The computation of excited states within the Complete Active Space Self-Consistent Field (CASSCF) framework remains a significant challenge in quantum chemistry, both theoretically and algorithmically. In this work, we extend the Kähler manifold formalism introduced in Part I of this series to the CASSCF theory, and draw a geometrical connection from the time-dependent CASSCF equations to state-specific and linear response methodologies for excited states. This is achieved by first investigating the underlying CASSCF manifold and identifying its Kähler structure, which is complicated by the nontrivial coupling of CI and orbital degrees of freedom. Building on these theoretical findings, we derive the CASSCF linear response equations in a straightforward manner, and develop a robust state-specific method that relies solely on first-order derivatives of the CASSCF energy functional. Numerical results on representative molecular systems-water, formaldehyde, and ethylene-demonstrate the effectiveness of the proposed state-specific method, while revealing the difficulty of reliable identification of excited states due to nonlinearity induced by the CASSCF theory.

This article presents a mathematical analysis and numerical strategies for solving the optimization problem of minimizing the quadratic function J(P) = Tr(BP)- 1/2Tr(A P A P), where A,B ∈ℝ^M × M_ sym, with A ≽ 0, over the Grassmann manifold Gr(m,ℝ^M). While this problem is non-convex and typically admits non-global local minima - posing challenges for Riemannian optimization and self-consistent field (SCF) algorithms - we identify cases where the global minimizer can be obtained by solving an auxiliary convex problem. When this approach is not directly applicable, the solution to the auxiliary problem still serves as an effective initialization for Riemannian optimization methods and SCF algorithms, significantly improving their performance. This work is motivated by applications in quantum embedding methods, particularly in the construction of bath orbitals, where such optimization problems naturally arise.

We provide a new formulation of time-dependent density functional theory (TDDFT) based on the geometric structure of the set of states constrained to have a fixed density. Orbital-free TDDFT is formulated using a hydrodynamics equation involving a new density-to-current functional map. In the corresponding Kohn-Sham equation, the density is reproduced using a nonlocal operator. Finally, we present numerical simulations for one-dimensional soft-Coulomb systems.

Time-dependent density-functional theory (TDDFT) is a central tool for studying the dynamical electronic structure of molecules and solids, yet aspects of its mathematical foundations remain insufficiently understood. In this work, we revisit the foundations of TDDFT within a finite-dimensional setting by developing a general geometric framework for Schr & ouml;dinger dynamics subject to prescribed expectation values of selected observables. We show that multiple natural definitions of such constrained dynamics arise from the underlying geometry of the state manifold. The conventional TDDFT formulation emerges from demanding stationarity of the action functional, while an alternative, purely geometric construction leads to a distinct form of constrained Schr & ouml;dinger evolution. This alternative dynamics may provide a more mathematically robust route to TDDFT and may suggest alternative strategies for constructing nonadiabatic approximations. Applying the theory to interacting fermions on finite lattices, we derive Kohn-Sham schemes in which the density constraint is enforced via an imaginary potential or, equivalently, a nonlocal Hermitian operator. Numerical illustrations for the Hubbard dimer demonstrate the behavior of these approaches.

Computing excited states of many-body quantum Hamiltonians is a fundamental challenge in computational physics and chemistry, with state-of-the-art methods broadly classified into variational (critical point search) and linear response approaches. The Kähler manifold formalism provides a uniform framework that naturally accommodates both strategies for a wide range of variational models, including Hartree-Fock, complete active space self-consistent field, full CI, and adiabatic time-dependent density functional theory. In particular, this formalism leads to a systematic and straightforward way to obtain the final equations of linear response theory for nonlinear models, which provides, in the case of mean-field models (Hartree-Fock and density functional theory), a simple alternative to Casida's derivations. We detail the mathematical structure of Hamiltonian dynamics on Kähler manifolds, establish connections to standard quantum chemistry equations, and provide theoretical and numerical comparisons of excitation energy computation schemes at the Hartree-Fock level.

