7. Quantum Monte Carlo (QMC)

Transcription

1 Molecular Simulations with Chemical and Biological Applications (Part I) 7. Quantum Monte Carlo (QMC) Dr. Mar(n Steinhauser 1 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 1 Introduc5on Dr. Mar(n Steinhauser 2 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 2 1

2 The quantum many- body problem Fundamental object in quantum mechanics: Want to find special wave func(ons such that Ψ(r 1,r 2,...,r N ) HΦ ( r, r,..., r ) = EΦ ( r, r,..., r ) n 1 2 N n 1 2 N where 1 Z 1 H = + 2 r 2 I i i ii rii i< j ij This is a fundamentally many- body equa(on! We want to find an accurate method that will work in general to solve this 3 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 3 The Problem of Dimensionality Direct quantum methods are too slow! Suppose, we represent the complete N- body wavefunc(on and treat it as strictly a problem in linear algebra then find the exact solu(on If each dimension takes 100 complex numbers, then: N par(cles in 3 dimensions will take 10 6N numbers. Even with computer (me and memory increasing exponen(ally, the size of system we can treat will only grow linearly in (me! 4 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 4 2

3 High Standards for Accuracy Set by Nature! 5 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 5 Algorithms for Solving these Problems came in HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 6 3

4 Monte Carlo Monte Carlo is solving a problem using random numbers! Evaluate integrals: expecta(on value of a random variable is just the integral over its probability distribu(on generate a bunch of random numbers and average to get the integral Simulate random processes- - random walks sample configura(on space Number of dimensions doesn t mazer for MC Efficiency That s why usually MC is the best method for high- dimensional problems 7 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 7 SUMMARY: IMPORTANT ENSEMBLES Microcanonical (NVE) Canonical (NVT) Isobaric/isothermal (NPT) Probability distributions ( { }) = P { r i }, p i 1 Ω E,V, N ( ) H ri ( { }) = e P { r i }, p i ( { }) { }, p i kt Z ( T,V, N ) ( ) e H ri P { r i },{ p i },V Z P ({ },{ p i }) PV kt ( T,P, N ) Ω( E,V, N) = δ E H micro ( ({ r i },{ p i })) Z P ( T,V, N ) = e micro E kt Z P ( T,P, N ) = e V micro E PV kt Free energies (atomistic macroscopic thermodynamics) S = k log Ω(E,V, N ) F (T,V, N ) = kt log Z G(T, P, N ) = kt log Z p 8 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 8 4

5 THE IDEA OF MONTE CARLO Simple vs. Importance Sampling ü The realiza(on that one should not sample the microstates completely at random (simple sampling), but according to to their actual Boltzmann weights (this is the equilibrium distribu(on!), is the first step towards the celebrated Metropolis algorithm (importance sampling). ü One can actually only speak of a temperature of a system (factor β in the exponent), when the system has reached equilibrium, i.e. when the microstate occupa(on probabili(es form a Boltzmann distribu(on. g( x) g( x) x x 9 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 9 MC AS A MARKOV PROCESS ü So far we haven t said anything about how we have to choose each state so it appears with its correct Boltzmann probability. One way to answer to this ques(on is to consider MC as a Markov process. ü A Markov process is a mechanism by which an ini(al state X is transformed into a new state Y in a stochas(c fashion using a set of transi(on probabili(es P( X Y ) which sa(sfy certain condi(ons (for details, see Tutorial on the Monte Carlo (MC) Method on the lecture website). X : p X,T X ( ) P X Y ( ) Y :( p Y,T Y ) P( Y ) 10 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 10 5

6 MC AS A MARKOV PROCESS ü Transi(on probabili(es of a true Markov process should sa(sfy the following condi(ons: They should vary with (me, They should neither depend on the proper(es of the ini(al or final state, nor on any other state through which the system passes, For a given ini(al state, the sum of transi(on probabili(es over all final states must be equal to one. ü In a MC simula(on we repeatedly apply a Markov process to generate a Markov chain of states. ü The Markov process is chosen such that it produces a succession of states which appear with their Boltzmann probabili(es. This is known as equilibrium of the system. 11 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 11 EXAMPLE OF A MARKOV PROCESS OF LAST LECTURE The Simple Random Walk (Brownian Mo5on) 12 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 12 6

