40 problems

Let $e_{\rm UEG}$ denote the energy per unit volume of the three-dimensional uniform electron gas at unit density, and let $\zeta_{\rm BCC}(1)$ be the Epstein zeta value at $1$ for…

Let $c_{\rm LO}(1,3)$ be the smallest constant for which the Lieb–Oxford inequality is valid in dimension $d=3$ with Coulomb interaction $w(\mathbf r)=|\mathbf r|^{-1}$. Lieb–Oxfor…

Distributional-potential conjecture. Some sort of distributional potentials are also needed for higher-dimensional domains.

Let $V$ be the electron-electron interaction matrix, let $X_t$ denote the density-matrix iterate along the optimization trajectory, and let $\rho(X_t)$ be its associated density. A…

Pure-state density representability conjecture. One may naturally conjecture that

Let $C_3$ denote the universal constant multiplying the term involving the deformation of the velocity field in the three-dimensional density-current representability bound. Optima…

Ionization conjecture. The number of electrons that can be bound to an atomic nucleus of charge $Z$ cannot exceed $Z+1$; mathematically, if $N\leq Z+1$, then $I_Z^{TFDW}$ has a min…

Hohenberg–Kohn conjecture. For every $\rho\in\mathcal{D}^N$, either

Let $E_{\scriptscriptstyle\rm X}[n]$ be the exact exchange energy of an $N$-electron system with density $n$, and let the two-electron spin-unpolarized lower-bound constant be…

Let $E_{\scriptscriptstyle\rm X}[n]$ denote the exact exchange energy of an $N$-electron, spin-unpolarized system with density $n({\bf r})$. Perdew–Ruzsinszky–Sun–Burke's conjectur…

The Müller functional assigns a ground-state energy to a quantum molecular system, while the true quantum molecular ground-state energy is the exact ground-state energy of that sys…

For a density $\rho$ on $\mathbb{R}^3$, let $E_{\rm ind}[\rho]$ be its indirect energy, and define the best Lieb–Oxford constant as the smallest constant $C$ such that … for all ad…

Let $\rho$ be an integrable, nonnegative radially symmetric density on $\mathbb{R}^d$ with $\int\rho=N$, and let $\rho_{\rm rad}(r)=\omega_d r^{d-1}\rho_0(r)$ when…

The Levy–Lieb functional is considered in the $\lambda\to\infty$ limit, where the leading correction due to kinetic energy is independent of particle statistics. Gori-Giorgi–Seidl–…

Local approximation conjecture. The local approximation for the XC energy becomes relatively exact in the Lieb–Simon limit.

Let $\mathcal{P}_{M,N}$ denote the space of densities for an $M$-vertex graph and $N$ particles, and call a density universally $v$-representable (uv) if it arises from the relevan…

Degeneracy-index conjecture. For every case considered,

Nonvanishing-minimizer conjecture. Any minimizer $\psi$ of $F^{\boldsymbol{\alpha},(0)}(r)$ satisfies

Arbitrary-particle-number representability conjecture. The pure excited $v$-representability theorem stated for $N=1$ holds for any $N$, with a sufficient condition that minimizers…

Continuity conjecture. If

Approximate inverse-potential convergence conjecture. If there exists a potential which exactly produces $\rho$, then the sequence $v_n$ converges to this exact inverse potential i…

Pure-state representability conjecture. The same pure representability result as in dimension $d=1$ holds for $d=2$, whereas the set of potential-representable pure-state densities…

Levy–Lieb upper-bound conjecture. The functional satisfies

Let $k$ electron-cloud distributions $\mu_1,\ldots,\mu_k$ be supported on points $x_1,\ldots,x_n\in\mathbb{R}^3$, and let the Coulomb cost be … A multimarginal optimal transport pr…

Let $Z$ denote the nuclear charge and consider the ground-state electron density of an atom in the large-$Z$ limit. On the length scale $Z^{-1}$, rescale the density around the nuc…