Nonadiabatic QED correction to the dissociation energy of the hydrogen   molecule

M. Puchalski; J. Komasa. P. Czachorowski; and K. Pachucki

arXiv:1812.02980·physics.atom-ph·April 8, 2019

Nonadiabatic QED correction to the dissociation energy of the hydrogen molecule

M. Puchalski, J. Komasa. P. Czachorowski, and K. Pachucki

PDF

TL;DR

This paper calculates the nonadiabatic QED correction to hydrogen molecule's dissociation energy with high precision, aligning theory with experimental data, and enhancing the molecule's role in fundamental constant determination.

Contribution

It presents a nonadiabatic QED correction calculation for hydrogen molecule's energy without mass ratio expansion, improving theoretical accuracy.

Findings

01

Achieved 1 MHz accuracy in dissociation energy prediction.

02

Excellent agreement with recent experimental measurements.

03

Enhanced the hydrogen molecule's utility in fundamental physics.

Abstract

The quantum electrodynamic correction to the energy of the hydrogen molecule has been evaluated without expansion in the electron-proton mass ratio. The obtained results significantly improve the accuracy of theoretical predictions reaching the level of 1 MHz for the dissociation energy, in a very good agreement with the parallel measurement [H\"olsch et al., Phys. Rev. Lett. 122, 103002 (2019)]. Molecular hydrogen has thus become a cornerstone of ultraprecise quantum chemistry, which opens perspectives for determination of fundamental physical constants from its spectra.

Tables3

Table 1. Table 1: Expectation values of the operators present in Eq. ( 8 ). Atomic units are used throughout the Table.

Operator	Expectation value
$E$	$- 1.164 025 030 86 (3)$
$⟨ J^{2} ⟩$	$2.518 270 507 19 (12)$
$4 π \sum_{a, x} ⟨ δ^{3} (r_{a x}) ⟩$	$11.346 476 34 (9)$
$4 π ⟨ δ^{3} (r_{23}) ⟩$	$0.202 830 306 (6)$
$\sum_{a, x} {⟨ r_{a x}^{- 3} ⟩}_{ϵ}$	$- 7.191 104 3 (10)$
${⟨ r_{23}^{- 3} ⟩}_{ϵ}$	$0.401 943 51 (6)$
${⟨ r_{01}^{- 3} ⟩}_{ϵ}$	$0.357 215 411 7 (3)$
$\sum_{a, x} {⟨ r_{a x}^{- 4} ⟩}_{ϵ}$	$- 5.712 727 (17)$
$\sum_{(a, x) < (b, y)} ⟨ \frac{{\vec{r}}_{a x}}{r_{a x}^{3}} \cdot \frac{{\vec{r}}_{b y}}{r_{b y}^{3}} ⟩$	$- 0.254 515 18 (18)$

Table 2. Table 2: Convergence of the nonrelativistic energy E 𝐸 E and the Bethe logarithm ln ⁡ k 0 subscript 𝑘 0 \ln k_{0} with the increasing size N 𝑁 N of the naECG basis set. The final uncertainty for ln ⁡ k 0 subscript 𝑘 0 \ln k_{0} is due to numerical inaccuracy of f ( t ) 𝑓 𝑡 f(t) at small t 𝑡 t .

$N$	$E$	$\ln k_{0}$
128	$- 1.164 023 669 155$	$3.016 586 1$
256	$- 1.164 024 987 878$	$3.018 137 0$
512	$- 1.164 025 027 334$	$3.018 258 91$
1024	$- 1.164 025 030 593$	$3.018 301 73$
2048	$- 1.164 025 030 843$	$3.018 303 90$
$\infty$	$- 1.164 025 030 86 (3)$	$3.018 304 9 (15)$

Table 3. Table 3: Theoretical predictions for the dissociation energy budget for the ground level of H 2 . E sec ( 6 ) subscript superscript 𝐸 6 sec E^{(6)}_{\rm sec} is a second order correction due to relativistic BO potential; E FS subscript 𝐸 FS E_{\mathrm{FS}} is the finite nuclear size correction with r p = 0.84087 ( 39 ) subscript 𝑟 𝑝 0.84087 39 r_{p}=0.84087(39) fm Antognini et al. ( 2013 ) . All the energy entries are given in cm − 1 superscript cm 1 \text{cm}^{-1} .