7 MC AS A MARKOV PROCESS ü In order to reach equilibrium, the Markov process has to be ergodic, i.e. it should be possible to reach any state of the system from any other state. ü The Markov process should sa(sfy the principle of detailed balance, i. e. the rates at which the system makes transi(ons into and out of any microstate should be the same: p X P( X Y ) = p Y P( Y X) ü The condi(on of detailed balance imposes a (me- reversal symmetry on the system and provides a sufficient (but not necessary) condi(on for the ra(o of the transi(on probabili(es to produce a Boltzmann equilibrium distribu(on of the occupancy of microstates. 13 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 13 MC AS A MARKOV PROCESS ü For the equilibrium distribu(on to be Boltzmann- like, the ra(on of the transi(on probabili(es should be (for details, see Sect in Tutorial on the Monte Carlo (MC) Method on the lecture website): p X p Y ( ) ( ) = exp β ( E E Y X ) = P Y X P X Y ü In prac(ce, this is done by tuning the actual transi(on probabili(es such that the acceptance ra5o, i.e. the probability of a randomly generated transi(on is as large as possible. ü Only Constraints: 1. The above equa(on 2. The sum over all transi(on probabili(es is equal to one. 14 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 14 7

8 THE METROPOLIS ALGORITHM ü Finally, we arrive at the Metropolis algorithm, which is a repe((on of three steps: 15 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 15 IMPROVING ON THE METROPOLIS ALGORITHM ü Achieving equilibrium of a system using the Metropolis algorithm can be greatly accelerated by introducing addi(onal MC moves into the system. ü The great advantage of the MC method is, that any MC trial moves that are thermodynamically permissible, are allowed they don t have to be physically realis:c, they only need to obey the acceptance rules which generate a Boltzmann- like distribu(on of microstates. However, one always has to make sure, that the moves are s:ll ergodic. For example: ü In a monoatomic fluid (see Monte Carlo NVT Fluid on lecture website), one can simply construct trial moves by randomly displacing par(cles. ü For flexible macromolecules (polymers), one must also consider internal degrees of freedom (s(ffness, bond length, bond angles, eigenvolume, etc.) by using orienta:onal MC trial moves. 16 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 16 8

9 IMPROVING ON THE METROPOLIS ALGORITHM Accelera5ng Equilibra5on in a binary mixture ü An exchange of par(cle iden(ty as a trial move, even though they might be separated by a large distance and other par(cles can greatly accelerate the simula(on of mixing of species in a dense system. trial move ü The price one pays for unphysical moves: Loss of informa(on about the dynamics ü In prac(ce: When using a reasonably physical set of trial moves, MC can also yield informa(on about the equilibrium dynamics of a system. Such MC methods are known as kine:c Monte Carlo (KMC) techniques. 17 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 17 IMPROVING ON THE METROPOLIS ALGORITHM Accelera5ng Equilibra5on of a Macromolecule Example from Research: ü Pivot moves which move part of a macromolecule about a randomly chosen monomer of the molecule can accelerate the large- scale equilibra(on of a polymer. 2 pivot moves 1 pivot move The algorithm is explained in: A molecular dynamics study on universal proper:es of polymer chains in different solvent quali:es. Part I. A review of linear chain proper:es, M. O. Steinhauser, J. Chem. Phys. 122, HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 18 9

10 Molecular Simulations with Chemical and Biological Applications (Part I) EXAMPLE: MC- NVT Ensemble of Hard Disks (2D, N=2000) Dr. Mar(n Steinhauser 19 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 19 EXAMPLE: MC- NVT Ensemble of Hard Spheres (N=400) Ini(al cubic setup Dr. Mar(n Steinhauser Ajer 104 MC steps 20 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 20 10

11 Quantum Monte Carlo (QMC) Premise: need to use simula(on techniques to solve many- body quantum problems just as you need them classically. Both the wavefunc(on and expecta(on values are determined by the simula(ons. Correla(ons are built in from the start. QMC gives most accurate method for general quantum many- body systems. QMC electronic energy is a standard for approximate DFT calcula(ons. (3rd largest cita(on in PRL.) Provide a new understanding of quantum phenomena and a prac(cal tool There are several different stochas(c methods (QMC): - Varia(onal Monte Carlo (VMC) - Projector Monte Carlo Methods for T=0: o Diffusion Monte Carlo (DMC) o Repta(on MD (RQMC) o Auxiliary Field QMC (AFQMC) - Path Integral Monte Carlo for T>0 (PICM) Goal is NOT large N, but higher accuracy and new capabili5es 21 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 21 Some of the Results obtained with QMC On 55 molecules, mean absolute devia(on of atomiza(on energy is 2.9 kcal/mol Successfully applied to organic molecules, transi(on metal oxides, solid state silicon, systems up to ~1000 electrons Can calculate accurate atomiza(on energies, phase energy differences, excita(on energies, one par(cle densi(es, correla(on func(ons, etc.. Scaling is from O(1) to O(N 3 ), depending on implementa(on and quan(ty. 22 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 22 11

12 Breakthrough Quantum Monte Carlo Simula5ons Hard- Core Bosons on a CDC 6600 (1974) Electron gas on CRAY- 1 (1980) Superfluid helium (1984) Ground state of solid hydrogen at high pressures, CRAY XMP and CYBER 205 (1987) Electronic and structure proper5es of carbon/silicon clusters on HP 9000/715 cluster and Cray Y- MP (1995) Coupled Electron- Ion Monte Carlo simula5ons of dense hydrogen on Linux Clusters (2000s) 23 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 23 VMC Varia(onal Monte Carlo 24 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 24 Wagner 12

