The enormous helium dimer - a quantum halo state

From Wikipedia:

"Based on molecular orbital theory, He₂ should not exist, and a chemical bond cannot form between the atoms. However, the van der Waals force exists between helium atoms as shown by the existence of liquid helium, and at a certain range of distances between atoms the attraction exceeds the repulsion. So a molecule composed of two helium atoms bound by the van der Waals force can exist. The existence of this molecule was proposed as early as 1930.

He₂ is the largest known molecule of two atoms when in its ground state, due to its extremely long bond length. The He₂ molecule has a large separation distance between the atoms of about 5,200 picometres. This is the largest for a diatomic molecule without ro-vibronic excitation. The binding energy is only about 1.3 mK, 10⁻⁷ eV or 1.1×10⁻⁵ kcal/mol. The bond is 5,000 times weaker than the covalent bond in the hydrogen molecule"

This news release in my Google Now feed intrigued me: the helium molecule is very, very big.

"Helium atoms are loners. Only if they are cooled down to an extremely low temperature do they form a very weakly bound molecule. In so doing, they can keep a tremendous distance from each other thanks to the quantum-mechanical tunnel effect. As atomic physicists in Frankfurt have now been able to confirm, over 75 percent of the time they are so far apart that their bond can be explained only by the quantum-mechanical tunnel effect.

The binding energy in the helium molecule amounts to only about a billionth of the binding energy in everyday molecules such as oxygen or nitrogen. In addition, the molecule is so huge that small viruses or soot particles could fly between the atoms. This is due, physicists explain, to the quantum-mechanical "tunnel effect."

They use a potential well to illustrate the bond in a conventional molecule. The atoms cannot move further away from each other than the "walls" of this well. However, in quantum mechanics the atoms can tunnel into the walls. "It's as if two people each dig a tunnel on their own side with no exit," explains Professor Reinhard Dörner of the Institute of Nuclear Physics at Goethe University Frankfurt."

We covered this in Volume III of my OU Quantum Mechanics course! (Chapter 6, page 162).

"The diatomic helium molecule

The He₂ molecule has four electrons, so you might think that the helium nuclei would be held together even more strongly than the protons in H₂. However, the He₂ molecule is unknown under normal conditions of temperature and pressure: at room temperature, helium gas contains only helium atoms.

We need to consider how the electrons occupy the available molecular orbitals. As with H₂, the two orbitals of lowest energy are 1σ_g and 1σ_u. In He₂, two electrons fill the bonding 1σ_g orbital, and the other two fill the antibonding 1σ_u orbital. The two electrons in the antibonding orbital do not help to bind the molecule together; on the contrary, they practically cancel out the effect of the electrons in the bonding orbital. Stable molecules generally have more electrons in bonding orbitals than in antibonding orbitals.

Accurate calculations indicate that there is a very shallow minimum in the energy curve of He₂ at a radius of R_equilibrium = 3×10⁻¹⁰ metres with a dissociation energy of D_equilibrium = 0 .0009 eV. This very small dissociation energy is close to the energy of the lowest vibrational state, so detection of He₂ molecules requires very low temperatures, and has only been achieved for a beam of helium atoms cooled to 10⁻³ K.

Because the energy curve has a very shallow minimum, the molecule samples a range of interatomic distances that are far from the equilibrium value.

Because the energy curve is asymmetric, the average separation of the two nuclei is much greater than the equilibrium separation, and has been estimated to be about 50×10⁻¹⁰ m."

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To get us up to speed, consider the hydrogen molecule ion: two protons and one electron. The electron wavefunction is decisive in this molecule. Below are the relevant diagrams for the hydrogen molecule ion using the Born-Oppenheimer approximation and LCAO trial functions for the molecular wave function (p. 148).

"The trial function is taken to be a linear combination of atomic orbitals centred on each of the nuclei that form the molecule. This method is known as the linear combination of atomic orbitals, frequently abbreviated to LCAO. The resulting one-electron eigenfunctions for the molecule are called molecular orbitals. We shall now apply the LCAO method to the electronic ground state of the hydrogen molecule ion."

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So now we're ready to look at the corresponding energy/probability-density vs separation graph for the helium dimer (from here). The weak binding requires that we consider the wave-function of the two helium nuclei.

 The weak helium-helium Van der Waals potential decrease leads to the particle probability density distribution leaking more into the classically forbidden region (i..e, tunneling). This effect allows the wavefunction to extend to sizes of fullerenes, the diameter of DNA and even small viruses, while the classical turning point is located at 13.6 Å, the overall wavefunction extends to more than 200 Å. 

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50 Å (Angstrom units) = 5,000 picometres. 

I suspect that R measures the distance from the nucleus-nucleus midpoint (in spherical coordinates), while the numbers quoted in the caption above are the nucleus-nucleus separation distances (2R). Ψ is the two-nuclei wave-function. Given the enormous nucleus-nucleus separation, the electrons will be quite tightly bound to their respective nuclei.

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Imaging the He₂ quantum halo state using a free electron laser (light edit).

"Quantum tunneling is a ubiquitous phenomenon in nature and crucial for many technological applications. It allows quantum particles to reach regions in space which are energetically not accessible according to classical mechanics.

"In this tunneling region the particle density is known to decay exponentially. This behavior is universal across all energy scales from MeV in nuclear physics, to eV in molecules and solids, and to neV in optical lattices. For bound matter the fraction of the probability density distribution in this classically forbidden region is usually small.

"For shallow short range potentials this can change dramatically: upon decreasing the potential depth excited states are expelled one after the other as they become unbound.

"A further decrease of the potential depth effects the ground state as well, as more and more of its wavefunction expands into the tunneling region. Consequently, at the threshold (i.e. in the limit of vanishing binding energy) the size of the quantum system expands to infinity.

"For short range potentials this expansion is accompanied by the fact that the system becomes less classical and more quantum-like. Systems existing near that threshold (and therefore being dominated by the tunneling part of their wave-function) are called quantum halo states.

"One of the most extreme examples of such a quantum halo state can be found in the realm of atomic physics: the helium dimer (He₂).

[More]."

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