Fundamental transitions and ionization energies of the hydrogen   molecular ions at the few ppt level

Vladimir I. Korobov; Laurent Hilico; Jean-Philippe Karr

arXiv:1703.07972·physics.atom-ph·June 14, 2017

Fundamental transitions and ionization energies of the hydrogen molecular ions at the few ppt level

Vladimir I. Korobov, Laurent Hilico, Jean-Philippe Karr

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TL;DR

This paper improves calculations of ionization energies and vibrational transitions for hydrogen molecular ions using NRQED, achieving unprecedented fractional uncertainties at the 10^-12 level.

Contribution

It introduces second-order vibrational corrections and evaluates higher-order effects, significantly enhancing the precision of theoretical predictions.

Findings

01

Fractional uncertainty reduced to 7×10^-12 for transitions.

02

Ionization energies calculated with 4×10^-12 uncertainty.

03

Enhanced theoretical framework for molecular ion spectroscopy.

Abstract

We calculate ionization energies and fundamental vibrational transitions for H $_{2}^{+}$ , D $_{2}^{+}$ , and HD $^{+}$ molecular ions. The NRQED expansion for the energy in terms of the fine structure constant $α$ is used. Previous calculations of orders $m α^{6}$ and $m α^{7}$ are improved by including second-order contributions due to the vibrational motion of nuclei. Furthermore, we evaluate the largest corrections at the order $m α^{8}$ . That allows to reduce the fractional uncertainty to the level of $7 \cdot 1 0^{- 12}$ for fundamental transitions and to $4 \cdot 1 0^{- 12}$ for the ionization energies.

Tables5

Table 1. Table 1: Summary of contributions at order m α 6 𝑚 superscript 𝛼 6 m\alpha^{6} to the fundamental transitions in H 2 + superscript subscript H 2 \mbox{H}_{2}^{+} , D 2 + superscript subscript D 2 \mbox{D}_{2}^{+} , and HD + superscript HD \mbox{HD}^{+} (in kHz). The first four contributions are defined as written in Eq. (1) of KorobovPRA08 , Δ E r e l − e l Δ subscript 𝐸 𝑟 𝑒 𝑙 𝑒 𝑙 \Delta E_{rel-el} is the electronic contribution from Eq. ( 8a ), Δ E r e l − v b Δ subscript 𝐸 𝑟 𝑒 𝑙 𝑣 𝑏 \Delta E_{rel-vb} is the newly obtained vibrational contribution from Eq. ( 8b ). The last contribution is the sum of the recoil and radiative-recoil corrections of order m α 6 ( m / M ) 𝑚 superscript 𝛼 6 𝑚 𝑀 m\alpha^{6}(m/M) (see text).

	$H_{2}^{+}$	$D_{2}^{+}$	${HD}^{+}$
$Δ E_{1 l o o p - SE}$	$- 1881.2$	$- 1362.3$	$- 1647.0$
$Δ E_{a n o m}$	$21.2$	$15.3$	$18.5$
$Δ E_{VP}$	$- 66.3$	$- 48.0$	$- 58.0$
$Δ E_{2 l o o p}$	$- 55.9$	$- 40.5$	$- 48.9$
$Δ E_{r e l - e l}$	$- 15.1$	$- 10.5$	$- 13.0$
$Δ E_{r e l - v b}$	$44.6$	$32.2$	$39.0$
$Δ E_{r e c o i l}$	$0.75 (3)$	$0.27 (1)$	$0.49 (2)$
$Δ E_{t o t}$	$- 1952.0 (1)$	$- 1413.4 (1)$	$- 1708.9 (1)$

Table 2. Table 2: Summary of contributions at order m α 7 𝑚 superscript 𝛼 7 m\alpha^{7} to the fundamental transitions in H 2 + superscript subscript H 2 \mbox{H}_{2}^{+} , D 2 + superscript subscript D 2 \mbox{D}_{2}^{+} , and HD + superscript HD \mbox{HD}^{+} (in kHz). The first two contributions are the one-loop self-energy and vacuum polarization corrections, which include the vibrational contribution (see text). The last three contributions are defined in Eqs. (23)–(25) of KorobovPRA14 .

	$H_{2}^{+}$	$D_{2}^{+}$	${HD}^{+}$
$Δ E_{1 l o o p - SE}$	$109.0 (1)$	$78.8 (1)$	$95.4 (1)$
$Δ E_{VP}$	$2.8$	$2.0$	$2.4$
$Δ E_{WK}$	$- 0.08$	$- 0.06$	$- 0.07$
$Δ E_{2 l o o p}$	$10.1$	$7.3$	$8.9$
$Δ E_{3 l o o p}$	$- 0.06$	$- 0.05$	$- 0.05$
$Δ E_{t o t}$	$121.8 (1)$	$88.1 (1)$	$106.4 (1)$

Table 3. Table 3: Summary of contributions at order m α 8 𝑚 superscript 𝛼 8 m\alpha^{8} to the fundamental transitions in H 2 + superscript subscript H 2 \mbox{H}_{2}^{+} , D 2 + superscript subscript D 2 \mbox{D}_{2}^{+} , and HD + superscript HD \mbox{HD}^{+} (in kHz).

