# Extraction of spin-averaged rovibrational transition frequencies in
$\text{HD}^{+}$ for the determination of fundamental constants

Jean-Philippe Karra,b and Jeroen C. J. Koelemeijc CONTACT J.-Ph. Karr. Email:
<EMAIL_ADDRESS>J. C. J. Koelemeij. Email<EMAIL_ADDRESS>aLaboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL,
Collège de France, 4 place Jussieu, F-75005 Paris, France; bUniversité d’Evry-
Val d’Essonne, Université Paris-Saclay, Boulevard François Mitterrand, F-91000
Evry, France; cLaserLaB, Department of Physics and Astronomy, Vrije
Universiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands.

###### Abstract

We present a comprehensive analysis of all currently available high-accuracy
frequency measurements of rotational and rovibrational transitions in the
hydrogen molecular ion $\text{HD}^{+}$. Our analysis utilises the
theoretically calculated hyperfine structure to extract the values of three
spin-averaged transition frequencies through a global linear least-squares
adjustment that takes into account theory-induced correlations between the
different transitions. We subsequently use the three spin-averaged transition
frequencies as input data in a second adjustment which employs precise
theoretical expressions for the transition frequencies, written as a function
of the proton, deuteron and electron relative atomic masses, the Rydberg
constant, and the proton and deuteron charge radii. Our analysis shows that
the $\text{HD}^{+}$ data may significantly improve the value of the electron
relative atomic mass and the proton-electron mass ratio, in particular if
combined with recent high-precision measurements of particle atomic masses and
mass ratios obtained from Penning traps.

###### keywords:

Deuterated hydrogen molecular ion; precision rotational-vibrational
spectroscopy; ab initio molecular theory; molecular hyperfine structure;
fundamental physical constants

##  1 Introduction

Fundamental physical constants are determined from precision measurements,
often performed on single particles or simple atomic quantum systems, by
adjusting the values of the constants such that theoretical predictions match
the experimental observations as closely as possible. Such adjustments,
implemented via least-squares optimization of the constants of interest with
respect to large bodies of input data, are carried out by standards
organizations such as the Committee on Data for Science and Technology
(CODATA) and the Atomic Mass Data Center (AMDC).

For example, the most precise determinations of the Rydberg constant,
$R_{\infty}$, and the electric charge radii of the proton, $r_{\rm p}$, and
the deuteron, $r_{\rm d}$, have been obtained from laser spectroscopy of both
electronic and muonic hydrogen and deuterium, complemented by high-precision
quantum electrodynamics (QED) theoretical predictions (see Ref. [1] and
references therein). By contrast, the relative atomic masses of particles like
the electron and light nuclei such as the proton and deuteron have
traditionally been determined using Penning-trap experiments and (for the
electron) theoretical calculations of bound-state $g$-factors; for recent
examples see Refs. [2, 3, 4, 5, 6, 7]. In 1976 Wing et al. already recognised
that vibrational spectroscopy of hydrogen molecular ions, such as
H${}_{2}^{+}$ and $\text{HD}^{+}$, could also be used for determinations of
particle mass ratios provided that both the measurements and the theory of
these three-body, two-center quantum systems would be sufficiently accurate
[8]. This concept was extended to laser spectroscopy of antiprotonic helium,
$\bar{\text{p}}\text{He}$, by Hori et al. who combined their results with
precise ab initio theory for a meaningful determination of the antiproton-
electron mass ratio which under the assumption of charge, parity and time
reversal (CPT) invariance must be the same as the proton-electron mass ratio,
$m_{\rm p}/m_{\rm e}$ [9, 10, 11]. As a result, $\bar{\text{p}}\text{He}$ data
were included in the 2006 and 2010 CODATA adjustments where they contributed
primarily to the value of the electron relative atomic mass, $A_{\rm r}({\rm
e})$ [12, 13]. However, the 2014 and 2018 CODATA adjustments no longer
considered $\bar{\text{p}}\text{He}$ after $A_{\rm r}({\rm e})$ was determined
using Penning-trap measurements and QED theory with over an order of magnitude
smaller uncertainty [2].

Over the past two decades, advances in laser and terahertz spectroscopy of
$\text{HD}^{+}$ stored in radiofrequency (rf) ion traps and sympathetically
cooled by laser-cooled ions, considerably improved the accuracy, culminating
in the measurement of various rovibrational transition frequencies with
uncertainties in the low parts-per-trillion (ppt) range [14, 15, 16].
Meanwhile, the precision of theoretical QED predictions of relevant transition
frequencies in $\text{HD}^{+}$ improved to below 10 ppt, such that the
uncertainty of theoretically predicted frequencies is now limited by
(primarily) the uncertainty of the 2018 CODATA value of $m_{\rm p}/m_{\rm e}$
[17, 18, 19]. As anticipated by Wing et al. [8] and more recently also by Karr
et al. [20], the availability of theoretical and experimental transition
frequencies with uncertainties in the low-ppt range enabled a determination of
$m_{\rm p}/m_{\rm e}$ with a precision of about 20 ppt [14, 15, 16]. These
values represented not only the most precise determinations of $m_{\rm
p}/m_{\rm e}$ to date, but they were also found to be consistent with a recent
precise measurement of the proton relative atomic mass, $A_{\rm r}({\rm p})$
[3, 4], that deviated significantly from determinations of $A_{\rm r}({\rm
p})$ included in the 2018 CODATA adjustment [1]. Furthermore, several
researchers have pointed out that results from $\text{HD}^{+}$ and
$\bar{\text{p}}\text{He}$ form a link between ’atomic’ fundamental constants
such as $R_{\infty}$, $r_{\rm p}$ and $r_{\rm d}$, and particle mass ratios
[21, 22, 23].

The recent advances in $\text{HD}^{+}$ experiments and theory and their
demonstrated potential for the determination of fundamental constants warrant
a closer investigation to prepare available $\text{HD}^{+}$ data for inclusion
in future CODATA adjustments. One crucial issue is that, in general, atomic
and molecular spectroscopic data include line shifts due to hyperfine
interactions, whereas CODATA adjustments have historically considered only
energy differences between hyperfine centroids. This implies that experimental
$\text{HD}^{+}$ spectroscopic data must undergo a preparatory analysis in
order to remove hyperfine shifts from measured rovibrational transition
frequencies. Here, we will use the term ’hyperfine components’ to indicate
rovibrational transitions between individual hyperfine levels. In principle,
the spin-averaged transition frequency can be determined from the measured
transition frequencies of a sufficient number of hyperfine components.
However, in the experiments conducted so far, the number of measured hyperfine
components has been too small, and no other independent experimental data on
the relevant $\text{HD}^{+}$ hyperfine structure exist. To derive spin-
averaged transition frequencies from limited sets of measured frequencies of
hyperfine components, several methods have been developed which overcome the
lack of experimental data by using the theoretically predicted hyperfine level
structure [14, 24, 25]. In this work we adopt the approach of [24], which is
based on a least-squares optimization and takes into account correlations
between theoretical hyperfine coefficients.

This article is organised as follows. The state-of-the-art $\text{HD}^{+}$
hyperfine theory used for the extraction of spin-averaged transition
frequencies is reviewed in Section 2, along with a discussion of correlated
hyperfine theory uncertainties. Section 3 summarises the available
experimental data (that include hyperfine shifts) related to three different
rotational-vibrational transitions in $\text{HD}^{+}$, followed by a global
adjustment of the three spin-averaged rotational-vibrational transition
frequencies in Section 4. In Section 5, the adjusted spin-averaged transition
frequencies and their covariances are subsequently compared to theoretical
expressions for these spin-averaged transition frequencies that are functional
expressions of several fundamental constants, including $A_{\rm r}({\rm e})$,
$A_{\rm r}({\rm p})$, and the deuteron relative atomic mass, $A_{\rm r}({\rm
d})$. We use the corresponding observational equations in a second adjustment,
aimed at determining the values of the fundamental constants. We consider
various sets of input data and observational equations (including recent
Penning-trap measurements of relative atomic masses), and compare values of
mass ratios such as $m_{\rm p}/m_{\rm e}$ derived from each adjustment, in
order to illustrate the potential impact of $\text{HD}^{+}$ on the precision
of fundamental particle mass ratios. Conclusions are presented in Section 6.

