---
title: Hyperfine van der Waals Interaction
url: https://www.emergentmind.com/topics/hyperfine-van-der-waals-interaction
type: topic
---

# Hyperfine van der Waals Interaction

Searching arXiv for recent and foundational papers on hyperfine van der Waals interactions.
Hyperfine van der Waals interaction denotes a class of intermolecular effects in which hyperfine structure, or a hyperfine-mediated coupling mechanism, qualitatively determines a van der Waals observable. In the literature represented here, the term is used in three closely related but technically distinct senses: a scalar nuclear spin-spin coupling transmitted across transient van der Waals contact in liquids; hyperfine-resolved van der Waals shifts in quasi-degenerate excited hydrogen; and a second-order \(R^{-6}\) interaction between open-shell polar molecules that emerges when resonant dipole-dipole exchange is forbidden by hyperfine selection rules [1112.5644] [1701.08813] [1701.08812] [2507.12167]. Taken together, these works suggest that hyperfine van der Waals physics is best understood not as a single universal Hamiltonian term, but as a family of hyperfine-conditioned long-range or contact-dominated interactions.

## 1. Conceptual scope and distinguishing features

The common theme is that hyperfine structure does not merely add a perturbative correction to an otherwise fixed van der Waals potential. Instead, it determines whether an interaction is present in first order or only in second order, which virtual states dominate the effective coupling, and which observable carries the signature of the interaction. In one setting, the relevant coupling is the ordinary NMR scalar interaction,
\[
H_J = h J\, \mathbf I_1 \cdot \mathbf I_2,
\]
whose microscopic origin is a second-order Fermi-contact hyperfine mechanism mediated by electrons. In another, the relevant operator is the dipole-dipole interaction, but hyperfine resolution determines whether degenerate exchange channels survive, producing \(1/R^3\) behavior, or are suppressed so that the leading effect is \(1/R^6\). In hydrogen, the decisive small denominators are the Lamb shift and hyperfine splittings; in open-shell molecules, they are hyperfine splittings within a rotational manifold; in hyperpolarized xenon solutions, the key object is an ensemble-averaged scalar coupling over transient encounters.

These distinctions are essential because the phrase “van der Waals” can otherwise be misleading. The repulsive interaction predicted for open-shell molecules is explicitly not ordinary electronic van der Waals attraction, and the scalar coupling observed in xenon-pentane is not a resolved line splitting of a bound complex. Likewise, the hydrogen calculations show that quasi-degenerate manifolds invalidate a naive nondegenerate dispersion treatment. A recurrent misconception addressed in these works is therefore that van der Waals physics is always isotropic, always attractive, or always describable by a single species-independent \(C_6\). The cited studies show instead that hyperfine structure can select the sign, magnitude, scaling law, and even the appropriate experimental observable.

## 2. Scalar nuclear spin-spin coupling across van der Waals contact

The most direct experimental realization of a hyperfine van der Waals interaction is the observation of scalar coupling between unbound nuclei in a solution of hyperpolarized \(^{129}\mathrm{Xe}\) and pentane. The central question was whether the scalar nuclear spin-spin coupling familiar from conventional NMR spectroscopy can survive in systems that are not covalently bound, but instead interact only transiently through van der Waals contact. The reported result was the first observation of such couplings in van der Waals molecules, establishing direct experimental evidence for a hyperfine van der Waals interaction of the scalar, electron-mediated type [1112.5644].

