Post on 06-Feb-2018
An introduction toQuantum Monte Carlo
Jose Guilherme VilhenaLaboratoire de Physique de la Matière Condensée et Nanostructures,
Université Claude Bernard Lyon 1
1) Quantum many body problem;
2) Metropolis Algorithm;
3) Statistical foundations of Monte Carlo methods;
4) Quantum Monte Carlo methods ; 4.1) Overview; 4.2) Variational Quantum Monte Carlo; 4.3) Diffusion Quantum Monte Carlo; 4.4) Typical QMC calculation;
Outline
1) Quantum many body problem;
2) Metropolis Algorithm;
3) Statistical foundations of Monte Carlo methods;
4) Quantum Monte Carlo methods ; 4.1) Overview; 4.2) Variational Quantum Monte Carlo; 4.3) Diffusion Quantum Monte Carlo; 4.4) Typical QMC calculation;
Outline
Consider of N electrons. Since me<<M
N , to a good
approximation, the electronic dynamics is governed by :
which is the Born Oppenheimer Hamiltonian.
We want to find the eigenvalues and eigenvectos of this hamiltonian, i.e. :
QMC allow us to solve numerically this, thus providing E0
and Ψ0 .
1 - Quantum many body problem
H=−12∑i
∇ i2−∑
i∑
Z∣r i−d∣
12∑i∑j≠i
Z∣ri−r j∣
Hr1,r 2,. .. ,rN =Er 1,r2,. .. , rN
Consider of N electrons. Since me<<M
N , to a good
approximation, the elotonic dynamics is governed by :
which is the Born Oppenheimer Hamiltonian.
We want to find the eigenvalues and eigenvectos of this hamiltonian, i.e. :
QMC allow us to solve numerically this, thus providing E0
and Ψ0 .
1 - Quantum many body problem
H=−12∑i
∇ i2−∑
i∑
Z∣r i−d∣
12∑i∑j≠i
Z∣ri−r j∣
Hr1,r 2,. .. ,rN =Er 1,r2,. .. , rN
Varionational principleFor any normalized wfn Ψ:
Zero Varience Property The variance of the “local energy” must vanish for Ψ
T=Ψ
0,
i.e.
1 - Quantum many body problem
⟨T R∣ H∣T R⟩E0
E L
2=∫
∣T∣2
∫ dR∣T∣2EL−⟨EL⟩
2=0 ;where EL=
H T R
T R
Example :
Harmonic Oscilator
EL x =HT x
T xthen ...∂ xEL x =0 ;if T=0
≠0 ; if T≠ 0
σEL
(x)
Varionational principleFor any normalized wfn Ψ:
Zero Varience Property The variance of the “local energy” must vanish for Ψ
T=Ψ
0,
i.e.
1 - Quantum many body problem
⟨T R∣ H∣T R⟩E0
E L
2=∫
∣T∣2
∫ dR∣T∣2EL−⟨EL⟩
2=0 ;where EL=
H T R
T R
Example :
Harmonic Oscilator
EL x =HT x
T xthen ...∂ xEL x =0 ;if T=0
≠0 ; if T≠ 0
σEL
(x)
Varionational principleFor any normalized wfn Ψ:
Zero Varience Property The variance of the “local energy” must vanish for Ψ
T=Ψ
0,
i.e.
1 - Quantum many body problem
⟨T R∣ H∣T R⟩E0
E L
2=∫
∣T∣2
∫ dR∣T∣2EL−⟨EL⟩
2=0 ;where EL=
H T R
T R
Example :
Harmonic Oscilator
EL x =HT x
T xthen ...∂ xEL x =0 ;if T=0
≠0 ; if T≠ 0
σEL
(x)
1) Quantum many body problem;
2) Metropolis Algorithm;
3) Statistical foundations of Monte Carlo methods;
4) Quantum Monte Carlo methods ; 4.1) Overview; 4.2) Variational Quantum Monte Carlo; 4.3) Diffusion Quantum Monte Carlo; 4.4) Typical QMC calculation
Outline
Markov Chain stocastic process (no memory).
Metropolis-Hastings algorith (Markov chain) randomly samples a “problematic” probability distribuition function;
Some definitions ...