	$H_{2}^{+}$	$D_{2}^{+}$	${HD}^{+}$
$Δ E_{2 l o o p}$	$- 1.34 (21)$	$- 0.97 (15)$	$- 1.17 (18)$
$Δ E_{1 l o o p - SE}$	$- 0.97 (48)$	$- 0.70 (35)$	$- 0.85 (42)$
$Δ E_{VP}$	$- 0.017$	$- 0.013$	$- 0.015$
$Δ E_{t o t}$	$- 2.3 (5)$	$- 1.7 (4)$	$- 2.0 (5)$

Table 4. Table 4: Fundamental transition frequencies ν 01 subscript 𝜈 01 \nu_{01} for H 2 + superscript subscript H 2 \mbox{H}_{2}^{+} , D 2 + superscript subscript D 2 \mbox{D}_{2}^{+} , and HD + superscript HD \mbox{HD}^{+} molecular ions (in kHz). CODATA14 recommended values of constants. The first error is the theoretical uncertainty, the second error is due to the uncertainty in mass ratios.

	$H_{2}^{+}$	$D_{2}^{+}$	${HD}^{+}$
$ν_{n r}$	$65 687 511 047.0$	$47 279 387 818.4$	$57 349 439 952.4$
$ν_{α^{2}}$	$1091 040.5$	$795 376.3$	$958 151.7$
$ν_{α^{3}}$	$- 276 545.1$	$- 200 278.0$	$- 242 126.3$
$ν_{α^{4}}$	$- 1952.0 (1)$	$- 1413.4 (1)$	$- 1708.9 (1)$
$ν_{α^{5}}$	$121.8 (1)$	$88.1 (1)$	$106.4 (1)$
$ν_{α^{6}}$	$- 2.3 (5)$	$- 1.7 (4)$	$- 2.0 (5)$
$ν_{t o t}$	$65 688 323 710.1 (5) (29)$	$47 279 981 589.8 (4) (8)$	$57 350 154 373.4 (5) (17)$

Table 5. Table 5: Ionization energies E I subscript 𝐸 𝐼 E_{I} for H 2 + superscript subscript H 2 \mbox{H}_{2}^{+} , D 2 + superscript subscript D 2 \mbox{D}_{2}^{+} , and HD + superscript HD \mbox{HD}^{+} molecular ions (in cm -1 ). CODATA14 recommended values of constants. The error is the theoretical uncertainty. The error due to the uncertainty in mass ratio is below 10 − 7 superscript 10 7 10^{-7} cm -1 .

	$H_{2}^{+}$	$D_{2}^{+}$	${HD}^{+}$
$E_{I, n r}$	$131 056.875 7465$	$131 418.947 7041$	$131 223.436 2578$
$E_{I, α^{2}}$	$1.599 4995$	$1.604 8306$	$1.601 9142$
$E_{I, α^{3}}$	$- 0.350 9300$	$- 0.352 5527$	$- 0.351 6791$
$E_{I, α^{4}}$	$- 0.002 4774 (1)$	$- 0.002 4897 (1)$	$- 0.002 4831 (1)$
$E_{I, α^{5}}$	$0.000 1569 (1)$	$0.000 1576 (1)$	$0.000 1409 (1)$
$E_{I, α^{6}}$	$- 0.000 0021 (6)$	$- 0.000 0021 (6)$	$- 0.000 0021 (6)$
$E_{I, t o t}$	$131 058.121 9937 (6)$	$131 420.197 6480 (6)$	$131 224.684 1650 (6)$

Equations46

Ψ^{BO} = ϕ_{el} (r; R) χ_{BO} (R)

Ψ^{BO} = ϕ_{el} (r; R) χ_{BO} (R)

[H_{el} - E_{el} (R)] ϕ_{el} = 0,

[H_{el} - E_{el} (R)] ϕ_{el} = 0,

(H_{v b} - E_{0}) χ_{BO} = [- \frac{\nabla _{R}^{2}}{2 μ _{n}} + E_{el} (R) - E_{0}] χ_{BO} = 0,

(H_{v b} - E_{0}) χ_{BO} = [- \frac{\nabla _{R}^{2}}{2 μ _{n}} + E_{el} (R) - E_{0}] χ_{BO} = 0,

\begin{array}[]{@{}l}\displaystyle H^{(6)}=\frac{p^{6}}{16m^{5}}+\frac{[\boldsymbol{\nabla}V]^{2}}{8m^{3}}-\frac{3\pi}{16m^{4}}\Bigl{\{}p^{2}\rho+\rho\,p^{2}\Bigr{\}}\\[5.69054pt] \displaystyle\hskip 28.45274pt+\frac{5}{128m^{4}}\left(p^{4}V+Vp^{4}\right)-\frac{5}{64m^{4}}\left(p^{2}Vp^{2}\right),\end{array}

\begin{array}[]{@{}l}\displaystyle H^{(6)}=\frac{p^{6}}{16m^{5}}+\frac{[\boldsymbol{\nabla}V]^{2}}{8m^{3}}-\frac{3\pi}{16m^{4}}\Bigl{\{}p^{2}\rho+\rho\,p^{2}\Bigr{\}}\\[5.69054pt] \displaystyle\hskip 28.45274pt+\frac{5}{128m^{4}}\left(p^{4}V+Vp^{4}\right)-\frac{5}{64m^{4}}\left(p^{2}Vp^{2}\right),\end{array}

Δ E_{B} = ⟨ H_{B} Q (E_{el} - H_{el})^{- 1} Q H_{B} ⟩ .

