Density Functionals in the Presence of Magnetic Field

In this paper density functionals for Coulomb systems subjected to electric and magnetic fields are developed. The density functionals depend on the particle density, $\rho$, and paramagnetic current density, $j^p$. This approach is motivated by an adapted version of the Vignale and Rasolt formulation of Current Density Functional Theory (CDFT), which establishes a one-to-one correspondence between the non-degenerate ground-state and the particle and paramagnetic current density. Definition of $N$-representable density pairs $(\rho,j^p)$ is given and it is proven that the set of $v$-representable densities constitutes a proper subset of the set of $N$-representable densities. For a Levy-Lieb type functional $Q(\rho,j^p)$, it is demonstrated that (i) it is a proper extension of the universal Hohenberg-Kohn functional, $F_{HK}(\rho,j^p)$, to $N$-representable densities, (ii) there exists a wavefunction $\psi_0$ such that $Q(\rho,j^p)=(\psi_0,H_0\psi_0)_{L^2}$, where $H_0$ is the Hamiltonian without external potential terms, and (iii) it is not convex. Furthermore, a convex and universal functional $F(\rho,j^p)$ is studied and proven to be equal the convex envelope of $Q(\rho,j^p)$. For both $Q$ and $F$, we give upper and lower bounds.Comment: 26 page

Unique Continuation for the Magnetic Schr\"odinger Equation

The unique-continuation property from sets of positive measure is here proven for the many-body magnetic Schr\"odinger equation. This property guarantees that if a solution of the Schr\"odinger equation vanishes on a set of positive measure, then it is identically zero. We explicitly consider potentials written as sums of either one-body or two-body functions, typical for Hamiltonians in many-body quantum mechanics. As a special case, we are able to treat atomic and molecular Hamiltonians. The unique-continuation property plays an important role in density-functional theories, which underpins its relevance in quantum chemistry

Guaranteed Convergence of a Regularized Kohn-Sham Iteration in Finite Dimensions

The exact Kohn-Sham iteration of generalized density-functional theory in finite dimensions witha Moreau-Yosida regularized universal Lieb functional and an adaptive damping step is shown toconverge to the correct ground-state density.Comment: 3 figures, contains erratum with additional author Paul E. Lammer

The Coupled-Cluster Formalism - a Mathematical Perspective

The Coupled-Cluster theory is one of the most successful high precision methods used to solve the stationary Schr\"odinger equation. In this article, we address the mathematical foundation of this theory with focus on the advances made in the past decade. Rather than solely relying on spectral gap assumptions (non-degeneracy of the ground state), we highlight the importance of coercivity assumptions - G\aa rding type inequalities - for the local uniqueness of the Coupled-Cluster solution. Based on local strong monotonicity, different sufficient conditions for a local unique solution are suggested. One of the criteria assumes the relative smallness of the total cluster amplitudes (after possibly removing the single amplitudes) compared to the G\aa rding constants. In the extended Coupled-Cluster theory the Lagrange multipliers are wave function parameters and, by means of the bivariational principle, we here derive a connection between the exact cluster amplitudes and the Lagrange multipliers. This relation might prove useful when determining the quality of a Coupled-Cluster solution. Furthermore, the use of an Aubin-Nitsche duality type method in different Coupled-Cluster approaches is discussed and contrasted with the bivariational principle

Coupled-Cluster Theory Revisited. Part II: Analysis of the single-reference Coupled-Cluster equations

In a series of two articles, we propose a comprehensive mathematical framework for Coupled-Cluster-type methods. In this second part, we analyze the nonlinear equations of the single-reference Coupled-Cluster method using topological degree theory. We establish existence results and qualitative information about the solutions of these equations that also sheds light on the numerically observed behavior. In particular, we compute the topological index of the zeros of the single-reference Coupled-Cluster mapping. For the truncated Coupled-Cluster method, we derive an energy error bound for approximate eigenstates of the Schrodinger equation.Comment: Published in: ESAIM: M2AN Volume 57, Number 2, March-April 2023, Pages 545-58

One-Dimensional Lieb-Oxford Bounds

We investigate and prove Lieb-Oxford bounds in one dimension by studying convex potentials that approximate the ill-defined Coulomb potential. A Lieb-Oxford inequality establishes a bound of the indirect interaction energy for electrons in terms of the one-body particle density $\rho_\psi$ of a wave function $\psi$. Our results include modified soft Coulomb potential and regularized Coulomb potential. For these potentials, we establish Lieb-Oxford-type bounds utilizing logarithmic expressions of the particle density. Furthermore, a previous conjectured form $I_\mathrm{xc}(\psi)\geq - C_1 \int_{\mathbb R} \rho_\psi(x)^{2} \mathrm{d}x$ is discussed for different convex potentials

Homotopy continuation methods for coupled-cluster theory in quantum chemistry

Homotopy methods have proven to be a powerful tool for understanding the multitude of solutions provided by the coupled-cluster polynomial equations. This endeavor has been pioneered by quantum chemists that have undertaken both elaborate numerical as well as mathematical investigations. Recently, from the perspective of applied mathematics, new interest in these approaches has emerged using both topological degree theory and algebraically oriented tools. This article provides an overview of describing the latter development