Mathematical Elements of Quantum Electronic Density Functions

Yangyi Lu, Jiali Gao

On the basis of recent advancements in the Hamiltonian matrix density functional for multiple electronic eigenstates, this study delves into the mathematical foundation of the multistate density functional theory (MSDFT). We extend a number of physical concepts at the core of Kohn-Sham DFT, such as density representability, to the matrix density functional. In this work, we establish the existence of the universal matrix functional for many states as a proper generalization of the Lieb universal functional for the ground state. Consequently, the variation principle of MSDFT can be rigorously defined within an appropriate domain of matrix densities, thereby providing a solid framework for DFT of both the ground state and excited states. We further show that the analytical structure of the Hamiltonian matrix functional is considerably constrained by the subspace symmetry and invariance properties, requiring and ensuring that all elements of the Hamiltonian matrix functional are variationally optimized in a coherent manner until the Hamiltonian matrix within the subspace spanned by the lowest eigenstates is obtained. This work solidifies the theoretical foundation to treat multiple electronic states using density functional theory.

Massimo V. Fischetti, William G. Vandenberghe

Lin Lin

This dissertation consists of two independent parts: density functional theory (Part I), and nuclear quantum effects (Part II). Kohn-Sham density functional theory (KSDFT) is by far the most widely used electronic structure theory in condensed matter systems. The computational time of KSDFT increases rapidly with respect to the number of electrons in the system, which hinders its practical application to systems of large size. The central quantities in KSDFT are the electron density and the electron energy, which can be fully characterized by the diagonal elements and the nearest off-diagonal elements of the single particle density matrix. However, methods that are currently available require the calculation of the full density matrix. This procedure becomes highly inefficient for systems of large size. Part I of this dissertation develops a new method for solving KSDFT, which directly targets at the calculation of the diagonal and the nearest offdiagonal elements of the single particle density matrix. The new method is developed under the framework of Fermi operator expansion. The new method achieves the optimal expansion cost in the operator level. The electron density and the electron energy is then evaluated from a serires of Green’s functions by a new fast algorithm developed in this dissertation. This dissertation also develops a novel method for discretizing the Hamiltonian of the system that achieves high accuracy with a very small number of basis functions. Combining all these components together, we obtain a unified, accurate, efficient method to solve KSDFT for insulating and metallic systems. Nuclear quantum effects play an important role in a large variety of hydrogen bonded systems such as water and ice due to the small mass of protons (the nuclei of hydrogen atoms). The equilibrium proton dynamics is reflected in the quantum momentum distribution and is the focus of intense research. The standard open path integral formalism for computing the quantum momentum distribution requires the

C Peltzer, Julius J. Brandstatter

Maralbek Ch. Oskonbaev, K. Osmonaliev, P.Zh. Kozhobekova +1 more

The process of mathematical modeling of the electron motion in a potential well of finite depth using a virtual laboratory is considered. Despite the fact that quantum physics has existed for more than a century, its laws were studied only with the help of mathematical formulas, and the probability density of the wave function describing the microparticle was determined. With the help of mathematical formulas, the values of discrete energy distributions, as well as the wave function of the microparticle, were determined. In this scientific article, a wave function of electron motion in a finite potential well and its discreteness relative to the values of energy quantum numbers are obtained using a special program. If the principal quantum number is equal to one, the wave function has only one curve, if the principal quantum number is equal to two, it has a curve with two peaks, if the principal quantum number is equal to three, it has a curve with three peaks, and if the principal quantum number is equal to four, it has a curve with four peaks. Between each quantum number, there are no values of the wave function density, which corresponds to forbidden zones, while in the areas where there is a w