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@ -1,9 +1,7 @@
#+TITLE: Quantum Monte Carlo #+TITLE: Quantum Monte Carlo
#+AUTHOR: Anthony Scemama, Claudia Filippi #+AUTHOR: Anthony Scemama, Claudia Filippi
# SETUPFILE: https://fniessen.github.io/org-html-themes/org/theme-readtheorg.setup
# SETUPFILE: https://fniessen.github.io/org-html-themes/org/theme-bigblow.setup
#+LANGUAGE: en #+LANGUAGE: en
#+INFOJS_OPT: toc:t mouse:underline path:http://orgmode.org/org-info.js #+INFOJS_OPT: toc:t mouse:underline path:org-info.js
#+STARTUP: latexpreview #+STARTUP: latexpreview
#+LATEX_CLASS: report #+LATEX_CLASS: report
#+LATEX_HEADER_EXTRA: \usepackage{minted} #+LATEX_HEADER_EXTRA: \usepackage{minted}
@ -47,7 +45,7 @@
computes a statistical estimate of the expectation value of the energy computes a statistical estimate of the expectation value of the energy
associated with a given wave function. associated with a given wave function.
Finally, we introduce the diffusion Monte Carlo (DMC) method which Finally, we introduce the diffusion Monte Carlo (DMC) method which
gives the exact energy of the hydrogen atom and of the $H_2$ molecule. gives the exact energy of the hydrogen atom and of the H_2 molecule.
Code examples will be given in Python and Fortran. You can use Code examples will be given in Python and Fortran. You can use
whatever language you prefer to write the program. whatever language you prefer to write the program.
@ -1156,17 +1154,17 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_uniform.f90 -o qmc_uniform
initial position $\mathbf{r}_0$, we will realize a random walk as follows: initial position $\mathbf{r}_0$, we will realize a random walk as follows:
$$ $$
\mathbf{r}_{n+1} = \mathbf{r}_{n} + \tau \mathbf{u} \mathbf{r}_{n+1} = \mathbf{r}_{n} + \delta t\, \mathbf{u}
$$ $$
where $\tau$ is a fixed constant (the so-called /time-step/), and where $\delta t$ is a fixed constant (the so-called /time-step/), and
$\mathbf{u}$ is a uniform random number in a 3-dimensional box $\mathbf{u}$ 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 $(-1,-1,-1) \le \mathbf{u} \le (1,1,1)$. We will then add the
accept/reject step that guarantees that the distribution of the accept/reject step that guarantees that the distribution of the
$\mathbf{r}_n$ is $\Psi^2$: $\mathbf{r}_n$ is $\Psi^2$:
1) Compute $\Psi$ at a new position $\mathbf{r'} = \mathbf{r}_n + 1) Compute $\Psi$ at a new position $\mathbf{r'} = \mathbf{r}_n +
\tau \mathbf{u}$ \delta t\, \mathbf{u}$
2) Compute the ratio $R = \frac{\left[\Psi(\mathbf{r'})\right]^2}{\left[\Psi(\mathbf{r}_{n})\right]^2}$ 2) Compute the ratio $R = \frac{\left[\Psi(\mathbf{r'})\right]^2}{\left[\Psi(\mathbf{r}_{n})\right]^2}$
3) Draw a uniform random number $v \in [0,1]$ 3) Draw a uniform random number $v \in [0,1]$
4) if $v \le R$, accept the move : set $\mathbf{r}_{n+1} = \mathbf{r'}$ 4) if $v \le R$, accept the move : set $\mathbf{r}_{n+1} = \mathbf{r'}$
@ -1213,15 +1211,15 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_uniform.f90 -o qmc_uniform
from hydrogen import * from hydrogen import *
