Exploring Zeta in Atomic Shells and Quantum Physics

[sryzdn]

Hi,

Simply, what is "zeta" in atomic shells and what is its relationship with orbital angular momentum (L)?
How is it used in quantum physics?

 

[cgk]

You need to provide context to your question, it is hard to guess what you mean. In quantum chemistry the exponents in atomic basis functions are sometimes called zeta, but the only semi-consistent use of that term I am aware of is in terms like "double zeta", "triple zeta", etc, which simply denote the number of shells of basis functions used for the valence atomic orbitals[*].

 

[sryzdn]

yes, my question exactly points out to what you have already mentioned.
What does "exponents in atomic basis function" mean?

And how does these triple and double zeta thing relate to quantum numbers?

 

[cgk]

The basis functions which are used almost exclusively are Gauss-type-orbitals. These have the form of some solid harmonic prefactor (effectively, a polynomial in the distance-to-the-atom-coordinates) multiplied with a Gaussian:
$$\mu(\vec r) = S^l_m(\vec r - \vec A) \exp(-\zeta(\vec r - \vec A)^2).$$
Here r is the electron coordinate and A the coordinate of the point the basis function is placed on. The exponent is the $\zeta$ (but often also called other things, and some functions are normally set into fixed linear combinations to form atomic orbitals).

Zeta, by itself, has no relationship to quantum numbers. But the double-zeta, triple-zeta etc. sets are designed such that the polarization and correlation of valence electrons can be described in a consistent and systematic quantitative way (you might want to read the first atomic natural orbital papers, or Dunning's first cc-pVnZ paper to get a clearer picture of how those sets are designed). Effectively, the only important point about this denomination is that a (n+1)-zeta basis set is a systematic improvement over a n-zeta basis set, by including in the larger set such functions of higher and the same angular momentum that they all produce similar energy improvements.