##  2 Status of hyperfine structure theory

###  2.1 Hyperfine Hamiltonian

The hyperfine structure of a rovibrational state of $\text{HD}^{+}$ can be
described by the following effective spin Hamiltonian [26]:

$\begin{split}H_{\rm eff}&=E_{1}(\mathbf{L}\\!\cdot\\!\mathbf{s}_{\rm
e})+E_{2}(\mathbf{L}\\!\cdot\\!\mathbf{I}_{\rm
p})+E_{3}(\mathbf{L}\\!\cdot\\!\mathbf{I}_{\rm d})+E_{4}(\mathbf{I}_{\rm
p}\\!\cdot\\!\mathbf{s}_{\rm e})\\\ &\quad+E_{5}(\mathbf{I}_{\rm
d}\\!\cdot\\!\mathbf{s}_{\rm e})+E_{6}\\{2\mathbf{L}^{2}(\mathbf{I}_{\rm
p}\\!\cdot\\!\mathbf{s}_{\rm e})\\!-\\!3[(\mathbf{L}\\!\cdot\\!\mathbf{I}_{\rm
p})(\mathbf{L}\\!\cdot\\!\mathbf{s}_{\rm e})\\\
&\quad+(\mathbf{L}\\!\cdot\\!\mathbf{s}_{\rm
e})(\mathbf{L}\\!\cdot\\!\mathbf{I}_{\rm
p})]\\}+E_{7}\\{2\mathbf{L}^{2}(\mathbf{I}_{\rm d}\\!\cdot\\!\mathbf{s}_{\rm
e})\\\ &\quad-3[(\mathbf{L}\\!\cdot\\!\mathbf{I}_{\rm
d})(\mathbf{L}\\!\cdot\\!\mathbf{s}_{\rm
e})\\!+\\!(\mathbf{L}\\!\cdot\\!\mathbf{s}_{\rm
e})(\mathbf{L}\\!\cdot\\!\mathbf{I}_{\rm d})]\\}\\\
&\quad+E_{8}\\{2\mathbf{L}^{2}(\mathbf{I}_{\rm p}\\!\cdot\\!\mathbf{I}_{\rm
d})\\!-\\!3[(\mathbf{L}\\!\cdot\\!\mathbf{I}_{\rm
p})(\mathbf{L}\\!\cdot\\!\mathbf{I}_{\rm
d})\\!+\\!(\mathbf{L}\\!\cdot\\!\mathbf{I}_{\rm
p})(\mathbf{L}\\!\cdot\\!\mathbf{I}_{\rm d})]\\}\\\
&\quad+E_{9}\\{\mathbf{L}^{2}\mathbf{I}_{\rm
d}^{2}\\!-\\!\frac{3}{2}(\mathbf{L}\\!\cdot\\!\mathbf{I}_{\rm
d})\\!-\\!3(\mathbf{L}\\!\cdot\\!\mathbf{I}_{\rm d})^{2}\\},\end{split}$ (1)

where $\mathbf{I}_{\rm d}$, $\mathbf{I}_{\rm p}$, and $\mathbf{s}_{\rm e}$ are
the spins of the deuteron, proton, and electron, respectively, and
$\mathbf{L}$ is the total angular momentum excluding electron and nuclear
spins (denoted by $\mathbf{N}$ in standard spectroscopic notation). Here and
throughout the paper, spin and angular momentum operators are written in units
of $\hbar$ and are thus dimensionless. The coefficients $E_{k}$,
$k=1,2,\ldots,9$ are then energies and depend on the rovibrational state
$(v,L)$, where $v$ stands for the vibrational quantum number. This Hamiltonian
includes all spin-dependent interactions that appear at the leading order
$m_{\rm e}c^{2}\alpha^{4}$, i.e. appear in the Breit-Pauli Hamiltonian, except
for the proton-deuteron spin-spin contact interaction, which is negligibly
small because of the strong Coulomb repulsion between the proton and deuteron.

Additional spin-dependent interactions appear at higher orders ($m_{\rm
e}c^{2}\alpha^{6}$ and above). The largest missing term in Equation (1) is a
coupling between proton and deuteron spins (proportional to $\mathbf{I}_{\rm
p}\\!\cdot\\!\mathbf{I}_{\rm d}$), mediated by the electron. The coupling
coefficient of this interaction was shown to be on the order of 100 Hz [25],
which is much smaller than the current theoretical uncertainty of $E_{4}$ (see
below). The effective spin Hamiltonian may thus be considered complete for our
purposes.

The largest terms in Equation (1) are the electron-proton ($E_{4}/h\sim 900$
MHz in the vibrational ground state) and electron-deuteron ($E_{5}/h\sim 140$
MHz) spin-spin contact interactions. They were calculated with relative
uncertainties below 1 parts per million (ppm) [27]. A key element to reach
this accuracy is that the sum of nuclear structure and recoil corrections to
$E_{4}$ ($E_{5}$), which we denote as ’nuclear correction’ in the following,
was determined from the difference between the experimental ground-state
hyperfine splitting of the hydrogen (deuterium) atom and the corresponding
nonrecoil, point-particle QED prediction, under the assumption that the
nuclear correction is entirely described by a contact (delta-function) term.
Uncertainties associated with nuclear corrections are thus suppressed, and the
theoretical precision is limited by unevaluated higher-order QED terms: (i)
nonrecoil corrections, dominated by the contribution of order
$\alpha(Z\alpha^{2})\ln(Z\alpha)E_{F}$ [28, 29], and (ii) recoil corrections
that deviate from the contact-term approximation, i.e. mainly the contribution
of order $(Z\alpha)^{2}(m/M)E_{F}$ [30]. Here, $E_{F}$ stands for the leading
Fermi contribution of order $m_{\rm e}c^{2}\alpha^{4}$.

The next largest terms are the spin-orbit ($E_{1}/h\sim 30$ MHz in the
vibrational ground state) and spin-spin tensor interactions ($E_{6}/h\sim 8$
MHz, $E_{7}/h\sim 1$ MHz for $L=1$). They were computed with few-ppm
uncertainties through calculation of higher-order corrections at orders
$m_{\rm e}c^{2}\alpha^{6}$ [31] and $m_{\rm e}c^{2}\alpha^{7}\ln(\alpha)$ [32,
33]. The uncertainty is dominated by unevaluated nonlogarithmic corrections at
the $m_{\rm e}c^{2}\alpha^{7}$ order.

The remaining hyperfine coefficients are much smaller, ranging from
$E_{8}/h\sim 3$ kHz to $E_{2}/h\sim 30$ kHz. They were calculated in the
framework of the Breit-Pauli Hamiltonian accounting for the electron’s
anomalous magnetic moment [26]. In general, the expected relative uncertainty
of this approximation is on the order of $\alpha^{2}$, but in this case
second-order contributions induced by the leading hyperfine interaction terms
can give larger contributions. Due to this, the relative uncertainty of
$E_{2}$, $E_{3}$, and $E_{8}$ is estimated to be $5\alpha^{2}$ [33]. In the
case of the $E_{9}$ term that describes the effect of the deuteron quadrupole
moment, second-order terms are too small to significantly influence the
uncertainty. The relative uncertainty of $E_{9}$ is estimated to be 98 ppm
[33], which accounts for both the theoretical uncertainty and the uncertainty
of the deuteron quadrupole moment from [34].

State-of-the art values of all the hyperfine coefficients, for rovibrational
levels involved in published high-precision measurements, are given in Table
1.

In our least-squares adjustment of $\text{HD}^{+}$ experimental data (see
Section 4), theoretical uncertainties in the hyperfine coefficients
$E_{k}(v,L)$ are taken into account by introducing additive corrections
$\delta E_{k}^{\rm th}(v,L)$ which are treated as adjusted constants,
following the approach of CODATA [1]. For each coefficient, an input datum
$\delta E_{k}(v,L)$ is included with a zero value and an uncertainty equal to
that given in Table 1, and an observational equation $\delta
E_{k}(v,L)\doteq\delta E_{k}^{\rm th}(v,L)$ is added to the adjustment.

Beyond the uncertainties themselves, it is also necessary to estimate the
correlation coefficients among the corrections $\delta E_{k}(v,L)$ due to
common sources of uncertainty. As shown in [24], the assumed level of
correlation can significantly influence the spin-averaged frequencies of
rovibrational transitions determined from an incomplete set of measurements of
their hyperfine components. Correlation coefficients are defined as follows:

$r(\delta E_{k}(v,L),\delta E_{l}(v^{\prime},L^{\prime}))=\frac{u(\delta
E_{k}(v,L),\delta E_{l}(v^{\prime},L^{\prime}))}{u(\delta
E_{k}(v,L))\,u(\delta E_{l}(v^{\prime},L^{\prime}))}\,,$ (2)

where $u(X,Y)$ is the covariance between quantities $X$ and $Y$, and
$u(X)=\sqrt{u(X,X)}$ is the uncertainty of $X$.

Here, we make assumptions similar to those presented in Appendix C of [33]:

  * •

Perfect positive correlations are assumed between $\delta E_{4}(v,L)$ and
$\delta E_{4}(v^{\prime},L^{\prime})$, as well as between $\delta E_{5}(v,L)$
and $\delta E_{5}(v^{\prime},L^{\prime})$.

  * •

For the other hyperfine coefficients ($k\neq 4,5$), correlation coefficients
between $\delta E_{k}(v,L)$ and $\delta E_{k}(v^{\prime},L^{\prime})$ are
expected to be positive, but are difficult to estimate. In our analysis, we
thus vary these coefficients between 0 and 1 to investigate their influence on
the spin-averaged frequencies, and additional uncertainties associated with
these correlations.

  * •

Perfect positive correlations are assumed between $\delta E_{2}(v,L)$ and
$\delta E_{3}(v,L)$, as well as between $\delta E_{6}(v,L)$ and $\delta
E_{7}(v,L)$. This stems from the fact that uncalculated QED corrections in
both coefficients are of the same nature.

  * •

The correlation coefficient between $\delta E_{4}(v,L)$ and $\delta
E_{5}(v,L)$ is estimated [33] to be $r(\delta E_{4}(v,L),\delta
E_{5}(v,L))=0.4016$.

  * •

No other correlations are assumed between $\delta E_{k}(v,L)$ and $\delta
E_{l}(v,L)$.

  * •

No correlations are assumed between $\delta E_{k}(v,L)$ and $\delta
E_{l}(v^{\prime},L^{\prime})$ with $k\neq l$ and
$(v,L)\neq(v^{\prime},L^{\prime})$, except for those that are implied by the
above hypotheses, e.g.: $r(\delta E_{4}(v,L),\delta
E_{5}(v^{\prime},L^{\prime}))=r(\delta E_{4}(v,L),\delta E_{5}(v,L))=0.4016$.