Δ E_{B} = ⟨ H_{B} Q (E_{el} - H_{el})^{- 1} Q H_{B} ⟩ .

H_{B} = - \frac{p ^{4}}{8 m ^{3}} + \frac{π ρ}{2 m ^{2}} + H_{B}^{so},

H_{B} = - \frac{p ^{4}}{8 m ^{3}} + \frac{π ρ}{2 m ^{2}} + H_{B}^{so},

\begin{array}[]{@{}l}\mathcal{E}_{rc}^{(6)}(R)=\alpha^{4}\Bigl{[}\Delta E_{B}(R)+\left\langle H^{(6)}\right\rangle(R)\Bigr{]}.\end{array}

\begin{array}[]{@{}l}\mathcal{E}_{rc}^{(6)}(R)=\alpha^{4}\Bigl{[}\Delta E_{B}(R)+\left\langle H^{(6)}\right\rangle(R)\Bigr{]}.\end{array}

Δ E_{e l}^{(6)} = ⟨ χ_{BO} ∣ E_{r c}^{(6)} (R) ∣ χ_{BO} ⟩ .

Δ E_{e l}^{(6)} = ⟨ χ_{BO} ∣ E_{r c}^{(6)} (R) ∣ χ_{BO} ⟩ .

\begin{array}[]{@{}l}\displaystyle\Delta E^{(6)}_{vb}=\\[5.69054pt] \displaystyle\hskip 14.22636pt\left\langle\chi_{\rm BO}|\mathcal{E}_{B}(R)Q^{\prime}(E_{0}\!-\!H_{vb})^{-1}Q^{\prime}\mathcal{E}_{B}(R)|\chi_{\rm BO}\right\rangle,\end{array}

\begin{array}[]{@{}l}\displaystyle\Delta E^{(6)}_{vb}=\\[5.69054pt] \displaystyle\hskip 14.22636pt\left\langle\chi_{\rm BO}|\mathcal{E}_{B}(R)Q^{\prime}(E_{0}\!-\!H_{vb})^{-1}Q^{\prime}\mathcal{E}_{B}(R)|\chi_{\rm BO}\right\rangle,\end{array}

\begin{array}[]{@{}l}\displaystyle\Delta E^{(7)}_{el}=\left\langle\chi_{\rm ad}|\mathcal{E}_{1loop-{\rm SE}}^{(7)}(R)|\chi_{\rm ad}\right\rangle,\\[5.69054pt] \displaystyle\Delta E^{(7)}_{vb}=\left\langle\chi_{\rm ad}|\mathcal{E}_{B}(R)Q^{\prime}(E_{0}\!-\!H_{vb})^{-1}Q^{\prime}\mathcal{E}^{(5)}_{\rm SE}(R)|\chi_{\rm ad}\right\rangle,\end{array}

\begin{array}[]{@{}l}\displaystyle\Delta E^{(7)}_{el}=\left\langle\chi_{\rm ad}|\mathcal{E}_{1loop-{\rm SE}}^{(7)}(R)|\chi_{\rm ad}\right\rangle,\\[5.69054pt] \displaystyle\Delta E^{(7)}_{vb}=\left\langle\chi_{\rm ad}|\mathcal{E}_{B}(R)Q^{\prime}(E_{0}\!-\!H_{vb})^{-1}Q^{\prime}\mathcal{E}^{(5)}_{\rm SE}(R)|\chi_{\rm ad}\right\rangle,\end{array}

\begin{array}[]{@{}l}\displaystyle\mathcal{E}_{\rm SE}^{(5)}(R)=\alpha^{3}\frac{4}{3}\left[\ln{\frac{1}{\alpha^{2}}}-{\beta(R)}+\frac{5}{6}-\frac{3}{8}\right]\\[8.53581pt] \displaystyle\hskip 85.35826pt\times\left\langle Z_{1}\delta(\mathbf{r}_{1})\!+\!Z_{2}\delta(\mathbf{r}_{2})\right\rangle\end{array}

\begin{array}[]{@{}l}\displaystyle\mathcal{E}_{\rm SE}^{(5)}(R)=\alpha^{3}\frac{4}{3}\left[\ln{\frac{1}{\alpha^{2}}}-{\beta(R)}+\frac{5}{6}-\frac{3}{8}\right]\\[8.53581pt] \displaystyle\hskip 85.35826pt\times\left\langle Z_{1}\delta(\mathbf{r}_{1})\!+\!Z_{2}\delta(\mathbf{r}_{2})\right\rangle\end{array}

\begin{array}[]{@{}l}\displaystyle E_{2loop}^{(8)}=\left(\frac{1}{\pi}\right)^{2}\!\frac{(Z\alpha)^{6}}{n^{3}}\Bigl{[}B_{63}L^{3}(Z\alpha)\!+\!B_{62}L^{2}(Z\alpha)\!\\[8.53581pt] \displaystyle\hskip 85.35826pt+B_{61}L(Z\alpha)+B_{60}\Bigr{]},\end{array}

\begin{array}[]{@{}l}\displaystyle E_{2loop}^{(8)}=\left(\frac{1}{\pi}\right)^{2}\!\frac{(Z\alpha)^{6}}{n^{3}}\Bigl{[}B_{63}L^{3}(Z\alpha)\!+\!B_{62}L^{2}(Z\alpha)\!\\[8.53581pt] \displaystyle\hskip 85.35826pt+B_{61}L(Z\alpha)+B_{60}\Bigr{]},\end{array}

Δ E (1 S) \approx \frac{( Z α ) ^{6}}{π ^{2}} [- 282 - 62 + 476 - 61] .

