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In most textbooks the precise Schrödinger euqation of a molecule is given (and the "beautiful thing" stops here), then born oppenheimer approximation is made, then layers of other approximations are made, the reason behind is to make the modal tractable with contemporary computuing power.

But are there examples where none of the major approximations are made at all and a "precise numerical solution" (finite element like? - I'm not sure if QM problems can be solved like this) is made? After a lot of search I never found such examples, I am not even sure they exists.

The reason for this curiosity is:

(1) It is very amazing if a solution without the major approximations is made and the result is indeed closer to experiment result than the predictios with approximations.

(2) Although the cost is very high and "doesn't worth it", but at least it can be done for one or a few times (in science history), not never. Although QM is verified in almost every experiments, but a direct prediction of molecule properties without any major approximations is more convincing since you witness the rightness of QM in this situation, not just "know it will work but can't try".

(3) At least it can be done on the most simple molecule, e.g. di-hydrogen molecule (not sure whether this is a trivial case, if so, then more complexer one is considered instead), won't the most powerful computer today give a precise prediction on these simple molecules?

Note:

A harder version is the relativisic prediction which is based on Dirac's equation. This one make sense on molecule since a lot of precision is lost without relativisic effects. But maybe only heavier elements can show the difference so this is not easy to setup for the computing power today, so it is not a major concern here. An even harder version is based on quantum electrodynamics which is even more amazing but also even more intractable even for the most simlest molecule I guess.

Updated

To make the question more clear:

(1) The title is changed, the old one may be misleading

(2) The focus of OP is the prediction method, not the result, although if a prediction method as described is used, the result should be very precise.

(3) The focus of OP is a general prediction of molecule which at least contains two atoms with a few electrons. To be general, it should give the "precise numerical prediction" of both the (multi particle) eigen functions (as listed in the "pure and beautiful" Schrödinger euqation mentioned in OP) and eigen values, not only some parameters (e.g. only eigen value) that can be measured in experiments. Here "precise numerical prediction" means a nemerical method that can get any degree of precision given enough computing effort (OP is not sure whether such method exist which is also a concern of OP).

(4) High precision is indeed not very important in OP. For example, some QED or RQM may give some prediction about "some parameters" mentioned in (3) with very high precision, but that is not a "general prediction" as described in (3). OP already mentioned to make such "general prediction" QED and RQM could be out of reach with today's computing power. A "general prediction" based on the multi particle Schrödinger euqation, without all the approximation methods, is very enough.

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  • $\begingroup$ Related: mattermodeling.stackexchange.com/a/555/5. But +1 and welcome to the site! Thank you for contributing your question here and we hope to see much more of you in the future!!! $\endgroup$ Dec 7 '20 at 1:22
  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Tyberius
    Dec 10 '20 at 4:33
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I wrote an answer to a similar question in the past, but focused in that question only on the state-of-the-art ultra-high precision calculations on atoms and the three most common isotopologues of $\ce{H_2}$.

I will first repeat those here:

Atomization energy of the H$_2$ molecule:

35999.582834(11) cm^-1 (present most accurate experiment)
35999.582820(26) cm^-1 (present most accurate calculation)

See here for more info.

Fundamental vibration of the H$_2$ molecule:

4161.16632(18) cm^-1 (present most accurate experiment)
4161.16612(90) cm^-1 (present most accurate calculation)

See here for HD and D$_2$.

Now I suppose you want to know about molecules with more electrons or more nucleons? Well you came to the right place.

$\ce{HeH^+}$: 2 electrons, 3-5 nucleons, 2 nuclei

$\ce{LiH^+}$: 3 electrons, 4 nucleons, 2 nuclei

$\ce{Li_2}$: 6 electrons, 6-8 nucleons, 2 nuclei

$\ce{BeH}$: 5 electrons, 9-12 nucleons, 2 nuclei

$\ce{BH}$: 6 electrons, 11 nucleons, 2 nuclei

"The results shown in this work represent a year of continuous calculations with the use of 6 processors/24 cores quadcore Intel Xeon 2.67 GHz or quadcore AMD Opteron 2.2 GHz"

$\ce{H_2O}$: 10 electrons (8 correlated), 3 nuclei

$\ce{O_3}$: 24 electrons (18 correlated), 3 nuclei

$\ce{He_{60}}$: 120 electrons, 60 nuclei (Helium Buckyball / Fullerene)

  • You asked about "finite element methods" but most of the above calculations, even for H$_2$ use basis set methods instead, However an extremely small number of people do solve the many-electron Schroedinger equation "on a grid", one of them being our very own Susi Lehtola who has written an entire review on such numerical calculations on atoms and diatomic molecules, and another one of them being Hiroshi Nakatsuji who famously once calculated the ground state electronic energy of the He atom to about 40 digits of precision. Even he uses basis set methods for larger systems though, for example in in this paper where he calculated energies for $\ce{He_{60}}$. You just can't do the explicitly correlated integrals efficiently for a 60-atom system, but if someone does decide to spend their entire year's CPU allocation on it, then I'm sure Nakatsuji would try to obtain the electronic Schroedinger equation using his famous explicitly correlated methods.
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