qml.qchem.hermite_moment

hermite_moment(alpha, beta, t, order, r)

Compute the Hermite moment integral recursively.

The Hermite moment integral in one dimension is defined as

$$M_{t}^{e} = \int_{-\infty }^{+\infty} q^e \Lambda_t dq,$$

where $e$ is a positive integer, that is represented by the order argument, $q = x, y, z$ is the coordinate at which the integral is evaluatedand and $\Lambda_t$ is the $t$ component of the Hermite Gaussian function. The integral can be computed recursively as [Helgaker (1995) p802]

$$M_{t}^{e+1} = t M_{t-1}^{e} + Q M_{t}^{e} + \frac{1}{2p} M_{t+1}^{e},$$

where $Q$ is the distance between the center of the Hermite Gaussian function and the origin, at dimension $q = x, y, z$ of the Cartesian coordinates system.

This integral is zero for $t > e$ and the base case solution is

$$M_t^0 = \delta _{t0} \sqrt{\frac{\pi}{p}},$$

where $p = \alpha + \beta$ and $\alpha, \beta$ are the exponents of the Gaussian functions that construct the Hermite Gaussian function $\Lambda$.

Parameters

• alpha (array[float]) – exponent of the left Gaussian function

• beta (array[float]) – exponent of the right Gaussian function

• t (integer) – order of the Hermite Gaussian function

• order (integer) – exponent of the position component

• r (array[float]) – distance between the center of the Hermite Gaussian function and the origin

Returns

the Hermite moment integral

Return type

array[float]

Example

>>> alpha = np.array([3.42525091])
>>> beta = np.array([3.42525091])
>>> t = 0
>>> order = 1
>>> r = 1.5
>>> hermite_moment(alpha, beta, t, order, r)
array([1.0157925])

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