The mechanism is the same second-order hyperfine process responsible for through-bond \(J\)-couplings. Each nucleus couples to the electronic spin density at its location, and second-order mixing through the electronic structure generates an effective bilinear interaction between the two nuclear spins. In the rapidly exchanging liquid, one does not resolve line splittings from individual Xe\(\cdots\)H encounters. Instead, if one spin species is strongly hyperpolarized, the averaged scalar interaction produces a mean-field frequency shift. For proton spins,
\[
H_Z = -\hbar \gamma_1\, \mathbf I_1 \cdot \mathbf B_0,
\]
and under rapid exchange the scalar term becomes
\[
H_J = h\langle J\rangle I_{1z}\langle I_{2z}\rangle,
\]
leading to
\[
\nu_1 = \frac{\gamma_1}{2\pi} B_z + \Delta \nu_1,\qquad
\Delta \nu_1 = -\langle J\rangle \langle I_{2z}\rangle.
\]
The shift can also be written in contact-shift form,
\[
\Delta \nu_1 = \frac{\gamma_1}{2\pi}\,\kappa\,\frac{8\pi}{3} M_{2z},
\]
with
\[
\langle J\rangle = -\frac{8}{3}\kappa \gamma_1 \mu_2 n_2.
\]

The experiment used hyperpolarized liquid \(^{129}\mathrm{Xe}\) and pentane in a spherical cell half-filled with pentane and then condensed xenon, giving an approximate xenon number density
\[
n_2 = 7\times 10^{21}\ \mathrm{cm^{-3}}.
\]
Typical xenon polarizations were \(2\)–\(3\%\). NMR was performed in a static field of about \(10\ \mathrm{mG}\) in a magnetically shielded environment, and high-\(T_c\) SQUID magnetometers provided a magnetometric sensitivity of about
\[
0.3\ \mathrm{nG/\sqrt{Hz}}.
\]
After a proton \(\pi/2\) pulse, the proton and xenon free induction decays were observed near \(39.2\ \mathrm{Hz}\) and \(10.8\ \mathrm{Hz}\), respectively. Proton \(T_2\) was typically about \(3.2\ \mathrm s\), xenon \(T_1\) about \(450\ \mathrm s\), and about \(50\) proton transients could be acquired from a single xenon batch.

A major issue was separation of the scalar shift from ordinary dipolar fields and SQUID-induced magnetic artifacts. The data were taken with \(\mathbf B_0\) along three orthogonal directions \(x\), \(y\), and \(z\), and the results were combined so that orientation-dependent dipolar demagnetization fields were compensated. After correction for SQUID-induced shifts, the \(^{129}\mathrm{Xe}\) frequency shift was consistent with zero, while the proton shift retained a nonzero component. Averaging over field directions and subtracting the SQUID contribution gave a net proton shift of \(-52 \pm 11\ \mathrm{Hz/G}\) per unit xenon magnetization, corresponding to
\[
\kappa = -0.0014 \pm 0.0003
\]
and
\[
\langle J\rangle = -2.7 \pm 0.6\ \mathrm{Hz}.
\]

The microscopic interpretation combined relativistic DFT and liquid-state averaging. The coupling was expressed as
\[
\langle J \rangle = n_2 \int J(r)\, e^{-V(r)/kT}\, d^3r
\]
or, equivalently,
\[
\langle J \rangle = n_2 \int_V 4\pi r^2 J(r) g(r)\, dr.
\]
Relativistic DFT with ZORA implemented in ADF at the scalar ZORA BP86/TZ2P level yielded \(1428\) coupling constants on a cubic grid around pentane. The calculated \(J(^{129}\mathrm{Xe},\,^{1}\mathrm H)\) values were mostly negative and ranged between about \(-8\) and \(0\ \mathrm{Hz}\), with a few positive values, and were fit over \(3.0\)–\(4.3\ \text{\AA}\) to
\[
J(r) = a e^{-br},
\qquad
a = -2.7867 \times 10^4\ \mathrm{Hz},
\qquad
b = 2.719\ \text{\AA}^{-1}.
\]
Molecular dynamics with OPLS-AA for pentane and a Lennard-Jones model for xenon gave the RDF, and the resulting average,
\[
\langle J\rangle = -3.2\ \mathrm{Hz},
\]
agreed well with the measured \(-2.7 \pm 0.6\ \mathrm{Hz}\).

