Predicting The Hydrogen Dissociation Energy: The 100-Year Battle of Quantum Chemistry vs. Experiment

I must say, lately, I’ve been having a blast writing about chemistry, quantum mechanics, and the London-Heitler’s prediction of the hydrogen dissociation energy. It’s a lot of work in my free time, but I love it. Just the other day, I was surfing the web for some new material on how awesome quantum chemistry is. I mean, quantum chemistry has been a powerful and reliable tool for decades, from the investigation of small molecules to advanced materials and even in the hunt for life beyond our world. And when quantum computing finally arrives, well, that will be the ultimate holy-grail combo. Now, all this reading got me thinking.

As pointed out by Mata and Suhm, other complex fields, such as, say, biology or even economics, hardly know what their fundamental equations are, since the conditions are too multi-faceted (read vague) to nail them down. Like, good luck with trying to give an exact monthly forecast of the weather. But us quantum chemists, we’re bold enough to think that our theoretical models can actually describe reaction mechanisms with hundreds of atoms, provide accurate energetics, and predict how spectra would look like. And while that is true, we sometimes get caught up in our own quantum egos. So, we must keep ourselves in check, avoid falling into the trap of over confidence, and maintain a healthy relationship between our experiments and theory.

Prediction of the Hydrogen Dissociation Energy - the battle between theory and experiment
Adapted from Mata and Suhm

I was just thinking about all this, when I stumbled upon a gem of a story that I believe is not well known to a lot of quantum chemists. It’s like a straight-up Hollywood flick, with all the right ingredients: success and suspense, failure, the hero at their worst moment, and a thought-provoking open end.

I’m talking about…

The theoretical prediction of the hydrogen dissociation energy

Before the scientific articles of the recent five years or so, if you asked any experts (including yours truly), we would’ve told you that the prediction of the hydrogen dissociation energy is the paramount proof of the validity of molecular quantum mechanics, aka our quantum chemistry.

This is exemplified by what Mata and Suhm wrote in Angewandte Chemie, a more-than respectable scientific journal, as late as 2017:

Already in the 1960’s, we observed examples of theory competing with experiment in terms of accuracy. A seminal example is the hydrogen adiabatic dissociation energy computed by Kolos and Wolniewicz. The authors carried out a variational calculation and found their value to exceed the best experimental estimate at the time. Given that the theoretical estimate would necessarily give an upper bound to the true energy of hydrogen and thus a lower bound to its dissociation energy, the experimental value was questioned. The episode was only concluded after a further experiment confirmed the theoretical result.

Mata and Suhm 2017

In other words, yay: theory was right all along and every quantum chemist is happy, right?

Not quite.

What Mata and Suhm argued was all true and dandy up to 2017. We should’ve just left it at that. But scientists, like writers, don’t know when to stop and keep pushing the envelope. And now, this whole success story seems to have crumbled before our very eyes.

Prediction of the Hydrogen Dissociation Energy - the hydrogen molecule
This is actually a representation of a hydrogen molecule – two protons, blue, and the cloud of two shared electrons, gray.

Why Hydrogen Dissociation matters

Picture this: you’ve got a hydrogen molecule. Just two little atoms holding hands, cruising through the universe. But then, something happens. Maybe it’s a jolt of radiation or a collision with a high energy particle: whatever the cause, those two little atoms suddenly let go of each other and head off in opposite directions in the great darkness. This, my friends, is the hydrogen dissociation – a process that’s got physicists and chemists all hot and bothered.

Since way back in 1927, quantum chemists have been using this process to test the Schrödinger equation.

Why, you ask? Because it’s simple.

The hydrogen molecule has got just two protons and two electrons, making it perfect for comparing our fancy calculations to measurements. Not only that, studying the hydrogen dissociation can help validate quantum chemistry methods, enhance our understanding of chemical bonds, and even lead to theoretical improvements that capture quantum electrodynamics effects.

Plus, the energy level structure of hydrogen is incredibly rich, with vibrational and rotational degrees of freedom that provide opportunities to explore new physics beyond the Standard Model, extra dimensions, and forces at the angstrom scale.

It sounds like science fiction, I know. But it is not.

So, why should you care about the hydrogen dissociation?

Because it’s a cornerstone of modern scientific research that could unlock new revelations in physics and chemistry far beyond our current grasp.

