class FourierPropagator(Propagator):
r"""This class can numerically propagate given initial values :math:`\Psi(x_0, t_0)` on
a potential hyper surface :math:`V(x)`. The propagation is done with a Strang splitting
of the time propagation operator :math:`\exp(-\frac{i}{\varepsilon^2} \tau H)`.
"""
def __init__(self, potential, initial_values, para):
r"""Initialize a new :py:class:`FourierPropagator` instance. Precalculate the
the kinetic operator :math:`T_e` and the potential operator :math:`V_e`
used or time propagation.
:param potential: The potential :math:`V(x)` governing the time evolution.
:type potential: A :py:class:`MatrixPotential` instance.
:param initial_values: The initial values :math:`\Psi(\Gamma, t_0)` given
in the canonical basis.
:type initial_values: A :py:class:`WaveFunction` instance.
:param para: The set of simulation parameters. It must contain at least
the semi-classical parameter :math:`\varepsilon` and the
time step size :math:`tau`.
:raise: :py:class:`ValueError` If the number of components of :math:`\Psi` does not match the
number of energy surfaces :math:`\lambda_i(x)` of the potential.
"""
# The embedded 'MatrixPotential' instance representing the potential 'V'.
self._potential = potential
# The initial values of the components '\psi_i' sampled at the given grid.
self._psi = initial_values
if self._potential.get_number_components() != self._psi.get_number_components():
raise ValueError("Potential dimension and number of components do not match.")
# The position space grid nodes '\Gamma'.
self._grid = initial_values.get_grid()
# The kinetic operator 'T' defined in momentum space.
self._KO = KineticOperator(self._grid, para["eps"])
# Exponential '\exp(-i/2*eps^2*dt*T)' used in the Strang splitting.
self._KO.calculate_exponential(-0.5j * para["dt"] * para["eps"]**2)
self._TE = self._KO.evaluate_exponential_at()
# Exponential '\exp(-i/eps^2*dt*V)' used in the Strang splitting.
self._potential.calculate_exponential(-0.5j * para["dt"] / para["eps"]**2)
VE = self._potential.evaluate_exponential_at(self._grid)
self._VE = tuple([ve.reshape(self._grid.get_number_nodes()) for ve in VE])
# TODO: Consider removing this, duplicate
def get_number_components(self):
r"""Get the number :math:`N` of components of :math:`\Psi`.
:return: The number :math:`N`.
"""
return self._potential.get_number_components()
def get_wavefunction(self):
r"""Get the wavefunction that stores the current data :math:`\Psi(\Gamma)`.
:return: The :py:class:`WaveFunction` instance.
"""
return self._psi
def get_operators(self):
r"""Get the kinetic and potential operators :math:`T(\Omega)` and :math:`V(\Gamma)`.
:return: A tuple :math:`(T, V)` containing two ``ndarrays``.
"""
# TODO: What kind of object exactly do we want to return?
self._KO.calculate_operator()
T = self._KO.evaluate_at()
V = self._potential.evaluate_at(self._grid)
V = tuple([v.reshape(self._grid.get_number_nodes()) for v in V])
return (T, V)
def propagate(self):
r"""Given the wavefunction values :math:`\Psi(\Gamma)` at time :math:`t`, calculate
new values :math:`\Psi^\prime(\Gamma)` at time :math:`t + \tau`. We perform exactly
one single timestep of size :math:`\tau` within this function.
"""
# How many components does Psi have
N = self._psi.get_number_components()
# Unpack the values from the current WaveFunction
values = self._psi.get_values()
tmp = [zeros(value.shape, dtype=complexfloating) for value in values]
# The first step with the potential
for row in range(N):
for col in range(N):
tmp[row] = tmp[row] + self._VE[row * N + col] * values[col]
# Go to Fourier space
tmp = [fftn(component) for component in tmp]
# Apply the kinetic operator
tmp = [self._TE * component for component in tmp]
# Go back to real space
tmp = [ifftn(component) for component in tmp]
# The second step with the potential
values = [zeros(component.shape, dtype=complexfloating) for component in tmp]
for row in range(N):
for col in range(N):
values[row] = values[row] + self._VE[row * N + col] * tmp[col]
# Pack values back to WaveFunction object
# TODO: Consider squeeze(.) of data before repacking
self._psi.set_values(values)
class ChinChenPropagator(Propagator):
r"""This class can numerically propagate given initial values :math:`\Psi(x_0, t_0)` on
a potential hyper surface :math:`V(x)`. The propagation is done with a Chin-Chen [1]_
splitting of the time propagation operator :math:`\exp(-\frac{i}{\varepsilon^2} \tau H)`.
