This document provides an introduction and overview of quantum Monte Carlo methods. It begins by reviewing the Metropolis algorithm and how it can be used to evaluate integrals and quantum mechanical operators. It then outlines the key topics which will be covered, including the path integral formulation of quantum mechanics, diffusion Monte Carlo, and calculating the one-body density matrix and excitation energies. The document proceeds to explain how the path integral formulation leads to the Schrodinger equation in the limit of small time steps, and how imaginary time evolution can be used to project out the ground state wavefunction. It concludes by providing examples of applying these methods to calculate properties of hydrogen, molecular hydrogen, and the one-body density matrix of silicon.

1. Introduction to
Quantum Monte Carlo Methods 2!
Claudio Attaccalite
http://attaccalite.com

2. What we learned last time
How to sample a given probability p(x) distribution with
Metropolis Algorithm:
Axx'=min1, px'T x, x'
pxT x', x
How to evaluate integrals in the form:
Evaluate Quantum Mechanical Operators:

〈 f 〉=∫ f xpxdx= 1N
Σ f xi
where xi are distributed
according to p(x)
〈 f  2 〉−〈 f 〉2
 f=N
〈 A〉=∫ A dx
∫ x2 dx
AL x=
x
 x
=∫ AL xpxdx px= ∣x∣2
∫∣x∣2dx

3. Outline
Path integral formulation of Quantum Mechanics
Diffusion Monte Carlo
One­Body
density matrix and excitation energies

4. Path Integral : classical action
The path followed by the particle is
the one that minimize:
t b dt Lxt , ˙xt ;t 
S=∫t a
where S is the Classical Action and L is the Lagrangian
Lxt  ,˙x
t ;t=m2
˙x
t2−V xt  ;t 
Only the extreme path contributes!!!!

5. Path Integral: Quantum Mechanics
In quantum mechanics non just the
extreme path contributes to the
probability amplitude
K B, A= Σ[xt]
over allpossiblepaths
PB, A=∣K 2,1∣2
where
[xt]=Aexp{iℏS[ xt]}
Feynman's path integral formula
B expiℏ
K B, A=∫A
S[B, A]Dxt

6. From Path Integral to Schrödinger equation 1
X2
X1
X4 X5
X3
X...
XM­1
XA XB
M m2
SM=Σ j=1
x j−x j−1
−V x j2
 2
It is possible to discretized the integral
on the continuum
into many intervals M slices of length
=∣xi1−xi∣
∞ K x2, t2 ; x1, t1 x1,t1
 x2,t2=∫−∞
K B, A=lim
∞
∫...∫expiℏ
SM[2,1]
dx1
A
...
dxM−1
A 
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 x2
 xi , t= 1
∞ expiℏ
A∫−∞
 , xi xi−1 , tdxi−1
Lxi−xi−1

7. From Path Integral to Schrödinger equation 2
xi , t= 1
Substituting the discretized action
∞ expiℏ
A∫−∞
m2
exp[−i
 ℏ
V xi ,t ] xi , tdxi−1
We call xi−1−xi= , then send
 , to zero
and compare left and right at the same order 
A=2 i ℏ 
m 1/2
−ℏ
i
∂
∂t =
−ℏ2
2m
∂2
∂ x2V x ,t 
At the order 0 we get the
normalization constant
At the order 1 we get the
Schroedinger equation!