We provide a mathematical analysis of the Dynamical Mean-Field Theory (DMFT), a celebrated representative of a class of approximations in quantum mechanics known as embedding methods. Within the simplest setting of this approximation, which we introduce briefly at this point, we prove that, under certain assumptions, the DMFT equations admit a solution for any set of physical parameters. Moreover, we establish some properties of the solution(s). For the reader’s convenience, we provide in appendix a pedagogical and self-contained mathematical formulation of the DMFT equations for the finite Hubbard model. After recalling the definition and properties of one-body time-ordered Green’s functions and self-energies, and the mathematical structure of the Hubbard and Anderson impurity models, we describe the specific impurity solver we have studied in this article, namely the Iterated Perturbation Theory (IPT) solver, which can be conveniently formulated using Matsubara’s Green’s functions.

We study Kohn-Sham Density Functional Theory (DFT) models describing the electronic structure of two-dimensional materials placed in a three-dimensional environment, encapsulated between two parallel conducting electrodes. In this geometry, the Dirichlet boundary conditions at the electrodes screen the Coulomb interaction, which becomes effectively short-ranged, of Yukawa type. We prove that some nonlinear Kohn-Sham DFT models are well-posed in this setting, both for periodic materials (such as graphene) and for quasi-periodic materials (such as twisted bilayer graphene and other moiré materials for generic incommensurate twist angles).

In this article, we derive fully guaranteed error bounds for the energy of convex nonlinear mean-field models. These results apply in particular to Kohn--Sham equations with convex density functionals, which includes the reduced Hartree--Fock (rHF) model, as well as the Kohn--Sham model with exact exchange-density functional (which is unfortunately not explicit and therefore not usable in practice). We then decompose the obtained bounds into two parts, one depending on the chosen discretization and one depending on the number of iterations performed in the selfconsistent algorithm used to solve the nonlinear eigenvalue problem, paving the way for adaptive refinement strategies. The accuracy of the bounds is demonstrated on a series of test cases, including a silicon crystal and a hydrogen fluoride molecule simulated with the rHF model and discretized with planewaves. We also show that, although not anymore guaranteed, the error bounds remain very accurate for a silicon crystal simulated with the Kohn--Sham model using nonconvex exchangecorrelation functionals of practical interest.

Quantum embedding methods enable the study of large, strongly correlated quantum systems by (usually self-consistent) decomposition into computationally manageable subproblems, in the spirit of divide-and-conquer methods. Among these, Density Matrix Embedding Theory (DMET) is an efficient approach that enforces self-consistency at the level of one-particle reduced density matrices (1-RDMs), facilitating applications across diverse quantum systems. However, conventional DMET is constrained by the requirement that the global 1-RDM (low-level descriptor) be an orthogonal projector, limiting flexibility in bath construction and potentially impeding accuracy in strongly correlated regimes. In this work, we introduce a generalized DMET framework in which the low-level descriptor can be an arbitrary 1-RDM and the bath construction is based on optimizing a quantitative criterion related to the maximal disentanglement between different fragments. This yields an alternative yet controllable bath space construction for generic 1-RDMs, lifting a key limitation of conventional DMET. We demonstrate its consistency with conventional DMET in appropriate limits and exploring its implications for bath construction, downfolding (impurity Hamiltonian construction), low-level solvers, and adaptive fragmentation. We expect that this more flexible framework, which leads to several new variants of DMET, can improve the robustness and accuracy of DMET.

This article provides the first mathematical analysis of the Density Matrix Embedding Theory (DMET) method. We prove that, under certain assumptions, (i) the exact ground-state density matrix is a fixed-point of the DMET map for non-interacting systems, (ii) there exists a unique physical solution in the weakly-interacting regime, and (iii) DMET is exact up to first order in the coupling parameter. We provide numerical simulations to support our results and comment on the physical meaning of the assumptions under which they hold true. We show that the violation of these assumptions may yield multiple solutions to the DMET equations. We moreover introduce and discuss a specific N$N$-representability problem inherent to DMET.

