# qml.qchem.gaussian_moment¶

gaussian_moment(li, lj, ri, rj, alpha, beta, order, r)[source]

Compute the one-dimensional multipole moment integral for two primitive Gaussian functions.

The multipole moment integral in one dimension is defined as

$S_{ij}^e = \left \langle G_i | q^e | G_j \right \rangle,$

where $$G$$ is a Gaussian function at dimension $$q = x, y, z$$ of the Cartesian coordinates system and $$e$$ is a positive integer that is represented by the order argument. The integrals can be evaluated as [Helgaker (1995) p803]

$S_{ij}^e = \sum_{t=0}^{\mathrm{min}(i+j, \ e)} E_t^{ij} M_t^e,$

where $$E$$ and $$M$$ are the Hermite Gaussian expansion coefficient and the Hermite moment integral, respectively, that can be computed recursively.

Parameters
• li (integer) – angular momentum for the left Gaussian function

• lj (integer) – angular momentum for the right Gaussian function

• ri (float) – position of the left Gaussian function

• rj (float) – position of the right Gaussian function

• alpha (array[float]) – exponent of the left Gaussian function

• beta (array[float]) – exponent of the right Gaussian function

• order (integer) – exponent of the position component

• r (array[float]) – distance between the center of the Hermite Gaussian function and origin

Returns

one-dimensional multipole moment integral between primitive Gaussian functions

Return type

array[float]

Example

>>> li, lj = 0, 0
>>> ri, rj = np.array([2.0]), np.array([2.0])
>>> alpha = np.array([3.42525091])
>>> beta = np.array([3.42525091])
>>> order = 1
>>> r = 1.5
>>> gaussian_moment(li, lj, ri, rj, alpha, beta, order, r)
array([1.0157925])


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