54 KiB
Quantum Monte Carlo
- Introduction
- Numerical evaluation of the energy
- Variational Monte Carlo
- Diffusion Monte Carlo
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
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*}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.
Exercise 1
Find the theoretical value of $a$ for which $\Psi$ is an eigenfunction of $\hat{H}$.
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 $$ V(\mathbf{r}) = -\frac{1}{\sqrt{x^2 + y^2 + z^2}} $$
Python
import numpy as np
def potential(r):
# TODO
Python solution
import numpy as np
def potential(r):
return -1. / np.sqrt(np.dot(r,r))
Fortran
double precision function potential(r)
implicit none
double precision, intent(in) :: r(3)
! TODO
end function potential
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
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.
Python
def psi(a, r):
# TODO
Python solution
def psi(a, r):
return np.exp(-a*np.sqrt(np.dot(r,r)))
Fortran
double precision function psi(a, r)
implicit none
double precision, intent(in) :: a, r(3)
! TODO
end function psi
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
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 $$ -\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) = -\frac{1}{2}\left(a^2 - \frac{2a}{\mathbf{|r|}} \right) $$
Python
def kinetic(a,r):
# TODO
Python solution
def kinetic(a,r):
return -0.5 * (a**2 - (2.*a)/np.sqrt(np.dot(r,r)))
Fortran
double precision function kinetic(a,r)
implicit none
double precision, intent(in) :: a, r(3)
! TODO
end function kinetic
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
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.
$$ E_L(\mathbf{r}) = -\frac{1}{2} \frac{\Delta \Psi}{\Psi} (\mathbf{r}) + V(\mathbf{r}) $$
Python
def e_loc(a,r):
#TODO
Python solution
def e_loc(a,r):
return kinetic(a,r) + potential(r)
Fortran
double precision function e_loc(a,r)
implicit none
double precision, intent(in) :: a, r(3)
! TODO
end function e_loc
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
Plot of the local energy along the $x$ axis
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.
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")
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")
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'
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'
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)
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}")
a = 0.1 E = -0.24518438948809218 a = 0.2 E = -0.26966057967803525 a = 0.5 E = -0.3856357612517407 a = 0.9 E = -0.49435709786716214 a = 1.0 E = -0.5 a = 1.5 E = -0.39242967082602226 a = 2.0 E = -0.08086980667844901
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
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.
Exercise (optional)
Prove that : $$\langle E - \langle E \rangle \rangle^2 = \langle E^2 \rangle - \langle E \rangle^2 $$
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}")
a = 0.1 E = -0.24518439 \sigma^2 = 0.02696522 a = 0.2 E = -0.26966058 \sigma^2 = 0.03719707 a = 0.5 E = -0.38563576 \sigma^2 = 0.05318597 a = 0.9 E = -0.49435710 \sigma^2 = 0.00577812 a = 1.0 E = -0.50000000 \sigma^2 = 0.00000000 a = 1.5 E = -0.39242967 \sigma^2 = 0.31449671 a = 2.0 E = -0.08086981 \sigma^2 = 1.80688143
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
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.
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
$$ E \pm \delta E, \text{ where } \delta E = \frac{\sigma}{\sqrt{M}} $$
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
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.
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}")
E = -0.4956255109300764 +/- 0.0007082875482711226
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
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.
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}")
E = -0.4950720838131573 +/- 0.00019089638602238043 A = 0.5172960000000001 +/- 0.0003443446549306529
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
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
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:
\[ T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) & = & \frac{1}{(2\pi\,\tau)^{3/2}} \exp \left[ - \frac{\left( \mathbf{r}_{n+1} - \mathbf{r}_{n} - \frac{\nabla \Psi(\mathbf{r}_n)}{\Psi(\mathbf{r}_n)} \right)^2}{2\,\tau} \right] \]
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
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}")
E = -0.4949730317138491 +/- 0.00012478601801760644 A = 0.7887163333333334 +/- 0.00026834549840347617
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
TODO Diffusion Monte Carlo
TODO Hydrogen atom
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}")
E = -0.49654807434947584 +/- 0.0006868522447409156 A = 0.9876193891840709 +/- 0.00041857361650995804
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
TODO Dihydrogen
We will now consider the H_2 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.