qml.qchem.contracted_norm

contracted_norm(l, alpha, a)

Compute the normalization constant for a contracted Gaussian function.

A contracted Gaussian function is defined as

\[\psi = a_1 G_1 + a_2 G_2 + a_3 G_3,\]

where \(a\) denotes the contraction coefficients and \(G\) is a primitive Gaussian function. The normalization constant for this function is computed as

\[N(l, \alpha, a) = [\frac{\pi^{3/2}(2l_x-1)!! (2l_y-1)!! (2l_z-1)!!}{2^{l_x + l_y + l_z}} \sum_{i,j} \frac{a_i a_j}{(\alpha_i + \alpha_j)^{{l_x + l_y + l_z+3/2}}}]^{-1/2}\]

where \(l\) and \(\alpha\) denote the angular momentum quantum number and the exponent of the Gaussian function, respectively.

Parameters

• l (tuple[int]) – angular momentum quantum number of the primitive Gaussian functions

• alpha (array[float]) – exponents of the primitive Gaussian functions

• a (array[float]) – coefficients of the contracted Gaussian functions

Returns

normalization coefficient

Return type

array[float]

Example

>>> l = (0, 0, 0)
>>> alpha = np.array([3.425250914, 0.6239137298, 0.168855404])
>>> a = np.array([1.79444183, 0.50032649, 0.18773546])
>>> n = contracted_norm(l, alpha, a)
>>> print(n)
0.39969026908800853

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