def compute_eigenstate(parameters, filename="eigenstates.hdf5", computepq=True, computePQ=True): r""" Special variables necessary in configuration: * eigenstate_of_level (default: 0) * eigenstates_indices (default: [0]) * starting_point (default: (2, ..., 2)) * hawp_template * innerproduct """ D = parameters["dimension"] if "eigenstate_of_level" in parameters: N = parameters["eigenstate_of_level"] else: # Upper-most potential surface N = 0 # Create output file now, in case this fails we did not waste computation time IOM = IOManager() IOM.create_file(filename) # Save the simulation parameters IOM.add_parameters() IOM.save_parameters(parameters) gid = IOM.create_group() BF = BlockFactory() # Create the potential V = BF.create_potential(parameters) V.calculate_local_quadratic() # Compute position and momentum if computepq: # Minimize the potential to find q0 f = lambda x: real((squeeze(V.evaluate_at(x)[N]))) # Start with an offset because exact 0.0 values can give # issues, especially with the Hessian evaluation. This way # the minimizer will always stay away from zero a tiny bit. # The current starting point can give issues if the potential # is stationary at the point (2, ..., 2) but that is less likely. if "starting_point" in parameters: x0 = atleast_1d(parameters["starting_point"]) else: x0 = 0.5 * ones(D) q0 = fmin(f, x0, xtol=1e-12) q0 = q0.reshape((D, 1)) # We are at the minimum with no momentum p0 = zeros_like(q0) else: if "q0" in parameters: q0 = atleast_2d(parameters["q0"]) else: q0 = zeros((D, 1)) if "p0" in parameters: p0 = atleast_2d(parameters["p0"]) else: p0 = zeros((D, 1)) # Compute spreads if computePQ: # Q_0 = H^(-1/4) H = V.evaluate_hessian_at(q0) Q0 = inv(sqrtm(sqrtm(H))) # P_0 = i Q_0^(-1) P0 = 1.0j * inv(Q0) else: if "Q0" in parameters: Q0 = atleast_2d(parameters["Q0"]) else: Q0 = identity(D) if "P0" in parameters: P0 = atleast_2d(parameters["P0"]) else: P0 = 1.0j * inv(Q0) # The parameter set Pi print(70 * "-") print("Parameter values are:") print("---------------------") print(" q0:") print(str(q0)) print(" p0:") print(str(p0)) print(" Q0:") print(str(Q0)) print(" P0:") print(str(P0)) # Consistency check print(" Consistency check:") print(" P^T Q - Q^T P =?= 0") print(dot(P0.T, Q0) - dot(Q0.T, P0)) print(" Q^H P - P^H Q =?= 2i") print( dot(transpose(conjugate(Q0)), P0) - dot(transpose(conjugate(P0)), Q0)) # Next find the new coefficients c' HAWP = BF.create_wavepacket(parameters["hawp_template"]) # Set the parameter values Pi = HAWP.get_parameters() Pi[0] = q0 Pi[1] = p0 Pi[2] = Q0 Pi[3] = P0 HAWP.set_parameters(Pi) # Next compute the matrix M_ij = <phi_i | T + V | phi_j> # The potential part HQ = BF.create_inner_product(parameters["innerproduct"]) opV = lambda x, q, entry: V.evaluate_at(x, entry=entry) MV = HQ.build_matrix(HAWP, operator=opV) # The kinetic part MT = zeros_like(MV, dtype=complexfloating) GR = GradientHAWP() BS = HAWP.get_basis_shapes(component=N) vects = {} for i in BS: z = zeros_like(HAWP.get_coefficient_vector(), dtype=complexfloating) HAWP.set_coefficient_vector(z) HAWP.set_coefficient(N, i, 1.0) Kn, cnew = GR.apply_gradient(HAWP, component=N, as_packet=False) vects[i] = cnew for j in BS: for k in BS: cj = vects[j] ck = vects[k] entry = 0.5 * squeeze(sum(conjugate(cj) * ck)) MT[BS[j], BS[k]] = entry # Find eigenvalues and eigenvectors of the whole matrix M = MT + MV ew, ev = eigh(M) ind = argsort(ew) # Build the requested energy levels and states if "eigenstates_indices" in parameters: states = parameters["eigenstates_indices"] else: # Groundstate