Computing the electronic structure of incommensurate materials is a central challenge in condensed matter physics, requiring efficient ways to approximate spectral quantities such as the density of states (DoS). In this paper, we numerically investigate two distinct approaches for approximating the DoS of incommensurate Hamiltonians for small values of the incommensurability parameters ε (e.g., small twist angle, or small lattice mismatch): the first employs a momentum-space decomposition, and the second exploits a semiclassical expansion with respect to ε. In particular, we compare these two methods using a 1D toy model. We check their consistency by comparing the asymptotic expansion terms of the DoS, and it is shown that, for full DoS, the two methods exhibit good agreement in the small ε limit, while discrepancies arise for less small ε, which indicates the importance of higher-order corrections in the semiclassical method for such regimes. We find these discrepancies to be caused by oscillations in the DoS at the semiclassical analogues of Van Hove singularities, which can be explained qualitatively, and quantitatively for ε small enough, by a semiclassical approach.

The aim of this article is to analyze from a mathematical perspective some existing schemes to partition a molecular density into several atomic contributions with a specific focus on Iterative Stockholder Atom (ISA) methods. We provide a unified mathematical framework to describe the latter family of methods and propose a new scheme, named L-ISA (for linear approximation of ISA), which generalizes the so-called additive variational Hirshfeld method. We prove several important mathematical properties of the ISA and L-ISA minimization problems and show that the so-called ISA algorithms can be viewed as alternating minimization schemes, which, in turn, enables us to obtain new convergence results for these numerical methods. Specific mathematical properties of the ISA decomposition for diatomic systems are also presented. Numerical results on diatomic systems illustrate the proven mathematical properties.

We study the decay of the interatomic force constants (equivalently, the smoothness properties of the dynamical matrix) in perfect crystals both at finite electronic temperature, and for insulators at zero temperature, within the reduced Hartree-Fock approximation (also called Random Phase Approximation). At finite temperature the electrons are mobile, leading to exponential decay of the force constants. In insulators, there is incomplete screening, leading to an algebraic decay of dipole-dipole interaction type.

In solid state physics, electronic properties of crystalline materials are often inferred from the spectrum of periodic Schrödinger operators. As a consequence of Bloch’s theorem, the numerical computation of electronic quantities of interest involves computing derivatives or integrals over the Brillouin zone of so-called energy bands, which are piecewise smooth, Lipschitz continuous periodic functions obtained by solving a parametrized elliptic eigenvalue problem on a Hilbert space of periodic functions. Classical discretization strategies for resolving these eigenvalue problems produce approximate energy bands that are either non-periodic or discontinuous, both of which cause difficulty when computing numerical derivatives or employing numerical quadrature. In this article, we study an alternative discretization strategy based on an ad hoc operator modification approach. While specific instances of this approach have been proposed in the physics literature, we introduce here a systematic formulation of this operator modification approach. We derive a priori error estimates for the resulting energy bands and we show that these bands are periodic and can be made arbitrarily smooth (away from band crossings) by adjusting suitable parameters in the operator modification approach. Numerical experiments involving a toy model in 1D, graphene in 2D, and silicon in 3D validate our theoretical results and showcase the efficiency of the operator modification approach.

We explore Riemannian optimization methods for Restricted-Open-shell Hartree-Fock (ROHF) and Complete Active Space Self-Consistent Field (CASSCF) methods. After showing that ROHF and CASSCF can be reformulated as optimization problems on so-called "flag manifolds", we review Riemannian optimization basics and their application to these specific problems. We compare these methods to traditional ones and find robust convergence properties without fine-tuning of numerical parameters. Our study suggests that Riemannian optimization is a valuable addition to orbital optimization for ROHF and CASSCF, warranting further investigation.

