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In this work, based on simple algebraic manipulations, the divergence-free description of molecular rotations is revisited using the axis-rotation formula for the rigid-body system. The so-called axis-rotation formula is useful in various fields of computational chemistry, including molecular simulations, graphical rendering and group theory, allowing more convenient ways to construct and to manipulate the atomic or fragment structures of rotations. It is shown that the analytical expression of the axis-rotation operator facilitates obtaining the symmetry operator in analytical form, which is useful in the determination of group symmetries of molecules and the adaptation to the symmetry of atomic and molecular orbitals.