When using the "even tempered method" for augmenting basis sets, what justification is there for the numerical factor which is used?

Question

Diffuse functions are often added to basis sets using "even-tempered" exponents, or sometimes I have heard the phrase "even tempering".

In this paper by Jacek Koput, the author says "the customary even-tempered exponents were calculated in this work by multiplying the exponent of the outermost primitive function of a given angular symmetry in the valence basis set by a factor of 0.4."

The factor of 0.4 (or factors with a very similar value, such as 0.3) have been used in various other papers, but in my experience the factor is chosen without any detailed explanation. Is there any reason why the numerical factor should be specifically 0.4 (or similar)?

Answer 1

Often it's the Stetson-Harrison method. Instead of multiplying, one often specifies the diffuse extrapolation by division. E.g. 0.4 corresponds to dividing the smallest exponent by 1/0.4 = 2.5. This is indeed in the usual ball park; a factor of 3 is quite commonly used.

The numerical value of the factor is related to the completeness of the basis set, which can be "measured", see e.g. my paper on completeness-optimization for a discussion. As the basis set becomes more and more complete, the spacing factor approaches the value 1.

If the basis set you are trying to augment has been optimized as even tempered, IMHO one should not change the spacing of the exponents, but rather re-use the even-tempered spacing of the optimized primitives. (I also used a similar scheme here)

Answer 2

I'm not completely satisfied with this answer, but I think it is worth including here what I have for now.

Going back to the original papers that defined the various aug-cc-pVXZ basis sets, the even-tempered exponents of the diffuse functions are obtained by an optimization to fit CI calculations for those atoms. The functions $\{\zeta_i\}$ are obtained as $$\zeta_i=\alpha\beta^{i-1},i=1,2,3,...,N$$ where $\alpha$ and $\beta$ are optimized. These optimizations all produced $\beta$ values right around 0.3-0.4. I would be surprised if this was entirely coincidental, but its difficult to parse the physical meaning of this value due to it stemming from a nonlinear optimization of a complicated function.

As to your paper, I think people have caught on to the recurrence of this value and so for the case considered (creating aug-cc-pV8Z diffuse functions), rather than going through an additional complicated optimization, one can just choose heuristically reasonable values for $\alpha$ and $\beta$ to arrive at a result sufficient enough for their purposes.