qml.qchem.gaussian_moment¶
-
gaussian_moment(li, lj, ri, rj, alpha, beta, order, r)[source]¶ Compute the one-dimensional multipole moment integral for two primitive Gaussian functions.
The multipole moment integral in one dimension is defined as
\[S_{ij}^e = \left \langle G_i | q^e | G_j \right \rangle,\]where \(G\) is a Gaussian function at dimension \(q = x, y, z\) of the Cartesian coordinates system and \(e\) is a positive integer that is represented by the
orderargument. The integrals can be evaluated as [Helgaker (1995) p803]\[S_{ij}^e = \sum_{t=0}^{\mathrm{min}(i+j, \ e)} E_t^{ij} M_t^e,\]where \(E\) and \(M\) are the Hermite Gaussian expansion coefficient and the Hermite moment integral, respectively, that can be computed recursively.
- Parameters
li (integer) – angular momentum for the left Gaussian function
lj (integer) – angular momentum for the right Gaussian function
ri (float) – position of the left Gaussian function
rj (float) – position of the right Gaussian function
alpha (array[float]) – exponent of the left Gaussian function
beta (array[float]) – exponent of the right Gaussian function
order (integer) – exponent of the position component
r (array[float]) – distance between the center of the Hermite Gaussian function and origin
- Returns
one-dimensional multipole moment integral between primitive Gaussian functions
- Return type
array[float]
Example
>>> li, lj = 0, 0 >>> ri, rj = np.array([2.0]), np.array([2.0]) >>> alpha = np.array([3.42525091]) >>> beta = np.array([3.42525091]) >>> order = 1 >>> r = 1.5 >>> gaussian_moment(li, lj, ri, rj, alpha, beta, order, r) array([1.0157925])