First-principles Molecular Dynamics Simulations François Gygi University of California, Davis [email protected] http://eslab.ucdavis.edu http://www.quantum-simulation.org MICCoM Computational School, Jul 17-19, 2017 1 Outline • Molecular dynamics simulations • Electronic structure calculations • First-Principles Molecular Dynamics (FPMD) • Applications 2 Molecular Dynamics • An atomic-scale simulation method – Compute the trajectories of all atoms – extract statistical information from the trajectories Atoms move according to Newton’s law: !! miRi = Fi Molecular dynamics: general principles • Integrate Newton’s equations of motion for N atoms !! miRi (t) = Fi (R1,…, RN ) i =1,…, N Fi (R1,…, RN ) = −∇iE(R1,…, RN ) • Compute statistical averages from time averages (ergodicity hypothesis) 1 T A = ∫ dr3N dp3N A(r, p)e−βH (r,p) ≅ ∫ A(t)dt Ω T 0 • Examples of A(t): potential energy, pressure, … Simple energy model • Model of the hydrogen molecule (H2): harmonic oscillator E(R1, R2 ) = E( R1 − R2 ) 2 = α( R1 − R2 − d0 ) • This model does not describe intermolecular interactions Simple energy model • Model of the hydrogen molecule including both intra- and intermolecular interactions: E(R1,…, RN ) = ∑ Eintra ( Ri − R j )+ ∑ Einter ( Ri − R j ) {i, j}∈M i∈M j∈M ' • This model does not describe adequately changes in chemical bonding Simple energy model • Description of the reaction H2+H→ H + H2 • The model fails! What is a good energy model? Atomistic simulation of complex structures • Complex structures – Nanoparticles – Assemblies of nanoparticles – Embedded nanoparticles – Liquid/solid interfaces 9 A difficult case: Structural phase transitions in CO2 Molecular phases polymeric phase The energy is determined by quantum mechanical properties • First-Principles Molecular Dynamics: Derive interatomic forces from quantum mechanics Ni-tris(2-aminoethylamine) First-Principles Molecular Dynamics Monte Carlo Electronic Molecular Dynamics Structure Theory … FPMD Quantum Chemistry Statistical Density Functional Theory Mechanics … R. Car and M. Parrinello (1985) 12 Electronic structure calculations • Problem: determine the electronic properties of an assembly of atoms using the laws of quantum mechanics. • Solution: solve the Schroedinger equation! The Schroedinger equation for N electrons • A partial differential equation for the wave function ψ: 3 2 3N ri ∈ R , ψ ∈ L (R ) ∂ i! ψ(r ,…, r ,t) = H(r ,…, r ,t) ψ(r ,…, r ,t) ∂t 1 N 1 N 1 N • H is the Hamiltonian operator: 2 ! 2 H(r , ,r ,t) V (r , ,r ,t) 1 … N = − ∑∇i + 1 … N 2m i The time-independent Schroedinger equation • If the Hamiltonian is time-independent, we have iEt/! ψ(r1,…, rN ,t) = ψ(r1,…, rN ) e • where ψ(r) is the solution of the time- independent Schroedinger equation: H(r1,…,rN )ψ (r1,…,rN ) = Eψ (r1,…,rN ) energy Solving the Schroedinger equation • The time-independent Schroedinger equation can have many solutions: H(r1,…, rN )ψn (r1,…, rN ) = En ψn (r1,…, rN ) n = 0,1, 2… • The ground state wave function ψ0 describes the state of lowest energy Ε0 • Excited states are described by ψ1, ψ2,.. and have energies Ε1, Ε2,.. > Ε0 Hamiltonian operator for N electrons and M nuclei • Approximation: treat nuclei as classical particles • Nuclei are located at positions Ri , electrons at ri H(r1,…, rN , R1,…, RM ) = 2 N N M 2 N 2 ! 2 Z je e − ∑∇i −∑∑ +∑ 2me i=1 i=1 j=1 ri − R j i< j ri − rj M Z Z e2 M i j 1 ! 