from qmc_stats import * from qmc_stats import *
def MonteCarlo(a,nmax,tau): def MonteCarlo(a,nmax,dt):
# TODO # TODO
return energy/nmax, N_accep/nmax return energy/nmax, N_accep/nmax
# Run simulation # Run simulation
a = 0.9 a = 0.9
nmax = 100000 nmax = 100000
tau = 1.3 dt = 1.3
X0 = [ MonteCarlo(a,nmax,tau) for i in range(30)] X0 = [ MonteCarlo(a,nmax,dt) for i in range(30)]
# Energy # Energy
X = [ x for (x, _) in X0 ] X = [ x for (x, _) in X0 ]
@ -1236,11 +1234,11 @@ print(f"A = {A} +/- {deltaA}")
*Fortran* *Fortran*
#+BEGIN_SRC f90 :tangle none #+BEGIN_SRC f90 :tangle none
subroutine metropolis_montecarlo(a,nmax,tau,energy,accep) subroutine metropolis_montecarlo(a,nmax,dt,energy,accep)
implicit none implicit none
double precision, intent(in) :: a double precision, intent(in) :: a
integer*8 , intent(in) :: nmax integer*8 , intent(in) :: nmax
double precision, intent(in) :: tau double precision, intent(in) :: dt
double precision, intent(out) :: energy double precision, intent(out) :: energy
double precision, intent(out) :: accep double precision, intent(out) :: accep
@ -1257,7 +1255,7 @@ end subroutine metropolis_montecarlo
program qmc program qmc
implicit none implicit none
double precision, parameter :: a = 0.9d0 double precision, parameter :: a = 0.9d0
double precision, parameter :: tau = 1.3d0 double precision, parameter :: dt = 1.3d0
integer*8 , parameter :: nmax = 100000 integer*8 , parameter :: nmax = 100000
integer , parameter :: nruns = 30 integer , parameter :: nruns = 30
@ -1266,7 +1264,7 @@ program qmc
double precision :: ave, err double precision :: ave, err
do irun=1,nruns do irun=1,nruns
call metropolis_montecarlo(a,nmax,tau,X(irun),Y(irun)) call metropolis_montecarlo(a,nmax,dt,X(irun),Y(irun))
enddo enddo
call ave_error(X,nruns,ave,err) call ave_error(X,nruns,ave,err)
@ -1288,13 +1286,13 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_metropolis.f90 -o qmc_metropolis
from hydrogen import * from hydrogen import *
from qmc_stats import * from qmc_stats import *
def MonteCarlo(a,nmax,tau): def MonteCarlo(a,nmax,dt):
energy = 0. energy = 0.
N_accep = 0 N_accep = 0
r_old = np.random.uniform(-tau, tau, (3)) r_old = np.random.uniform(-dt, dt, (3))
psi_old = psi(a,r_old) psi_old = psi(a,r_old)
for istep in range(nmax): for istep in range(nmax):
r_new = r_old + np.random.uniform(-tau,tau,(3)) r_new = r_old + np.random.uniform(-dt,dt,(3))
psi_new = psi(a,r_new) psi_new = psi(a,r_new)
ratio = (psi_new / psi_old)**2 ratio = (psi_new / psi_old)**2
v = np.random.uniform() v = np.random.uniform()
@ -1308,8 +1306,8 @@ def MonteCarlo(a,nmax,tau):
# Run simulation # Run simulation
a = 0.9 a = 0.9
nmax = 100000 nmax = 100000
tau = 1.3 dt = 1.3
X0 = [ MonteCarlo(a,nmax,tau) for i in range(30)] X0 = [ MonteCarlo(a,nmax,dt) for i in range(30)]
# Energy # Energy
X = [ x for (x, _) in X0 ] X = [ x for (x, _) in X0 ]
@ -1328,11 +1326,11 @@ print(f"A = {A} +/- {deltaA}")
*Fortran* *Fortran*
#+BEGIN_SRC f90 :exports code #+BEGIN_SRC f90 :exports code
