The internal energy is the population-weighted sum of the molecular energy levels:At fixed volume, assume the energy levels themselves are temperature-independent. Differentiating the molecular partition function givesChoosing the zero of energy changes $U$ by the corresponding constant per molecule, so the ground-level convention must be consistent throughout.

GaN has a substantial band-gap advantage for high-temperature operation. Its gap is larger than those of GaAs and GaP, suppressing thermally generated intrinsic carriers through $n_i\propto e^{-E_g/(2kT)}$. A doped junction therefore retains its intended carrier imbalance and can have lower thermally generated leakage over a wider temperature range, other factors being equal. The wide gap also supports emission at higher photon energies.

This is a materials-based expectation, not a guarantee for every finished device. Dopant activation and compensation, defects, contact stability, heat extraction and nonradiative recombination can set the practical limit. Increasing temperature also reduces carrier mobility and can broaden or quench light emission. The relevant advantage is reduced intrinsic excitation; a comparison of operating temperatures needs those other device properties too.

At equilibrium, electrons initially diffuse from the n side and holes from the p side. Recombination leaves ionized donors and acceptors in a depletion region, producing an internal electric field and band bending. The Fermi level is constant across the equilibrium p-n junction. With p material on the left, both band edges lie higher on the p side, as in the left panel.

Forward bias, with the p side positive relative to n, reduces the injection barrier. Electrons and holes enter the junction region and can recombine radiatively, emitting photons with energy close to the band gap. Under bias, separate electron and hole quasi-Fermi levels replace a single equilibrium Fermi level.

Both GaN polymorphs have a direct band gap, so a conduction-band electron and valence-band hole can recombine without a large crystal-momentum change. Silicon and germanium have indirect band gaps, requiring phonon assistance for near-edge radiative recombination, which makes efficient band-edge light emission much harder. A direct gap enables the process; defects, carrier confinement and competing nonradiative channels still determine device efficiency.

The extra electron occupies the lowest unoccupied molecular orbital of the neutral molecule, the $B_{3g}$ orbital at the middle root ${\lambda }_L=0.25410169\dots$ of ${\lambda }^3-4\lambda +1=0$. WriteThe first and third rows of the $B_{3g}$ block giveChoosing the overall phase so that site 1 is positive gives $(a,b,c)\simeq (0.664623,-0.495741,-0.559033)$. In the original site order,The squared coefficients sum to one. Within this fixed-orbital Hückel orbital charge density approximation, the added electron contributes site populationsThus the oxygen sites $\boxed{1\text{ and }8}$ receive the largest added negative charge, each receiving an increment of approximately $-0.221e$. Including the four doubly occupied neutral orbitals gives an anion $\pi$ population of about $1.621$ on each oxygen, $0.870$ on sites 2 and 7, and $1.005$ on each remaining carbon. Thus oxygen also has the largest total $\pi$ population in this model. A coefficient's sign is an orbital phase, not the sign of its electric charge. Predicting an actual charge density would additionally require orbital relaxation and a population-analysis convention; the stated result is the requested orbital model.

Set $E=\alpha +\lambda \beta$, with the usual bonding convention $\beta <0$. In the Hückel method, the dimensionless matrix $M=(H-\alpha I)/\beta$ has diagonal entries one on oxygen sites 1 and 8, zero on carbon, and unit entries between bonded neighbours. In the bases $(O_{+},C_{+},R_{+})$ and $(O_{-},C_{-},R_{-})$ above, its blocks areThe one-dimensional $B_{2g}$ and $A_u$ blocks have $\lambda =1$ and $-1$, respectively. The characteristic polynomial of the $B_{1u}$ block factors asso its three molecular orbital energies areFor the $B_{3g}$ block the matrix trace is zero. Thus its three $\lambda$ values sum to zero, and the pair supplied in the paper gives the remaining one as $-(1.86-2.12)\simeq 0.26$. Directly, its secular equation is ${\lambda }^3-4\lambda +1=0$, whose roots are approximately $1.860806$, $0.254102$ and $-2.114908$. The supplied pair is approximate, so $0.26$ from its trace and $0.2541$ from the exact block should not be mistaken for an additional physical splitting.

All eight energies, ordered from low to high for $\beta <0$, areThe eight $\pi$ electrons fill the four lowest molecular orbitals. The equality at $\lambda =1$ is an accidental degeneracy of this parameterized model, not a multidimensional representation of $D_{2h}$.

Write $p_i$ for the orthonormal $2p_z$ atomic orbital on numbered site $i$, with every positive lobe pointing along $+z$. A normalized set of symmetry-adapted linear combinations isEach has unit squared coefficient sum, and every distinct pair has zero inner product. The orbital representation therefore reduces to $\boxed{3B_{1u}+3B_{3g}+B_{2g}+A_u}$. For example, the fully symmetric site sum still has $B_{1u}$ orbital symmetry because a $p_z$ orbital itself changes sign under operations that reverse $z$. Ignoring that sign would assign the wrong representations.

The planar para-quinone framework has three mutually perpendicular twofold axes, inversion through its centre, and the associated mirror planes. With $x$ horizontal, $y$ along the two C=O bonds and $z$ perpendicular to the molecule, it belongs to the D2h point group, $\boxed{D_{2h}}$.