Wave Function Optimization in Quantum Monte Carlo Arne Lüchow Theoretical Chemistry Group, RWTH Aachen University Autumn School on Correlated Electrons, FZ Jülich Sept. 16, 2019 1 / 32 Outline Brief overview of quantum Monte Carlo methods (QMC) Optimization of wave functions Examples Recent developments 2 / 32 Two main variants: 1 solve stationary Schrödinger equation with stochastic process: Diffusion Monte Carlo (DMC) Simulate stochastic process with random numbers 2 evaluate E = hΨjHjΨi with Monte Carlo integration: Variational Monte Carlo (VMC) Sample the integrand with random numbers Quantum Monte Carlo – a brief overview Quantum Monte Carlo methods Numerical methods to solve the Schrödinger equation by means of Monte Carlo techniques (i.e. using random sampling) are called quantum Monte Carlo methods (QMC) electron structure QMC for electronic Schrödinger equation 3 / 32 2 evaluate E = hΨjHjΨi with Monte Carlo integration: Variational Monte Carlo (VMC) Sample the integrand with random numbers Quantum Monte Carlo – a brief overview Quantum Monte Carlo methods Numerical methods to solve the Schrödinger equation by means of Monte Carlo techniques (i.e. using random sampling) are called quantum Monte Carlo methods (QMC) electron structure QMC for electronic Schrödinger equation Two main variants: 1 solve stationary Schrödinger equation with stochastic process: Diffusion Monte Carlo (DMC) Simulate stochastic process with random numbers 3 / 32 Quantum Monte Carlo – a brief overview Quantum Monte Carlo methods Numerical methods to solve the Schrödinger equation by means of Monte Carlo techniques (i.e. using random sampling) are called quantum Monte Carlo methods (QMC) electron structure QMC for electronic Schrödinger equation Two main variants: 1 solve stationary Schrödinger equation with stochastic process: Diffusion Monte Carlo (DMC) Simulate stochastic process with random numbers 2 evaluate E = hΨjHjΨi with Monte Carlo integration: Variational Monte Carlo (VMC) Sample the integrand with random numbers 3 / 32 Mathematically, this is a diffusion equation In diffusion Monte Carlo (DMC), this diffusion process is simulated with a stochastic process1 Diffusion Monte Carlo (DMC) The starting point: Schrödinger equation in imaginary time t = it ¶ 1 Ψ = −HΨ = ∆Ψ − V Ψ ¶t 2 Long-time solution is the ground state wave function Ψ0 −Ht lim e Ψ ∝ Ψ0 t!¥ 1A. Lüchow, WIREs Comput Mol Sci 2011, 1, 388-402 4 / 32 Diffusion Monte Carlo (DMC) The starting point: Schrödinger equation in imaginary time t = it ¶ 1 Ψ = −HΨ = ∆Ψ − V Ψ ¶t 2 Long-time solution is the ground state wave function Ψ0 −Ht lim e Ψ ∝ Ψ0 t!¥ Mathematically, this is a diffusion equation In diffusion Monte Carlo (DMC), this diffusion process is simulated with a stochastic process1 1A. Lüchow, WIREs Comput Mol Sci 2011, 1, 388-402 4 / 32 2 stationary distribution of drift-diffusion process is jΨG j for ∆t ! 0. DMC: weights 0 − 1 (E (x)+E (x 0))−E ∆t W (∆t;x;x ) = e [ 2 L L ref ] fluctuate around 1 if local energies EL = Hˆ ΨG =ΨG are close to Eref (nonsingular!) Importance Sampling: Drift-Diffusion process discretized stochastic process: time step ∆t p Xk+1 = Xk + b(Xk )∆t + ∆tx (drift-diffusion process) drift term b(x) = ∇ΨG =ΨG : non-random moves to large values of jΨG j (guide function ΨG ). 