subroutine metropolis_montecarlo(a,nmax,tau,energy,accep) subroutine metropolis_montecarlo(a,nmax,dt,energy,accep)
implicit none implicit none
double precision, intent(in) :: a double precision, intent(in) :: a
integer*8 , intent(in) :: nmax integer*8 , intent(in) :: nmax
double precision, intent(in) :: tau double precision, intent(in) :: dt
double precision, intent(out) :: energy double precision, intent(out) :: energy
double precision, intent(out) :: accep double precision, intent(out) :: accep
@ -1346,11 +1344,11 @@ subroutine metropolis_montecarlo(a,nmax,tau,energy,accep)
energy = 0.d0 energy = 0.d0
n_accep = 0_8 n_accep = 0_8
call random_number(r_old) call random_number(r_old)
r_old(:) = tau * (2.d0*r_old(:) - 1.d0) r_old(:) = dt * (2.d0*r_old(:) - 1.d0)
psi_old = psi(a,r_old) psi_old = psi(a,r_old)
do istep = 1,nmax do istep = 1,nmax
call random_number(r_new) call random_number(r_new)
r_new(:) = r_old(:) + tau * (2.d0*r_new(:) - 1.d0) r_new(:) = r_old(:) + dt * (2.d0*r_new(:) - 1.d0)
psi_new = psi(a,r_new) psi_new = psi(a,r_new)
ratio = (psi_new / psi_old)**2 ratio = (psi_new / psi_old)**2
call random_number(v) call random_number(v)
@ -1368,7 +1366,7 @@ end subroutine metropolis_montecarlo
program qmc program qmc
implicit none implicit none
double precision, parameter :: a = 0.9d0 double precision, parameter :: a = 0.9d0
double precision, parameter :: tau = 1.3d0 double precision, parameter :: dt = 1.3d0
integer*8 , parameter :: nmax = 100000 integer*8 , parameter :: nmax = 100000
integer , parameter :: nruns = 30 integer , parameter :: nruns = 30
@ -1377,7 +1375,7 @@ program qmc
double precision :: ave, err double precision :: ave, err
do irun=1,nruns do irun=1,nruns
call metropolis_montecarlo(a,nmax,tau,X(irun),Y(irun)) call metropolis_montecarlo(a,nmax,dt,X(irun),Y(irun))
enddo enddo
call ave_error(X,nruns,ave,err) call ave_error(X,nruns,ave,err)
@ -1477,12 +1475,12 @@ end subroutine random_gauss
Now, if instead of drawing uniform random numbers we Now, if instead of drawing uniform random numbers we
choose to draw Gaussian random numbers with zero mean and variance choose to draw Gaussian random numbers with zero mean and variance
$\tau$, the transition probability becomes: $\delta t$, the transition probability becomes:
\[ \[
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) =
\frac{1}{(2\pi\,\tau)^{3/2}} \exp \left[ - \frac{\left( \frac{1}{(2\pi\,\delta t)^{3/2}} \exp \left[ - \frac{\left(
\mathbf{r}_{n+1} - \mathbf{r}_{n} \right)^2}{2\tau} \right] \mathbf{r}_{n+1} - \mathbf{r}_{n} \right)^2}{2\delta t} \right]
\] \]
@ -1500,19 +1498,19 @@ end subroutine random_gauss
The numerical scheme becomes a drifted diffusion: The numerical scheme becomes a drifted diffusion:
\[ \[
\mathbf{r}_{n+1} = \mathbf{r}_{n} + \tau \frac{\nabla \mathbf{r}_{n+1} = \mathbf{r}_{n} + \delta t\, \frac{\nabla
\Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi \Psi(\mathbf{r})}{\Psi(\mathbf{r})} + \chi
\] \]
where $\chi$ is a Gaussian random variable with zero mean and where $\chi$ is a Gaussian random variable with zero mean and
variance $\tau$. variance $\delta t$.
