The hydrogen molecule H2 is the simplest stable molecule, with two protons and two electrons bound together. Scientists have studied it for more than a century because it’s small enough that theory can try to predict its behaviour from basic physics yet it’s rich enough to include many features found in larger molecules.
This said, after all this time, can scientists predict H2’s energy levels so accurately that the predictions match with today’s best measurements?
Four challenges
An experimental technique called spectroscopy can measure the distances between the different possible energy levels of H2 by observing which frequencies of light the molecule can absorb or emit. Modern setups can measure some of these gaps with a relative accuracy of about one part in 100 billion. At this precision, experiments are sensitive not only to the main predictions of quantum mechanics but also to extremely small effects due to quantum electrodynamics (QED). QED is the theory of how charged particles like electrons interact with electromagnetic fields.
A calculation that tries to predict the effects of quantum mechanics as well as QED effects has to get many things right at once. According to a new study by researchers from Poland, there are four big challenges. First, the two electrons in H2 affect each other strongly, so the calculation must capture their togetherness, or what physicists call correlation. Second, the nuclei don’t sit still: the electrons and the nuclei (the clumps of protons) influence each others’ motions. Third, the electrons move fast enough that the special theory of relativity makes a small but measurable difference to their energy. Fourth, there are smaller QED effects that are measurable with today’s instruments.
Full monty
Over the last decade, experiments have improved dramatically. Earlier, theory and experiment would agree to within about 10 MHz of each other; newer measurements can reach accuracies of the order of 10 kHz. In these experiments, scientists measure the frequency of light that H2 absorbs or emits when it jumps between two specific energy levels. The frequency of that light equals the energy difference between the levels (divided by Planck’s constant). So 10 MHz and 10 kHz refer to the uncertainty in the measured transition frequency, i.e. how precisely scientists know the difference between two energy levels.
But when measurements got that good, scientists realised that some theoretical predictions were off by several MHz. According to the authors of the new study, from the University of Warsaw and the Adam Mickiewicz University, the scientists suspected one particular reason: the older calculations didn’t completely account for recoil effects inside the relativistic and QED parts of the theory. ‘Recoil’ here refers to the nuclei having a finite mass and responding to the electrons, in slight ways. Ignoring that response could shift the predicted frequency by very small, yet nonzero, amounts — and which matter at today’s precision.
To catch up, the new study’s authors skipped a common shortcut in molecular physics: the Born-Oppenheimer approximation. It assumes the nuclei to be almost fixed while the electrons move, allowing physicists to ignore the effects of the nuclei’s small movements. The authors instead solved the Schrödinger equation for H2, treating the two electrons and two protons together, without using the Born-Oppenheimer approximation. This is called the direct nonadiabatic approach.
Computationally intensive
The Schrödinger equation is the basic rule in quantum mechanics that says how a quantum system behaves. Put differently, it’s an important mathematical statement that accepts as its inputs which particles and forces are present and spits out which energy levels and quantum states are possible.
If that sounds simple, the truth is that solving the Schrödinger equation for four particles at once is extremely difficult (which is one of the reasons Max Born and J. Robert Oppenheimer came up with their approximation). The whole four-particle system has a single wavefunction, a kind of mathematical master description that physicists use to predict where the particles are likely to be found. And it depends on the positions of two electrons and two protons simultaneously. That means it lives in a very high-dimensional space and physicists need lots of computing power to represent it accurately.
Each particle also interacts with all the others: electrons and protons attract, electron and electron repel, proton and proton repel. As a result physicists can’t cleanly split the problem into ‘the part with the electrons’ and the ‘part with the protons’. In this picture, without the Born-Oppenheimer approximation, physicists must handle electronic motion and nuclear motion together from the start, which is also computationally demanding.
Finally, the goal here was to predict the light’s frequencies with a precision of under 1 MHz. That means the calculation couldn’t skip subtle details of the wavefunction, especially in regions where the particles got close.
Exponential functions
To deal with these challenges, the authors used a special kind of wavefunction. A wavefunction is ‘good’ if it can do a good job of describing the stronger attraction when an electron is closer to a proton, the stronger repulsion when two electrons are closer to each other, and the electrons’ situation depending on how close the protons are to each other. The team’s special wavefunction thus used exponential functions. These functions are designed to describe things that change quickly with distance.
For example, an electron’s ‘influence’ around a nucleus tends to be much stronger nearby and drops off as it gets farther away. Exponential functions can naturally describe this sort of behaviour.
After calculating a very accurate ‘baseline’ energy from quantum mechanics, the authors then made smaller adjustments due to the special theory of relativity and QED.
Exam for physics to pass
Finally the authors reported their results in two ways. One was the dissociation energy, the energy required to split one H2 molecule into two separate hydrogen atoms. The authors reported the dissociation energy at the lowest rotational and vibrational states of H2 in its ground electronic state with a relative accuracy of 7 × 10-10. Second, they predicted the frequency corresponding to the gap in energy between these states with a relative accuracy of about 3 × 10-9.
Next, they compared their theoretical predictions to nine recent measurements of these energy levels and found they agreed almost perfectly.
Their findings were published in the Journal of Chemical Theory and Computation on December 5.
Reaching this level of accuracy is important for physicists to test QED in molecular systems and to interpret any future mismatch as a possible sign of an unknown force rather than as a gap in the existing theory. The study’s authors also pointed to the next bottleneck: further progress for excited states will require fully nonadiabatic calculations of some especially difficult QED ingredients.
More broadly, thanks to the new work, the H2 molecule is now an ‘exam’ that fundamental physics has to pass because experiments and the theory now agree at a level such that any disagreements between them, if any, would have to be extraordinarily small. And physicists developing new theories predicated on such disagreements also have to come up with ways to identify them.
mukunth.v@thehindu.co.in
Published – January 06, 2026 05:30 am IST
