This document provides an introduction to quantum Monte Carlo methods. It discusses using Monte Carlo integration to evaluate multi-dimensional integrals that arise in quantum mechanical problems. Variational Monte Carlo is introduced as using a trial wavefunction to sample configuration space and estimate observables, like the energy. The Metropolis algorithm is described as a way to generate Markov chains that sample a given probability distribution. This allows using Monte Carlo methods to solve the electronic structure problem by approximating many-body wavefunctions and integrals over configuration space.

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

2. Outline
A bit of probability theory
Variational Monte Carlo
Wave­Function
and Optimization

3. Definition of probability
P(Ei)= pi= Number of successful events
Total Number of experiments
In the limit of a large number of experiments
N
pi=1
Σi
=1
probability of composite events
pi=Σk
joint probability: pi,j
marginal probability:
conditional probability: p(i|j)
probability for j whatever the second
pi,k
event may be or not
probability for occurrence of j give
that the event i occurred

4. More Definitions
Mean Value: 〈x〉=Σi
xipi
The mean value <x> is the expected average value after repeating
several times the same experiment
Variance: var x=〈 x2 〉−〈x〉2=Σi
xi−〈x〉 2pi
The variance is a positive quantity that is zero only if all the events having
a non­vanishing
probability give the same value for the variable xi
Standard deviation: =var x
The standard deviation is assumed as a measure of the dispersion of the
variable x

5. Chebyshev's inequality
P=P [x−〈 x〉 2var x
 ] for ≤1
If the variance is small the random variable x became
“more” predictable, in the sense
that is value xi at each event is close to <x> with a
non­negligible
probability

6. Extension to Continues Variables
Cumulative probability : F y=P{x≤y}
Clearly F(y) is a monotonically increasing function and
Density probability: y= dFy
dy
And for discrete distributions:
F ∞=1
Obviously: y≥0
y=Σi
pi y−xEi

7. The law of large number
x=1N
Σi
xi
The average of x is obtained averaging over a
large number N of independent realizations of the
same experiment
〈 x〉=〈x〉 var  x=〈  x2 〉−〈 x〉=1N
var x
Central Limit Theorem
 x= 1
2 2/N
e
− x−〈x 〉2
22/N
The average of x is Gaussian distributed
for large N and its standard deviation decrease
as 1/sqrt(N)

8. Monte Carlo Example:
Estimating p
If you are a very poor dart player, it is easy to imagine
throwing darts randomly at the above figure, and it should be
apparent that of the total number of darts that hit within the
square, the number of darts that hit the shaded part is
proportional to the area of that part.

9. In other words:
Pinside=r 2
4r2=4
and:

10. A Simple Integral
Consider the simple integral:
This can be evaluated in the same
way as the pi example. By
randomly tossing darts in the
interval a-b and evaluating the
function f(x) on these points

11. The electronic structure problem
P.A.M. Dirac:The fundamental laws necessary for the
mathematical treatment of a large part of physics and the
whole of chemistry are thus completely known, and the
difficulty lies only in the fact that application of these laws
leads to equations that are too complex to be solved.

12. Variational Monte Carlo
Monte Carlo integration is necessary because the wave­function
contains
explicit particle correlations that
leads to non­factoring
multi­dimension
integrals.

13. How to sample a given probability
distribution?

14. Solution Markov chain:
random walk in configuration space
A Markov chain is a stochastic dynamics for which
a random variable xn evolves according to
xn1=Fxn ,n
xn and xn+1 are not independent so we can define a
joint probability to go from first to the second
f nxn1 ,xn=K xn1∣xn n xn 
Marginal probability to be in xn Conditional probability to go from xn to xn+1
n1xn1=Σx
Master equation: K xn1∣xnn xn
n

15. Limit distribution
of the Master equation
n1xn1=Σxn
K xn1∣xnn xn
1) Does It exist a limiting distribution? x
2) Starting form a given arbitrary configuration under
which condition we converge?

