def H(W):
'''
H dot W where H is Harmiton operator
@input:
W: Expansion coefficients for Ns unconstrained wave functions, stored as
an (S0 * S1 * S2, Ns) matrix
@global variables
Vdual: Dual potential coefficients stored as a (S0 * S1 * S2, 1) column vector.
@return:
matrix of shape (S0 * S1 * S2, Ns)
'''
f = 2
U = np.dot(W.conj().T, op.O(W))
U_inv = np.linalg.inv(U)
# each col of W_real represents W's value in sampled real space
W_real = op.cI(W)
n = f * op.diagouter(np.dot(W_real, U_inv), W_real)
exc = excVWN(n)
exc_prime = excpVWN(n)
# electrostatic potential (phi) in freqency space
phi = op.poisson(n, real_phi=False)
Veff = gb_Vdual \
+ op.cJdag(op.O(phi)) \
+ op.cJdag(op.O(op.cJ(exc))) \
+ op.diagprod(exc_prime, op.cJdag(op.O(op.cJ(n))))
HW_potential = op.cIdag(op.diagprod(Veff, op.cI(W)))
HW_kinetic = -0.5 * op.L(W)
HW = HW_kinetic + HW_potential
return HW
def getE(W):
'''
get the energy from expansion coefficients for Ns unconstrained wave function
@input:
W: Expansion coefficients for Ns unconstrained wave functions, stored as
an (S0 * S1 * S2, Ns) matrix
@global variables
Vdual: Dual potential coefficients stored as a (S0 * S1 * S2, 1) column vector.
@return:
E: Energies summed over Ns states
'''
f = 2
U = np.dot(W.conj().T, op.O(W))
U_inv = np.linalg.inv(U)
# each col of W_real represents W's value in sampled real space
W_real = op.cI(W)
n = f * op.diagouter(np.dot(W_real, U_inv), W_real)
E_potential = np.dot(gb_Vdual.T, n)
E_kinetic = -0.5 * np.trace(np.dot(W.conj().T, op.L(np.dot(W, U_inv)))) * f
# electrostatic potential (phi) in freqency space
phi = op.poisson(n, real_phi=False)
# E_hartree = 0.5 * np.dot(n.conj().T, op.cJdag(op.O(phi)))
E_hartree = 0.5 * np.dot(op.cJ(n).conj().T, op.O(phi))
exc = excVWN(n)
# the Exc is negative here, which seems problematic.
Exc = np.dot(op.cJ(n).conj().T, op.O(op.cJ(exc)))
E = (E_potential + E_kinetic + E_hartree + Exc)[0, 0]
E = E.real
return E
def getgrad(W):
'''
get gradient dE/dW
'''
U = np.dot(W.conj().T, op.O(W))
U_inv = np.linalg.inv(U)
HW = H(W)
t1 = np.dot(op.O(W), U_inv) # (#. sample points, Ns)
t2 = np.dot(t1, np.dot(W.conj().T, HW))
t3 = HW - t2
t4 = np.dot(t3, U_inv)
return t4
def orthonormalizeW(W):
'''
orthonormalize W so that Wdag dot O(W) is identity matrix
'''
U = np.dot(W.conj().T, op.O(W))
U_inv = np.linalg.inv(U)
U_inv_sqrt = linalg.sqrtm(U_inv)
W_orthonormal = np.dot(W, U_inv_sqrt)
return W_orthonormal
def test_orthonormalizeW():
ns = 4
W = np.random.rand(np.prod(global_vars.S), ns) + 1j * \
np.random.rand(np.prod(global_vars.S), ns)
W = orthonormalizeW(W)
# Overlap matrix should equal to identity matrix up to machine precison
Overlap = np.dot(W.conj().T, op.O(W))
assert np.linalg.norm(
np.sum(Overlap - np.identity(ns))) < 1e-6, 'ERROR in orthonormalizeW'
def getE(W):
'''
get the energy from expansion coefficients for Ns unconstrained wave function
@input:
W: Expansion coefficients for Ns unconstrained wave functions, stored as
an (S0 * S1 * S2, Ns) matrix
@global variables
Vdual: Dual potential coefficients stored as a (S0 * S1 * S2, 1) column vector.
@return:
E: Energies summed over Ns states
'''
U = np.dot(W.conj().T, op.O(W))
U_inv = np.linalg.inv(U)
W_real = op.cI(
W) # each col of W_real represents W's value in sampled real space
n = op.diagouter(np.dot(W_real, U_inv), W_real)
E_potential = np.dot(gb_Vdual.T, n)
E_kinetic = -0.5 * np.trace(np.dot(W.conj().T, op.L(np.dot(W, U_inv))))
E = (E_potential + E_kinetic)[0, 0]
E = E.real
return E
sum_over_cell = lambda x: np.sum(x) * np.linalg.det(global_vars.R) / \
(global_vars.S[0] * global_vars.S[1] * global_vars.S[2]) # sum(xdv)
sigma1, sigma2 = 0.75, 0.5
g1 = gen_gaussian3(dr, sigma1)
g2 = gen_gaussian3(dr, sigma2)
n = g2 - g1
if global_vars.is_debug:
print(sum_over_cell(g1))
print(sum_over_cell(g2))
net_charge = sum_over_cell(n)
print('net charge is {:.4f}'.format(net_charge))
assert net_charge < 1e-3, 'Current charge is not zero'
phi = op.cI(op.Linv(-4 * np.pi * op.O(op.cJ(n))))
# Due to rounding, tiny imaginary parts creep into the solution. Eliminate
# by taking the real part.
phi = np.real(phi)
# Unum = (0.5 * np.real(np.dot(op.cJ(phi).conj().T, op.O(op.cJ(n)))))[0, 0]
Unum = (0.5 * np.dot(phi.T, n)[0]) * (np.linalg.det(global_vars.R) /
np.prod(global_vars.S))
Uanal = ((1 / sigma1 + 1 / sigma2) / 2 -
np.sqrt(2) / np.sqrt(sigma1**2 + sigma2**2)) / np.sqrt(np.pi)
res_deviation = np.abs(Unum - Uanal)
print('Deviation between numerical and analytical results is {:.4f}'.format(
res_deviation))
if global_vars.is_debug:
print(Unum)
print(Uanal)
import numpy as np
import matplotlib.pyplot as plt
import global_vars
import op
from scipy import linalg
import vis
lattice_center = np.reshape(np.sum(global_vars.R / 2, axis=1), (1, -1))
dr = np.sqrt(np.sum((global_vars.r - lattice_center)**2, axis=1))
V = 2 * (dr**2)
gb_Vdual = op.cJdag(op.O(op.cJ(V)))
def getE(W):
'''
get the energy from expansion coefficients for Ns unconstrained wave function
@input:
W: Expansion coefficients for Ns unconstrained wave functions, stored as
an (S0 * S1 * S2, Ns) matrix
@global variables
Vdual: Dual potential coefficients stored as a (S0 * S1 * S2, 1) column vector.
@return:
E: Energies summed over Ns states
'''
U = np.dot(W.conj().T, op.O(W))
U_inv = np.linalg.inv(U)
W_real = op.cI(
W) # each col of W_real represents W's value in sampled real space
n = op.diagouter(np.dot(W_real, U_inv), W_real)
E_potential = np.dot(gb_Vdual.T, n)
E_kinetic = -0.5 * np.trace(np.dot(W.conj().T, op.L(np.dot(W, U_inv))))