qml.qchem.nuclear_attraction

nuclear_attraction(la, lb, ra, rb, alpha, beta, r)[source]

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 the first Gaussian function

  • rb (array[float]) – position vector of the 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]