The transition probability becomes: The transition probability becomes:
\[ \[
T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) =
\frac{1}{(2\pi\,\tau)^{3/2}} \exp \left[ - \frac{\left( \frac{1}{(2\pi\,\delta t)^{3/2}} \exp \left[ - \frac{\left(
\mathbf{r}_{n+1} - \mathbf{r}_{n} - \frac{\nabla \mathbf{r}_{n+1} - \mathbf{r}_{n} - \frac{\nabla
\Psi(\mathbf{r}_n)}{\Psi(\mathbf{r}_n)} \right)^2}{2\,\tau} \right] \Psi(\mathbf{r}_n)}{\Psi(\mathbf{r}_n)} \right)^2}{2\,\delta t} \right]
\] \]
@ -1570,14 +1568,14 @@ end subroutine drift
from hydrogen import * from hydrogen import *
from qmc_stats import * from qmc_stats import *
def MonteCarlo(a,nmax,tau): def MonteCarlo(a,nmax,dt):
# TODO # TODO
# Run simulation # Run simulation
a = 0.9 a = 0.9
nmax = 100000 nmax = 100000
tau = 1.3 dt = 1.3
X0 = [ MonteCarlo(a,nmax,tau) for i in range(30)] X0 = [ MonteCarlo(a,nmax,dt) for i in range(30)]
# Energy # Energy
X = [ x for (x, _) in X0 ] X = [ x for (x, _) in X0 ]
@ -1592,14 +1590,14 @@ print(f"A = {A} +/- {deltaA}")
*Fortran* *Fortran*
#+BEGIN_SRC f90 :tangle none #+BEGIN_SRC f90 :tangle none
subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate) subroutine variational_montecarlo(a,dt,nmax,energy,accep_rate)
implicit none implicit none
double precision, intent(in) :: a, tau double precision, intent(in) :: a, dt
integer*8 , intent(in) :: nmax integer*8 , intent(in) :: nmax
double precision, intent(out) :: energy, accep_rate double precision, intent(out) :: energy, accep_rate
integer*8 :: istep integer*8 :: istep
double precision :: sq_tau, chi(3) double precision :: sq_dt, chi(3)
double precision :: psi_old, psi_new double precision :: psi_old, psi_new
double precision :: r_old(3), r_new(3) double precision :: r_old(3), r_new(3)
double precision :: d_old(3), d_new(3) double precision :: d_old(3), d_new(3)
@ -1612,7 +1610,7 @@ end subroutine variational_montecarlo
program qmc program qmc
implicit none implicit none
double precision, parameter :: a = 0.9 double precision, parameter :: a = 0.9
double precision, parameter :: tau = 1.0 double precision, parameter :: dt = 1.0
integer*8 , parameter :: nmax = 100000 integer*8 , parameter :: nmax = 100000
integer , parameter :: nruns = 30 integer , parameter :: nruns = 30
@ -1621,7 +1619,7 @@ program qmc
double precision :: ave, err double precision :: ave, err
do irun=1,nruns do irun=1,nruns
call variational_montecarlo(a,tau,nmax,X(irun),accep(irun)) call variational_montecarlo(a,dt,nmax,X(irun),accep(irun))
enddo enddo
call ave_error(X,nruns,ave,err) call ave_error(X,nruns,ave,err)
print *, 'E = ', ave, '+/-', err print *, 'E = ', ave, '+/-', err
@ -1641,23 +1639,23 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
from hydrogen import * from hydrogen import *
from qmc_stats import * from qmc_stats import *
def MonteCarlo(a,nmax,tau): def MonteCarlo(a,nmax,dt):
energy = 0. energy = 0.
accep_rate = 0. accep_rate = 0.