only states = [0] BS = HAWP.get_basis_shapes(component=0) KEY = ("q", "p", "Q", "P", "S", "adQ") print(70 * "-") for state in states: if state > BS.get_basis_size(): print( "Warning: can not compute energy level {} with basis size of {}" .format((state, BS))) continue index = ind[state] coeffs = ev[:, index] energy = ew[index] # Try to resolve ambiguities in sign imax = argmax(abs(coeffs)) a = abs(angle(coeffs[imax])) if a > pi / 2.0: coeffs *= -1 print("State: {}".format(state)) print("Energy: {}".format(energy)) print("Coefficients: \n") print(str(coeffs)) print(70 * "-") HAWP.set_coefficient_vector(coeffs.reshape((-1, 1))) # Save all the wavepacket data bid = IOM.create_block(groupid=gid) IOM.add_wavepacket(parameters, blockid=bid, key=KEY) IOM.save_wavepacket(HAWP, 0, blockid=bid, key=KEY) IOM.finalize() # TODO: Find better criterion if norm(q0) > 1000: print("+----------------------------------+") print("| Run-away minimum? |") print("| Maybe try different: |") print("| starting_point = [x0, y0, ...] |") print("+----------------------------------+")
P = iomc.load_parameters() iome.create_file(P, filename=filename[:-5]+"_eigen.hdf5") # Iterate over all groups for groupid in iomc.get_group_ids(): # Create the group if necessary if not groupid in iome.get_group_ids(): iome.create_group(groupid=groupid) for blockid in iomc.get_block_ids(groupid=groupid): print("Computing eigentransformation of data in block '"+str(blockid)+"'") # Create the block if necessary if not blockid in iome.get_block_ids(groupid=groupid): iome.create_block(blockid=blockid, groupid=groupid) # See if we have a wavefunction if iomc.has_wavefunction(blockid=blockid): from EigentransformWavefunction import transform_wavefunction_to_eigen transform_wavefunction_to_eigen(iomc, iome, blockidin=blockid, blockidout=blockid) # See if we have a homogeneous wavepacket next if iomc.has_wavepacket(blockid=blockid): from EigentransformHagedornWavepacket import transform_hawp_to_eigen transform_hawp_to_eigen(iomc, iome, blockidin=blockid, blockidout=blockid) # See if we have an inhomogeneous wavepacket next if iomc.has_inhomogwavepacket(blockid=blockid): from EigentransformHagedornWavepacket import transform_hawpih_to_eigen transform_inhawp_to_eigen(iomc, iome, blockidin=blockid, blockidout=blockid)
def compute_eigenstate(parameters): r""" Special variables necessary in configuration: * eigenstate_of_level (default: 0) * states_indices (default: [0]) """ D = parameters["dimension"] if parameters.has_key("eigenstate_of_level"): N = parameters["eigenstate_of_level"] else: # Upper-most potential surface N = 0 # Create output file now, in case this fails we did not waste computations IOM = IOManager() IOM.create_file("eigenstates.hdf5") # Save the simulation parameters IOM.add_parameters() IOM.save_parameters(parameters) gid = IOM.create_group() BF = BlockFactory() # Create the potential V = BF.create_potential(parameters) V.calculate_local_quadratic() # Minimize the potential to find q0 f = lambda x: real((squeeze(V.evaluate_at(x)[N]))) # Start with an offset because exact 0.0 values can give # issues, especially with the Hessian evaluation. This way # the minimizer will always stay away from zero a tiny bit. # The current starting point can give issues if the potential # is stationary at the point (2, ..., 2) but that is less likely. x0 = 2.0*ones(D) q0 = fmin(f, x0, xtol=1e-12) q0 = q0.reshape((D,1)) # We are at the minimum with no momentum p0 = zeros_like(q0) # Compute spreads now # Q_0 = H^(-1/4) H = V.evaluate_hessian_at(q0) Q0 = inv(sqrtm(sqrtm(H))) # Take P_00 = i Q_0^(-1) P0 = 1.0j * inv(Q0) # print(70*"-") print("Parameter values are:") print("---------------------") print(" q0:") print(str(q0)) print(" p0:") print(str(p0)) print(" Q0:") print(str(Q0)) print(" P0:") print(str(P0)) # Consistency check print(" consistency:") print(str(conj(Q0)*P0 - conj(P0)*Q0)) print(70*"-") # Next find the new coefficients c' HAWP = BF.create_wavepacket(parameters["hawp_template"]) # Set the parameter values Pi = HAWP.get_parameters() Pi[0] = q0 Pi[1] = p0 Pi[2] = Q0 Pi[3] = P0 HAWP.set_parameters(Pi) # Next compute the matrix M_ij = <phi_i | T + V | phi_j> # The potential part HQ = BF.create_inner_product(parameters["innerproduct"]) opV = lambda x, q, entry: V.evaluate_at(x, entry=entry) MV = HQ.build_matrix(HAWP, operator=opV) # The kinetic part MT = zeros_like(MV, dtype=complexfloating) GR = GradientHAWP() BS = HAWP.get_basis_shapes(N) vects = {} for i in BS: z = zeros_like(HAWP.get_coefficient_vector(), dtype=complexfloating) HAWP.set_coefficient_vector(z) HAWP.set_coefficient(N, i, 1.0) Kn, cnew = GR.apply_gradient(HAWP, N) vects[i] = cnew for j in BS: for k in BS: cj = vects[j] ck = vects[k] entry = 0.5 * squeeze(sum(conj(cj) * ck)) MT[BS[j], BS[k]] = entry # Find eigenvalues and eigenvectors of the whole matrix M = MT + MV ew, ev = eigh(M) ind = argsort(ew) # Build the requested energy levels and states if parameters.has_key("eigenstates_indices"): states = parameters["eigenstates_indices"] else: # Groundstate only states = [0] BS = HAWP.get_basis_shapes(component=0) KEY = ("q","p","Q","P","S","adQ") print(70*"-") for state in states: if state > BS.get_basis_size(): print("Warning: can not compute energy level "+state+" with basis size of "+str(BS)) continue index = ind[state] coeffs = ev[:,index] energy = ew[index] print("Level: "+str(state)) print("Energy: "+str(energy)) print("Coefficients: \n") print(str(coeffs)) print(70*"-") HAWP.set_coefficient_vector(coeffs.reshape((-1, 1))) # Save all the wavepacket data bid = IOM.create_block(groupid=gid) IOM.add_wavepacket(parameters, blockid=bid, key=KEY) IOM.save_wavepacket_description(HAWP.get_description(), blockid=bid) for shape in HAWP.get_basis_shapes(): IOM.save_wavepacket_basisshapes(shape, blockid=bid) IOM.save_wavepacket_parameters(HAWP.get_parameters(key=KEY), timestep=0, blockid=bid, key=KEY) IOM.save_wavepacket_coefficients(HAWP.get_coefficients(), HAWP.get_basis_shapes(), timestep=0, blockid=bid) IOM.finalize()
def compute_eigenstate(parameters, filename="eigenstates.hdf5", computepq=True, computePQ=True): r""" Special variables necessary in configuration: * eigenstate_of_level (default: 0) * eigenstates_indices (default: [0]) * starting_point (default: (2, ..., 2)) * hawp_template * innerproduct """ D = parameters["dimension"] if "eigenstate_of_level" in parameters: N = parameters["eigenstate_of_level"] else: # Upper-most potential surface N = 0 # Create output file now, in case this fails we did not waste computation time IOM = IOManager() IOM.create_file(filename) # Save the simulation parameters IOM.add_parameters() IOM.save_parameters(parameters) gid = IOM.create_group() BF = BlockFactory() # Create the potential V = BF.create_potential(parameters) V.calculate_local_quadratic() # Compute position and momentum if computepq: # Minimize the potential to find q0 f = lambda x: real((squeeze(V.evaluate_at(x)[N]))) # Start with an offset because exact 0.0 values can give # issues, especially with the Hessian evaluation. This way # the minimizer will always stay