qml.qchem.fermionic_dipole

fermionic_dipole(mol, cutoff=1e-18, core=None, active=None)

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

code/api/pennylane.qchem.fermionic_dipole

 

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