Quantum Monte Carlo

Table of Contents

1 Introduction

We propose different exercises to understand quantum Monte Carlo (QMC) methods. In the first section, we propose to compute the energy of a hydrogen atom using numerical integration. The goal of this section is to introduce the local energy. Then we introduce the variational Monte Carlo (VMC) method which computes a statistical estimate of the expectation value of the energy associated with a given wave function. Finally, we introduce the diffusion Monte Carlo (DMC) method which gives the exact energy of the \(H_2\) molecule.

Code examples will be given in Python and Fortran. Whatever language can be chosen.

We consider the stationary solution of the Schrödinger equation, so the wave functions considered here are real: for an \(N\) electron system where the electrons move in the 3-dimensional space, \(\Psi : \mathbb{R}^{3N} \rightarrow \mathbb{R}\). In addition, \(\Psi\) is defined everywhere, continuous and infinitely differentiable.

Note

In Fortran, when you use a double precision constant, don't forget to put d0 as a suffix (for example 2.0d0), or it will be interpreted as a single precision value

2 Numerical evaluation of the energy

In this section we consider the Hydrogen atom with the following wave function:

\[ \Psi(\mathbf{r}) = \exp(-a |\mathbf{r}|) \]

We will first verify that, for a given value of \(a\), \(\Psi\) is an eigenfunction of the Hamiltonian

\[ \hat{H} = \hat{T} + \hat{V} = - \frac{1}{2} \Delta - \frac{1}{|\mathbf{r}|} \]

To do that, we will check if the local energy, defined as

\[ E_L(\mathbf{r}) = \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}, \]

is constant.

The probabilistic expected value of an arbitrary function \(f(x)\) with respect to a probability density function \(p(x)\) is given by

\[ \langle f \rangle_p = \int_{-\infty}^\infty p(x)\, f(x)\,dx. \]

Recall that a probability density function \(p(x)\) is non-negative and integrates to one:

\[ \int_{-\infty}^\infty p(x)\,dx = 1. \]

The electronic energy of a system is the expectation value of the local energy \(E(\mathbf{r})\) with respect to the 3N-dimensional electron density given by the square of the wave function:

\begin{eqnarray*} E & = & \frac{\langle \Psi| \hat{H} | \Psi\rangle}{\langle \Psi |\Psi \rangle} = \frac{\int \Psi(\mathbf{r})\, \hat{H} \Psi(\mathbf{r})\, d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}} \\ & = & \frac{\int \left[\Psi(\mathbf{r})\right]^2\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}} = \frac{\int \left[\Psi(\mathbf{r})\right]^2\, E_L(\mathbf{r})\,d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}} = \langle E_L \rangle_{\Psi^2} \end{eqnarray*}

2.1 Local energy

Write all the functions of this section in a single file : hydrogen.py if you use Python, or hydrogen.f90 is you use Fortran.

2.1.1 Exercise 1

Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(\hat{H}\).

2.1.2 Exercise 2

Write a function which computes the potential at \(\mathbf{r}\). The function accepts a 3-dimensional vector r as input arguments and returns the potential.

\(\mathbf{r}=\left( \begin{array}{c} x \\ y\\ z\end{array} \right)\), so \[ V(\mathbf{r}) = -\frac{1}{\sqrt{x^2 + y^2 + z^2}} \]

2.1.2.1 Python
import numpy as np

def potential(r):
    # TODO
2.1.2.2 Python   solution
import numpy as np

def potential(r):
    return -1. / np.sqrt(np.dot(r,r))
2.1.2.3 Fortran
double precision function potential(r)
  implicit none
  double precision, intent(in) :: r(3)
  ! TODO
end function potential
2.1.2.4 Fortran   solution
double precision function potential(r)
  implicit none
  double precision, intent(in) :: r(3)
  potential = -1.d0 / dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) )
end function potential

2.1.3 Exercise 3

Write a function which computes the wave function at \(\mathbf{r}\). The function accepts a scalar a and a 3-dimensional vector r as input arguments, and returns a scalar.

2.1.3.1 Python
def psi(a, r):
    # TODO
2.1.3.2 Python   solution
def psi(a, r):
    return np.exp(-a*np.sqrt(np.dot(r,r)))
2.1.3.3 Fortran
double precision function psi(a, r)
  implicit none
  double precision, intent(in) :: a, r(3)
  ! TODO
end function psi
2.1.3.4 Fortran   solution
double precision function psi(a, r)
  implicit none
  double precision, intent(in) :: a, r(3)
  psi = dexp(-a * dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) ))
end function psi

2.1.4 Exercise 4

Write a function which computes the local kinetic energy at \(\mathbf{r}\). The function accepts a and r as input arguments and returns the local kinetic energy.

