diff --git a/QMC.org b/QMC.org index 3ce77d7..5e6faf7 100644 --- a/QMC.org +++ b/QMC.org @@ -469,7 +469,7 @@ gfortran hydrogen.f90 energy_hydrogen.f90 -o energy_hydrogen $$ which can be simplified as - $$ \sigma^2(E_L) = \langle E^2 \rangle - \langle E \rangle^2 $$ + $$ \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 @@ -804,88 +804,79 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_uniform.f90 -o qmc_uniform #+RESULTS: : E = -0.49588321986667677 +/- 7.1758863546737969E-004 -** Gaussian sampling +** Metropolis sampling with $\Psi^2$ :PROPERTIES: - :header-args:python: :tangle qmc_gaussian.py - :header-args:f90: :tangle qmc_gaussian.f90 + :header-args:python: :tangle qmc_metropolis.py + :header-args:f90: :tangle qmc_metropolis.f90 :END: - We will now improve the sampling and allow to sample in the whole - 3D space, correcting the bias related to the sampling in the box. + We will now use the square of the wave function to sample random + points distributed with the probability density + \[ + P(\mathbf{r}) = \left[\Psi(\mathbf{r})\right]^2 + \] - Instead of drawing uniform random numbers, we will draw Gaussian - random numbers centered on 0 and with a variance of 1. + 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 : - To obtain Gaussian-distributed random numbers, you can apply the - [[https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform][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*} - - Here is a Fortran implementation returning a Gaussian-distributed - n-dimensional vector $\mathbf{z}$; - - *Fortran* - #+BEGIN_SRC f90 :tangle qmc_stats.f90 -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 - #+END_SRC - - - Now the sampling probability can be inserted into the equation of the energy: + $$ + E \approx \frac{1}{M}\sum_{i=1}^M E_L(\mathbf{r}_i) + $$ - \[ - E = \frac{\int P(\mathbf{r}) \frac{\left[\Psi(\mathbf{r})\right]^2}{P(\mathbf{r})}\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int P(\mathbf{r}) \frac{\left[\Psi(\mathbf{r}) \right]^2}{P(\mathbf{r})} d\mathbf{r}} - \] - with the Gaussian probability - - \[ - P(\mathbf{r}) = \frac{1}{(2 \pi)^{3/2}}\exp\left( -\frac{\mathbf{r}^2}{2} \right). - \] - - As the coordinates are drawn with probability $P(\mathbf{r})$, the - average energy can be computed as + To sample a chosen probability density, an efficient method is the + [[https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm][Metropolis-Hastings sampling algorithm]]. Starting from a random + initial position $\mathbf{r}_0$, we will realize a random walk as follows: $$ - E \approx \frac{\sum_i w_i E_L(\mathbf{r}_i)}{\sum_i w_i}, \;\; - w_i = \frac{\left[\Psi(\mathbf{r}_i)\right]^2}{P(\mathbf{r}_i)} \delta \mathbf{r} + \mathbf{r}_{n+1} = \mathbf{r}_{n} + \tau \mathbf{u} $$ + 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 $\mathbf{r}_{n+1}$ + + #+begin_note + 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. + #+end_note + + 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 - + #+begin_exercise - Modify the exercise of the previous section to sample with - Gaussian-distributed random numbers. Can you see an reduction in - the statistical error? + 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? #+end_exercise *Python* @@ -893,91 +884,110 @@ end subroutine random_gauss from hydrogen import * from qmc_stats import * -norm_gauss = 1./(2.*np.pi)**(1.5) -def gaussian(r): - return norm_gauss * np.exp(-np.dot(r,r)*0.5) - -def MonteCarlo(a,nmax): +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 = np.random.normal(loc=0., scale=1.0, size=(3)) - w = psi(a,r) - w = w*w / gaussian(r) - N += w - E += w * e_loc(a,r) - return E/N + 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 -X = [MonteCarlo(a,nmax) for i in range(30)] +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}") #+END_SRC #+RESULTS: - : E = -0.49511014287471955 +/- 0.00012402022172236656 + : E = -0.4950720838131573 +/- 0.00019089638602238043 + : A = 0.5172960000000001 +/- 0.0003443446549306529 - *Fortran* #+BEGIN_SRC f90 -double precision function