sq_tau = np.sqrt(tau) sq_dt = np.sqrt(dt)
r_old = np.random.normal(loc=0., scale=1.0, size=(3)) r_old = np.random.normal(loc=0., scale=1.0, size=(3))
d_old = drift(a,r_old) d_old = drift(a,r_old)
d2_old = np.dot(d_old,d_old) d2_old = np.dot(d_old,d_old)
psi_old = psi(a,r_old) psi_old = psi(a,r_old)
for istep in range(nmax): for istep in range(nmax):
chi = np.random.normal(loc=0., scale=1.0, size=(3)) chi = np.random.normal(loc=0., scale=1.0, size=(3))
r_new = r_old + tau * d_old + sq_tau * chi r_new = r_old + dt * d_old + sq_dt * chi
d_new = drift(a,r_new) d_new = drift(a,r_new)
d2_new = np.dot(d_new,d_new) d2_new = np.dot(d_new,d_new)
psi_new = psi(a,r_new) psi_new = psi(a,r_new)
# Metropolis # Metropolis
prod = np.dot((d_new + d_old), (r_new - r_old)) prod = np.dot((d_new + d_old), (r_new - r_old))
argexpo = 0.5 * (d2_new - d2_old)*tau + prod argexpo = 0.5 * (d2_new - d2_old)*dt + prod
q = psi_new / psi_old q = psi_new / psi_old
q = np.exp(-argexpo) * q*q q = np.exp(-argexpo) * q*q
if np.random.uniform() < q: if np.random.uniform() < q:
@ -1673,8 +1671,8 @@ def MonteCarlo(a,nmax,tau):
# Run simulation # Run simulation
a = 0.9 a = 0.9
nmax = 100000 nmax = 100000
tau = 1.3 dt = 1.3
X0 = [ MonteCarlo(a,nmax,tau) for i in range(30)] X0 = [ MonteCarlo(a,nmax,dt) for i in range(30)]
# Energy # Energy
X = [ x for (x, _) in X0 ] X = [ x for (x, _) in X0 ]
@ -1693,20 +1691,20 @@ print(f"A = {A} +/- {deltaA}")
*Fortran* *Fortran*
#+BEGIN_SRC f90 #+BEGIN_SRC f90
subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate) subroutine variational_montecarlo(a,dt,nmax,energy,accep_rate)
implicit none implicit none
double precision, intent(in) :: a, tau double precision, intent(in) :: a, dt
integer*8 , intent(in) :: nmax integer*8 , intent(in) :: nmax
double precision, intent(out) :: energy, accep_rate double precision, intent(out) :: energy, accep_rate
integer*8 :: istep integer*8 :: istep
double precision :: sq_tau, chi(3), d2_old, prod, u double precision :: sq_dt, chi(3), d2_old, prod, u
double precision :: psi_old, psi_new, d2_new, argexpo, q double precision :: psi_old, psi_new, d2_new, argexpo, q
double precision :: r_old(3), r_new(3) double precision :: r_old(3), r_new(3)
double precision :: d_old(3), d_new(3) double precision :: d_old(3), d_new(3)
double precision, external :: e_loc, psi double precision, external :: e_loc, psi
sq_tau = dsqrt(tau) sq_dt = dsqrt(dt)
! Initialization ! Initialization
energy = 0.d0 energy = 0.d0
@ -1718,7 +1716,7 @@ subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
do istep = 1,nmax do istep = 1,nmax
call random_gauss(chi,3) call random_gauss(chi,3)
r_new(:) = r_old(:) + tau * d_old(:) + chi(:)*sq_tau r_new(:) = r_old(:) + dt * d_old(:) + chi(:)*sq_dt
call drift(a,r_new,d_new) 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) 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) psi_new = psi(a,r_new)
@ -1726,7 +1724,7 @@ subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
prod = (d_new(1) + d_old(1))*(r_new(1) - r_old(1)) + & 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(2) + d_old(2))*(r_new(2) - r_old(2)) + &
(d_new(3) + d_old(3))*(r_new(3) - r_old(3)) (d_new(3) + d_old(3))*(r_new(3) - r_old(3))
argexpo = 0.5d0 * (d2_new - d2_old)*tau + prod argexpo = 0.5d0 * (d2_new - d2_old)*dt + prod
q = psi_new / psi_old q = psi_new / psi_old
q = dexp(-argexpo) * q*q q = dexp(-argexpo) * q*q
call random_number(u) call random_number(u)
@ -1746,7 +1744,7 @@ end subroutine variational_montecarlo
program qmc program qmc
implicit none implicit none
double precision, parameter :: a = 0.9 double precision, parameter :: a = 0.9
double precision, parameter :: tau = 1.0 double precision, parameter :: dt = 1.0
integer*8 , parameter :: nmax = 100000 integer*8 , parameter :: nmax = 100000
integer , parameter :: nruns = 30 integer , parameter :: nruns = 30
@ -1755,7 +1753,7 @@ program qmc
double precision :: ave, err double precision :: ave, err
do irun=1,nruns do irun=1,nruns
call variational_montecarlo(a,tau,nmax,X(irun),accep(irun)) call variational_montecarlo(a,dt,nmax,X(irun),accep(irun))
enddo enddo
call ave_error(X,nruns,ave,err) call ave_error(X,nruns,ave,err)
print *, 'E = ', ave, '+/-', err print *, 'E = ', ave, '+/-', err
@ -1845,7 +1843,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
- a rate equation for the potential. - a rate equation for the potential.