The local kinetic energy is defined as \[-\frac{1}{2}\frac{\Delta \Psi}{\Psi}.\]

We differentiate \(\Psi\) with respect to \(x\):

\[\Psi(\mathbf{r}) = \exp(-a\,|\mathbf{r}|) \] \[\frac{\partial \Psi}{\partial x} = \frac{\partial \Psi}{\partial |\mathbf{r}|} \frac{\partial |\mathbf{r}|}{\partial x} = - \frac{a\,x}{|\mathbf{r}|} \Psi(\mathbf{r}) \]

and we differentiate a second time:

\[ \frac{\partial^2 \Psi}{\partial x^2} = \left( \frac{a^2\,x^2}{|\mathbf{r}|^2} - \frac{a(y^2+z^2)}{|\mathbf{r}|^{3}} \right) \Psi(\mathbf{r}). \]

The Laplacian operator \(\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\) applied to the wave function gives:

\[ \Delta \Psi (\mathbf{r}) = \left(a^2 - \frac{2a}{\mathbf{|r|}} \right) \Psi(\mathbf{r}) \]

So the local kinetic energy is \[ -\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right) \]

2.1.4.1 Python
def kinetic(a,r):
    # TODO
2.1.4.2 Python   solution
def kinetic(a,r):
    return -0.5 * (a**2 - (2.*a)/np.sqrt(np.dot(r,r)))
2.1.4.3 Fortran
double precision function kinetic(a,r)
  implicit none
  double precision, intent(in) :: a, r(3)
  ! TODO
end function kinetic
2.1.4.4 Fortran   solution
double precision function kinetic(a,r)
  implicit none
  double precision, intent(in) :: a, r(3)
  kinetic = -0.5d0 * (a*a - (2.d0*a) / &
       dsqrt( r(1)*r(1) + r(2)*r(2) + r(3)*r(3) ) ) 
end function kinetic

2.1.5 Exercise 5

Write a function which computes the local energy at \(\mathbf{r}\), using the previously defined functions. The function accepts a and r as input arguments and returns the local kinetic energy.

\[ E_L(\mathbf{r}) = -\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) + V(\mathbf{r}) \]

2.1.5.1 Python
def e_loc(a,r):
    #TODO
2.1.5.2 Python   solution
def e_loc(a,r):
    return kinetic(a,r) + potential(r)
2.1.5.3 Fortran
double precision function e_loc(a,r)
  implicit none
  double precision, intent(in) :: a, r(3)
  ! TODO
end function e_loc
2.1.5.4 Fortran   solution
double precision function e_loc(a,r)
  implicit none
  double precision, intent(in) :: a, r(3)
  double precision, external   :: kinetic, potential
  e_loc = kinetic(a,r) + potential(r)
end function e_loc

2.2 Plot of the local energy along the \(x\) axis

2.2.1 Exercise

For multiple values of \(a\) (0.1, 0.2, 0.5, 1., 1.5, 2.), plot the local energy along the \(x\) axis. In Python, you can use matplotlib for example. In Fortran, it is convenient to write in a text file the values of \(x\) and \(E_L(\mathbf{r})\) for each point, and use Gnuplot to plot the files.

2.2.1.1 Python
import numpy as np
import matplotlib.pyplot as plt

from hydrogen import e_loc

x=np.linspace(-5,5)
plt.figure(figsize=(10,5))

# TODO

plt.tight_layout()
plt.legend()
plt.savefig("plot_py.png")
2.2.1.2 Python   solution
import numpy as np
import matplotlib.pyplot as plt

from hydrogen import e_loc

x=np.linspace(-5,5)
plt.figure(figsize=(10,5))

for a in [0.1, 0.2, 0.5, 1., 1.5, 2.]:
  y=np.array([ e_loc(a, np.array([t,0.,0.]) ) for t in x])
  plt.plot(x,y,label=f"a={a}")

plt.tight_layout()
plt.legend()
plt.savefig("plot_py.png")

plot_py.png

2.2.1.3 Fortran
program plot
  implicit none
  double precision, external :: e_loc

  double precision :: x(50), dx
  integer :: i, j

  dx = 10.d0/(size(x)-1)
  do i=1,size(x)
     x(i) = -5.d0 + (i-1)*dx
  end do

  ! TODO

end program plot

To compile and run: 

gfortran hydrogen.f90 plot_hydrogen.f90 -o plot_hydrogen
./plot_hydrogen > data

To plot the data using gnuplot: 

set grid
set xrange [-5:5]
set yrange [-2:1]
plot './data' index 0 using 1:2 with lines title 'a=0.1', \
     './data' index 1 using 1:2 with lines title 'a=0.2', \
     './data' index 2 using 1:2 with lines title 'a=0.5', \
     './data' index 3 using 1:2 with lines title 'a=1.0', \
     './data' index 4 using 1:2 with lines title 'a=1.5', \
     './data' index 5 using 1:2 with lines title 'a=2.0'
2.2.1.4 Fortran   solution
program plot
  implicit none
  double precision, external :: e_loc

  double precision :: x(50), energy, dx, r(3), a(6)
  integer :: i, j

  a = (/ 0.1d0, 0.2d0, 0.5d0, 1.d0, 1.5d0, 2.d0 /)

  dx = 10.d0/(size(x)-1)
  do i=1,size(x)
     x(i) = -5.d0 + (i-1)*dx
  end do

  r(:) = 0.d0

  do j=1,size(a)
     print *, '# a=', a(j)
     do i=1,size(x)
        r(1) = x(i)
        energy = e_loc( a(j), r )
        print *, x(i), energy
     end do
     print *, ''
     print *, ''
  end do

end program plot

To compile and run: 

gfortran hydrogen.f90 plot_hydrogen.f90 -o plot_hydrogen
./plot_hydrogen > data

To plot the data using gnuplot: 

set grid
set xrange [-5:5]
set yrange [-2:1]
plot './data' index 0 using 1:2 with lines title 'a=0.1', \
     './data' index 1 using 1:2 with lines title 'a=0.2', \
     './data' index 2 using 1:2 with lines title 'a=0.5', \
     './data' index 3 using 1:2 with lines title 'a=1.0', \
     './data' index 4 using 1:2 with lines title 'a=1.5', \
     './data' index 5 using 1:2 with lines title 'a=2.0'

plot.png

2.3 TODO Numerical estimation of the energy

If the space is discretized in small volume elements \(\mathbf{r}_i\) of size \(\delta \mathbf{r}\), the expression of \(\langle E_L \rangle_{\Psi^2}\) becomes a weighted average of the local energy, where the weights are the values of the probability density at \(\mathbf{r}_i\) multiplied by the volume element:

\[ \langle E \rangle_{\Psi^2} \approx \frac{\sum_i w_i E_L(\mathbf{r}_i)}{\sum_i w_i}, \;\; w_i = \left[\Psi(\mathbf{r}_i)\right]^2 \delta \mathbf{r} \]

