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A: ethoxy CH${}_2$ ↔ CH${}_3$, $J\simeq 7\mathrm{Hz}$; vinyl CH ↔ each alkene methyl, small ${}^{4}J\simeq 1$–$1.5\mathrm{Hz}$.

C: ethoxy CH${}_2$ ↔ CH${}_3$, about $7\mathrm{Hz}$; the two OCH${}_2$ proton types have a geminal coupling of about $9$–$10\mathrm{Hz}$; acetal CH ↔ C≡CH has ${}^{4}J\simeq 2\mathrm{Hz}$.

B: internal vinyl CH ↔ trans terminal H, about $17\mathrm{Hz}$; ↔ cis terminal H, about $10\mathrm{Hz}$; terminal CH${}_2$ geminal coupling about $1$–$2\mathrm{Hz}$. Vinyl CH ↔ OCH${}_2$ is about $6\mathrm{Hz}$. Both vicinal links along the butanoate alkyl chain are about $7$–$8\mathrm{Hz}$.

Use $J=300\Delta \delta$ when a separation is measured in ppm. Thus $0.057\mathrm{ppm}$ corresponds to about $17\mathrm{Hz}$, whereas $0.023\mathrm{ppm}$ corresponds to about $7\mathrm{Hz}$. These are approximate readings, not a fitted spin simulation.

The small four-bond splittings illustrate allylic scalar coupling and acetal-alkyne long-range coupling.

N${}_2$: Homo ${\sigma }_{2p_z}$; LUMO the degenerate ${\pi }_{2p}^{\ast }$ pair.

O${}_2$: the two ${\pi }_{2p}^{\ast }$ orbitals are the highest occupied, singly occupied spatial orbitals. The next completely empty spatial orbital is ${\sigma }_{2p_z}^{\ast }$. An added electron can instead occupy the opposite-spin vacancy in a singly occupied pi-star orbital; this distinction matters if “LUMO” is being used for an electron-addition spin orbital.

Li${}_2$: Homo ${\sigma }_{2s}$; next empty sigma orbital conventionally denoted ${\sigma }_{2s}^{\ast }$ in this elementary diagram. Be${}_2$: Homo ${\sigma }_{2s}^{\ast }$; the low empty 2p manifold supplies the LUMO candidates, drawn as ${\pi }_{2p}$ in the stated light-diatomic ordering. Mixing makes these labels descriptions of dominant atomic-orbital character, not strictly unmixed atomic functions.

This distinction is the open-shell frontier orbital convention.

Suppress the filled 1s core pair, which contributes zero net bond order. For N${}_2$, use the valence orderingand occupy it as $({\sigma }_{2s}{)}^2({\sigma }_{2s}^{\ast }{)}^2({\pi }_{2p}{)}^4({\sigma }_{2p_z}{)}^2$.

For O${}_2$, the ordering of the bonding 2p levels is reversed: ${\sigma }_{2p_z}<{\pi }_{2p}$. Occupancy is $({\sigma }_{2s}{)}^2({\sigma }_{2s}^{\ast }{)}^2({\sigma }_{2p_z}{)}^2({\pi }_{2p}{)}^4({\pi }_{2p_x}^{\ast }{)}^1({\pi }_{2p_y}^{\ast }{)}^1$, with parallel spins in the two degenerate antibonding pi orbitals.

Li${}_2$ has $({\sigma }_{2s}{)}^2$; Be${}_2$ has $({\sigma }_{2s}{)}^2({\sigma }_{2s}^{\ast }{)}^2$ in the elementary valence description. Empty 2p-derived levels must also be included to discuss mixing and Be${}_2$'s weak bonding; their precise energies are not set by this qualitative diagram.

There are $\boxed{4+2(1)+2(5)=16}$ valence electrons, all paired, so diazomethane exhibits diamagnetism. A localized orbital occupancy consistent with the double-bond construction in 2(c)(i) isHere $c,t$ mean central and terminal. Each listed occupied spatial orbital contains two electrons of opposite spin. The central N's two p atomic orbitals are perpendicular, so they can support the two perpendicular pi bonds. No central-N lone pair is left over: its positive formal charge accompanies four bonding pairs in this Lewis structure.

Allowing resonance mixes a terminal-N lone pair into the C–N–N $\pi$ orbitals. In the alternative terminal-$sp$ basis, the same electron count is four occupied $\sigma$ orbitals ($8$ electrons), one terminal-$sp$ lone pair ($2$), one perpendicular N–N $\pi$ orbital ($2$), and two occupied orbitals of a three-centre C–N–N $\pi$ system ($4$). The $16$-electron count and the paired ground state survive this change of description; the localized picture does not assert two perfectly isolated double bonds.

