qml.qchem.scf

scf(mol, n_steps=50, tol=1e-08)

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

$$\phi_i(r) = \sum_{\mu} C_{\mu i} \chi_{\mu}(r),$$

with coefficients $C_{\mu i}$ 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

$$FC = SCE,$$

where $E$ is a diagonal matrix of eigenvalues, representing the molecular orbital energies, $C$ is the matrix of molecular orbital coefficients, $S$ is the overlap matrix and $F$ is the Fock matrix, which also depends on the coefficients. Fixing an initial guess $C_0$, the corresponding $F_0$ is built and the system $F_0C_0 = SC_0E$ is solved to obtain a solution $C_1$. 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

$$F = H + \frac{1}{2} J - K,$$

where $H$, $J$ and $K$ are the core Hamiltonian matrix, Coulomb matrix and exchange matrix, respectively. The entries of $H$ are computed from the electronic kinetic energy and the electron-nuclear attraction integrals, which are integrals over atomic basis functions. The elements of the $J$ and $K$ matrices are obtained from the Coulomb and exchange integrals over the basis functions.

Following the procedure in [Lehtola et al. Molecules 2020, 25, 1218], we express the molecular orbital coefficients in terms of a matrix $X$ as $C = X \tilde{C}$ which gives the following transformed equation

$$\tilde{F} \tilde{C} = \tilde{S} \tilde{C} E,$$

where $\tilde{F} = X^T F X$, $\tilde{S} = X^T S X$ and $S$ is the overlap matrix. We chose $X$ such that $\tilde{S} = 1$ as

$$X = V \Lambda^{-1/2} V^T,$$

where $V$ and $\Lambda$ are the eigenvectors and eigenvalues of $S$, respectively. This gives the eigenvalue equation

$$\tilde{F}\tilde{C} = \tilde{C}E,$$

which is solved with conventional methods iteratively.

Parameters

• mol (Molecule) – the molecule object

• n_steps (int) – the number of iterations

• tol (float) – convergence tolerance

Returns

function that performs the self-consistent-field calculations

Return type

function

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])

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