The energy is biased because:

  • The volume elements are not infinitely small (discretization error)
  • The energy is evaluated only inside the box (incompleteness of the space)

2.3.1 Exercise

Compute a numerical estimate of the energy in a grid of \(50\times50\times50\) points in the range \((-5,-5,-5) \le \mathbf{r} \le (5,5,5)\).

Python 

import numpy as np
from hydrogen import e_loc, psi

interval = np.linspace(-5,5,num=50)
delta = (interval[1]-interval[0])**3

r = np.array([0.,0.,0.])

for a in [0.1, 0.2, 0.5, 0.9, 1., 1.5, 2.]:
    E = 0.
    norm = 0.
      for x in interval:
          r[0] = x
            for y in interval:
                r[1] = y
                  for z in interval:
                      r[2] = z
                      w = psi(a,r)
                      w = w * w * delta
                      E    += w * e_loc(a,r)
                      norm += w 
    E = E / norm
    print(f"a = {a} \t E = {E}")

Fortran 

program energy_hydrogen
  implicit none
  double precision, external :: e_loc, psi
  double precision :: x(50), w, delta, energy, dx, r(3), a(6), norm
  integer :: i, k, l, j

  a = (/ 0.1d0, 0.2d0, 0.5d0, 1.d0, 1.5d0, 2.d0 /)

  dx = 10.d0/(size(x)-1)
  do i=1,size(x)
     x(i) = -5.d0 + (i-1)*dx
  end do

  delta = dx**3

  r(:) = 0.d0

  do j=1,size(a)
     energy = 0.d0
     norm = 0.d0
     do i=1,size(x)
        r(1) = x(i)
        do k=1,size(x)
           r(2) = x(k)
           do l=1,size(x)
              r(3) = x(l)
              w = psi(a(j),r)
              w = w * w * delta
              energy = energy + w * e_loc(a(j), r)
              norm   = norm   + w 
           end do
        end do
     end do
     energy = energy / norm
     print *, 'a = ', a(j), '    E = ', energy
  end do

end program energy_hydrogen

To compile the Fortran and run it: 

gfortran hydrogen.f90 energy_hydrogen.f90 -o energy_hydrogen
./energy_hydrogen 

a =   0.10000000000000001          E =  -0.24518438948809140     
a =   0.20000000000000001          E =  -0.26966057967803236     
a =   0.50000000000000000          E =  -0.38563576125173815     
a =    1.0000000000000000          E =  -0.50000000000000000     
a =    1.5000000000000000          E =  -0.39242967082602065     
a =    2.0000000000000000          E =   -8.0869806678448772E-002

2.4 TODO Variance of the local energy

The variance of the local energy is a functional of \(\Psi\) which measures the magnitude of the fluctuations of the local energy associated with \(\Psi\) around the average:

\[ \sigma^2(E_L) = \frac{\int \left[\Psi(\mathbf{r})\right]^2\, \left[ E_L(\mathbf{r}) - E \right]^2 \, d\mathbf{r}}{\int \left[\Psi(\mathbf{r}) \right]^2 d\mathbf{r}} \] which can be simplified as

\[ \sigma^2(E_L) = \langle E_L^2 \rangle - \langle E_L \rangle^2 \]

If the local energy is constant (i.e. \(\Psi\) is an eigenfunction of \(\hat{H}\)) the variance is zero, so the variance of the local energy can be used as a measure of the quality of a wave function.

2.4.1 Exercise (optional)

Prove that : \[\langle E - \langle E \rangle \rangle^2 = \langle E^2 \rangle - \langle E \rangle^2 \]

2.4.2 Exercise

Add the calculation of the variance to the previous code, and compute a numerical estimate of the variance of the local energy in a grid of \(50\times50\times50\) points in the range \((-5,-5,-5) \le \mathbf{r} \le (5,5,5)\) for different values of \(a\).

Python 

import numpy as np
from hydrogen import e_loc, psi

interval = np.linspace(-5,5,num=50)
delta = (interval[1]-interval[0])**3

r = np.array([0.,0.,0.])

for a in [0.1, 0.2, 0.5, 0.9, 1., 1.5, 2.]:
    E = 0.
    E2 = 0.
    norm = 0.
    for x in interval:
        r[0] = x
        for y in interval:
            r[1] = y
            for z in interval:
                r[2] = z
                w = psi(a, r)
                w = w * w * delta
                El = e_loc(a, r)
                E  += w * El
                E2 += w * El*El
                norm += w 
    E = E / norm
    E2 = E2 / norm
    s2 = E2 - E*E
    print(f"a = {a} \t E = {E:10.8f}  \t  \sigma^2 = {s2:10.8f}")

