C–F Exchange Interactions in Quantum Materials

• C–F exchange interactions are quantum processes where carbon and fluorine atoms exchange electrons via direct and indirect mechanisms, impacting magnetic and reactivity properties.
• In functionalized graphene derivatives such as C₂F and C₂H, these interactions induce geometric frustration, stabilize quantum spin liquids, and support skyrmion formation.
• In CaF+Ca systems, barrierless atom-exchange and isotope-dependent effects enable tunable ultracold reactions and provide precise benchmarks for quantum chemical dynamics.

C-f exchange interactions refer to the quantum mechanical exchange processes involving carbon (C) and fluorine (F) atoms and ions. These interactions govern the magnetic coupling in functionalized graphene derivatives and the intermolecular atom-exchange pathways in CaF+Ca systems. Exchange interactions can be characterized by the competition between direct and indirect mechanisms, by their impact on geometric frustration and magnetism, and by their role in quantum dynamics such as skyrmion formation or ultracold chemical reactivity.

1. Fundamental Principles of C–F Exchange Interactions

At the quantum many-body level, exchange interactions between C and F orbitals arise from two primary mechanisms:

• Direct Exchange: This ferromagnetic coupling originates from the Coulomb integral overlapping neighboring atomic-like Wannier orbitals, most notably the $p_z$ functions on carbon sites in C$_2$F. The magnitude of the direct exchange $J^{\rm dir}_{ij}$ is calculated as

$$J^F_{ij} = \int d\mathbf r\,d\mathbf r'\; \frac{W_i^*(\mathbf r)\,W_j^*(\mathbf r')\,W_i(\mathbf r')\,W_j(\mathbf r)} {|\mathbf r-\mathbf r'|},$$

where $W_i$ denotes the Wannier function for site $i$.

• Indirect (Superexchange) Interaction: This antiferromagnetic contribution is mediated by virtual electron hops involving intermediate orbitals, for example through fluorine $2p$ states in graphene derivatives or metal–ligand bridges in transition metal compounds.

The competition between these mechanisms determines the net isotropic exchange coupling $J_{ij}$, which appears in effective Heisenberg Hamiltonians for the low-energy degrees of freedom:

$$\hat H_{\rm Heis} = -\sum_{i \neq j} J_{ij} \mathbf s_i \cdot \mathbf s_j.$$

2. C–F Exchange in Functionalized Graphene: C$_2$F and C$_2$H

In single-side fluorinated graphene (C$_2$F), the magnetism can be mapped onto a triangular lattice of unsaturated carbon $p_z$ orbitals. The key exchange channels and their numerical values (Rudenko et al., 2013, Mazurenko et al., 2016) are:

Coupling Type	Magnitude (meV)	Nature
$p_z$–$p_z$ (C–C, C$_2$F)	$J_1^{p_z-p_z} \approx -10$	Nearest-neighbor AFM (triangular frustration)
$p_z$–$\sigma_{CF}$ (C–F hybrid)	$|J|<1$	Ferromagnetic, negligible beyond first shell
$p_z$–$\sigma_{CH}$ (C$_2$H)	$J_1^{p_z-\sigma_{CH}}\approx 30$–40 (FM)	Strong leading coupling in semihydrogenated graphene
$p_z$–$p_z$ (C–C, C$_2$H)	Weak AFM	Subdominant to FM $p_z$–$\sigma_{CH}$

Numerical evaluation of the exchange integrals in C$_2$F (DFT with magnetic force theorem, $10^6$ k-points, $\eta=0.02$ eV) reveals rapid decay of AFM coupling beyond nearest neighbor, excluding long-range RKKY oscillations.

The triangular geometry of unsaturated C sites in C$_2$F, combined with the dominance of $J_1 <0$, leads to geometrical frustration. In the classical limit, the system prefers a $120^\circ$ Néel arrangement of spins. Quantum mechanically, moderate itinerancy ($M_{p_z}\sim 0.59\,\mu_B$), a small effective Hubbard $U$, and a $S\approx1/2$ local moment regime allow for quantum-spin-liquid ground states.

In C$_2$H, the strong direct $p_z$–$\sigma_{CH}$ ferromagnetic exchange suppresses long-range magnetic order at finite temperature, consistent with the Mermin-Wagner theorem.

3. Mechanisms of Direct and Indirect C–F Exchange

Direct exchange in C–F-related systems is quantified via constrained random phase approximation (cRPA) and real-space Coulomb integrals. The competition with kinetic (superexchange) terms is critical. For C$_2$F (Mazurenko et al., 2016):

Quantity	Value (meV)
$J^{\rm kin}_{01}$ (AFM)	40
$J^F_{01}$ (screened FM)	18
$J^F_{01}$ (bare FM)	44
$J_{01}$ (net, screened)	22
$J_{01}$ (net, bare)	$-4$
$|\mathbf D_{01}|$ (DM vector)	$\sim$0.98

The net exchange $J_{01}=J^{\rm kin}_{01}-J^F_{01}$ is antiferromagnetic unless the screened direct ferromagnetic term exceeds $J^{\rm kin}_{01}$. Screening by environment or strain can tune $J_{01}$ through zero, placing the system at the AFM–FM instability threshold.