Contribution	$D_{0, 0}$	$D_{0, 1}$	$(0, 1) \to (0, 0)$	Remarks and references
$E^{(2)}$	$36 118.797 746 10 (3)$	$36 000.312 485 66 (2)$	$118.485 260 44 (4)$	naJC; Pachucki and Komasa (2018)
$E^{(4)}$	$- 0.531 215 6 (5)$	$- 0.533 799 2 (5)$	$0.002 583 56 (1)$	naECG; Puchalski et al. (2018), this work
$E^{(5)}$	$- 0.194 910 43 (15)$	$- 0.193 887 7 (11)$	$- 0.001 022 7 (11)$	naECG; Puchalski et al. (2017), this work
$E^{(6)}$	$- 0.002 067 (6)$	$- 0.002 058 (6)$	$- 0.000 008 9$	BO; Puchalski et al. (2016)
$E_{\sec}^{(6)}$	$0.000 009 2$	$0.000 009 1$	$0.000 000 1$	BO; this work
$E^{(7)}$	$0.000 101 (25)$	$0.000 101 (25)$	$0.000 000 5 (1)$	BO; Puchalski et al. (2016, 2017)
$E_{FS}^{(4)}$	$- 0.000 031$	$- 0.000 031$	$- 0.000 000 2$	BO; Puchalski et al. (2016, 2017)
Total	$36 118.069 632 (26)$	$35 999.582 820 (26)$	$118.486 812 7 (11)$
Exp.	$36 118.069 62 (37)$	$35 999.582 894 (25)$	$118.486 8 (1)$	Liu et al. (2009), Cheng et al. (2018), Jennings et al. (1984)
Diff.	$- 0.000 01 (37)$	$0.000 074 (36)$	$0.000 0 (1)$
Exp.	$36 118.069 45 (31)$	$35 999.582 834 (11)$		Altmann et al. (2018), Holsch:19
Diff.	$- 0.000 18 (31)$	$0.000 014 (28)$

Equations61

E (α) = α^{2} E^{(2)} + α^{4} E^{(4)} + α^{5} E^{(5)} + α^{6} E^{(6)} + α^{7} E^{(7)} + \dots,

E (α) = α^{2} E^{(2)} + α^{4} E^{(4)} + α^{5} E^{(5)} + α^{6} E^{(6)} + α^{7} E^{(7)} + \dots,

H = T + V,

H = T + V,

T

T

V

Ψ

Ψ

ψ_{i}

ϕ = r_{01}^{n} e^{- a_{1} r_{01}^{2} - a_{2} r_{02}^{2} - a_{3} r_{03}^{2} - a_{4} r_{12}^{2} - a_{5} r_{13}^{2} - a_{6} r_{23}^{2}} .

ϕ = r_{01}^{n} e^{- a_{1} r_{01}^{2} - a_{2} r_{02}^{2} - a_{3} r_{03}^{2} - a_{4} r_{12}^{2} - a_{5} r_{13}^{2} - a_{6} r_{23}^{2}} .

E^{(5)} =

E^{(5)} =

\displaystyle\ +\frac{m^{2}}{m_{p}^{2}}\biggl{[}\ln\biggl{(}\frac{m_{p}}{m}\biggr{)}+\ln(\alpha^{-2})+4\biggr{]}\biggr{\}}\sum_{a,x}\big{\langle}\delta^{3}(r_{ax})\big{\rangle}+\left(\frac{164}{15}+\frac{14}{3}\,\ln\alpha\right)\,\big{\langle}\delta^{3}(r_{23})\big{\rangle}-2\,E^{(5)}_{\rm H}

E_{H}^{(5)} =

\displaystyle\ +2\,\frac{m^{2}}{m_{p}^{2}}\biggl{(}1+2\,\ln\frac{m_{p}}{m}\biggr{)}\biggr{\}},

ln k_{0} =

ln k_{0} =

ln k_{0}^{H} =

J

J

D

μ

δ E_{strong} =

δ E_{strong} =

=

δ E_{vp} =

δ E_{vp} =

=

⟨ 4 π δ^{3} (r_{ab})⟩ =

⟨ 4 π δ^{3} (r_{ab})⟩ =

\displaystyle\ \times\bigg{\langle}\frac{2}{r_{ab}}(E-V)-\sum_{c}\frac{1}{m_{c}}\vec{p}_{c}\,\frac{1}{r_{ab}}\,\vec{p}_{c}\biggr{\rangle},