The significance of this result is specific. The measured quantity is an ensemble average over rapidly exchanging configurations, not a resolved pairwise coupling for a single van der Waals complex. The experiment therefore established that electron-mediated scalar coupling can persist across transient noncovalent contact and be read out as a magnetization-dependent NMR line shift, rather than as a conventional splitting pattern.

## 3. Hyperfine-resolved van der Waals structure in the hydrogen \(2S\)-\(2S\) system

In the excited hydrogen problem, hyperfine van der Waals interaction appears in a different form. The system consists of two hydrogen atoms restricted to the \(n=2\), \(J=1/2\) manifold, namely \(2S_{1/2}\) and \(2P_{1/2}\), each split into \(F=0\) and \(F=1\) hyperfine levels. There are therefore \(8\) one-atom states and \(64\) two-atom product states before symmetry reduction. The total Hamiltonian is
\[
H = H_{{\rm LS},A} + H_{{\rm LS},B} + H_{{\rm HFS},A} + H_{{\rm HFS},B} + H_{\rm vdW},
\]
with the nonretarded dipole-dipole interaction
\[
H_{\rm vdW} = \alpha \hbar c\,\frac{x_A x_B+y_A y_B-2 z_A z_B}{R^3}.
\]
The key scales are
\[
\mathcal H \equiv h\,59.1856114(22)\,{\rm MHz},\qquad
\mathcal L \equiv h\,1057.845(9)\,{\rm MHz},\qquad
\mathcal V \equiv 3\alpha \hbar c\,\frac{a_0^2}{R^3},
\]
with the hierarchy \(E_{\rm vdW}\sim E_{\rm HFS}\sim \mathcal L \ll E_{\rm FS}\) valid for \(R>100\,a_0\) [1701.08813].

The paper’s main conceptual point is that hyperfine resolution determines whether a given long-range interaction is first-order or second-order. The conserved quantum number is
\[
F_z = F_{z,A}+F_{z,B},
\]
which decomposes the \(64\)-dimensional space into \(F_z=\pm2\), \(F_z=\pm1\), and \(F_z=0\) manifolds of dimensions \(4\), \(16\), and \(24\), respectively. Each further splits into two irreducible uncoupled subspaces, one containing \(S\)-\(S\) and \(P\)-\(P\) combinations and one containing \(S\)-\(P\) and \(P\)-\(S\) combinations. The most complicated irreducible block is \(12\)-dimensional.

For states involving one \(2S\) atom and one \(2P\) atom, exact or near-exact degeneracy under interchange \(A\leftrightarrow B\) allows direct coupling by \(H_{\rm vdW}\), giving first-order shifts linear in
\[
\mathcal V \propto \frac{1}{R^3}.
\]
For example, certain \(F_z=+1\) states have energies
\[
E^{(A)}_\pm = \mathcal L-2\mathcal H \pm \mathcal V.
\]
By contrast, when both atoms are metastable \(2S\) states, the hyperfine-resolved states are not directly coupled to energetically degenerate partners, so the leading interaction is second order,
\[
\Delta E \sim \frac{\mathcal V^2}{\Delta E}\propto \frac{1}{R^6}.
\]
The characteristic scale is
\[
E_{2S;2S}(R)\sim \frac{\mathcal V^2}{\mathcal L}
\sim E_h\left(\frac{a_0}{R}\right)^6 \frac{E_h}{\mathcal L}
= E_h\,\chi\left(\frac{a_0}{R}\right)^6,
\qquad
\chi = \frac{E_h}{\mathcal L}\approx 6.22\times 10^6.
\]