Dzyaloshinskii–Moriya interaction (DMI) arises from spin–orbit coupling and superexchange processes. For C$_2$F, the DMI reaches magnitudes $\sim$1 meV, rendering $|\mathbf D|/J$ of order unity and enabling stabilization of Néel-type skyrmion lattices under moderate fields ($B_c\sim1.6$ T, $T\sim0.5$ K when $J_{01}\sim2$ meV).

4. Atom–Exchange Pathways in CaF+Ca Systems

Ab-initio quantum chemistry methods (CCSD(T), MRCI) reveal the structure of ground and excited-state potential energy surfaces (PES) for CaF+Ca, which determine the C–F atom-exchange dynamics (Sardar et al., 27 Oct 2025). The nuclear Hamiltonian is

$$\hat H = -\frac{\hbar^2}{2\mu} \frac{\partial^2}{\partial R^2} + \frac{\hat L_\theta^2}{2\mu R^2} + V(R,\theta),$$

with $V$ parametrized in Legendre polynomials. The ground-state X $^2A'$ surface is deeply bound ($D_e=7401$ cm$^{-1}$, bent geometry $$R_{\rm eq}=5.873\,a_0,\,\theta_{\rm eq}=137.49^\circ$$), with a strong angular anisotropy from $V_1(R)$. Excited (2) $^2A'$ surface associated with CaF ($^2\Sigma^+$)+Ca ($^3P$) is even deeper ($>10\,000$ cm$^{-1}$ along linear $\theta=0^\circ$).

Atom–exchange is barrierless for the ground channel:

• No transition-state barrier is found in 2D $V(R,\theta)$ scans.
• The process is exothermic or isoenergetic (isotope-dependent zero-point energy differences).
• Long-range van der Waals coefficients for $^{40}$CaF+$^{40}$Ca: $C_{6,0} = 1778\,E_h\,a_0^6$, $C_{6,2}=95\,E_h\,a_0^6$.

Excited-state atom-exchange is governed by deep PES wells and strong Ca($^3P$) spin–orbit coupling, supporting possible nonadiabatic transitions.

5. Isotope-Dependent Exchange and Ultracold Reaction Dynamics

Isotope-exchange reactions of the form

$$^A{\rm CaF} + {}^B{\rm Ca} \rightarrow {}^B{\rm CaF} + {}^A{\rm Ca}$$

are controlled entirely by the zero‐point energy difference. The $Q$-values, ranging 1–8 cm$^{-1}$, are well below vibrational spacings ($\sim581$ cm$^{-1}$) and above rotational spacings ($0.68$ cm$^{-1}$), ensuring product CaF in $v=0$ across multiple $J$ levels. Reaction is exothermic if $B > A$, fully tunable via isotope selection. The absence of an activation barrier implies near-unit reaction probabilities at ultracold collision energies.

	$^{40}$CaF	$^{42}$CaF	$^{43}$CaF	$^{44}$CaF	$^{46}$CaF	$^{48}$CaF
$^{40}$Ca	0	+2.247	+3.301	+4.305	+6.199	+7.947
$^{42}$Ca	−2.247	0	+1.053	+2.059	+3.953	+5.701
...

6. Charge-Exchange Excitations and Continuum Effects

Charge-exchange excitations involving C–F pairs in nuclear and solid-state systems are modeled by self-consistent continuum RPA with finite-range Gogny-like interactions (Donno et al., 2016). The inclusion of both direct (Fock) and tensor–isospin channels ensures:

• Proper treatment of Fermi, Gamow–Teller, and spin–dipole operators.
• Preservation of energy-weighted sum rules.
• Strong dependence of SD($0^-$) modes on tensor contributions, evidenced by centroid shifts of 2–9 MeV.

Continuum coupling yields smooth, physical strength distributions above threshold, in contrast to discretized RPA approaches. Finite-range exchange (Fock) explicitly shapes the charge-exchange spectra, particularly for spin-dependent modes.

7. Implications and Tuning of C–F Exchange in Contemporary Research

C–F exchange interactions underpin several modern quantum phenomena:

• Geometric Frustration: Dominant AFM exchange on triangular lattices leads to nontrivial spin textures and suppresses conventional magnetic order.
• Quantum Spin Liquids: Small $U/t$ ratios and moderate moments in C$_2$F shift ground states from classical order to highly entangled nonmagnetic quantum phases.
• Skyrmion Formation: The interplay of near-cancelled AFM/FM couplings and strong DMI in C$_2$F enables stabilization of skyrmion crystals under experimentally accessible fields and temperatures.
• Ultracold Chemistry: Deep anisotropic PES and barrierless exchange in CaF+Ca systems allow for tunable synthesis, isotope-selective reactions, and precision benchmarking of quantum chemical dynamics.
• Spectroscopic and Dynamical Probes: The detailed mapping of exchange parameters, decay rates, and mode sensitivities informs experimental design in scanning probe, neutron scattering, and cold-molecule research.

A plausible implication is that environmental tuning (screening via substrates or strain, isotope engineering) can steer C–F exchange interactions across quantum phase boundaries, thereby enabling controlled exploration of frustrated magnetism and exotic reactivity regimes.