Source code for pennylane.qchem.hartree_fock

# Copyright 2018-2021 Xanadu Quantum Technologies Inc.

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"""
This module contains the functions needed for performing the self-consistent-field calculations.
"""

import itertools

import autograd.numpy as anp

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] + [anp.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) s = s + anp.diag(anp.random.rand(len(s)) * 1.0e-12) w, v = anp.linalg.eigh(s) x = v @ anp.diag(anp.array([1 / anp.sqrt(i) for i in w])) @ v.T eigvals, w_fock = anp.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 = anp.einsum("pqrs,rs->pq", rep_tensor, p) k = anp.einsum("psqr,rs->pq", rep_tensor, p) fock_matrix = h_core + 2 * j - k eigvals, w_fock = anp.linalg.eigh(x.T @ fock_matrix @ x) coeffs = x @ w_fock p_update = mol_density_matrix(n_electron, coeffs) if anp.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 = anp.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] / anp.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 = anp.einsum( "pq,qp", fock_matrix + h_core, mol_density_matrix(mol.n_electrons, coeffs) ) return e_elec + e_rep return _hf_energy