The hyperfine-resolved treatment makes the denominators state dependent. For \(2S\)-\(2S\) states, virtual transitions involve combinations such as \(\mathcal L\pm \mathcal H\), \(2\mathcal L\pm \mathcal H\), and \(2\mathcal L-5\mathcal H\). In the \(F_z=+1\) manifold, the effective Hamiltonian for a degenerate \(2S\)-\(2S\) pair yields eigenvalue shifts
\[
\epsilon_{1,3}^{(1)\pm}= \dfrac52 \, \dfrac{\mathcal V^2}{\mathcal L-\mathcal H} + \dfrac{\mathcal V^2}{2\mathcal L-\mathcal H} \pm \dfrac{2\mathcal V^2}{\mathcal L-\mathcal H},
\]
and analogous expressions with \(\mathcal L+\mathcal H\) for the companion pair. In the \(F_z=0\) sector, even some exactly degenerate states remain unshifted in first order because the direct van der Waals matrix elements vanish; their splittings arise only through second-order effective interactions.

The physical consequence is that hyperfine structure reorganizes the long-range interaction landscape. A hyperfine-unresolved treatment would miss which channels possess resonant \(1/R^3\) exchange and which remain purely dispersive at \(1/R^6\). For spectroscopy, the practically important statement is explicit: all states with both atoms in metastable \(2S\) levels are shifted only in second order, while states with one atom in a \(P\) state can acquire large first-order hyperfine-resolved van der Waals splittings.

## 4. Dirac-\(\delta\) hyperfine modification of the hydrogen \(2S\)-\(1S\) interaction

The \(2S\)-\(1S\) problem isolates another aspect of hyperfine van der Waals physics: a hyperfine-induced modification of an already existing dispersion interaction. The reference configurations,
\[
|2S\rangle_A |1S\rangle_B,\qquad |1S\rangle_A |2S\rangle_B,
\]
are degenerate under exchange. The leading interaction comes in second-order perturbation theory through virtual \(P\)-state excitations, governed by
\[
H_{\rm vdW} = \frac{e^2}{4\pi\epsilon_0 \, R^3} \left( \delta_{k\ell} - 3 \hat R_k \hat R_\ell \right) r_{Ak} r_{B\ell}.
\]
The crucial feature is that the excited \(2S\) atom has quasi-degenerate \(2P_{1/2}\) and \(2P_{3/2}\) virtual states, separated only by the Lamb shift \({\cal L}\) and fine-structure interval \({\cal F}\). If \({\cal L}\) and \({\cal F}\) are set to zero too early, an important contribution to \(C_6(2S;1S)\) is lost; the paper resolves a discrepancy in the literature precisely on this point [1701.08812].

In the nonretarded van der Waals range,
\[
a_0 \ll R \ll \frac{a_0}{\alpha},
\]
the interaction has the symmetry-dependent form
\[
E_\pm=E_0-\frac{D_6\pm M_6}{R^6},
\qquad
C_6 = D_6 \pm M_6.
\]
The direct and exchange coefficients are
\[
D_6(2S;1S)=176.752\,266\,285\,E_h a_0^6,
\qquad
M_6(2S;1S)=27.983\,245\,543\,E_h a_0^6,
\]
so
\[
E_{2S;1S}(R)\approx -\left(176.752\,266 \pm 27.983\,245\right)E_h\left(\frac{a_0}{R}\right)^6.
\]
The large quasi-degenerate contribution
\[
\overline D_6(2S;1S)=\frac{243}{2}E_h a_0^6
\]
is the missing piece in earlier calculations that obtained only about \(56\).

The hyperfine modification is modeled by a local Dirac-\(\delta\) perturbation. For \(S\)-states, the relevant term in the hyperfine Hamiltonian is the Fermi contact interaction,
\[
H_{\rm HFS}\supset \frac{\mu_0}{4\pi}\mu_B\mu_N g_s g_p \sum_{i=A,B} \frac{8\pi}{3}\,\vec S_i\!\cdot\!\vec I_i\,\delta^{(3)}(\vec r_i).
\]
To treat it generically, the paper introduces
\[
\delta V=\alpha mc^2\left(\frac{\hbar}{mc}\right)^3\pi\,\delta^{(3)}(\vec r_A),
\]
with
\[
\langle nS|\delta V|nS\rangle=\frac{\alpha^4mc^2}{n^3},
\]
and then substitutes
\[
\delta H_{\rm HFS}= \frac{2}{3}g_s g_p \frac{m}{M}\,\delta V(\vec r_A)\,\vec S_A\cdot\vec I_A.
\]
This gives the same physics in two equivalent representations: a correction to the intermolecular potential, and the shift of the \(2S\) hyperfine frequency caused by nearby \(1S\) atoms.

