The HO functions in Cartesian coordinates

In our discussion of the HO functions and the GTOs so far, we have worked in polar coordinates in order to separate the functions in radial and angular parts. However, a different symmetry is also present in the Hamiltonian of the three-dimensional isotropic HO. Writing the Hamiltonian (6.6.1) in Cartesian coordinates, we obtain 

From this expression, it is easy to verify the degeneracy (n + l)(n + 2)/2 of the HO system. 

The Hermite polynomials of degree n, H x), that appear in the HO functions in Cartesian form fulfil the orthogonality relation [11,12] 

We note that the Hermite polynomials are either symmetric or antisymmetric with respect to inversion through the origin. The reader should be aware that a different form of the Hermite 

generalizations of the H x) to nonintegral complex n exist, called the Hermite functions. The lowest-degree Hermite polynomials are listed in Table 6.7. 

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