Solutions in C up to 3.4

.gitignore

@@ -1,2 +1,7 @@
+__pycache__
 plt_dat
-data
+data
+get_energ
+get_var
+qmc_uni
+qmc_met

energy_hydrogen.c

@@ -0,0 +1,35 @@
+#include "hydrogen.h"
+
+int main() {
+      const double a[6] = {0.1, 0.2, 0.5, 1., 1.5, 2.};
+      const int sizex = 50;
+      double energy, r[3];
+      double x[sizex], dx, w, wsum, dV;
+
+      dx = 10.0 / (sizex - 1);
+      for (int i = 0; i < sizex; ++i) {
+	    x[i] = -5.0 + (i - 1) * dx;
+      }
+      dV = dx * dx * dx;
+
+      for (int l = 0; l < 6; ++l) {
+	    energy = 0.0;
+	    wsum = 0.0;
+	    for (int i = 0; i < sizex; ++i) {
+		  r[0] = x[i];
+		  for (int j = 0; j < sizex; ++j) {
+			r[1] = x[j];
+			for (int k = 0; k < sizex; ++k) {
+			      r[2] = x[k];
+			      w = psi(a[l], r, 3) * psi(a[l], r, 3) * dV;
+			      wsum += w;
+			      energy += w * e_loc(a[l], r, 3);
+			}
+		  }
+	    }
+	    printf("a = %.2f  E = %.5f \n", a[l], energy / wsum); 
+      }
+
+      
+      return 0;
+}

hydrogen.h

@@ -12,7 +12,7 @@ double potential(double *r, const int l) {
       if (pot > 0.0) {
 	   return -1.0 / sqrt(pot); 
       }
-      return -1;
+      return 1e6;
 
 }
 

qmc_metropolis.c

@@ -0,0 +1,73 @@
+#include "hydrogen.h"
+#include "qmc_stats.h"
+#include <stdlib.h>
+#include <time.h>
+
+void metropolis_montecarlo(double a, const int nmax, double dt,
+			   double *energy, double *accept) {
+      int n_accept;
+      double r_old[3], r_new[3], psi_old, psi_new, rnd;
+      double aval, u;
+
+      // Initial position
+      for (int j = 0; j < 3; ++j) {
+	    rnd = (double) rand() / RAND_MAX;
+	    r_old[j] = dt * (2.0 * rnd - 1.0);
+      }
+      psi_old = psi(a, r_old, 3) * psi(a, r_old, 3);
+
+      *energy = 0.0;
+      *accept = 0.0;
+      n_accept = 0;
+      for (int i = 0; i < nmax; ++i) {
+	    // Compute and accumulate the local energy
+	    *energy += e_loc(a, r_old, 3);
+
+	    // Compute new position
+	    for (int j = 0; j < 3; ++j) {
+		 rnd = (double) rand() / RAND_MAX;
+		 r_new[j] = r_old[j] + dt * (2.0 * rnd - 1.0);
+	    }
+
+	    // New WF and acceptance probability
+	    psi_new = psi(a, r_new, 3) * psi(a, r_new, 3);
+	    aval = psi_new / psi_old;
+
+	    u = (double) rand() / RAND_MAX;
+
+	    if (u <= aval) {
+		  for (int j = 0; j < 3; ++j) {
+			r_old[j] = r_new[j];
+		  }
+		  psi_old = psi_new;
+		  n_accept += 1;
+
+	    }
+
+      }
+      *energy /= nmax;
+      *accept = (double) n_accept / nmax;
+}
+
+int main() {
+      const double a = 1.2;
+      const double dt = 1.0;
+      const long nmax = 1e5;
+      int nruns = 30;
+
+      srand(time(NULL));
+
+      double x[nruns], y[nruns], obs[2];
+
+      for (int i = 0; i < nruns; ++i) {
+	    metropolis_montecarlo(a, nmax, dt, &x[i], &y[i]);
+      }
+
+      ave_error(x, nruns, obs);
+      printf("E = %.5lf +/- %.5lf\n", obs[0], obs[1]);
+
+      ave_error(y, nruns, obs);
+      printf("A = %.5lf +/- %.5lf\n", obs[0], obs[1]);
+      
+      return 0;
+}