\displaystyle\bigg{\langle}\frac{1}{r_{ab}^{3}}\bigg{\rangle}_{\!\epsilon}=

\displaystyle\bigg{\langle}\frac{1}{r_{ab}^{3}}\bigg{\rangle}_{\!\epsilon}=

\displaystyle\ \times\bigg{\langle}\frac{2\,\ln r_{ab}}{r_{ab}}(E-V)-\sum_{c}\frac{1}{m_{c}}\vec{p}_{c}\,\frac{\ln r_{ab}}{r_{ab}}\,\vec{p}_{c}\biggr{\rangle},

\displaystyle\bigg{\langle}\frac{1}{r_{ab}^{4}}\bigg{\rangle}_{\!\epsilon}=

\displaystyle\bigg{\langle}\frac{1}{r_{ab}^{4}}\bigg{\rangle}_{\!\epsilon}=

\displaystyle-2\,(E-V)\,\frac{1}{r_{ab}^{2}}\pm 12\,\pi\,\delta^{3}(r_{ab})\bigg{\rangle}_{\!\epsilon}.

ln k_{0} = \frac{1}{D} \int_{0}^{1} d t \frac{f ( t ) - f _{0} - f _{2} t ^{2}}{t ^{3}}

ln k_{0} = \frac{1}{D} \int_{0}^{1} d t \frac{f ( t ) - f _{0} - f _{2} t ^{2}}{t ^{3}}

f(t)=\biggl{\langle}\vec{J}\,\frac{k}{k+H-E}\,\vec{J}\biggr{\rangle},\qquad t=\frac{1}{\sqrt{1+2\,k}}

f(t)=\biggl{\langle}\vec{J}\,\frac{k}{k+H-E}\,\vec{J}\biggr{\rangle},\qquad t=\frac{1}{\sqrt{1+2\,k}}

f (t) = f_{0} + f_{2} t^{2} + f_{3} t^{3} + (f_{4 l} ln t + f_{4}) t^{4} + O (t^{5}) .

f (t) = f_{0} + f_{2} t^{2} + f_{3} t^{3} + (f_{4 l} ln t + f_{4}) t^{4} + O (t^{5}) .

f_{0} = ⟨ J^{2} ⟩, f_{2} = - 2 D .

f_{0} = ⟨ J^{2} ⟩, f_{2} = - 2 D .

f_{3} =

f_{3} =

f_{4 l} =

f_{4} =

E^{(7)} \approx

E^{(7)} \approx

\displaystyle\ +A_{62}\,\ln^{2}\alpha^{-2}\big{]}+\frac{1}{\pi^{2}}\,B_{50}+\frac{1}{\pi^{3}}\,C_{40}\biggr{\}}-2\,E^{(7)}({\rm H}),

E (H_{2}, IP) = D_{0} (H_{2}) + E (H, IP) - D_{0} (H_{2}^{+})

E (H_{2}, IP) = D_{0} (H_{2}) + E (H, IP) - D_{0} (H_{2}^{+})

\frac{E ( H _{2} , IP )}{E ( H , 2 S - 1 S )} = 1.512 \dots - 1.4 \cdot 1 0^{- 10} r_{p}^{2} / fm^{2} .

\frac{E ( H _{2} , IP )}{E ( H , 2 S - 1 S )} = 1.512 \dots - 1.4 \cdot 1 0^{- 10} r_{p}^{2} / fm^{2} .

δ E (H_{2}, IP)

δ E (H_{2}, IP)

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Full text

Nonadiabatic QED Correction to the Dissociation Energy of the Hydrogen Molecule

Mariusz Puchalski

Faculty of Chemistry, Adam Mickiewicz University, Umultowska 89b, 61-614 Poznań, Poland

Jacek Komasa

Faculty of Chemistry, Adam Mickiewicz University, Umultowska 89b, 61-614 Poznań, Poland

Paweł Czachorowski

Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland

Krzysztof Pachucki

Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland

Abstract

The quantum electrodynamic correction to the energy of the hydrogen molecule has been evaluated without expansion in the electron-proton mass ratio. The obtained results significantly improve the accuracy of theoretical predictions reaching the level of 1 MHz for the dissociation energy, in a very good agreement with the parallel measurement [Hölsch et al., Phys. Rev. Lett. 122, 103002 (2019)]. Molecular hydrogen has thus become a cornerstone of ultraprecise quantum chemistry, which opens perspectives for determination of fundamental physical constants from its spectra.

