qml.qchem.electron_repulsion

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

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]

code/api/pennylane.qchem.electron_repulsion

 

Download Python script

 

Download Notebook

 

View on GitHub