Fortran 

program variance_hydrogen
  implicit none
  double precision, external :: e_loc, psi
  double precision :: x(50), w, delta, energy, dx, r(3), a(6), norm, s2
  double precision :: e, energy2
  integer :: i, k, l, j

  a = (/ 0.1d0, 0.2d0, 0.5d0, 1.d0, 1.5d0, 2.d0 /)

  dx = 10.d0/(size(x)-1)
  do i=1,size(x)
     x(i) = -5.d0 + (i-1)*dx
  end do

  delta = dx**3

  r(:) = 0.d0

  do j=1,size(a)
     energy = 0.d0
     energy2 = 0.d0
     norm = 0.d0
     do i=1,size(x)
        r(1) = x(i)
        do k=1,size(x)
           r(2) = x(k)
           do l=1,size(x)
              r(3) = x(l)
              w = psi(a(j),r)
              w = w * w * delta
              e = e_loc(a(j), r)
              energy  = energy  + w * e
              energy2 = energy2 + w * e * e
              norm   = norm   + w 
           end do
        end do
     end do
     energy  = energy  / norm
     energy2 = energy2 / norm
     s2 = energy2 - energy*energy
     print *, 'a = ', a(j), ' E = ', energy, ' s2 = ', s2
  end do

end program variance_hydrogen

To compile and run: 

gfortran hydrogen.f90 variance_hydrogen.f90 -o variance_hydrogen
./variance_hydrogen 

a =   0.10000000000000001       E =  -0.24518438948809140       s2 =    2.6965218719722767E-002
a =   0.20000000000000001       E =  -0.26966057967803236       s2 =    3.7197072370201284E-002
a =   0.50000000000000000       E =  -0.38563576125173815       s2 =    5.3185967578480653E-002
a =    1.0000000000000000       E =  -0.50000000000000000       s2 =    0.0000000000000000     
a =    1.5000000000000000       E =  -0.39242967082602065       s2 =   0.31449670909172917     
a =    2.0000000000000000       E =   -8.0869806678448772E-002  s2 =    1.8068814270846534

3 TODO Variational Monte Carlo

Numerical integration with deterministic methods is very efficient in low dimensions. When the number of dimensions becomes large, instead of computing the average energy as a numerical integration on a grid, it is usually more efficient to do a Monte Carlo sampling.

Moreover, a Monte Carlo sampling will alow us to remove the bias due to the discretization of space, and compute a statistical confidence interval.

3.1 TODO Computation of the statistical error

To compute the statistical error, you need to perform \(M\) independent Monte Carlo calculations. You will obtain \(M\) different estimates of the energy, which are expected to have a Gaussian distribution by the central limit theorem.

The estimate of the energy is

\[ E = \frac{1}{M} \sum_{i=1}^M E_M \]

The variance of the average energies can be computed as

\[ \sigma^2 = \frac{1}{M-1} \sum_{i=1}^{M} (E_M - E)^2 \]

And the confidence interval is given by

\[ E \pm \delta E, \text{ where } \delta E = \frac{\sigma}{\sqrt{M}} \]

3.1.1 Exercise

Write a function returning the average and statistical error of an input array.

Python 

from math import sqrt
def ave_error(arr):
    M = len(arr)
    assert (M>1)
    average = sum(arr)/M
    variance = 1./(M-1) * sum( [ (x - average)**2 for x in arr ] )
    return (average, sqrt(variance/M))

Fortran 

subroutine ave_error(x,n,ave,err)
  implicit none
  integer, intent(in)           :: n 
  double precision, intent(in)  :: x(n) 
  double precision, intent(out) :: ave, err
  double precision :: variance
  if (n == 1) then
     ave = x(1)
     err = 0.d0
  else
     ave = sum(x(:)) / dble(n)
     variance = sum( (x(:) - ave)**2 ) / dble(n-1)
     err = dsqrt(variance/dble(n))
  endif
end subroutine ave_error

3.2 TODO Uniform sampling in the box

We will now do our first Monte Carlo calculation to compute the energy of the hydrogen atom.

At every Monte Carlo step:

  • Draw a random point \(\mathbf{r}_i\) in the box \((-5,-5,-5) \le (x,y,z) \le (5,5,5)\)
  • Compute \([\Psi(\mathbf{r}_i)]^2\) and accumulate the result in a variable normalization
  • Compute \([\Psi(\mathbf{r}_i)]^2 \times E_L(\mathbf{r}_i)\), and accumulate the result in a variable energy

One Monte Carlo run will consist of \(N\) Monte Carlo steps. Once all the steps have been computed, the run returns the average energy \(\bar{E}_k\) over the \(N\) steps of the run.

To compute the statistical error, perform \(M\) runs. The final estimate of the energy will be the average over the \(\bar{E}_k\), and the variance of the \(\bar{E}_k\) will be used to compute the statistical error.

3.2.1 Exercise

Parameterize the wave function with \(a=0.9\). Perform 30 independent Monte Carlo runs, each with 100 000 Monte Carlo steps. Store the final energies of each run and use this array to compute the average energy and the associated error bar.