Except for the unknown correlation between $\delta E_{k}(v,L)$ and $\delta
E_{k}(v^{\prime},L^{\prime})$ ($k\neq 4,5$), we do not consider the effect of
uncertainties in other correlation coefficients. Indeed, although these
uncertainties are not strictly zero, they are estimated to be too small to
have any significant impact on the spin-averaged frequencies.

$(v,L)$ | $E_{1}/h$ | $E_{2}/h$ | $E_{3}/h$ | $E_{4}/h$ | $E_{5}/h$ | $E_{6}/h$ | $E_{7}/h$ | $E_{8}/h$ | $E_{9}/h$
---|---|---|---|---|---|---|---|---|---
$(0,0)$ |  |  |  | $925\,394.159(860)$ |  $142\,287.556(84)$ |  |  |  |
$(0,1)$ | $31\,985.417(116)$ | $-31.345(8)$ | $-4.809(1)$ | $924\,567.718(859)$ | $142\,160.670(84)$ | $8611.299(18)$ | $1321.796(3)$ | $-3.057(1)$ | $5.660(1)$
$(1,1)$ | $30\,280.736(109)$ | $-30.463(8)$ | $-4.664(1)$ | $903\,366.501(839)$ | $138\,910.266(82)$ | $8136.858(17)$ | $1248.963(3)$ | $-2.945(1)$ | $5.653(1)$
$(0,3)$ | $31\,628.097(114)$ | $-30.832(8)$ | $-4.733(1)$ | $920\,479.981(855)$ | $141\,533.075(83)$ | $948.542(2)$ | $145.597$ | $-0.335$ | $0.613$
$(9,3)$ | $18\,270.853(62)$ | $-21.304(6)$ | $-3.225(1)$ | $775\,706.122(721)$ | $119\,431.933(73)$ | $538.999(1)$ | $82.726$ | $-0.219$ | $0.501$

Table 1: Theoretical hyperfine coefficients of rovibrational states of
$\text{HD}^{+}$ involved in high-precision measurements, in kHz. Missing
numbers in the first line imply a zero value. The values of $E_{4}/h$,
$E_{5}/h$ ($E_{1}/h$, $E_{6}/h$, $E_{7}/h$) were calculated in Ref. [27] (Ref.
[33]). Those of $E_{2}/h$, $E_{3}/h$, $E_{8}/h$, and $E_{9}/h$ are taken from
[26]. The value of $E_{9}/h$ has been updated using the latest determination
of the deuteron’s quadrupole moment [34].

###  2.2 Hyperfine splitting

The hyperfine structure of a rovibrational state $(v,L)$ is obtained by
diagonalizing the effective spin Hamiltonian (1). Given the relative sizes of
the $E_{k}$, the natural coupling scheme of angular momenta is [26]

$\mathbf{F}=\mathbf{I}_{\rm p}+\mathbf{s}_{\rm
e},\quad\mathbf{S}=\mathbf{F}+\mathbf{I}_{\rm
d},\quad\mathbf{J}=\mathbf{L}+\mathbf{S}.$ (3)

The matrix elements of the effective spin Hamiltonian in the basis of coupled
states $|vLFSJ\rangle$ are calculated using standard angular algebra
procedures. $J$ is an exact quantum number, whereas $F$ and $S$ are only
approximate quantum numbers. A spin eigenstate is denoted by
$|vL\tilde{F}\tilde{S}J\rangle$, which is the state having the largest weight
in its decomposition over the basis states. In the following, we omit the
$\tilde{}$ symbols for simplicity.

Since the matrix of $H_{\rm eff}$ is block diagonal with blocks of size
$4\times 4$ at most, its eigenvalues $E_{\rm hfs}(v,L,F,S,J)$ can be expressed
analytically as a function of the hyperfine interaction coefficients $E_{k}$.
These expressions are then linearised around the theoretical values of the
coefficients using the sensitivity coefficients

$\gamma_{k}(v,L,F,S,J)=\frac{\partial E_{\rm hfs}(v,L,F,S,J)}{\partial
E_{k}}.$ (4)

The theoretical hyperfine shifts $E_{\rm hfs}$ and sensitivity coefficients
$\gamma_{k}$ of all the levels involved in the measurements are given in Table
2.

$(v,L)$ | $(F,S,J)$ |  $E_{\rm hfs}/h$ (kHz) | $\gamma_{1}$ | $\gamma_{2}$ | $\gamma_{3}$ | $\gamma_{4}$ | $\gamma_{5}$ | $\gamma_{6}$ | $\gamma_{7}$ | $\gamma_{8}$ | $\gamma_{9}$
---|---|---|---|---|---|---|---|---|---|---|---
$(0,0)$ | $(0,1,1)$ |  $-705\,735.655(639)$ |  |  |  | $-0.7367$ |  $-0.1688$ |  |  |  |
$(1,0,0)$ |  $89\,060.984(197)$ |  |  |  | $0.2500$ |  $-1.0000$ |  |  |  |
$(1,1,1)$ |  $171\,894.797(194)$ |  |  |  | $0.2367$ |  $-0.3312$ |  |  |  |
$(1,2,2)$ |  $302\,492.318(235)$ |  |  |  | $0.2500$ |  $0.5000$ |  |  |  |
$(0,1)$ | $(0,1,2)$ | $-707\,870.694(637)$ | $-0.1070$ | $0.1110$ | $0.9953$ | $-0.7338$ | $-0.1845$ | $0.0100$ | $0.1434$ | $-0.1572$ | $-0.4930$
$(1,0,1)$ | $79\,986.308(204)$ | $-0.4286$ | $-0.3861$ | $0.6539$ | $0.2494$ | $-0.9084$ | $-1.0279$ | $0.8343$ | $0.7206$ | $-0.5019$
$(1,1,2)$ | $183\,682.795(196)$ | $0.3319$ | $0.1539$ | $0.5142$ | $0.2341$ | $-0.3152$ | $0.2555$ | $-0.5821$ | $-0.4019$ | $0.2287$
$(1,2,1)$ | $269\,283.177(241)$ | $-0.5693$ | $-0.5592$ | $-1.7183$ | $0.2500$ | $0.4251$ | $0.0392$ | $-3.3471$ | $-3.2839$ | $-2.9444$
$(1,2,3)$ | $312\,567.273(242)$ | $0.5000$ | $0.5000$ | $1.0000$ | $0.2500$ | $0.5000$ | $-0.5000$ | $-1.0000$ | $-1.0000$ | $-0.5000$
$(1,2,2)$ | $314\,228.466(238)$ | $-0.2249$ | $-0.2649$ | $-0.5096$ | $0.2497$ | $0.4997$ | $1.7345$ | $3.4387$ | $3.5591$ | $1.7643$
$(1,1)$ | $(1,2,1)$ | $263\,806.300(235)$ | $-0.5753$ | $-0.5654$ | $-1.7152$ | $0.2500$ | $0.4296$ | $-0.0105$ | $-3.3687$ | $-3.3063$ | $-2.9087$
$(1,2,3)$ | $305\,099.957(236)$ | $0.5000$ | $0.5000$ | $1.0000$ | $0.2500$ | $0.5000$ | $-0.5000$ | $-1.0000$ | $-1.0000$ | $-0.5000$
$(0,3)$ | $(0,1,4)$ | $-711\,832.144(631)$ | $-0.4142$ | $0.4242$ | $2.9867$ | $-0.7267$ | $-0.2156$ | $0.2750$ | $3.0796$ | $-3.3289$ | $-7.4000$
$(1,2,5)$ | $338\,970.635(290)$ | $1.5000$ | $1.5000$ | $3.0000$ | $0.2500$ | $0.5000$ | $-7.5000$ | $-15.0000$ | $-15.0000$ | $-7.5000$
$(9,3)$ | $(0,1,4)$ | $-596\,833.613(533)$ | $-0.3590$ | $0.3727$ | $2.9847$ | $-0.7303$ | $-0.2017$ | $0.1934$ | $2.4743$ | $-2.7221$ | $-7.3853$
$(1,2,5)$ | $275\,723.281(219)$ | $1.5000$ | $1.5000$ | $3.0000$ | $0.2500$ | $0.5000$ | $-7.5000$ | $-15.0000$ | $-15.0000$ | $-7.5000$

Table 2: Theoretical hyperfine shifts (column 3) and sensitivity coefficients
(columns 4-12) for all the levels involved in high-precision measurements.
Missing numbers for the $(v,L)=(0,0)$ state imply a zero value. The
uncertainties of the hyperfine shifts are calculated assuming no correlation
between $\delta E_{k}(v,L)$ and $\delta E_{k}(v^{\prime},L^{\prime})$ for
$k\neq 4,5$, which provides an upper bound of the uncertainty.

##  3 Available experimental data

At the time of writing, ppt-range frequency measurements of three rotational-
vibrational transitions in $\text{HD}^{+}$ had been reported in peer-reviewed
publications. These are the $(v,L)$: $(0,0)\rightarrow(0,1)$ rotational
transition at 1.31 THz [14], the $(v,L)$: $(0,0)\rightarrow(1,1)$
rovibrational transition at 58.6 THz [16], and the $(v,L)$:
$(0,3)\rightarrow(9,3)$ vibrational overtone at 415 THz [15]. All frequency
measurements were made on $\text{HD}^{+}$ ions stored in an rf trap, and
sympathetically cooled to millikelvin temperatures via laser-cooled Be+ ions
stored in the same trap. For further details on the experimental setups and
the justification of the achieved measurement uncertainties we refer to the
original publications.