.. note:: This propagator is implemented for single-level potentials (:py:class:`MatrixPotential1S`)
only. More precisely, the other potential implementations do not provide
some functionality needed here.
.. [1] S. A. Chin and C. R. Chen, "Fourth order gradient symplectic integrator methods
for solving the time-dependent Schroedinger equation",
J. Chem. Phys. Volume 114, Issue 17, (2001) 7338-7341.
"""
def __init__(self, parameters, potential, initial_values):
r"""Initialize a new :py:class:`ChinChenPropagator` instance. Precalculate the
the kinetic operator :math:`T_e` and the potential operators :math:`V_e` and
:math:`\tilde{V}_e` used for time propagation.
:param parameters: The set of simulation parameters. It must contain at least
the semi-classical parameter :math:`\varepsilon` and the
time step size :math:`\tau`.
:param potential: The potential :math:`V(x)` governing the time evolution.
:type potential: A :py:class:`MatrixPotential` instance.
:param initial_values: The initial values :math:`\Psi(\Gamma, t_0)` given
in the canonical basis.
:type initial_values: A :py:class:`WaveFunction` instance.
:raise: :py:class:`ValueError` If the number of components of :math:`\Psi` does not match the
number of energy surfaces :math:`\lambda_i(x)` of the potential.
"""
# The embedded 'MatrixPotential' instance representing the potential 'V'.
self._potential = potential
# The initial values of the components '\psi_i' sampled at the given grid.
self._psi = initial_values
if self._potential.get_number_components() != self._psi.get_number_components():
raise ValueError("Potential dimension and number of components do not match.")
# The position space grid nodes '\Gamma'.
self._grid = initial_values.get_grid()
# The kinetic operator 'T' defined in momentum space.
self._KO = KineticOperator(self._grid, parameters["eps"])
# Exponential '\exp(-i/4*eps^2*dt*T)' used in the Chin-Chen splitting.
self._KO.calculate_exponential(-0.25j * parameters["dt"] * parameters["eps"]**2)
self._TE = self._KO.evaluate_exponential_at()
# Exponential '\exp(-i/(6*eps^2)*dt*V)' used in the Chin-Chen splitting.
self._potential.calculate_exponential(-1.0j / 6.0 * parameters["dt"] / parameters["eps"]**2)
VE = self._potential.evaluate_exponential_at(self._grid)
self._VE = tuple([ve.reshape(self._grid.get_number_nodes()) for ve in VE])
# Operator V-tilde
N = self._potential.get_number_components()
n = self._grid.get_number_nodes(overall=True)
# Memory for storing temporary values
tmp = ndarray((n, N, N), dtype=complexfloating)
# Evaluate potential and gradient
self._potential.calculate_jacobian_canonical()
Vx = self._potential.evaluate_at(self._grid)
Jx = self._potential.evaluate_jacobian_canonical_at(self._grid)
Jx = norm(vstack(Jx), axis=0).reshape(1, n)
# Exponential '\exp(-2i/(3*eps^2)*dt * (V + 1/48*dt^2*JV))' used in the Chin-Chen splitting.
# Fill in values
for row in range(N):
for col in range(N):
tmp[:, row, col] = (-2.0j / (3.0 * parameters["eps"]**2) *
(parameters["dt"] * Vx[N * row + col] +
parameters["dt"]**3 / 48.0 * Jx[N * row + col]**2))
# Calculate exponential
for i in range(n):
tmp[i, :, :] = expm(tmp[i, :, :])
# Split the data into different components
self._VEtilde = tuple([tmp[:, row, col].reshape((1, n)) for row in range(N) for col in range(N)])
# TODO: Consider removing this, duplicate
def get_number_components(self):
r"""Get the number :math:`N` of components of :math:`\Psi`.
:return: The number :math:`N`.
"""
return self._potential.get_number_components()
def get_wavefunction(self):
r"""Get the wavefunction that stores the current data :math:`\Psi(\Gamma)`.
:return: The :py:class:`WaveFunction` instance.
