Quantum Electronic StructureDecember 5, 2025

From First-Principles to Quantum Electrodynamics: Pushing the Limits of Theory with the Hydrogen Molecule
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Krzysztof Pachucki*
Krzysztof Pachucki
Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
*Email: [email protected]
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Jacek Komasa*
Jacek Komasa
Faculty of Chemistry, Adam Mickiewicz University, Uniwersytetu Poznanskiego 8, 61-614 Poznań, Poland
*Email: [email protected]
More by Jacek Komasa
https://orcid.org/0000-0002-9876-1304

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Journal of Chemical Theory and Computation

Cite this: J. Chem. Theory Comput. 2025, 21, 24, 12664–12673

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https://doi.org/10.1021/acs.jctc.5c01702

Published December 5, 2025

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Abstract

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Modern spectroscopic techniques enable the determination of the spacing between rovibrational levels of H₂ with a relative accuracy of approximately 10^–11. At this extreme level of precision, subtle quantum electrodynamic (QED) effects, such as electron self-interaction and vacuum polarization, are probed. A theoretical model aiming to achieve similar accuracy must precisely describe not only these relatively small QED effects but also the more significant contributions related to electron correlation, coupling between electronic and nuclear motions, and relativistic effects. Although the hydrogen molecule exhibits most of the phenomena found in larger molecules, it is simple enough to meet the requirements mentioned above. In this article, we report on enhancements to the current capabilities of quantum mechanical calculations for the hydrogen molecule. We present a method based on exponential functions that fully captures electron correlation or, more broadly, interparticle correlation, enabling a comprehensive description of effects related to nuclear motion. Specifically, we solve the four-particle Schrödinger equation without invoking commonly used approximations such as the one-electron or the Born–Oppenheimer approximation. The only source of nonrelativistic energy error comes from the finite size of the basis set. The explicitly correlated nonadiabatic wave function used here is then employed to determine the relativistic and QED effects. As a result, the dissociation energy for the lowest rovibrational levels in the electronic ground state of H₂ has been obtained with a relative accuracy of 7 × 10^–10, while the frequencies of intervals between these levels have been determined with sub-MHz accuracy, corresponding to a relative accuracy of 3 × 10^–9. In consequence, the discrepancies between the highest precision measurements and earlier theoretical predictions have been resolved.

This publication is licensed for personal use by The American Chemical Society.

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Special Issue

Published as part of Journal of Chemical Theory and Computation special issue “100 Years of Quantum Mechanics-All About Molecules”.

1. Introduction

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The hydrogen molecule has served as a fundamental benchmark for theoretical calculations since the advent of quantum mechanics more than a hundred years ago. In 1927, Heitler and London (1) provided the first qualitative explanation of molecular binding based on quantum mechanical principles. A few years later, James and Coolidge (2) reached an agreement with the then-available experimental data using an explicitly correlated wave function. Early calculations, including those by Kołos and Roothaan, (3) were based on the clamped nucleus approximation. Efforts to incorporate nuclear motion began with the evaluation of adiabatic and nonadiabatic corrections, initiated by Kołos and Wolniewicz in the early 1960s. (4−6) The most accurate calculations of these corrections to date have been performed within the framework of nonadiabatic perturbation theory (NAPT), (7) employing explicitly correlated exponential wave functions and expanding the perturbational series up to the second order in the electron-to-nucleus mass ratio. (8) The first direct nonadiabatic calculation of molecular energy, bypassing the intermediate step of evaluating internuclear potentials, is also attributed to Kołos and Wolniewicz, who used a product of their electronic wave function and a harmonic oscillator nuclear function. (9) A breakthrough in accurate direct nonadiabatic calculations of the nonrelativistic component of the ground-state energy was achieved by Kinghorn and Adamowicz, (10) who employed explicitly correlated Gaussian (ECG) functions, reaching the relative accuracy of 7 × 10^–9. The accuracy was later improved to 3 × 10^–11 by Puchalski et al., (11) also using ECGs. Currently, the most precise direct nonadiabatic results are obtained with exponential-type wave functions, yielding the nonrelativistic dissociation energy with an accuracy of 10^–8 cm^–1, corresponding to the relative precision of 10^–13. (12)

In parallel with studies on the coupling of nuclear and electronic motion, attempts were made to estimate the contribution of relativistic effects to the energy of the hydrogen molecule. In 1961, Kołos and Wolniewicz published a pioneering work (4) that provided the first estimation of these relativistic effects within the framework of the Born–Oppenheimer (BO) approximation. Relativistic calculations were not pursued until 1993, when Wolniewicz presented ground-state relativistic energies augmented by approximate QED corrections. (13) A significant advancement occurred in 2009 with the introduction of explicitly correlated Gaussian (ECG) functions for evaluating relativistic matrix elements, which enabled accurate determination of the dissociation energy, including the complete leading-order quantum electrodynamic (QED) correction. (14,15) The development of the so-called rECG functions, incorporating the cusp 1 + r₁₂/2 prefactor, facilitated efficient quadrature of nonstandard two-center integrals (16) and enabled evaluation of the higher-order QED correction potential. Subsequent studies addressed finite nuclear mass relativistic corrections using NAPT, leading to the derivation of the relativistic recoil potential, (17) followed by improvements in the numerical accuracy of the QED correction components. (18)

A fully nonadiabatic treatment of the recoil effect in the relativistic correction for H₂, incorporating finite nuclear mass terms directly into the relativistic operators, was first rigorously implemented in 2018. (11,19) The complete expression for the leading nonadiabatic QED correction was subsequently introduced in 2019. (11) Nonetheless, the numerical evaluation of both types of corrections remained confined to the ground rovibrational state.