> P(R): normalized probability distribuition function> R=(r
1,..., r
N) : configuration or walker (can be vector with
the positions of all electrons)> T(R'←R): Proposal density (can be 1, a gaussian, ...)> q=rand(0,1): random number in [0,1]
2 - Metropolis Algorithm
We seek a set {R1, R
2, ..., R
M} of sample points of P(R).
How can we get them ?!? Metropolis Algorithm:
1) Start in a point Rt ;2) Propose a point R', with a probability of T(R'←R) ;3) Calculate the “acceptance rate” : 4) if A > 1 then : Rt+1=R' else if A > q then : Rt+1=R' else : Rt+1=Rt
5) Repeat all from setp 2;6) stop at k>M , and take away the first k-M points.
The sampled points match P(R). The mixing is controled by T(R'←R).
2 - Metropolis Algorithm
A R 'R=T R 'RPRT RR ' PR '
We seek a set {R1, R
2, ..., R
M} of sample points of P(R).
How can we get them ?!? Metropolis Algorithm:
1) Start in a point Rt ;2) Propose a point R', with a probability of T(R'←R) ;3) Calculate the “acceptance rate” : 4) if A > 1 then : Rt+1=R' else if A > q then : Rt+1=R' else : Rt+1=Rt
5) Repeat all from setp 2;6) stop at k>M , and take away the first k-M points.
The sampled points match P(R). The mixing is controled by T(R'←R).
2 - Metropolis Algorithm
A R 'R=T R 'RPRT RR ' PR '
We seek a set {R1, R
2, ..., R
M} of sample points of P(R).
How can we get them ?!? Metropolis Algorithm:
1) Start in a point Rt ;2) Propose a point R', with a probability of T(R'←R) ;3) Calculate the “acceptance rate” : 4) if A > 1 then : Rt+1=R' else if A > q then : Rt+1=R' else : Rt+1=Rt
5) Repeat all from setp 2;6) stop at k>M , and take away the first k-M points.
The sampled points match P(R). The mixing is controled by T(R'←R).
2 - Metropolis Algorithm
A R 'R=T R 'RPRT RR ' PR '
We seek a set {R1, R
2, ..., R
M} of sample points of P(R).
How can we get them ?!? Metropolis Algorithm:
1) Start in a point Rt ;2) Propose a point R', with a probability of T(R'←R) ;3) Calculate the “acceptance rate” : 4) if A > 1 then : Rt+1=R' else if A > q then : Rt+1=R' else : Rt+1=Rt
5) Repeat all from setp 2;6) stop at k>M , and take away the first k-M points.
The sampled points match P(R). The mixing is controled by T(R'←R).
2 - Metropolis Algorithm
A R 'R=T R 'RPRT RR ' PR '
We seek a set {R1, R
2, ..., R
M} of sample points of P(R).
How can we get them ?!? Metropolis Algorithm:
1) Start in a point Rt ;2) Propose a point R', with a probability of T(R'←R) ;3) Calculate the “acceptance rate” : 4) if A > 1 then : Rt+1=R' else if A > q then : Rt+1=R' else : Rt+1=Rt
5) Repeat all from setp 2;6) stop at k>M , and take away the first k-M points.
The sampled points match P(R). The mixing is controled by T(R'←R).
2 - Metropolis Algorithm
A R 'R=T R 'RPRT RR ' PR '
We seek a set {R1, R
2, ..., R
M} of sample points of P(R).
How can we get them ?!? Metropolis Algorithm:
1) Start in a point Rt ;2) Propose a point R', with a probability of T(R'←R) ;3) Calculate the “acceptance rate” : 4) if A > 1 then : Rt+1=R' else if A > q then : Rt+1=R' else : Rt+1=Rt
5) Repeat all from setp 2;6) stop at k>M , and take away the first k-M points.
The sampled points match P(R). The mixing is controled by T(R'←R).
2 - Metropolis Algorithm
A R 'R=T R 'RPRT RR ' PR '
We seek a set {R1, R
2, ..., R
M} of sample points of P(R).
How can we get them ?!? Metropolis Algorithm:
1) Start in a point Rt ;2) Propose a point R', with a probability of T(R'←R) ;3) Calculate the “acceptance rate” : 4) if A > 1 then : Rt+1=R' else if A > q then : Rt+1=R' else : Rt+1=Rt
5) Repeat all from setp 2;6) stop at k>M , and take away the first k-M points.
