Forces converged? Peter Kratzer Abeokuta School 5/2/ / 34

16 Variational properties At the stationary point, the Kohn-Sham equations lead us to T 0 [n] = N j =1 k BZ ε j,k dr n(r)v eff [n](r) For the non-consistent case, we may introduce the generalization T 0 [n, V eff ] = N j =1 k BZ ε j,k [V eff ] dr n(r)v eff (r) This can be used to define the double functional E D tot [n, V eff ] = T 0[n, V eff ] + E e e [n] + E XC [n] + V ion ion R I with the variational properties E D tot [n 0 + δn, V eff [n 0 ]] E D tot [n 0, V eff [n 0 ]] = c 1 (δn) 2 E D tot [n 0, V eff [n 0 ] + δv] E D tot [n 0, V eff [n 0 ]] = c 2 (δv) 2. Normally, one has c 1 > 0, c 2 < 0. Then, E D tot [ n (i 1), V eff [n (i 1) ] ] is a lower bound for the (coverged) total energy. Peter Kratzer Abeokuta School 5/2/ / 34

17 Total-energy and thermodynamics The thermodynamic ground state is determined by the global minimum of the free energy of the system (for fixed volume) F (T, V ) = E tot (T, V ) TS e (T, V ) +Ekin ion (T, V ) Structural relaxation: Minimize F (T, V ) (in practice: E tot ) Caution: ionic contribution to the free energy important if, e.g.,anharmonic effects come into play or by the Gibbs free energy (for fixed pressure) G(T, p) = F (T, V ) + pv Examples: equilibrium between two structurally different crystalline phases equilibrium between a solid and its vapor Peter Kratzer Abeokuta School 5/2/ / 34

18 Thermal decay of GaAs GaAs Ga metal As 2 GaAs bulk GGa (p, T ) = G GaAs (T ) G As (p, T ). G As (p, T ) from ideal two-atomic gas, SGaAs ion and S ion Ga from Debye model of the solid. 2.9 g Ga [ev/particle] 3.1 µ Ga (GaAs, p(as 2 )=10 3 Pa) µ Ga (GaAs, p(as 2 )=10 4 Pa) µ Ga (bulk Ga) temperature [K] Free enthalpy per Ga atom in bulk GaAs in thermodynamic equilibrium with As 2 vapor at two pressures, compared to the free enthalpy per Ga atom in elemental Ga. [P. K., C.G. Morgan and M. Scheffler, Phys. Rev. B 59, (1999)] Peter Kratzer Abeokuta School 5/2/ / 34

19 Finite electronic temperature occupation numbers from Fermi distribution f j,k = [exp((ε j,k ɛ F )/kt ) 1] 1 Computer codes actually calculate the free energy F! entropy of the electronic system S e = 2k B n B [f j,k lnf j,k + (1 f j,k )ln(1 f j,k )] j =1 k BZ For a (free-electron-like) metal, we have F (T ) = E(T = 0) γ 2 T 2 + O(T 3 ) E(T ) = E(T = 0) + γ 2 T 2 + O(T 3 ) extrapolation to zero electronic temperature E(T = 0) [F (T ) + E(T )]/2 = E(T ) S e T /2 [see M. J. Gillan, J. Phys. Cond. Mat. 1, 689 (1989)] Peter Kratzer Abeokuta School 5/2/ / 34

20 How accurate is DFT? Peter Kratzer Abeokuta School 5/2/ / 34

Technical set-up determines precision Early pseudopotential methods involved significant approximations. Methods have evolved to more uniform precision by employing PAW or all-electron calculations.

21 Technical set-up determines precision Early pseudopotential methods involved significant approximations. Methods have evolved to more uniform precision by employing PAW or all-electron calculations. Nowadays, results are reproducible by all codes (if correct settings are used). codes all-electron PAW US-PPs NC-PPs [K. Lejaeghere et al., Science 351 (6280) 1415 (2016) DOI /science.aad3000.] Peter Kratzer Abeokuta School 5/2/ / 34

22 How accurate is DFT? K. Lejaeghere et al., Crit. Rev. Solid State Mat. Sci (2014). DOI / Peter Kratzer Abeokuta School 5/2/ / 34

23 Prediction of crystal structure A. Bialon et al., Chem. Mater (2016) Peter Kratzer Abeokuta School 5/2/ / 34

24 Domain composition in low-dimensional space A. Bialon et al., Chem. Mater. 28, 2550 (2016) Peter Kratzer Abeokuta School 5/2/ / 34

with 18 valence electrons discovered! [A. Zakutayev et al. J. Am. Chem. Soc.