away from zero a tiny bit. # The current starting point can give issues if the potential # is stationary at the point (2, ..., 2) but that is less likely. if "starting_point" in parameters: x0 = atleast_1d(parameters["starting_point"]) else: x0 = 0.5 * ones(D) q0 = fmin(f, x0, xtol=1e-12) q0 = q0.reshape((D, 1)) # We are at the minimum with no momentum p0 = zeros_like(q0) else: if "q0" in parameters: q0 = atleast_2d(parameters["q0"]) else: q0 = zeros((D, 1)) if "p0" in parameters: p0 = atleast_2d(parameters["p0"]) else: p0 = zeros((D, 1)) # Compute spreads if computePQ: # Q_0 = H^(-1/4) H = V.evaluate_hessian_at(q0) Q0 = inv(sqrtm(sqrtm(H))) # P_0 = i Q_0^(-1) P0 = 1.0j * inv(Q0) else: if "Q0" in parameters: Q0 = atleast_2d(parameters["Q0"]) else: Q0 = identity(D) if "P0" in parameters: P0 = atleast_2d(parameters["P0"]) else: P0 = 1.0j * inv(Q0) # The parameter set Pi print(70 * "-") print("Parameter values are:") print("---------------------") print(" q0:") print(str(q0)) print(" p0:") print(str(p0)) print(" Q0:") print(str(Q0)) print(" P0:") print(str(P0)) # Consistency check print(" Consistency check:") print(" P^T Q - Q^T P =?= 0") print(dot(P0.T, Q0) - dot(Q0.T, P0)) print(" Q^H P - P^H Q =?= 2i") print(dot(transpose(conjugate(Q0)), P0) - dot(transpose(conjugate(P0)), Q0)) # Next find the new coefficients c' HAWP = BF.create_wavepacket(parameters["hawp_template"]) # Set the parameter values Pi = HAWP.get_parameters() Pi[0] = q0 Pi[1] = p0 Pi[2] = Q0 Pi[3] = P0 HAWP.set_parameters(Pi) # Next compute the matrix M_ij = <phi_i | T + V | phi_j> # The potential part HQ = BF.create_inner_product(parameters["innerproduct"]) opV = lambda x, q, entry: V.evaluate_at(x, entry=entry) MV = HQ.build_matrix(HAWP, operator=opV) # The kinetic part MT = zeros_like(MV, dtype=complexfloating) GR = GradientHAWP() BS = HAWP.get_basis_shapes(component=N) vects = {} for i in BS: z = zeros_like(HAWP.get_coefficient_vector(), dtype=complexfloating) HAWP.set_coefficient_vector(z) HAWP.set_coefficient(N, i, 1.0) Kn, cnew = GR.apply_gradient(HAWP, component=N, as_packet=False) vects[i] = cnew for j in BS: for k in BS: cj = vects[j] ck = vects[k] entry = 0.5 * squeeze(sum(conjugate(cj) * ck)) MT[BS[j], BS[k]] = entry # Find eigenvalues and eigenvectors of the whole matrix M = MT + MV ew, ev = eigh(M) ind = argsort(ew) # Build the requested energy levels and states if "eigenstates_indices" in parameters: states = parameters["eigenstates_indices"] else: # Groundstate only states = [0] BS = HAWP.get_basis_shapes(component=0) KEY = ("q", "p", "Q", "P", "S", "adQ") print(70 * "-") for state in states: if state > BS.get_basis_size(): print("Warning: can not compute energy level {} with basis size of {}".format((state, BS))) continue index = ind[state] coeffs = ev[:, index] energy = ew[index] # Try to resolve ambiguities in sign imax = argmax(abs(coeffs)) a = abs(angle(coeffs[imax])) if a > pi / 2.0: coeffs *= -1 print("State: {}".format(state)) print("Energy: {}".format(energy)) print("Coefficients: \n") print(str(coeffs)) print(70 * "-") HAWP.set_coefficient_vector(coeffs.reshape((-1, 1))) # Save all the wavepacket data bid = IOM.create_block(groupid=gid) IOM.add_wavepacket(parameters, blockid=bid, key=KEY) IOM.save_wavepacket(HAWP, 0, blockid=bid, key=KEY) IOM.finalize() # TODO: Find better criterion if norm(q0) > 1000: print("+----------------------------------+") print("| Run-away minimum? |") print("| Maybe try different: |") print("| starting_point = [x0, y0, ...] |") print("+----------------------------------+")