The sampled points match P(R). The mixing is controled by T(R'←R).
2 - Metropolis Algorithm
A R 'R=T R 'RPRT RR ' PR '
1) Quantum many body problem;
2) Metropolis Algorithm;
3) Statistical foundations of Monte Carlo methods;
4) Quantum Monte Carlo methods ; 4.1) Overview; 4.2) Variational Quantum Monte Carlo; 4.3) Diffusion Quantum Monte Carlo; 4.4) Typical QMC calculation
Outline
Monte Carlo methods: solve problems 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 with random walkers.
Number of dimensions doesn’t matter.- Simpsons rule (error ~ O (M-4/d))- Monte Carlo method (error ~ O (M-1/2))
3 – Statistical foundations of MC methods
Monte Carlo methods: solve problems 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 with random walkers.
Number of dimensions doesn’t matter.- Simpsons rule (error ~ O (M-4/d))- Monte Carlo method (error ~ O (M-1/2))
3 – Statistical foundations of MC methods
Monte Carlo methods: solve problems 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 with random walkers.
Number of dimensions doesn’t matter.- Simpsons rule (error ~ O (M-4/d))- Monte Carlo method (error ~ O (M-1/2))
3 – Statistical foundations of MC methods
Monte Carlo methods: solve problems 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 with random walkers.
Number of dimensions doesn’t matter.- Simpsons rule (error ~ O (M-4/d))- Monte Carlo method (error ~ O (M-1/2))
3 – Statistical foundations of MC methods
Condider a well behaved f(R) such that:
where P(R) is any normed probability density function.
Central limit theorem:
For any p.d.f. , and a large enough ramdom sample {R
1,R
2,...R
M} of it :
i.e., F(R) is normally distributed with F=μf and σ
F=σ
fM-1/2
3 – Statistical foundations of MC methods
F R=1M∑k=0
M
f Rk ≈f ; limM∞
F R=∫ d R f RP R
F2M =⟨F R ;M −f
2⟩= f
2
M
f=∫d R f R PR ; f=∫ d R [ f R−f ]2PR
Condider a well behaved f(R) such that:
where P(R) is any normed probability density function.
Central limit theorem:
For any p.d.f. , and a large enough ramdom sample {R
1,R
2,...R
M} of it :
i.e., F(R) is normally distributed with F=μf and σ
F=σ
fM-1/2
3 – Statistical foundations of MC methods
F R=1M∑k=0
M
f Rk ≈f ; limM∞
F R=∫ d R f RP R
F2M =⟨F R ;M −f
2⟩= f
2
M
f=∫d R f R PR ; f=∫ d R [ f R−f ]2PR
Consider the integral:
P(R) is a normed p.d.f. and f(R)=g(R)/P(R)
i) Metropolis algorithm samples P(R) to get {R1,R
2,...R
M} ;
ii) Monte Carlo estimative of I is
with a statistical error of σIMC
=σfM-1/2
iii) We can estimate σIMC
using MC again, i.e. :
3 – Statistical foundations of MC methods
I=∫ g Rd R=∫P R f RdR
IMC=1M∑k=1
M
f Rk ≈I
f=1M∑i=1
M
[ f Ri−1M∑j=1
M
f R j]2
Consider the integral:
P(R) is any normed p.d.f. and f(R)=g(R)/P(R)
i) Metropolis algorithm samples P(R) to get {R1,R
2,...R
M} ;
ii) Monte Carlo estimative of I is
with a statistical error of σIMC
=σfM-1/2
iii) The σf can be estimated using MC again, i.e. :
3 – Statistical foundations of MC methods
I=∫ g Rd R=∫P R f RdR
IMC=1M∑k=1
M
f R k ≈I
f=1M∑i=1
M
[ f Ri−1M∑j=1
M
f R j]2
Consider the integral:
P(R) is any normed p.d.f. and f(R)=g(R)/P(R)
i) Metropolis algorithm samples P(R) to get {R1,R
2,...R
M} ;
ii) Monte Carlo estimative of I is
with a statistical error of σIMC
=σfM-1/2
iii) The σf can be estimated using MC again, i.e. :
3 – Statistical foundations of MC methods
I=∫ g Rd R=∫P R f RdR
IMC=1M∑k=1
M
f R k ≈I
f=1M∑i=1
M
[ f Ri−1M∑j=1
M
f R j]2
Consider the integral:
P(R) is any normed p.d.f. and f(R)=g(R)/P(R)
i) Metropolis algorithm samples P(R) to get {R1,R
2,...R
M} ;
ii) Monte Carlo estimative of I is
with a statistical error of σIMC
=σfM-1/2
iii) The σf can be estimated using MC again, i.e. :
3 – Statistical foundations of MC methods
I=∫ g Rd R=∫P R f RdR
IMC=1M∑k=1
M
f R k ≈I
f=1M∑i=1
M
[ f Ri−1M∑j=1
M
f R j]2
1) Quantum many body problem;
2) Metropolis Algorithm;
3) Statistical foundations of Monte Carlo methods;
4) Quantum Monte Carlo methods ; 4.1) Overview; 4.2) Variational Quantum Monte Carlo; 4.3) Diffusion Quantum Monte Carlo; 4.4) Typical QMC calculation
Outline
Quantum Monte Carlo simulate the quantum MB problem;(Variational MC; Difusion MC; Path integral MC; Auxiliary field MC; Reptation MC ; Gaussian quantum MC; etc ...)