Table 3 provides an overview of the frequencies (and their uncertainties) of
the 10 measured hyperfine components belonging to the three rotational-
vibrational transitions of interest. The measured frequencies are denoted by
$f^{\rm exp}[(v,L,a)\to(v^{\prime},L^{\prime},a^{\prime})]$, where $a=(F,S,J)$
and $a^{\prime}=(F^{\prime},S^{\prime},J^{\prime})$ are the initial and final
hyperfine states.

In all cases the experimental uncertainties receive appreciable contributions
from statistical as well as systematic sources of uncertainty. In the
uncertainty evaluation of the $(v,L)$: $(0,0)\rightarrow(0,1)$ and $(v,L)$:
$(0,0)\rightarrow(1,1)$ transitions, no mention is made of possible correlated
systematic errors [14, 16]. In the case of the $(v,L)$:
$(0,3)\rightarrow(9,3)$ transition, a correlation between two negligibly small
(sub-0.01-kHz) uncertainties in the estimate of the Zeeman shift was
identified [15]. In what follows we assume that the uncertainties of all
experimental transition frequencies are uncorrelated.

$(v\\!,\\!L)\\!\to\\!(v^{\prime}\\!,\\!L^{\prime})$ | Label | $(F\\!,\\!S\\!,\\!J)$ | $(F^{\prime}\\!,\\!S^{\prime}\\!,\\!J^{\prime})$ |  $f^{\rm exp}[(v\\!,\\!L\\!,\\!a)\\!\to\\!(v^{\prime}\\!,\\!L^{\prime}\\!,\\!a^{\prime})]$ (kHz) | Alt. label | Ref.
---|---|---|---|---|---|---
$(0,0)\rightarrow(0,1)$ | A1 | (1,2,2) | (1,2,1) | 1,314,892,544.276(40) | 12 | [14]
A2 | (1,0,0) | (1,0,1) | 1,314,916,678.487(64) | 14
A3 | (0,1,1) | (0,1,2) | 1,314,923,618.028(17) | 16
A4 | (1,2,2) | (1,2,3) | 1,314,935,827.695(37) | 19
A5 | (1,2,2) | (1,2,2) | 1,314,937,488.614(60) | 20
A6 | (1,1,1) | (1,1,2) | 1,314,937,540.762(46) | 21
$(0,0)\rightarrow(1,1)$ | A7 | (1,2,2) | (1,2,1) | 58,605,013,478.03(19) | 12 | [16]
A8 | (1,2,2) | (1,2,3) | 58,605,054,772.08(26) | 16
$(0,3)\rightarrow(9,3)$ | A9 | (0,1,4) | (0,1,4) | 415,265,040,503.57(59) | $F=0$ | [15]
A10 | (1,2,5) | (1,2,5) | 415,264,862,249.16(66) | $F=1$

Table 3: Experimentally determined transition frequencies, $f^{\rm
exp}[(v,L,a)\to(v^{\prime},L^{\prime},a^{\prime})]$ (where $a=(F,S,J)$ and
$a^{\prime}=(F^{\prime},S^{\prime},J^{\prime})$), of various hyperfine
components used in the adjustment of the spin-averaged transition frequencies
(Section 4). The alternate labels for the hyperfine components in column 6 are
those used in the original references (column 7).

##  4 Global adjustment of spin-averaged transition frequencies

###  4.1 Adjustment details

The spin-averaged transition frequencies are determined from a least-squares
adjustment of the available theoretical and experimental data presented in
Secs. 2 and 3, respectively. We follow the fitting procedure described in
Appendix E of [35].

All experimental data listed in Table 3 are included as input data for the
adjustment. The associated “observational equations” (see [35] for details)
read

$\begin{split}&\,f^{\rm exp}[(v,L,a)\to(v^{\prime},L^{\prime},a^{\prime})]\\\
&\doteq f_{\rm SA}[(v,L)\\!\to\\!(v^{\prime},L^{\prime})]\\!+\\!\left(E_{\rm
hfs}(v^{\prime}\\!,\\!L^{\prime}\\!,\\!a^{\prime})\\!-\\!E_{\rm
hfs}(v\\!,\\!L\\!,\\!a)\right)/h\\\
&\quad+\frac{1}{h}\sum_{k=1}^{9}(\gamma_{k}(v^{\prime}\\!,\\!L^{\prime}\\!,\\!a^{\prime})\,\delta
E_{k}^{\rm
th}(v^{\prime}\\!,\\!L^{\prime})\\!-\\!\gamma_{k}(v\\!,\\!L\\!,\\!a)\,\delta
E_{k}^{\rm th}(v\\!,\\!L)),\end{split}$ (5)

where the dotted equality sign means that the left and right hand sides are
not equal in general (since the set of equations is overdetermined) but should
agree within estimated uncertainties. Here, $f_{\rm
SA}[(v,L)\to(v^{\prime},L^{\prime})]$ are the spin-averaged frequencies of the
rovibrational transitions. Moreover, there are observational equations
associated with the additive corrections $\delta E_{k}^{\rm th}(v,L)$ to
theoretical hyperfine coefficients, as explained in Section 2.1:

$\delta E_{k}(v,L)\doteq\delta E_{k}^{\rm th}(v,L).$ (6)

The adjusted constants are $f_{\rm SA}[(v,L)\\!\to\\!(v^{\prime},L^{\prime})]$
and $\delta E_{k}^{\rm th}(v,L)$. However, some of the $\delta E_{k}(v,L)$ are
redundant because certain pairs of coefficients are perfectly correlated with
each other (see Section 2.1). This leads to a singular covariance matrix,
which cannot be inverted as required for the least-squares adjustment. To
solve this, redundant variables are eliminated using the relationships:

$\displaystyle\delta E_{k}(v,L)=\delta
E_{k}(0,0)\frac{u_{k}(v,L)}{u_{k}(0,0)},\quad k=4,5,$ (7a)
$\displaystyle\delta E_{l}(v,L)=\delta
E_{k}(v,L)\frac{u_{l}(v,L)}{u_{k}(v,L)},\;(k,l)=(2,3)\;\mbox{or}\;(6,7),$ (7b)

where $u_{k}(v,L)$ is the uncertainty of $\delta E_{k}(v,L)$. There are a
total of $N=32$ input data: 10 experimental frequencies (Table 3), and 22
additive energy corrections (Table 4). The total number of adjusted constants
is $M=25$: 3 spin-averaged transition frequencies and the 22 energy
corrections.

| Input datum | Value (kHz) |  | Input datum | Value (kHz)
---|---|---|---|---|---
B1 | $\delta E_{4}(0,0)/h$ |  $0.000(860)$ | B12 | $\delta E_{9}(1,1)/h$ | $0.000\,00(55)$
B2 | $\delta E_{5}(0,0)/h$ |  $0.000(84)$ | B13 | $\delta E_{1}(0,3)/h$ | $0.000(114)$
B3 | $\delta E_{1}(0,1)/h$ |  $0.000(116)$ | B14 | $\delta E_{2}(0,3)/h$ | $0.0000(82)$
B4 | $\delta E_{2}(0,1)/h$ |  $0.0000(83)$ | B15 | $\delta E_{6}(0,3)/h$ | $0.0000(20)$
B5 | $\delta E_{6}(0,1)/h$ |  $0.000(18)$ | B16 | $\delta E_{8}(0,3)/h$ | $0.000\,000(89)$
B6 | $\delta E_{8}(0,1)/h$ |  $0.000\,00(81)$ | B17 | $\delta E_{9}(0,3)/h$ | $0.000\,000(60)$
B7 | $\delta E_{9}(0,1)/h$ |  $0.000\,00(55)$ | B18 | $\delta E_{1}(9,3)/h$ | $0.000(62)$
B8 | $\delta E_{1}(1,1)/h$ |  $0.000(109)$ | B19 | $\delta E_{2}(9,3)/h$ | $0.0000(57)$
B9 | $\delta E_{2}(1,1)/h$ |  $0.0000(81)$ | B20 | $\delta E_{6}(9,3)/h$ | $0.0000(12)$
B10 | $\delta E_{6}(1,1)/h$ |  $0.000(17)$ | B21 | $\delta E_{8}(9,3)/h$ | $0.000\,000(58)$
B11 | $\delta E_{8}(1,1)/h$ |  $0.000\,00(78)$ | B22 | $\delta E_{9}(9,3)/h$ | $0.000\,000(49)$
Table 4: Input data for the additive energy corrections to account for the
theoretical uncertainties of hyperfine interaction coefficients.