††preprint: Version 2.0

Introduction – The spectra of hydrogenic atoms are being used for determination of physical constants and for precision tests of the fundamental interactions theory. However, the finite lifetime of excited states makes further progress in accuracy very challenging. It would require, among others, determination of the resonance frequency to at least one part in 10,000 of the observed line width Beyer et al. (2017). In contrast, the hydrogen molecule (H2) has many narrow lines, which in principle can be measured very accurately Liu et al. (2009); Sprecher et al. (2011); Cheng et al. (2012); Niu et al. (2014); Cheng et al. (2018); Altmann et al. (2018). In this work, we demonstrate that they can also be calculated very accurately, namely with 1 MHz uncertainty or better.

Although the hydrogen molecule is one of the simplest molecular systems, the high-precision calculations of its energy levels have been difficult to perform, even in the nonrelativistic limit. The standard Born-Oppenheimer (BO) approximation gives a relative accuracy of the order of $10^{-3}-10^{-4}$ only as a consequence of the omission of the coupling between electrons and nuclei movements. In principle, the finite nuclear mass corrections to the BO potential can be included systematically within the nonadiabatic perturbation theory (NAPT) Pachucki and Komasa (2009). However, evaluation of the higher order terms of the NAPT becomes complicated Pachucki and Komasa (2015). For this reason the direct nonadiabatic methods have recently been developed in which two electrons and two protons are treated on the same footing. This allowed the inaccuracy of the nonrelativistic energy $E^{(2)}$ to be reduced to the limit of $4\times 10^{-12}$ resulting from the uncertainty in the proton mass Pachucki and Komasa (2016).

Regarding subsequent terms in the expansion of energy in the fine structure constant $\alpha$ ,

[TABLE]

the relativistic correction, $E^{(4)}$ , has recently been calculated to a high numerical precision both with direct nonadiabatic treatment Wang and Yan (2018a); Puchalski et al. (2018); Wang and Yan (2018b) and with the NAPT Puchalski et al. (2017); Czachorowski et al. (2018). Moreover, the quantum electrodynamic (QED) $E^{(5)}$ Piszczatowski et al. (2009) and the higher order $E^{(6)}$ Puchalski et al. (2016) corrections had been calculated within the BO approximation only, while $E^{(7)}$ is known approximately from the atomic hydrogen theory Eides et al. (2001). Neglected nonadiabatic effects of the order $O(\alpha^{5})$ had been the largest source of the uncertainty in theoretical predictions, which for the ground state dissociation energy equals to $2\times 10^{-4}\,\text{cm}^{-1}$ (6 MHz) Puchalski et al. (2018).

In this work, we report on calculation of the QED correction $E^{(5)}$ in the framework of the direct nonadiabatic approach, improving the accuracy of theoretical prediction for the dissociation energies by an order of magnitude down to $2.6\times 10^{-5}$ cm*-1* (0.78 MHz), the best ever theoretical prediction for any molecule, which becomes now sensitive to the nuclear charge radii.

Nonrelativistic wave function – The basis of the accurate theoretical predictions is the precise nonrelativistic wave function. In the nonadiabatic approach all particles are treated on an equal footing, and the wave function is an eigenstate of the nonrelativistic H2 Hamiltonian of the form

[TABLE]

where

[TABLE]

The indices 0,1 denote protons of mass $m_{p}$ and 2,3—electrons of mass $m$ . In the center of mass frame the nonrelativistic wave function depends only on the interparticle distances $r_{ij}$ . The most convenient approach Puchalski et al. (2018) to calculate this function is based on a variational principle with the nonadiabatic explicitly correlated Gaussian (naECG) functions. In this approach, the wave function is represented as

[TABLE]

where $P_{i\leftrightarrow j}$ is the particle exchange operator, which accounts for the fact that the ground state H2 wave function $\Psi$ is symmetric with respect to the exchange of nuclear and electronic variables. The spatial functions $\phi_{i}$ in Eq. (6) are naECG functions of the form

[TABLE]

The nonlinear $a$ parameters are optimized variationally and the internuclear coordinate $r_{01}^{n}$ prefactor ensures proper representation of the vibrational part of the wave function. The powers $n$ of this coordinate are restricted to even integers within the range $0-80$ and are generated following the log-normal distribution. The nonrelativistic wave function $\Psi$ has been optimized for several basis sizes to observe the convergence of the nonrelativistic energy and expectation values.