Δ E (1 S) \approx \frac{( Z α ) ^{6}}{π ^{2}} [- 282 - 62 + 476 - 61] .

\begin{array}[]{@{}l}\displaystyle Z^{6}B_{63}=-\frac{8}{27}\,Z^{3}\pi\bigl{\langle}\delta(\mathbf{r}_{1})\!+\!\delta(\mathbf{r}_{2})\bigr{\rangle},\\[7.11317pt] \displaystyle Z^{6}B_{62}=\frac{1}{9}\left\langle\boldsymbol{\nabla}^{2}V\>Q(E_{0}-H)^{-1}Q\>\boldsymbol{\nabla}^{2}V\right\rangle_{\rm fin}\\[4.2679pt] \hskip 56.9055pt\displaystyle+\frac{1}{18}\left\langle\boldsymbol{\nabla}^{4}V\right\rangle_{\rm fin}\\[5.69054pt] \hskip 39.83385pt\displaystyle+\frac{16}{9}\left[\frac{31}{15}+2\ln{2}\right]\,Z^{3}\pi\bigl{\langle}\delta(\mathbf{r}_{1})\!+\!\delta(\mathbf{r}_{2})\bigr{\rangle},\end{array}

\begin{array}[]{@{}l}\displaystyle Z^{6}B_{63}=-\frac{8}{27}\,Z^{3}\pi\bigl{\langle}\delta(\mathbf{r}_{1})\!+\!\delta(\mathbf{r}_{2})\bigr{\rangle},\\[7.11317pt] \displaystyle Z^{6}B_{62}=\frac{1}{9}\left\langle\boldsymbol{\nabla}^{2}V\>Q(E_{0}-H)^{-1}Q\>\boldsymbol{\nabla}^{2}V\right\rangle_{\rm fin}\\[4.2679pt] \hskip 56.9055pt\displaystyle+\frac{1}{18}\left\langle\boldsymbol{\nabla}^{4}V\right\rangle_{\rm fin}\\[5.69054pt] \hskip 39.83385pt\displaystyle+\frac{16}{9}\left[\frac{31}{15}+2\ln{2}\right]\,Z^{3}\pi\bigl{\langle}\delta(\mathbf{r}_{1})\!+\!\delta(\mathbf{r}_{2})\bigr{\rangle},\end{array}

\begin{array}[]{@{}l}\displaystyle Z^{6}B_{61}=-2\Bigl{[}\frac{1}{9}\left\langle\boldsymbol{\nabla}^{2}V\>Q(E_{0}-H)^{-1}Q\>\boldsymbol{\nabla}^{2}V\right\rangle_{\rm fin}\\[8.53581pt] \displaystyle\hskip 34.1433pt+\frac{1}{18}\left\langle\boldsymbol{\nabla}^{4}V\right\rangle_{\rm fin}\Bigr{]}\ln{2}+\frac{4}{3}N(R)\\[8.53581pt] \displaystyle\hskip 34.1433pt+\frac{19}{135}\left\langle\boldsymbol{\nabla}^{2}V\>Q(E_{0}-H)^{-1}Q\>\boldsymbol{\nabla}^{2}V\right\rangle_{\rm fin}\\[8.53581pt] \displaystyle\hskip 34.1433pt+\frac{19}{270}\left\langle\boldsymbol{\nabla}^{4}V\right\rangle_{\rm fin}+\frac{1}{24}\left\langle 2\mathrm{i}\sigma^{ij}p^{i}\boldsymbol{\nabla}^{2}Vp^{j}\right\rangle\\[11.38109pt] \displaystyle\hskip 34.1433pt+\Bigl{[}\frac{48781}{64800}\!+\!\frac{2027\pi^{2}}{864}\!+\!\frac{56}{27}\ln{2}\!-\!\frac{2\pi^{2}}{3}\ln{2}\!\\[8.53581pt] \displaystyle\hskip 42.67912pt+\!8\ln^{2}{2}\!+\!\zeta(3)\Bigr{]}Z^{3}\pi\bigl{\langle}\delta(\mathbf{r}_{2})\!+\!\delta(\mathbf{r}_{2})\bigr{\rangle}.\end{array}