Python 

from hydrogen  import *
from qmc_stats import *

def MonteCarlo(a, nmax):
     E = 0.
     N = 0.
     for istep in range(nmax):
          r = np.random.uniform(-5., 5., (3))
          w = psi(a,r)
          w = w*w
          N += w
          E += w * e_loc(a,r)
   return E/N

a = 0.9
nmax = 100000
X = [MonteCarlo(a,nmax) for i in range(30)]
E, deltaE = ave_error(X)
print(f"E = {E} +/- {deltaE}")

Fortran

When running Monte Carlo calculations, the number of steps is usually very large. We expect nmax to be possibly larger than 2 billion, so we use 8-byte integers (integer*8) to represent it, as well as the index of the current step. 

subroutine uniform_montecarlo(a,nmax,energy)
  implicit none
  double precision, intent(in)  :: a
  integer*8       , intent(in)  :: nmax 
  double precision, intent(out) :: energy

  integer*8 :: istep

  double precision :: norm, r(3), w

  double precision, external :: e_loc, psi

  energy = 0.d0
  norm   = 0.d0
  do istep = 1,nmax
     call random_number(r)
     r(:) = -5.d0 + 10.d0*r(:)
     w = psi(a,r)
     w = w*w
     norm = norm + w
     energy = energy + w * e_loc(a,r)
  end do
  energy = energy / norm
end subroutine uniform_montecarlo

program qmc
  implicit none
  double precision, parameter :: a = 0.9
  integer*8       , parameter :: nmax = 100000
  integer         , parameter :: nruns = 30

  integer :: irun
  double precision :: X(nruns)
  double precision :: ave, err

  do irun=1,nruns
     call uniform_montecarlo(a,nmax,X(irun))
  enddo
  call ave_error(X,nruns,ave,err)
  print *, 'E = ', ave, '+/-', err
end program qmc

gfortran hydrogen.f90 qmc_stats.f90 qmc_uniform.f90 -o qmc_uniform
./qmc_uniform

E =  -0.49588321986667677      +/-   7.1758863546737969E-004

3.3 TODO Metropolis sampling with \(\Psi^2\)

We will now use the square of the wave function to sample random points distributed with the probability density \[ P(\mathbf{r}) = \left[\Psi(\mathbf{r})\right]^2 \]

The expression of the average energy is now simplified to the average of the local energies, since the weights are taken care of by the sampling :

\[ E \approx \frac{1}{M}\sum_{i=1}^M E_L(\mathbf{r}_i) \]

To sample a chosen probability density, an efficient method is the Metropolis-Hastings sampling algorithm. Starting from a random initial position \(\mathbf{r}_0\), we will realize a random walk as follows:

\[ \mathbf{r}_{n+1} = \mathbf{r}_{n} + \tau \mathbf{u} \]

where \(\tau\) is a fixed constant (the so-called time-step), and \(\mathbf{u}\) is a uniform random number in a 3-dimensional box \((-1,-1,-1) \le \mathbf{u} \le (1,1,1)\). We will then add the accept/reject step that will guarantee that the distribution of the \(\mathbf{r}_n\) is \(\Psi^2\):

  • Compute a new position \(\mathbf{r}_{n+1}\)
  • Draw a uniform random number \(v \in [0,1]\)
  • Compute the ratio \(R = \frac{\left[\Psi(\mathbf{r}_{n+1})\right]^2}{\left[\Psi(\mathbf{r}_{n})\right]^2}\)
  • if \(v \le R\), accept the move (do nothing)
  • else, reject the move (set \(\mathbf{r}_{n+1} = \mathbf{r}_n\))
  • evaluate the local energy at \(\mathbf{r}_{n+1}\)

A common error is to remove the rejected samples from the calculation of the average. Don't do it!

All samples should be kept, from both accepted and rejected moves.

If the time step is infinitely small, the ratio will be very close to one and all the steps will be accepted. But the trajectory will be infinitely too short to have statistical significance.

On the other hand, as the time step increases, the number of accepted steps will decrease because the ratios might become small. If the number of accepted steps is close to zero, then the space is not well sampled either.

The time step should be adjusted so that it is as large as possible, keeping the number of accepted steps not too small. To achieve that we define the acceptance rate as the number of accepted steps over the total number of steps. Adjusting the time step such that the acceptance rate is close to 0.5 is a good compromise.

3.3.1 Exercise

Modify the program of the previous section to compute the energy, sampling with \(Psi^2\). Compute also the acceptance rate, so that you can adapt the time step in order to have an acceptance rate close to 0.5 . Can you observe a reduction in the statistical error?

Python 

from hydrogen  import *
from qmc_stats import *

def MonteCarlo(a,nmax,tau):
    E = 0.
    N = 0.
    N_accep = 0.
    r_old = np.random.uniform(-tau, tau, (3))
    psi_old = psi(a,r_old)
    for istep in range(nmax):
        r_new = r_old + np.random.uniform(-tau,tau,(3))
        psi_new = psi(a,r_new)
        ratio = (psi_new / psi_old)**2
        v = np.random.uniform(0,1,(1))
        if v < ratio:
            N_accep += 1.
            r_old = r_new
            psi_old = psi_new
        N += 1.
        E += e_loc(a,r_old)
    return E/N, N_accep/N

a = 0.9
nmax = 100000
tau = 1.3
X0 = [ MonteCarlo(a,nmax,tau) for i in range(30)]
X = [ x for x, _ in X0 ]
A = [ x for _, x in X0 ]
E, deltaE = ave_error(X)
A, deltaA = ave_error(A)
print(f"E = {E} +/- {deltaE}")
print(f"A = {A} +/- {deltaA}")