The spin isomers of hydrogen illustrate how nuclear spin statistics constrain the product of electronic, vibrational, rotational and nuclear-spin exchange symmetries. In the ground electronic and vibrational states of these homonuclear molecules, the first two factors are symmetric. Exchanging the nuclei reverses the internuclear direction, so a rotational state has parity $(-1{)}^J$.

For ${}^1{\mathrm{H}}_2$, the protons are spin-one-half fermions and the total nuclear-exchange wavefunction must be antisymmetric. The antisymmetric spin singlet pairs with even $J$ (para-hydrogen, one spin state); the symmetric spin triplet pairs with odd $J$ (ortho-hydrogen, three spin states). For ${}^2{\mathrm{H}}_2$, spin-one deuterons are bosons, so the total must be symmetric. Nuclear spins $I=0,2$ supply six symmetric states paired with even $J$ (ortho-deuterium); $I=1$ supplies three antisymmetric states paired with odd $J$ (para-deuterium). The high-temperature ortho/para statistical ratios are $3:1$ for hydrogen and $2:1$ for deuterium, while low-temperature equilibrium populations also depend on rotational energies and whether spin conversion can occur.

At high temperature, $\chi T$ tends towards the nearly constant Curie constant of weakly correlated $S=\frac{1}{2}$ centers. On cooling, $\chi T$ falls because neighboring spins increasingly correlate antiparallel. The broad maximum in $\chi$ marks the crossover from the high-temperature Curie-like regime to substantial short-range order.

At still lower temperature, the Magnetic susceptibility becomes small because the antiferromagnetic ladder favors a singlet-like ground state and has few thermally accessible magnetic excitations. An even-leg spin ladder can have a spin gap, which reinforces this suppression. A broad Magnetic susceptibility maximum and falling $\chi T$ support short-range antiferromagnetic correlations; they do not establish a three-dimensional Néel transition. Such a transition would require independent evidence, rather than identifying the broad maximum itself as the ordering temperature.

Formal ionic counting gives La(III), N(−III), and therefore Re(VI), with a $5d^1$ configuration. An isolated localized $d^1$ center would carry a saturation magnetic moment of order $1{\mu }_{\mathrm{B}}$, much larger than the value quoted. It is not a $d^0$ ion made diamagnetic simply by an octahedral ligand field.

The spatially extended Re $5d$ orbitals overlap strongly with N $2p$ orbitals. Re–N $\sigma$ interactions raise the $e_g$-derived antibonding states, while $\pi$ interactions mix the $t_{2g}$ states with the nitrogen network. Repetition through corner-sharing octahedra turns these combinations into broad energy bands. The nominal $d$ electron occupies a delocalized, partially filled band rather than an isolated magnetic orbital.

Only electrons close to the Fermi energy change their occupations appreciably in an applied magnetic field. Consequently a Pauli paramagnetism has a small, nonzero field-induced magnetization:when $g$ is the density of states including both spin directions, expressed per formula unit and per unit energy. This band model explains a very small moment without requiring the absence of electrons in the Re-derived band. The reported material has metallic transport and a Pauli-like response.^[ref]

The description “saturated” in the question should not be interpreted as complete alignment of one localized spin per Re: a nearly linear metallic response over a laboratory field range need not be a true fully spin-polarized saturation moment.

Using the spin values above and accounting for two metal ions per formula unit,The deviations from 2 indicate spin-orbit coupling and incomplete quenching of orbital angular momentum. The partly filled $e^{\prime \prime }$ manifold in Fe and $e^{\prime }$ manifold in Ni permit orbital degeneracy in the ideal geometry. Co's high-spin orbital occupation is more orbitally balanced, but mixing with excited ligand-field states still produces a substantial effective $g$ shift. These fitted scalar values summarize a thermally averaged response; they do not determine the individual components of an anisotropic g tensor.

For two independent identical ions at sufficiently high temperature, Curie's law gives $\chi T=2(N_A{\mu }_B^2/3k_B)g^{2}S(S+1)$. In the cgs units used here, $N_A{\mu }_B^2/3k_B\simeq 0.125\mathrm{c}{\mathrm{m}}^3\mathrm{K}\mathrm{mo}{\mathrm{l}}^{-1}$.