A more detailed account of the centennial history behind the effort to determine the precise total energy of the hydrogen molecule can be found in the review. (20) A key aspect of this theoretical research has been the ongoing interaction with the experimental results. Over the years, the fruitful interplay between theory and experiment has led to an increased precision in both fields. Of particular interest are measurable separations between energy levels including the dissociation energy. Thanks to the long lifetimes of rovibrational states, counted in days, (21) H₂ is an attractive subject for highly precise spectroscopic measurements. The highly accurate rovibrational ladder of energy levels and the infrared (IR) spectrum of the hydrogen molecule are widely utilized across various disciplines, including astronomy, (22−26) plasma physics, (27) physical chemistry, (28,29) and metrology. (30,31)

Throughout a century of research on the hydrogen molecule, the agreement between theory and experiment has undergone continuous, albeit diminishing, fluctuations. For instance, just a decade ago, the theoretical predictions and measurements of the energy of rovibrational transitions in H₂ isotopologues agreed to within 10 MHz (≈0.00033 cm^–1). (32−52) This precision opened the possibility for testing QED effects on rotational transitions, (53) setting bounds on a hypothetical fifth force, (54) and on the number of extra spatial dimensions. (55,56) Recent advancements in spectroscopic techniques have enabled measurements to achieve an accuracy of the order of 10 kHz. (57−73) As a result, discrepancies between theory and experiment at the level of several megahertz have been revealed. It was suspected that the incomplete treatment of relativistic and QED recoil effects could be the main reason for these discrepancies.

In this paper, we report the results of our calculations, which were performed using a nonadiabatic exponential wave function. Such functions possess correct asymptotic properties but are extremely rarely used in molecular calculations due to difficulties in evaluating corresponding integrals. We have overcome these difficulties (74,75) and developed a method applicable to arbitrary rovibrational excitation levels of diatomic two-electron molecules, like H₂, HeH⁺, and their isotopologues. (12,76−78) We also present results from using the direct nonadiabatic method to estimate the relativistic and QED corrections. With this approach, we were able to bring the theoretical predictions back into agreement with the measurements at a new sub-MHz level of accuracy.

2. Theoretical Background

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Our calculations are based on the nonrelativistic quantum electrodynamic (NRQED) theory, which describes the energy of a bound rovibrational level of a light molecule as a function of fine-structure constant α. This energy can be expanded in powers of α and its logarithm (atomic units are used for E⁽ⁿ⁾ coefficients)

E (α) = m α^{2} E^{(2)} + m α^{4} E^{(4)} + m α^{4} E_{f s}^{(4)} + m α^{5} E^{(5)} + m α^{6} E^{(6)} + m α^{7} E^{(7)} + ...

(1)

Subsequent terms of this expansion can be identified as the nonrelativistic energy, relativistic correction, finite nuclear size correction, QED correction, and higher-order corrections. An important feature of NRQED is that the coefficients E⁽ⁿ⁾ of this expansion can be expressed as expectation values of some effective Hamiltonians H⁽ⁿ⁾ with the nonrelativistic wave function Ψ. These expectation values can be evaluated using a fully four-body wave function or, if this is infeasible, using approximations that involve the separation of nuclear and electronic variables, as in, e.g., NAPT. Obviously, the former approach, referred to as the direct nonadiabatic method, is preferable because it fully accounts for the effects of the finite nuclear mass.

2.1. The Wave Function

The wave function must reflect the rotational and electronic degrees of freedom present in the ground electronic state (

{}^{1}{Σ_{g}^{+}}

) of the hydrogen molecule. Ψ should also account for the coupling between the rotational angular momentum of nuclei and orbital angular momentum of electrons

\vec{L}

. This coupling results in the total spatial angular momentum, represented by

\vec{J}

. Considering these factors, we can construct the wave function Ψ as follows

Ψ^{J, M} = \sum_{Λ} Ψ_{Λ}^{J, M}

(2)

where J and M are the total rotational and magnetic quantum numbers, respectively. The latter is the projection of

\vec{J}

onto the laboratory frame Z axis. The summation index Λ is an eigenvalue of the projection of

\vec{L}

on the internuclear axis and runs over subsequent electronic states Σ_g, Π_g, Δ_g, ..., which correspond to |Λ| = 0, 1, 2, ..., respectively.

The functions Ψ_Λ^J,M are represented as the linear combination

Ψ_{Λ}^{J, M} = \sum_{{k}} c_{{k}} (1 + P_{12}) ψ_{{k}, Λ}^{J, M}

(3)

of four-particle basis functions

ψ_{{k}, Λ}^{J, M} = Q_{Λ} e^{- α_{Λ} R - β_{Λ} (ζ_{1} + ζ_{2})} R^{k_{0}} r_{12}^{k_{1}} η_{1}^{k_{2}} η_{2}^{k_{3}} ζ_{1}^{k_{4}} ζ_{2}^{k_{5}}

(4)

with elliptic-like coordinates ζ₁ = r₁₃ + r₁₄, η₁ = r₁₃ – r₁₄, ζ₂ = r₂₃ + r₂₄, η₂ = r₂₃ – r₂₄, and

\vec{R} = {\vec{r}}_{34}

. From this point onward, indices 1 and 2 assigned to the distance between particles r, and later also to the particle momentum

\vec{p}

and mass m, will refer to the electrons, while indices 3 and 4 will correspond to protons. The exponents k_i are non-negative integers collectively denoted as {k}. The basis function is symmetric with respect to the inversion when k₂ + k₃ is even and antisymmetric when this sum is odd. The operator

P_{12}

permutes electrons, while the factor

Q_{Λ}

determines the electronic angular momentum associated with a given basis function. Explicit expressions for

Q_{Λ}

corresponding to the lowest angular momentum values are provided below

Q_{Σ} = Y_{J}^{M}, for J \geq 0

(5)