This article provides the first mathematical analysis of the Density Matrix Embedding Theory (DMET) method. We prove that, under certain assumptions, (i) the exact ground-state density matrix is a fixed-point of the DMET map for non-interacting systems, (ii) there exists a unique physical solution in the weakly-interacting regime, and (iii) DMET is exact at first order in the coupling parameter. We provide numerical simulations to support our results and comment on the physical meaning of the assumptions under which they hold true. We show that the violation of these assumptions may yield multiple solutions of the DMET equations. We moreover introduce and discuss a specific N-representability problem inherent to DMET.

This paper is concerned with the numerical analysis of linear and nonlinear Schrödinger equations with periodic analytic potentials. We prove that, for linear equations, when the potential is analytic in a strip of width A of the complex plane, the solution is analytic in the same strip, ensuring an exponential convergence of the planewave discretization of the equation with rate A. On the other hand, for nonlinear equations, we find that the solution may be analytic only in a strip of width smaller than A. This behavior is illustrated by two examples using a combination of numerical and analytical arguments.

Response calculations in density functional theory aim at computing the change in ground-state density induced by an external perturbation. At finite temperature, these are usually performed by computing variations of orbitals, which involve the iterative solution of potentially badly conditioned linear systems, the Sternheimer equations. Since many sets of variations of orbitals yield the same variation of density matrix, this involves a choice of gauge. Taking a numerical analysis point of view, we present the various gauge choices proposed in the literature in a common framework and study their stability. Beyond existing methods, we propose a new approach, based on a Schur complement using extra orbitals from the self-consistent field calculations, to improve the stability and efficiency of the iterative solution of Sternheimer equations. We show the success of this strategy on nontrivial examples of practical interest, such as Heusler transition metal alloy compounds, where savings of around 40% in the number of required cost-determining Hamiltonian applications have been achieved.

We provide a formal derivation of a reduced model for twisted bilayer graphene (TBG) from Density Functional Theory. Our derivation is based on a variational approximation of the TBG Kohn-Sham Hamiltonian and asymptotic limit techniques. In contrast with other approaches, it does not require the introduction of an intermediate tight-binding model. The so-obtained model is similar to that of the Bistritzer-MacDonald (BM) model but contains additional terms. Its parameters can be easily computed from Kohn-Sham calculations on single-layer graphene and untwisted bilayer graphene with different stackings. It allows one in particular to estimate the parameters $w_{\rm AA}$ and $w_{\rm AB}$ of the BM model from first-principles. The resulting numerical values, namely $w_{\rm AA}= w_{\rm AB} \simeq 126$ meV for the experimental interlayer mean distance are in good agreement with the empirical values $w_{\rm AA}= w_{\rm AB}=110$ meV obtained by fitting to experimental data. We also show that if the BM parameters are set to $w_{\rm AA}= w_{\rm AB} \simeq 126$ meV, the BM model is an accurate approximation of our reduced model.

This paper investigates the mathematical properties of independent-electron models for twisted bilayer graphene by examining the density-of-states of corresponding single-particle Hamiltonians using tools from semiclassical analysis. This study focuses on a specific atomic-scale Hamiltonian $H_{d,\theta}$ constructed from Density-Functional Theory, and a family of moir\'e-scale Hamiltonians $H_{d,K,\theta}^{\rm eff}$ containing the Bistritzer-MacDonald model. The parameter $d$ represents the interlayer distance, and $\theta$ the twist angle. It is shown that the density-of-states of $H_{d,\theta}$ and $H_{d,K,\theta}^{\rm eff}$ admit asymptotic expansions in the twist angle parameter $\epsilon:=\sin(\theta/2)$. The proof relies on a twisted version of the Weyl calculus and a trace formula for an exotic class of pseudodifferential operators suitable for the study of twisted 2D materials. We also show that the density-of-states of $H_{d,\theta}$ admits an asymptotic expansion in $\eta:=\tan(\theta/2)$ and comment on the differences between the expansions in $\epsilon$ and $\eta$.