Source code for pennylane.qchem.dipole
# Copyright 2018-2022 Xanadu Quantum Technologies Inc.
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
# http://www.apache.org/licenses/LICENSE-2.0
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
"""
This module contains the functions needed for computing the dipole moment.
"""
import pennylane as qml
from pennylane.fermi import FermiSentence, FermiWord
from .basis_data import atomic_numbers
from .hartree_fock import scf
from .matrices import moment_matrix
from .observable_hf import fermionic_observable, qubit_observable
[docs]def dipole_integrals(mol, core=None, active=None):
r"""Return a function that computes the dipole moment integrals over the molecular orbitals.
These integrals are required to construct the dipole operator in the second-quantized form
.. math::
\hat{D} = -\sum_{pq} d_{pq} [\hat{c}_{p\uparrow}^\dagger \hat{c}_{q\uparrow} +
\hat{c}_{p\downarrow}^\dagger \hat{c}_{q\downarrow}] -
\hat{D}_\mathrm{c} + \hat{D}_\mathrm{n},
where the coefficients :math:`d_{pq}` are given by the integral of the position operator
:math:`\hat{{\bf r}}` over molecular orbitals
:math:`\phi`
.. math::
d_{pq} = \int \phi_p^*(r) \hat{{\bf r}} \phi_q(r) dr,
and :math:`\hat{c}^{\dagger}` and :math:`\hat{c}` are the creation and annihilation operators,
respectively. The contribution of the core orbitals and nuclei are denoted by
:math:`\hat{D}_\mathrm{c}` and :math:`\hat{D}_\mathrm{n}`, respectively.
The molecular orbitals are represented as a linear combination of atomic orbitals as
.. math::
\phi_i(r) = \sum_{\nu}c_{\nu}^i \chi_{\nu}(r).
Using this equation the dipole moment integral :math:`d_{pq}` can be written as
.. math::
d_{pq} = \sum_{\mu \nu} C_{p \mu} d_{\mu \nu} C_{\nu q},
where :math:`d_{\mu \nu}` is the dipole moment integral over the atomic orbitals and :math:`C`
is the molecular orbital expansion coefficient matrix. The contribution of the core molecular
orbitals is computed as
.. math::
\hat{D}_\mathrm{c} = 2 \sum_{i=1}^{N_\mathrm{core}} d_{ii},
where :math:`N_\mathrm{core}` is the number of core orbitals.
Args:
mol (~qchem.molecule.Molecule): the molecule object
core (list[int]): indices of the core orbitals
active (list[int]): indices of the active orbitals
Returns:
function: function that computes the dipole moment integrals in the molecular orbital basis
**Example**
>>> symbols = ['H', 'H']
>>> geometry = np.array([[0.0, 0.0, 0.0], [0.0, 0.0, 1.0]], requires_grad = False)
>>> alpha = np.array([[3.42525091, 0.62391373, 0.1688554],
>>> [3.42525091, 0.62391373, 0.1688554]], requires_grad=True)
>>> mol = qml.qchem.Molecule(symbols, geometry, alpha=alpha)
>>> args = [alpha]
>>> constants, integrals = dipole_integrals(mol)(*args)
>>> print(integrals)
(array([[0., 0.],
[0., 0.]]),
array([[0., 0.],
[0., 0.]]),
array([[ 0.5 , -0.8270995],
[-0.8270995, 0.5 ]]))
"""
def _dipole_integrals(*args):
r"""Compute the dipole moment integrals in the molecular orbital basis.