Change in electron density as the hydrogen molecule dissociates

The Hydrogen Dissociation Energy Over the Years

The battle to estimate the hydrogen dissociation energy (aka, D0) is a long-standing scientific dance in the worlds of physics and chemistry. It’s a tale filled with history, experiments, and theoretical advancements that have kept scientists on the edge of their seats for decades.

C’mon, let me take you for a short historical stroll.

1927 and the movie Metropolis is released

The brilliant work of Heitler and London made a groundbreaking progress in the field. They approximated the Schrödinger equation for two electrons in the Coulomb field of two protons and demonstrated that the hydrogen molecule was stable against dissociation. This was a big deal, as it was the first time the Schrödinger equation had been applied to a chemical problem. And get this, they did it all by hand! Sure, their approximation was only about 60% of the observed D0, but let’s take a moment to appreciate the majesty of the human mind.

1933 and the electron microscope is invented

Fast forward to the era when human calculators, also known as experienced computers, were a common sight. It was during this time that James and Coolidge took on the challenge. They applied the Schrödinger equation to the dissociation of hydrogen, obtaining results that were within the error margin of the experimental value of their time, and astonishingly, just 0.5% off from current measurements. Truly remarkable that they achieved this feat using nothing more than a pencil and paper.

1964 and The Beatles go for their first world tour

As the experimental techniques improved over the years, theoreticians didn’t simply sit around scratching their heads; rather, they worked diligently to develop a sharp and reliable weapon for their arsenal: the digital computer. As immense computing power became available, a new duo, Kolos and Wolniewicz, entered the scene of the hydrogen dissociation saga. They created methods and computer codes to account for the coupling of electronic and nuclear motion, as well as the effects of relativity.

Their theoretical determination of D0 could finally be compared to the most recent and accurate experimental figures: 36117.4 cm−1 for theory vs. 36113.6 cm−1 for experiment. This was a bit underwhelming and fell short of the expectations of our demanding theoreticians. In fact, the theoretical community was thrown into a bit of a panic as the value calculated by Kolos and Wolniewicz was significantly higher than the measurements.

Now, in our daily lives, this difference would be negligible, but when it comes to the minuscule effects of quantum mechanics (and spectroscopy), we’ve got to be super picky. After all, if quantum mechanics is meant to work, we should be able to calculate the “exact” value, right?

1970 and Paul McCartney announces that The Beatles have disbanded

But fear not, my quantum buddies, because a delicious win was in store for theory. Ironically, it came from two experiments by Herzberg and Stwalley, which improved on the previous estimations and produced some sweet new figures: 36118.3 cm−1 and 36118.6 cm−1. These put the latest theoretical prediction right between the lower and upper limits: 36113.6 cm−1 < Kolos&Wolniewicz < 36118.6 cm−1.

Take that, experiment.

From the 1980’s to 1990’s

This confirmed that the Schrödinger equation was a powerful tool, which can predict molecular energy levels with incredible precision when solved accurately and corrected for the tiny effects of relativity. That was the deal for quite some time. In fact, throughout the 1980’s and 1990’s, the distance between predictions and measurements was reduced to just a few hundredths of a cm−1, with Kolos, Rychlewski, Wolniewicz setting the bar for prediction to 36118.049 cm−1 and 36118.069 cm−1.

2009 and Barack Obama becomes president

In 2009, the battle between experiment and theory appeared to come to an end, when Liu and co. brought a new pivotal point to the table. Their hybrid approach determined the hydrogen dissociation energy to be 36118.06962 cm−1, with an uncertainty of only ±0.00037 cm−1. That’s almost two orders of magnitude more accurate than previously reported.

If truth exists, Liu and co. got very close.

Now, the most reliable theoretical predictions up to this point (36118.049 cm−1 from Kolos and Rychlewski, and 36118.069 cm−1 from Wolniewicz) were spot on. But just when everyone thought the issue was finally solved, along came Piszczatowski and co. to say, “Nope, not yet.”

It turned out, the previous theoretical models didn’t take into account some quantum electrodynamics corrections. Which means, that the wild good agreement between calculation and experiment might have just been a lucky coincidence.

So, Piszczatowski and co. did their thing, included these effects, and bam! They announced their dazzling prediction to be 36118.0695(10) cm−1. And guess what? The difference between the experiment, 36118.0696(4) cm−1, and theory was only 1 unit at the ninth decimal place.

That’s like being able to weigh two objects and detect a difference of only a few electrons, or measure the distance between two points on Earth and be accurate to within a few atoms.

Let’s face it: it’s quantumly amazing!