\begin{array}[]{@{}l}\displaystyle Z^{6}B_{61}=-2\Bigl{[}\frac{1}{9}\left\langle\boldsymbol{\nabla}^{2}V\>Q(E_{0}-H)^{-1}Q\>\boldsymbol{\nabla}^{2}V\right\rangle_{\rm fin}\\[8.53581pt] \displaystyle\hskip 34.1433pt+\frac{1}{18}\left\langle\boldsymbol{\nabla}^{4}V\right\rangle_{\rm fin}\Bigr{]}\ln{2}+\frac{4}{3}N(R)\\[8.53581pt] \displaystyle\hskip 34.1433pt+\frac{19}{135}\left\langle\boldsymbol{\nabla}^{2}V\>Q(E_{0}-H)^{-1}Q\>\boldsymbol{\nabla}^{2}V\right\rangle_{\rm fin}\\[8.53581pt] \displaystyle\hskip 34.1433pt+\frac{19}{270}\left\langle\boldsymbol{\nabla}^{4}V\right\rangle_{\rm fin}+\frac{1}{24}\left\langle 2\mathrm{i}\sigma^{ij}p^{i}\boldsymbol{\nabla}^{2}Vp^{j}\right\rangle\\[11.38109pt] \displaystyle\hskip 34.1433pt+\Bigl{[}\frac{48781}{64800}\!+\!\frac{2027\pi^{2}}{864}\!+\!\frac{56}{27}\ln{2}\!-\!\frac{2\pi^{2}}{3}\ln{2}\!\\[8.53581pt] \displaystyle\hskip 42.67912pt+\!8\ln^{2}{2}\!+\!\zeta(3)\Bigr{]}Z^{3}\pi\bigl{\langle}\delta(\mathbf{r}_{2})\!+\!\delta(\mathbf{r}_{2})\bigr{\rangle}.\end{array}

N (1 S) = 17.8556720362 (1),

N (1 S) = 17.8556720362 (1),

\begin{array}[]{@{}l}\displaystyle\Delta E^{(8)}_{2loop}=\alpha^{6}\left[B_{63}L^{3}(\alpha)\!+\!B_{62}L^{2}(\alpha)\!+\!B_{61}L(\alpha)\!+\!B_{60}\right]\\[8.53581pt] \displaystyle\hskip 42.67912pt\approx 37.0-17.3-52.9+7.8.\end{array}

\begin{array}[]{@{}l}\displaystyle\Delta E^{(8)}_{2loop}=\alpha^{6}\left[B_{63}L^{3}(\alpha)\!+\!B_{62}L^{2}(\alpha)\!+\!B_{61}L(\alpha)\!+\!B_{60}\right]\\[8.53581pt] \displaystyle\hskip 42.67912pt\approx 37.0-17.3-52.9+7.8.\end{array}

\begin{array}[]{@{}l}\displaystyle\Delta\nu^{(8)}_{2loop}=\alpha^{6}\left[B_{63}^{\nu}L^{3}(\alpha)\!+\!B_{62}^{\nu}L^{2}(\alpha)\!+\!B_{61}^{\nu}L(\alpha)\!+\!B_{60}^{\nu}\right]\\[8.53581pt] \displaystyle\hskip 42.67912pt\approx 0.97-1.68-0.84+0.21,\end{array}

\begin{array}[]{@{}l}\displaystyle\Delta\nu^{(8)}_{2loop}=\alpha^{6}\left[B_{63}^{\nu}L^{3}(\alpha)\!+\!B_{62}^{\nu}L^{2}(\alpha)\!+\!B_{61}^{\nu}L(\alpha)\!+\!B_{60}^{\nu}\right]\\[8.53581pt] \displaystyle\hskip 42.67912pt\approx 0.97-1.68-0.84+0.21,\end{array}

E_{1 l oo p}^{(8)} = \frac{α ^{6}}{π n ^{3}} Z^{7} [A_{71} ln (Z α)^{- 2} + A_{70}],

E_{1 l oo p}^{(8)} = \frac{α ^{6}}{π n ^{3}} Z^{7} [A_{71} ln (Z α)^{- 2} + A_{70}],

Δ μ_{p} / μ_{p} = 1.5 \times 1 0^{- 11} .

Δ μ_{p} / μ_{p} = 1.5 \times 1 0^{- 11} .

\begin{array}[]{@{}l}\displaystyle\nu(\mbox{H}_{2}^{+})=\nu_{0}(\mbox{H}_{2}^{+})+{\textstyle\frac{\Delta R_{\infty}}{R_{\infty}}}\,\nu_{0}(\mbox{H}_{2}^{+})+2(R_{\infty}c)\,\times\\[5.69054pt] \displaystyle\hskip 14.22636pt\Bigl{[}-2.55528\cdot 10^{-6}\Delta\mu_{p}-8.117\cdot 10^{-12}\Delta r_{p}\Bigr{]},\end{array}

\begin{array}[]{@{}l}\displaystyle\nu(\mbox{H}_{2}^{+})=\nu_{0}(\mbox{H}_{2}^{+})+{\textstyle\frac{\Delta R_{\infty}}{R_{\infty}}}\,\nu_{0}(\mbox{H}_{2}^{+})+2(R_{\infty}c)\,\times\\[5.69054pt] \displaystyle\hskip 14.22636pt\Bigl{[}-2.55528\cdot 10^{-6}\Delta\mu_{p}-8.117\cdot 10^{-12}\Delta r_{p}\Bigr{]},\end{array}

\begin{array}[]{@{}l}\displaystyle\nu(\mbox{D}_{2}^{+})=\nu_{0}(\mbox{D}_{2}^{+})+{\textstyle\frac{\Delta R_{\infty}}{R_{\infty}}}\,\nu_{0}(\mbox{D}_{2}^{+})+2(R_{\infty}c)\,\times\\[5.69054pt] \displaystyle\hskip 14.22636pt\Bigl{[}-9.37686\cdot 10^{-7}\Delta\mu_{d}-5.877\cdot 10^{-12}\Delta r_{d}\Bigr{]},\end{array}