MC handles directly the many dimention electronic wave function ( R=[r
1,r
2,…,r
N] )
We’ll cover two main flavors: Variational Monte Carlo (integration over (3N)D space); Diffusion Monte Carlo (projector approach)
4 – Quantum Monte Carlo methods 4.1 - Overview
On 55 molecules, mean absolute deviation of atomization energy is 2.9 kcal/mol (MPPT and CI ~ 2kcal/mol ;O(N4-N6) )
Same accuraccy and better scaling than post Hartree-Fock methods (~ N3). (Non Fixed node DMC scales as eN)
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..
No free lunch!! Statistical error ~ (Computational time)-1/2
4 – Quantum Monte Carlo methods 4.1 - Overview
1) Quantum many body problem;
2) Metropolis Algorithm;
3) Statistical foundations of Monte Carlo methods;
4) Quantum Monte Carlo methods ; 4.1) Overview; 4.2) Variational Quantum Monte Carlo; 4.3) Diffusion Quantum Monte Carlo; 4.4) Typical QMC calculation
Outline
Consider: R=(r1, r
2, ...r
N); r
i position of the ith e- ; and a trial wfn Ψ
T(R)
such that:i) Ψ
T(R) and Ψ∇
T(R) exist and are continous in all* R;
ii) and exist;
The <H> with respect to ΨT(R) is:
where EL={HΨ
T Ψ
T} and P(R)=|Ψ
T|2 |Ψ
T|2dR
Using metropolis MC method we sample P(R) ( {R1,..., R
M} ) and
average the EL to get:
Optimizing T to get acceptance ~ 50% we improve the mixing of the
sample;
4 – Quantum Monte Carlo methods 4.2 – Variational Quantum Monte Carlo
EV= ⟨T∣ H∣T ⟩=∫∣T∣
2
∫d R∣T∣2 EL Rd R
EV= ⟨ H ⟩= 1M∑k=1
M
EL Rk
∫∣T∣2d R ∫T
∗ HT d R
Consider: R=(r1, r
2, ...r
N); r
i position of the ith e- ; and a trial wfn Ψ
T(R)
such that:i) Ψ
T(R) and Ψ∇
T(R) exist and are continous in all* R;
ii) and exist;
The <H> with respect to ΨT(R) is:
where EL={HΨ
T Ψ
T} and P(R)=|Ψ
T|2 |Ψ
T|2dR
Using metropolis MC method we sample P(R) ( {R1,..., R
M} ) and
average the EL to get:
Optimizing T to get acceptance ~ 50% we improve the mixing of the
sample;
4 – Quantum Monte Carlo methods 4.2 – Variational Quantum Monte Carlo
EV= ⟨T∣ H∣T ⟩=∫∣T∣
2
∫d R∣T∣2 EL Rd R
EV= ⟨ H ⟩= 1M∑k=1
M
EL Rk
∫∣T∣2d R ∫T
∗ HT d R
Consider: R=(r1, r
2, ...r
N); r
i position of the ith e- ; and a trial wfn Ψ
T(R)
such that:i) Ψ
T(R) and Ψ∇
T(R) exist and are continous in all* R;
ii) and exist;
The <H> with respect to ΨT(R) is:
where EL={HΨ
T Ψ
T} and P(R)=|Ψ
T|2 |Ψ
T|2dR
Using metropolis MC method we sample P(R) ( {R1,..., R
M} ) and
average the EL to get:
Optimizing T to get acceptance ~ 50% we improve the mixing of the
sample;
4 – Quantum Monte Carlo methods 4.2 – Variational Quantum Monte Carlo
EV= ⟨T∣ H∣T ⟩=∫∣T∣
2
∫d R∣T∣2 EL Rd R
EV= ⟨ H ⟩= 1M∑k=1
M
EL Rk
∫∣T∣2d R ∫T
∗ HT d R
Consider: R=(r1, r
2, ...r
N); r
i position of the ith e- ; and a trial wfn Ψ
T(R)
such that:i) Ψ
T(R) and Ψ∇
T(R) exist and are continous in all* R;
ii) and exist;
The <H> with respect to ΨT(R) is:
where EL={HΨ
T Ψ
T} and P(R)=|Ψ
T|2 |Ψ
T|2dR
Using metropolis MC method we sample P(R) ( {R1,..., R