5 / 32 DMC: weights 0 − 1 (E (x)+E (x 0))−E ∆t W (∆t;x;x ) = e [ 2 L L ref ] fluctuate around 1 if local energies EL = Hˆ ΨG =ΨG are close to Eref (nonsingular!) Importance Sampling: Drift-Diffusion process discretized stochastic process: time step ∆t p Xk+1 = Xk + b(Xk )∆t + ∆tx (drift-diffusion process) drift term b(x) = ∇ΨG =ΨG : non-random moves to large values of jΨG j (guide function ΨG ). 2 stationary distribution of drift-diffusion process is jΨG j for ∆t ! 0. 5 / 32 Importance Sampling: Drift-Diffusion process discretized stochastic process: time step ∆t p Xk+1 = Xk + b(Xk )∆t + ∆tx (drift-diffusion process) drift term b(x) = ∇ΨG =ΨG : non-random moves to large values of jΨG j (guide function ΨG ). 2 stationary distribution of drift-diffusion process is jΨG j for ∆t ! 0. DMC: weights 0 − 1 (E (x)+E (x 0))−E ∆t W (∆t;x;x ) = e [ 2 L L ref ] fluctuate around 1 if local energies EL = Hˆ ΨG =ΨG are close to Eref (nonsingular!) 5 / 32 Schrödinger equation is solved with additional boundary condition given by nodes of a guide function ΨG : ΨG (R) = 0 R: all electron coords: R = (x1;y1;z1;:::;zn)! Nodes (nodal set): 3n − 1 hypersurface, unrelated to orbital nodes Fermion sign problem Perfect for vibrational ground state (vibrational DMC). BUT: lowest eigenstate of H is not the electronic ground state (but a bosonic state) for n > 2 The stochastic process is forced to an antisymmetric state by fixed node approximation 6 / 32 Fermion sign problem Perfect for vibrational ground state (vibrational DMC). BUT: lowest eigenstate of H is not the electronic ground state (but a bosonic state) for n > 2 The stochastic process is forced to an antisymmetric state by fixed node approximation Schrödinger equation is solved with additional boundary condition given by nodes of a guide function ΨG : ΨG (R) = 0 R: all electron coords: R = (x1;y1;z1;:::;zn)! Nodes (nodal set): 3n − 1 hypersurface, unrelated to orbital nodes 6 / 32 Monte Carlo integration with sample fRk g ∼ p(R): 1 N EVMC = hELip = lim EL(Rk ) VMC N! ∑ ¥ N k=1 Metropolis-Hastings often: drift-diffusion step Variational Monte Carlo (VMC) Variational energy of a trial function ΨT : hΨT jHjΨT i Z EVMC = = EL(R) · p(R)dR hΨT jΨT i with local energy: HΨT (R) EL(R) = ΨT (R) and probability distribution: jΨ(R)j2 p(R) = R jΨ(R)j2dR 7 / 32 Variational Monte Carlo (VMC) Variational energy of a trial function ΨT : hΨT jHjΨT i Z EVMC = = EL(R) · p(R)dR hΨT jΨT i with local energy: HΨT (R) EL(R) = ΨT (R) and probability distribution: jΨ(R)j2 p(R) = R jΨ(R)j2dR Monte Carlo integration with sample fRk g ∼ p(R): 1 N EVMC = hELip = lim EL(Rk ) VMC N! ∑ ¥ N k=1 Metropolis-Hastings often: drift-diffusion step 7 / 32 Variational quantum Monte Carlo (VMC): VMC energy: average of local energies hΨT jHjΨT i EVMC = hELipVMC = hΨT jΨT i DMC more accurate, VMC more efficient VMC and DMC energies Diffusion Monte Carlo (DMC): (FN) weighted/branched drift-diffusion: pDMC (R) ∝ ΨG Ψ0 DMC energy: average of local energies hΨ(FN)jHjΨ i E = hE i = 0 G DMC L pDMC (FN) hΨ0 jΨG i (after time step extrapolation to ∆t ! 