The diffusion equation can be simulated by a Brownian motion: The diffusion equation can be simulated by a Brownian motion:
\[ \mathbf{r}_{n+1} = \mathbf{r}_{n} + \sqrt{\tau}\, \chi \] \[ \mathbf{r}_{n+1} = \mathbf{r}_{n} + \sqrt{\delta t}\, \chi \]
where $\chi$ is a Gaussian random variable, and the rate equation where $\chi$ is a Gaussian random variable, and the rate equation
can be simulated by creating or destroying particles over time (a can be simulated by creating or destroying particles over time (a
so-called branching process). so-called branching process).
@ -1871,7 +1869,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
\left[ -\frac{1}{2} \Delta \psi(\mathbf{r},\tau) + V(\mathbf{r}) \psi(\mathbf{r},\tau) \right] \Psi_T(\mathbf{r}) \left[ -\frac{1}{2} \Delta \psi(\mathbf{r},\tau) + V(\mathbf{r}) \psi(\mathbf{r},\tau) \right] \Psi_T(\mathbf{r})
\] \]
Defining $\Pi(\mathbf{r},t) = \psi(\mathbf{r},\tau) Defining $\Pi(\mathbf{r},\tau) = \psi(\mathbf{r},\tau)
\Psi_T(\mathbf{r})$, \Psi_T(\mathbf{r})$,
\[ \[
@ -1881,10 +1879,15 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
\right] + E_L(\mathbf{r}) \Pi(\mathbf{r},\tau) \right] + E_L(\mathbf{r}) \Pi(\mathbf{r},\tau)
\] \]
The new "kinetic energy" can be simulated by the drifted diffusion
scheme presented in the previous section (VMC).
The new "potential" is the local energy, which has smaller fluctuations The new "potential" is the local energy, which has smaller fluctuations
as $\Psi_T$ tends to the exact wave function. The new "kinetic energy" when $\Psi_T$ gets closer to the exact wave function. It can be simulated by
can be simulated by the drifted diffusion scheme presented in the changing the number of particles according to $\exp\left[ -\delta t\,
previous section (VMC). \left(E_L(\mathbf{r}) - E_\text{ref}\right)\right]$
where $E_{\text{ref}}$ is a constant introduced so that the average
of this term is close to one, keeping the number of particles rather
constant.
This equation generates the /N/-electron density $\Pi$, which is the This equation generates the /N/-electron density $\Pi$, which is the
product of the ground state with the trial wave function. It product of the ground state with the trial wave function. It
@ -1895,7 +1898,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
systems where the wave function has nodes (systems with 3 or more systems where the wave function has nodes (systems with 3 or more
electrons), this scheme introduces an error known as the /fixed node electrons), this scheme introduces an error known as the /fixed node
error/. error/.
*** Appendix : Details of the Derivation *** Appendix : Details of the Derivation
\[ \[
@ -1942,15 +1945,87 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
\right] + E_L(\mathbf{r}) \Pi(\mathbf{r},\tau) \right] + E_L(\mathbf{r}) \Pi(\mathbf{r},\tau)
\] \]
** Fixed-node DMC energy
Now that we have a process to sample $\Pi(\mathbf{r},\tau) =
\psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})$, we can compute the exact
energy of the system, within the fixed-node constraint, as:
\[
E = \lim_{\tau \to \infty} \frac{\int \Pi(\mathbf{r},\tau) E_L(\mathbf{r}) d\mathbf{r}}
{\int \Pi(\mathbf{r},\tau) d\mathbf{r}} = \lim_{\tau \to
\infty} E(\tau).
