Counting the atomic orbitals in a basis set

[Quantamania]

Greetings.

I have a question for you:

I read one of the old threads about the basis set for Virtual NanoLab.  One of the responses described the atomic orbital components in hydrogen and gold for different basis sets.  I want to know how can we quickly determine the number of pseudopotential orbitals in a particular atom, based on its valence electron configuration.

In my instance, I have carbon, which has s and p orbitals available for describing the valence shell.  This is also true for boron and nitrogen.  How many UNIQUE atomic pseudopotential orbitals are in a carbon atom at SZ, SZP, DZ, DZP, and DZDP?  This will help me with my dissertation defense, particularly when discussing the number of primitives in the cells of my calculations.  It will also help me identify the polarization orbitals for these atoms.

Are there general formulas that work for counting basis functions in elements using valence shells only?

[Anders Blom]

First of all, the definition of the valence is not fixed. Different pseudopotentials can use a different valence configuration; sometimes it's desirable to open up a closed shell, sometimes to keep it as part of the core.

For ATK, you can find, for each element (and exchange-correlation functional) from the pseudopotential files themselves, in the installation tree. For platinum we can for instance see that ATK uses this definition in LDA:

Wavefunctions         nl  l   occ
                       6S  0  1
                       6P  1  0
                       5D  2  9
                       5F  3  0

So 6s+5d would be the SingleZeta set, that is 6s, 5d(-2), 5d(-1), 5d(0), 5d(+1), 5d(+2).

Going to the SingleZetaPolarized, we add the three missing 6p orbitals.

In the DoubleZeta, we double the SingleZeta set, so we have two orbitals of each kind (but with different radial profile).

DoubleZetaPolarized we get the DoubleZeta plus 6p.

Finally, DoubleZetaDoublePolarized is DoubleZeta plus two sets of 6p orbitals.

So, the number of basis functions becomes

SZ: 6
SZP: 9
DZ: 12
DZP: 15
DZDP: 18

Note that in the newly released package ATK 2009.11, there is a functionality for visualizing the basis orbitals!

[Quantamania]

How do I find this information in the Virtual NanoLab or ATK file directory tree?  I currently have Virtual NanoLab 2008.10 on my computers.  Even though this does answer some of my questions, it does not fully help me in the case of carbon, boron and nitrogen.  These are the elements I only use in my dissertation studies, and will need the types of orbitals so I can describe them in my draft.

[zh]

The pseudopotentials are stored in the subdirectory of VNL:
atk/share/pseudopotentials/trunk

[Anders Blom]

The number of basis orbitals is not to be confused with the orbitals of the actual atom, in a sense. ATK uses atomic orbitals (sort of) as a basis, but other software packages use e.g. Gaussians instead, and clearly in that case the nature, and number, of the basis orbitals is not very relevant when you're analyzing the "orbitals" (to the extent that such can be identified) of the results.

What primarily matters is rather the number of valence electrons. Secondary, of course, the occupation of the basis orbitals will tell you something about the atom, but for all five elements that you refer to, the basis orbitals are essentially the same (the occupation is not, however). In SingleZeta, all these elements have one s-like basis function and three p-like. With polarization you add five d-like basis functions, while the "double" means that we double up the single set, so DoubleZetaPolarized has two s-like, six p-like, and five d-like basis functions (the polarization set is not doubled in this case, only in DoubleZetaDoublePolarized).