0) 8 / 32 VMC and DMC energies Diffusion Monte Carlo (DMC): (FN) weighted/branched drift-diffusion: pDMC (R) ∝ ΨG Ψ0 DMC energy: average of local energies hΨ(FN)jHjΨ i E = hE i = 0 G DMC L pDMC (FN) hΨ0 jΨG i (after time step extrapolation to ∆t ! 0) Variational quantum Monte Carlo (VMC): VMC energy: average of local energies hΨT jHjΨT i EVMC = hELipVMC = hΨT jΨT i DMC more accurate, VMC more efficient 8 / 32 Computational Considerations VMC and DMC require guide or trial function Ψ for local energy no integrals! Only 2nd derivatives! Guide and Trial Functions Compact and physically motivated approximations to exact wave function. Explicitly correlated! Scaling of local energy EL = HΨ=Ψ with number of electrons 2 Variance h(EL − hELi) i determines sample size Scales well to 10,000 of cpu cores 9 / 32 Small parameter changes p0 to p: weighted mean K 2 ∑k=1 wk EL(Rk ;p) jΨ(Rk ;p)j EVMC = Ewm(p) = K ; wk = 2 : ∑k=1 wk jΨ(Rk ;p0)j Stochastic optimization of wave function: minimization of energy Optimization of wave functions: energy minimization large sample of electron configurations fRk g; k = 1;:::;K with p(R) = jΨ(R)j2=R jΨ(R)j2dt parameter vector p wave function Ψ(R;p) ≡ Ψ(p) local energy EL(R;p) ≡ EL(p) 1 K EVMC = Em(p) = ∑ EL(Rk ;p): K k=1 10 / 32 Optimization of wave functions: energy minimization large sample of electron configurations fRk g; k = 1;:::;K with p(R) = jΨ(R)j2=R jΨ(R)j2dt parameter vector p wave function Ψ(R;p) ≡ Ψ(p) local energy EL(R;p) ≡ EL(p) 1 K EVMC = Em(p) = ∑ EL(Rk ;p): K k=1 Small parameter changes p0 to p: weighted mean K 2 ∑k=1 wk EL(Rk ;p) jΨ(Rk ;p)j EVMC = Ewm(p) = K ; wk = 2 : ∑k=1 wk jΨ(Rk ;p0)j Stochastic optimization of wave function: minimization of energy 10 / 32 Optimization of wave functions: variance minimization variance of the local energy K 1 2 Vm(p) = ∑ (EL(Rk ;p) − Em) K k=1 exact eigenfunctions of H have Vm = 0 Advantage: stable, each term positive Variance minimization: very stable and efficient less accurate energies (VMC and DMC) 11 / 32 BUT: (FN) projected Ψ0 depends on (guide) function parameters (FN) pDMC = Ψ0 ΨG only statistically known derivatives? AND: use optimized VMC trial function as guide functions in DMC! VMC optimization can optimize DMC energy parameters modifying nodes of wave function required! FN-DMC energy: optimal wf with given nodes flexible wf: optimal VMC energy close to DMC energy VMC vs DMC wave function optimization DMC more accurate than VMC: DMC optimization desired! 12 / 32 AND: use optimized VMC trial function as guide functions in DMC! VMC optimization can optimize DMC energy parameters modifying nodes of wave function required! FN-DMC energy: optimal wf with given nodes flexible wf: optimal VMC energy close to DMC energy VMC vs DMC wave function optimization DMC more accurate than VMC: DMC optimization desired! BUT: (FN) projected Ψ0 depends on (guide) function parameters (FN) pDMC = Ψ0 ΨG only statistically known derivatives? 12 / 32 VMC vs DMC wave function optimization DMC more accurate than VMC: DMC optimization desired! BUT: (FN) projected Ψ0 depends on (guide) function parameters (FN) pDMC = Ψ0 ΨG only statistically known derivatives? AND: use optimized VMC trial function as guide functions in DMC! VMC optimization