"""
return self._psi
def get_operators(self):
r"""Get the kinetic and potential operators :math:`T(\Omega)` and :math:`V(\Gamma)`.
:return: A tuple :math:`(T, V)` containing two ``ndarrays``.
"""
# TODO: What kind of object exactly do we want to return?
self._KO.calculate_operator()
T = self._KO.evaluate_at()
V = self._potential.evaluate_at(self._grid)
V = tuple([v.reshape(self._grid.get_number_nodes()) for v in V])
return (T, V)
def propagate(self):
r"""Given the wavefunction values :math:`\Psi(\Gamma)` at time :math:`t`, calculate
new values :math:`\Psi^\prime(\Gamma)` at time :math:`t + \tau`. We perform exactly
one single timestep of size :math:`\tau` within this function.
"""
# How many components does Psi have
N = self._psi.get_number_components()
# Unpack the values from the current WaveFunction
values = self._psi.get_values()
# First step with the potential
tmp = [zeros(value.shape, dtype=complexfloating) for value in values]
for row in range(0, N):
for col in range(0, N):
tmp[row] = tmp[row] + self._VE[row * N + col] * values[col]
# Go to Fourier space
tmp = [fftn(component) for component in tmp]
# First step with the kinetic operator
tmp = [self._TE * component for component in tmp]
# Go back to real space
tmp = [ifftn(component) for component in tmp]
# Central step with V-tilde
tmp2 = [zeros(value.shape, dtype=complexfloating) for value in values]
for row in range(0, N):
for col in range(0, N):
tmp2[row] = tmp2[row] + self._VEtilde[row * N + col] * tmp[col]
# Go to Fourier space
tmp = [fftn(component) for component in tmp2]
# Second step with the kinetic operator
tmp = [self._TE * component for component in tmp]
# Go back to real space
tmp = [ifftn(component) for component in tmp]
# Second step with the potential
values = [zeros(component.shape, dtype=complexfloating) for component in tmp]
for row in range(0, N):
for col in range(0, N):
values[row] = values[row] + self._VE[row * N + col] * tmp[col]
# Pack values back to WaveFunction object
# TODO: Consider squeeze(.) of data before repacking
self._psi.set_values(values)
class ChinChenPropagator(Propagator):
r"""This class can numerically propagate given initial values :math:`\Psi(x_0, t_0)` on
a potential hyper surface :math:`V(x)`. The propagation is done with a Chin-Chen [1]_
splitting of the time propagation operator :math:`\exp(-\frac{i}{\varepsilon^2} \tau H)`.
.. note:: This propagator is implemented for single-level potentials (:py:class:`MatrixPotential1S`)
only. More precisely, the other potential implementations do not provide
some functionality needed here.
.. [1] S. A. Chin and C. R. Chen, "Fourth order gradient symplectic integrator methods
for solving the time-dependent Schroedinger equation",
J. Chem. Phys. Volume 114, Issue 17, (2001) 7338-7341.
"""
def __init__(self, parameters, potential, initial_values):
r"""Initialize a new :py:class:`ChinChenPropagator` instance. Precalculate the
the kinetic operator :math:`T_e` and the potential operators :math:`V_e` and
:math:`\tilde{V}_e` used for time propagation.
:param parameters: The set of simulation parameters. It must contain at least
the semi-classical parameter :math:`\varepsilon` and the
time step size :math:`\tau`.
:param potential: The potential :math:`V(x)` governing the time evolution.
:type potential: A :py:class:`MatrixPotential` instance.
:param initial_values: The initial values :math:`\Psi(\Gamma, t_0)` given
in the canonical basis.
:type initial_values: A :py:class:`WaveFunction` instance.
:raise: :py:class:`ValueError` If the number of components of :math:`\Psi` does not match the
number of energy surfaces :math:`\lambda_i(x)` of the potential.
"""
# The embedded 'MatrixPotential' instance representing the potential 'V'.
self._potential = potential
# The initial values of the components '\psi_i' sampled at the given grid.
self._psi = initial_values
if self._potential.get_number_components(
) != self._psi.get_number_components():
raise ValueError(
"Potential dimension and number of components do not match.")
