Symmetry-Adapted Linear Combinations of Hydrogen Orbitals in Ammonia

1 Symmetry-Adapted Linear Combinations of Hydrogen Orbitals in Ammonia 

From now on we shall no longer be working with the nuclei, but with electronic orbital functions that are anchored on the nuclei. As an example, we take the Is orbitals on the hydrogen atoms in ammonia. The 1sa is defined in Ekj. (4.1), where Ra denotes the position vector of atom A with respect to the Cartesian origin ao is the Bohr radius (0.529 A), which is the atomic unit of length. 

Following Sect. 1.2, the transformation of this function by the threefold axis is given 

The distance between the electron at position and nucleus A is equal to the distance between the electron at position r and nucleus B. This can be established by working out the distances as functions of the Cartesian coordinates, but a straightforward demonstration is based on the fact that the distance does not change if we rotate both nuclei and electrons  

we have denoted the bodily rotation of the entire molecule as Qs in order to indicate that its action also involves the nuclei, as opposed to C3, which is reserved for electrons. In the expression of Eq. (4.1), Ra is replaced by Rb, or 

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