I have a question about the basis functions of Gaussian. When the basis function is set (e.g., 6-31G), I would like to know the specific value of Gaussian's parameters for each atom.
In particular, I would like to know the value of $\zeta$ (i.e., the orbital exponent) in $\exp(-\zeta r^2)$, where $r$ is the distance. Is this fixed or optimized in the energy minimization process?
Go to Basis Set Exchange and click on the elements in which you're interested, then click the basis set in which you're interested (in your question, you mentioned 6-31G) and then for "Format" you can choose "Gaussian" or any of various other programs. After all that, when you click "Get Basis Set" you will get the basis set parameters for those elements, for the specific basis set you chose, and in the format you chose.
Here's the basis set parameters for H, He, and Li for 6-31G in Gaussian format (which I obtained by following my above instructions):
H 0
S 3 1.00
0.1873113696D+02 0.3349460434D-01
0.2825394365D+01 0.2347269535D+00
0.6401216923D+00 0.8137573261D+00
S 1 1.00
0.1612777588D+00 1.0000000
****
He 0
S 3 1.00
0.3842163400D+02 0.4013973935D-01
0.5778030000D+01 0.2612460970D+00
0.1241774000D+01 0.7931846246D+00
S 1 1.00
0.2979640000D+00 1.0000000
****
Li 0
S 6 1.00
0.6424189150D+03 0.2142607810D-02
0.9679851530D+02 0.1620887150D-01
0.2209112120D+02 0.7731557250D-01
0.6201070250D+01 0.2457860520D+00
0.1935117680D+01 0.4701890040D+00
0.6367357890D+00 0.3454708450D+00
SP 3 1.00
0.2324918408D+01 -0.3509174574D-01 0.8941508043D-02
0.6324303556D+00 -0.1912328431D+00 0.1410094640D+00
0.7905343475D-01 0.1083987795D+01 0.9453636953D+00
SP 1 1.00
0.3596197175D-01 0.1000000000D+01 0.1000000000D+01
****
In the above data, for each row, the first number is the exponent, and the rest of the numbers in that row are the contraction coefficients.
I also host an open-source basis set repository which is easy for any member of the public to edit, though the exponents and contraction coefficients are in GENBAS format (the format used by at least ACES, CFOUR, and MRCC) instead of Gaussian format; but in this format it might be clearer to see what's going on (and you can see all basis sets at the same time without clicking anything or worrying about new windows popping up). For the H atom, the same basis set as above, is displayed this way:
H:6-31G-EMSL
EMSL 30_08_2019
1
0
2
4
0.1873113696D+02 0.2825394365D+01 0.6401216923D+00 0.1612777588D+00
0.3349460434D-01 0.00000000
0.2347269535D+00 0.00000000
0.8137573261D+00 0.00000000
0.00000000 1.0000000
0 2 4, to indicate the number of exponents and contractions for s-type (hence the 0) basis functions.4 exponents: 0.1873113696D+02 0.2825394365D+01 0.6401216923D+00 0.1612777588D+00If you find the Gaussian format confusing, you can always check the corresponding numbers in the GENBAS format, which organizes it differently (and vice versa if you're confused about the GENBAS format). You can see that the exponents and contraction coefficients in the Gaussian-formatted basis set from Basis Set Exchange are the same as in my GENBAS file, for which I've given what I hope to be clear and thorough explanations for what each part represents.
As Nike noted in his answer, Gaussian (and most gaussian orbital based electronic structure programs) does use fixed orbital exponents throughout a calculation. These are either from previously designed basis sets that Gaussian stores (like the Pople or Dunning basis sets) or could be entered in manually using the Gen keyword.
These exponents are often obtained from an optimization using a single atom computation. This is often good enough to make a reasonable basis for a molecule composed of these atoms, but it could still be improved. It's possible to perform calculations where you do optimize these orbital exponents for the particular molecule they are in.
Gaussian provides a utility for this called gauopt which triggers multiple calculations starting from a template input file defining properties to be optimized. Depending on your programming savvy, you could also make a Python function that creates/runs Gaussian jobs given orbital exponents and you could pass this function to an optimization procedure like scipy's minimize.
This paper[1] describes the DiffiQult program, which can compute gradients of various parameters of an HF calculation using automatic differentiation. They demonstrated this with fully variational calculations on $\ce{H2O}$, where in addition to the MO coefficients, they also optimized the contraction coefficients and orbital exponents.
This idea of a fully optimized basis was looked at more extensively (and specifically for solid state systems) in [2], where they use a DIIS-like procedure to optimize all the basis parameters. They point out how optimizing the basis can lead to greater accuracy than would be expected for a fixed basis of the same (or even larger) size.
A few points to add to Nike and Tyberius' answers.
Literature basis sets $R_{nl}(r) =\sum_i {d_i^{nl}} \exp (-\alpha_{il} r^2) $ indeed have fixed exponents $\alpha_i$ and contraction coefficients $d_i$. A great paper I just started reading is, for instance, the first paper on the development of the earlier Karlsruhe basis set series, J. Chem. Phys. 97, 2571 (1992). You can find over 600 Gaussian basis sets on the Basis Set Exchange.
Although work on system-optimized basis sets have been recently published in the articles cited by Tyberius but also by some of my collaborators and myself in arXiv:2110.11678 (and I think a few more recent works), the idea is quite old. I think its selling point is just as a "cool" demo for automatic differentiation. You don't really want to optimize the basis set for your system, since you don't know how it will affect the quality of your results. Even if your energy improves, your dipole moment may suffer since they are famously sensitive to diffuse functions that energy optimization doesn't favor. Also, it was shown already several decades ago that the optimization takes more time than just using a bigger basis set in the first place! [I am trying to remember the reference but I have to dig it up.]
addendum: the early days of quantum chemistry even used floating orbitals, so the idea of system adapted basis sets was obvious already in the 1960s.
addendum II: I finally found the reference for system-optimized basis sets I wrote about above. It took non-trivial effort to find, and this was not the first time I tried to dig it out again... I hope next time I'll be able to find it on here ;) The work by Tachikawa, Taneda, and Mori published in Int. J. Quantum Chem. 75, 497 (1999) discusses the "fully variational molecular orbital" (FVMO) method, in which the exponents and centers of the Gaussian basis functions are optimized.
If you are using the software Gaussian, gfinput can print out the basis set coefficients/exponents. For more information,
https://gaussian.com/gfinput/
One example is at http://www.ccl.net/cca/documents/dyoung/topics-orig/gaussian.html with a sample input
#n test rohf/sto-3g pop=full GFINPUT
O sto-3g triplet
0 3
O