Q_{Π} = \sqrt{\frac{2}{J (J + 1)}} ρ^{i} R (\nabla_{R}^{i} Y_{J}^{M}), for J \geq 1

(6)

Q_{Δ} = \sqrt{\frac{4}{(J - 1) J (J + 1) (J + 2)}} \times \frac{1}{2} (ρ^{i} ρ^{' j} + ρ^{j} ρ^{' i} - δ_{⊥}^{i j} \vec{ρ} \cdot {\vec{ρ}}^{'}) \times R^{2} (\nabla_{R}^{i} \nabla_{R}^{j} Y_{J}^{M}), for J \geq 2

(7)

In addition to a J-dependent normalization constant, the

Q_{Λ}

consists of the electronic and the nuclear parts. The electronic part is constructed from combinations of the vectors

\vec{ρ} \equiv {\vec{ρ}}_{a}

(a = 1 or 2), which correspond to the components of the electron-nucleus vectors

{\vec{r}}_{a 3}

that are perpendicular to the molecular axis. These components are defined as ρ_aⁱ = (δ^ij – nⁱn^j)r_a3^j, where

\vec{n} = \vec{R} / R

denotes the normalized internuclear vector. In this context, we utilize the Einstein summation convention. The nuclear part, in turn, involves derivatives with respect to nuclear coordinates of the solid harmonic

Y_{J}^{M} (\vec{n})

The basis function described in eq 4 is termed the nonadiabatic James–Coolidge (naJC) function. This name reflects its similarity to the classic two-electron function introduced initially by James and Coolidge. (2) Their inspiration came from the Hylleraas function, which was designed for the helium atom in 1929. (79)

The nonlinear (α_Λ, β_Λ) and linear c_{k} parameters are determined variationally by solving the four-body Schrödinger equation H⁽²⁾Ψ^J,M = E⁽²⁾Ψ^J,M with the nonrelativistic Hamiltonian

H^{(2)} = T + V

(8)

T = \sum_{a = 1}^{4} \frac{m}{2 m_{a}} p_{a}^{2}

(9)

V = \frac{1}{r_{12}} - \frac{1}{r_{13}} - \frac{1}{r_{23}} - \frac{1}{r_{14}} - \frac{1}{r_{24}} + \frac{1}{r_{34}}

(10)

According to the notation introduced earlier, for H₂, m₁ = m₂ = m is the electron mass, and m₃ = m₄ = m_p denotes the mass of the proton. A more detailed description of the Ansatz and the results of nonrelativistic calculations performed using this wave function can be found in refs (12and74). At this point, we merely note that the nonrelativistic energy can be determined with an accuracy limited only by the uncertainties in physical constants such as the proton-to-electron mass ratio and the Rydberg constant.

The nonrelativistic wave function described above is utilized to calculate the expectation values of operators present in relativistic and QED Hamiltonians H⁽ⁿ⁾. Relevant formulas are provided in the following subsections.

2.2. Relativistic Correction, E⁽⁴⁾

The relativistic correction is evaluated as an expectation value of the spin-independent Breit–Pauli Hamiltonian H⁽⁴⁾. It consists of the mass velocity term, the Breit orbit–orbit terms, and the Dirac delta terms

\begin{array}{l} E^{(4)} = - ⟨ Ψ | \sum_{a = 1}^{4} \frac{m^{3}}{8 m_{a}^{3}} p_{a}^{4} | Ψ ⟩ \\ - \frac{1}{2} ⟨ Ψ | \sum_{a = 1}^{3} \sum_{b = a + 1}^{4} \frac{m^{2} s_{a, b}}{m_{a} m_{b}} p_{a}^{i} (\frac{δ^{i j}}{r_{a b}} + \frac{r_{a b}^{i} r_{a b}^{j}}{r_{a b}^{3}}) p_{b}^{j} | Ψ ⟩ \\ - π ⟨ Ψ | δ^{3} (r_{12}) | Ψ ⟩ + \frac{π}{2} (1 + \frac{m^{2}}{m_{p}^{2}}) ⟨ Ψ | \sum_{a = 1}^{2} \sum_{b = 3}^{4} δ^{3} (r_{a b}) | Ψ ⟩ \end{array}

(11)

In the above, s_a,b = 2[δ_a,1·δ_b,2 + δ_a,3·δ_b,4] – 1 provides a proper sign. To achieve the target of sub-MHz accuracy, it is essential to consider another small relativistic effect arising from the spatial distribution of the proton charge. The leading correction due to the finite nuclear size, which is formally of the order of α⁴, is given by (80)

E_{f s}^{(4)} = \frac{2 π}{3} ⟨ Ψ | \sum_{a, X} δ^{3} (r_{a X}) | Ψ ⟩ \frac{r_{p}^{2}}{ƛ^{2}}

(12)

where r_p is the root-mean-square charge radius of the proton and ƛ is the reduced electron Compton wavelength. All of the expectation values mentioned above were calculated using the naJC wave function. Detailed information regarding the newly developed integration methods, which are suitable for evaluating these expectation values, along with their regularization and numerical convergence, can be found in refs (75and81). Preliminary results for the lowest rotational levels of H₂ are also shown therein.

In eq 11, the nucleus–nucleus Dirac delta interaction is omitted due to its negligible value. Contributions from the coupling between the nuclear spin and rotation as well as between nuclear spins are small because they depend quadratically on the electron-to-nucleus mass ratio. These effects are usually undetectable in infrared spectroscopy. Nonetheless, they were measured by Ramsey in 1971, (82) which allowed us, using the four-body Gaussian wave function, to determine the quadrupole moment of the deuteron to an accuracy better than for any other element of the periodic table. (30) In this work, the energy levels are averaged over the nuclear spin orientations; therefore, no hyperfine coupling is considered.

