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Use the ordered computational basis $(|00⟩,|01⟩,|10⟩,|11⟩)$, with $Z|0⟩=|0⟩$ and $Z|1⟩=-|1⟩$. Write $X,Y,Z$ for the Pauli matrices, $I$ for the identity operator, $J={\omega }_{12}$, $\Sigma ={\omega }_1+{\omega }_2$, and $\Delta ={\omega }_1-{\omega }_2$. As in the question, the Hamiltonian operators have angular frequency units, so the time-evolution operator is $e^{-iHt}$. The Heisenberg coupling matrix for two spins isConsequently the full Heisenberg model Hamiltonian operator and its proposed Ising coupling approximation are

For nondegenerate perturbation theory, choose $H_{\mathrm{I}}$ as the unperturbed Hamiltonian operator andas the perturbation. Only $|01⟩$ and $|10⟩$ are coupled, and their unperturbed energy levels differ by $\Delta$. All first-order energy corrections vanish. The first-order eigenvectors and second-order energy corrections areThe states $|00⟩,|11⟩$ are exact eigenvectors throughout. Hence the weak coupling condition for two NMR spins isA large common Zeeman energy alone does not ensure this condition. Crossings involving the uncoupled parallel-spin states do not invalidate the argument: the relevant denominator is the separation of the two states connected by $V$.

As an exact check, the middle $2\times 2$ matrix has eigenvaluesFor $\Delta \ne 0$, labeling by the limiting computational basis states reproduces the shifts above. The quantum transition probability from $|01⟩$ to $|10⟩$ isThus unwanted exchange has a uniformly small amplitude when $|J/\Delta |\ll 1$. If accumulated phases must also be approximated over a duration $t$, neglecting the second-order shifts additionally requires $J^{2}t/|\Delta |\ll 1$, or those shifts must be retained in an improved effective Hamiltonian operator.

The same result is the secular approximation for scalar-coupled spins. In the interaction picture of the Zeeman energy terms,These terms average rapidly on the coupling timescale $1/|J|$ when $|\Delta |\gg |J|$, while $JZ\otimes Z/2$ survives.

For fixed nuclei and electronic environments, the individual NMR angular frequencies scale with the static magnetic field $B_0$: schematically ${\omega }_i={\kappa }_i{B}_0$, where ${\kappa }_i$ includes the gyromagnetic ratio and magnetic shielding. The NMR scalar coupling $J$ in angular frequency units is approximately independent of $B_0$. Therefore a nonzero separation $|\Delta |=|{\kappa }_1-{\kappa }_2|B_0$ increases with magnetic field, improving the approximation. In particular, a fixed difference of chemical shifts gives a larger separation in hertz at higher magnetic field. If the two resonances remain exactly equal, increasing their common frequency does not suppress their exchange.

In heteronuclear liquid-state NMR quantum computing, different gyromagnetic ratios give well-separated resonance frequencies. A resonant transverse radio-frequency pulse selects one nucleus; its phase sets the rotation axis in the transverse plane and its duration and amplitude set the rotation angle. Detuning, free precession and rotating-frame phase changes supply $z$ rotations. For homonuclear spins, differing chemical shifts can provide frequency selectivity, but selective pulses must be narrow enough not to rotate the other spin. Exactly magnetically equivalent spins cannot be independently addressed by a spatially uniform radio-frequency field merely because they are labelled as different qubits.

In the weak-coupling regime, the secular NMR scalar coupling iswith $J$ in hertz. Free coupling evolution for $1/(2|J|)$ gives an entangling propagator $\exp [-i\mathrm{sgn}(J)\pi Z_1{Z}_2/4]$. Suitable single-qubit $z$ rotations convert it to a Controlled-Z gate, and Hadamard conjugation on the target converts controlled-$Z$ to a controlled-NOT gate. Refocusing pulses remove unwanted Zeeman evolution and unwanted couplings. This construction applies to resolved, weakly coupled homonuclear spins as well as heteronuclear spins; strongly coupled or equivalent spins need a different control analysis.

These controls supply a universal gate set on a finite addressable connected register, but conventional liquid-state molecular NMR is not an architecture that scales straightforwardly to arbitrary register size. A finite chemical-shift bandwidth causes spectral crowding as more spins and coupling-split transitions are added; increasingly selective pulses become slow relative to coherence times, and complicated coupling networks require extensive refocusing. The exponentially small pseudo-pure-state signal further limits the complete ensemble scheme. Formal unitary universality therefore does not establish scalable physical addressability, gate speed or initialization.

Use two-dimensional NOESY, or ROESY where molecular tumbling makes NOESY less favorable, to look for through-space correlations between proton sites on the two ends that are brought together in the closed conformation. Assign resonances first and compare predicted ring contacts with an open reference or model.

The nuclear Overhauser effect involves NMR cross-relaxation through magnetic dipole–dipole coupling. Its strong approximate $r^{-6}$ dependence makes it sensitive to short interproton distances. A NOESY cross-peak therefore reports spatial proximity, including between sites far apart along the covalent skeleton; it is not a NMR scalar coupling transmitted along that skeleton.

Run a dilution series: end-to-end contacts that persist at concentrations where intermolecular association is negligible support intramolecular ring closure. Hydrogen-bonded NH chemical shifts and diffusion measurements can support the assignment, but neither alone proves ring topology. A flexible open chain can transiently bring ends together, and intermolecular contacts can also give NOEs. Several geometry-consistent cross-peaks, concentration behavior and controlled mixing times make the case much stronger than one peak.

nuclear magnetic resonance of the spin-one-half isotope ${}^{29}\mathrm{Si}$. Equivalent silicon sites give the same isotropic chemical shift, while condensation and silanol substitution can distinguish environments. Line heights alone are not quantitative populations.

Topicity classifies the relationship between apparently similar atoms, groups, or molecular faces by a replacement test. Replacing one member and then the other gives identical products for homotopic sites, enantiomers for enantiotopic sites, and diastereomers for diastereotopic sites.

For example, replacing either hydrogen on a suitable prochiral methylene carbon by deuterium can produce opposite enantiomers. Enantiotopic sites are equivalent in an achiral environment but can be distinguished by a chiral reagent or enzyme. Diastereotopic hydrogens can give different chemical shifts even in an achiral solvent. Topicity therefore connects molecular symmetry to spectroscopy and stereoselective reactions.

For the two-spin Hamiltonian operator $H=({\omega }_{1}ZI+{\omega }_{2}IZ+J(XX+YY+ZZ))/2$, juxtaposition of two Pauli matrices denotes their tensor product. Put $\Delta ={\omega }_1-{\omega }_2$. The exchange matrix element between $|01⟩$ and $|10⟩$ is $J$, whereas their diagonal separation is $\Delta$. Nondegenerate perturbation theory therefore gives admixtures $\pm J/\Delta$ and second-order frequency shifts $\pm J^2/\Delta$. Neglecting the exchange terms is justified to leading order when $|J/\Delta |\ll 1$. A large common Zeeman energy is insufficient. To approximate accumulated phases by this leading-order Hamiltonian operator, one also needs $J^{2}t/|\Delta |\ll 1$ over the observation time, or must include the second-order shifts.

For fixed nonidentical chemical shifts, increasing the static magnetic field increases $|\Delta |$ in angular-frequency units while the NMR scalar coupling changes little. If $\Delta =0$ remains exact, the approximation fails at every nonzero $J$.