QED correction – The formula for the leading quantum electrodynamic correction $E^{(5)}$ , derived here on the basis of QED theory in agreement with the known formulas for H Eides et al. (2001) and He Pachucki and Sapirstein (2000), is

[TABLE]

where $a=0,1$ , $x=2,3$ , the Bethe logarithm is Drake and Swainson (1990)

[TABLE]

and where

[TABLE]

In all the above formulas the expectation values are taken with the nonrelativistic wave function $\Psi$ . Moreover, the expectation values of singular terms $\langle 1/r^{3}\rangle_{\epsilon}$ are obtained by integration from $\epsilon$ to $\infty$ and subtraction of $\ln\epsilon+\gamma$ , where the symbol $\gamma$ indicates the Euler-Mascheroni constant.

There are certain ambiguities regarding the molecular QED correction of Eq. (8), which need to be explained. The first one is due to the lack of the contact term between protons $\delta^{3}(r_{01})$ . In fact, such a term exists, for example from the strong interaction $V_{\rm strong}$ between protons Wiringa et al. (1995)

[TABLE]

It is of the same order as the electron vacuum polarization correction to the Coulomb interaction between protons

[TABLE]

However, both contributions are totally negligible, because $\langle\delta^{3}(r_{01})\rangle\sim 10^{-50}\,(m\,\alpha)^{3}$ for the ground state of H2.

Another subtle point to be clarified is the proton self-energy correction and the corresponding definition of the proton charge radius. This correction is insignificant for a regular hydrogen atom but non-negligible for muonic hydrogen ( $\mu$ H). So, for consistency with the determination of the proton charge radius $r_{p}$ in $\mu$ H Pohl et al. (2010), we chose to include this effect into the total energy of H2. Following (Pachucki, 1999), we do so in a minimal way, by including only logarithmic terms, and the nonlogarithmic terms are absorbed into the mean square charge radius $r_{p}^{2}$ .

Numerical calculation – QED corrections involve several nontrivial terms: the Bethe logarithm $\ln k_{0}$ , the interparticle Dirac delta $\langle\delta(r_{ab})\rangle$ , and the Araki-Sucher term $\langle 1/r_{ab}^{3}\rangle_{\epsilon}$ . Because the naECG basis does not reproduce the electron-electron and electron-nucleus cusps of the wave function, the two latter terms exhibit a slow convergence if calculated directly from their definitions. Therefore, to increase numerical performance, it is crucial to transform singular operators to a more regular form, whose behavior is less sensitive to the local inaccuracies of the wave function. For this purpose, we generalized known relations Pachucki et al. (2005); Puchalski et al. (2017) and obtained the following expectation value identities with $a,b=0,1,2,3$

[TABLE]

where $+$ is for particles with the same and $-$ with opposite charges respectively. Results for the expectation values extrapolated to the infinite basis size, along with their estimated uncertainty, are presented in Table 1.

Bethe logarithm – Among all the terms in Eq. (8), the calculation of the Bethe logarithm $\ln k_{0}$ is the most complicated one. We express $\ln k_{0}$ in terms of the one-dimensional integral

[TABLE]

with the function $f(t)$ defined as

[TABLE]

which has the following Taylor expansion

[TABLE]

with the first two coefficients

[TABLE]

The integrand in Eq. (20), as a smooth function of $t$ was evaluated at 200 equally spaced points in the range $t\in[0,1]$ , which enabled relative accuracy higher than $10^{-7}$ . In the numerical calculation of $f(t)$ , the resolvent in Eq. (21) was represented in terms of pseudostates of the form $\phi^{\Pi}=\vec{r}_{ab}\,\phi$ for all interparticle coordinates. The nonlinear parameters of $\phi^{\Pi}$ are found by a maximization of $f$ . In the particular case of $t=1$ $(k=0)$ , $f$ can be evaluated analytically using the generalized Thomas-Reiche-Kuhn sum rule Zhou et al. (2006) $\langle\vec{J}\,(H-E)^{-1}\,\vec{J}\rangle=3\,(1+m/m_{p})$ . We used this opportunity to assess the completeness of the pseudostates space and to estimate uncertainties.