2.3. QED Correction, E⁽⁵⁾

The next term of the α-expansion (1) is the leading spin-independent quantum electrodynamic correction represented by the following expression, including terms up to the first order in the electron-to-nucleus mass ratio (11)

\begin{array}{l} \begin{array}{l} E^{(5)} = - \frac{2}{3 π} \frac{m^{2}}{μ^{2}} D \ln k_{0} \\ - \frac{7}{6 π} ⟨ Ψ | \frac{1}{r_{12}^{3}} | Ψ ⟩ - \frac{7}{6 π} \sum_{a = 1}^{2} \sum_{b = 3}^{4} \frac{m}{m_{b}} ⟨ Ψ | \frac{1}{r_{a b}^{3}} | Ψ ⟩ \end{array} \\ + \frac{4}{3} \sum_{a = 1}^{2} \sum_{b = 3}^{4} {\frac{19}{30} - 2 \ln α + \frac{m}{m_{p}} (\frac{31}{6} - \frac{1}{2} \ln α)} ⟨ Ψ | δ^{3} (r_{a b}) | Ψ ⟩ \\ + (\frac{164}{15} + \frac{14}{3} \ln α) ⟨ Ψ | δ^{3} (r_{12}) | Ψ ⟩ \end{array}

(13)

The atomic reduced mass, defined as μ = m_pm/(m_p + m), is used in this context. Equation 13 consists of five terms. The first term contains the so-called Bethe logarithm

\ln k_{0} = \frac{1}{D} ⟨ Ψ | (\sum_{a = 1}^{2} {\vec{p}}_{a}) (H - E) \ln [2 (H - E)] (\sum_{a = 1}^{2} {\vec{p}}_{a}) | Ψ ⟩

(14)

where

D = 2 π \sum_{a = 1}^{2} \sum_{b = 3}^{4} ⟨ Ψ | δ^{3} (r_{a b}) | Ψ ⟩

(15)

The next two expressions represent the electron–electron and electron–nucleus Araki–Sucher terms, respectively. These expectation values are defined as

⟨ Ψ | \frac{1}{r^{3}} | Ψ ⟩ = \lim_{ϵ \to 0} [⟨ Ψ | \frac{θ (r - ϵ)}{r^{3}} | Ψ ⟩ + 4 π (γ + \ln ϵ) ⟨ Ψ | δ^{3} (r) | Ψ ⟩]

(16)

where the symbol γ denotes the Euler–Mascheroni constant, and θ is the Heaviside function. Finally, there are two terms in eq 13 that involve electron–electron and electron–nucleus Dirac deltas. Currently, only these two numerically dominant terms have been evaluated by using the four-body naJC wave function. The remaining three terms of eq 13 were estimated within the framework of the BO approximation, which significantly influenced the overall error budget. The Bethe logarithm and the electron–electron Araki–Sucher BO potentials were taken from ref (18), while the electron–nucleus Araki–Sucher potential was from ref (83).

2.4. Higher-Order QED Correction, E⁽⁶⁾

The higher-order QED contribution, calculated within the BO approximation using the wave function

Ψ (\vec{r}, \vec{R}) = ϕ (\vec{r}) χ (\vec{R})

, is denoted as E^(6,0). The energy E^(6,0) is expressed in terms of two internuclear potentials: the Breit–Pauli potential

E^{(4, 0)} (R)

and the higher-order relativistic potential

E^{(6, 0)} (R)

, which are defined as follows (16)

E^{(4, 0)} (R) = {⟨ ϕ | H^{(4, 0)} | ϕ ⟩}_{e l}

(17)

\begin{array}{l} E^{(6, 0)} (R) = {⟨ ϕ | H^{(6, 0)} | ϕ ⟩}_{e l} \\ + {⟨ ϕ | H^{(4, 0)} \frac{1}{{(E_{e l} - H_{e l})}^{'}} H^{(4, 0)} | ϕ ⟩}_{e l} \end{array}

(18)

The Hamiltonians H^(4,0) and H^(6,0) are the leading and higher-order relativistic Hamiltonians, respectively, both in the nonrecoil limit indicated by 0 in the superscript. This limit entails neglecting all terms that depend on the nuclear mass. The expectation value symbol ⟨···⟩_el means integration over electronic variables only. H_el ≡ H^(2,0) and H_n represent the electronic and nuclear components of nonrelativistic Hamiltonian H⁽²⁾, respectively. The explicit formulas for

E^{(6, 0)} (R)

are too extensive to be presented here─they can be found in ref (16). The total correction is obtained by integration over the nuclear variables

\begin{array}{l} E^{(6, 0)} = ⟨ χ_{v J} | E^{(6, 0)} (R) | χ_{v J} ⟩ \\ + ⟨ χ_{v J} | E^{(4, 0)} (R) \frac{1}{{(E_{v J}^{(2, 0)} - H_{n})}^{'}} E^{(4, 0)} (R) | χ_{v J} ⟩ \end{array}

(19)

where χ is the nuclear wave function obtained by solving the radial Schrödinger equation. More details on how to evaluate this correction can be found in the original article. (16)

2.5. Estimation of E⁽⁷⁾

The E⁽⁷⁾ term is a particularly problematic correction, because its complete form is still unknown. It is therefore estimated by a dominating term inferred from the atomic hydrogen theory (80)

E^{(7)} \approx π ⟨ Ψ | \sum_{a = 1}^{2} \sum_{b = 3}^{4} δ^{3} (r_{a b}) | Ψ ⟩ {\frac{1}{π} [A_{60} + A_{61} \ln α^{- 2} + A_{62} \ln^{2} α^{- 2}] + \frac{1}{π^{2}} B_{50} + \frac{1}{π^{3}} C_{40}}

(20)

This approximation adds another significant contribution to the error budget.