M} ) and
average the EL to get:
Optimizing T to get acceptance ~ 50% we improve the mixing of the
sample;
4 – Quantum Monte Carlo methods 4.2 – Variational Quantum Monte Carlo
EV= ⟨T∣ H∣T ⟩=∫∣T∣
2
∫d R∣T∣2 EL Rd R
EV= ⟨ H ⟩= 1M∑k=1
M
EL Rk
∫∣T∣2d R ∫T
∗ HT d R
4 – Quantum Monte Carlo methods 4.2 – Variational Quantum Monte Carlo
Evaluate Probability ratio
Propose a move
Initial Setup
Update electrons positions
Metropolis accept/reject
Calculate local energy
Output the result
acceptreject
" How many graduate students lives have been lost optimizing wavefuctions ... " D. Ceperley
Good news !!! T can have any functional form !!!
Most commonly used T is a Slater-Jastrow wavefunction:
4 – Quantum Monte Carlo methods 4.2 – Variational Quantum Monte Carlo
X =eJ X ; 1,. ..,i {∑j=1
jD jX }> X=x1, ... ,X N , with x i= {ri ,i }> j i , coefficients to be optimized ;
> DnX =∣ 1x1 ⋯ 1xN ⋮ ⋱ ⋮
N x1 ⋯ N xN ∣ hxg are single particle Slater orbs.
> eJ X ;1,. ..,i , is the Jastrow correlation function;> J=F d ,r i , r ij , the Jastrow factor: usually sums of Chebyshev polynomials
How do we acctually perform the optimization of the wfn ? (minimize
E() or E
v or a mixing of both )
The most used is the minimization of E() , because:
i) We know it's exact value for the g.s.ii) Better numerical stability;
The variance of the energy is given by:
where: parameters to optimize ; Ev variational energy, E
L
local energy.
4 – Quantum Monte Carlo methods 4.2 – Variational Quantum Monte Carlo
E2 =∫
∣∣2
∫∣∣2d R
[EL −EV ]2d R
Correlated sampling method1) Start form a guessed set of paramters {
} ;
2) Sample the P(R,) using MMC method;
3) With this sampling minimize E() like this :
Calculate EL ; E
V and
E() , for a different set of parameters {
}
(choosen in a way to minimize E()), like this:
4) Once E reaches a minimum, set {
}={
}
5) Repeat all the steps 2-5 until E()~0
4 – Quantum Monte Carlo methods 4.2 – Variational Quantum Monte Carlo
EL Ri ,N =HRi ,N Ri ,N
; N =N
T
EV=∫∣T ∣
2N
∫∣T ∣2N d R
E LN d R≈1
∑k=1
M
Rk ,N ∑i=1
M
Ri ,N EL Ri ,N
E2N =∫
∣T ∣2N
∫∣T ∣2N d R
[EL N −EV N ]2d R≈
1
∑k=1
M
R k ,N ∑i=1
M
Ri ,N [EL Ri ,N −EV ]2
PR= ∣R,T ∣2
∫∣ R,T ∣2d R
Correlated sampling method1) Start form a guessed set of paramters {
} ;
2) Sample the P(R,) using MMC method;
3) With this sampling minimize E() like this :
Calculate EL ; E
V and
E() , for a different set of parameters {
}
(choosen in a way to minimize E()), like this:
4) Once E reaches a minimum, set {
}={
}
5) Repeat all the steps 2-5 until E()~0
4 – Quantum Monte Carlo methods 4.2 – Variational Quantum Monte Carlo
EL Ri ,N =HRi ,N Ri ,N
; N =N
T
EV=∫∣T ∣
2N
∫∣T ∣2N d R
E LN d R≈1
∑k=1
M
Rk ,N ∑i=1
M
Ri ,N EL Ri ,N
E2N =∫
∣T ∣2N
∫∣T ∣2N d R
[EL N −EV N ]2d R≈
1
∑k=1
M
R k ,N ∑i=1
M
Ri ,N [EL Ri ,N −EV ]2
PR= ∣R,T ∣2
∫∣ R,T ∣2d R
Correlated sampling method1) Start form a guessed set of paramters {
} ;