###  4.2 Effect of unknown hyperfine correlation coefficients

As discussed in Section 2.1, the level of correlation between some of the
theoretical hyperfine coefficients (namely, $\delta E_{k}(v,L)$ and $\delta
E_{k}(v^{\prime},L^{\prime})$ with $k\neq 4,5)$ is not well known. We thus
varied their correlation coefficients between 0 and 1 to study their influence
on spin-averaged frequencies. These frequencies were found to be mostly
sensitive to the correlation between $\delta E_{1}(v,L)$ and $\delta
E_{1}(v^{\prime},L^{\prime})$, with a smaller effect from the correlation
between $\delta E_{6}(v,L)$ and $\delta E_{6}(v^{\prime},L^{\prime})$ (which
themselves are fully correlated with $\delta E_{7}(v,L)$ and $\delta
E_{7}(v^{\prime},L^{\prime})$, respectively). Correlations involving $E_{2}$,
$E_{3}$, $E_{8}$, and $E_{9}$ have a negligible impact. In order to estimate
the additional shift, $f_{\rm corr}(v,L,v^{\prime},L^{\prime})$, and its
uncertainty, $\delta f_{\rm corr}(v,L,v^{\prime},L^{\prime})$, caused by the
unknown correlation coefficients, we carried out a Monte-Carlo simulation
similar to [24], as further detailed in Appendix A. We find that the mean
shift of the simulated spin-averaged transition frequencies is well accounted
for when carrying out a single adjustment using correlation coefficients of
0.5. Furthermore, we determine the full widths of the simulated frequency
distributions to be 2.09 Hz, 144 Hz, and 237 Hz for the rotational, $v=0\to
1$, and $v=0\to 9$ transitions, respectively. We conservatively define $\delta
f_{\rm corr}(v,L,v^{\prime},L^{\prime})$ to be equal to the full width divided
by two. As will be shown below, these correlation-induced uncertainties are
considerably smaller than the frequency uncertainties resulting from the
adjustment itself.

The question is whether the uncertainties $\delta f_{\rm
corr}(v,L,v^{\prime},L^{\prime})$ are correlated (which would introduce
correlations between different transition frequencies $f_{\rm
SA}[(v,L)\\!\to\\!(v^{\prime},L^{\prime})]$ in addition to the correlations
that may result from the adjustment itself). We empirically find either strong
correlations (i.e. $r(\delta f_{\rm corr}(v,L,v^{\prime},L^{\prime}),\delta
f_{\rm
corr}(v^{\prime\prime},L^{\prime\prime},v^{\prime\prime\prime},L^{\prime\prime\prime}))=+1$)
or strong anticorrelations (i.e. $r(\delta f_{\rm
corr}(v,L,v^{\prime},L^{\prime}),\delta f_{\rm
corr}(v^{\prime\prime},L^{\prime\prime},v^{\prime\prime\prime},L^{\prime\prime\prime}))=-1$);
see Appendix A. We can thus construct a ’correlation-induced’ covariance
matrix from the correlation-induced uncertainties defined above, in
combination with the empirically found correlation coefficients.

###  4.3 Results and discussion

In Table 5, we compare our derived spin-averaged frequencies with those in the
literature. These literature values are based on the same experimental data as
used in our analysis; literature values based on specific subsets of the
experimental data (such as the value presented in [25]) are not considered
here. The rotational transition is shifted by about four standard deviations
with respect to the literature value of [14]. This is mainly due to the fact
that we make different assumptions regarding correlations between theoretical
hyperfine coefficients, especially those between $E_{4}(0,0)$ and
$E_{4}(0,1)$. As justified in Ref. [33], we assume perfect positive
correlation, whereas zero correlation was assumed in the initial analysis
[14]. The sensitivity of the rotational transition on the assumed level of
correlation was already observed in [24] (see Fig 4 (b) in that reference).
For the other transitions, the frequencies determined in this work are well
within the error bars of the previous determinations.

As can be seen in Table 5, the improvement of hyperfine structure theory [33]
has allowed the reduction of the uncertainties of the $v=0\to 1$ and $v=0\to
9$ transition frequencies with respect to their initial determinations [15,
16]. For the rotational transition, the precision has only slightly improved,
because in this case 6 spin components have been measured, making the
extracted frequency much less sensitive to theoretical uncertainties.

The normalised residuals of the adjustment (assuming a value of 0.5 for the
unknown correlation coefficients) are shown in Figure 1. They reveal
substantial discrepancies, with normalised residuals larger than two, between
the experimental data and the theoretical hyperfine structure:

  * •

the hyperfine components A1, A3, and A5 of the rotational transition (with
labels Ai defined in Table 3) have discrepancies of $3.0\sigma$ (121 Hz),
$2.1\sigma$ (35 Hz), and $-7.1\sigma$ ($-427$ Hz), respectively;

  * •

both components of the $v=0\to 9$ transition, A9 and A10, exhibit
discrepancies of $-6.9\sigma$ ($-4.5$ kHz) and $6.1\sigma$ (3.6 kHz),
respectively;

  * •

several theoretical spin coefficients, in particular $E_{1}(v=0,L=1)$ and
$E_{6}(v=0,L=1)$, with deviations of $4.2\sigma$ ($h\times 484$ Hz) and
$4.7\sigma$ ($h\times 85$ Hz), respectively. Deviations slightly above
2$\sigma$ are also found in $E_{6}(v=1,L=1)$, $E_{6}(v=0,L=3)$,
$E_{1}(v=9,L=3)$, and $E_{6}(v=9,L=3)$.

Discrepancies in the hyperfine structure of the $L=0\to 1$ and $v=0\to 9$
transitions had already been discussed in [27, 33]. Their origin is presently
unknown; they could be due to a problem in the theory, in the experiments, or
both. It is furthermore worth mentioning that experimental data on the
hyperfine structure of H${}_{2}^{+}$ are in good agreement with theoretical
predictions; see [33] and references therein.

We treat these deviations following the procedure used in the CODATA
adjustments (see, e.g., [1]). A multiplicative expansion factor is applied to
the initially assigned uncertainties of all (experimental and theoretical)
input data, such that the absolute values of all normalised residuals are
smaller than two. The required expansion factor is $\eta=3.56$. As a result,
the uncertainties of the spin-averaged frequencies are multiplied by the same
factor (third line in Table 5). The expanded uncertainties are one order of
magnitude larger than the frequency uncertainties caused by the unknown
correlation coefficients.

The spin-averaged frequencies obtained by the least-squares adjustment have
small but nonzero correlations between them due to correlations between
$\delta E_{k}$ in different rovibrational levels. The magnitude of the
corresponding correlation coefficients is less than 0.01, and their values
(and in some case also their signs) depend on the choice of hyperfine
correlation coefficients (Section 2.1). Therefore, the correlations between
the three adjusted spin-averaged transition frequencies have an uncertainty
due to the unknown hyperfine theory correlations. In Section 5 we show that
these correlations have a negligible impact on the fundamental constants that
may be derived from the $\text{HD}^{+}$ data, and we do not investigate this
particular source of uncertainty further here. However, for completeness we do
combine the covariance matrix resulting from the adjustment with the
correlation-induced covariance matrix that was constructed as described in
Section 4.2. The resulting (combined) uncertainties of the spin-averaged
transition frequencies are listed in the last line of Table 5, and their
(combined) correlation coefficients are presented in Table 8.

| $(0,0)\to(0,1)$ | $(0,0)\to(1,1)$ | $(0,3)\to(9,3)$
---|---|---|---
$f_{\rm SA}$ (previous) | 1 314 925 752.910(17) | 58 605 052 164.24(86) | 415 264 925 500.5(1.2)
$f_{\rm SA}$ ($\eta=1$) | 1 314 925 752.978(14) | 58 605 052 164.14(16) | 415 264 925 501.3(0.4)
$f_{\rm SA}$ ($\eta=3.56$) | 1 314 925 752.978(48) | 58 605 052 164.14(55) | 415 264 925 501.3(1.6)
$f_{\rm SA}$ ($\eta=3.56$ and corr. ind. unc.) | 1 314 925 752.978(48) | 58 605 052 164.14(56) | 415 264 925 501.3(1.6)

Table 5: Spin-averaged transition frequencies $f_{\rm SA}$ (in kHz). Previous
determinations from the original publications [14, 15, 16] are given in the
first line. The results of the least-squares adjustment performed in the
present work are given in line 2. These results were obtained with an
expansion factor $\eta=1$ (i.e. no expansion factor applied). The spin-
averaged frequencies in line 3 are obtained by applying an expansion factor
$\eta=3.56$ to the uncertainties of all (experimental and theoretical) input
data. Finally, line 4 shows the frequencies for $\eta=3.56$ and including the
hyperfine-correlation-induced uncertainty; see text for details. The
frequencies on line 4 are our recommended values.  Figure 1:  Normalised
residuals of the 32 input data for the adjustment of spin-averaged transition
frequencies. Labels Ai and Bi follow those defined in Tables 3 and 4.

##  5 Implications for the determination of particle masses

Using the spin-averaged frequencies determined in this work, we can now assess
the potential contribution of $\text{HD}^{+}$ spectroscopy to the
determination of fundamental constants, in particular the relative atomic
masses $A_{\rm r}({\rm e})$, $A_{\rm r}({\rm p})$, $A_{\rm r}({\rm d})$, and
related mass ratios. To do this, we use a least-squares adjustment following
the same procedure as in the previous section, and compare three different
scenarios:

  * •

Adjustment 1 only includes Penning trap measurements related to the electron,
proton, and deuteron relative atomic masses;

  * •

Adjustment 2 only includes $\text{HD}^{+}$ spectroscopic data and values of
fundamental constants (other than atomic masses) involved in the theoretical
energy levels of $\text{HD}^{+}$;

  * •

Adjustment 3 includes all input data of Adjustments 1 and 2.