2.6. Atomic Limits

This section provides a comprehensive list of atomic thresholds relevant to the subsequent terms of eq 1. These atomic limits can be used in the calculation of dissociation energy D_v,J = 2 E(H) – E(H₂) for a specific rovibrational level (v, J).

E^{(2)} (H) = - \frac{μ}{2 m}

(21)

E^{(4)} (H) = - \frac{μ^{3}}{8 m^{3}} [1 + 3 \frac{m}{m_{p}} + \frac{m^{2}}{m_{p}^{2}}]

(22)

E_{f s}^{(4)} (H) = \frac{2 μ^{3}}{3 m^{3}} \frac{r_{p}^{2}}{ƛ^{2}}

(23)

E^{(5)} (H) = - \frac{4}{3 π} \frac{μ}{m} (\ln k_{0}^{H} + \ln \frac{μ}{m}) + \frac{7}{6 π} \frac{m}{m_{p}} 4 \ln 2 + \frac{4}{3 π} \frac{μ^{3}}{m^{3}} [\frac{19}{30} - 2 \ln α + \frac{m}{m_{p}} (\frac{31}{6} - \frac{1}{2} \ln α)]

(24)

E^{(6)} (H) = - \frac{1}{16} + (\frac{427}{96} - 2 \ln 2) + (- \frac{9 ζ (3)}{4 π^{2}} - \frac{2179}{648 π^{2}} + \frac{3 \ln 2}{2} - \frac{10}{27})

(25)

E^{(7)} (H) = \frac{μ^{3}}{m^{3}} [\frac{1}{π} (A_{60} + A_{61} \ln α^{- 2} + A_{62} \ln^{2} α^{- 2}) + \frac{1}{π^{2}} B_{50} + \frac{1}{π^{3}} C_{40}]

(26)

ζ is the Riemann zeta function. The atomic Bethe logarithm is defined as follows

\ln k_{0}^{H} = \frac{1}{2} ⟨ \vec{p} (H - E) \ln [2 (H - E)] \vec{p} ⟩

(27)

with the atomic nonrelativistic Hamiltonian H = p²/2 – 1/r. The numerical value of the hydrogenic Bethe logarithm is known with high accuracy, (84)

\ln k_{0}^{H} = 2.984128555765498

. In the present work, eqs 21–23 were employed in molecular calculations performed using the nonadiabatic wave function. The Bethe logarithm, in turn, was evaluated in the BO approximation, which means that the first term in eqs 13 and 24 is modified by taking the infinite nuclear mass limit. The expression E⁽⁶⁾(H) represents the atomic limit in the BO approximation, too. Three distinct components of eq 25 arise from the Dirac theory and one-loop and two-loop QED corrections, respectively. Finally, the factor μ³/m³ in E⁽⁷⁾(H) reflects the usage of the Dirac delta calculated with the fully nonadiabatic molecular wave function.

3. Results

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Numerical calculations were performed using the CODATA 2022-recommended physical constants. (80) Specifically, the following constants were used: the proton-to-electron mass ratio, m_p/m = 1836.152 673 426(32), the inverse fine structure constant, 1/α = 137.035999177(21), the Rydberg constant,

R_{\infty} = 109,737.31568157 (12) {c m}^{- 1}

, the proton charge radius, r_p = 0.84075(64) fm, the reduced electron Compton wavelength, ƛ = 386.15926744(12) fm, and the speed of light in vacuum, c = 299,792,458 m s^–1. To convert our numerical results from atomic units to wavenumbers, we multiply the E⁽ⁿ⁾ by

2 R_{\infty} α^{n - 2}

, while conversion to MHz requires multiplication by

2 \times 1 0^{- 4} R_{\infty} c α^{n - 2}

In accordance with eq 1, the total energy of a rovibrational level is calculated as the sum of the nonrelativistic energy and successive corrections of increasing order in α, as outlined in the previous section. The convergence of individual relativistic expectation values, evaluated within the framework of the direct nonadiabatic approach, is demonstrated in the Supporting Information tables. Expectation values involving the electron–nucleus Dirac delta, which appear in corrections E_fs⁽⁴⁾, E⁽⁵⁾, and E⁽⁷⁾, can also be inferred from this data. Analyzing this convergence provides a foundation for estimating the uncertainties associated with individual corrections and the total energy. Furthermore, the results presented in these tables may serve as benchmarks for other calculation methods.

3.1. Dissociation Energy

Table 1 provides the final results of the dissociation energy D_v,J for the lowest vibrational (v = 0, 1, 2) and rotational (J = 0, ..., 4) quantum numbers. In addition to the total dissociation energy, the table includes all components of D_v,J that pertain to subsequent terms E⁽ⁿ⁾ of the α-expansion. These components were calculated as 2 E⁽ⁿ⁾(H) – E⁽ⁿ⁾(H₂). The numerical values of E⁽ⁿ⁾(H), discussed in Sec. 2.6, are considered exact, meaning they do not affect the uncertainty of D_v,J.