gaussian(r) - implicit none - double precision, intent(in) :: r(3) - double precision, parameter :: norm_gauss = 1.d0/(2.d0*dacos(-1.d0))**(1.5d0) - gaussian = norm_gauss * dexp( -0.5d0 * (r(1)*r(1) + r(2)*r(2) + r(3)*r(3) )) -end function gaussian - - -subroutine gaussian_montecarlo(a,nmax,energy) +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(3), w - + 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_gauss(r,3) - w = psi(a,r) - w = w*w / gaussian(r) - norm = norm + w - energy = energy + w * e_loc(a,r) + 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 -end subroutine gaussian_montecarlo + accep = n_accep / norm +end subroutine metropolis_montecarlo program qmc implicit none - double precision, parameter :: a = 0.9 + 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) + double precision :: X(nruns), Y(nruns) double precision :: ave, err do irun=1,nruns - call gaussian_montecarlo(a,nmax,X(irun)) + 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 #+END_SRC #+begin_src sh :results output :exports both -gfortran hydrogen.f90 qmc_stats.f90 qmc_gaussian.f90 -o qmc_gaussian -./qmc_gaussian +gfortran hydrogen.f90 qmc_stats.f90 qmc_metropolis.f90 -o qmc_metropolis +./qmc_metropolis #+end_src - #+RESULTS: - : E = -0.49517104619091717 +/- 1.0685523607878961E-004 + : E = -0.49478505004797046 +/- 2.0493795299184956E-004 + : A = 0.51737800000000000 +/- 4.1827406733181444E-004 + ** Sampling with $\Psi^2$ @@ -1206,7 +1216,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc #+RESULTS: : E = -0.48584030499187431 +/- 1.0411743995438257E-004 -*** Metropolis algorithm +*** Generalized Metropolis algorithm :PROPERTIES: :header-args:python: :tangle vmc_metropolis.py :header-args:f90: :tangle vmc_metropolis.f90 @@ -1225,17 +1235,19 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc entering $\mathbf{r}_{n+1}$: \[ A(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = \min \left( 1, - \frac{g(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) P(\mathbf{r}_{n+1})} - {g(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) P(\mathbf{r}_{n})} + \frac{T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n}) P(\mathbf{r}_{n+1})} + {T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) P(\mathbf{r}_{n})} \right) \] + 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 our Hydrogen atom example, $P$ is $\Psi^2$ and $g$ is a solution of the discretized Fokker-Planck equation: \begin{eqnarray*} P(r_{n}) &=& \Psi^2(\mathbf{r}_n) \\ - g(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) & = & + T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) & = & \frac{1}{(4\pi\,D\,\tau)^{3/2}} \exp \left[ - \frac{\left( \mathbf{r}_{n+1} - \mathbf{r}_{n} - 2D \frac{\nabla \Psi(\mathbf{r}_n)}{\Psi(\mathbf{r}_n)} \right)^2}{4D\,\tau} \right] @@ -1247,8 +1259,7 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc.f90 -o vmc - if $u \le A(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1})$, accept the move - if $u>A(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1})$, reject - the move: set $\mathbf{r}_{n+1} = \mathbf{r}_{n}$, but *don't - remove the sample from the average!* + the move: set $\mathbf{r}_{n+1} = \mathbf{r}_{n}$, but *don't remove the sample from the average!* The /acceptance rate/ is the ratio of the number of accepted step over the total number of steps. The time step should be adapted so @@ -1567,3 +1578,180 @@ gfortran hydrogen.f90 qmc_stats.f90 vmc_metropolis.f90 -o vmc_metropolis + + + +* Appendix + +** Gaussian sampling :noexport: + :PROPERTIES: + :header-args:python: :tangle qmc_gaussian.py + :header-args:f90: :tangle qmc_gaussian.f90 + :END: + + We will now improve the sampling and allow to sample in the whole + 3D space, correcting the bias related to the sampling in the box. + + Instead of drawing uniform random numbers, we will draw Gaussian + random numbers centered on 0 and with a variance of 1. + + To obtain Gaussian-distributed random numbers, you can apply the + [[https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform][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*} + + Here