qml.qchem.dipole_integrals

dipole_integrals(mol, core=None, active=None)

Return a function that computes the dipole moment integrals over the molecular orbitals.

These integrals are required to construct the dipole operator in the second-quantized form

$$\hat{D} = -\sum_{pq} d_{pq} [\hat{c}_{p\uparrow}^\dagger \hat{c}_{q\uparrow} + \hat{c}_{p\downarrow}^\dagger \hat{c}_{q\downarrow}] - \hat{D}_\mathrm{c} + \hat{D}_\mathrm{n},$$

where the coefficients $d_{pq}$ 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.

The molecular orbitals are represented as a linear combination of atomic orbitals as

$$\phi_i(r) = \sum_{\nu}c_{\nu}^i \chi_{\nu}(r).$$

Using this equation the dipole moment integral $d_{pq}$ can be written as

$$d_{pq} = \sum_{\mu \nu} C_{p \mu} d_{\mu \nu} C_{\nu q},$$

where $d_{\mu \nu}$ is the dipole moment integral over the atomic orbitals and $C$ is the molecular orbital expansion coefficient matrix. The contribution of the core molecular orbitals is computed as

$$\hat{D}_\mathrm{c} = 2 \sum_{i=1}^{N_\mathrm{core}} d_{ii},$$

where $N_\mathrm{core}$ is the number of core orbitals.

Parameters

• mol (Molecule) – the molecule object

• core (list[int]) – indices of the core orbitals

• active (list[int]) – indices of the active orbitals

Returns

function that computes the dipole moment integrals in the molecular orbital basis

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]
>>> constants, integrals = dipole_integrals(mol)(*args)
>>> print(integrals)
(array([[0., 0.],
        [0., 0.]]),
 array([[0., 0.],
        [0., 0.]]),
 array([[ 0.5      , -0.8270995],
        [-0.8270995,  0.5      ]]))

code/api/pennylane.qchem.dipole_integrals

 

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