Fortran 

subroutine metropolis_montecarlo(a,nmax,tau,energy,accep)
  implicit none
  double precision, intent(in)  :: a
  integer*8       , intent(in)  :: nmax 
  double precision, intent(in)  :: tau
  double precision, intent(out) :: energy
  double precision, intent(out) :: accep

  integer*8 :: istep

  double precision :: norm, r_old(3), r_new(3), psi_old, psi_new
  double precision :: v, ratio, n_accep
  double precision, external :: e_loc, psi, gaussian

  energy = 0.d0
  norm   = 0.d0
  n_accep = 0.d0
  call random_number(r_old)
  r_old(:) = tau * (2.d0*r_old(:) - 1.d0)
  psi_old = psi(a,r_old)
  do istep = 1,nmax
     call random_number(r_new)
     r_new(:) = r_old(:) + tau * (2.d0*r_new(:) - 1.d0)
     psi_new = psi(a,r_new)
     ratio = (psi_new / psi_old)**2
     call random_number(v)
     if (v < ratio) then
        r_old(:) = r_new(:)
        psi_old = psi_new
        n_accep = n_accep + 1.d0
     endif
     norm = norm + 1.d0
     energy = energy + e_loc(a,r_old)
  end do
  energy = energy / norm
  accep  = n_accep / norm
end subroutine metropolis_montecarlo

program qmc
  implicit none
  double precision, parameter :: a = 0.9d0
  double precision, parameter :: tau = 1.3d0
  integer*8       , parameter :: nmax = 100000
  integer         , parameter :: nruns = 30

  integer :: irun
  double precision :: X(nruns), Y(nruns)
  double precision :: ave, err

  do irun=1,nruns
     call metropolis_montecarlo(a,nmax,tau,X(irun),Y(irun))
  enddo
  call ave_error(X,nruns,ave,err)
  print *, 'E = ', ave, '+/-', err
  call ave_error(Y,nruns,ave,err)
  print *, 'A = ', ave, '+/-', err
end program qmc

gfortran hydrogen.f90 qmc_stats.f90 qmc_metropolis.f90 -o qmc_metropolis
./qmc_metropolis

E =  -0.49478505004797046      +/-   2.0493795299184956E-004
A =   0.51737800000000000      +/-   4.1827406733181444E-004

3.4 TODO Gaussian random number generator

To obtain Gaussian-distributed random numbers, you can apply the Box Muller transform to uniform random numbers:

\begin{eqnarray*} z_1 &=& \sqrt{-2 \ln u_1} \cos(2 \pi u_2) \\ z_2 &=& \sqrt{-2 \ln u_1} \sin(2 \pi u_2) \end{eqnarray*}

Below is a Fortran implementation returning a Gaussian-distributed n-dimensional vector \(\mathbf{z}\). This will be useful for the following sections.

Fortran 

subroutine random_gauss(z,n)
  implicit none
  integer, intent(in) :: n
  double precision, intent(out) :: z(n)
  double precision :: u(n+1)
  double precision, parameter :: two_pi = 2.d0*dacos(-1.d0)
  integer :: i

  call random_number(u)
  if (iand(n,1) == 0) then
     ! n is even
     do i=1,n,2
        z(i)   = dsqrt(-2.d0*dlog(u(i))) 
        z(i+1) = z(i) * dsin( two_pi*u(i+1) )
        z(i)   = z(i) * dcos( two_pi*u(i+1) )
     end do
  else
     ! n is odd
     do i=1,n-1,2
        z(i)   = dsqrt(-2.d0*dlog(u(i))) 
        z(i+1) = z(i) * dsin( two_pi*u(i+1) )
        z(i)   = z(i) * dcos( two_pi*u(i+1) )
     end do
     z(n)   = dsqrt(-2.d0*dlog(u(n))) 
     z(n)   = z(n) * dcos( two_pi*u(n+1) )
  end if
end subroutine random_gauss

3.5 TODO Generalized Metropolis algorithm

One can use more efficient numerical schemes to move the electrons. But in that case, the Metropolis accepation step has to be adapted accordingly: the acceptance probability \(A\) is chosen so that it is consistent with the probability of leaving \(\mathbf{r}_n\) and the probability of entering \(\mathbf{r}_{n+1}\):

\[ A(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = \min \left( 1, \frac{T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) P(\mathbf{r}_{n+1})} {T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) P(\mathbf{r}_{n})} \right) \] where \(T(\mathbf{r}_n \rightarrow \mathbf{r}_{n+1})\) is the probability of transition from \(\mathbf{r}_n\) to \(\mathbf{r}_{n+1}\).

In the previous example, we were using uniform random numbers. Hence, the transition probability was

\[ T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) & = & \text{constant} \]

So the expression of \(A\) was simplified to the ratios of the squared wave functions.

Now, if instead of drawing uniform random numbers choose to draw Gaussian random numbers with mean 0 and variance \(\tau\), the transition probability becomes:

\[ T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) & = & \frac{1}{(2\pi\,\tau)^{3/2}} \exp \left[ - \frac{\left( \mathbf{r}_{n+1} - \mathbf{r}_{n} \right)^2}{2\tau} \right] \]

To sample even better the density, we can "push" the electrons into in the regions of high probability, and "pull" them away from the low-probability regions. This will mechanically increase the acceptance ratios and improve the sampling.

To do this, we can add the drift vector

\[ \frac{\nabla [ \Psi^2 ]}{\Psi^2} = 2 \frac{\nabla \Psi}{\Psi} \].