is a Fortran implementation returning a Gaussian-distributed + n-dimensional vector $\mathbf{z}$; + + *Fortran* + #+BEGIN_SRC f90 :tangle qmc_stats.f90 +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 + #+END_SRC + + Now the sampling probability can be inserted into the equation of the energy: + + \[ + E = \frac{\int P(\mathbf{r}) \frac{\left[\Psi(\mathbf{r})\right]^2}{P(\mathbf{r})}\, \frac{\hat{H} \Psi(\mathbf{r})}{\Psi(\mathbf{r})}\,d\mathbf{r}}{\int P(\mathbf{r}) \frac{\left[\Psi(\mathbf{r}) \right]^2}{P(\mathbf{r})} d\mathbf{r}} + \] + + with the Gaussian probability + + \[ + P(\mathbf{r}) = \frac{1}{(2 \pi)^{3/2}}\exp\left( -\frac{\mathbf{r}^2}{2} \right). + \] + + As the coordinates are drawn with probability $P(\mathbf{r})$, the + average energy can be computed as + + $$ + E \approx \frac{\sum_i w_i E_L(\mathbf{r}_i)}{\sum_i w_i}, \;\; + w_i = \frac{\left[\Psi(\mathbf{r}_i)\right]^2}{P(\mathbf{r}_i)} \delta \mathbf{r} + $$ + + +*** Exercise + + #+begin_exercise + Modify the program of the previous section to sample with + Gaussian-distributed random numbers. Can you see an reduction in + the statistical error? + #+end_exercise + + *Python* + #+BEGIN_SRC python :results output +from hydrogen import * +from qmc_stats import * + +norm_gauss = 1./(2.*np.pi)**(1.5) +def gaussian(r): + return norm_gauss * np.exp(-np.dot(r,r)*0.5) + +def MonteCarlo(a,nmax): + E = 0. + N = 0. + for istep in range(nmax): + r = np.random.normal(loc=0., scale=1.0, size=(3)) + w = psi(a,r) + w = w*w / gaussian(r) + 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}") + #+END_SRC + + #+RESULTS: + : E = -0.49511014287471955 +/- 0.00012402022172236656 + + *Fortran* + #+BEGIN_SRC f90 +double precision function gaussian(r) + implicit none + double precision, intent(in) :: r(3) + double precision, parameter :: norm_gauss = 1.d0/(2.d0*dacos(-1.d0))**(1.5d0) + gaussian = norm_gauss * dexp( -0.5d0 * (r(1)*r(1) + r(2)*r(2) + r(3)*r(3) )) +end function gaussian + + +subroutine gaussian_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, gaussian + + energy = 0.d0 + norm = 0.d0 + do istep = 1,nmax + call random_gauss(r,3) + w = psi(a,r) + w = w*w / gaussian(r) + norm = norm + w + energy = energy + w * e_loc(a,r) + end do + energy = energy / norm +end subroutine gaussian_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 gaussian_montecarlo(a,nmax,X(irun)) + enddo + call ave_error(X,nruns,ave,err) + print *, 'E = ', ave, '+/-', err +end program qmc + #+END_SRC + + #+begin_src sh :results output :exports both +gfortran hydrogen.f90 qmc_stats.f90 qmc_gaussian.f90 -o qmc_gaussian +./qmc_gaussian + #+end_src + + #+RESULTS: + : E = -0.49517104619091717 +/- 1.0685523607878961E-004 diff --git a/qmc_metropolis.f90 b/qmc_metropolis.f90 new file mode 100644 index 0000000..96fc1b3 --- /dev/null +++ b/qmc_metropolis.f90 @@ -0,0 +1,57 @@ +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 diff --git a/qmc_metropolis.py b/qmc_metropolis.py new file mode 100644 index 0000000..c2091d7 --- /dev/null +++ b/qmc_metropolis.py @@ -0,0 +1,32 @@ +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}") diff --git a/variance_hydrogen.py b/variance_hydrogen.py index 4de7300..40f9a27 100644 --- a/variance_hydrogen.py +++ b/variance_hydrogen.py @@ -8,6 +8,7 @@ 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 @@ -19,18 +20,9 @@ for a in [0.1, 0.2, 0.5, 0.9, 1., 1.5, 2.]: w = w * w * delta El = e_loc(a, r) E += w * El + E2 += w * El*El norm += w E = E / norm - s2 = 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) - s2 += w * (El - E)**2 - s2 = s2 / norm + E2 = E2 / norm + s2 = E2 - E*E print(f"a = {a} \t E = {E:10.8f} \t \sigma^2 = {s2:10.8f}") diff --git a/vmc_metropolis.py b/vmc_metropolis.py index 78771b0..6521e58 100644 --- a/vmc_metropolis.py +++ b/vmc_metropolis.py @@ -34,8 +34,8 @@ def MonteCarlo(a,tau,nmax): a = 0.9 nmax = 100000 -tau = 2.5 +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} {A} +/- {deltaA}") +print(f"E = {E} +/- {deltaE}\nA = {A} +/- {deltaA}")