Generating functions for oscillator matrix elements

Two general harmonic oscillator elements \documentclass{article}\pagestyle{empty}\begin{document}$$ \left\langle m \right|\hat x^s \hat p\left| n \right\rangle $$\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$$ \left\langle m \right|\exp \left[ { - \alpha \left( {\hbar /m*\omega } \right)\hat x^2 } \right]\exp \left( {\beta \hat x} \right)\left| n \right\rangle $$\end{document} are derived by a generating function method using operator techniques which contain practically all one‐ and two‐centre integrals with equal frequencies of chemical physics.