The basis set project continues

In the last blog post, we were discussing both the saturation and the contraction. In the end, I've settled on a (26s16p12d2f) basis set for the aug-cc-pVTZ-Juc basis set (uc is for uncontracted) and now I turn to the contraction of the basis set.

So why do we contract the basis set? - well – for every primitive s-function have in our system, we increase the size of the Fock matrix by one. For p-functions, we have \(p_x\), \(p_y\) and \(p_z\) and thus for every primitive p-function we increase the size of the Fock matrix by three. For d-fucntions it's five and for f-functions is 10. So, by just including one atom using our newly constructed aug-cc-pVTZ-Juc basis set increases the size of the Fock matix by

\[ 26 \times 1 + 3 \times 16 + 12 \times 5 + 2 \times 10 = 156 \]

and remember that the methods you want to use scales really bad with the number of basis functions.

So a contraction of the basis set can be seen as a reasonable reduction of the basis set size without compromising its quality (too much).

Above you see a figure I 've made with the error in percent to aug-cc-pVTZ-Juc as the contraction level is increased. The further you go to the right, the more s-functions (see the number on the x-axis) you have in your contracted basis set. Obviously, the fewer the better because of computational speed but as you go further to the left, the error starts increasing. Initially not so much, but below 10 some errors are > 100 %.

I've decided that an error below 1 % in the contraction when compared to the uncontracted basis set is acceptable (at least for the s-functions), but as observed from the figure above, this threshold is reached at quite different points depending on the which element is investigated.

And this is where it stops being: there is one answer only and it turns into voodoo. So personally, I think that a contraction level of 14 would be quite nice, but since that is the first point where all five elements are below or very close to 1 % (Bromine is not quite there) I don't feel that this should be the contraction level. Going to a contraction level of 15, all calculations give results that are below 1 % but somehow it saddens me that the error for Gallium shoots up to 1 % while Bromine goes down. Obviously I should not concern myself with Germanium, Arsenic or Selenium as their errors are well below 0.2 % for the same contraction level. I guess for s-functions I could probably settle with a contraction level of 17 since reduces the error to 0.6 % in the worst case (Gallium) without adding a lot of computational expense, but then again – a difference of 0.4 % is not really a whole lot.

Revival of an old basis set project

During the stillness in the summer months I've revived an old project regarding the construction of a new basis set optimized for calculating spin-spin coupling constants. It is the aug-cc-pVTZ-J basis set family I hope to contribute to and I am deriving basis sets for the elements Gallium, Germanium, Arsenic, Selenium and Bromine.

I've already constructed the uncontracted version of the basis set (aptly named aug-cc-pVTZ-Juc) by saturating it with additional primitive s, p and d functions following the even-tempered approach were the ratio between exponents are kept constant. It is a pretty straightforward and standard way of increasing the size of a basis set.

If you do this for the simplest hydride of Bromine, I.e. HBr, you end up with something that looks like the following figure when calculating spin-spin coupling constants as the basis is saturated with tight s-functions (black), five tight s-functions and tight p-functions (medium grey) and five tight s-functions, two tight p-functions and tight d-functions (light gray):

It can be seen that there is a nice convergence as the basis set is saturated with s-functions (owing to a convergence of the Fermi contact term). Adding tight p- and d-functions are observed to yield a small increase and a bit of wobbling back and forth as convergence is reached.

I've settled on the the following primitive set of basis functions for the basis sets across all tested elements: [26s16p14d2f] which means I've added 5s, 2p and 4d primitive basis functions to the original uncontracted aug-cc-pVTZ basis set.

However, this is much too large to be useful in any sort of real calculation except for the very basic hydrides I'm using. So we need to contract the basis set. So far, the approach with the J-family of the aug-cc-pVTZ basis sets has been to use the molecular obital coefficients from the simple hydrides as contraction coefficients up to some level (I should rather use down to I suppose but it feels wrong).

Contracting the 26 primitive s-functions first into ns contracted s-functions (while letting the p and d-functions remain uncontracted) gives the solid black line with cirles in the following figure where the error compared to the fully uncontracted basis set is given in percent.

Here we observe that we must use a rather loose form of contraction of at least 15 (I.e. the 26 primitives are fixed in such a way that there is only 15 contracted basis functions left) to obtain an error below 1 % compared to the uncontracted level.

I've furthermore shown data for using a contraction of either 17 or 18 for the s-functions (to get a really low error) and contracted the p-functions on top of that (orange for 17 and blue for 18, respectively). Here we observe that saturation in both cases is obtained by contracting the 16 primitive p-functions into 10 contracted p-functions. Notice that the difference between the orange and blue curve is equal to the difference between the error we make in the contraction of the s-functions. Going above contraction level 10 for the p-functions is not giving us much improvement overall.

Finally, the dashed gray lines represent the contraction of d-functions on top of either the 17s10p or 18s10p results. We observe that a contraction level of 10 is needed for convergence (although it could be argued that a contraction level 9 is more than enough). Again, the difference between 17s10p10d and 18s10p10d is equal to the error from the contraction of the s-functions and p-functions.

As a final note, we see from the 17s11p and 17s12p that using either of these as a base for the contraction of the d-functions, the errors amounts to the difference betwen the contraction level of the p-functions.