2 +∑ + ∑MiRi i< j Ri − R j 2 i=1 The adiabatic approximation • The Hamiltonian describing an assembly of atoms is time-dependent because atoms move: 2 ! 2 H (r,t) V (r,t) = − ∑∇i + 2m i V (r,t) V (r R (t)) V (r) = ∑ ion − j + e-e j time-dependence through ionic positions The adiabatic approximation • If ions move sufficiently slowly, we can assume that electrons remain in the electronic ground state at all times ψ (r,t) =ψ 0 (r) H(r,{Ri (t)})ψ 0 (r) = E0ψ 0 (r) Ground state Ground state energy wave function Mean-field approximation • The problem of solving the N-electron Schroedinger equation is formidable (N! complexity). H(r1,…,rN )ψ n (r1,…,rN ) = Enψ n (r1,…,rN ) – Wave functions must be antisymmetric (Pauli principle) ψ (r1,…,ri ,…,rj ,…,rN ) = −ψ (r1,…,rj ,…,ri ,…,rN ) exchanged • Assuming that electrons are independent (i.e. feel the same potential) reduces this complexity dramatically. – The potential is approximated by an average effective potential Independent particles, solutions are Slater determinants • A Slater determinant is a simple form of antisymmetric wave function : ψ (r1,…,rN ) = det{ϕi (rj )} • The one-particle wave functions ϕi satisfy the one-particle Schroedinger equation: h(r)ϕi (r) = εiϕi (r) 2 ! 2 h(r) = − ∇ +V (r) 2m eff Note: effective potential Electron-electron interaction H(r1,…, rN , R1,…, RM ) = 2 N N M 2 N 2 ! 2 Z je e − ∑∇i −∑∑ +∑ 2me i=1 i=1 j=1 ri − R j i< j ri − rj M Z Z e2 M i j 1 ! 2 +∑ + ∑MiRi i< j Ri − R j 2 i=1 Density Functional Theory • Introduced by Hohenberg & Kohn (1964) • Chemistry Nobel prize to W.Kohn (1999) • The electronic density is the fundamental quantity from which all electronic properties can be derived E = E [ρ] E [ρ] = T [ρ] + ∫ V(r)ρ(r)dr + Exc [ρ] • Problem: the functional E[ρ] is unknown! The Local Density Approximation • Kohn & Sham (1965) E [ρ] = T [ρ] + ∫ V(r)ρ(r)dr + Exc [ρ] • Approximations: – The kinetic energy is that of a non-interacting electron gas of same density – The exchange-correlation energy density depends locally on the electronic density Exc = Exc [ρ(r)] = ∫ εxc (ρ(r))ρ(r)dr The Local Density Approximation ρ(r!) V = dr!+V (ρ(r)) e-e ∫ r − r! XC • The mean-field approximation is sometimes not accurate, in particular for – strongly correlated electrons – excited state properties The Kohn-Sham equations • Coupled, non-linear, integro-differential equations: (−Δφi +V(ρ, r)φi = εiφi i =1…Nel * * ρ(r#) V(ρ, r) = Vion (r)+ ∫ dr#+VXC (ρ(r), ∇ρ(r)) * r − r# ) Nel * 2 ρ(r) = ∑ φi (r) * i=1 * φ ∗(r)φ (r)dr = δ +* ∫ i j ij Numerical methods • Basis sets: solutions are expanded on a basis of N orthogonal functions N φi (r) = ∑cij ϕ j (r) j=1 ∗ 3 ∫ ϕ j (r)ϕk (r) = δ jk Ω ⊂ R Ω • The solution of the Schroedinger equation reduces to a linear algebra problem Numerical methods: choice of basis • Gaussian basis (non-orthogonal) 2 −αi r−R ϕi (r) = e • Plane wave basis (orthogonal) iq⋅R ϕq (r) = e • Other representations of solutions: – values on a grid – finite element basis Numerical methods: choice of basis • Hamiltonian matrix: ∗ 3 Hij = ϕi H ϕ j = ∫ ϕi (r) Hϕ j (r) d r Ω • Schroedinger equation: an algebraic eigenvalue problem N Hcn = εncn cn ∈ C Numerical methods: choice of basis • Non-orthogonal basis sets lead to generalized eigenvalue problems S ∗ (r) (r) d 3r ij = φi φ j = ∫φi φ j ≠ δ ij Ω N Hcn = εnScn cn ∈ C Solving large eigenvalue problems • The size of the matrix H often exceeds 103-104 • Direct diagonalization