qmc_stats.h

@@ -0,0 +1,27 @@
+#include <math.h>
+#include <stdio.h>
+
+void ave_error(double *x, const int n, double obs[2]) {
+      // obs[1] --> mean and obs[2] --> err
+      double var;
+
+      if (n == 1) {
+	    obs[0] = x[0];
+	    obs[1] = 0.0;
+      }
+      else {
+	    for (int i = 0; i < n; ++i) {
+		  obs[0] += x[i];
+	    }
+            obs[0] = obs[0] / n;
+
+            var = 0.0;
+            for (int i = 0; i < n; ++i) {
+            	  var += (x[i] - obs[0]) * (x[i] - obs[0]);
+            }
+            var = var / (n - 1);
+
+            obs[1] = sqrt(var / n);
+      }
+
+}

qmc_uniform.c

@@ -0,0 +1,44 @@
+#include "hydrogen.h"
+#include "qmc_stats.h"
+#include <stdlib.h>
+#include <time.h>
+
+double uniform_montecarlo(double a, const int nmax) {
+      double r[3], energy, norm, w, rnd;
+
+      norm = 0.0;
+      energy = 0.0;
+      for (int i = 0; i < nmax; ++i) {
+	    for (int j = 0; j < 3; ++j) {
+		  rnd = (double) rand() / RAND_MAX;
+		  r[j] = -5.0 + 10.0 * rnd;
+	    }
+	    w = psi(a, r, 3) * psi(a, r, 3);
+	    norm += w;
+	    energy += w * e_loc(a, r, 3);
+      }
+
+      return energy / norm;
+
+}
+
+int main() {
+      const double a = 1.2;
+      const long nmax = 1e5;
+      int nruns = 30;
+
+      srand(time(NULL));
+
+      double x[nruns], obs[2];
+
+      for (int i = 0; i < nruns; ++i) {
+	    x[i] = uniform_montecarlo(a, nmax);
+      }
+
+      ave_error(x, nruns, obs);
+
+      printf("E = %.5lf +/- %.5lf\n", obs[0], obs[1]);
+      
+      return 0;
+}
+

variance_hydrogen.c

@@ -0,0 +1,41 @@
+#include "hydrogen.h"
+
+int main() {
+      const double a[6] = {0.1, 0.2, 0.5, 1., 1.5, 2.};
+      const int sizex = 50;
+      double energy, r[3], sigma;
+      double x[sizex], dx, w, wsum, dV, e_tmp;
+
+      dx = 10.0 / (sizex - 1);
+      for (int i = 0; i < sizex; ++i) {
+	    x[i] = -5.0 + i * dx;
+      }
+      dV = dx * dx *dx;
+
+      for (int l = 0; l < 6; ++l) {
+	    energy = 0.0;
+	    sigma = 0.0;
+	    wsum = 0.0;
+	    for (int i = 0; i < sizex; ++i) {
+		  r[0] = x[i];
+		  for (int j = 0; j < sizex; ++j) {
+			r[1] = x[j];
+			for (int k = 0; k < sizex; ++k) {
+			      r[2] = x[k];
+			      w = psi(a[l], r, 3) * psi(a[l], r, 3) * dV;
+			      wsum += w;
+			      e_tmp = e_loc(a[l], r, 3);
+			      energy += w * e_tmp;
+			      sigma += w * e_tmp * e_tmp;
+			}
+		  }
+	    }
+	    energy = energy / wsum;
+	    sigma = sigma / wsum;
+	    sigma -= energy * energy;
+	    printf("a = %.2lf  E = %.5lf  Sigma^2 = %.5lf\n", a[l], energy, sigma); 
+      }
+
+      
+      return 0;
+}

variance_hydrogen.py

@@ -4,6 +4,8 @@ import numpy as np
 from hydrogen import e_loc, psi
 
 interval = np.linspace(-5,5,num=50)
+print(interval
+     )
 
 delta = (interval[1]-interval[0])**3
 
@@ -34,5 +36,5 @@ for a in [0.1, 0.2, 0.5, 0.9, 1., 1.5, 2.]:
     E  = E  / norm
     E2 = E2 / norm
 
-    s2 = E2 - E**2
+    s2 = E2 #- E**2
     print(f"a = {a} \t E = {E:10.8f} \t \sigma^2 = {s2:10.8f}")