Table 1. Theoretically Predicted Dissociation Energy D_v,J for the Five Lowest Rotational Levels in the Vibrational States v = 0, 1, 2 of H₂ [cm^–1]b

component	D_0,0	D_1,0	D_2,0
E⁽²⁾	36,118.797744716(4)a	31,957.63367420(2)	28,031.79525303(2)
E⁽⁴⁾	–0.531217263(61)	–0.554783333(62)	–0.572886764(67)
E_fs⁽⁴⁾	–0.000030900(33)	–0.000027745(30)	–0.000024797(27)
E⁽⁵⁾	–0.19491021(15)	–0.173629(10)	–0.153814(20)
E⁽⁶⁾	–0.0020577(66)	–0.0018664(60)	–0.0016868(54)
E⁽⁷⁾	0.000101(25)	0.000090(23)	0.000081(20)
total	36,118.069630(26)	31,956.903458(26)	28,031.066922(29)

component	D_0,1	D_1,1	D_2,1
E⁽²⁾	36,000.31248423(1)	31,845.06061442(1)	27,925.00480794(2)
E⁽⁴⁾	–0.53380004(31)	–0.55714556(38)	–0.57502713(41)
E_fs⁽⁴⁾	–0.000030749(33)	–0.000027603(30)	–0.000024664(27)
E⁽⁵⁾	–0.19388906(52)	–0.172669(11)	–0.152912(20)
E⁽⁶⁾	–0.0020488(66)	–0.0018581(60)	–0.0016789(54)
E⁽⁷⁾	0.000100(25)	0.000090(22)	0.000080(20)
total	35,999.582816(26)	31,844.329004(25)	27,924.275245(29)

component	D_0,2	D_1,2	D_2,2
E⁽²⁾	35,764.42923080(1)	31,620.96551340(1)	27,712.44065580(2)
E⁽⁴⁾	–0.53890776(38)	–0.56181401(38)	–0.57925386(41)
E_fs⁽⁴⁾	–0.000030451(33)	–0.000027322(29)	–0.000024399(26)
E⁽⁵⁾	–0.1918636(15)	–0.170765(12)	–0.151123(21)
E⁽⁶⁾	–0.0020312(65)	–0.0018415(59)	–0.0016633(53)
E⁽⁷⁾	0.000099(25)	0.000089(22)	0.000079(20)
total	35,763.696497(26)	31,620.231155(26)	27,711.708670(29)

component	D_0,3	D_1,3	D_2,3
E⁽²⁾	35,413.28801899(1)	31,287.41651194(2)	27,396.10305521(2)
E⁽⁴⁾	–0.54642754(38)	–0.56868046(38)	–0.58546271(41)
E_fs⁽⁴⁾	–0.000030009(32)	–0.000026906(29)	–0.000024007(26)
E⁽⁵⁾	–0.1888668(30)	–0.167948(13)	–0.148477(22)
E⁽⁶⁾	–0.0020052(64)	–0.0018169(58)	–0.0016401(53)
E⁽⁷⁾	0.000098(24)	0.000088(22)	0.000078(20)
total	35,412.550787(25)	31,286.678128(26)	27,395.367529(30)

component	D_0,4	D_1,4	D_2,4
E⁽²⁾	34,950.01523845(2)	30,847.43371054(2)	26,978.91200753(2)
E⁽⁴⁾	–0.55619744(35)	–0.57758917(37)	–0.59350370(40)
E_fs⁽⁴⁾	–0.000029431(32)	–0.000026362(28)	–0.000023495(25)
E⁽⁵⁾	–0.1849460(49)	–0.164264(15)	–0.145018(24)
E⁽⁶⁾	–0.0019711(63)	–0.0017848(57)	–0.0016098(52)
E⁽⁷⁾	0.000096(24)	0.000086(21)	0.000076(19)
total	34,949.272190(25)	30,846.690132(26)	26,978.171929(31)

^{^a}

This energy differs from that in our previous studies (11,12,74) due to differences in the physical constants used.

^{^b}

Components of D_v,J related to subsequent terms E⁽ⁿ⁾ of the α-expansion (1) are also included.

As mentioned in the previous section, there are three components of the leading QED correction, E⁽⁵⁾, that were determined within the BO approximation. To enhance the precision of this correction’s final value, we utilized the precise values of these components obtained for the ground level (0,0) from the direct nonadiabatic calculations with the naECG wave function. (11) For all levels (v, J), we applied a uniform shift to these three components based on the difference between the nonadiabatic value and the value obtained from the BO approximation. Thus, the shifted contribution to the dissociation energy from the component Q can be expressed as Q_v,J^BO – Q_0,0^BO + Q_0,0^naECG. The total shift from all three components is −1.1 × 10^–5 cm^–1. Because this is a uniform shift applied to all energy levels, it does not alter the spacing between them.

The uncertainties related to E⁽²⁾ and E⁽⁴⁾ are purely numerical; they arise from the finite size of the naJC basis set and have a negligible influence on the overall error budget. The most considerable contributions to the uncertainty of D_v,J come from the E⁽⁵⁾ and E⁽⁷⁾ terms. In the first case, the primary source of error is the omission of finite mass effects in the three aforementioned components. Their uncertainties were estimated as (Q_v,J^BO – Q_0,0^BO)/μ_n, where μ_n = m_p/2 is the nuclear reduced mass. In the latter case, the error comes from the incompleteness of the expression for E⁽⁷⁾. In summary, the uncertainties associated with D_v,J are less than 3.3 × 10^–5 cm^–1(≈1 MHz).

3.2. Rovibrational Transition Energy

Establishing reliable uncertainty for a transition frequency ν is challenging due to the partial error cancellation between two rovibrational levels. The overall uncertainty u is calculated as the root-mean-square of the individual uncertainties u⁽ⁿ⁾, which arise from the terms ν⁽ⁿ⁾ = ΔE⁽ⁿ⁾ in the α-expansion. To determine the u⁽ⁿ⁾ values, we applied distinct approaches depending on how each ν⁽ⁿ⁾ term was evaluated. For ν⁽ⁿ⁾ obtained with the direct nonadiabatic approach (as ν⁽²⁾ and ν⁽⁴⁾), the uncertainty was estimated as the maximum uncertainty of both levels. In contrast, for ν⁽ⁿ⁾ calculated using the BO approximation (as ν⁽⁵⁾ and ν⁽⁶⁾), the uncertainty was estimated by the formula ν⁽ⁿ⁾/μ_n. This scaling accounts for the missing finite nuclear mass effects. Since the complete ν⁽⁷⁾ is currently unknown, and its dominant term from the atomic hydrogen theory only provides an estimate, we assumed a conservative error margin of 25% for this contribution. The uncertainty of ν_fs⁽⁴⁾ is determined by that of the proton charge radius and is relatively small compared to the main error components.