\begin{array}[]{@{}l}\displaystyle\nu(\mbox{HD}^{+})=\nu_{0}(\mbox{HD}^{+})+{\textstyle\frac{\Delta R_{\infty}}{R_{\infty}}}\,\nu_{0}(\mbox{HD}^{+})+2(R_{\infty}c)\,\times\\[5.69054pt] \displaystyle\hskip 11.38109pt\Bigl{[}-\!1.49998\!\cdot\!10^{-6}\Delta\mu_{p}\!-\!3.75470\!\cdot\!10^{-7}\Delta\mu_{d}\\[5.69054pt] \displaystyle\hskip 22.76219pt-3.555\cdot 10^{-12}\Delta r_{p}-3.550\cdot 10^{-12}\Delta r_{d}\Bigr{]}.\end{array}

r_{d}^{2} - r_{p}^{2} = 3.81948 (37) \mbox f m^{2}

r_{d}^{2} - r_{p}^{2} = 3.81948 (37) \mbox f m^{2}

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Full text

Fundamental transitions and ionization energies of the hydrogen molecular ions at the few ppt level.

Vladimir I. Korobov

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia

L. Hilico

J.-Ph. Karr

Laboratoire Kastler Brossel, UPMC-Univ. Paris 6, ENS, CNRS, Collège de France

4 place Jussieu, F-75005 Paris, France

Université d’Evry-Val d’Essonne, Boulevard François Mitterrand, F-91000 Evry, France

Abstract

We calculate ionization energies and fundamental vibrational transitions for H ${}_{2}^{+}$ , D ${}_{2}^{+}$ , and HD+ molecular ions. The NRQED expansion for the energy in terms of the fine structure constant $\alpha$ is used. Previous calculations of orders $m\alpha^{6}$ and $m\alpha^{7}$ are improved by including second-order contributions due to the vibrational motion of nuclei. Furthermore, we evaluate the largest corrections at the order $m\alpha^{8}$ . That allows to reduce the fractional uncertainty to the level of $7.6\times 10^{-12}$ for fundamental transitions and to $4.5\times 10^{-12}$ for the ionization energies.

The hydrogen molecular ions (HMI) play an essential role in testing molecular quantum mechanics Carrington89 ; Leach95 . From the theoretical point of view the HMI is one of the simplest nonintegrable quantum system, which still allows very accurate numerical treatment. As was pointed out already some time ago Wing76 , and recently discussed more extensively Karr16 , if the theory would be sufficiently precise, the spectroscopy of HMI may be used for determining of the fundamental physical constants such as the proton-to-electron mass ratio. The ionization energy of HMI is also of high importance for the determination of ionization and dissociation energies of the hydrogen molecule from spectroscopic studies MerktH2 ; MerktD2 ; MerktHD as well as for the determination of atomic masses of light nuclei Myers15 ; VanDyck15 ; codata14 .

On the experimental side there are many new projects started, which are now oriented towards Doppler-free spectroscopy with accuracy targeted to 1 ppt (one part per trillion) or better Karr16 ; Tran13 ; Schiller14 ; Schiller17 . These perspectives bring strong motivation for theory.

The aim of this Letter is to improve the theoretical precision of spin-averaged energies and ro-vibrational transition frequencies in HMI. To this end we consider the largest QED contributions which had not been evaluated in our previous works KorobovPRA08 ; KorobovPRA14 ; KorobovPRL14 , namely, corrections to orders $m\alpha^{6}$ and $m\alpha^{7}$ due to vibrational motion of nuclei and the leading contributions to order $m\alpha^{8}$ . As it was shown recently KorobovPRL16 , taking into account the vibrational motion of nuclei is essential for accurate theoretical description. It has allowed to resolve the longstanding discrepancy between theory and experiment in the hyperfine structure of H ${}_{2}^{+}$ ion. These new achievements reduce the relative uncertainty in the fundamental vibrational transitions of HMI to the level of $7.6\times 10^{-12}$ and allow to obtain the most precise theoretical values for the ionization energies of H ${}_{2}^{+}$ , D ${}_{2}^{+}$ , and HD+ molecular ions. In conclusion we discuss how these new results may have impact on fundamental physical constants such as the Rydberg constant, proton-to-electron mass ratio, and proton charge radius.

We use atomic units throughout this paper.

The terms of $m\alpha^{6}$ and higher orders are calculated in the adiabatic approximation. For this purpose we use the Born-Oppenheimer formalism. In this approach the states of the molecule are taken in the form

[TABLE]

The electronic wave function obeys the clamped nuclei Schrödinger equation for a bound electron

[TABLE]

where $H_{el}=p^{2}/(2m)+V+Z_{1}Z_{2}/R$ is the electronic Hamiltonian, $V=-Z_{1}/r_{1}-Z_{2}/r_{2}$ , where $Z_{1}$ and $Z_{2}$ are the charges of the nuclei and $r_{1}$ , $r_{2}$ are the distances from the electron to nuclei 1 and 2, respectively. The wave function $\chi_{\rm BO}(R)$ describes the relative nuclear motion, and is a solution of

[TABLE]

where $\mu_{n}$ is the reduced mass of the nuclei.