\]
Indeed, this is equivalent to
\[
E(\tau) = \frac{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) E_L(\mathbf{r}) d\mathbf{r}}
{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) d\mathbf{r}}
= \frac{\int \psi(\mathbf{r},\tau) \hat{H} \Psi_T(\mathbf{r}) d\mathbf{r}}
{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) d\mathbf{r}}
= \frac{\langle \psi_\tau | \hat{H} | \Psi_T \rangle}
{\langle \psi_\tau | \Psi_T \rangle}
\]
As $\hat{H}$ is Hermitian,
\[
E(\tau) = \frac{\langle \psi_\tau | \hat{H} | \Psi_T \rangle}
{\langle \psi_\tau | \Psi_T \rangle}
= \frac{\langle \Psi_T | \hat{H} | \psi_\tau \rangle}
{\langle \Psi_T | \psi_\tau \rangle}
= E[\psi_\tau] \frac{\langle \Psi_T | \psi_\tau \rangle}
{\langle \Psi_T | \psi_\tau \rangle}
= E[\psi_\tau]
\]
So computing the energy within DMC consists in generating the
density $\Pi$, and simply averaging the local energies computed
with the trial wave function.
** Pure Diffusion Monte Carlo ** Pure Diffusion Monte Carlo
Instead of having a variable number of particles to simulate the
branching process, one can choose to sample $[\Psi_T(\mathbf{r})]^2$ instead of
$\psi(\mathbf{r},\tau) \Psi_T(\mathbf{r})$, and consider the term
$\exp \left( -\delta t\,( E_L(\mathbf{r}) - E_{\text{ref}} \right)$ as a
cumulative product of branching weights:
\[
W(\mathbf{r}_n, \tau)
= \exp \left( \int_0^\tau - (E_L(\mathbf{r}_t) - E_{\text{ref}}) dt \right)
\approx \prod_{i=1}^{n} \exp \left( -\delta t\, (E_L(\mathbf{r}_i) - E_{\text{ref}}) \right).
\]
The wave function becomes
\[
\psi(\mathbf{r},\tau) = \Psi_T(\mathbf{r}) W(\mathbf{r},\tau)
\]
and the expression of the fixed-node DMC energy is
\begin{eqnarray*}
E(\tau) & = & \frac{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) E_L(\mathbf{r}) d\mathbf{r}}
{\int \psi(\mathbf{r},\tau) \Psi_T(\mathbf{r}) d\mathbf{r}} \\
& = & \frac{\int \left[ \Psi_T(\mathbf{r}) \right]^2 W(\mathbf{r},\tau) E_L(\mathbf{r}) d\mathbf{r}}
{\int \left[ \Psi_T(\mathbf{r}) \right]^2 W(\mathbf{r},\tau) d\mathbf{r}} \\
\end{eqnarray*}
This algorithm is less stable than the branching algorithm: it
requires to have a value of $E_\text{ref}$ which is close to the
fixed-node energy, and a good trial wave function. Its big
advantage is that it is very easy to program starting from a VMC code.
** TODO Hydrogen atom ** TODO Hydrogen atom
*** Exercise *** Exercise
#+begin_exercise #+begin_exercise
Modify the Metropolis VMC program to introduce the PDMC weight. Modify the Metropolis VMC program to introduce the PDMC weight.
In the limit $\tau \rightarrow 0$, you should recover the exact In the limit $\delta t \rightarrow 0$, you should recover the exact
energy of H for any value of $a$. energy of H for any value of $a$.
#+end_exercise #+end_exercise
@ -2094,7 +2169,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
: A = 0.78861366666666655 +/- 3.5096729498002445E-004 : A = 0.78861366666666655 +/- 3.5096729498002445E-004
** TODO Dihydrogen ** TODO H_2
We will now consider the H_2 molecule in a minimal basis composed of the We will now consider the H_2 molecule in a minimal basis composed of the
$1s$ orbitals of the hydrogen atoms: $1s$ orbitals of the hydrogen atoms:
@ -2108,7 +2183,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis
the nuclei. the nuclei.
* Appendix :noexport: * Old sections to be removed :noexport:
** Gaussian sampling :noexport: ** Gaussian sampling :noexport:
:PROPERTIES: :PROPERTIES:

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