qml.qchem.electron_repulsion

electron_repulsion(la, lb, lc, ld, ra, rb, rc, rd, alpha, beta, gamma, delta)[source]

Compute the electron-electron repulsion integral between four primitive Gaussian functions.

The electron repulsion integral between four Gaussian functions denoted by \(a\), \(b\) , \(c\) and \(d\) is computed as [Helgaker (1995) p820]

\[g_{abcd} = \frac{2\pi^{5/2}}{pq\sqrt{p+q}} \sum_{tuv} E_t^{o_a o_b} E_u^{m_a m_b} E_v^{n_a n_b} \sum_{rsw} (-1)^{r+s+w} E_r^{o_c o_d} E_s^{m_c m_d} E_w^{n_c n_d} R_{t+r, u+s, v+w},\]

where \(E\) and \(R\) are the Hermite Gaussian expansion coefficients and the Hermite Coulomb integral, respectively. The sums go over the angular momentum quantum numbers \(o_i + o_j + 1\), \(m_i + m_j + 1\) and \(n_i + n_j + 1\) respectively for \(t, u, v\) and \(r, s, w\). The exponents of the Gaussian functions are used to compute \(p\) and \(q\) as \(p = \alpha + \beta\) and \(q = \gamma + \delta\).

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

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

  • lc (tuple[int]) – angular momentum for the third Gaussian function

  • ld (tuple[int]) – angular momentum for the forth Gaussian function

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

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

  • rc (array[float]) – position vector of the the third Gaussian function

  • rd (array[float]) – position vector of the the forth Gaussian function

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

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

  • gamma (array[float]) – exponent of the third Gaussian function

  • delta (array[float]) – exponent of the forth Gaussian function

Returns

electron-electron repulsion integral between four Gaussian functions

Return type

array[float]