The 2-matrix of the spin-polarized electron gas: contraction sum rules and spectral resolutions

The spin-polarized homogeneous electron gas with densities $\rho_\uparrow$ and $\rho_\downarrow$ for electrons with spin `up' ($\uparrow$) and spin `down' ($\downarrow$), respectively, is systematically analyzed with respect to its lowest-order reduced densities and density matrices and their mutual relations. The three 2-body reduced density matrices $\gamma_{\uparrow\uparrow}$, $\gamma_{\downarrow\downarrow}$, $\gamma_a$ are 4-point functions for electron pairs with spins $\uparrow\uparrow$, $\downarrow\downarrow$, and antiparallel, respectively. From them, three functions $G_{\uparrow\uparrow}(x,y)$, $G_{\downarrow\downarrow}(x,y)$, $G_a(x,y)$, depending on only two variables, are derived. These functions contain not only the pair densities but also the 1-body reduced density matrices. The contraction properties of the 2-body reduced density matrices lead to three sum rules to be obeyed by the three key functions $G_{ss}$, $G_a$. These contraction sum rules contain corresponding normalization sum rules as special cases. The momentum distributions $n_\uparrow(k)$ and $n_\downarrow(k)$, following from $f_\uparrow(r)$ and $f_\downarrow(r)$ by Fourier transform, are correctly normalized through $f_s(0)=1$. In addition to the non-negativity conditions $n_s(k),g_{ss}(r),g_a(r)\geq 0$ [these quantities are probabilities], it holds $n_s(k)\leq 1$ and $g_{ss}(0)=0$ due to the Pauli principle and $g_a(0)\leq 1$ due to the Coulomb repulsion. Recent parametrizations of the pair densities of the spin-unpolarized homogeneous electron gas in terms of 2-body wave functions (geminals) and corresponding occupancies are generalized (i) to the spin-polarized case and (ii) to the 2-body reduced density matrix giving thus its spectral resolutions.Comment: 32 pages, 4 figure

Methods for electronic-structure calculations - an overview from a reduced-density-matrix point of view

The methods of quantum chemistry and solid state theory to solve the many-body problem are reviewed. We start with the definitions of reduced density matrices, their properties (contraction sum rules, spectral resolutions, cumulant expansion, $N$-representability), and their determining equations (contracted Schr\"odinger equations) and we summarize recent extensions and generalizations of the traditional quantum chemical methods, of the density functional theory, and of the quasi-particle theory: from finite to extended systems (incremental method), from density to density matrix (density matrix functional theory), from weak to strong correlation (dynamical mean field theory), from homogeneous (Kimball-Overhauser approach) to inhomogeneous and finite systems. Measures of the correlation strength are discussed. The cumulant two-body reduced density matrix proves to be a key quantity. Its spectral resolution contains geminals, being possibly the solutions of an approximate effective two-body equation, and the idea is sketched of how its contraction sum rule can be used for a variational treatment.Comment: 27 pages, conference contributio

Reduced density matrices, their spectral resolutions, and the Kimball-Overhauser approach

Recently, it has been shown, that the pair density of the homogeneous electron gas can be parametrized in terms of 2-body wave functions (geminals), which are scattering solutions of an effective 2-body Schr\"odinger equation. For the corresponding scattering phase shifts, new sum rules are reported in this paper. These sum rules describe not only the normalization of the pair density (similar to the Friedel sum rule of solid state theory), but also the contraction of the 2-body reduced density matrix. This allows one to calculate also the momentum distribution, provided that the geminals are known from an appropriate screening of the Coulomb repulsion. An analysis is presented leading from the definitions and (contraction and spectral) properties of reduced density matrices to the Kimball-Overhauser approach and its generalizations. Thereby cumulants are used. Their size-extensivity is related to the thermodynamic limit.Comment: 15 pages, conference contributio

New sum rules relating the 1-body momentum distribution of the homogeneous electron gas to the Overhauser 2-body wave functions of its pair density

The recently derived sum rules for the scattering phase shifts of the Overhauser geminals (being 2-body-wave functions which parametrize the pair density together with an appropriately chosen occupancy) are generalized to integral equations which allow in principle to calculate the momentum distribution, supposed the phase sifts of the Overhauser geminals are known from an effective parity-dependent interaction potential (screened Coulomb repulsion).Comment: 10 page

Growth and thermal stability of TiN/ZrAlN: Effect of internal interfaces

Wear resistant hard films comprised of cubic transition metal nitride (c-TMN) and metastable c-AlN with coherent interfaces have a confined operating envelope governed by the limited thermal stability of metastable phases. However, equilibrium phases (c-TMN and wurtzite(w)-AlN) forming semicoherent interfaces during film growth offer higher thermal stability. We demonstrate this concept for a model multilayer system with TiN and ZrAlN layers where the latter is a nanocomposite of ZrN- and AlN- rich domains. The interfaces between the domains are tuned by changing the AlN crystal structure by varying the multilayer architecture and growth temperature. The interface energy minimization at higher growth temperature leads to formation of semicoherent interfaces between w-AlN and c-TMN during growth of 15 nm thin layers. Ab initio calculations predict higher thermodynamic stability of semicoherent interfaces between c-TMN and w-AlN than isostructural coherent interfaces between c-TMN and c-AlN. The combination of a stable interface structure and confinement of w-AlN to nm-sized domains by its low solubility in c-TMN in a multilayer, results in films with a stable hardness of 34 GPa even after annealing at 1150 °C.Peer ReviewedPostprint (author's final draft

Nucleus segmentation: towards automated solutions

Single nucleus segmentation is a frequent challenge of microscopy image processing, since it is the first step of many quantitative data analysis pipelines. The quality of tracking single cells, extracting features or classifying cellular phenotypes strongly depends on segmentation accuracy. Worldwide competitions have been held, aiming to improve segmentation, and recent years have definitely brought significant improvements: large annotated datasets are now freely available, several 2D segmentation strategies have been extended to 3D, and deep learning approaches have increased accuracy. However, even today, no generally accepted solution and benchmarking platform exist. We review the most recent single-cell segmentation tools, and provide an interactive method browser to select the most appropriate solution

3D-Cell-Annotator: an open-source active surface tool for single-cell segmentation in 3D microscopy images

Segmentation of single cells in microscopy images is one of the major challenges in computational biology. It is the first step of most bioimage analysis tasks, and essential to create training sets for more advanced deep learning approaches. Here, we propose 3D-Cell-Annotator to solve this task using 3D active surfaces together with shape descriptors as prior information in a semi-automated fashion. The software uses the convenient 3D interface of the widely used Medical Imaging Interaction Toolkit (MITK). Results on 3D biological structures (e.g. spheroids, organoids, embryos) show that the precision of the segmentation reaches the level of a human expert