In the van der Waals regime, the correction renormalizes the \(R^{-6}\) coefficient:
\[
\delta D_6(2S;1S) = 367.914\,605\,710\,\alpha^2 a_0^6 E_h,
\qquad
\delta M_6(2S;1S) = -58.095\,351\,093\,\alpha^2 a_0^6 E_h,
\]
so
\[
\delta_{2S}E_{2S;1S}(R)\approx -\left(367.914\,605 \mp 58.095\,351\right)\alpha^2 E_h\left(\frac{a_0}{R}\right)^6.
\]

In the intermediate regime,
\[
\frac{a_0}{\alpha}\ll R\ll \frac{\hbar c}{\cal L},
\]
the hyperfine correction changes character. The energy-type contribution becomes more singular than the unperturbed leading term and scales as \(R^{-5}\):
\[
\delta_{2S}E_{2S;1S}(R) \approx - \left( \frac{891}{32}\mp\frac{630\,784}{59\,049} \right) \frac{\alpha^3}{\pi} E_h \left(\frac{a_0}{R}\right)^5.
\]
The paper notes that logarithmic terms generated by individual retardation contributions cancel in the final result. In the extreme Lamb-shift range,
\[
R\gg \frac{\hbar c}{\cal L},
\]
the residual interaction is negligible, below \(10^{-36}\,\mathrm{Hz}\).

The experimental connection is explicit. The \(2S\) hyperfine splitting was measured as
\[
177\,556\,834.3(6.7)\ \mathrm{Hz},
\]
and the theory gives shifts of the \(2S\) hyperfine interval due to a neighboring \(1S\) atom of
\[
-\left(3.592\mp 0.567\right)\times 10^2\ \mathrm{Hz}\quad \text{at }20\,\text{\AA},
\]
\[
-\left(5.612\mp 0.886\right)\times 10^0\ \mathrm{Hz}\quad \text{at }40\,\text{\AA},
\]
and
\[
-\left(8.769\mp 1.441\right)\times 10^{-2}\ \mathrm{Hz}\quad \text{at }80\,\text{\AA}.
\]
The paper therefore frames the hyperfine van der Waals effect both as a modification of the interatomic potential and as a measurable hyperfine-frequency shift in precision spectroscopy.

## 5. Hyperfine van der Waals repulsion in open-shell polar molecules

A newer use of the term refers to a distinct long-range interaction mechanism between ultracold open-shell polar molecules in rotational states differing by one quantum. The ordinary expectation for a pair such as \(j+j'=0+1\) is resonant dipole-dipole exchange, because \(|0,1\rangle\) is degenerate with \(|1,0\rangle\), giving a first-order interaction
\[
\Delta E \sim \pm \frac{C_3}{R^3}.
\]
For open-shell molecules, however, the rotational levels are split into hyperfine sublevels by rotation, electron spin, and nuclear spin couplings. For certain hyperfine choices, dipole selection rules forbid the exchange process
\[
j,F + j',F' \;\leftrightarrow\; j',F' + j,F.
\]
The paper states the relevant dipole selection rule as
\[
\Delta F = 0,\pm1,\qquad F=0\to F'=0 \text{ forbidden}.
\]
When the exchange matrix element vanishes, the first-order resonant \(R^{-3}\) interaction is removed, and the leading term becomes second order in the dipole-dipole operator,
\[
\sum_{n'\neq n}\frac{\left|\langle n'|\hat{V}_\mathrm{dd}(R)|n\rangle\right|^2}{E_n-E_{n'}}
=-\frac{C_6}{R^6}.
\]
Because the denominators are only hyperfine splittings, typically tens to hundreds of MHz, the resulting \(R^{-6}\) interaction can be unusually strong and can be either attractive or repulsive [2507.12167].