Alright, so if we just leave it at that, we’d be all like “woo-hoo, happy ending!” But life’s always throwing curveballs, and this time it’s no different. Just when we thought things were settled, scientists went and shook things up again like they’re some kind of hollywood storytellers who can’t resist a good plot twist.

And boy, did they deliver – the latest data left everyone breathless and turned all cards on the table.

Prediction of the Hydrogen Dissociation Energy - the most recent estimations
Hydrogen dissociation energy – Theoretical predictions are diverging from experimental values – Adapted from Cheng 2018

The Recent Twist in the Hydrogen Dissociation Saga

So, here’s where we’re at with this never-ending saga. In 2017, Puchalski and co. came up with this super advanced model that was supposed to provide the most accurate hydrogen dissociation energy ever. Their theoretical prediction, while sophisticated, didn’t quite match up with previous calculations nor experiments.

Our friends tried to improve things, but ended up with poorer results. What a bummer.

Of course, the history of science is riddled with instances where theories have been proven right and experiments have been proven wrong. In the case of the hydrogen dissociation energy, the work of Puchalski raised more questions than it could answer.

Could previous experiments have had major flaws, or perhaps the theoretical approach was heading in the wrong direction? More advanced experimental data was needed to determine who was right and put to rest once and for all the battle between experiments and theory.

Unfortunately for theory, novel experimental data did come in and they were far from ideal. In 2018, Altmann and Cheng conducted two experimental studies that supported the validity of Liu’s measurements, confirming a significant disagreement (of 0.0017 cm−1) with Puchalski’s flashy theoretical model.

So, where does this leave us? After one hundred years of successes, is it now time to freak out and start questioning the fundamental laws of physics? Did the theoreticians become a bunch of goofballs who miss the mark?

Well no and no. Let’s take a take a deep breath and remember that we are talking about miniscule effects. The theoretical prediction is still incredibly accurate, with a difference in kcal/mol of 0.0000049! Think about it, you would need an energy 20 million times larger to break a standard covalent bond. To many chemists, myself included, this degree of accuracy approaches perfection.

Now, If we want to be very picky (and in the case of quantum mechanics and spectroscopy, we should), then we can identify one potential issue in some key relativistic corrections, like the nuclear recoil effects, which may have not been properly included into the calculations. This is where the solution possibly hides.

Solving this puzzle is not going to be a piece of cake, but we need to keep building on the gigantic shoulders of our scientific predecessors and come up with even smarter approaches. It’s up to the next generation of theoreticians to tackle these challenges with a fresh perspective and push the boundaries of what we know about quantum mechanics.

A Final Personal Touch

You know what I find awesome in all this hydrogen-dissociation-energy story? It beautifully captures the heart and soul of science – that constant back-and-forth between theory and experiments, the never-ending cycle of discovery and refinement that inches us closer to the scientific truth every single day.

Will theory have its come back? And how will experiments respond?

Like many of my colleagues, I used to tell this story to show how badass quantum chemistry is. And you know what? Despite the recent turns of events, the story still holds up. Quantum chemistry gives the right answer to many decimal places when it comes to the hydrogen dissociation energy, and that still blows my mind.

But here’s the thing: I’m positive that our human creativity will add even more entertaining chapters to this saga.

Sure, failure can be a real bummer. But those moments of sudden realization, when we scream “Eureka!,” always come. Whether through theory or experiment, science rocks on.

10.04.23 – UPDATE

After I published this story on Linkedin, Prof. Michał Tomza of the University of Warsaw brought to my attention the most recent theoretical prediction by Puchalski. I will write another story about this – It’s a double twist. The theoretical results of the new study align perfectly with the latest experimental measurements. Is the 100-year battle for the determination of the hydrogen dissociation energy finally over?

Scientific Reading List:

1) Theoretical Determination of the Dissociation Energy of Molecular Hydrogen – Piszczatowski et al. 2009
2) Relativistic Corrections for the Ground Electronic State of Molecular Hydrogen – Puchalski et al. 2017
3) Deep-Ultraviolet Frequency Metrology of H_{2} for Tests of Molecular Quantum Theory – Altmann et al. 2018
4) Dissociation Energy of the Hydrogen Molecule at 10−9 accuracy – Cheng et al. 2018
5) Improved determination of the dissociation energy of H2, HD and D2 – Hussels 2021


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A collection of my thoughts and musings on science, writing, and the intersection of the two.

Arturo Robertazzi - Me, my room. My books.