Args:
*args (array[array[float]]): initial values of the differentiable parameters
Returns:
tuple[array[float]]: tuple containing the core orbital contributions and the dipole
moment integrals
"""
_, coeffs, _, _, _ = scf(mol)(*args)
# x, y, z components
d_x = qml.math.einsum(
"qr,rs,st->qt", coeffs.T, moment_matrix(mol.basis_set, 1, 0)(*args), coeffs
)
d_y = qml.math.einsum(
"qr,rs,st->qt", coeffs.T, moment_matrix(mol.basis_set, 1, 1)(*args), coeffs
)
d_z = qml.math.einsum(
"qr,rs,st->qt", coeffs.T, moment_matrix(mol.basis_set, 1, 2)(*args), coeffs
)
# x, y, z components (core orbitals contribution)
core_x, core_y, core_z = qml.math.array([0]), qml.math.array([0]), qml.math.array([0])
if core is None and active is None:
return (core_x, core_y, core_z), (d_x, d_y, d_z)
for i in core:
core_x = core_x + 2 * d_x[i][i]
core_y = core_y + 2 * d_y[i][i]
core_z = core_z + 2 * d_z[i][i]
d_x = d_x[qml.math.ix_(active, active)]
d_y = d_y[qml.math.ix_(active, active)]
d_z = d_z[qml.math.ix_(active, active)]
return (core_x, core_y, core_z), (d_x, d_y, d_z)
return _dipole_integrals
[docs]def fermionic_dipole(mol, cutoff=1.0e-18, core=None, active=None):
r"""Return a function that builds the fermionic dipole moment observable.
The dipole operator in the second-quantized form is
.. math::
\hat{D} = -\sum_{pq} d_{pq} [\hat{c}_{p\uparrow}^\dagger \hat{c}_{q\uparrow} +
\hat{c}_{p\downarrow}^\dagger \hat{c}_{q\downarrow}] -
\hat{D}_\mathrm{c} + \hat{D}_\mathrm{n},
where the matrix elements :math:`d_{pq}` are given by the integral of the position operator
:math:`\hat{{\bf r}}` over molecular orbitals :math:`\phi`
.. math::
d_{pq} = \int \phi_p^*(r) \hat{{\bf r}} \phi_q(r) dr,
and :math:`\hat{c}^{\dagger}` and :math:`\hat{c}` are the creation and annihilation operators,
respectively. The contribution of the core orbitals and nuclei are denoted by
:math:`\hat{D}_\mathrm{c}` and :math:`\hat{D}_\mathrm{n}`, respectively, which are computed as
.. math::
\hat{D}_\mathrm{c} = 2 \sum_{i=1}^{N_\mathrm{core}} d_{ii},
and
.. math::
\hat{D}_\mathrm{n} = \sum_{i=1}^{N_\mathrm{atoms}} Z_i {\bf R}_i,
where :math:`Z_i` and :math:`{\bf R}_i` denote, respectively, the atomic number and the
nuclear coordinates of the :math:`i`-th atom of the molecule.
Args:
mol (~qchem.molecule.Molecule): the molecule object
cutoff (float): cutoff value for discarding the negligible dipole moment integrals
core (list[int]): indices of the core orbitals
active (list[int]): indices of the active orbitals
Returns:
function: function that builds the fermionic dipole moment observable
**Example**
>>> symbols = ['H', 'H']
>>> geometry = np.array([[0.0, 0.0, 0.0], [0.0, 0.0, 1.0]], requires_grad = False)
>>> alpha = np.array([[3.42525091, 0.62391373, 0.1688554],
>>> [3.42525091, 0.62391373, 0.1688554]], requires_grad=True)
>>> mol = qml.qchem.Molecule(symbols, geometry, alpha=alpha)
>>> args = [alpha]
>>> fermionic_dipole(mol)(*args)[2]
-0.4999999988651487 * a⁺(0) a(0)
+ 0.82709948984052 * a⁺(0) a(2)
+ -0.4999999988651487 * a⁺(1) a(1)
+ 0.82709948984052 * a⁺(1) a(3)
+ 0.82709948984052 * a⁺(2) a(0)
+ -0.4999999899792451 * a⁺(2) a(2)
+ 0.82709948984052 * a⁺(3) a(1)
+ -0.4999999899792451 * a⁺(3) a(3)
+ 1.0 * I
"""
def _fermionic_dipole(*args):
r"""Build the fermionic dipole moment observable.