2) Sample the P(R,) using MMC method;
3) With this sampling minimize E() like this :
Calculate EL ; E
V and
E() , for a different set of parameters {
}
(choosen in a way to minimize E()), like this:
4) Once E reaches a minimum, set {
}={
}
5) Repeat all the steps 2-5 until E()~0
4 – Quantum Monte Carlo methods 4.2 – Variational Quantum Monte Carlo
EL Ri ,N =HRi ,N Ri ,N
; N =N
T
EV=∫∣T ∣
2N
∫∣T ∣2N d R
E LN d R≈1
∑k=1
M
Rk ,N ∑i=1
M
Ri ,N EL Ri ,N
E2N =∫
∣T ∣2N
∫∣T ∣2N d R
[EL N −EV N ]2d R≈
1
∑k=1
M
R k ,N ∑i=1
M
Ri ,N [EL Ri ,N −EV ]2
PR= ∣R,T ∣2
∫∣ R,T ∣2d R
Correlated sampling method1) Start form a guessed set of paramters {
} ;
2) Sample the P(R,) using MMC method;
3) With this sampling minimize E() like this :
Calculate EL ; E
V and
E() , for a different set of parameters {
}
(choosen in a way to minimize E()), like this:
4) Once E reaches a minimum, set {
}={
}
5) Repeat all the steps 2-5 until E()~0
4 – Quantum Monte Carlo methods 4.2 – Variational Quantum Monte Carlo
EL Ri ,N =HRi ,N Ri ,N
; N =N
T
EV=∫∣T ∣
2N
∫∣T ∣2N d R
E LN d R≈1
∑k=1
M
Rk ,N ∑i=1
M
Ri ,N EL Ri ,N
E2N =∫
∣T ∣2N
∫∣T ∣2N d R
[EL N −EV N ]2d R≈
1
∑k=1
M
R k ,N ∑i=1
M
Ri ,N [EL Ri ,N −EV ]2
PR= ∣R,T ∣2
∫∣ R,T ∣2d R
Correlated sampling method1) Start form a guessed set of paramters {
} ;
2) Sample the P(R,) using MMC method;
3) With this sampling minimize E() like this :
Calculate EL ; E
V and
E() , for a different set of parameters {
}
(choosen in a way to minimize E()), like this:
4) Once E reaches a minimum, set {
}={
}
5) Repeat all the steps 2-5 until E()~0
4 – Quantum Monte Carlo methods 4.2 – Variational Quantum Monte Carlo
EL Ri ,N =HRi ,N Ri ,N
; N =N
T
EV=∫∣T ∣
2N
∫∣T ∣2N d R
E LN d R≈1
∑k=1
M
Rk ,N ∑i=1
M
Ri ,N EL Ri ,N
E2N =∫
∣T ∣2N
∫∣T ∣2N d R
[EL N −EV N ]2d R≈
1
∑k=1
M
R k ,N ∑i=1
M
Ri ,N [EL Ri ,N −EV ]2
PR= ∣R,T ∣2
∫∣ R,T ∣2d R
Correlated sampling method1) Start form a guessed set of paramters {
} ;
2) Sample the P(R,) using MMC method;
3) With this sampling minimize E() like this :
Calculate EL ; E
V and
E() , for a different set of parameters {
}
(choosen in a way to minimize E()), like this:
4) Once E reaches a minimum, set {
}={
}
5) Repeat all the steps 2-5 until E()~0
4 – Quantum Monte Carlo methods 4.2 – Variational Quantum Monte Carlo
EL Ri ,N =HRi ,N Ri ,N
; N =N
T
EV=∫∣T ∣
2N
∫∣T ∣2N d R
E LN d R≈1
∑k=1
M
Rk ,N ∑i=1
M
Ri ,N EL Ri ,N
E2N =∫
∣T ∣2N
∫∣T ∣2N d R
[EL N −EV N ]2d R≈
1
∑k=1
M
R k ,N ∑i=1
M
Ri ,N [EL Ri ,N −EV ]2
PR= ∣R,T ∣2
∫∣ R,T ∣2d R
1) Quantum many body problem;
2) Metropolis Algorithm;
3) Statistical foundations of Monte Carlo methods;
4) Quantum Monte Carlo methods ; 4.1) Overview; 4.2) Variational Quantum Monte Carlo; 4.3) Diffusion Quantum Monte Carlo; 4.4) Typical QMC calculation;
Outline
4 – Quantum Monte Carlo methods 4.3 – Diffusion Quantum Monte Carlo
General strategy: stochastically simulate a differential equation that converges to the eigenstate