methods cannot be used • Iterative methods: – Lanczos type methods – subspace iteration methods • Many algorithms focus on one (or a few) eigenpairs • Electronic structure calculations involve many eigenpairs (~ # of electrons) • robust methods are necessary Solving the Kohn-Sham equations: fixed-point iterations • The Hamiltonian depends on the electronic density (−Δφi +V(ρ, r)φi = εiφi i =1…Nel * * ρ(r#) V(ρ, r) = Vion (r)+ ∫ dr#+VXC (ρ(r), ∇ρ(r)) * r − r# ) Nel * 2 ρ(r) = ∑ φi (r) * i=1 * φ ∗(r)φ (r)dr = δ +* ∫ i j ij Self-consistent iterations • For k=1,2,… – Compute the density ρk – Solve the Kohn-Sham equations • The iteration may converge to a fixed point Simplifying the electron-ion interactions: Pseudopotentials • The electron-ion interaction is singular Ze2 V (r) = − e-ion r − R • Only valence electrons play an important role in chemical bonding core electrons Valence electrons Simplifying the electron-ion interactions: Pseudopotentials • The electron-ion potential can be replaced by a smooth function near the atomic core " Ze2 $− r − R > rcut Ve-ion (r) = # r − R $ f ( r − R ) r − R < r % cut • Core electrons are not included in the calculation (they are assumed to be "frozen") Pseudopotentials: Silicon • Solutions of the Schroedinger equation for Si including all electrons (core+valence): ψ3s ψ3p Potential = -Z/r Core Valence wavefunctions Pseudopotentials: Silicon • Solutions of the Schroedinger equation for Si including all electrons (zoom on core region): rψ1s rψ2s rψ2p Potential = -Z/r Core region Pseudopotentials: Silicon • The electron-ion potential can be replaced by a smooth function near the atomic core ψ3s ψ3p pseudopotentials Core Valence -Z/r wavefunctions Summary: First-principles electronic structure • Time-independent Schroedinger equation • Mean-field approximation • Simplified electron-electron interaction: – Density Functional Theory, Local Density Approximation • Simplified electron-ion interaction: – Pseudopotentials Molecular dynamics: Computation of ionic forces • Hamiltonian: H(λ) • Hellman-Feynman theorem: if ψ0(λ) is the electronic ground state of H(λ) ∂E ∂ ∂H(λ) = ψ0 (λ) H(λ) ψ0 (λ) = ψ0 (λ0 ) ψ0 (λ0 ) ∂λ λ0 ∂λ ∂λ λ0 • For ionic forces: λ=Ri (ionic positions) ∂E ∂H ∂ Fi = − = ψ0 ψ0 = ψ0 ∑Ve-ion (r − Rj ) ψ0 ∂Ri ∂Ri ∂Ri j Integrating the equations of motion: the Verlet algorithm • The equations of motion are coupled, second order ordinary differential equations • Any ODE integration method can be used • Time-reversible integrators are preferred • The Verlet algorithm (or leapfrog method) is time-reversible Δt 2 x(t + Δt) = 2x(t) − x(t + Δt) + F(x(t)) m Integrating the equations of motion: the Verlet algorithm • Derivation of the Verlet algorithm: Taylor expansion of x(t) 2 2 3 3 dx Δt d x Δt d x 4 x(t + Δt) = x(t) + Δt + + +O(Δt ) dt 2 dt 2 6 dt3 2 2 3 3 dx Δt d x Δt d x 4 x(t t) x(t) t O( t ) − Δ = − Δ + 2 − 3 + Δ dt 2 dt 6 dt • Add the two Taylor expansions: d 2 x x(t + Δt) + x(t − Δt) = 2x(t) + Δt 2 +O(Δt 4 ) dt 2 Integrating the equations of motion: the Verlet algorithm • use Newton’s law d 2 x m = f (x(t)) dt 2 d 2 x x(t + Δt) + x(t − Δt) = 2x(t) + Δt 2 +O(Δt 4 ) dt 2 2 Δt 4 x(t + Δt) = 2x(t) − x(t − Δt) + F(x(t)) +O(Δt ) m First-Principles Molecular Dynamics Molecular Dynamics Density Functional Theory V (x) (x) 2 (−Δ + eff )ϕi = εiϕi d m R F n i 2 i = i FPMD 2 dt x (x) ρ ( ) = ∑ ϕi i=1 Newton equations Kohn-Sham equations R.