Download - Introduction to Diffusion Monte Carlo
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Introduction to Introduction to Quantum Monte Carlo Methods Quantum Monte Carlo Methods 22!!
Claudio Attaccalitehttp://attaccalite.com
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What we learned last timeWhat we learned last time
How to sample a given probability p(x) distribution with Metropolis Algorithm:
How to evaluate integrals in the form:
Evaluate Quantum Mechanical Operators:
A xx ' =min 1,p x ' T x, x 'p x T x ', x
⟨ f ⟩=∫ f x p x dx=1N ∑ f x i
where xi are distributed
according to p(x)
f = ⟨ f 2⟩−⟨ f ⟩
2
N
⟨ A⟩=∫ A dx
∫ x 2 dx
=∫ AL xp xdx p x=∣ x ∣
2
∫∣ x ∣2dx
AL x=x
x
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OutlineOutline
Path integral formulation of Quantum Mechanics
Diffusion Monte Carlo
OneBody density matrix and excitation energies
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Path Integral : classical action
where S is the Classical Action and L is the Lagrangian
S=∫t a
t b
dt Lx t , x t ;t
The path followed by the particle is the one that minimize:
Lxt , x t ;t =m2
x t 2−V xt ;t
Only the extreme path contributes!!!!
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Path Integral: Quantum Mechanics
K B, A= ∑ [x t ]over allpossiblepaths
In quantum mechanics non just the extreme path contributes to the
probability amplitude
P B, A=∣K 2,1∣2
where
[x t ]=Aexp { iℏ
S[x t ]}
K B, A=∫A
Bexp i
ℏS[B, A ]Dx t
Feynman's path integral formula
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From Path Integral to Schrödinger equation 1
XA
XB
X1
X2
X3
X4 X
5
X...
XM1
SM=∑
j=1
M m2 x j−x j−1
2
−V x j2
It is possible to discretized the integral on the continuum
into many intervals M slices of length
x2,t2=∫−∞
∞
K x2, t2 ; x1,t1 x1,t1
K B, A =lim ∞
∫ ...∫exp iℏ
SM[2,1 ]
dx1
A...
dxM−1
A
=∣x i1−xi∣
On each path the discretized classical action can be written as
We want use this propagator in order to obtain the wavefunction at time t2 in the position x
2
x i , t=1A∫
−∞
∞
exp iℏ
L x i−x i−1
, xi xi−1 , t dxi−1
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From Path Integral to Schrödinger equation 2
x i , t=1A∫−∞
∞
exp iℏ m2
exp [−iℏ V x i ,t ] x i , t dxi−1
We call xi−1−x i= , then send
Substituting the discretized action
and compare left and right at the same order
A=2 iℏ
m 1 /2
−ℏ
i∂
∂t=
−ℏ2
2m∂2
∂x2V x ,t
, to zero
At the order 0 we get the normalization constant
At the order 1 we get the Schroedinger equation!
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Cafe Moment
x ,t =∫−∞
∞
dx0K x ,t ,x0,0 x0,0
I=∫ f x1,. .. , xN p x1,. .. , xN dx1. ..dxNWhat we want: >
What we have: >
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Imaginary Time Evolution
=itWe want to solve the Schrödinger equation in imaginary time: ℏ
∂
∂=
ℏ
2m∂2
∂x2[V x −Er ]
The formal solution is: x , =exp[− H−ER
ℏ ] x0 ,0
If we expand in a eigenfunction of H: x , =∑n=0
∞
cnnx e−
En−ER
ℏ
if ER > E
0
if E
R < E
0
if ER = E
0
limt∞ =∞
limt∞ =o
limt∞ =0
Tree Possibility:
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From Path Integral to DMC: 1
x ,=∫−∞
∞
dx0K x , , x0,0 x0,0
Using Feynman path integralthe imaginary time evolution can
be rewritten as
limN∞
∫−∞
∞
...∫−∞
∞ m2ℏ
N /2
exp{−
ℏ∑j=1
N
[ m2
x i−x j−12V x i−En ]}
K x, ,x0 ,0 is equal to
and as usual we rewrite this integral as
K x, ,x0 ,0= limN ∞
∫−∞
∞ ∏j=1
N−1
dx j ∑n=1
N
W xn×P xn , xn1 x0,0
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From Path Integral to DMC: 2
K x, ,x0 ,0= limN ∞
∫−∞
∞ ∏j=1
N−1
dx j ∑n=1
N
W xn×P xn , xn−1 x0,0
P xn ,xn−1= m2 ℏ exp [−mxn−xn−1
2
2 ℏ ]
W xn=exp[−[V xn −ER ]
2ℏ ]
A Gaussian probability distribution
A Weight Function
P x0, x1,...xn , xN = x0,0∏i=1
N
P xn , xn−1
f x1,... xn , xN =∏i=1
N
W xN
If we define:
I=∫ f x1,. .. , xN P x1,. .. , xN dx1. ..dxNwe have
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The Algorithm
P x0, x1,...xn , xN = x0,0∏i=1
N
P xn , xn−1
f x1,...xn , xN =∏i=1
N
W xN
We want generate the probability distribution
and sample
Generate points distributed on (x
0,0)
x1 is generate from x
0
sampling P(xn,x
n1) (a Gaussian)
the weight function is evaluated W(x1)
x ,∞=0
X
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An example H and H2
Convergence of the Energy H molecule versus
H atom wavefunction and energy
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Application to Silicon:one body density matrix
r ,r '=∑i , j
i , ji r j r ' i r LDA local orbitals
i , j=N∫∗ir i jr ' r ' ,r 2,.... , rN
r1,. .. , rN
∣r1,. .. , rN 2∣dr 'dr 1. ..drN
The matrix elements are calculated as:
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Results on Silicon
Max difference between ii
QMC and LDA is 0.00625
Max offdiagonal element 0.0014(1)
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Results on Silicon: 2
QMC onebodydensity matrix on the 110 plane where r is fixed
on the center of the bonding
Difference between QMC and LDAfor r=r' is 1.7%
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ReferenceSISSA Lectures on Numerical methods for strongly correlated
electrons 4th draft S. Sorella G. E. Santoro and F. Becca (2008)
Introduction to Diffusion Monte Carlo MethodI. Kostin, B. Faber and K. Schulten, physics/9702023v1 (1995)
Quantum Monte Carlo calculations of the onebody density matrix and excitation energies of siliconP. R. C. Kent et al. Phys. Rev. B 57 15293 (1998)
FreeScience.info> Quantum Monte Carlohttp://www.freescience.info/books.php?id=35
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From Path Integral to Schrödinger equation: 1+1/2
x i , t=1A∫
−∞
∞
exp iℏ m x i−xi−12
exp[−iℏ V x i , t ] xi−1 , t dxi−1
We call xi−1−x i= and send to zero
Substituting the discretized action
x i , t ∂∂t
x i , t =1A∫−∞
∞
exp im2
ℏ [1−iℏV x i , t ...]
[ x i ,t ∂∂x i
x i , t 12
2 ∂2
∂x i2 x i , t ]dxi−1
1A∫−∞
∞
exp[ imℏ2
2ℏ ]d=1 and ∂
∂t=−
iℏV −
ℏ
2m∂2
∂x2