13 Varia5onal Monte Carlo (McMillan, 1965) VMC is concep5onally the simplest QMC method Put correla(on directly into the wavefunc(on Integrals are hard to do: need Monte Carlo Take sequence of increasingly bezer wavefunc(ons Stochas(c op(miza(on is important! Can we make arbitrarily accurate func(ons? Method of residuals says how to do this No sign problem or with classical complexity 25 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 25 Varia5onal Monte Carlo I Rewrite expecta(on value E( P) = 2 Ψ( RP, ) HΨ( RP, ) 2 Ψ( R, P) dr Ψ( RP, ) Probability distribu(on dr func(on Idea: generate random walkers with probability equal to the above pdf and average (Metropolis method) Minimize the variance of the local energy with respect to the parameters 26 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 26 13

14 Varia5onal Monte Carlo II Take!a!wave!function!Ψ ( R, P)! ( ) 2!with!random!walk Sample! Ψ R, P! Minimize!the!energy!(or!variance)!of!Ψ ( R, P)!with!respect!to!P! ( ) = "walker"! R r 1,r 2,...r 3N Ψ 2 (R) = Det[ϕ i (r j )]exp(u ) Ψ n+1 (R) Ψ n (R)exp φ n 1 Hφ n ( ) 27 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 27 Trial Wave Func5ons General form of wave func(on Slater determinant (Hartree- Fock) Two- body Jastrow Three- body Jastrow Ψ T (R) = Det[φ i (r j )]exp( U ) We op(mize only the c coefficients U = 0 U = c ei k a k (r ii ) + c ee k b k (r ij ) ii k ij k U = two body + c eei [a (r ) klm a k ii l (r ji ) + a k (r ji )a l (r ii )]b k (r ij ) iji klm 28 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 28 14

15 Projector Monte Carlo Projector Monte Carlo 29 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 29 Projector Monte Carlo, e.g. Diffusion Monte Carlo (DMC) 30 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 30 15

16 Diffusion Monte Carlo 31 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 31 Diffusion Monte Carlo General strategy: stochas(cally simulate a differen(al equa(on that converges to the eigenstate Equa(on: dψ(r,t) dt = (H E)Ψ(R,t) Must propagate an en(re func(on forward in (me <=> distribu(on of walkers 32 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 32 16

17 Diffusion Monte Carlo: Imaginary Time Dynamics We want to find a wave func(on so that decreases Kine(c energy (curvature) decreases, poten(al energy stays the same HΨ = EΨ dψ( Rt,) Our differen(al equa(on is = ( H E) Ψ( Rt,) dt Suppose that V (R) Ψ > EΨ Ψ t=0 Time deriva(ve is zero when HΨ = EΨ t=infinity 33 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 33 Diffusion Monte Carlo: Harmonic Oscillator dψ(r,t) dt = Ψ(R,t) + (V (R) E)Ψ(R,t) Ini(al Diffusion Birth/death Generate walkers with a guess distribu(on Each (me step: Take a random step (diffuse) A walker can either die, give birth, or just keep going t Keep following rules, and we find the ground state! Works in an arbitrary number of dimensions Final 34 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 34 17

18 DMC: Importance of a Good Trial Func5on Importance sampling: mul(ply the differen(al equa(on by a trial wave func(on Converges to Ψ T Φ 0 instead of Φ 0 The bezer the trial func(on, the faster DMC is feed it a wave func(on from VMC Fixed node approxima(on: for fermions, ground state has nega(ve and posi(ve parts Not a pdf can t sample it Approxima(on: Ψ T Φ 0 > 0 35 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 35 A Typical QMC Calcula5on Choose system and get one- par(cle orbitals Op(mize wave func(on using VMC, evaluate energy and proper(es of wave func(on Use op(mized wave func(on in DMC for most accurate, lowest energy calcula(ons 36 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 36 18

19 Interes5ng things to look at in QMC Calcula5ons The pair correla(on func(on for the Slater determinant and op(mized two- body wave func(ons The rela(ve energies of Slater determinant/two- body wave func(on/dmc. The fluctua(ons in the trace of the Slater determinant versus two- body wave func(on. 37 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 37 References Hammond, Lester, and Reynolds. Monte Carlo Methods in Quantum Chemistry (book) Foulkes, Mitas, Needs, and Rajagopal. Rev. Mod. Phys. 73, 33 (2001) Umrigar, Nigh(ngale, and Runge. J. Chem. Phys. 99, 2865 (1993) 38 HS 2014 Molecular Simula(ons with Chemical and Biological Applica(ons 38 19