Args:
*args (array[array[float]]): initial values of the differentiable parameters
Returns:
FermiSentence: fermionic dipole moment
"""
constants, integrals = dipole_integrals(mol, core, active)(*args)
nd = [qml.math.array([0]), qml.math.array([0]), qml.math.array([0])]
for i, s in enumerate(mol.symbols): # nuclear contributions
nd[0] = nd[0] + atomic_numbers[s] * mol.coordinates[i][0]
nd[1] = nd[1] + atomic_numbers[s] * mol.coordinates[i][1]
nd[2] = nd[2] + atomic_numbers[s] * mol.coordinates[i][2]
d_ferm = []
for i in range(3):
f = fermionic_observable(constants[i], integrals[i], cutoff=cutoff)
d_ferm.append(FermiSentence({FermiWord({}): nd[i][0]}) - f)
return d_ferm
return _fermionic_dipole
[docs]def dipole_moment(mol, cutoff=1.0e-16, core=None, active=None):
r"""Return a function that computes the qubit dipole moment observable.
The dipole operator in the second-quantized form is
.. math::
\hat{D} = -\sum_{pq} d_{pq} [\hat{c}_{p\uparrow}^\dagger \hat{c}_{q\uparrow} +
\hat{c}_{p\downarrow}^\dagger \hat{c}_{q\downarrow}] -
\hat{D}_\mathrm{c} + \hat{D}_\mathrm{n},
where the matrix elements :math:`d_{pq}` are given by the integral of the position operator
:math:`\hat{{\bf r}}` over molecular orbitals :math:`\phi`
.. math::
d_{pq} = \int \phi_p^*(r) \hat{{\bf r}} \phi_q(r) dr,
and :math:`\hat{c}^{\dagger}` and :math:`\hat{c}` are the creation and annihilation operators,
respectively. The contribution of the core orbitals and nuclei are denoted by
:math:`\hat{D}_\mathrm{c}` and :math:`\hat{D}_\mathrm{n}`, respectively, which are computed as
.. math::
\hat{D}_\mathrm{c} = 2 \sum_{i=1}^{N_\mathrm{core}} d_{ii},
and
.. math::
\hat{D}_\mathrm{n} = \sum_{i=1}^{N_\mathrm{atoms}} Z_i {\bf R}_i,
where :math:`Z_i` and :math:`{\bf R}_i` denote, respectively, the atomic number and the
nuclear coordinates of the :math:`i`-th atom of the molecule.
The fermonic dipole operator is then transformed to the qubit basis which gives
.. math::
\hat{D} = \sum_{j} c_j P_j,
where :math:`c_j` is a numerical coefficient and :math:`P_j` is a ternsor product of
single-qubit Pauli operators :math:`X, Y, Z, I`.
Args:
mol (~qchem.molecule.Molecule): the molecule object
cutoff (float): cutoff value for discarding the negligible dipole moment integrals
core (list[int]): indices of the core orbitals
active (list[int]): indices of the active orbitals
Returns:
function: function that computes the qubit dipole moment observable
**Example**
>>> symbols = ['H', 'H']
>>> geometry = np.array([[0.0, 0.0, 0.0], [0.0, 0.0, 1.0]], requires_grad = False)
>>> alpha = np.array([[3.42525091, 0.62391373, 0.1688554],
>>> [3.42525091, 0.62391373, 0.1688554]], requires_grad=True)
>>> mol = qml.qchem.Molecule(symbols, geometry, alpha=alpha)
>>> args = [alpha]
>>> dipole_moment(mol)(*args)[2].ops
[Identity(wires=[0]),
PauliZ(wires=[0]),
PauliY(wires=[0]) @ PauliZ(wires=[1]) @ PauliY(wires=[2]),
PauliX(wires=[0]) @ PauliZ(wires=[1]) @ PauliX(wires=[2]),
PauliZ(wires=[1]),
PauliY(wires=[1]) @ PauliZ(wires=[2]) @ PauliY(wires=[3]),
PauliX(wires=[1]) @ PauliZ(wires=[2]) @ PauliX(wires=[3]),
PauliZ(wires=[2]),
PauliZ(wires=[3])]
"""
def _dipole(*args):
r"""Compute the qubit dipole moment observable.
Args:
*args (array[array[float]]): initial values of the differentiable parameters
Returns:
(list[Hamiltonian]): x, y and z components of the dipole moment observable
"""
d = []
d_ferm = fermionic_dipole(mol, cutoff, core, active)(*args)
for i in d_ferm:
d.append(qubit_observable(i, cutoff=cutoff))
return d
return _dipole
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