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Update variance
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QMC.org
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QMC.org
@ -478,7 +478,7 @@ gfortran hydrogen.f90 energy_hydrogen.f90 -o energy_hydrogen
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*** Exercise (optional)
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#+begin_exercise
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Prove that :
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$$ \sigma^2(E_L) = \langle E^2 \rangle - \langle E \rangle^2 $$
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$$\langle E - \langle E \rangle \rangle^2 = \langle E^2 \rangle - \langle E \rangle^2 $$
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#+end_exercise
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*** Exercise
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@ -537,6 +537,7 @@ program variance_hydrogen
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implicit none
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double precision, external :: e_loc, psi
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double precision :: x(50), w, delta, energy, dx, r(3), a(6), norm, s2
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double precision :: e, energy2
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integer :: i, k, l, j
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a = (/ 0.1d0, 0.2d0, 0.5d0, 1.d0, 1.5d0, 2.d0 /)
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@ -552,6 +553,7 @@ program variance_hydrogen
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do j=1,size(a)
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energy = 0.d0
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energy2 = 0.d0
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norm = 0.d0
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do i=1,size(x)
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r(1) = x(i)
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@ -561,29 +563,16 @@ program variance_hydrogen
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r(3) = x(l)
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w = psi(a(j),r)
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w = w * w * delta
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energy = energy + w * e_loc(a(j), r)
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e = e_loc(a(j), r)
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energy = energy + w * e
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energy2 = energy2 + w * e * e
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norm = norm + w
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end do
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end do
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end do
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energy = energy / norm
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s2 = 0.d0
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norm = 0.d0
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do i=1,size(x)
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r(1) = x(i)
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do k=1,size(x)
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r(2) = x(k)
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do l=1,size(x)
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r(3) = x(l)
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w = psi(a(j),r)
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w = w * w * delta
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s2 = s2 + w * ( e_loc(a(j), r) - energy )**2
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norm = norm + w
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end do
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end do
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end do
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s2 = s2 / norm
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energy = energy / norm
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energy2 = energy2 / norm
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s2 = energy2 - energy*energy
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print *, 'a = ', a(j), ' E = ', energy, ' s2 = ', s2
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end do
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@ -598,12 +587,12 @@ gfortran hydrogen.f90 variance_hydrogen.f90 -o variance_hydrogen
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#+end_src
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#+RESULTS:
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: a = 0.10000000000000001 E = -0.24518438948809140 s2 = 2.6965218719733813E-002
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: a = 0.20000000000000001 E = -0.26966057967803236 s2 = 3.7197072370217653E-002
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: a = 0.50000000000000000 E = -0.38563576125173815 s2 = 5.3185967578488862E-002
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: a = 0.10000000000000001 E = -0.24518438948809140 s2 = 2.6965218719722767E-002
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: a = 0.20000000000000001 E = -0.26966057967803236 s2 = 3.7197072370201284E-002
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: a = 0.50000000000000000 E = -0.38563576125173815 s2 = 5.3185967578480653E-002
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: a = 1.0000000000000000 E = -0.50000000000000000 s2 = 0.0000000000000000
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: a = 1.5000000000000000 E = -0.39242967082602065 s2 = 0.31449670909180444
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: a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814270851303
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: a = 1.5000000000000000 E = -0.39242967082602065 s2 = 0.31449670909172917
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: a = 2.0000000000000000 E = -8.0869806678448772E-002 s2 = 1.8068814270846534
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* Variational Monte Carlo
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@ -2,6 +2,7 @@ program variance_hydrogen
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implicit none
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double precision, external :: e_loc, psi
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double precision :: x(50), w, delta, energy, dx, r(3), a(6), norm, s2
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double precision :: e, energy2
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integer :: i, k, l, j
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a = (/ 0.1d0, 0.2d0, 0.5d0, 1.d0, 1.5d0, 2.d0 /)
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@ -17,6 +18,7 @@ program variance_hydrogen
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do j=1,size(a)
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energy = 0.d0
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energy2 = 0.d0
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norm = 0.d0
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do i=1,size(x)
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r(1) = x(i)
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@ -26,29 +28,16 @@ program variance_hydrogen
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r(3) = x(l)
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w = psi(a(j),r)
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w = w * w * delta
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energy = energy + w * e_loc(a(j), r)
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e = e_loc(a(j), r)
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energy = energy + w * e
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energy2 = energy2 + w * e * e
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norm = norm + w
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end do
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end do
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end do
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energy = energy / norm
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s2 = 0.d0
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norm = 0.d0
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do i=1,size(x)
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r(1) = x(i)
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do k=1,size(x)
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r(2) = x(k)
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do l=1,size(x)
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r(3) = x(l)
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w = psi(a(j),r)
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w = w * w * delta
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s2 = s2 + w * ( e_loc(a(j), r) - energy )**2
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norm = norm + w
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end do
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end do
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end do
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s2 = s2 / norm
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energy = energy / norm
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energy2 = energy2 / norm
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s2 = energy2 - energy*energy
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print *, 'a = ', a(j), ' E = ', energy, ' s2 = ', s2
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end do
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@ -12,7 +12,7 @@ subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
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double precision, external :: e_loc, psi
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sq_tau = dsqrt(tau)
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! Initialization
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energy = 0.d0
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norm = 0.d0
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@ -30,8 +30,8 @@ subroutine variational_montecarlo(a,tau,nmax,energy,accep_rate)
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psi_new = psi(a,r_new)
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! Metropolis
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prod = (d_new(1) + d_old(1))*(r_new(1) - r_old(1)) + &
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(d_new(2) + d_old(2))*(r_new(2) - r_old(2)) + &
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(d_new(3) + d_old(3))*(r_new(3) - r_old(3))
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(d_new(2) + d_old(2))*(r_new(2) - r_old(2)) + &
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(d_new(3) + d_old(3))*(r_new(3) - r_old(3))
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argexpo = 0.5d0 * (d2_new - d2_old)*tau + prod
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q = psi_new / psi_old
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q = dexp(-argexpo) * q*q
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@ -27,8 +27,8 @@ def MonteCarlo(a,tau,nmax):
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d_old = d_new
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d2_old = d2_new
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psi_old = psi_new
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N += 1.
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E += e_loc(a,r_old)
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N += 1.
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E += e_loc(a,r_old)
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return E/N, accep_rate/N
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