Having determined the uncertainties, we are now in a position to compare our results with experimental values─a crucial step in validating the accuracy and reliability of the theoretical model. Among the numerous rovibrational intervals measured for H₂, only nine are known with sub-MHz precision. These measurements, taken within the last two years, (69−71,73,85) encompass both purely rotational transitions and those belonging to the fundamental vibrational band as well as the first overtone band. Table 2 presents a compilation of these experimental frequencies alongside their theoretical counterparts. Each panel in the table is dedicated to a specific transition line and includes all components that contribute to the final theoretical frequency along with their individual error bars. The bottom row of each panel displays the difference between the theoretical and experimental frequencies, accompanied by the combined uncertainty, σ. We observe agreement for all nine lines with the largest difference being 1.4 σ for the Q₁(1) line. Figure 1 illustrates these differences relative to the ±1 MHz band, clearly demonstrating that the theoretical predictions and spectroscopic measurements are in sub-MHz agreement. In some cases, the experimental results are an order of magnitude more accurate than the theoretical predictions, which highlights the need for further development of computational methods, particularly for ν⁽⁷⁾.

Table 2. Comparison of Theoretically Predicted Transition Frequencies (in MHz) for H₂ with Nine Experimental Results That Were Measured with Sub-MHz Accuracyf

component	S₀(0): (0, 2) → (0, 0)	S₀(1): (0, 3) → (0, 1)	Q₁(1): (1, 1) → (0, 1)
ν⁽²⁾	10,623,700.7825(3)	17,598,550.7339(4)	124,571,317.1657(4)
ν⁽⁴⁾	230.555(11)	378.562(11)	699.881(11)
ν_fs⁽⁴⁾	–0.01346(2)	–0.02219(3)	–0.0943(1)
ν⁽⁵⁾	–91.335(47)	–150.565(77)	–636.16(33)
ν⁽⁶⁾	–0.793(3)	–1.308(4)	–5.719(18)
ν⁽⁷⁾	0.040(10)	0.070(17)	0.310(78)
theory	10,623,839.229(49)	17,598,777.469(80)	124,571,375.38(34)
experiment	10,623,839.09(39)a	17,598,777.46(75)a	124,571,374.73(31)b
difference	0.14(39)	0.01(75)	0.65(46)

component	S₁(0): (1, 2) → (0, 0)	Q₂(1): (2, 1) → (0, 1)	Q₂(2): (2, 2) → (0, 2)
ν⁽²⁾	134,841,618.0297(4)	242,091,633.7378(5)	241,392,544.6686(5)
ν⁽⁴⁾	917.267(11)	1235.957(12)	1209.546(12)
ν_fs⁽⁴⁾	–0.1072(2)	–0.1824(3)	–0.1814(3)
ν⁽⁵⁾	–723.86(37)	–1228.46(63)	–1221.38(63)
ν⁽⁶⁾	–6.482(21)	–11.089(36)	–11.031(35)
ν⁽⁷⁾	0.340(85)	0.59(15)	0.59(15)
theory	134,841,805.19(38)	242,091,630.55(65)	241,392,522.22(65)
experiment	134,841,805.102(15)c	242,091,630.140(9)d	241,392,522.00(34)a
difference	0.09(38)	0.41(65)	0.22(73)

component	Q₂(3): (2, 3) → (0, 3)	S₂(0): (2, 2) → (0, 0)	S₂(1): (2, 3) → (0, 1)
ν⁽²⁾	240,349,158.6533(5)	252,016,245.4511(5)	257,947,709.3872(5)
ν⁽⁴⁾	1170.245(12)	1440.101(12)	1548.807(12)
ν_fs⁽⁴⁾	–0.1799(3)	–0.1949(3)	–0.2021(3)
ν⁽⁵⁾	–1210.85(62)	–1312.71(68)	–1361.42(70)
ν⁽⁶⁾	–10.945(35)	–11.824(38)	–12.253(39)
ν⁽⁷⁾	0.59(15)	0.63(16)	0.66(17)
theory	240,349,107.52(64)	25,2016,361.45(69)	25,7947,884.98(72)
experiment	240,349,107.15(71)a	25,2016,361.164(8)e	25,7947,884.597(30)a
difference	0.36(96)	0.28(69)	0.39(72)

^{^a}

Fleurbaey et al., 2023. (69)

^{^b}

Lamperti et al., 2023. (70)

^{^c}

Stankiewicz et al., 2025. (73)

^{^d}

Diouf et al., 2024. (85)

^{^e}

Cozijn et al., 2023. (71)

^{^f}

The components ν⁽ⁿ⁾ = ΔE⁽ⁿ⁾ of the theoretical frequency, corresponding to subsequent terms of the α-expansion (1), are also included. CODATA 2022-recommended physical constants were used. (80)

Figure 1. Differences between theoretically predicted and experimental line positions shown against the ±1 MHz error band (in blue) and the theoretical error band (in green). The error bars around the dots come from the experiment.

Unlike earlier calculations (86) that estimated the relativistic and quantum electrodynamic corrections using the Born–Oppenheimer approximation, the current direct nonadiabatic results include the finite nuclear mass effects in both the relativistic and QED corrections. For frequencies in the first overtone, the difference between current and previous calculations reaches 3 MHz; in the fundamental band, it is approximately 1.5 MHz, and for purely rotational transitions, it is about 0.1 MHz. The small third value demonstrates the effective cancellation of recoil effects within a single oscillation band. The change in the total recoil effect is primarily due to relativistic and leading QED components of the frequency.

