qml.qchem.gaussian_overlap

gaussian_overlap(la, lb, ra, rb, alpha, beta)

Compute overlap integral for two primitive Gaussian functions.

The overlap integral between two Gaussian functions denoted by $a$ and $b$ can be computed as [Helgaker (1995) p803]:

$$S_{ab} = E^{ij} E^{kl} E^{mn} \left (\frac{\pi}{p} \right )^{3/2},$$

where $E$ is a coefficient that can be computed recursively, $i-n$ are the angular momentum quantum numbers corresponding to different Cartesian components and $p$ is computed from the exponents of the two Gaussian functions as $p = \alpha + \beta$.

Parameters

• la (integer) – angular momentum for the first Gaussian function

• lb (integer) – angular momentum for the second Gaussian function

• ra (float) – position vector of the first Gaussian function

• rb (float) – position vector of the second Gaussian function

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

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

Returns

overlap integral between primitive Gaussian functions

Return type

array[float]

Example

>>> la, lb = (0, 0, 0), (0, 0, 0)
>>> ra, rb = np.array([0., 0., 0.]), np.array([0., 0., 0.])
>>> alpha = np.array([np.pi/2])
>>> beta = np.array([np.pi/2])
>>> o = gaussian_overlap(la, lb, ra, rb, alpha, beta)
>>> o
array([1.])

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