WagnerNCN

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The basics of quantum Monte Carlo

Lucas K. WagnerComputational Nanosciences Group

University of California, Berkeley

In collaboration with Jeffrey C. Grossman

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Introduction

Introduction

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The quantum many-body problem

• Fundamental object in quantum mechanics:

• Want to find special wave functions such that

where

• This is a fundamentally many-body equation!

• We want to find an accurate method that will work in general to solve this

(r1,r2,...,rN )

1 2 1 2( , ,..., ) ( , ,..., )n N n NH r r r E r r r

21 1

2I

ii iI i jiI ij

ZH

r r

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Fluctuation minimization

• Eigenvalue problem

•

• Non-eigenfunction andeigenfunction for simple harmonic osc.

• One way to approximatethe eigenfunction: minimizedeviation from a constant

( )

( )n

nn

H xE

x

( )( )

( )

H xE x

x

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Monte Carlo

• Monte Carlo is solving a problem using random numbers!

• Evaluate integrals: – expectation value of a random variable is just the integral over its

probability distribution– generate a bunch of random numbers and average to get the

integral

• Simulate random processes--random walks sample configuration space

• Number of dimensions doesn’t matter

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Quantum Monte Carlo

• Comes in many flavors

• Deals with the many body wave function directlywork with R=[r1,r2,…,rn]

• We’ll cover two main flavors: variational Monte Carlo and diffusion Monte Carlo

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Results

• On 55 molecules, mean absolute deviation of atomization energy is 2.9 kcal/mol

• Successfully applied to organic molecules, transition metal oxides, solid state silicon, systems up to ~1000 electrons

• Can calculate accurate atomization energies, phase energy differences, excitation energies, one particle densities, correlation functions, etc..

• Scaling is from O(1) to O(N3), depending on implementation and quantity.

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VMC

Variational Monte Carlo

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Variational Monte Carlo

• Rewrite expectationvalue

2

2

( , ) ( , )( )

( , )( , )

R P H R PE P dR

R PR P dR

Probability distribution function

• 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

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Trial wave functions in the tool

• General form of wave function

– Slater determinant (Hartree-Fock)

– Two-body Jastrow

– Three-body Jastrow

• We optimize only the c coefficients

T (R) Det[ i(rj )]exp(U)

U 0

U ckeiak (riI )

k

iI

ckeebk (rij )

k

ij

U two body cklmeei [ak (riI )

klm

ijI

al (rjI ) ak (rjI )al (riI )]bk (rij )

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DMC

Diffusion Monte Carlo

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• General strategy: stochastically simulate a differential equation that converges to the eigenstate

• Equation:

• Must propagate an entire function forward in time <=> distribution of walkers

Diffusion Monte Carlo

( , )( ) ( , )

d R tH E R t

dt

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Diffusion Monte Carlo: imaginary time dynamics

• We want to find a wave function so

• Our differential equation

• Suppose that

• decreases

• Kinetic energy (curvature)decreases, potential energystays the same

• Time derivative is zero when

t=0

t=infinity

( , )( ) ( , )

d R tH E R t

dt

H E

H E

1

22 V (R)

E

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Diffusion Monte Carlo: Harmonic Oscillator

d(R,t)

dt

1

22(R, t) (V (R) E)(R, t)

Diffusion Birth/death

•Generate walkers with a guess distribution

•Each time step:•Take a random step (diffuse)•A walker can either die, give birth, or just keep going

•Keep following rules, and we find the groundstate!

•Works in an arbitrary number of dimensions

t

Initial

Final

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Diffusion Monte Carlo: importance of a good trial

function

• Importance sampling: multiply the differential equation by a trial wave function– Converges to instead of– The better the trial function, the faster DMC is-- feed it

a wave function from VMC

• Fixed node approximation: for fermions, ground state has negative and positive parts– Not a pdf--can’t sample it– Approximation:

T0

0

T0 0

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Typical calculation

• Choose system and get one-particle orbitals (we’ve already prepared the orbitals for you)

• Optimize wave function using VMC, evaluate energy and properties of wave function

• Use optimized wave function in DMC for most accurate, lowest energy calculations

• Check the tooltips for explanations of the few settings

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Interesting things to look at

• The pair correlation function for the Slater determinant and optimized two-body wave functions, also compare with DMC

• The relative energies of Slater determinant/two-body wave function/DMC.

• The fluctuations in the trace of the Slater determinant versus two-body wave function.

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References

• Hammond, Lester, and Reynolds. Monte Carlo Methods in Quantum Chemistry (book)

• Wikipedia, www.qwalk.org, www.qmcwiki.org

• Foulkes, Mitas, Needs, and Rajagopal. Rev. Mod. Phys. 73, 33 (2001)

• Umrigar, Nightingale, and Runge. J. Chem. Phys. 99, 2865 (1993)