8. Cafe Moment
∞ dx0K x ,t ,x0,0 x0,0
x ,t =∫−∞
What we have: ­>
I=∫ f x1,. .., xNpx1,. .., xNdx1. ..dxN What we want: ­>

9. Imaginary Time Evolution
We want to solve the Schrödinger equation
in imaginary time: ℏ ∂
=it
∂ 
= ℏ
2m
∂2
∂ x2 [V x−Er ]
The formal solution is: x , =exp[−H −ER
ℏ ] x0 ,0
∞
cnnxe−
If we expand in a eigenfunction of H: x , =Σ
n=0
En−ER
ℏ
if ER > E0
if ER < E0
if ER = E0
limt ∞   =∞
limt ∞  =o
limt ∞  =0
Tree Possibility:

10. From Path Integral to DMC: 1
∞ dx0K x , , x0,0 x0,0
 x ,=∫−∞
Using Feynman path integral
the imaginary time evolution can
be rewritten as
lim
N∞
∞ ...∫−∞
∫−∞
∞  m
2ℏ   N/2
exp{− 
N [ m
ℏ Σj
=1
2 
xi−x j−12V xi−En ]}
K x, ,x0 ,0 is equal to
and as usual we rewrite this integral as
K x, ,x0 ,0=lim
N ∞
∞ Π j=1
∫−∞
N−1
dx j Σ
N
Wxn×Pxn , xn1 x0,0
n=1

11. From Path Integral to DMC: 2
K x, ,x0 ,0=lim
N ∞
∞ Π j=1
∫−∞
N−1
dx j Σ
N
Wxn×Pxn , xn−1 x0,0
n=1
2 ℏ exp[−mxn−xn−12
Pxn ,xn−1= m
2 ℏ  ]
Wxn=exp[−[V xn −ER ] 
2ℏ  ]
A Gaussian
probability distribution
A Weight Function
N
Pxn , xn−1
Px0, x1,... xn , xN= x0,0Π i=1
N
WxN
f x1,... xn , xN=Π i=1
If we define:
I=∫ f x1,. .., xN Px1,. .., xN dx1. ..dxN we have

12. The Algorithm
We want generate the
probability distribution
N
Pxn , xn−1
Px0, x1,... xn , xN= x0,0Π i=1
and sample
N
WxN
f x1,... xn , xN=Π i=1
Generate points
distributed on (x0,0)
x1 is generate from x0
sampling P(xn,xn­1
) (a Gaussian)
the weight function is evaluated W(x1)
x ,∞=0
X

14. An example H and H2
Convergence of the Energy
H molecule versus 
H atom wave­function
and energy

17. Application to Silicon:
one body density matrix
r , r '=Σi
, j
i , ji r j r ' i r  LDA local orbitals
The matrix elements are calculated as:
i , j=N∫∗iri jr '
r ' ,r 2,.... , r N
r1,. .., r N
∣r1,. .., r N2∣dr 'dr 1. ..dr N

18. Results on Silicon
Max difference between ii
QMC and LDA is 0.00625
Max off­diagonal
element
0.0014(1)

19. 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 LDA
for r=r' is 1.7%

20. Reference
SISSA Lectures on Numerical methods for strongly correlated
electrons 4th draft
S. Sorella G. E. Santoro and F. Becca (2008)
Introduction to Diffusion Monte Carlo Method
I. Kostin, B. Faber and K. Schulten, physics/9702023v1 (1995)
Quantum Monte Carlo calculations of the one­body
density matrix
and excitation energies of silicon
P. R. C. Kent et al. Phys. Rev. B 57 15293 (1998)
FreeScience.info­>
Quantum Monte Carlo
http://www.freescience.info/books.php?id=35

21. From Path Integral to Schrödinger equation: 1+1/2
xi , t= 1
Substituting the discretized action
∞ expiℏ
A∫−∞
 exp[−i
mxi−xi−12
ℏ
V xi ,t]xi−1 , tdxi−1
We call xi−1−xi= and send to zero
∂t xi , t= 1
A∫−∞
xi , t ∂
∞ expim2
ℏ  [1−i
ℏ V xi ,t...]
[ xi ,t  ∂
∂ xi
 xi ,t 12
2  xi , t]dxi−1
2 ∂2
∂ xi
∞ exp[imℏ2
1
A∫−∞
2ℏ  ]d=1 and  ∂
∂t =−i
ℏ V − ℏ 
2m
∂2
∂ x2