Hermite Gaussians
Hermite Gaussians (T. Helgaker, Jørgensen, and Olsen 2000) are functions of the form \[\begin{equation} \require{physics} \Lambda_{tuv}(\vb{r}; p, \vb{P}) = \qty( \pdv{P_x} )^t \qty( \pdv{P_y} )^u \qty( \pdv{P_z} )^v \exp( -p \thinspace \norm{\vb{r}_P}^2 ) \thinspace , \end{equation}\] where \(p\) is the Gaussian exponent, \[\begin{equation} \vb{r}_P = \vb{r} - \vb{P} \end{equation}\] and \(\vb{P}\) is the center of the Hermite Gaussian. Looking at the form of the Cartesian GTO, we can see that Hermite Gaussians are some sort of derivative Gaussians. Due to their form, we may also factorize them into their Cartesian components: \[\begin{equation} \Lambda_{tuv}(\vb{r}; p, \vb{P}) = \Lambda_t(x; p, P_x) \thinspace \Lambda_u(y; p, P_y) \thinspace \Lambda_v(z; p, P_z) \thinspace , \end{equation}\] where for example the \(x\)-component is given by \[\begin{equation} \Lambda_t(x; p, P_x) = \qty( \pdv{P_x} )^t \exp( -p x_P^2 ) \thinspace , \end{equation}\] which is a one-dimensional Gaussian times a polynomial of degree \(t\).
We can calculate the integral \[\begin{equation} \int_{-\infty}^{+\infty} \dd{x} \Lambda_t(x; p, P_x) = \delta_{t0} \sqrt{\frac{\pi}{p}} \thinspace . \end{equation}\]
We have the recurrence relations \[\begin{equation} \qty( \pdv{P_x} ) \Lambda_t(x; p, P_x) = \Lambda_{t+1}(x; p, P_x) \end{equation}\] and \[\begin{equation} x_P \Lambda_t(x; p, P_x) = \frac{1}{2p} \Lambda_{t+1}(x; p, P_x) + t \Lambda_{t-1}(x; p, P_x) \thinspace . \end{equation}\]
Hermite Gaussians are used in the McMurchie-Davidson integration scheme.