For the given size $N$ of the wave function $\Psi$ expansion, the size of pseudostate basis set was chosen as $N^{\prime}=\frac{3}{2}N$ , which appeared to be sufficient for most of the $t$ points. There are also additional factors taken into account for the accurate representation of the resolvent in Eq. (21). The powers of internuclear coordinate $r_{01}$ , analogously to the wave function, are restricted to even integers and are generated randomly for each basis function from the log-normal distribution within the $0-80$ range. However, for small values of $t$ ( $\leq 0.1$ ), due to a cancellation in the numerator of Eq. (20), an additional tuning of the distribution was made and $N^{\prime}=2\,N$ was set to achieve high accuracy. Moreover, in this critical region of small $t$ , the function $f(t)$ was expanded in a power series with $f_{3}$ , $f_{4}$ , and $f_{4l}$ coefficients deduced from the known high- $k$ expansion by Korobov Korobov (2012) ( $\mu^{\prime}=\mu/m$ )

[TABLE]

The higher order expansion terms are obtained from the fit to numerical values of $f(t)$ . As a test, $f_{4}=72.114\,86(7)$ calculated using the above formula, agrees well with $72.0(3)$ obtained from the numerical fit. In order to perform integration in Eq. (20), we use a polynomial interpolation of the integrand for $t>0.1$ , and power expansion for the critical region $t\in[0,0.1]$ .

The convergence of the Bethe logarithm with the increasing size of the naECG basis is shown in Table 2. Six significant figures can be considered stable and the estimated relative accuracy is a half ppm. Our recommended value $\ln k_{0}=3.018\,304\,9(15)$ is consistent with $3.0188$ obtained in the framework of adiabatic approximation Piszczatowski et al. (2009), and the difference between them divided by $\ln k_{0}$ is smaller than the electron-proton mass ratio. An analogous difference of $0.0005$ between adiabatic and nonadiabatic $\ln k_{0}$ has been noted for H ${}_{2}^{+}$ Korobov (2006); Piszczatowski et al. (2009).

Higher order QED – Because of the significant increase in the accuracy of the QED correction achieved in this work, the dominating contribution to the uncertainty comes from the higher order $E^{(7)}$ correction. Currently, an explicit form of this correction is unknown, which prevents its accurate evaluation. Its first estimation, made within the BO approximation framework, was reported in Ref. Puchalski et al. (2016). Here, we account for several additional terms, namely

[TABLE]

and assume a conservative 25% uncertainty. All the coefficients $A$ , $B$ , and $C$ can be found in Ref. Eides et al. (2001), and we use the values for the $1S$ state of H. The dominating term is the one containing $A_{62}=-1$ and inclusion of all the other terms decreases $E^{(7)}$ by about 14%.

Summary – Theoretical predictions for all the known contributions to $D_{0,0}$ and $D_{0,1}$ are assembled in Table 3. By $D_{v,J}$ we denote there the dissociation energy of the hydrogen molecule in the state with the vibrational number $v$ and rotational $J$ .

The nonrelativistic contribution was calculated directly for $D_{0,0}$ and $D_{0,1}$ . All the corrections were calculated for $D_{0,0}$ and separately for the rotational excitation energy using NAPT. However, the QED correction for this excitation energy was calculated within the BO approximation only, but the related uncertainty is small. Finally, the dissociation energy $D_{0,1}$ was obtained as the difference between $D_{0,0}$ and the rotational excitation.

The improved theoretical result for the ground state dissociation energy $D_{0,0}$ of the hydrogen molecule is in very good agreement with the most recent measurements Liu et al. (2009); Altmann et al. (2018), but their uncertainties are an order of magnitude larger. The situation is more intriguing for the dissociation energy $D_{0,1}$ of the first rotationally excited state. Although, our theoretical prediction differs by 2 $\,\sigma$ from the equally accurate recent measurement Cheng et al. (2018), it is in very good agreement with the twice as accurate measurement reported in parallel to this work Holsch:19 , see Table III.

