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Cross-posted on Reddit.

I have been reading through [1] to get a better understanding of geminal-based methods. Some short passages are included below:

The occupation of each orbital in the expansion above (Eq. 5) is indicated by the summation index $m_i$, while the total number of occupied orbitals in each Slater determinant has to equal the number of electron pairs. The APIG wave function thus includes $K\choose P$ determinants ($K$ is number of orbitals, $P$ the number of electron pairs).

The AGP ansatz in eqn (8), a tensor product of the same geminal, represents a computationally attractive correlated wave function with very few parameters; in the most general case, the AGP scales as $\frac{K(K-1)}{2}$ where $K$ is the number of spin-orbitals.

Why does this 2nd paragraph (shortly after Expression (10) in the paper) say that AGP has a complexity of $O(K^2)$, which contradicts its own sentences for the early paragraph after expression (5) about having binomial complexity?

References

  1. P. Tecmer & K. Boguslawski Phys. Chem. Chem. Phys., 2022,24, 23026-23048 DOI
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    $\begingroup$ I did not read carefully all of this. But one expansion is called APIG and the other APG, so I guess it is not the same ansatz?! Compare for example eq. 4 and 8. $\endgroup$
    – Jakob
    Feb 24 at 8:13
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    $\begingroup$ Linking to the paper and quoting the relevant passage are helpful. But copying large sections of the text as an image can be problematic for a number of reasons. 1. Requires users to scan through an image to find what you are talking about. 2. Images can't be processed by screen readers. 3. Copying too much of an article starts to straddle a line of what is fair use. $\endgroup$
    – Tyberius ♦
    Feb 24 at 16:40
  • $\begingroup$ I agree with Tyberius. I've removed the copy-pasted copyrighted material. You're welcome to type Expressions 10 and 5 in your question if you want to include that part, but screenshots are inappropriate because they're not searchable, they're bulky and load more slowly, they don't load on browsers that don't load images (e.g. terminal-based browsers), and they can't be read by blind people who use screen readers. $\endgroup$
    – Nike Dattani
    Feb 24 at 16:50
  • $\begingroup$ 12 seconds later I see that @Tyberius typed some of the relevant material for you. Thanks so much Tyberius !!! $\endgroup$
    – Nike Dattani
    Feb 24 at 16:51
  • $\begingroup$ @Jakob see this pictorial comparison , I am still checking out how the equations map to the graph connections. Referenced from twitter.com/QPratz_chem/status/1630417794675163140 $\endgroup$
    – kevin
    Feb 28 at 5:12

1 Answer 1

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"Why does the paragraph after expression (10) says that AGP has a complexity of O(K^2) which contradicts its own sentences just after expression (5) about having binomial complexity?

Expression 10 and Expression 5 are about different things. Expression 10 is about AGP (antisymmetrized geminal power) and Expression 5 is about APIG (antisymmetric product of interacting geminals).

After Expression 5, it says that APIG includes $K\choose P$ Slater determinants. After Expression 10, it implies that APG has $K(K-1)/2$ or $K\choose2$ parameters. The key point is that APIG and APG are not the same.

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  • $\begingroup$ I am actually quite curious and I am reading on how eq(1) could be modeled SIMILARLY using APIG an AGP $\endgroup$
    – kevin
    Feb 24 at 17:47
  • $\begingroup$ @kevin is there any part of the original version of this question, as it was written, that was not answered? If not, please click the green checkmark and upvote button. $\endgroup$
    – Nike Dattani
    Feb 24 at 21:59
  • $\begingroup$ Should I ask about the differences between APIG and AGP here, or in a new separate question thread ? $\endgroup$
    – kevin
    Feb 25 at 13:58
  • $\begingroup$ Please say "hello" to me here: chat.stackexchange.com/rooms/119831/… to get started and we can discuss APIG vs APG there. $\endgroup$
    – Nike Dattani
    Feb 25 at 14:00

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