Fe(II), Co(II) and Ni(II) have $S=2$, $3/2$ and 1, respectively. The corresponding high-spin occupancies, ordered as $(e^{\prime \prime },e^{\prime },a_1^{\prime })$, areHund filling puts one electron in each orbital before additional pairing. Spin-only dimer values at $g=2$ would be 6.0, 3.75 and 2.0; the observed larger values anticipate orbital contributions. The spin assignments require the chemically appropriate high-spin states, rather than rounding the measured $\chi T$ directly to a spin-only integer.

Adopt $\mathcal{H}=-2J({\mathbf{S}}_1+{\mathbf{S}}_2)\cdot \mathbf{s}-2J^{\prime }{\mathbf{S}}_1\cdot {\mathbf{S}}_2$, with $S_1=S_2=1$ and ligand spin $s=1/2$. Let ${\mathbf{S}}_{12}={\mathbf{S}}_1+{\mathbf{S}}_2$ and $\mathbf{S}={\mathbf{S}}_{12}+\mathbf{s}$. The angular momentum addition identities giveThe five allowed multiplets areTheir degeneracies $2,2,4,4,6$ sum to 18, matching $3\times 3\times 2$. For dominant ferromagnetic $J$, the ground multiplet has $S_{12}=2$, $S=5/2$. A weak antiferromagnetic $J^{\prime }$ does not overcome the two strong metal–radical bonds.

For an axial spin Hamiltonian ${\mathcal{H}}_D=D{S}_z^2$ with $D/k_B=-25$ K, the minima are $m_S=\pm 5/2$. The highest levels along the reversal ladder are the $m_S=\pm 1/2$ doublet. Henceabout 1.25 kJ mol${}^{-1}$ or 12.9 meV per molecule. Using $|D|S^2$ would incorrectly put a half-integer spin at a nonexistent $m_S=0$ level. Transverse anisotropy and quantum tunnelling of magnetization can create faster relaxation pathways, so this axial ladder barrier need not equal an experimentally fitted effective barrier.

At high temperature, approximately independent paramagnetic ions give the appropriate Curie constant. The negative Weiss temperatures and the reduction of $\chi T$ on cooling initially indicate predominantly antiferromagnetic correlations. The sharp rise near the ordering temperature nevertheless reveals a strongly enhanced low-field magnetic response.

A net moment could arise from a canted antiferromagnet, a ferrimagnet, or a ferromagnet with competing antiferromagnetic interactions. Differences between field-cooled magnetization and zero-field-cooled magnetization suggest history-dependent domain selection or slow relaxation, but are not alone proof of a particular microscopic spin arrangement. An ordinary compensated antiferromagnet would not account for a large ordering-related ferromagnetic-like low-field contribution without an additional mechanism.

Weak ferromagnetism from canted antiferromagnetic order is the consistent model. Importantly, the plot has different magnetization scales: the left axis belongs to Mn and Fe; the right belongs to Co and Ni. Approximate low-temperature field-cooled readings are 40, 50, 450 and 390 G cm${}^3$ mol${}^{-1}$, respectively. Dividing by 5585 givesThese moments are tiny compared with fully aligned moments of about 5.09, 4.40, 4.25 and 2.47 ${\mu }_B$, calculated using the Curie-derived $g$ values below. A conventional ferromagnet of the full high-spin ions would have much larger spontaneous magnetization; a simple bipartite array of identical metal sites also supplies no obvious unequal sublattice moments for ferrimagnetism. Opposed large sublattice moments with a small transverse uncompensated component reconcile the net response with negative $\Theta$.

If the 10 Oe field has aligned the weak-ferromagnetic domains, $M_{\mathrm{weak}}=gS{\mu }_B\sin \alpha$ estimates sublattice canting angles of roughly 0.08°, 0.12°, 1.1° and 1.6°. These are illustrative, not precision structure determinations: the low field may not fully align the domains, and the readings are graphical. Bulk magnetization strongly supports this model but does not uniquely establish every spin direction; magnetic neutron diffraction would test the microscopic arrangement.

Define $\mathcal{H}=-2J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$. Each site has $z=6$ exchange-connected neighbours. In mean-field theory, replacing neighbours by their mean spin produces the effective field that yieldsThusfor Mn, Fe, Co and Ni. Negative $J$ means antiferromagnetic exchange in this convention; a Hamiltonian written with $-J$ instead of $-2J$ would have twice these numerical values. The increasingly negative values indicate a growing effective exchange energy, although different orbital occupancy and anisotropy mean that they are not a simple universal function of $d$-electron number.

