Source code for pennylane.qchem.hartree_fock
# Copyright 2018-2021 Xanadu Quantum Technologies Inc.
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# you may not use this file except in compliance with the License.
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# 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 performing the self-consistent-field calculations.
"""
import itertools
import pennylane as qml
from .matrices import core_matrix, mol_density_matrix, overlap_matrix, repulsion_tensor
[docs]def scf(mol, n_steps=50, tol=1e-8):
r"""Return a function that performs the self-consistent-field calculations.
In the Hartree-Fock method, molecular orbitals are typically constructed as a linear combination
of atomic orbitals
.. math::
\phi_i(r) = \sum_{\mu} C_{\mu i} \chi_{\mu}(r),
with coefficients :math:`C_{\mu i}` that are initially unknown. The self-consistent-field
iterations are performed to find a converged set of molecular orbital coefficients that minimize
the total energy of the molecular system. This optimization problem can be reduced to solving a
linear system of equations which are usually written as
.. math::
FC = SCE,
where :math:`E` is a diagonal matrix of eigenvalues, representing the molecular orbital
energies, :math:`C` is the matrix of molecular orbital coefficients, :math:`S` is the overlap
matrix and :math:`F` is the Fock matrix, which also depends on the coefficients. Fixing an
initial guess :math:`C_0`, the corresponding :math:`F_0` is built and the system
:math:`F_0C_0 = SC_0E` is solved to obtain a solution :math:`C_1`. This process is iteratively
repeated until the coefficients are converged.
The key step in in this process is constructing the Fock matrix which is defined as
.. math::
F = H + \frac{1}{2} J - K,
where :math:`H`, :math:`J` and :math:`K` are the core Hamiltonian matrix, Coulomb matrix and
exchange matrix, respectively. The entries of :math:`H` are computed from the electronic kinetic
energy and the electron-nuclear attraction integrals, which are integrals over atomic basis
functions. The elements of the :math:`J` and :math:`K` matrices are obtained from the Coulomb
and exchange integrals over the basis functions.
Following the procedure in
[`Lehtola et al. Molecules 2020, 25, 1218 <https://www.mdpi.com/1420-3049/25/5/1218>`_], we
express the molecular orbital coefficients in terms of a matrix :math:`X` as
:math:`C = X \tilde{C}` which gives the following transformed equation
.. math::
\tilde{F} \tilde{C} = \tilde{S} \tilde{C} E,
where :math:`\tilde{F} = X^T F X`, :math:`\tilde{S} = X^T S X` and :math:`S` is the overlap
matrix. We chose :math:`X` such that :math:`\tilde{S} = 1` as
.. math::
X = V \Lambda^{-1/2} V^T,
where :math:`V` and :math:`\Lambda` are the eigenvectors and eigenvalues of :math:`S`,
respectively. This gives the eigenvalue equation
.. math::
\tilde{F}\tilde{C} = \tilde{C}E,
which is solved with conventional methods iteratively.
Args:
mol (~qchem.molecule.Molecule): the molecule object
n_steps (int): the number of iterations
tol (float): convergence tolerance
Returns:
function: function that performs the self-consistent-field calculations
**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]
>>> v_fock, coeffs, fock_matrix, h_core, rep_tensor = scf(mol)(*args)
>>> v_fock
array([-0.67578019, 0.94181155])
"""
def _scf(*args):
r"""Perform the self-consistent-field iterations.
