How accurate are the most accurate calculations?

Question

Taking into account the fact that the theory of quantum gravity does not exist and the QED calculations are not possible for most realistic chemical systems, what levels of accuracy can we expect from a theoretical calculation on simple (small) materials? Examples that come to mind are:

Simplest molecular materials: $^3ce{He}$, $^4ce{He}$, $ce{H_2}$.
Simplest periodic systems, i.e. metallic $ce{Li}$.

Nike Dattani · Answer

Exhibit 1: Ground state hyperfine splitting of the H atom:
1420405751767(1) mHz (present most accurate experiment)
142045199        mHz (present most accurate theory)

The error in the theory is due to the difficulty in treating the nuclear structure (2 up quarks + 1 down).
Exhibit 2: Ground state hyperfine splitting of the muonium atom:
4463302780(050) Hz (present most accurate experiment)
4463302880(550) Hz (present most accurate theory)

Why does it agree so well? μ$^+$ is a fundamental particle and therefore has no nuclear structure. QED is the correct theory for describing the interaction between pure electric charges (e$^-$ and μ$^+$). The only QFD (quantum flavordynamics) needed is for the electro-weak interaction between the particles (not for interactions within sub-nuclear particles), and QFD calculations were done here in anticipation of more accurate experiments to come.
Exhibit 3: Ground state hyperfine splitting of the He atom:
6739701177(0016) Hz (present most accurate experiment)
6739699930(1700) Hz (present most accurate theory)

Notice how much harder it is when you add an electron.
Exhibit 5: $Srightarrow P$ transition in the Li atom:
14903.632061014(5003) cm^-1 (present most accurate experiment)
14903.631765(0006670) cm^-1 (present most accurate theory)

Exhibit 6: Ionization energy of the Li atom:
43487.15940(18) cm^-1 (present most accurate experiment)
43487.1590(080) cm^-1 (present most accurate theory)

Exhibit 7: Ionization energy of the Be atom:
76192.64(0060) cm^-1 (present most accurate experiment)
76192.699(007) cm^-1 (present most accurate theory)

Notice that theory is 1 order of magnitude more accurate than experiment!!!
Exhibit 8: Atomization energy of the H$_2$ molecule:
35999.582834(11) cm^-1 (present most accurate experiment)
35999.582820(26) cm^-1 (present most accurate theory)

See here for more info.
Exhibit 9: Fundamental vibration of the H$_2$ molecule:
4161.16632(18) cm^-1 (present most accurate experiment)
4161.16612(90) cm^-1 (present most accurate theory)

See here for HD and D$_2$.

How accurate are the most accurate calculations?

One Answer

Exhibit 1: Ground state hyperfine splitting of the H atom:

Exhibit 2: Ground state hyperfine splitting of the muonium atom:

Exhibit 3: Ground state hyperfine splitting of the He atom:

Exhibit 5: $Srightarrow P$ transition in the Li atom:

Exhibit 6: Ionization energy of the Li atom:

Exhibit 7: Ionization energy of the Be atom:

Exhibit 8: Atomization energy of the H$_2$ molecule:

Exhibit 9: Fundamental vibration of the H$_2$ molecule:

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