25 Filling the gaps in materials tableworks Most common structures in V IX IV compounds Competing low-energy crystal structures New ternary semiconductors with 18 valence electrons discovered! [A. Zakutayev et al. J. Am. Chem. Soc (2013). DOI /ja311599g ] Peter Kratzer Abeokuta School 5/2/ / 34

26 How to compute observables A) energy differences between different structures, formation energy of defects, heat of adsorption,... B) derivatives of the thermodynamic potentials E tot, F or G For simplicity s sake, we consider a system with constant volume at T = 0 : F (T = 0, V ) = E tot (V ) pressure p = bulk modulus B = V 2 E tot (V ) V 2 forces F I on atom I (in electronic ground state) F I = E tot R I Etot (V ) V = N ϕ j,k H / R I ϕ j,k due to the Hellmann-Feynman theorem j Peter Kratzer Abeokuta School 5/2/ / 34

27 How to compute observables C) second derivatives Examples: 1. force constant matrix E tot R i R j calculation of phonon spectrum, vibrational entropy, particle number fluctuations chemical softness and hardness ( ) ( n(r) δ 2 ) 1 E tot s(r) = = µ δn(r)δn(r dr ) ( µ v,t = δ 2 E tot n(r ) δn(r)δn(r ) 2N dr h(r) = 1 2 n(r)) v,t Note: When calculating second derivatives, the response of the density to the perturbation must be taken into account. Density Functional Perturbation Theory Peter Kratzer Abeokuta School 5/2/ / 34

28 Matter under extreme conditions Calculated equation of state of iron in the earth s core [D. Alfé et al., Nature (1999).] [M. S. T. Bukowinski, Nature (1999).] Peter Kratzer Abeokuta School 5/2/ / 34

29 What materials properties are accessible to calculation?. structural properties Examples: structural phase transitions, surface reconstructions yes elastic properties Examples: bulk modulus, C 11, C 12, C 44,... yes chemical properties Examples: thermochemical stability of compounds, reactivity of surfaces yes transport properties Examples: conductance of nanowires, magneto-resistence developing field optical/spectroscopic properties Examples: photoemission spectra, cross sections for light absorption topic beyond Kohn-Sham theory time-dependent DFT many other applications Peter Kratzer Abeokuta School 5/2/ / 34

30 Electronic properties of materials From the Kohn-Sham variational principle, we have δe tot δn(r) = µ = ɛ F in a metal However, one cannot proof that the Fermi surface is represented correctly in DFT. Further, in a Kohn-Sham theory, there is no straightforward relationship between ɛ j,k and the excitation energy of quasiparticles and quasiholes. One can only show Janak s theorem E tot f j,k = ɛ j,k. [supposing that E XC is a continuously differentiable function of particle number, which is not true in a finite system] As a first approximation to the effective mass tensor, one may use ( ɛj,k k n k m ) ɛ=ɛ F = (M 1 ) nm For a better treatment, use time-dependent DFT! Peter Kratzer Abeokuta School 5/2/ / 34

31 Simulation techniques Challenges: length scales much larger than the atomic scale (e.g. in phase transitions, line defects, plastic deformation,... ). dynamic properties, evolution on time scales as long as seconds (e.g. melting, crystal growth) combine the reliability of the first-principles approach with molecular dynamics to follow dynamical processes in time, e.g. chemical reactions scale: several hundred atoms, pico-seconds for calculation of time correlation functions and phonons (using Fourier transformations) as a way to calculate the free energy (as a time average) statistical physics kinetic modelling Peter Kratzer Abeokuta School 5/2/ / 34

32 Equilibrium statistical physics To treat phenomena on large scales, map your problem to a model spin Hamiltonian, with parameters determined from DFT calculations. Examples: order-disorder transition in alloys ordered ad-layers on surfaces 1 Calculate E DFT (σ, V ) at T = 0 for many structures 2 Construct E CE (σ, V ) = k,m d k,m Π k,m (σ)j k,m (V ) Π k,m (σ) lattice-averaged spin product in k-atom-motif of type m J k,m (V ) interaction parameters, fitted to DFT energies 3 Calculate partition function Z = exp( E CE (σ, V )/k B T ) Peter Kratzer Abeokuta School 5/2/ / 34

Kinetic Modeling of rare events Typical time interval τ between activated processes is too long to be covered by a Molecular Dynamics simulation.

33 Kinetic Modeling of rare events Typical time interval τ between activated processes is too long to be covered by a Molecular Dynamics simulation. Transition State Theory: τ 1 = Γ = Γ 0 exp( E/k B T ) Γ 0 = N ν (i) k N 1 ν (TS) k Calculate these quantities using input from DFT calculations! ν (i) k : normal mode frequencies at energy minimum ν (TS) k : normal mode frequencies at transition state E: energy barrier Peter Kratzer Abeokuta School 5/2/ / 34

34 THE END Enjoy the workshop! Peter Kratzer Abeokuta School 5/2/ / 34

1 Density functional theory (DFT)

1 Density functional theory (DFT) 1.1 Introduction Density functional theory is an alternative to ab initio methods for solving the nonrelativistic, time-independent Schrödinger equation H Φ = E Φ. The