qml.qchem.fermionic_dipole¶
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fermionic_dipole(mol, cutoff=1e-18, core=None, active=None)[source]¶ Return a function that builds the fermionic dipole moment observable.
The dipole operator in the second-quantized form is
\[\hat{D} = -\sum_{pq} d_{pq} [\hat{c}_{p\uparrow}^\dagger \hat{c}_{q\uparrow} + \hat{c}_{p\downarrow}^\dagger \hat{c}_{q\downarrow}] - \hat{D}_\mathrm{c} + \hat{D}_\mathrm{n},\]where the matrix elements \(d_{pq}\) are given by the integral of the position operator \(\hat{{\bf r}}\) over molecular orbitals \(\phi\)
\[d_{pq} = \int \phi_p^*(r) \hat{{\bf r}} \phi_q(r) dr,\]and \(\hat{c}^{\dagger}\) and \(\hat{c}\) are the creation and annihilation operators, respectively. The contribution of the core orbitals and nuclei are denoted by \(\hat{D}_\mathrm{c}\) and \(\hat{D}_\mathrm{n}\), respectively, which are computed as
\[\hat{D}_\mathrm{c} = 2 \sum_{i=1}^{N_\mathrm{core}} d_{ii},\]and
\[\hat{D}_\mathrm{n} = \sum_{i=1}^{N_\mathrm{atoms}} Z_i {\bf R}_i,\]where \(Z_i\) and \({\bf R}_i\) denote, respectively, the atomic number and the nuclear coordinates of the \(i\)-th atom of the molecule.
- Parameters
mol (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 that builds the fermionic dipole moment observable
- 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] >>> 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