Equation:
Must propagate an entire function forward in time <=> distribution of walkers
−dR , t d t
= H−E R , t
4 – Quantum Monte Carlo methods 4.3 – Diffusion Quantum Monte Carlo
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
€
Ψ
−12∇ 2V R E
H=E
H=E
−dR , t d t
= H−E R , t
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 ground state!
Works in an arbitrary number of dimensions
4 – Quantum Monte Carlo methods 4.3 – Diffusion Quantum Monte Carlo
−dR , t d t
=−12∇
2R V R−E R , t
Diffusion Birth/death
Final
Initial
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 ground state!
Works in an arbitrary number of dimensions
4 – Quantum Monte Carlo methods 4.3 – Diffusion Quantum Monte Carlo
−dR , t d t
=−12∇
2R V R−E R , t
Diffusion Birth/death
Final
Initial
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 ground state!
Works in an arbitrary number of dimensions
4 – Quantum Monte Carlo methods 4.3 – Diffusion Quantum Monte Carlo
−dR , t d t
=−12∇
2R V R−E R , t
Diffusion Birth/death
Final
Initial
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 ground state!
Works in an arbitrary number of dimensions
4 – Quantum Monte Carlo methods 4.3 – Diffusion Quantum Monte Carlo
−dR , t d t
=−12∇
2R V R−E R , t
Diffusion Birth/death
Initial
Final
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 ground state!
Works in an arbitrary number of dimensions
4 – Quantum Monte Carlo methods 4.3 – Diffusion Quantum Monte Carlo
Diffusion Birth/deathInitial
Final
t
−dR , t d t
=−12∇
2R V R−E R , t
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:
4 – Quantum Monte Carlo methods 4.3 – Diffusion Quantum Monte Carlo
T0 0
T00
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:
4 – Quantum Monte Carlo methods 4.3 – Diffusion Quantum Monte Carlo
T0
T00
0
1) Quantum many body problem;
2) Metropolis Algorithm;
3) Statistical foundations of Monte Carlo methods;
4) Quantum Monte Carlo methods ; 4.1) Overview; 4.2) Variational Quantum Monte Carlo; 4.3) Diffusion Quantum Monte Carlo; 4.4) Typical QMC calculation;
Outline
Choose system and get one-particle orbitals
Optimize wave function using VMC, evaluate energy and properties of wave function
Use optimized wave function in DMC for most accurate, lowest energy calculations
4 – Quantum Monte Carlo methods 4.4 – Typical QMC calculation
Choose system and get one-particle orbitals
Optimize wave function using VMC, evaluate energy and properties of wave function
Use optimized wave function in DMC for most accurate, lowest energy calculations
4 – Quantum Monte Carlo methods 4.4 – Typical QMC calculation
Choose system and get one-particle orbitals
Optimize wave function using VMC, evaluate energy and properties of wave function
Use optimized wave function in DMC for most accurate, lowest energy calculations
4 – Quantum Monte Carlo methods 4.4 – Typical QMC calculation