Theoretical rovibrational transition frequencies of $\text{HD}^{+}$ depend on
[20]: the Rydberg constant ($R_{\infty}$), nucleus-to-electron mass ratios
$\lambda_{\rm p}\equiv m_{\rm p}/m_{\rm e}$ and $\lambda_{\rm d}\equiv m_{\rm
d}/m_{\rm e}$, the charge radii of the proton $r_{\rm p}$ and deuteron $r_{\rm
d}$, and the fine-structure constant $\alpha$. The dependence on $\alpha$ is
very weak because it only enters in the relativistic and QED corrections, and
can be safely neglected without loss of precision. As noted in previous works,
e.g., [17, 14], the mass dependence can to a good approximation be reduced to
a dependence on a single parameter $\lambda_{\rm pd}\equiv\mu_{\rm pd}/m_{\rm
e}$, where $\mu_{\rm pd}$ is the proton-deuteron reduced mass. This is exactly
true in the adiabatic approximation, but, due to nonadiabatic effects, there
is actually a much weaker dependence on a second mass ratio, for example
$\lambda_{\rm d}$. In our Adjustment 2, we neglect this weak dependence,
allowing $\lambda_{\rm pd}$ to be determined from $\text{HD}^{+}$
spectroscopic data alone, as done in [14, 16]. Although $\lambda_{\rm pd}$ is
not an adjustable parameter in Adjustment 1, it is possible to infer a value
of $\lambda_{\rm pd}$ from the adjusted values of $A_{\rm r}({\rm e})$,
$A_{\rm r}({\rm p})$, and $A_{\rm r}({\rm d})$. The relative impact of
$\text{HD}^{+}$ spectroscopy and Penning-trap measurements on future values of
particle mass ratios can then be assessed by comparing the determinations of
$\lambda_{\rm pd}$ (and their uncertainties) obtained from Adjustments 1 and
2. To further illustrate the effect of including $\text{HD}^{+}$ data, we also
compare the values of $\lambda_{\rm p}$ (= $m_{\rm p}/m_{\rm e}$) inferred
from Adjustments 1 and 3.

In the perspective of a global adjustment of fundamental constants, it is
preferable to take the dependence of the theoretical $\text{HD}^{+}$
transition frequencies on both parameters $\lambda_{\rm pd}$, $\lambda_{\rm
d}$ into account in order to avoid any unnecessary loss of precision.
Furthermore, for the global Adjustment 3, we have found it more convenient to
parametrise the mass dependence of $\text{HD}^{+}$ transition frequencies in
terms of the quantities that are treated as adjusted parameters in the CODATA
adjustments [1], i.e., the relative atomic masses $A_{\rm r}({\rm e})$,
$A_{\rm r}({\rm p})$, and $A_{\rm r}({\rm d})$.

###  5.1 Adjustment details

Let us briefly describe Adjustment 1. It includes the 2018 CODATA value of
electron’s relative atomic mass, for which no new measurement has been
reported since then. For the proton relative atomic mass, we use the value of
$A_{\rm r}(^{1}{\rm H})$ from the latest (2020) Atomic Mass Evaluation (AME)
[36, 37], corrected for the electron mass and for the theoretical binding
energy. This value takes into account recent high-precision Penning trap data
[5, 6], except for a very precise measurement of the deuteron-to-proton mass
ratio, $m_{\rm d}/m_{\rm p}$, performed in 2021 [7]. We thus also include the
latter, together with the value of the deuteron mass deduced from the 2020 AME
value of $A_{\rm r}(^{2}{\rm H})$ (again corrected for the electron mass and
the binding energy), improving the proton mass determination. Adjustment 1
thus comprises $N=4$ input data and $M=3$ adjusted parameters, the relative
atomic masses of the electron, proton and deuteron.

The other adjustments include $\text{HD}^{+}$ spectroscopic data. Similarly to
the hyperfine structure, theoretical uncertainties of spin-averaged transition
frequencies are accounted for by introducing additive corrections $\delta
f^{\rm th}(v,L,v^{\prime},L^{\prime})$ to the theoretical frequencies. They
are treated as adjusted constants, and input data with a zero value and an
uncertainty equal to the theoretical uncertainty of $f_{\rm SA}^{\rm
th}[(v,L)\to(v^{\prime},L^{\prime})]$ are included (B1-B3 in Table 7). Their
correlation coefficients (listed in Table 8) are estimated using the results
of [19]. Correlation coefficients are found to be close to 1, because the
uncertainties are dominated by unevaluated QED terms, whereas numerical
uncertainties are negligibly small.

In Adjustment 2, each of the three spin-averaged frequencies determined in the
previous section (inputs A1-A3 in Table 7) is associated with an observational
equation of the form

$\begin{split}&\,f_{\rm SA}[(v,L)\to(v^{\prime},L^{\prime})]\\\ &\doteq f_{\rm
SA}^{\rm th}[(v,L)\to(v^{\prime},L^{\prime})]+\delta f^{\rm
th}(v,L,v^{\prime},L^{\prime})\\\
&\quad+\beta_{R_{\infty}}c(R_{\infty}\\!-\\!R_{\infty 0})+\eta_{r_{\rm
p}}(r_{\rm p}^{2}\\!-\\!r_{{\rm p}0}^{2})+\eta_{r_{\rm d}}(r_{\rm
d}^{2}\\!-\\!r_{{\rm d}0}^{2})\\\ &\quad+\beta_{\lambda_{\rm pd}}(\lambda_{\rm
pd}\\!-\\!\lambda_{\rm pd0})+\beta_{\lambda_{\rm d}}(\lambda_{\rm
d}\\!-\\!\lambda_{\rm d0}).\end{split}$ (8)

Here, $f_{\rm SA}^{\rm th}[(v,L)\to(v^{\prime},L^{\prime})]$ is the reference
value of the theoretical transition frequency [19], calculated with 2018
CODATA values of fundamental constants. The reference (2018 CODATA) values of
$A_{\rm r}({\rm e})$, $R_{\infty}$, $r_{\rm p}$, and $r_{\rm d}$ are given in
Table 7. Those of the proton and deuteron atomic masses are: $A^{\rm
CODATA2018}_{\rm r}({\rm p})=1.007\,276\,466\,621$ and $A^{\rm
CODATA2018}_{\rm r}({\rm d})=2.013\,553\,212\,745$. The mass ratios
$\lambda_{\rm pd}$ and $\lambda_{\rm d}$ are calculated using $\lambda_{\rm
pd}=A_{\rm r}({\rm p})A_{\rm r}({\rm d})/(A_{\rm r}({\rm e})[A_{\rm r}({\rm
p})+A_{\rm r}({\rm d})])$ and $\lambda_{\rm d}=A_{\rm r}({\rm d})/A_{\rm
r}({\rm e})$ (and similar for the reference values $\lambda_{\rm pd0}$ and
$\lambda_{\rm d0}$).

As explained above, we furthermore have observational equations of the form

$\delta f(v,L,v^{\prime},L^{\prime})\doteq\delta f^{\rm
th}(v,L,v^{\prime},L^{\prime}),$ (9)

with input value $\delta f(v,L,v^{\prime},L^{\prime})=0$, to deal with
theoretical uncertainties.

Finally, we require observational equations for $R_{\infty}$, $r_{\rm p}$,
$r_{\rm d}$ as well as relative atomic masses and mass ratios. They are
written as

$\displaystyle R_{\infty}^{\rm CODATA2018}\doteq R_{\infty}\,,$ (10a)
$\displaystyle r_{\rm p}^{\rm CODATA2018}\doteq r_{\rm p}\,,$ (10b)

etc., where the input data $R_{\infty}^{\rm CODATA2018}$, $r_{\rm p}^{\rm
CODATA2018}$, $\ldots$ are found in Table 7 under labels C1-C7 (note that only
C5-C7 are used in Adjustment 2). The correlation coefficients among these
input data are given in Table 8.

The coefficients $\beta$, $\eta$ in Eq. (8) depend on
$(v,L,v^{\prime},L^{\prime})$, but for brevity this dependence is not
indicated. $\beta_{R_{\infty}}$ is the linear sensitivity coefficient to the
Rydberg constant, and is simply given by

$\beta_{R_{\infty}}(v,L,v^{\prime},L^{\prime})=\frac{f_{\rm SA}^{\rm
th}[(v,L)\to(v^{\prime},L^{\prime})]}{cR_{\infty}}.$ (11)

In the dependence on the nuclear charge radii, following CODATA practice we
only take into account the leading-order finite-size correction (see Eq. (6)
of [38]). Higher-order finite-size corrections for the deuteron, which are
included in the theoretical predictions [19], are taken as fixed, and the
corresponding uncertainties are taken into account in $\delta f$. This yields
a quadratic dependence on $r_{\rm p}$ and $r_{\rm d}$, with sensitivity
coefficients

$\displaystyle\eta_{r_{i}}(v,L,v^{\prime},L^{\prime})$
$\displaystyle=\eta_{r_{i}}(v^{\prime},L^{\prime})-\eta_{r_{i}}(v,L),$ (12a)
$\displaystyle\eta_{r_{i}}(v,L)$ $\displaystyle=(2cR_{\infty})\frac{2\pi
a_{0}}{3}\left\langle\delta(\mathbf{x}_{i})\right\rangle\,$ (12b)

with $i=\text{p,d}$. $a_{0}$ is Bohr’s radius, and $\mathbf{x}_{\rm p}$
($\mathbf{x}_{\rm d}$) is the position vector of the electron with respect to
the proton (deuteron). Brackets denote expectation values over the
nonrelativistic wavefunction of the $(v,L)$ rovibrational state, which are
calculated numerically with high precision using variational wavefunctions
[18].