The paper focuses on laser-coolable \(^{2}\Sigma\) molecules, especially CaF, while also showing the effect for MgF, SrF, BaF, and YO. In CaF, the \(j=0\) manifold has \(F=0,1\), and \(j=1\) has \(F=1^-,0,1^+,2\), with splittings on the order of \(25\)–\(123\) MHz in the low rotational states shown. The dimer Hamiltonian contains radial kinetic energy, end-over-end rotation, the two single-molecule Hamiltonians, and the intermolecular dipole-dipole operator
\[
\hat{V}_\mathrm{dd}(R)=\frac{1}{4\pi\epsilon_0R^3}\left(\hat{\bm{d}}^{(A)}\!\cdot\!\hat{\bm{d}}^{(B)}-3\hat{d}_z^{(A)}\hat{d}_z^{(B)}\right),
\]
equivalently written in spherical-tensor form.

CaF provides the clearest example of hyperfine van der Waals repulsion. In the \(j+j'=0+1\) manifold, the pair states \(0,0+1,0\) and \(0,0+1,2\) do not undergo resonant dipole-dipole exchange. For \(0,0+1,2\), the transition
\[
0,0 \to 1,2
\]
is forbidden because \(\Delta F=+2\), so the exchange process
\[
0,0+1,2 \leftrightarrow 1,2+0,0
\]
does not occur. The dominant second-order coupling is then to a lower-lying \(1+1^-\) pair state, so the induced \(R^{-6}\) interaction is repulsive. The lowest \(s\)-wave adiabat for the \(F+F'=0+2\) channel develops a long-range repulsive tail, then becomes strongly attractive at shorter distance, producing a barrier of order mK in height. Collisions below the barrier are reflected, and loss can occur only by tunneling to short range or by inelastic transitions.

The scattering calculations are full coupled-channels calculations with absorbing boundary conditions at short range. Most hyperfine combinations in CaF are highly lossy because resonant dipole-dipole interactions dominate. The attractive \(0+0\) channel approaches universal loss at low temperature. By contrast, the repulsive \(0+2\) channel can have loss rates as low as
\[
10^{-14}\ \mathrm{cm^3/s},
\]
whereas the universal loss benchmark for CaF is
\[
5.4\times 10^{-10}\ \mathrm{cm^3/s}.
\]
The suppression therefore reaches about five orders of magnitude, and the paper emphasizes that merely changing the fluorine nuclear spin state can switch between strongly dipolar and strongly shielded behavior.

The same mechanism appears broadly across species, but the optimal channel is species dependent. For MgF, CaF, SrF, and BaF, the paper mainly examines the \(0,0+1,2\) channel. All show suppression at low temperature, but CaF gives the lowest loss among that series because it best balances tunneling to short range and inelastic hyperfine-changing loss. YO is qualitatively different because its hyperfine constants have opposite signs, giving an inverted hyperfine structure relative to CaF. In YO the repulsive channel is \(0+0\), not \(0+2\), and predicted rates can be as low as
\[
10^{-17}\ \mathrm{cm^3/s},
\]
about eight orders of magnitude below universal.