Args:
*args (array[array[float]]): initial values of the differentiable parameters
Returns:
tuple(array[float]): eigenvalues of the Fock matrix, molecular orbital coefficients,
Fock matrix, core matrix
"""
basis_functions = mol.basis_set
charges = mol.nuclear_charges
r = mol.coordinates
n_electron = mol.n_electrons
if r.requires_grad:
args_r = [[args[0][i]] * mol.n_basis[i] for i in range(len(mol.n_basis))]
args_ = [*args] + [qml.math.vstack(list(itertools.chain(*args_r)))]
rep_tensor = repulsion_tensor(basis_functions)(*args_[1:])
s = overlap_matrix(basis_functions)(*args_[1:])
h_core = core_matrix(basis_functions, charges, r)(*args_)
else:
rep_tensor = repulsion_tensor(basis_functions)(*args)
s = overlap_matrix(basis_functions)(*args)
h_core = core_matrix(basis_functions, charges, r)(*args)
rng = qml.math.random.default_rng(2030)
s = s + qml.math.diag(rng.random(len(s)) * 1.0e-12)
w, v = qml.math.linalg.eigh(s)
x = v @ qml.math.diag(1.0 / qml.math.sqrt(w)) @ v.T
eigvals, w_fock = qml.math.linalg.eigh(
x.T @ h_core @ x
) # initial guess for the scf problem
coeffs = x @ w_fock
p = mol_density_matrix(n_electron, coeffs)
for _ in range(n_steps):
j = qml.math.einsum("pqrs,rs->pq", rep_tensor, p)
k = qml.math.einsum("psqr,rs->pq", rep_tensor, p)
fock_matrix = h_core + 2 * j - k
eigvals, w_fock = qml.math.linalg.eigh(x.T @ fock_matrix @ x)
coeffs = x @ w_fock
p_update = mol_density_matrix(n_electron, coeffs)
if qml.math.linalg.norm(p_update - p) <= tol:
break
p = p_update
mol.mo_coefficients = coeffs
return eigvals, coeffs, fock_matrix, h_core, rep_tensor
return _scf
[docs]def nuclear_energy(charges, r):
r"""Return a function that computes the nuclear-repulsion energy.
The nuclear-repulsion energy is computed as
.. math::
\sum_{i>j}^n \frac{q_i q_j}{r_{ij}},
where :math:`q`, :math:`r` and :math:`n` denote the nuclear charges (atomic numbers), nuclear
positions and the number of nuclei, respectively.
Args:
charges (list[int]): nuclear charges in atomic units
r (array[float]): nuclear positions
Returns:
function: function that computes the nuclear-repulsion energy
**Example**
>>> symbols = ['H', 'F']
>>> geometry = np.array([[0.0, 0.0, 0.0], [0.0, 0.0, 2.0]], requires_grad = True)
>>> mol = qml.qchem.Molecule(symbols, geometry)
>>> args = [mol.coordinates]
>>> e = nuclear_energy(mol.nuclear_charges, mol.coordinates)(*args)
>>> print(e)
4.5
"""
def _nuclear_energy(*args):
r"""Compute the nuclear-repulsion energy.
Args:
*args (array[array[float]]): initial values of the differentiable parameters
Returns:
array[float]: nuclear-repulsion energy
"""
if r.requires_grad:
coor = args[0]
else:
coor = r
e = qml.math.array([0.0])
for i, r1 in enumerate(coor):
for j, r2 in enumerate(coor[i + 1 :]):
e = e + (charges[i] * charges[i + j + 1] / qml.math.linalg.norm(r1 - r2))
return e
return _nuclear_energy
[docs]def hf_energy(mol):
r"""Return a function that computes the Hartree-Fock energy.
Args:
mol (~qchem.molecule.Molecule): the molecule object
Returns:
function: function that computes the Hartree-Fock energy
**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]
>>> hf_energy(mol)(*args)
-1.065999461545263
"""
def _hf_energy(*args):
r"""Compute the Hartree-Fock energy.
Args:
*args (array[array[float]]): initial values of the differentiable parameters
Returns:
float: the Hartree-Fock energy
"""
_, coeffs, fock_matrix, h_core, _ = scf(mol)(*args)
e_rep = nuclear_energy(mol.nuclear_charges, mol.coordinates)(*args)
e_elec = qml.math.einsum(
"pq,qp", fock_matrix + h_core, mol_density_matrix(mol.n_electrons, coeffs)
)
energy = e_elec + e_rep
return energy.reshape(())
return _hf_energy
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