Finally, in order to evaluate the linear sensitivity coefficients with respect
to the mass ratios, $\beta_{\lambda_{\rm pd}}$ and $\beta_{\lambda_{\rm d}}$,
it suffices to consider the nonrelativistic energy levels, which yields a
relative accuracy on the order of $\alpha^{2}$ for these coefficients.
Following the approach of [39], they can be expressed as

$\displaystyle\beta_{\lambda_{i}}(v,L,v^{\prime},L^{\prime})$
$\displaystyle=\beta_{\lambda_{i}}(v^{\prime},L^{\prime})-\beta_{\lambda_{i}}(v,L),$
(13a) $\displaystyle\beta_{\lambda_{\rm pd}}(v,L)$
$\displaystyle=(2cR_{\infty})\frac{a_{0}^{2}}{2\hbar\lambda_{\rm
pd}^{2}}\left\langle\nabla_{\mathbf{X}}^{2}\right\rangle,$ (13b)
$\displaystyle\beta_{\lambda_{\rm d}}(v,L)$
$\displaystyle=(2cR_{\infty})\frac{a_{0}^{2}}{\hbar\lambda_{\rm
d}^{2}}\left(\frac{1}{2}\left\langle\nabla_{\mathbf{x}_{\rm
d}}^{2}\right\rangle\\!+\\!\left\langle\nabla_{\mathbf{X}}\cdot\nabla_{\mathbf{x}_{\rm
d}}\right\rangle\right),$ (13c)

where $\mathbf{X}$ is the position of the proton with respect to the deuteron.
The expectation values are calculated numerically with high precision using
variational wavefunctions [39].

For Adjustment 3, the mass dependence is parametrised in terms of the relative
atomic masses, $A_{\rm r}({\rm e})$, $A_{\rm r}({\rm p})$, and $A_{\rm r}({\rm
d})$. The observational equation (8) becomes

$\begin{split}&\,f_{\rm SA}[(v,L)\to(v^{\prime},L^{\prime})]\\\ &\doteq f_{\rm
SA}^{\rm th}[(v,L)\to(v^{\prime},L^{\prime})]+\delta
f(v,L,v^{\prime},L^{\prime})\\\
&\quad+\beta_{R_{\infty}}c(R_{\infty}\\!-\\!R_{\infty 0})+\eta_{r_{\rm
p}}(r_{\rm p}^{2}\\!-\\!r_{{\rm p}0}^{2})+\eta_{r_{\rm d}}(r_{\rm
d}^{2}\\!-\\!r_{{\rm d}0}^{2})\\\ &\quad+\beta_{A_{\rm r}({\rm e})}(A_{\rm
r}({\rm e})\\!-\\!A_{\rm r}({\rm e})_{0})+\beta_{A_{\rm r}({\rm p})}(A_{\rm
r}({\rm p})\\!-\\!A_{\rm r}({\rm p})_{0})+\beta_{A_{\rm r}({\rm d})}(A_{\rm
r}({\rm d}))\\!-\\!A_{\rm r}({\rm d})_{0})\,.\end{split}$ (14)

Here, the observational equations, values, and correlation coefficients for
$\delta f$ are the same as for Adjustment 2. The sensitivity coefficients
$\beta_{A_{\rm r}({\rm\it i})}$ ($i={\rm e,p,d}$) with respect to the relative
atomic masses are given by

$\beta_{A_{\rm r}({\rm\it
i})}(v,L,v^{\prime},L^{\prime})=\frac{\partial\lambda_{\rm pd}}{\partial
A_{\rm r}({\rm\it i})}\beta_{\lambda_{\rm
pd}}(v,L,v^{\prime},L^{\prime})+\frac{\partial\lambda_{\rm d}}{\partial A_{\rm
r}({\rm\it i})}\beta_{\lambda_{\rm d}}(v,L,v^{\prime},L^{\prime}).$ (15)

The theoretical frequencies and all the sensitivity coefficients appearing in
Eqs. (8) and (14) are given in Table 6. The dependence of the sensitivity
coefficients on fundamental constants is weak, so that the values of Table 6
can be safely used during all iterations of the least-squares adjustment
procedure. Note that the theoretical frequencies differ very slightly from
those given in [19] because they are calculated with 2018 CODATA values of
$A_{\rm r}({\rm e})$, $A_{\rm r}({\rm p})$ and $A_{\rm r}({\rm d})$, whereas
the values of [19] were obtained with 2018 CODATA values of $m_{\rm p}/m_{\rm
e}$ and $m_{\rm d}/m_{\rm e}$. When switching between these two
parametrizations, round-off errors in the published recommended values cause
tiny shifts in the nonrelativistic energy levels.

In total, Adjustment 2 contains $N=9$ input data and $M=7$ adjusted
parameters: $\lambda_{\rm pd}$, $R_{\infty}$, $r_{\rm p}$, $r_{\rm d}$, and
the additive corrections for the three rovibrational transition frequencies.

Adjustment 3 combines the input data of the first two adjustments, with $N=13$
input data and $M=9$ adjusted parameters.

Transition | $f_{\rm SA}^{\rm theor}$ | $\beta_{cR_{\infty}}$ | $\eta_{r_{p}}$ | $\eta_{r_{d}}$
---|---|---|---|---
| (kHz) |  | (kHz.fm-2) | (kHz.fm-2)
$(0,0)\to(0,1)$ | 1 314 925 752.929 | 3.9969[-04] | -9.0991[-01] | -9.0991[-01]
$(0,0)\to(1,1)$ | 58 605 052 163.88 | 1.7814[-02] | -2.4253[+01] | -2.4220[+01]
$(0,3)\to(9,3)$ | 415 264 925 502.7 | 1.2623[-01] | -1.5940[+02] | -1.5850[+02]

Transition | $\beta_{\lambda_{pd}}$ | $\beta_{\lambda_{d}}$ | $\beta_{A_{\rm r}({\rm e})}$ | $\beta_{A_{\rm r}({\rm p})}$ | $\beta_{A_{\rm r}({\rm d})}$
---|---|---|---|---|---
| (kHz) | (kHz) | (kHz.u-1) | (kHz.u-1) | (kHz.u-1)
$(0,0)\to(0,1)$ | -1.0601[+06] | -3.2126[+01] | 2.3653[+12] | -8.5857[+08] | -2.1492[+08]
$(0,0)\to(1,1)$ | -2.3201[+07] | -7.0874[+02] | 5.1767[+13] | -1.8790[+10] | -4.7036[+09]
$(0,3)\to(9,3)$ | -1.2580[+08] | -3.9569[+03] | 2.6664[+14] | -9.6785[+10] | -2.4227[+10]

Table 6: Reference theoretical values and sensitivity coefficients of
$\text{HD}^{+}$ transition frequencies. The theoretical frequencies are
calculated using 2018 CODATA values of $A_{\rm r}({\rm e})$, $A_{\rm r}({\rm
p})$, $A_{\rm r}({\rm d})$, $cR_{\infty}$, $r_{p}$, $r_{d}$, and $\alpha$. The
sensitivity coefficient for the Rydberg constant $\beta_{cR_{\infty}}$ is
dimensionless. u stands for the relative atomic mass unit.

Label | Input datum | Value | Rel. Uncertainty | Reference
---|---|---|---|---
A1 | $f_{\rm SA}[(0\\!,\\!0)\\!\to\\!(0\\!,\\!1)]$ | 1 314 925 752.978(48) kHz | $3.7\times 10^{-11}$ | this work
A2 | $f_{\rm SA}[(0\\!,\\!0)\\!\to\\!(1\\!,\\!1)]$ | 58 605 052 164.14(56) kHz | $9.6\times 10^{-12}$ | this work
A3 | $f_{\rm SA}[(0\\!,\\!3)\\!\to\\!(9\\!,\\!3)]$ | 415 264 925 501.3(1.6) kHz | $3.9\times 10^{-12}$ | this work
B1 | $\delta f(0,0,0,1)$ | 0.000(19) kHz |  | [19]
B2 | $\delta f(0,0,1,1)$ | 0.00(49) kHz |  | [19]
B3 | $\delta f(0,3,9,3)$ | 0.0(3.2) kHz |  | [19]
C1 |  $A_{\rm r}({\rm e})$ (CODATA 2018) | 5.485 799 090 65(16)$\times 10^{-4}$ u | $2.9\times 10^{-11}$ | [1]
C2 |  $A_{\rm r}({\rm d})$ (AMDC 2020) | 2.013 553 212 537(15) u | $7.5\times 10^{-12}$ | [36]
C3 |  $A_{\rm r}({\rm p})$ (AMDC 2020) | 1.007 276 466 587(14) u | $1.4\times 10^{-11}$ | [36]
C4 |  $m_{\rm d}/m_{\rm p}$ (Fink 2021) | 1.999 007 501 272(9) | $4.5\times 10^{-12}$ | [7]
C5 |  $cR_{\infty}$ (CODATA 2018) | 3 289 841 960 250.8(6.4) kHz | $1.9\times 10^{-12}$ | [1]
C6 |  $r_{\rm p}$ (CODATA 2018) | 0.8414(19) fm | $2.2\times 10^{-3}$ | [1]
C7 |  $r_{\rm d}$ (CODATA 2018) | 2.12799(74) fm | $3.5\times 10^{-4}$ | [1]

Table 7: Input data for the adjustments discussed in Section 5. Entries C1 and
C5-C7 also serve as reference values in equations (8) and (14).