Relativistic corrections of order $m\alpha^{6}$ to the energy of a bound electron in the two-center problem are determined Pachucki97 ; KorobovJPB07 by the effective Hamiltonian:

[TABLE]

and the second order contribution of the Breit-Pauli Hamiltonian,

[TABLE]

Here $\rho=\boldsymbol{\nabla}^{2}V/(4\pi)$ , $Q$ is a projection operator onto a subspace orthogonal to $\phi_{\rm el}$ from Eq. (2). $H_{B}$ is the Breit–Pauli relativistic correction for a bound electron:

[TABLE]

$H_{B}^{so}$ is the electron spin-orbit contribution (see details in KorobovJPB07 ). Both terms are divergent but their sum is finite

[TABLE]

The leading contribution was obtained in KorobovPRA08 by averaging this effective potential over $R$ :

[TABLE]

The next step is to consider the three-body correction to the energy $E_{0}$ of a molecular state, which we derive within the framework of the adiabatic approximation defined by Eqs. (1)-(3). This correction stems from the insertion of the Breit-Pauli effective potential $\mathcal{E}_{B}(R)=\alpha^{2}\langle H_{B}\rangle$ into Eq. (3) and in the order $m\alpha^{6}$ is expressed by

[TABLE]

$Q^{\prime}$ is a projection operator onto a subspace orthogonal to $\chi_{\rm BO}(R)$ .

Obviously, instead of the Born-Oppenheimer solution $\chi_{\rm BO}(R)$ one may use the adiabatic solution $\chi_{\rm ad}(R)$ , which includes as well the adiabatic corrections (see for definitions Ref. Wolniewicz80 , or a review by Carrington et al. Carrington89 ).

A complete set of the contributions at order $m\alpha^{6}$ is presented in Table 1. We include here as well the relativistic recoil contribution at order $m(Z\alpha)^{6}(m/M)$ PachuckiPRL97 and the radiative recoil contribution Pachucki95 ; Melnikov01 . In the former case the part, which depends on the state wave function, was evaluated with the help of LCAO approximation and its value had been used as an error bar for the recoil term.

The total contribution to the one-loop self energy correction at order $m\alpha^{7}$ similarly should be written

[TABLE]

where $\mathcal{E}_{1loop-{\rm SE}}^{(7)}(R)$ is an effective potential of the $m\alpha^{7}$ -order correction (see Eq. (11), in KorobovPRA14 ), to the energy of the bound electron in the two-center problem, and

[TABLE]

is the one-loop self-energy correction of order $m\alpha^{5}$ . $\beta(R)$ is the nonrelativistic Bethe logarithm for the bound electron in the two-center problem, whose values as a function of $R$ may be found in the Supplemental Material to Ref. Korobov13 or in Kolos .

A similar separation between electronic and vibrational contributions also occurs for the one-loop vacuum polarization term, which was obtained in Karr17 .

Contributions to order $m\alpha^{7}$ without the vibrational second-order term were considered in KorobovPRA14 ; KorobovPRL14 . Here we present final results which appear in Table 2. We have managed to significantly improve precision of the relativistic correction to the Bethe logarithm (see, for details, KorobovHI15 ), which allowed to reduce the theoretical uncertainty in the one-loop self-energy by an order of magnitude.

Finally, we turn to the evaluation of $m\alpha^{8}$ -order corrections.

For hydrogen-like atoms, the two-loop correction at order $m\alpha^{8}$ may be written in the form

[TABLE]

where $L(Z\alpha)\equiv\ln(Z\alpha)^{-2}$ . It is useful to recall the numerical values of the various terms for the ground state of the hydrogen atom Eides_book :

[TABLE]

This shows that the third term (linear in $\ln(Z\alpha)^{-2}$ ) is the largest one, contrary to our intuition on hierarchy of the consecutive terms in the $Z\alpha$ expansion.

In case of a two-center system the corrections can still be written in the form of Eq. (11) (with $n\!=\!1$ and $Z_{1}\!=\!Z_{2}\!=\!Z$ ). The first three coefficients $B_{6k}$ can be obtained from the results of JCP05 as

[TABLE]

and

[TABLE]

Among the terms presented in Eqs. (12), (13) all the distributions were determined and calculated in KorobovPRA14 except $N$ , which is defined in Eq. (4.21.a) of JCP05 . On the other hand the expression for $N$ is similar to the one of Eq. (10) in Ref. Korobov13 . Using the same technique, which has been used for calculations of the relativistic Bethe logarithm we were able to get $N$ for the hydrogen atom ground state:

[TABLE]

which is in a good agreement with the value given in JCP05 and even adds two more significant digits. Having validated our approach, we then did calculations of the $N(R)$ ”effective” potential for the two-center problem. Putting it into Eq. (13) and then averaging over $R$ we get for the ionization energy of H ${}_{2}^{+}$ (in kHz)

[TABLE]

The last term in the second line has been evaluated in the LCAO approximation using the atomic hydrogen ground state value for $B_{60}$ . We take the error bar on the two-loop contribution as equal to this approximate value of nonlogarithmic term. In our previous studies we used the same kind of estimates for the uncertainty resulted from the yet uncalculated terms and further improvements of the theory showed the good relevance of this approach.

