What do you exactly want to do? Please elaborate so that I can try to answer.

Thank you

I want to get the explicit formula that describes the molecular orbitals in the Gaussian standard basis set.

Molecular orbitals are linear combinations of normalized gaussian contracted (or uncontracted) orbitals associated with each atom.

Each orbital is multiplied by it's corresponding "real" spherical harmonic.

The explicit formula can be found even in wiikipedia http://en.wikipedia.org/wiki/Gaussian_orbital

Otherwise I suggest :

Jensen F. "Introduction to computational chemistry" 1999

or

 Helgaker, Jorgensen, Olsen, "Mol. Electronic Structure Theory" 2000

Thank you,  Matteo, 

However the question is the following: 

How should I combine these gaussians if I have the basis set 6-31g? A big number of basis sets use gaussian orbitals, but their analytical form is different - different combinations of Gaussians and their coefficients. 

If you want to find this info on the web, please look for instance here, Eqs. 1 and 2:

http://www.chm.davidson.edu/ronutt/che401/Gaussian/Gaussian.htm

You should take care also on the values dwy (1) and N in (2). The convention N=1 is probably the most popular one. In the other convention, g in Eq.2 is normalized (N differs from 1), and the contraction coefficients dwy are changed properly.

Thank you!

Is there any online publication with characteristics of different basis sets, like this one, but more wide?

I know that there is a book "Handbook of Gaussian basis sets", but I didn't find it in the web.

Dear Alexander, 

the list of exponents and coefficients of the basis set expansions can be found in the https://bse.pnl.gov/bse/portal 

database.

Dear Matteo,

Thank you. It strikes me that I should use the original articles introducing basis sets cause the information I need is there. I thought that it could be compiled in one publication, this is why I asked.

As Matteo Barborini have noted above, everything is compiled in one publication, which is

Helgaker, Jorgensen, Olsen, "Mol. Electronic Structure Theory" 2000

Alexander: You should be careful with gaussian basis sets. Notation like 6-31G is just that, notation that give you a good idea of the main details of the basis set. However, if you want to play properties written in the primitive basis set, then you better check the exponents and contraction coefficients your program is using. Also check whether  you are using  Cartesian or pure d,f,g… functions.  The EMSL quoted by Matteo is very comprehensive and "standard", but nothing tells you that the basis set in your program uses is exactly the same (this helps to understand why there is not such a book with "all" basis sets). I have found some differences even in main elements in, for instance, Gaussian 09/03. If you you are using a program with periodic boundary conditions, you should pay more attention because basis sets are less "standard" in this case.

People have gave you good references. But just in case you are interested in molecular integrals over gaussian functions, I recommend you this chapter by Peter Giil.

Thank you, Carlos

I'm gonna use Gaussian 09W.

Which chapter?

Hello again,

in order to let Gaussian print the basis set used in the output (in a usable Gaussian form), you must use the flag GFInput, for example:

# BLYP/AUG-cc-pVTZ gfinput Opt=VeryTight Int=fine pop=full SP

Moreover in the output file you will find a section starting with the line:

"Molecular Orbital Coefficients"

There in columns you can find the coefficients of the molecular expansions associated with the basis set orbitals, for each atom.

A book that can help with Gussian is:

 Foresman,  Frisch  "Exploring Chemistry With Electronic Structure Methods: A Guide to Using Gaussian" 3rd ed.

I don't understand why you actually need those numbers, but, if you want to see those orbitals I recommend the use of the cubegen tool to obtain a graphical representation of the orbitals, or density.

http://www.gaussian.com/g_tech/g_ur/u_cubegen.htm

If you need to study the charge or spin localization, then the pop=full key will enable the calculation of Mulliken charges.

Sorry that I can't help you more, but I don't use the Gaussian code anymore so I don't know if there have been some changes.

Thank you very much, Matteo! This is really helpful.