The numerical scheme becomes a drifted diffusion:

\[ \mathbf{r}_{n+1} = \mathbf{r}_{n} + \tau \frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi \]

where \(\chi\) is a Gaussian random variable with zero mean and variance \(\tau\). The transition probability becomes:

\[ T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) & = & \frac{1}{(2\pi\,\tau)^{3/2}} \exp \left[ - \frac{\left( \mathbf{r}_{n+1} - \mathbf{r}_{n} - \frac{\nabla \Psi(\mathbf{r}_n)}{\Psi(\mathbf{r}_n)} \right)^2}{2\,\tau} \right] \]

3.5.1 Exercise 1

Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\).

Python 

def drift(a,r):
  ar_inv = -a/np.sqrt(np.dot(r,r))
  return r * ar_inv

Fortran 

subroutine drift(a,r,b)
  implicit none
  double precision, intent(in)  :: a, r(3)
  double precision, intent(out) :: b(3)
  double precision :: ar_inv
  ar_inv = -a / dsqrt(r(1)*r(1) + r(2)*r(2) + r(3)*r(3))
  b(:) = r(:) * ar_inv
end subroutine drift

3.5.2 Exercise 2

Modify the previous program to introduce the drifted diffusion scheme. (This is a necessary step for the next section).

Python 

from hydrogen  import *
from qmc_stats import *

def MonteCarlo(a,tau,nmax):
    E = 0.
    N = 0.
    accep_rate = 0.
    sq_tau = np.sqrt(tau)
    r_old = np.random.normal(loc=0., scale=1.0, size=(3))
    d_old = drift(a,r_old)
    d2_old = np.dot(d_old,d_old)
    psi_old = psi(a,r_old)
    for istep in range(nmax):
        chi = np.random.normal(loc=0., scale=1.0, size=(3))
        r_new = r_old + tau * d_old + sq_tau * chi
        d_new = drift(a,r_new)
        d2_new = np.dot(d_new,d_new)
        psi_new = psi(a,r_new)
        # Metropolis
        prod = np.dot((d_new + d_old), (r_new - r_old))
        argexpo = 0.5 * (d2_new - d2_old)*tau + prod
        q = psi_new / psi_old
        q = np.exp(-argexpo) * q*q
        if np.random.uniform() < q:
            accep_rate += 1.
            r_old = r_new
            d_old = d_new
            d2_old = d2_new
            psi_old = psi_new
        N += 1.
        E += e_loc(a,r_old)
    return E/N, accep_rate/N


a = 0.9
nmax = 100000
tau = 1.0
X = [MonteCarlo(a,tau,nmax) for i in range(30)]
E, deltaE = ave_error([x[0] for x in X])
A, deltaA = ave_error([x[1] for x in X])
print(f"E = {E} +/- {deltaE}\nA = {A} +/- {deltaA}")

Fortran 

subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
  implicit none
  double precision, intent(in)  :: a, tau
  integer*8       , intent(in)  :: nmax 
  double precision, intent(out) :: energy, accep_rate

  integer*8 :: istep
  double precision :: norm, sq_tau, chi(3), d2_old, prod, u
  double precision :: psi_old, psi_new, d2_new, argexpo, q
  double precision :: r_old(3), r_new(3)
  double precision :: d_old(3), d_new(3)
  double precision, external :: e_loc, psi

  sq_tau = dsqrt(tau)

  ! Initialization
  energy = 0.d0
  norm   = 0.d0
  accep_rate = 0.d0
  call random_gauss(r_old,3)
  call drift(a,r_old,d_old)
  d2_old = d_old(1)*d_old(1) + d_old(2)*d_old(2) + d_old(3)*d_old(3)
  psi_old = psi(a,r_old)

  do istep = 1,nmax
     call random_gauss(chi,3)
     r_new(:) = r_old(:) + tau * d_old(:) + chi(:)*sq_tau
     call drift(a,r_new,d_new)
     d2_new = d_new(1)*d_new(1) + d_new(2)*d_new(2) + d_new(3)*d_new(3)
     psi_new = psi(a,r_new)
     ! Metropolis
     prod = (d_new(1) + d_old(1))*(r_new(1) - r_old(1)) + &
          (d_new(2) + d_old(2))*(r_new(2) - r_old(2)) + &
          (d_new(3) + d_old(3))*(r_new(3) - r_old(3))
     argexpo = 0.5d0 * (d2_new - d2_old)*tau + prod
     q = psi_new / psi_old
     q = dexp(-argexpo) * q*q
     call random_number(u)
     if (u<q) then
        accep_rate = accep_rate + 1.d0
        r_old(:) = r_new(:)
        d_old(:) = d_new(:)
        d2_old = d2_new
        psi_old = psi_new
     end if
     norm = norm + 1.d0
     energy = energy + e_loc(a,r_old)
  end do
  energy = energy / norm
  accep_rate = accep_rate / norm
end subroutine variational_montecarlo

program qmc
  implicit none
  double precision, parameter :: a = 0.9
  double precision, parameter :: tau = 1.0
  integer*8       , parameter :: nmax = 100000
  integer         , parameter :: nruns = 30

  integer :: irun
  double precision :: X(nruns), accep(nruns)
  double precision :: ave, err

  do irun=1,nruns
     call variational_montecarlo(a,tau,nmax,X(irun),accep(irun))
  enddo
  call ave_error(X,nruns,ave,err)
  print *, 'E = ', ave, '+/-', err
  call ave_error(accep,nruns,ave,err)
  print *, 'A = ', ave, '+/-', err
end program qmc

gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
./vmc_metropolis

E =  -0.49499990423528023      +/-   1.5958250761863871E-004
A =   0.78861366666666655      +/-   3.5096729498002445E-004

4 TODO Diffusion Monte Carlo

4.1 TODO Hydrogen atom

4.1.1 Exercise

Modify the Metropolis VMC program to introduce the PDMC weight. In the limit \(\tau \rightarrow 0\), you should recover the exact energy of H for any value of \(a\).