16. Sufficient and necessary
conditions for the convergence
The answer to the first question requires that:
xn1=Σx
n Kxn1∣xn xn 
In order to satisfy this requirement it is sufficient but not necessary that
the so­called
detailed balance holds:
K x'∣x x=K x∣x' x '
The answer to the second question requires ergodicity!
Namely that every configuration x' can be reached
in a sufficient large number of Markov interactions,
starting from any initial configuration x

17. 17
Nicholas Metropolis (1915­1999)
The algorithm by Metropolis
(and A Rosenbluth, M
Rosenbluth, A Teller and E
Teller, 1953) has been cited as
among the top 10 algorithms
having the "greatest influence
on the development and
practice of science and
engineering in the 20th
century."

18. 18
Metropolis Algorithm
We want
1) a Markov chain such that, for large n, converge to (x)
2) a condition probability K(x'|x) that satisfy the detailed balance with this
probability distribution
Solution! (by Metropolis and friends)
K x'∣x= Ax '∣xT x'∣x
Ax'∣x=min{1, x'Tx∣x '
xT x '∣x }
where T(x'|x) is a general and reasonable transition probability from x to x'

19. 19
The Algorithm
start from a random configuration x'
generate a new one according to T(x'|x)
accept or reject according to Metropolis rule
evaluate our function
Important
It not necessary to have a normalized probability
distribution (or wave­function!)

20. 20
More or less we have arrived
we can evaluate this integral
〈 A〉=∫ R A RdR
∫ R2dR
=∫ AL R2RdR
∫ R2dR
and its variance
var  A=∫ AL
2 R2RdR
∫R2dR
−〈 A〉2
but we just need a wave function . . . . . . .

21. The trial wave­function
The trial­function
completely determines
quality of the approximation for the physical observables
The simplest WF is the Slater­Jastrow
r1,r2,. ..,rn=Det∣A∣expUcorr 
Det from DFT, CI, HF, scratch, etc..
other functional forms: pairing BCS, multi­determinant,
pfaffian

22. Optimization strategies
In order to obtain a good variational wave­function,
it is possible to optimize the WF minimizing one of the following functionals
or a linear combination of both
EV a ,b,c=∫ a ,b,c..H a ,b,c...dRn
∫¿
The Variational Energy
The Variance of the Energy:
2
2−Ev
(always positive and 0 for exact 2 a ,b,c...=∫[ ground state!) H

]
2

23. And finally an application!!!

24. 2D electron gas
H=
The Hamiltonian :
−1
2rs 2
N
∇i
Σi
2 1
r s
N 1
∣ri−r j∣
Σi
 j
rS= 1
naB
Unpolarized phase Unpolarized phase Wigner Crystal

25. 2D electron gas: the phase diagram
We found a new phase
of the 2D electron gas at
low density
a stable spin polarized
phase
before the Wigner
crystallization.

26. Difficulties With VMC
The many­electron
wavefunction is unknown
Has to be approximated
May seem hopeless to have to actually guess
the wavefunction
But is surprisingly accurate when it works

27. The Limitation of VMC
Nothing can really be done if the trial
wavefunction isn’t accurate enough
Moreover it favours simple states over more
complicated ones
Therefore, there are other methods
Example: Diffusion QMC

28. Next Monday
Diffusion Monte Carlo and Sign­Problem
Applications
Then . . . .
Finite Temperature Path­Integral
Monte Carlo
One­dimensional
electron gas
Excited States
One­body
density matrix
Diagramatic Monte Carlo

29. 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)
FreeScience.info­>
Quantum Monte Carlo
http://www.freescience.info/books.php?id=35

30. Exact conditions
Electron­Nuclei
cusp conditions
When one electron approach a nuclei the wave­function
reduce
to a simple hydrogen like, namely:
The same condition holds when two
electron meet electron­electron
cusp
condition and can be satisfied with a
two­body
Jastrow factor