In addition to the nine transition frequencies listed in Table 2, other energy gaps of interest for future measurements can be evaluated by calculating the difference in dissociation energies from Table 1. An upper bound for the uncertainty associated with these transitions can be inferred from Table 2: 0.1 MHz for rotational transitions, 0.4 MHz for the fundamental band, and 0.8 MHz for the first overtone. The energies for both the levels and transitions will be included in the new release of the publicly accessible H2Spectre program. (86)

4. Conclusion

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We have shown that the relative uncertainty of the theoretically predicted rovibrational intervals in H₂ is about three parts per billion. This indicates that the calculated vibrational transition frequencies have an uncertainty at the tenth significant figure. The results obtained demonstrate that finite nuclear mass effects significantly impact not only nonrelativistic energy but also relativistic and QED corrections. By incorporation of these effects, the theoretical predictions matched the experimental results with sub-MHz accuracy.

Attaining the current level of experimental accuracy is essential for testing QED within molecular systems as well as for interpreting any discrepancies in terms of hypothetical unknown forces. There are only a few instances of physical systems where such a level of agreement between molecular or atomic spectroscopy and first-principles calculations has been achieved. Examples include the transitions in hydrogen (87,88) and helium atoms, (89) as well as those in the hydrogen molecular ion. (90)

Further advancements will require a fully nonadiabatic calculation of the Bethe logarithm and an Araki–Sucher correction for excited rovibrational levels. We believe this task is feasible since it has already been accomplished at the ground rovibrational level. (11)

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.5c01702.

Convergence of relativistic expectation values for individual rovibrational levels of H₂ (PDF)

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Author Information

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Corresponding Authors
- Krzysztof Pachucki - Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland; Email: [email protected]
- Jacek Komasa - Faculty of Chemistry, Adam Mickiewicz University, Uniwersytetu Poznanskiego 8, 61-614 Poznań, Poland; https://orcid.org/0000-0002-9876-1304; Email: [email protected]
Notes
The authors declare no competing financial interest.

Acknowledgments

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This research was supported by the National Science Center (Poland) Grant No. 2021/41/B/ST4/00089. A computer grant from the Poznań Supercomputing and Networking Center was used to carry out the numerical calculations.

References

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K. Stankiewicz, M. Makowski, M. Słowiński, K. L. Sołtys, B. Bednarski, H. J. Jóźwiak, N. Stolarczyk, M. Narożnik, D. Kierski, S. Wójtewicz, A. Cygan, G. Kowzan, P. Masłowski, M. Piwiński, D. Lisak, P. Wcisło. Cavity-enhanced spectroscopy in the deep cryogenic regime for quantum sensing and metrology. Nature Physics 2026, 47 https://doi.org/10.1038/s41567-026-03204-8
Michał Siłkowski, Paweł Czachorowski, Jacek Komasa, Mariusz Puchalski. Progress toward spectroscopic accuracy: Relativistic and QED corrections for HeH + . Physical Review A 2026, 113 (3) https://doi.org/10.1103/k4qf-thhm
Jacek Komasa. Electron-nucleus Araki-Sucher correction in the hydrogen molecule isotopologues. Physical Review A 2026, 113 (1) https://doi.org/10.1103/qyz4-n1tc

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Abstract
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Figure 1
Figure 1. Differences between theoretically predicted and experimental line positions shown against the ±1 MHz error band (in blue) and the theoretical error band (in green). The error bars around the dots come from the experiment.
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Supporting Information
Supporting Information
The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.5c01702.
- Convergence of relativistic expectation values for individual rovibrational levels of H₂ (PDF)
- ct5c01702_si_001.pdf (140.25 kb)
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From First-Principles to Quantum Electrodynamics: Pushing the Limits of Theory with the Hydrogen Molecule
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Journal of Chemical Theory and Computation

Publication History

Abstract

Subjects

Special Issue

1. Introduction

2. Theoretical Background

2.1. The Wave Function

2.2. Relativistic Correction, E⁽⁴⁾

2.3. QED Correction, E⁽⁵⁾

2.4. Higher-Order QED Correction, E⁽⁶⁾

2.5. Estimation of E⁽⁷⁾

2.6. Atomic Limits

3. Results

3.1. Dissociation Energy

3.2. Rovibrational Transition Energy

Figure 1

4. Conclusion

Supporting Information

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Acknowledgments

References

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Abstract

Figure 1

References

Supporting Information

Supporting Information

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From First-Principles to Quantum Electrodynamics: Pushing the Limits of Theory with the Hydrogen MoleculeClick to copy article linkArticle link copied!

Journal of Chemical Theory and Computation

Abstract

Special Issue

1. Introduction

2. Theoretical Background

2.1. The Wave Function

2.2. Relativistic Correction, E(4)

2.3. QED Correction, E(5)

2.4. Higher-Order QED Correction, E(6)

2.5. Estimation of E(7)

2.6. Atomic Limits

3. Results

3.1. Dissociation Energy

3.2. Rovibrational Transition Energy

Figure 1

4. Conclusion

Supporting Information

Terms & Conditions

Author Information

Acknowledgments

References

Cited By

Journal of Chemical Theory and Computation

Article Views

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Citations

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Abstract

Figure 1

References

Supporting Information

Supporting Information

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From First-Principles to Quantum Electrodynamics: Pushing the Limits of Theory with the Hydrogen Molecule
Click to copy article linkArticle link copied!

2.2. Relativistic Correction, E⁽⁴⁾

2.3. QED Correction, E⁽⁵⁾

2.4. Higher-Order QED Correction, E⁽⁶⁾

2.5. Estimation of E⁽⁷⁾