# qml.qchem.hermite_moment¶

hermite_moment(alpha, beta, t, order, r)[source]

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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