A three-dimensional six-connected framework is more favourable to mean-field reasoning than isolated chains, but the theory neglects short-range correlations, anisotropic exchange and fluctuations. The ratios $|\Theta |/T_C\simeq 1.65,1.69,3.75,2.10$ show substantial departures from the simplest unfrustrated isotropic mean-field relation. The values are therefore effective high-temperature exchange estimates, not a complete microscopic fit to the ordered phases.

Below the ordering temperature, coherent spin correlations produce magnetic Bragg intensity. Depending on the magnetic propagation vector, this can appear as additional peaks or as added intensity at positions already occupied by nuclear reflections. A magnetic unit cell larger than the chemical one gives extra reciprocal-space positions; a zero-propagation-vector canted structure need not do so.

Magnetic intensity generally decreases at large scattering vector through the magnetic form factor and depends on moment components perpendicular to the scattering vector. Above the transition the long-range magnetic Bragg contribution disappears, though short-range correlations can leave diffuse scattering. The nuclear pattern remains approximately unchanged unless magnetic ordering also drives a structural distortion.

For a positive gyromagnetic ratio the magnetic moment is parallel to the spin angular momentum; for a negative ratio it is antiparallel. The Hamiltonian ${\hat{H}}_Z=-\gamma B_0{\hat{I}}_z$ gives $E_m=-\gamma \hslash B_{0}m$. Thus $m=+1/2$ is lower for the proton, whereas $m=-1/2$ is lower for nitrogen-15. Their spin precession senses are opposite for a common positive field convention; measurable absorption frequencies and energy separations remain positive.

The energy separation is $\Delta E=\hslash |\gamma |B_0$. At 10 T this givesAt 300 K, $k_{B}T=4.142\times 10^{-21}\mathrm{J}=0.02585\mathrm{eV}$, so the splitting-to-thermal-energy ratios are $6.80\times 10^{-5}$ and $6.87\times 10^{-6}$. The two spin levels therefore have nearly equal populations. Their equilibrium population polarizations are approximately half these ratios, $3.40\times 10^{-5}$ and $3.44\times 10^{-6}$, explaining weak thermal NMR sensitivity despite well-defined resonant transition energies.

NMR probes the chosen isotope's local environments and motion in the bulk without electrical contacts. Line narrowing, relaxation and exchange measurements can expose hopping across a broad range of timescales, separate sites with resolved shifts, and provide microscopic activation energies. Pulsed-field-gradient measurements can sometimes measure longer-range displacement directly; local relaxation measurements usually infer it through a model.

Limitations include low sensitivity for dilute or low-abundance nuclei, overlap of inequivalent sites, quadrupolar broadening for spins above one-half, and contributions from paramagnetic impurities or other relaxation mechanisms. Heating can change defect populations or phase composition. Most importantly, local hopping need not produce long-range ionic conductivity: back-and-forth motion, correlated jumps and trapped ions can relax spins without carrying charge over macroscopic distances. Connecting $T_1$ to a diffusion coefficient requires an explicit jump model and comparison with a bulk transport measurement.

For a one-dimensional free-electron gas of length $L$, allowed wave vectors have spacing $2\pi /L$. Including two spin states and occupied wave vectors from $-k_F$ to $+k_F$,The ground-state kinetic energy is found by integrating rather than assigning every electron the Fermi energy:Charge neutrality gives $N_e=ZN$ because each positive ion has charge $+Ze$. Thus $K_e=ZN{E}_F/3$. Since the ion separation is $2R$, $L\simeq 2RN$ and $n_e=Z/(2R)$. These are one-dimensional number densities and state counts; using a three-dimensional $k_F\propto n^{1/3}$ relation here would change the model. The results concern the ground-state Fermi-gas contribution, neglecting electron interactions beyond the supplied electrostatic energy.

In the SI linear-response convention $\mathbf{M}={\chi }_H\mathbf{H}$, a diamagnetic contribution has negative susceptibility: the induced moment opposes the applied field. Orbital motion supplies a familiar microscopic mechanism, and closed-shell atoms can be diamagnetic even without permanent moments. Paramagnetism has positive susceptibility, with induced magnetization along the field; thermal alignment of permanent localized moments is one mechanism, while conduction-electron spin response is another.

Neither sign alone is a definition of ferromagnetic ordering. An ordinary paramagnetic phase has no spontaneous zero-field magnetization. A diamagnetic background can coexist with a larger paramagnetic contribution in the same specimen. Curie's inverse-temperature law describes an appropriate localized-moment model, not every possible paramagnet.