Conclusions – This work concludes efforts to increase the accuracy of theoretical predictions for the dissociation energy of H2, feasible within the framework of the existing theory. Further progress is likely, provided an explicit formula for the $E^{(7)}$ term in the expansion Eq. (1) is found. Currently, the main uncertainty of $25\times 10^{-6}\,\text{cm}^{-1}$ (0.75 MHz) comes from this correction, which for the time being, is estimated using the known atomic hydrogen formula, see Eq. (25). Despite this approximation, the new results for dissociation energies of the hydrogen molecule become the most accurate ever obtained for any molecule.

Regarding the possibility of determination of the Rydberg constant or the proton charge radius, let us point out that in the atomic hydrogen, apart from $1S\text{-}2S$ transition, there is no other narrow transition, and the present charge radius determination relies on an average of many transitions with much larger natural linewidth than the accuracy of individual measurements. The alternative route suggested by Merkt fmerkt is to use the ionization energy of the hydrogen molecule, as a second transition, because its natural width is exactly zero. The determination of the dissociation energy in Refs. Liu et al. (2009); Cheng et al. (2018); Altmann et al. (2018); Holsch:19 is in fact the measurement of the ionization energy $E$$({\rm H}_{2},{\rm IP})$ ,

[TABLE]

which for ortho-H2 amounts to about $124\,357$ cm*-1* Cheng et al. (2018). The ratio with the precisely known $2S$ - $1S$ transition $82\,259$ cm*-1* Parthey et al. (2011) is independent of the Rydberg constant, but depends on the proton charge radius through

[TABLE]

Consequently, one needs to achieve

[TABLE]

accuracy for the ionization energy of H2 to obtain the proton radius with 1% precision. Among the contributions to $E({\rm H}_{2},{\rm IP})$ in Eq. (26), the last two are known with much higher precision Korobov et al. (2017) than required. Therefore, it is only $D_{0}({\rm H}_{2})$ which needs to be improved. This can be achieved by the evaluation of the nonadiabatic $E^{(6)}$ , $E^{(7)}$ in the BO approximation, and the $E^{(8)}$ contribution using the atomic hydrogen theory. Among them, the calculation of $E^{(7)}$ is the most demanding task, since it has not yet been accomplished for helium or for any other system except the hydrogen atom and H ${}_{2}^{+}$ ion Eides et al. (2001); Korobov et al. (2014), but it is feasible using present-day technologies.

Acknowledgments – This research was supported by National Science Center (Poland) Grant No. 2014/15/B/ST4/05022 as well as by a computing grant from Poznań Supercomputing and Networking Center and by PL-Grid Infrastructure.

Bibliography35

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1Beyer et al. (2017) A. Beyer, L. Maisenbacher, A. Matveev, R. Pohl, K. Khabarova, A. Grinin, T. Lamour, D. C. Yost, T. W. Hänsch, N. Kolachevsky, et al., Science 358 , 79 (2017).
2Liu et al. (2009) J. Liu, E. J. Salumbides, U. Hollenstein, J. C. J. Koelemeij, K. S. E. Eikema, W. Ubachs, and F. Merkt, J. Chem. Phys. 130 , 174306 (2009).
3Sprecher et al. (2011) D. Sprecher, C. Jungen, W. Ubachs, and F. Merkt, Faraday Discuss. 150 , 51 (2011).
4Cheng et al. (2012) C.-F. Cheng, Y. R. Sun, H. Pan, J. Wang, A.-W. Liu, A. Campargue, and S.-M. Hu, Phys. Rev. A 85 , 024501 (2012).
5Niu et al. (2014) M. Niu, E. Salumbides, G. Dickenson, K. Eikema, and W. Ubachs, J. Mol. Spectrosc. 300 , 44 (2014).
6Cheng et al. (2018) C. Cheng, J. Hussels, M. Niu, H. L. Bethlem, K. S. E. Eikema, E. J. Salumbides, W. Ubachs, M. Beyer, N. J. Hölsch, J. A. Agner, et al., Phys. Rev. Lett. 121 , 013001 (2018).
7Altmann et al. (2018) R. K. Altmann, L. S. Dreissen, E. J. Salumbides, W. Ubachs, and K. S. E. Eikema, Phys. Rev. Lett. 120 , 043204 (2018).
8Pachucki and Komasa (2009) K. Pachucki and J. Komasa, J. Chem. Phys. 130 , 164113 (2009).