Similarly, for the fundamental transition $(L\!=\!0,v\!=\!0)\to(L^{\prime}\!=\!0,v^{\prime}\!=\!1)$ (in kHz)

[TABLE]

and for the uncertainty we take $u_{r}(E_{2loop})=0.21$ kHz.

The other significant contribution at the $m\alpha^{8}$ order is the one-loop self-energy,

[TABLE]

where the leading term has analytic result Eides_book ; Karshenboim97 : $A_{71}(nS)=\pi\left[139/64-\ln{2}\right]$ . For the hydrogen atom the nonlogarithmic contribution $A_{70}$ of order $m\alpha(Z\alpha)^{7}$ has never been calculated directly. By extrapolation of the $G_{se}(1S,Z\alpha)$ function A70 with the expansion over $Z\alpha$ (see Eq. (5.1) from A70 ) one may get $A_{70}=44.4$ . Similarly to the two-loop corrections above, we take the nonlogarithmic term as estimate of the theoretical uncertainty..

The second order contributions due to vibrational motion, both from one- and two-loop diagrams, were evaluated as well. The total frequency shift is about 100 Hz and may be neglected for the time being.

The main results of our work, frequencies for the fundamental transitions $(L=0,v=0)\to(0,1)$ and ionization energies of the HMI, are presented in Tables 4 and 5, respectively. To get precision data for the relativistic corrections of order $m\alpha^{4}$ we have used the expectation values of the Breit-Pauli operators, which were obtained in YanH2+ ; YanHD+ ; YanD2+ with 15 or even more significant digits. As it may be extracted from the Tables the new theoretical relative uncertainty for the fundamental transition frequency is $u_{r}(\nu(\mbox{H}_{2}^{+}))=0.5/(66.\!\times\!10^{9})\approx 7.6\times 10^{-12}$ , and accordingly for the ionization energy one gets $u_{r}(E_{I})=4.5\times 10^{-12}$ . The CODATA14 uncertainty of the Rydberg constant is $u_{r}(R_{\infty})=5.9\times 10^{-12}$ . Since this constant enters in the data of the Tables as a multiplier, an uncertainty in the energies due to the uncertainty in the Rydberg constant can be easily evaluated and is not shown.

These results have direct impact on the potential determination of the fundamental constants. For example, the theoretical uncertainty on the fundamental transition in H ${}_{2}^{+}$ sets the following limit on the achievable precision of the proton-to-electron mass ratio ( $\mu_{p}=m_{p}/m_{e}$ ) to

[TABLE]

This uncertainty is smaller by a factor of 6 with respect to the present CODATA, $u_{r}(\mu_{p})=9.5\times 10^{-11}$ codata14 , which is currently limited by uncertainty on the proton’s atomic mass. The electron’s atomic mass has been recently improved ( $u_{r}(A_{r}(e))=3.1\times 10^{-11}$ ) by a high-precision measurement of the $g$ -factor of a bound electron in a 12C5+ ion Sturm14 . In terms of ultimate accuracy limits, the $1.5\times 10^{-11}$ theoretical uncertainty that we have achieved for HMI spectroscopy is comparable to the current theoretical uncertainty of $1.3\times 10^{-11}$ on the $g$ -factor of 12C5+ codata14 ; Pachucki05 .

The proton rms charge radius ( $r_{p}$ ) uncertainty as determined in the CODATA14 adjustment has a much smaller contribution $\sim\!5\times 10^{-12}$ to the uncertainty in the fundamental transitions. However, replacing the CODATA value of $r_{p}$ with that obtained from muonic hydrogen spectroscopy Pohl10 ; Antognini13 leads to a 3 kHz blueshift of the transition, i.e. a relative shift of $5\times 10^{-11}$ . If we assume that the muonic hydrogen adjusted Rydberg constant should be used as proper constant when using the muonic hydrogen proton radius Pohl13 , then we get a global shift of 1.1 kHz, which is still feasible for detection (see also the more detailed discussion in Karr16 ).

Finally, since the fundamental transitions have potentiality to be used for adjustment of the fundamental constants we present here in explicit form the frequency dependence of transition lines on the masses and on the proton and deuteron charge radii:

[TABLE]

In the last equation $\Delta r_{d}$ may be in principle eliminated since the measured H-D isotope shift of the 1 $S$ -2 $S$ transition Parthey10 determines the deuteron-proton charge radius difference Jentschura11 ; codata14

[TABLE]

with much smaller error than the CODATA14 uncertainties for $r_{p}$ and $r_{d}$ .

In summary, we have considered several new contributions to the binding energies of HMI, which result in an essential improvement of the theoretical uncertainty both for the ionization energies and for the transition frequencies of the HMI. This level of precision allows to use the HMI spectroscopy as an alternative way for determination of the fundamental physical constants.

Acknowledgements. This work was supported by UPMC, which is gratefully acknowledged. J.-Ph. Karr acknowledges support as a fellow of the Institut Universitaire de France. V.I.K. acknowledges support of the Russian Foundation for Basic Research under Grant No. 15-02-01906-a. Authors also very much appreciate valuable discussions with K. Pachucki.

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