The basics of quantum Monte Carlo
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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)