Correlation coefficients
---
$r({\rm A1},{\rm A2})=0.00036$ | $r({\rm A1},{\rm A3})=0.00644$ | $r({\rm A2},{\rm A3})=0.00741$ | $r({\rm B1},{\rm B2})=0.99570$
$r({\rm B1},{\rm B3})=0.95733$ | $r({\rm B2},{\rm B3})=0.97998$ | $r({\rm C1},{\rm C5})=0.00704$ | $r({\rm C1},{\rm C6})=-0.00133$
$r({\rm C1},{\rm C7})=0.00317$ | $r({\rm C2},{\rm C3})=0.31986$ | $r({\rm C5},{\rm C6})=0.88592$ | $r({\rm C5},{\rm C7})=0.90366$
$r({\rm C6},{\rm C7})=0.99165$ |  |  |

Table 8: Values of all nonzero correlation coefficients in the input data of
Table 7.

###  5.2 Results and discussion

Adjustment | Input data | Adjusted value |  $u_{\rm r}$ ($10^{-11}$)
---|---|---|---
1 | C1-C4 | $\lambda_{\rm pd}=1\,223.899\,228\,646(37)$ | 3.0
2 | A,B,C5-C7 | $\lambda_{\rm pd}=1\,223.899\,228\,719(26)$ | 2.1
1 | C1-C4 | $\lambda_{\rm p}=1\,836.152\,673\,353(56)$ | 3.0
3 | All | $\lambda_{\rm p}=1\,836.152\,673\,423(33)$ | 1.8
3 | All |  $A_{\rm r}({\rm e})=5.485\,799\,090\,46(10)\times 10^{-4}$ u | 1.8
Table 9:  Adjustment results. $u_{\rm r}$ stands for relative uncertainty.

The determinations of $\lambda_{\rm pd}$ and of the proton-electron mass
ratio, $\lambda_{\rm p}$, from the adjustments described above are shown in
Table 9.

Adjustment 1 illustrates the potential precision permitted by Penning trap
measurements alone. The relative uncertainty of $\lambda_{\rm pd}$ is found to
be $3.0\times 10^{-11}$, mainly limited by the electron’s relative atomic mass
that has a relative uncertainty of $2.9\times 10^{-11}$ (see Table 7). The
contribution from the proton and deuteron relative atomic masses to the
uncertainty of $\lambda_{\rm pd}$ is $7.6\times 10^{-12}$.

Adjustment 2 provides an alternative determination of $\lambda_{\rm pd}$ from
$\text{HD}^{+}$ spectroscopic data. Its uncertainty is slightly smaller
($2.1\times 10^{-11}$), and the obtained values are in reasonable agreement,
with a mild (1.6$\sigma$) tension. Since the precision of the first adjustment
is limited by the electron’s atomic mass, the smaller uncertainty from
$\text{HD}^{+}$ data indicates that the impact of $\text{HD}^{+}$ on the
global CODATA adjustment would primarily be to improve the precision of
$A_{\rm r}({\rm e})$.

This is verified in the last adjustment (Adjustment 3), which illustrates the
potential improvement that can be obtained by combining Penning trap with
$\text{HD}^{+}$ spectroscopy data. Whereas the proton and deuteron atomic
masses are very precisely determined by mass spectrometry, $\text{HD}^{+}$
spectroscopy, being sensitive to the nucleus-to-electron mass ratios, provides
a link to the electron mass, allowing to improve the precision of $A_{\rm
r}({\rm e})$ and $m_{\rm p}/m_{\rm e}$ to $1.8\times 10^{-11}$, which for the
latter would represent an improvement by more than a factor of three over its
2018 CODATA uncertainty. It is worth noting that the adjustment reveals no
significant tension in the data, with all normalised residuals being smaller
than 1.2 in absolute value. The found values of $A_{\rm r}({\rm e})$ and
$m_{\rm p}/m_{\rm e}$ are furthermore in good agreement with the 2018 CODATA
values.

We carried out Adjustment 3 taking into account the weak correlations between
the adjusted experimental spin-averaged frequencies, discussed in Section 4.3.
To assess the influence of these correlations, we repeated Adjustment 3
assuming zero correlation between the spin-averaged frequencies. The value of
$m_{\rm p}/m_{\rm e}$ obtained from this adjustment is identical to the one
shown in the last lines of Table 9, indicating a negligible impact of these
correlations on the determination of fundamental constants from recent
$\text{HD}^{+}$ theory and measurements.

##  6 Conclusion

In conclusion, we have carried out a comprehensive analysis of the available
high-precision spectroscopic data of rotational and vibrational transitions in
$\text{HD}^{+}$, with the aim of providing a set of spin-averaged rotational-
vibrational transition frequencies that can serve as input data for future
CODATA adjustments of the fundamental physical constants. A crucial part of
the analysis involves the use of theoretically computed hyperfine structure to
remove hyperfine shifts from the measured transition frequencies, taking into
account the correlations between theroretical spin coefficients. The
experimental input data are complemented by the most recent theoretical
predictions of the spin-averaged transition frequencies, and by linearised
expressions that parametrise these theoretical predictions in terms of
relevant fundamental constants. Uncertainties and correlation coefficients
pertaining to the input data were also provided, which completes the set of
observational equations and covariances needed for future CODATA adjustments.
To illustrate the potential of the $\text{HD}^{+}$ data, we have carried out
adjustments of the electron, proton and deuteron relative atomic mass using
state-of-the-art data from Penning-trap measurements as well as the
$\text{HD}^{+}$ data presented here. We found that the $\text{HD}^{+}$ data
have a particularly large impact on the precision of the electron relative
atomic mass which, together with the precise values of the proton and deuteron
relative atomic masses from Penning-trap measurements, may improve the
precision of $m_{\rm p}/m_{\rm e}$ by more than a factor of three compared to
the current 2018 CODATA value.

## Acknowledgement(s)

The authors warmly thank Wim Ubachs for his invaluable contributions and
continued support to the Amsterdam-Paris HD+ research collaboration. Eite
Tiesinga and Fabian Heiße are acknowledged for helpful comments to the
manuscript.

## Disclosure statement

The authors report there are no competing interests to declare.

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##  Appendix A Monte-Carlo simulation

To assess the uncertainty associated with the unknown correlation coefficients
$r[E_{k}(v,L),E_{k}(v^{\prime},L^{\prime})]\equiv r_{k}$ (with $k\neq 4,5$) we
carry out a Monte-Carlo simulation analogous to [24]. As explained in Section
4.1, we set $r_{2}=r_{3}$ and $r_{6}=r_{7}$. We subsequently choose random
values from the interval $[0,1)$ for each of the coefficients of the vector
$(r_{1},r_{2},r_{6},r_{8},r_{9})$, and carry out the adjustment of the spin-
averaged frequencies and $\delta E_{k}(v\\!,\\!L)$. This is repeated 500
times, and in addition for the vectors $(0,0,0,0,0)$ and
$(1-\epsilon,1-\epsilon,1-\epsilon,1-\epsilon,1-\epsilon)$, with
$\epsilon=10^{-6}$ to avoid singular covariance matrices. Adjustments are
carried out numerically using 100 digits of precision. Histograms of the
Monte-Carlo distributions of adjusted transition frequencies are shown in
Figure A1. The full widths of the frequency distributions are 2.09 Hz, 144 Hz,
and 237 Hz for the $(v,L)$: $(0,0)\rightarrow(0,1)$, $(v,L)$:
$(0,0)\rightarrow(1,1)$ and $(v,L)$: $(0,3)\rightarrow(9,3)$ transitions,
respectively; the deviation of the mean spin-averaged frequency from the
frequency obtained for the $(0.5,0.5,0.5,0.5,0.5)$ case is 0.05 Hz, $-3$ Hz,
and $-2$ Hz (i.e. less than $3\%$ of the full width). Figure A1 furthermore
shows the dependence of the correlation-induced shift if all correlation
coefficients are varied as $(\xi,\xi,\xi,\xi,\xi)$, with $\xi$ between 0 and
1. Comparing the shifts of the three transitions shows that the shift of the
rotational transition is anticorrelated with the shift of the $v=0\to 1$ and
$v=0\to 9$ transitions, whereas the shifts for the latter two are correlated.

Figure A1: Distribution of correlation-induced shifts of spin-averaged
frequencies obtained from a Monte-Carlo simulation consisting of 502 runs.
Here the components of the vector $(r_{1},r_{2},r_{6},r_{8},r_{9})$ are varied
together as $(\xi,\xi,\xi,\xi,\xi)$; for example, $\xi=0.5$ stands for the
vector $(0.5,0.5,0.5,0.5,0.5)$ (a) Distribution for the $(v,L)$:
$(0,0)\rightarrow(0,1)$ transition. (b) Distribution for the $(v,L)$:
$(0,0)\rightarrow(1,1)$ transition. (c) Distribution for the $(v,L)$:
$(0,3)\rightarrow(9,3)$ transition. Dashed vertical lines indicate the mean
value of the frequency distribution. (d) Frequency shift of the $(v,L)$:
$(0,0)\rightarrow(0,1)$ transition versus $\xi$, with all components of the
vector $(r_{1},r_{2},r_{6},r_{8},r_{9})$ being varied together as
$(\xi,\xi,\xi,\xi,\xi)$. (e) Same as (d), but for the $(v,L)$:
$(0,0)\rightarrow(1,1)$ transition. (f) Same as (d), but for the $(v,L)$:
$(0,3)\rightarrow(9,3)$ transition.