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 first Gaussian function
rb (array[float]) – position vector of the second Gaussian function
rc (array[float]) – position vector of the third Gaussian function
rd (array[float]) – position vector of 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]