Python 

from hydrogen  import *
from qmc_stats import *

def MonteCarlo(a,tau,nmax,Eref):
    E = 0.
    N = 0.
    accep_rate = 0.
    sq_tau = np.sqrt(tau)
    r_old = np.random.normal(loc=0., scale=1.0, size=(3))
    d_old = drift(a,r_old)
    d2_old = np.dot(d_old,d_old)
    psi_old = psi(a,r_old)
    w = 1.0
    for istep in range(nmax):
        chi = np.random.normal(loc=0., scale=1.0, size=(3))
        el = e_loc(a,r_old)
        w *= np.exp(-tau*(el - Eref))
        N += w
        E += w * el

        r_new = r_old + tau * d_old + sq_tau * chi
        d_new = drift(a,r_new)
        d2_new = np.dot(d_new,d_new)
        psi_new = psi(a,r_new)
        # Metropolis
        prod = np.dot((d_new + d_old), (r_new - r_old))
        argexpo = 0.5 * (d2_new - d2_old)*tau + prod
        q = psi_new / psi_old
        q = np.exp(-argexpo) * q*q
        # PDMC weight
        if np.random.uniform() < q:
            accep_rate += w
            r_old = r_new
            d_old = d_new
            d2_old = d2_new
            psi_old = psi_new
    return E/N, accep_rate/N


a = 0.9
nmax = 10000
tau = .1
X = [MonteCarlo(a,tau,nmax,-0.5) for i in range(30)]
E, deltaE = ave_error([x[0] for x in X])
A, deltaA = ave_error([x[1] for x in X])
print(f"E = {E} +/- {deltaE}\nA = {A} +/- {deltaA}")

Fortran 

subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
  implicit none
  double precision, intent(in)  :: a, tau
  integer*8       , intent(in)  :: nmax 
  double precision, intent(out) :: energy, accep_rate

  integer*8 :: istep
  double precision :: norm, sq_tau, chi(3), d2_old, prod, u
  double precision :: psi_old, psi_new, d2_new, argexpo, q
  double precision :: r_old(3), r_new(3)
  double precision :: d_old(3), d_new(3)
  double precision, external :: e_loc, psi

  sq_tau = dsqrt(tau)

  ! Initialization
  energy = 0.d0
  norm   = 0.d0
  accep_rate = 0.d0
  call random_gauss(r_old,3)
  call drift(a,r_old,d_old)
  d2_old = d_old(1)*d_old(1) + d_old(2)*d_old(2) + d_old(3)*d_old(3)
  psi_old = psi(a,r_old)

  do istep = 1,nmax
     call random_gauss(chi,3)
     r_new(:) = r_old(:) + tau * d_old(:) + chi(:)*sq_tau
     call drift(a,r_new,d_new)
     d2_new = d_new(1)*d_new(1) + d_new(2)*d_new(2) + d_new(3)*d_new(3)
     psi_new = psi(a,r_new)
     ! Metropolis
     prod = (d_new(1) + d_old(1))*(r_new(1) - r_old(1)) + &
            (d_new(2) + d_old(2))*(r_new(2) - r_old(2)) + &
            (d_new(3) + d_old(3))*(r_new(3) - r_old(3))
     argexpo = 0.5d0 * (d2_new - d2_old)*tau + prod
     q = psi_new / psi_old
     q = dexp(-argexpo) * q*q
     call random_number(u)
     if (u<q) then
        accep_rate = accep_rate + 1.d0
        r_old(:) = r_new(:)
        d_old(:) = d_new(:)
        d2_old = d2_new
        psi_old = psi_new
     end if
     norm = norm + 1.d0
     energy = energy + e_loc(a,r_old)
  end do
  energy = energy / norm
  accep_rate = accep_rate / norm
end subroutine variational_montecarlo

program qmc
  implicit none
  double precision, parameter :: a = 0.9
  double precision, parameter :: tau = 1.0
  integer*8       , parameter :: nmax = 100000
  integer         , parameter :: nruns = 30

  integer :: irun
  double precision :: X(nruns), accep(nruns)
  double precision :: ave, err

  do irun=1,nruns
     call variational_montecarlo(a,tau,nmax,X(irun),accep(irun))
  enddo
  call ave_error(X,nruns,ave,err)
  print *, 'E = ', ave, '+/-', err
  call ave_error(accep,nruns,ave,err)
  print *, 'A = ', ave, '+/-', err
end program qmc

gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
./vmc_metropolis

E =  -0.49499990423528023      +/-   1.5958250761863871E-004
A =   0.78861366666666655      +/-   3.5096729498002445E-004

4.2 TODO Dihydrogen

We will now consider the H2 molecule in a minimal basis composed of the \(1s\) orbitals of the hydrogen atoms:

\[ \Psi(\mathbf{r}_1, \mathbf{r}_2) = \exp(-(\mathbf{r}_1 - \mathbf{R}_A)) + \] where \(\mathbf{r}_1\) and \(\mathbf{r}_2\) denote the electron coordinates and \(\mathbf{R}_A\) and \(\mathbf{R}_B\) the coordinates of the nuclei.

Author: Anthony Scemama, Claudia Filippi

Created: 2021-01-21 Thu 22:25

Validate