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]

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Development

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