A re-statement of the Hohenberg–Kohn theorem and its extension to finite subspaces

Bearing in mind the insight into the Hohenberg–Kohn theorem for Coulomb systems provided recently by Kryachko (Int J Quantum Chem 103:818, 2005), we present a re-statement of this theorem through an elaboration on Lieb’s proof as well as an extension of this theorem to finite subspaces. Contribution to the Serafin Fraga Memorial Issue.     

A Liouville-type theorem on halfspaces for the Kohn Laplacian

Let be the Kohn Laplacian on the Heisenberg group and let be a halfspace of whose boundary is parallel to the center of . In this paper we prove that if is a non-negative -superharmonic function such that

then in .

     

The theorem of hohenberg and kohn for subdomains of a quantum system

The theorem of Hohenberg and Kohn is extended to subdomains of a bounded quantum system. It is shown that the ground state particle density of an arbitrary subdomain uniquely determines the ground state properties of this subdomain, of any other subdomain, and of the total domain of the system.     

Hohenberg–Kohn theorems in the presence of magnetic field

In this article, we examine Hohenberg–Kohn theorems for Current Density Functional Theory, that is, generalizations of the classical Hohenberg–Kohn theorem that includes both electric and magnetic fields. In the Vignale and Rasolt formulation (Vignale and Rasolt, Phys. Rev. Lett. 1987, 59, 2360), which uses the paramagnetic current density, we address the issue of degenerate ground states and prove that the ensemble‐representable particle and paramagnetic current density determine the degenerate ground states. For the formulation that uses the total current density, we note that the proof suggested by Diener (Diener, J. Phys.: Condens. Matter. 1991, 3, 9417) is unfortunately not correct. Furthermore, we give a proof that the magnetic field and the ensemble‐representable particle density determine the scalar and vector potentials up to a gauge transformation. This generalizes the result of Grayce and Harris (Grayce and Harris, Phys. Rev. A 1994, 50, 3089) to the case of degenerate ground states. We moreover prove the existence of a positive wavefunction that is the ground state of infinitely many different Hamiltonians. © 2014 Wiley Periodicals, Inc.     

Investigation of dephasing in an open quantum system under chaotic influence via a fractional Kohn–Sham scheme

In this work, the dynamics of dephasing (without relaxation) in the presence of a chaotic oscillator is theoretically investigated. The time‐dependent density functional theory framework was used in tandem with the Lindblad master equation approach for modeling the dissipative dynamics. Using the Kohn–Sham (K–S) scheme under certain approximations, the exact model for the potentials was acquired. In addition, a space‐fractional K–S scheme was developed (using the modified Riemann–Liouville operator) for modeling the dephasing phenomenon. Extensive analyses and comparative studies were then done on the results obtained using the space‐fractional K–S system and the conventional K–S system. © 2014 Wiley Periodicals, Inc.     

Electron and hole states in a quantum ring grown by droplet epitaxy: Influence of the layer inside the ring opening

The electronic structure of the conduction and valence bands of a quantum ring containing a layer inside the ring opening is modeled. This structure (nanocup) consists of a GaAs nanodisk (the cup’s bottom) and a GaAs nanoring (the cup’s rim) which encircles the disk. The whole system is embedded in an (Al,Ga)As matrix, and its shape resembles realistic ring structures grown by the droplet epitaxy technique. The conduction-band states in the structure are modeled by the single-band effective-mass theory, while the 4-band Luttinger–Kohn model is adopted to compute the valence-band states. We analyze how the electronic structure of the nanocup evolves from the one of a quantum ring when the size of either the nanodisk or the nanoring is changed. For that purpose, (1) the width of the ring, (2) the disk radius, and (3) the disk height are separately varied. For dimensions typical for experimentally realized structures, we find that the electron wavefunctions are mainly localized inside the ring, even when the thickness of the inner layer is 90% of the ring thickness. These calculations indicate that topological phenomena, like the excitonic Aharonov–Bohm effect, are negligibly affected by the presence of the layer inside the ring.     

Several logarithmic Caffarelli–Kohn–Nirenberg inequalities and applications

In this paper, based on the Caffarelli–Kohn–Nirenberg inequalities on the Euclidean space and the weighted Hölder inequality, we establish the logarithmic Caffarelli–Kohn–Nirenberg inequalities and parameter type logarithmic Caffarelli–Kohn–Nirenberg inequalities, and give applications for the weighted ultracontractivity of positive strong solutions to a kind of evolution equations. We also prove corresponding logarithmic Caffarelli–Kohn–Nirenberg inequalities and parameter type logarithmic Caffarelli–Kohn–Nirenberg inequalities on the Heisenberg group and related to generalized Baouendi–Grushin vector fields. Some applications are provided.     

Heisernberg群上低阶特征值的估计

研究Heisernberg群Hn上的Kohn Laplacian算子的特征值问题,通过构造合适的测试函数,给出低阶特征值估计的一个一致不等式.     

Alternative Representations of the Correlation Energy in Density‐Functional Theory: A Kinetic‐Energy Based Adiabatic Connection

The adiabatic‐connection framework has been widely used to explore the properties of the correlation energy in density‐functional theory. The integrand in this formula may be expressed in terms of the electron–electron interactions directly, involving intrinsically two‐particle expectation values. Alternatively, it may be expressed in terms of the kinetic energy, involving only one‐particle quantities. In this work, we explore this alternative representation for the correlation energy and highlight some of its potential for the construction of new density functional approximations. The kinetic‐energy based integrand is effective in concentrating static correlation effects to the low interaction strength regime and approaches zero asymptotically, offering interesting new possibilities for modeling the correlation energy in density‐functional theory     

Neural network and ReaxFF comparison for Au properties

We have studied how ReaxFF and Behler–Parrinello neural network (BPNN) atomistic potentials should be trained to be accurate and tractable across multiple structural regimes of Au as a representative example of a single‐component material. We trained these potentials using subsets of 9,972 Kohn‐Sham density functional theory calculations and then validated their predictions against the untrained data. Our best ReaxFF potential was trained from 848 data points and could reliably predict surface and bulk data; however, it was substantially less accurate for molecular clusters of 126 atoms or fewer. Training the ReaxFF potential to more data also resulted in overfitting and lower accuracy. In contrast, BPNN could be fit to 9,734 calculations, and this potential performed comparably or better than ReaxFF across all regimes. However, the BPNN potential in this implementation brings significantly higher computational cost. © 2016 Wiley Periodicals, Inc.