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In the paper "High Precision Theory of Atomic Helium", Drake lists the then best-known ground state energy of the Schrödinger equation for helium to 22 digits as: $$\lambda_0 \approx -2.90372437703411959382$$ in atomic units.

Korobov gives the ground state to 24 digits as: $$\lambda_0 \approx -2.903724377034119598311159.$$

In 2007, we get ~45 digits with Nakashima and Nakasuji's calculation of: $$\lambda_0 = −2.90372437703411959831115924519440444669690537.$$

I am looking for the highest precision calculation of the helium ground state, but with Nakashima and Nakasuji's paper, the trail goes cold. The closest citing article is again from Korobov, which suggests perhaps not all of Nakasuji's digits are correct.

What is the most accurate computation of the helium ground state?

I am also interested in the first excited state.

Korobov lists the first excited state at $\lambda_1 \approx -2.145974046054417415805028975461921$, but is this the best calculation?

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The 2006 variational calculation by Schwartz is lower (more accurate) than Nakashima & Nakatsuji's 2007 energy:

2006 Schwartz:              -2.903724377034119598311159245194404446696925309838
2007 Nakashima & Nakatsuji: -2.90372437703411959831115924519440444669690537

The lowest variational upper bounds ground state energies for the first 6 elements, based on the non-relativistic Schrödinger equation are given in Table 1 of my paper about the carbon atom, as well as the relevant citations.

For the first excited state you have referred to a paper by Korobov that was published in 2018, and has been cited 10 times according to Google Scholar, but none of those papers were about excited states of He, which is enough for me to conclude that no better calculation has been done since 2018 (there is only a very, very, small number of people that are interested in anything smaller than a pico-Hartree, so if anyone were do better than what was state-of-the-art in 2018 they would have cited the Korobov paper).

Keep in mind that QED corrections to the Schrödinger equation have errors as big as 0.00001 in the units that you and I are using to present these numbers, so the differences between the numbers for the ground state energies that you have given by Drake (1999), Korobov (2002), Nakashima & Nakatsuji (2007) and the value I've given by Schwartz (2006), is of no practical relevance.

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  • $\begingroup$ Good point on the physics, but perhaps I should clarify that I'm interested in this as a pure math problem. $\endgroup$ – user14717 Sep 18 at 12:35
  • $\begingroup$ Taking old physics problems and enjoying them as pure math problems is of practical relevance . . . to mathletes . . . $\endgroup$ – user14717 Sep 18 at 12:56
  • $\begingroup$ What is the pure math problem here? $\endgroup$ – Nike Dattani Sep 18 at 14:46
  • $\begingroup$ The algorithms for recovering eigenvalues from Sturm-Liouville PDEs. $\endgroup$ – user14717 Sep 18 at 14:47
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    $\begingroup$ When dealing with the 20th digit and beyond, the fact that one author has an energy slightly more accurate than another, might have nothing to do with the algorithm and a lot to do with how many years of CPU time they burned. Here there is also a matter of using different basis sets. Schwartz used Hylleraas+Log while others just used Hylleraas, but both of these are catered towards the He atom problem and wouldn't be relevant for some different Sturm-Liouville PDE. $\endgroup$ – Nike Dattani Sep 18 at 14:53

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