# The position space grid nodes '\Gamma'.
self._grid = initial_values.get_grid()
# The kinetic operator 'T' defined in momentum space.
self._KO = KineticOperator(self._grid, parameters["eps"])
# Exponential '\exp(-i/4*eps^2*dt*T)' used in the Chin-Chen splitting.
self._KO.calculate_exponential(-0.25j * parameters["dt"] *
parameters["eps"]**2)
self._TE = self._KO.evaluate_exponential_at()
# Exponential '\exp(-i/(6*eps^2)*dt*V)' used in the Chin-Chen splitting.
self._potential.calculate_exponential(-1.0j / 6.0 * parameters["dt"] /
parameters["eps"]**2)
VE = self._potential.evaluate_exponential_at(self._grid)
self._VE = tuple(
[ve.reshape(self._grid.get_number_nodes()) for ve in VE])
# Operator V-tilde
N = self._potential.get_number_components()
n = self._grid.get_number_nodes(overall=True)
# Memory for storing temporary values
tmp = ndarray((n, N, N), dtype=complexfloating)
# Evaluate potential and gradient
self._potential.calculate_jacobian_canonical()
Vx = self._potential.evaluate_at(self._grid)
Jx = self._potential.evaluate_jacobian_canonical_at(self._grid)
Jx = norm(vstack(Jx), axis=0).reshape(1, n)
# Exponential '\exp(-2i/(3*eps^2)*dt * (V + 1/48*dt^2*JV))' used in the Chin-Chen splitting.
# Fill in values
for row in range(N):
for col in range(N):
tmp[:, row, col] = (
-2.0j / (3.0 * parameters["eps"]**2) *
(parameters["dt"] * Vx[N * row + col] +
parameters["dt"]**3 / 48.0 * Jx[N * row + col]**2))
# Calculate exponential
for i in range(n):
tmp[i, :, :] = expm(tmp[i, :, :])
# Split the data into different components
self._VEtilde = tuple([
tmp[:, row, col].reshape((1, n)) for row in range(N)
for col in range(N)
])
# TODO: Consider removing this, duplicate
def get_number_components(self):
r"""Get the number :math:`N` of components of :math:`\Psi`.
:return: The number :math:`N`.
"""
return self._potential.get_number_components()
def get_wavefunction(self):
r"""Get the wavefunction that stores the current data :math:`\Psi(\Gamma)`.
:return: The :py:class:`WaveFunction` instance.
"""
return self._psi
def get_operators(self):
r"""Get the kinetic and potential operators :math:`T(\Omega)` and :math:`V(\Gamma)`.
:return: A tuple :math:`(T, V)` containing two ``ndarrays``.
"""
# TODO: What kind of object exactly do we want to return?
self._KO.calculate_operator()
T = self._KO.evaluate_at()
V = self._potential.evaluate_at(self._grid)
V = tuple([v.reshape(self._grid.get_number_nodes()) for v in V])
return (T, V)
def propagate(self):
r"""Given the wavefunction values :math:`\Psi(\Gamma)` at time :math:`t`, calculate
new values :math:`\Psi^\prime(\Gamma)` at time :math:`t + \tau`. We perform exactly
one single timestep of size :math:`\tau` within this function.
"""
# How many components does Psi have
N = self._psi.get_number_components()
# Unpack the values from the current WaveFunction
values = self._psi.get_values()
# First step with the potential
tmp = [zeros(value.shape, dtype=complexfloating) for value in values]
for row in range(0, N):
for col in range(0, N):
tmp[row] = tmp[row] + self._VE[row * N + col] * values[col]
# Go to Fourier space
tmp = [fftn(component) for component in tmp]
# First step with the kinetic operator
tmp = [self._TE * component for component in tmp]
# Go back to real space
tmp = [ifftn(component) for component in tmp]
# Central step with V-tilde
tmp2 = [zeros(value.shape, dtype=complexfloating) for value in values]
for row in range(0, N):
for col in range(0, N):
tmp2[row] = tmp2[row] + self._VEtilde[row * N + col] * tmp[col]
# Go to Fourier space
tmp = [fftn(component) for component in tmp2]
# Second step with the kinetic operator
tmp = [self._TE * component for component in tmp]
# Go back to real space
tmp = [ifftn(component) for component in tmp]
# Second step with the potential
values = [
zeros(component.shape, dtype=complexfloating) for component in tmp
]
for row in range(0, N):
for col in range(0, N):
values[row] = values[row] + self._VE[row * N + col] * tmp[col]
# Pack values back to WaveFunction object
# TODO: Consider squeeze(.) of data before repacking
self._psi.set_values(values)