qml.qchem.nuclear_attraction

nuclear_attraction(la, lb, ra, rb, alpha, beta, r)

Compute nuclear attraction integral between primitive Gaussian functions.

The nuclear attraction integral between two Gaussian functions denoted by $a$ and $b$ can be computed as [Helgaker (1995) p820]

$$V_{ab} = \frac{2\pi}{p} \sum_{tuv} E_t^{ij} E_u^{kl} E_v^{mn} R_{tuv},$$

where $E$ and $R$ represent the Hermite Gaussian expansion coefficients and the Hermite Coulomb integral, respectively. The sum goes over $i + j + 1$, $k + l + 1$ and $m + n + 1$ for $t$, $u$ and $v$, respectively, and $p$ is computed from the exponents of the two Gaussian functions as $p = \alpha + \beta$.

Parameters

• la (tuple[int]) – angular momentum for the first Gaussian function

• lb (tuple[int]) – angular momentum for the second Gaussian function

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

• rb (array[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

• r (array[float]) – position vector of nucleus

Returns

nuclear attraction integral between two Gaussian functions

Return type

array[float]

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