The paper also gives a scaling law in terms of the hyperfine energy scale relative to the dipolar energy scale, with
\[
a_\mathrm{dd}=2\mu C_3/\hbar^2,\qquad
E_\mathrm{dd}=\hbar^2/(2\mu a_\mathrm{dd}^2).
\]
In the \(T\to0\) limit, the loss rate becomes a universal function of \(E_\mathrm{hf}/E_\mathrm{dd}\). This explains why the effect is strong in open-shell molecules with hyperfine splittings of tens to hundreds of MHz, but not in typical closed-shell bialkali molecules with hyperfine splittings of only tens of kHz. Magnetic fields up to about \(10\) G preserve the suppression in CaF for the repulsive \(F+F'=0+2\) channel, with the lowest Zeeman component \(M_F=-2\) optimal below \(10\) G. The proposed experimental test is a merged-optical-tweezer collision experiment in which one prepares one molecule in \(j=0,F=0\), the other in \(j=1\) in different hyperfine states, merges the tweezers, and measures survival or loss.

## 6. Unifying structure, observables, and limitations

Taken together, these studies indicate a unifying structural principle: hyperfine degrees of freedom decide whether the dominant intermolecular effect is first-order or second-order, and therefore whether the observable scales as \(R^{-3}\), \(R^{-6}\), a contact-shift-like mean field, or a hyperfine-frequency correction. In xenon-pentane, fast exchange converts an underlying scalar coupling into a xenon-magnetization-dependent proton frequency shift. In the hydrogen \(2S\)-\(2S\) problem, hyperfine resolution decides which channels are resonant and which remain dispersive. In the hydrogen \(2S\)-\(1S\) problem, the hyperfine interaction appears as a Dirac-\(\delta\) modification of the dispersion potential and of the \(2S\) hyperfine interval. In open-shell molecules, hyperfine selection rules can eliminate first-order resonant dipolar exchange altogether, leaving a second-order hyperfine van der Waals interaction that may be repulsive [1112.5644] [1701.08813] [1701.08812] [2507.12167].

The observables are correspondingly different. The liquid-state scalar-coupling experiment measures a line shift linear in xenon longitudinal magnetization, not a resolved splitting. The hydrogen calculations predict state-resolved level shifts and hyperfine-interval shifts in beam or gas environments. The ultracold-molecule analysis predicts adiabatic potential barriers, tunneling suppression, inelastic loss, and hyperfine-state-dependent collision rates. This suggests that “hyperfine van der Waals interaction” is an umbrella for several Hamiltonian reductions rather than a single measurement protocol.

The limitations are equally system specific. In the xenon-pentane experiment, the measured \(\langle J\rangle\) is an ensemble average over rapidly exchanging configurations, and the analysis assumes fast exchange and near-parallel xenon polarization. The structural averaging in
\[
\langle J \rangle = n_2 \int 4\pi r^2 J(r) g(r)\, dr
\]
neglects explicit angular structure, even though the underlying DFT results show orientation dependence. In the \(2S\)-\(2S\) hydrogen treatment, the neglect of the \(2P_{3/2}\) manifold requires \(R>100\,a_0\), and the interaction is intrinsically quasi-degenerate rather than an ordinary nondegenerate van der Waals calculation. In the \(2S\)-\(1S\) problem, the most distinctive hyperfine correction, the \(R^{-5}\) term, belongs only to the intermediate regime \(\frac{a_0}{\alpha}\ll R\ll \frac{\hbar c}{\cal L}\); in the extreme Lamb-shift range the effect is negligible. In open-shell molecules, the shielding depends on species-specific hyperfine constants and on the ratio \(E_\mathrm{hf}/E_\mathrm{dd}\), and residual loss may be dominated either by tunneling to short range or by inelastic coupling to other hyperfine states.

A final point of interpretation follows directly from the cited works. Hyperfine van der Waals interactions are not confined to covalent frameworks, not necessarily attractive, and not necessarily small. They can arise from second-order Fermi-contact physics across transient van der Waals contact, from hyperfine-resolved quasi-degenerate dipole-dipole coupling in atoms, or from hyperfine-selected second-order dipolar interactions in ultracold molecules. What unifies these cases is that hyperfine structure controls the effective intermolecular coupling channel, and therefore the observable long-range physics.

Source: https://www.emergentmind.com/topics/hyperfine-van-der-waals-interaction